Structural investigation of La(2-x)Sr(x)CuO(4+y) - Following staging as a function of temperature

Abstract

The cuprate La(2-x)Sr(x)CuO(4+y) – a high-temperature superconductor – was discovered almost three decades ago. However the mechanisms behind the superconductivity in the material for different doping values x and y are still not fully understood. A small part of this large puzzle is added to the pile with this thesis, where results on the structure for several different samples are presented.

The emphasis in this thesis is on a specific superstructure thought to be connected to the ordering of interstitial oxygen, known from the isostructural compound La(2)NiO(4+y) as staging. Four single crystal samples with different co-doping values are investigated by the use of both X-rays and neutrons.

Staging is observed for all four samples at low temperatures with X-ray measurements. The sample with strontium doping x = 0.00 shows several coexisting staging levels with staging numbers between 2 and 8, with the highest contribution from a staging level between 4 and 5. The co-doped samples show increasing staging number with increasing x. It is found that the staging belongs to a structural phase assumed in space group Fmmm, while the unstaged fraction of the samples are in the Bmab space group. These two structural phases are found to have significantly different lengths of the long crystal axis for the two low x samples, in the order of a fraction of a percent, while the two higher x samples had a difference of only a small fraction of a permille.

The temperature dependent phase transitions for both the Bmab structure and the staging reflections are investigated between 5 and 300 K. The critical exponents for the Bmab reflections are found to be significantly lower than results from similar materials in literature, although with transition temperatures consistent with literature for comparable sample compositions. It is found that the critical exponents for the staging reflections increase for increasing doping while the transition temperatures decrease, both consistent with results on the isostructural La(2)NiO(4+y).

Results from previous neutron measurements are found to be consistent with the X-ray measurements in this work, and measured reciprocal space maps from this work show a large variety of other superstructure reflections which will be interesting to investigate in the future.

Full text

F A C U L T Y O F S C I E N C E
U N I V E R S I T Y O F C O P E N H A G E N
Master’s thesis
Pia Jensen Ray
Structural investigation of La
2-x
Sr
x
CuO
4+y
Following staging as a function of temperature
Academic advisors: Kim Lefmann and Linda Udby
Submitted: 19 November 2015

ii
buffer
University of Copenhagen
A thesis
submitted to the Faculty of Science
in partial fulfilment of the requirements for the
degree of Master of Science in physics
Niels Bohr Institute
Copenhagen, Denmark
November 2015
Date CPR number Pia Jensen Ray

iii
Contents
Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
List of figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
List of tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
List of abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii
Resumé . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix
I Introductions and theory 1
1 Introduction to the project 3
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 The project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Reading guide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Superconductivity 5
2.1 Conventional superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.1 The Meissner effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.2 BCS theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 High-temperature superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2.1 Type-I and type-II superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2.2 Cuprates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.3 Other high-temperature superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3 Crystallography 13
3.1 Atomic lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.1.1 Crystal planes and Miller indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.1.2 Reciprocal space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.2 Symmetries and space groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2.1 Crystal systems and Bravais lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2.2 The two La
2−x
Sr
x
CuO
4+y
crystal systems . . . . . . . . . . . . . . . . . . . . . . . . . . 15

iv Contents
4 Scattering 19
4.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.1.1 Scattering cross section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.1.2 Elastic scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.2 Diffraction from crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.2.1 The unit cell structure factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.2.2 The Laue condition and Bragg’s law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.2.3 The structure factor for specific crystal symmetries . . . . . . . . . . . . . . . . . . 23
4.3 Diffraction from disordered crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.3.1 Twinning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.3.2 Modulated structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.4 X-rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.4.1 Extinction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.4.2 X-ray production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.5 Neutrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.5.1 Incoherent scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.5.2 Inelastic scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.5.3 Neutron production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.6 Choosing between X-rays and neutrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
5 Phase transitions 33
5.1 The order parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
5.2 First order phase transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
5.3 Continuous phase transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5.3.1 The critical exponent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5.3.2 Universality classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5.3.3 Saturation of the order parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.3.4 The critical scattering region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
6 The La
2-x
Sr
x
CuO
4+y
compound 37
6.1 The mother compound La
2
CuO
4
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
6.2 Doping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
6.2.1 Doping with strontium to La
2−x
Sr
x
CuO
4
. . . . . . . . . . . . . . . . . . . . . . . . . . . 39
6.2.2 Doping with oxygen to La
2
CuO
4+y
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
6.2.3 Co-doping to La
2−x
Sr
x
CuO
4+y
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
6.3 Crystallographic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
6.3.1 Twinning from HTT to LTO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
6.3.2 Staging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
II Experiments 45
7 Measuring staging patterns 47
7.1 The samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
7.2 The BW5 instrument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
7.2.1 An overview of the instrument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
7.2.2 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
7.3 The RITA-II instrument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
7.3.1 An overview of the instrument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
7.3.2 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

Contents v
8 Mapping out reciprocal space 55
8.1 The DMC instrument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
8.1.1 An overview of the instrument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
8.1.2 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
III Results and analysis 57
9 Staging measurements 59
9.1 Initial treatment of BW5 staging data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
9.1.1 Correcting for changing lattice parameter . . . . . . . . . . . . . . . . . . . . . . . . . . 59
9.2 Fitting to find integrated intensities for BW5 staging data . . . . . . . . . . . . . . . . . . . 67
9.2.1 Fitting La
1.96
Sr
0.04
CuO
4+y
peaks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
9.2.2 Fitting La
1.935
Sr
0.065
CuO
4+y
peaks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
9.2.3 Fitting La
1.91
Sr
0.09
CuO
4+y
peaks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
9.2.4 Fitting La
2
CuO
4+y
peaks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
9.3 Alternative ways to find the transition temperature . . . . . . . . . . . . . . . . . . . . . . . . 80
9.3.1 Taking extinction and twinning at the phase transition into account . . . . . 80
9.3.2 Slope analysis of the integrated intensities . . . . . . . . . . . . . . . . . . . . . . . . . . 83
9.4 Analysing the Bmab and staging phase transitions . . . . . . . . . . . . . . . . . . . . . . . . . 85
9.4.1 Choosing which part of the data to fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
9.4.2 Fit results for the phase transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
9.5 Results from RITA-II measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
10 Reciprocal space maps 93
10.1 Raw data from DMC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
10.1.1 Combining many datasets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
10.2 Transformed data as reciprocal planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
10.3 Observations in the reciprocal space maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
10.3.1 Staging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
10.3.2 Other superstructures and forbidden reflections . . . . . . . . . . . . . . . . . . . . . 98
IV Discussion of results 99
11 Discussions 101
11.1 Lattice parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
11.1.1 Two different structural phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
11.1.2 Lattice length as a function of doping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
11.2 Comparing measured intensities for X-ray and neutron data . . . . . . . . . . . . . . . . . 104
11.2.1 About absolute scale intensities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
11.3 Discussion of phase transition results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
11.3.1 Critical exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
11.3.2 Transition temperatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
11.4 Further comments on staging in literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
11.4.1 Phase transition behavior for varying staging number . . . . . . . . . . . . . . . . . 109
11.5 Other superstructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
11.6 Peak widths and shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
12 Conclusions 111
12.1 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

vi Contents
Appendixes 113
A Calculations 115
A.1 Derivation of the Bragg condition for scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
A.2 Proof that the Bragg and Laue conditions are equivalent . . . . . . . . . . . . . . . . . . . . 116
A.3 Loss of flux due to typical sample environments on BW5 . . . . . . . . . . . . . . . . . . . . 117
A.4 Typical twinning separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
A.5 X-ray attenuation lengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
B Additional crystallographic information 119
B.1 Overview of space group investigations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
B.2 Examples of crystal refinement results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
C Mathematical functions 123
C.1 Gaussian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
C.2 Lorentzian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
C.3 Voigt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
C.4 Power law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
D Additional stag ing figures 127
D.1 All raw BW5 staging measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
D.2 Extra BW5 data analysis figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
E Additional reciprocal space figures 149
E.1 All DMC maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
Bibliography I
Books . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I
Articles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I
Websites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII
Lookup tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VIII
Reports and theses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VIII
Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IX
Other . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IX

vii
List of Figures
Below is a list of figures in the report, with short descriptions and their page locations.
2.1 Kamerlingh Onnes’ measurements of resistance in mercury . . . . . . . . . . . . . . . . . . . 6
2.2 Magnetic field lines for a superconductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Mediation o f a bosonic pair of electrons via phonons . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.4 Timeline for the history of superconducting compounds . . . . . . . . . . . . . . . . . . . . . . 8
2.5 The magnetization of a superconductor as a function of applied field . . . . . . . . . . . 9
2.6 Magnetic field lines for a superconductor in the vortex state . . . . . . . . . . . . . . . . . . . 9
2.7 Examples of typical cuprate unit cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.8 Typical phase diagram for a cuprate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.1 The basis vectors and angles i n a crystal lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.2 Simple examples of crystal planes in a crystal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.3 The P, I, F, and C lattice centerings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.4 Symmetries o f tetragonal crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.5 Symmetries o f ortorhombic crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.1 Particle energies versus wavelengths for differ ent scattering probes . . . . . . . . . . . . . 20
4.2 A sim ple look at scattering vectors and angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.3 Requirements for constructive interference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.4 Examples of reciprocal space for twinned and modulated crystals . . . . . . . . . . . . . . 26
4.5 Relating di ffere nt energy units for X-rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.6 Relating di ffere nt energy units for neutrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.7 X-ray and neutron scatteri ng lengths compared . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5.1 A typic al phase transition with saturated order parameter and critical scattering . . 36
6.1 The Bm ab crystal structure of La
2
CuO
4
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
6.2 Comparison o f the tetragonal and ortorhombic unit cells . . . . . . . . . . . . . . . . . . . . . 38
6.3 Phase diagrams following dopi ng for La
2−x
Sr
x
CuO
4
and La
2
CuO
4+y
. . . . . . . . . . . . . 39
6.4 The twinning of La
2−x
Sr
x
CuO
4+y
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
6.5 Angular distance between twinning reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
6.6 CuO
6
tilt structures and staging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
7.1 Photos of the samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
7.2 Overview o f the BW5 instrument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

viii List of figures
7.3 Overview of the RITA-II instrument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
8.1 Overview of the DMC instrument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
9.1 Raw BW5 data on a linear scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
9.2 Raw BW5 data on a logarithmic scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
9.3 Examples of fits done to find the correcti on on the l axis . . . . . . . . . . . . . . . . . . . . . 62
9.4 Overview of the peak center positions before the l axis correction . . . . . . . . . . . . . . 64
9.5 Found c axis lengths from the l axis corrections on BW5 data . . . . . . . . . . . . . . . . . 64
9.6 Corrected BW5 data on a linear scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
9.7 Corrected BW5 data on a logarithmic scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
9.8 Example of fits done for the La
1.96
Sr
0.04
CuO
4+y
BW5 data . . . . . . . . . . . . . . . . . . . . . 67
9.9 Found peak positions for the La
1.96
Sr
0.04
CuO
4+y
sample . . . . . . . . . . . . . . . . . . . . . . 68
9.10 Found integrated peak intensities for the La
1.96
Sr
0.04
CuO
4+y
sample . . . . . . . . . . . . 69
9.11 Found peak widths for the La
1.96
Sr
0.04
CuO
4+y
sample . . . . . . . . . . . . . . . . . . . . . . . . 69
9.12 Example of fit done for the La
1.935
Sr
0.065
CuO
4+y
BW5 data . . . . . . . . . . . . . . . . . . . . 70
9.13 Found peak positions for the La
1.935
Sr
0.065
CuO
4+y
sample . . . . . . . . . . . . . . . . . . . . 70
9.14 Found integrated peak intensities for the La
1.935
Sr
0.065
CuO
4+y
sample . . . . . . . . . . . 71
9.15 Found peak widths for the La
1.935
Sr
0.065
CuO
4+y
sample . . . . . . . . . . . . . . . . . . . . . . . 71
9.16 Example of fit done for the La
1.91
Sr
0.09
CuO
4+y
BW5 data . . . . . . . . . . . . . . . . . . . . . . 72
9.17 Found peak positions for the La
1.91
Sr
0.09
CuO
4+y
sample . . . . . . . . . . . . . . . . . . . . . . 73
9.18 Found integrated peak intensities for the La
1.91
Sr
0.09
CuO
4+y
sample . . . . . . . . . . . . 73
9.19 Found peak widths for the La
1.91
Sr
0.09
CuO
4+y
sample . . . . . . . . . . . . . . . . . . . . . . . . 74
9.20 Example of fit done for the La
2
CuO
4+y
BW5 data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
9.21 Found peak positions for the La
2
CuO
4+y
sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
9.22 Found integrated peak intensities for the La
2
CuO
4+y
sample . . . . . . . . . . . . . . . . . . 77
9.23 Found peak widths for the La
2
CuO
4+y
sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
9.24 (020) scans from BW5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
9.25 Integrated peak intensities for the La
1.935
Sr
0.065
CuO
4+y
(020) peak . . . . . . . . . . . . . . 82
9.26 Integrated peak intensities for the La
1.91
Sr
0.09
CuO
4+y
(020) peaks . . . . . . . . . . . . . . 82
9.27 Slope analysis for some of the integrated intensities . . . . . . . . . . . . . . . . . . . . . . . . . 84
9.28 Additional slope analysis for some of the integrated intensities . . . . . . . . . . . . . . . . 84
9.29 Integrated intensities fitted with power laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
9.30 Results from RITA-II for La
2
CuO
4+y
and La
1.96
Sr
0.04
CuO
4+y
. . . . . . . . . . . . . . . . . . . 90
9.31 Results from RITA-II for La
1.935
Sr
0.065
CuO
4+y
and La
1.91
Sr
0.09
CuO
4+y
. . . . . . . . . . . . 91
10.1 Example of (2θ, ω) plane for L a
2
CuO
4+y
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
10.2 Example of a kl reciprocal space map for La
2
CuO
4+y
. . . . . . . . . . . . . . . . . . . . . . . . . 94
10.3 Low-temperature reciprocal space maps for the La
2
CuO
4+y
sample . . . . . . . . . . . . . 95
10.4 Low-temperature reciprocal space maps for the La
1.94
Sr
0.06
CuO
4+y
sample . . . . . . . 96
10.5 Low-temperature reciprocal space maps for the La
1.91
Sr
0.09
CuO
4+y
sample . . . . . . . 97
11.1 Lattice constant c as a function of Sr doping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
11.2 Transition temperatures and critical exponents compared to theoretical values . . . . 107
11.3 Temperature dependence of the scattered intensity seen in La
2
NiO
4.06
. . . . . . . . . . 108
A.1 Illustration for derivation of Bragg’s law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
A.2 Illustration for proof of equivalence of Laue and Bragg conditions (version 1) . . . . . 116
A.3 Illustration for proof of equivalence of Laue and Bragg conditions (version 2) . . . . . 116
A.4 Calculated X-ray flux after absorption of Al . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
A.5 X-ray attenuation lengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
C.1 A Gaussian and Lorentzian peak for comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

List of figures ix
D.1 Temperature and l interval overvie w of all the BW5 staging scans . . . . . . . . . . . . . . . 128
D.2 Raw BW5 data for La
2
CuO
4+y
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
D.3 Raw BW5 data for La
1.96
Sr
0.04
CuO
4+y
, continued in Fig. D.4 . . . . . . . . . . . . . . . . . . . . 130
D.5 Raw BW5 data for La
1.935
Sr
0.065
CuO
4+y
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
D.6 Raw BW5 data for La
1.91
Sr
0.09
CuO
4+y
, continued in Fig. D.7 . . . . . . . . . . . . . . . . . . . . 132
D.8 Raw BW5 data for La
2
CuO
4+y
(log scale) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
D.9 Raw BW5 data for La
1.96
Sr
0.04
CuO
4+y
(log scale), continued in Fig. D.10 . . . . . . . . . . . 135
D.11 Raw BW5 data for La
1.935
Sr
0.065
CuO
4+y
(log scale) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
D.12 Raw BW5 data for La
1.91
Sr
0.09
CuO
4+y
(log scale), continued in Fig. D.13 . . . . . . . . . . . 137
D.14 Raw BW5 data on a linear scale, all on the same l scale . . . . . . . . . . . . . . . . . . . . . . . 139
D.15 Raw BW5 data on a logarithmic scale, all on the same l scale . . . . . . . . . . . . . . . . . . 140
D.16 Corrected BW5 data on a linear scale, all on the same l scale . . . . . . . . . . . . . . . . . . 141
D.17 Corrected BW5 data on a logarithmic scale, all on the same l scale . . . . . . . . . . . . . . 142
D.18 The central peak for La
2
CuO
4+y
before and after correction . . . . . . . . . . . . . . . . . . . 144
D.19 The central peaks for La
1.96
Sr
0.04
CuO
4+y
before and after correction . . . . . . . . . . . . 144
D.20 The central peak for La
1.935
Sr
0.065
CuO
4+y
before and after correction . . . . . . . . . . . 145
D.21 The central peak for La
1.91
Sr
0.09
CuO
4+y
before and after correction . . . . . . . . . . . . . 145
D.22 (004) and (022) scans from BW5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
D.23 Additional slope analysis not shown in the main thesis . . . . . . . . . . . . . . . . . . . . . . 147
D.24 Double logarithmic plots of the integrated intensities with fits . . . . . . . . . . . . . . . . . 148
E.1 DMC reciprocal kl plane for La
2
CuO
4+y
at T = 10 K . . . . . . . . . . . . . . . . . . . . . . . . . . 150
E.2 DMC reciprocal kl plane for La
2
CuO
4+y
at T = 100 K . . . . . . . . . . . . . . . . . . . . . . . . . 150
E.3 DMC reciprocal kl plane for La
2
CuO
4+y
at T = 300 K . . . . . . . . . . . . . . . . . . . . . . . . . 151
E.4 DMC reciprocal hk plane for La
2
CuO
4+y
at T = 10 K . . . . . . . . . . . . . . . . . . . . . . . . . 151
E.5 DMC reciprocal hk plane for La
2
CuO
4+y
at T = 300 K . . . . . . . . . . . . . . . . . . . . . . . . 152
E.6 DMC reciprocal kl plane for La
2
CuO
4+y
(other sample) at T = 5 K . . . . . . . . . . . . . . 152
E.7 DMC reciprocal kl plane for La
1.94
Sr
0.06
CuO
4+y
at T = 1.5-100 K . . . . . . . . . . . . . . . . 153
E.8 DMC reciprocal kl plane for La
1.94
Sr
0.06
CuO
4+y
at T = 100 K . . . . . . . . . . . . . . . . . . . 153
E.9 DMC reciprocal kl plane for La
1.94
Sr
0.06
CuO
4+y
at T = 300 K . . . . . . . . . . . . . . . . . . . 154
E.10 DMC reciprocal hk plane for La
1.94
Sr
0.06
CuO
4+y
at T = 2 K . . . . . . . . . . . . . . . . . . . . 154
E.11 DMC reciprocal hk plane for La
1.94
Sr
0.06
CuO
4+y
at T = 300 K . . . . . . . . . . . . . . . . . . 155
E.12 DMC reciprocal kl plane for La
1.91
Sr
0.09
CuO
4+y
at T = 100 K . . . . . . . . . . . . . . . . . . . 155
E.13 DMC reciprocal kl plane for La
1.91
Sr
0.09
CuO
4+y
at T = 300 K . . . . . . . . . . . . . . . . . . . 156
E.14 DMC reciprocal hk plane for La
1.91
Sr
0.09
CuO
4+y
at T = 100 K . . . . . . . . . . . . . . . . . . 156

x List of figures

xi
List of Tables
Below is a list of tables in the report, with short descriptions and their page locations.
2.1 An overview of cuprates and their critical temper atures
. . . . . . . . . . . . . . . . . . . . . . 10
3.1 The 7 crystal systems and 14 Bravais lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.2 The seven tetragonal and three orto rhombic crystal classes . . . . . . . . . . . . . . . . . . . 17
4.1 Systematic reflecti on conditions due to translational symmetry elements . . . . . . . . . 24
4.2 Reflection conditi ons for specific space groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.3 Pros and cons of choosing eit her X-rays or neutrons . . . . . . . . . . . . . . . . . . . . . . . . . 32
5.1 Critical ex ponents for various models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
6.1 Allowed reflec tions for different space group choices . . . . . . . . . . . . . . . . . . . . . . . . 41
7.1 Information about the samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
7.2 Overview o f staging measurement experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
8.1 Overview o f reciprocal space mapping experi ments . . . . . . . . . . . . . . . . . . . . . . . . . . 56
9.1 Comparison o f lattice parameter lengths c for neutrons and X-rays . . . . . . . . . . . . . 63
9.2 Overview o f transition temperatures, critical exponents etc. for BW5 data . . . . . . . . 88
9.3 Overview o f transition temperatures observed for RITA-II data . . . . . . . . . . . . . . . . 92
11.1 Lattice constants for the two structural phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
11.2 Observed critical temperatures and critical exponents from literature . . . . . . . . . . . 107
B.1 Overview of space group results from thorough investigations . . . . . . . . . . . . . . . . . 120
B.2 First room temperature structure solution for La
2
CuO
4
. . . . . . . . . . . . . . . . . . . . . . . 120
B.3 Room temperature structure solutions for La
2
CuO
4
and La
2
CuO
4+y
. . . . . . . . . . . . . 121
B.4 Room and low temperature structure solutions for La
2−x
Sr
x
CuO
4(+y)
. . . . . . . . . . . 122

xii List of tables

xiii
List of abbreviations
Below is a list of abbreviations used in the repo rt, with explanations of their meaning. The given
page numbers are locations of their first mention.
N
A
Avogadro’s constant, equal to 6.022 × 10
23
1/mol . . . . . . . . . . . . . . . . . . . . . . . . 118
amu Atomic mass unit, equal to 1 g/(mol·N
A
) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
BW5 X-ray triple-axis instrument at DORIS, HASYLAB, D ESY, Germany . . . . . . . . . . . . 47
DESY Deutsches Elektronen Synchrotron in Hamburg, Germany . . . . . . . . . . . . . . . . . . 47
DMC Neutron powder instrument at SINQ, PSI, Switzerland . . . . . . . . . . . . . . . . . . . . . 55
DORIS Doppel-Ring-Speicher storage ring at DESY, Germany . . . . . . . . . . . . . . . . . . . . . . 47
FWHM Full width at half maximum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
HASYLAB Hamburger Synchrotronstrahlungslabor at DORIS, DESY, Germany . . . . . . . . . . 47
HTSC High-temperature superconductor/superconductivity . . . . . . . . . . . . . . . . . . . . . 7
HTT High-temperature tetragonal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
HWHM Half width at half maximum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
ILL Institute Laue Langevin in Grenoble, France . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
ITfC International Tables of Crystallography, [143] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
LTO Low-temperature ortorhombic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
LTSC Low-temperature superconductor/superconductivity . . . . . . . . . . . . . . . . . . . . . . 7
PG Pyrolytic graphite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
PSI Paul Scherrer Institute in Villigen, Switzerland . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
r.l.u. Reciprocal lattice units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
RITA-II Neutron triple-axis instrument at SINQ, PSI, Switzerland . . . . . . . . . . . . . . . . . . . 52
RTSC Room-temperature superconductor/superconductivity . . . . . . . . . . . . . . . . . . . . 11
SINQ Spallation neutron source at PSI, Switzerland . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
Å Ångstrøm, a unit of length equal to 10
−10
m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

xiv List of abbreviations

xv
Preface
This thesis for the degree of MSc (Master in Science) is written as part of Pia Jensen Ray’s
4+4 PhD studies, where MSc studies and PhD studies are combined. A s such, this is not a
finished work, and will be expanded with the remaining two years of the PhD time.
The work presented in this thesis is an analysis of staging data for several different samples
of La
2−x
Sr
x
CuO
4+y
, taken over several years at the hard X-ray instrument BW5 at DESY, Ham-
burg, Germany, as well as a comparison with data from the triple-axis neutron spectromete r
RITA-II at SINQ, Paul Scherrer Institute, Villigen, Switzerland. A preliminary comparison with
reciprocal space mappings of several different superstructures taken at the neutron powder
diffractometer D MC at SINQ, Paul Scherrer Institute, Villigen, Switzerland, will also be shown.
Acknowledgements
I will first of all thank Linda Udby and Kim Lefmann for their excellent guidance and supervision
through my work for this thesis and my work for the rest of my PhD. I also thank Linda Udby, as
well as Niels Hessel Andersen, Christof Niedermayer, Samuel B. Emery, and Thomas B. S. Jensen
for doing the measurements at BW5 that I analyze in this thesis. I also thank Linda Udby, and
a long list of other people, who did measurements and analysis at RITA-II, which I compare
my results to. I thank Linda Udby, Ursula Hansen, and Kim Lefmann for helping me do the
reciprocal space mapping measurements on DMC.
I express gratitude to the instrument responsibles at the different instruments; Martin von
Zimmerman, instrument responsible at BW5; Christof Niedermayer, instrument responsible at
RITA-II; and Matthias Frontzek, instrument responsible at DMC.
Finally, I would like to thank my husband Tyler Ray and my colleagues in the research group –
especially Marcella Berg, Julie Hougaard, Henrik Jacobsen, Mads Bertelsen, Sonja Holm, Morten
Sales, and Ursula Hansen – for keeping me sane throughout this process and putting up with
my silly moods. Special gratitude goes Henrik Jacobsen for reading through the manuscript at
his usual insane speed and providing valuable feedback.
— Pia Jensen Ray
November 2015

xvi Preface

xvii
Abstract
The cuprate La
2−x
Sr
x
CuO
4+y
– a high-temperature superconductor – was discovered almost
three decades ago. However the mechanisms behind the superconductivity in the material for
different doping values x and y are still not fully understood. A small part of this large puzzle
is added to the pile with this thesis, where results on the structure for several different samples
are presented.
The emphasis in this thesis is on a specific superstructure thought to be connected to the
ordering of interstitial oxygen, known from the i sostructural compound La
2
NiO
4+y
as staging.
Four single crystal samples with different co-doping values are investigated by the use of both
X-rays and neutrons.
Staging is observed for all four samples at low temperatures with X-ray measurements. The
sample with strontium dopi ng x = 0.00 shows several co-existing staging levels with staging
numbers between 2 and 8, with the highest contribution from a staging level between 4 and
5. The co-doped samples show increasing staging number with increasing x. It is found that
the staging belongs to a structural phase assumed in space group Fmmm, while the unstaged
fraction of the samples are in the Bmab space group. These two structural phases are found
to have significantly different lengths of the long crystal axis for the two low x samples, in the
order of a fraction of a percent, while the two higher x samples had a difference of only a small
fraction of a permille.
The temperature dependent phase transitions for both the Bmab structure and the stag-
ing reflections are investigated between 5 and 300 K. The critical exponents for the Bmab
reflections are found to be significantly lower than results from similar materials in literature,
although with transition temperatures consistent with literature for comparable sample compo-
sitions. It is found that the critical exponents for the staging reflections increase for increasing
doping while the transition temperatures decrease, both consistent with results on the isostruc-
tural La
2
NiO
4+y
.
Results from previous neutron measurements are found to be consistent wit h the X-ray
measurements in this work, and measured reciprocal space maps from this work show a large
variety of other superstructure reflections which will be interesting to investigate i n the future.

xviii Abstract

xix
Resumé
Den kobber-holdige perovskit La
2−x
Sr
x
CuO
4+y
– en højtemperatur superleder – blev opdaget
for snart tre årtier siden. Til trods for dette er mekanismerne bag superledningen i materia-
let for forskellige dopingværdier x og y stadig ikke fuldt forstået. Denne afhandling tilføjer
nu, med de præsenterede resultater for strukturen af flere forskellige La
2−x
Sr
x
CuO
4+y
prøver,
endnu en brik til dette puslespil.
Der fokuseres her på en specifik superstruktur, der formodentlig er forbundet til ordnin-
gen af dopede oxygenatomer i mellempositioner af grundmaterialets krystalstruktur. Denne
superstruktur er kendt som staging fra materialet La
2
NiO
4+y
, der har samme grundstruktur
som La
2−x
Sr
x
CuO
4+y
. Fire forskellige enkeltkrystaller m ed forskellige co-dopingværdier bliver
undersøgt med både røntgenstråling og neutroner.
Staging observeres i alle fire prøver ved lav temperatur med røntgenmålingerne. For prøven
med strontiumdoping x = 0.00 ses der staging med niveau mellem 2 og 8, med det højeste
signal fra stagingen mellem 4 og 5. Der ses staging med højere niveau for højere strontium
doping x i de tre andre prøver. Det bliver fundet at staging-strukturen hører til en strukturel
fase, formodentlig i rumgruppen Fmmm, mens den ikke-stagede del af prøverne er i Bmab
rumgruppen. Det blev fundet at disse to strukturelle faser har signifikant forskellige længder
af den lange krystalakse for de to prøver med lavest x, i størrelsesordenen af en brøkdel af
en procent, mens de to prøver med højere x kun har en længdeforskel på en lille brøkdel af
en promille.
De temperaturafhængige faseovergange for refleksionerne fra både Bmab strukturen og
staging strukturen bliver undersøgt mellem 5 og 300 K. De kritiske eksponenter for Bmab re-
fleksionerne blev fundet til at være signifikant lavere end resultater fra lignende materialer i lit-
teraturen, dog med overgangstemperaturer der er konsistente med disse. For stagingstrukturen
findes kritiske eksponenter der stiger med stigende doping x, mens overgangstemperaturerne
falder for stigende x – hvilket er i fin overensstemmelse med resultater for La
2
NiO
4+y
.
Det blev fundet at resultater fra tidligere målinger med neutroner er konsistente med re-
sultaterne fra røntgenmålingerne i denne afhandling. Målte planer i det reciprokke rum fra
denne afhandling viser desuden en stor samling af andre superstrukturer der vil være yderst
interessante at undersøge i fremtiden.

xx Resumé

1
I.
Introductions and theory

2
.
“
Physics is still the discipline farthest away from the art of stamp collecting.
— Ekaterina Shamonina
Metamaterials 1:1, p. 12 (2007)
”

3
1
Introduction to the project
This thesis is written such that a physics master student should be able to understand it, given
that they have some background in solid state physics.
The general books that I have used throughout this project have been Superconductivity:
Physics and Applications by Fossheim et al. [
1] for superconductivity, Magnetism in Condensed
Matter by Blundell [
2] for magnetism, Crystal Structure Determination by Massa [3] and chap-
ter 8.1 of International Tables for Crystallography Volume A: Space-group symmetry by Won-
dratschek [
4] for crystallography, Elements of Modern X-Ray Physics by McMorrow et al. [5 ] for
X-ray scattering, Neutron Scattering: Theory, Instrumentation, and Simulat ion by Lefmann [
6]
and Introduction to the Theory of Thermal Neutron Scattering by Squires [
7] for neutron scatter-
ing, and Structural Phase Transitions: I. Landau Theory by Cowley [
8] and chapter 18 of volume
23 (part C) of Methods of Experimental Physics by Cowley [
9] for phase transitions.
1.1 Motivation
The ceramic material La
2−x
Sr
x
CuO
4+y
is a superconductor, capable of transporting electricity
with no resistance below a critical temperature T
c
of around 40 K, [
12]. The parent compound,
La
2
CuO
4
, can be doped by chemical substitution of Sr
2+
for La
3+
and/or by intercalation of O
2–
into the structure, and this family of materials has been investigated for both its superconduct-
ing and magnetic properties since the 1980’s.
New superconducting compounds have since been found to have higher critical tempera-
tures – for Hg
1−x
Tl
x
Ba
2
Ca
2
Cu
3
O
8+δ
, another cuprate, up to T
c
= 138 K, [
13, 14], and even
164 K for HgBa
2
Ca
2
Cu
3
O
8+δ
at high pressure, [
15]. The naturally simpler single-layer structure
of La
2−x
Sr
x
CuO
4+y
however, makes it a good model system to investigate the cuprates. Adding
the fact that the intricate details of the workings of the superconductivity in La
2−x
Sr
x
CuO
4+y
have still not been understood, it makes it an even more interesting system to work with.
In conventional superconductors working at lower temperatures, there are no important
structural effects since the coherence length ξ is much longer than the penetration depth λ,
[
1]. This is not the case for these high-temperature superconductors, since these have ξ ≪ λ.
Because of this, knowing the structure of the HTSCs is an important part of understanding their
physical properties and their superconductivity fully – which is why this thesis will work with
exactly this.

4 1.2. The project
1.2 The project
This project deals with the superstructure known as staging, which has been observed in
La
2−x
Sr
x
CuO
4+y
both by the use of neutrons and X-rays.
The main emphasis of the thesis will be the analysis of X-ray data, with a comparison to
similar already part-analyzed neutron data taken on the same samples. Neutron data measured
to obtain overview reciprocal space maps of the relevant reciprocal space regions will also be
presented, mainly as a base for comparison and as an outlook for work to come.
The X-ray data will be analyzed in order to understand the phase transitions of both the
underlying structure and the staging superstructure, giving an insight into how the materials
behave over temperature changes and how they form long range order.
1.3 Reading guide
This thesis is separated in parts, reflecting the type of content in the collected chapters. In
part
I, I focus on introducing concepts used in this project:
In Chap.
2, I introduce superconductivity and give a motivation as to why this subject is
still important and interesting to work with even a century after its discovery. In Chap.
3, I
explain relevant parts of crystallography, and go through the lattice systems that are important
for this work. In Chap.
4, I go through neutron and X-ray scattering, focusing on scattering
from crystals. In Chap.
5, I briefly explain phase transitions. In Chap. 6, I write more about the
La
2−x
Sr
x
CuO
4+y
compound, and give a small review of earlier results.
In part
II, I go through the instruments and experimental methods used to gather data:
In Chap.
7, I introduce the X-ray instrument BW5 (DORIS, HASYLAB, DESY, Germany) and
the neutron instrument RITA-II (SINQ, PSI, Switzerland), and explain what was done to measure
staging patterns on four different La
2−x
Sr
x
CuO
4+y
samples. In Chap.
8, I introduce the neutron
instrument DMC (SINQ, PSI, Switzerland) and explain how it was used to measure planes in
reciprocal space for several different samples.
In part
III, I go through measurements and explain my analysis of the data in detail:
In Chap.
9, I focus on the staging measurements from the different experiments, and go from
raw data to temperature-dependent overviews of the patterns. In Chap.
10, I present some of
the measured reciprocal space planes, and relate them to the staging measurements.
In part
IV, I discuss and conclude on the analyzed data:
In Chap.
11, I go through the results in an overview, discussing what they could mean. In
Chap.
12, I conclude on the overall project and give an outlook of what to do next.
Finally, in the appendices I show relevant material that did not make it into the main text –
such as calculations and more detailed figures of the data. References to these appendixes are
in the main text where relevant.

5
2
Superconductivity
Superconductivity is a state of matter where the DC electrical resistance abruptly goes to zero
and magnetic fields are expelled below a critical tem perature T
c
.
In this chapter, I will go through a bit of the history of the discovery of superconductivity,
and briefly elaborate o n the different types of superconductors and how they are thought to
work. Finally, I will mention a few applications of superconductors.
2.1 Conventional superconductivity
Superconductivity was first discovered by Heike Kamerlingh Onnes in 1911, [
16, 17]. After his
successful experiments to liquify helium in 1908, he continued to work with pure metals at
low temperatures, and hence found that the electrical resistance of extremely pure mercury
dropped to zero below a critical temperature T
c
of 4.2 K, see Fig.
2.1. Onnes continued working
on other pure metals, and soon added both lead and tin – as well as many others – to the list,
bringing the maximum T
c
further up to 7 K.
2.1.1 The Meissner effect
The magnetic effects in superconductors were extensively studied after the discoveries of Onnes.
In 1933, Walther Meissner and Robert Ochsenfeld found that superconductors are perfect dia-
magnets, meaning that they have magnetic susceptibility χ = −1, below a certain critical field
H
c
, [
18].
This second characteristic defining property of superconductors, now called the Meissner
effect, means that magnetic flux lines will be expelled from the superconductor, keeping the
magnetic field B inside the material at zero, as illustrated in Fig.
2.2. The expulsion happens by
induced surface currents, so-called shielding currents, that cancel o ut the applied field inside
the sample, and since the resistance in a superconductor is zero, these currents can run freely
as long as they are below a critical current I
c
.
The magnetization M of the superconducting material will, in this instance, be the perfect
opposite of the applied field H inside the material, since
B = µ
0
(H + M) = 0 ⇒ M = −H, (2.1.1)
which can also be seen directly from the perfect diamagnetism, χ = dM/dH = −1.
When the applied field exceeds H
c
, the superconducting state will break down, and the
material returns to its metallic state.

6 2.1. Conventional superconductivity
4.00 4.10 4.20 4.30 4.40
0.00
0.025
0.05
0.075
0.10
0.125
0.15
Temperature T [K]
Resistance R [Ω]
10
−5
Ω
Figure 2.1: Kamerlingh Onnes’ measurements of resistance versus temperature in mercury from 1911,
showing the superconductivity below 4.2 K. Adapted from Onnes [
16].
(a)
B ≠ 0
(b)
B = 0
Figure 2.2: Illustration of magnetic field lines for a superconductor in the (a) normal state, where the flux
lines go through the sample, and the (b) Meissner state, where the shielding currents (shown in yellow)
make it so the resulting magnetic field inside the sample is zero.
2.1.2 BCS theory
Up until 1957, superconductors had not been understood on a microscopic level, but this was
changed by John Bardeen, Leon Neil Cooper, and John Robert Schrieffer with their quantum
mechanical theory now known as the BCS theory, [
19, 20].
This theory describes how electrons couple in bosonic pairs, so-called Cooper pairs, [
21],
mediated by lattice vibrations in the crystal they travel within, phonons. In simple terms, as
sketched in Fig.
2.3, the positively charged ions in the lattice are attracted to the negatively
charged electron traveling through the lattice, creating an overall area of positive charge density
after the electron has passed by. Since the ions are heavy, they are relatively slow at moving
back to their o riginal arrangement, and a second electron is attracted to the net positive charge,
hence following the first electron through the lattice. In practice, the two electrons are then
correlated with each other via phonons in the lattice.
Electrons with opposite spin and momentum (as to not violate the Pauli exclusion principle)
pair up in Cooper pairs, and since electrons are fermions, the collected pair behaves as a boson.
In a superconductor, there will be a lot of these pairs, and they can overlap strongly and form a

Chapter 2. Superconductivity 7
(a) (b)
+
(c)
Figure 2.3: Very simple illustration of mediation of a bosonic pair of electrons via phonons. (a) An electron
travels through a lattice of positively charged ions, attracting them. (b) While the electron travels, the ions
displace slightly towards it, creating an area of positive charge density. (c) After the electron has left, the
ions are still slightly displaced for a short period, attracting a second electron.
highly collective Bose- Einstein-like condensate, behaving with phase-coherence below T
c
. Since
they act as a condensate, breaking up even one of the pairs would require that all the pairs be
broken up. This is also called the energy gap, since there is a certain minimum energy needed to
excite the system above the Fermi level, as opposed to electrons in a normal metal, which can be
excited with an arbitrarily small energy. Hence, the usual resistance that single electrons would
feel from moving through the lattice will not be felt by electrons in the condensate, resulting in
superconductivity – where the electron flow as a whole does not experience any resi stance.
2.2 High-temperature superconductivity
For a long time, it was believed that superconductivity could not occur above 30 K, [
22, 23],
since the BSC theory does not allow Cooper pairs to form above this temperature. The Coulomb
repulsion between the electrons become too large compared to the binding through the phonons.
However, in 1986 – following investigations of oxide superconductors such as the Nb-doped
perovskite SrTiO
3
, [24–26] – Johannes G. Bednorz and Karl A. Müller discovered a new class of
superconductor, [
27]. They found that the ceramic material La
2−x
Ba
x
CuO
4
was superconduct-
ing below T
c
of 30 K, later precised to 35 K, [
28]. Not only was this the first time a ceramic –
which is generally a very bad conductor – had been seen to be superconducting, but this was
outside the range of superconductivity at the time, and well above the previous record-setting
value T
c
= 22.3 K for Nb
3
Ge from 1973, [
29].
This new type of superconductor, the first of the high-temperature superconducting (HTSC)
cuprates, started a worldwide race of discoveries of new higher and higher temperature super-
conductors, both for copper-oxi de based and other types of materials – as illustrated on the
timeline in Fig.
2.4. It seems that the mediation of superconductivity is still Cooper pairs, but
the “glue” that keeps them together seems of a different nature than the phonons used in the
BCS theory, [
30].
2.2.1 Type-I and type-II superconductors
The HTSCs are type-II superconductors, as opposed to the conventional low-temperature super-
conductors (LTSC), which are type-I. This classification comes from the Ginzburg-Landau theory,
which is a macroscopic theory describing superconductivity from the point of free energies, fol-
lowing Landau’s theory of phase transitions, [31 , 32]. It was originally published in 1950 by
Vitaly L. Ginzburg and Lev D. Landau, [
33, 34], and later derived again from the BCS theory by
Lev P. Go r’kov, [
35, 36]. This theory defines two characteristic lengths: the penetration depth
λ and the superconducting coherence length ξ. The penetration depth
*
determines how an
*
λ is sometimes called the London penetration depth, since it originates from the London equations for electro-
magnetic fields in superconductors, [
37].

8 2.2. High-temperature superconductivity
1900 1940 1980 1985 1990 1995 2000 2005 2010 2015
0
10
20
30
40
50
100
150
200
Year
Critical temperature T
c
[K]
liq. He
liq. H
2
liq. N
2
liq. CF
4
Hg
Pb
Nb
NbN
V
3
Si
Nb
3
Sn
Nb
3
Ge
BKBO
YbPd
2
B
2
C
MgB
2
Li @ 33 GPa
H
2
S @ 155 GPa
CeCu
2
Si
2
UBe
13
UPt
3
UPd
2
Al
3
CeCoIn
5
PuCoGa
5
PuRhGa
5
K
3
C
60
RbCsC
60
Cs
3
C
60
@ 1.4 GPa
LaBaCuO
YBaCuO
BiSrCaCuO
TlBaCaCuO
HgBaCaCuO
HgTlBaCaCuO
HgBaCaCuO @ 30 GPa
LaSrCuO
CNT
diamond
CNT
LaOFeP
LaOFFeAs
SrFFeAs
FeSe film
SmOFFeAs
Figure 2.4: The history of some of the discovered superconducting compounds. Note in particular the
BCS superconductors (green circles), the cuprates (blue diamonds), and the iron-based superconductors
(yellow squares). Note also the change in the axes at around year 1980 and T
c
= 50 K. Partly based on
figures from [
137] and [1, Fig. 2.1].
external magnetic field decays inside a superconductor (or, in turn, how far into the supercon-
ductor the screening currents occur), while the coherence length determines how the density
of electrons in the superconducting state can be excited and recover again without breaking
superconductivity.
The ratio of λ and ξ is the Ginzburg-Landau parameter,
κ =
λ
ξ
, (2.2.1)
and type-I and type-II superconductors are split up by the value of κ by
type-I: κ <
1
√
2
, type-II: κ >
1
√
2
. (2.2.2)
The HTSCs are furthermore classified as extreme type-II superconductors, meaning that they
have κ ≫ 1, or ξ ≪ λ, and they are not well described by the BCS or Ginzburg-Landau theories.
As illustrated on Fig.
2.5, type-II superconductors do not follow the Meissner effect all the way
to H
c
, but instead break into a mixed state when the applied field reaches H
c1
, [
38, 39]. In t his
state, the magnetic flux penetrates the material parti ally, and continously penetrates more and
more until the sample is in the normal state at H
c2
.
The mixed state is also called the vortex state, since the field lines tend to penetrate in
vortices of quantized units of flux Φ
0
= h/2e (h being Planck’s constant, and e the electron
charge), as shown in Fig.
2.6, [40, 41]. The density of these vortex lines rises with rising external
field, until the superconducting areas of the sample collapse at H
c2
. Interestingly, these vortices
can order into lattices, see e.g. [
42].
2.2.2 Cuprates
After Bednorz and Müller’s discovery of superconductivity in La
2−x
Ba
x
CuO
4
, an important mile-
stone was reached in 1987, when Paul Ching-Wu Chu et al. found that T
c
could be raised to
above 40 K when applying pressure, [
43]. This effect can also be achieved by replacing some

Chapter 2. Superconductivity 9
H
−M
H
c
H
c
(a)
H
−M
H
c
H
c1
H
c2
superconducting vortex
(b)
Figure 2.5: The magnetization M of a superconductor as a function of applied field H. (a) For a type-I
superconductor the magnetization is the opposite of the direct field until the critical applied field H
c
(the
Meissner effect), while (b) for a type-II superconductor it will start to break down in a vortex state from
H
c1
until it breaks down fully at H
c2
.
Figure 2.6: Illustration of magnetic field lines for a type-II superconductor in the vortex state, where the
flux lines enter the sample in a vortex lattice, where shielding currents (shown in yellow) shield the bulk
of the superconductor from both the field outside and the penetrating field lines.
of the atoms in the material with smaller ones with the same chemical properties,
*
leading to
the discovery that T
c
could go as high as 38 K at ambient pressure when replacing Ba with Sr,
[
12, 44], for the first time observing superconductivity in the oxygen-stoichiometric version of
the compound that I work with in this thesis. I will go into more detail on the oxygen-codoped
La
2−x
Sr
x
CuO
4+y
compound in Chap.
6.
During the following year, it was found that structures containing Y instead of La results in
the critical temperature leaping to above 90 K for YBa
2
Cu
3
O
6 + δ
, [
45, 46]. This was a break-
through of historic proportions – superconductivity at temperatures above liquid nitrogen
(77 K) suddenly made applications of superconductivity much more realistic.
Now, decades later, thousands of scientists have published an impressively large number
of research papers on the subject,
†
exploring new compounds to push T
c
higher and trying
to understand the HTSCs. A superconducting transition temperature as high as 13 8 K was
observed for Hg
1−x
Tl
x
Ba
2
Ca
2
Cu
3
O
8+δ
in 1994, [
13, 14], and even 164 K for HgBa
2
Ca
2
Cu
3
O
8+δ
under pressure, [
15], still holding the official record for cuprates to this day. It was very recently
discovered that H
2
S can exhibit BCS superconductivity at extreme pressures with a T
c
of up to
203 K, [
47] – hence giving H
2
S the overall critical temperature record for now.
All of the HTSC compounds mentioned so far belong to the same family of materials, the
cuprates. These materials are ceramics containing parallel weakly coupled copper-oxide (CuO
2
)
layers. In these layers, each Cu atom can r elinquish one loosely bound electron which can
then move around in the layer as a conduction electron. These electrons order into an anti-
ferromagnetic Mott insulator, where electron-electron interactions do not allow conduction
*
Applying pressure to a crystal will compress the unit cell slightly. Replacing atoms with smaller atoms will instead
cause the unit cell to shrink slightly from within, res ult in g in the sam e overall effect.
†
A simple search on Web of Science (
http://webofknowledge.com) for publications with topics superconduct*
suggests almost 250 000 articles at the time of wr it i ng this.

10 2.2. High-temperature superconductivity
Compound T
c
[K] .... Planes .. Reference Fig.
La
2−x
Sr
x
CuO
4
39 1 Dover et al. [12] (a)
YBa
2
Cu
3
O
6+δ
93 2 Wu et al. [
45], Cava et al. [46] (b)
Bi
2
Sr
2
CaCu
2
O
9
120 1 Hazen et al. [
48]
Tl
2
Ba
2
CuO
6
81 1 Sheng et al. [
49] (c)
Tl
2
Ba
2
CaCu
2
O
8+δ
110 2 Hazen et al. [
50]
Tl
2
Ba
2
Ca
2
Cu
3
O
10+δ
125 3 Parkin et al. [
51]
HgBa
2
CuO
4+δ
94 1 Putilin et al. [
52] (d)
HgBa
2
Ca
2
Cu
3
O
8+δ
135 3 S chilli ng et al. [
53], Huang et al. [54] (e)
Hg
1−x
Tl
x
Ba
2
Ca
2
Cu
3
O
8+δ
138 3 S un et al. [
13], Dai et al. [14]
Table 2.1: An overview of a selection of cuprates and their critical temperatures at ambient pressure,
along with the number of CuO
2
planes in the unit cell. The figure letters refer to the unit cells shown in
Fig.
2.7.
Cu
O
La/Sr
(a)
Ba
Y
Cu
O
(b)
Cu
Tl
Ba
O
(c)
Ba
Cu
O
Hg
(d)
Ba
Ca
Hg
O
Cu
(e)
Figure 2.7: Unit cells of some typical cuprates, to show the CuO
2
layers for comparison with Tab.
2.1.
(a) La
2−x
Sr
x
CuO
4
, (b) YBa
2
Cu
3
O
6
, (c) Tl
2
Ba
2
CuO
6
, (d) HgBa
2
CuO
4
, (e) HgBa
2
Ca
2
Cu
3
O
8
.
electrons to move. The CuO
2
layers are separated by neighboring layers of atoms, e.g. lan-
thanum, yttrium, strontium, or barium, which help stabilize the structure, and at the same time
dope extra electrons or holes (“stealing” electrons) into (from) the CuO
2
layers, thus changing
the conduction properties. Because these layers provide the charge carriers to the CuO
2
layers,
they are also often called the charge reservoirs.
A list of selected superconducting cuprates and their critical temperatures – as well as the
number of CuO
2
layers in the structure – is shown in Tab. 2.1, with a few examples of the
structures shown in Fig.
2.7.
Note that the number of unit cells given in Tab.
2.1 on first glance looks like it is too low for
some of the structures. For e.g. La
2−x
Sr
x
CuO
4
in Fig.
2.7(a), i t is seen that there is seemingly
three layers in the unit cell. The top layer is simply the bottom for the unit cell on top, but the
structure is said to only have one layer, not two. This apparent discrepancy is explained by the
fact that the cuprates follow the Ruddlesden-Popper structure AO—n(ABO
3
), in which A and
B are cations and O oxygen anions. The unit cell is then a stacking of n layers of perovskite
ABO
3
between rocksalt AO layers, and the number of layers in the cuprates refer to the stacking
number n, [
55]. For La
2−x
Sr
x
CuO
4
this is exactly one, as expected.
The superconducting cuprates are very different from the conventional superconductors, in
the fact that they are not traditional metals, but instead doped oxides that behave like bad
metals. Often, the pairing for superconduction does not happen with electrons, but instead

Chapter 2. Superconductivity 11
Insulator
Antiferromagnet
Pseudogap
T
N
Underdoped
T
c
T
∗
Optimally doped
Overdoped
Superconductor
Strange metal
(non-Fermi liquid)
Metal
(Fermi liquid)
Doping p
0 0.1 0.2 0.3
Temperature T [K]
0
100
200
300
Figure 2.8: Typical phase diagram for a cuprate with doping parameter p, with the antiferromagnetic
phase at low doping, a pseudogap above this separated by T
∗
to a metallic state, which is again split into
a strange non-Fermi liquid metal and a normal Fermi liquid metal phase. The superconducting state is
split into underdoped, optimally doped, and overdoped.
with the doped holes – which act as quasiparticles that pair up and behave like the Cooper
pairs, but with opposite charge, [
56]. The un-doped parent compounds are Mott insulators
with antiferromagnetic order at low temperatures, and upon doping, the superconductivity
(amongst several other interesting phases) appear, as shown in the phase diagram in Fig.
2.8.
The complexity of the HTSC phase diagram is still the source of debate and research.
As seen in the timeline in Fig.
2.4, the maximum T
c
of the cuprates (the blue points) skyrock-
eted in the late 1980’s, but has since slowly stagnated, with the last record-setting T
c
found
more than two decades ago. Despite this, it is still not fully known what drives the pairing
mechanism to get superconductivity in these materials – although multiple models have been
presented, e.g. spin fluctuations [57, 58], spin-charge separation [59], stripe formation [60], and
Fermi surface nesting [
61].
2.2.3 Other high-temperature superconductors
For a long time, it seemed like all HTSC materials had copper-oxide planes. However, in the
effort to realize higher transition temperatures, new novel superconducting materials such as
Sr
2
RuO
4
(T
c
= 0.93 K), [
62], LaFe
4
Pn
12
(T
c
= 4.1 K), [63], and KOs
2
O
6
(T
c
= 9.6 K), [64], were
discovered.
In 2008, a break-through happened when Yoichi Kamihara and colleagues discovered su-
perconductivity at 26 K for the iron pnictide LaO
1−x
F
x
FeAs, [65]. This work of finding a fairly
high critical temperature was based on their earlier work on LaOFeP, which was found to be
superconducting at 3 .2 K, [
66]. This demonstrated a promising new platform for HTSCs apart
from the cuprates, and since then the critical temperature for the iron pnictides has been rising
steadily, as seen on the yellow points in Fig.
2.4.
Finding new materials in the HTSC line-up is extremely interesting, since it is possible to
compare differences and similarities in the properties of the materials with the already known
HTSCs – hence po ssibly getting closer to a full understanding of the superconductivity in all
the materials, and getting closer to the ultimate goal of room-temperature superconductivity
(RTSC).
2.3 Applications
The zero resistance of superconductors has useful and interesting consequences for applica-
tions ranging from electric cables and windmills to medical scanners and scientific equipment.

12 2.3. Applications
Even though RTSC has still not been achieved, superconductors are still used in a variety of
applications already.
The largest current application of superconductors is in the field of NMR and MRI instruments,
where superconductors are used to produce large and stable high magnetic fields. Such high
field magnets are also used in large scale research facilities. Typically, these instruments use
LTSCs cooled with helium, since the current known HTSCs are too expensive for the instru-
ments to be cost-effective even when including the possible lower cost of cooling. A further
complication is the fact that the currently known HTSC materials are often too brittle to use
in large-scale applications. A RTSC, or even just a cheaper HTSC, would make the need for
helium unnecessary in these instruments, and would be a great benefit for both hospitals and
researchers.
There are of course many other current applications of superconductors – both LTSCs and
HTSCs have been used to make cables for transport of electricity over long distances, [
67, 68],
and applications in transformers, [
69], generators, [70], energy storage, [7 1], and many others
have also been seen. Most of these would greatly gain in both the form of lower cost and si mpler
maintenance, i f cheaper, less brittle, and easier malleable higher temperature superconductors
were to be found.

13
3
Crystallography
A crystal is a solid material consisting of atoms or molecules arranged into a repeating pattern
in three dimensions. Crystallography is the science of investigating these patterns in terms of
a unit cell that describes the lattice.
In this chapter, I will explain the terminology of crystallography and briefly elaborate on
symmetries and space groups. These results will be used further when explaining scattering
in Chap.
4.
3.1 Atomic lattices
In order to describe a crystal, a lattice and unit cell have to be set up. The lattice is a coordinate
system spanned by the lattice vectors a, b, c, such that each lattice point r can be written as
r = n
1
a + n
2
b + n
3
c, (3.1.1)
where the n
i
’s are integers. The lattice parameters
*
are the lengths and interaxial angles of
the lattice vectors, a, b, c, α, β, and γ, as shown in Fig.
3.1. Typical inter-atomic distances
in crystallography are in the order of Ångstrøm (shortened Å), which is 10
−10
m, whereas the
lattice vector lengths are up to a couple of hundred times this length for protein crystals.
The lattice vectors span a single unit cell, and repeating this unit cell in each lattice point will
construct the crystal on an infinite lattice, ignoring the crystal surface as well as any defects
and imperfections. The lattice can always be chosen in such a way that the unit cell is the
smallest possible repeating volume (giving the primitive unit cell), but this is not always the
most convenient.
The atom positions in the unit cell are described in terms of fractional coordinates x, y,
z of the lattice parameters a, b, c – e.g. an atom in the center of the unit cell is written as
(1/2, 1/2, 1/2). The atom position vector is written for atom i as
∆
i
= x
i
a + y
i
b + z
i
c, (3.1.2)
such that the absolute position of atom i in unit cell j is
r
i,j
= ∆
i
+ r
j
(3.1.3)
with r
j
from Eq. (
3.1.1). The position of each of the N atoms in the unit cell, the basis ∆
1
,
∆
2
, . . ., ∆
N
, is then folded with the lattice (by translating to all r
j
) to create the full infinitely
repeating crystal.
*
The lattice parameters are also called cell parameters and lattice constants/angles.

14 3.1. Atomic lattices
a
b
c
α
β
γ
Figure 3.1: The basis vectors a, b, c and their angles α, β, γ in a crystal lattice, here with lattice points
shown in green. A single unit cell is spanned between the three vectors.
a
b
c
(a) (b) (c)
Figure 3.2: Simple examples of two neighboring planes in a single unit cell of a crystal with lattice vectors
a, b, and c: (a) (111), (b) (102), and (c) (
111).
3.1.1 Crystal planes and Miller indices
When the crystal has been defined with the lattice and basis, crystal planes can be defined
as families of parallel planes with the same distance to each other in the crystal, intersecting
atoms and lattice points in the crystal.
A family of crystal planes is specified by three integers as (hkl), the Miller indices. These
correspond to the inverse values of the intersections between the lattice vectors and the plane
closest to the origin,
*
e.g. a plane that intersects in a/2, b/3, c/2 will have Miller indices (232)
– see also the example in Fig.
3.2. Negative values are often written by overstriking the re levant
index, e.g. (
201).
3.1.2 Reciprocal space
When working with scattering on crystals, it is often simpler to work in the so-called reciprocal
space as opposed to real-space. Here, the reciprocal lattice vectors are defined as
a
∗
=
2π
V
0
b × c, b
∗
=
2π
V
0
c × a, c
∗
=
2π
V
0
a × b, (3.1.4)
where V
0
= a · b × c is the volume of the real-space unit cell.
Each point in the reciprocal space corresponds to a family of planes in the real space, known
by their Miller indices (hkl). A point in the reciprocal space is then written as the reciprocal
vector
τ
hkl
= ha
∗
+ kb
∗
+ lc
∗
. (3.1.5)
*
In case the plane intersects origo, the nearest plane away from this is chosen.

Chapter 3. Crystallography 15
The real-space distance between two nearest planes can be calculated from the Miller indices
(hkl) and the corresponding reciprocal lattice point as
d
hkl
=
2π
|τ
hkl
|
. (3.1.6)
The Miller indices can be used to refer to the orientation of a crystal, in which case they are
written as [hkl]. With this notation, [001] would indicate the direction of the c
∗
axis.
Often, positions in reciprocal space will be given in reciprocal lattice units, or r.l.u. for short,
instead of the other typical unit, Å
−1
. This simply means that the units are in the lengths of
the reciproc al lattice vectors, a
∗
, b
∗
, and c
∗
, and it will be stated which lattice vector is used
for the unit in the given context.
3.2 Symmetries and space groups
It turns out that the arrangement of atoms or molecules in a crystal lattice follows certain
symmetries. Mathematically, these can divide all crystal structures into a collection of similar
structures with same structural properties. Several symmetry operations are used to describe
these structures: reflec tions, rotations, improper rotations, screw axes, and glide planes.
All the different possible crystal symmetries are divided into 230 space groups, which rep-
resent the symmetry operations and translations that build up the crystal. The full list with
names, numberings, and descriptions of all these space groups are kept in the International
Tables of Crystallography (ITfC) published by the International Union of Crystallography (IUCr),
[
143].
3.2.1 Crystal systems and Bravais lattices
The 230 space groups are divided into seven crystal systems, differentiated by how the symme-
try elements of the space groups restrict the shape of the unit cell. These are cubic, tetragonal,
ortorhombic, monoclinic, triclinic, rhombohedral,
*
and hexagonal, as listed in Tab. 3.1.
The crystal systems are again split up into the 14 Bravais lattices, which combine the seven
crystal systems with the different types of possible centerings of lattice points. These are, as
illustrated in Fig.
3.3, primitive (shortened as P), which simply has lattice points at the edges of
the unit cell; body-centered (shortened I), which has an extra lattice point
†
in the center of the
unit cell; face-centered (shortened F), which has extra lattice points at the center of each of the
faces of the unit cell; and base-centered (shortened A, B, or C), which has two extra lattice points
at the center of a pair of parallel faces. Auguste Bravais found in 1850 that the combination of
crystal systems and lattices, as shown in Tab.
3.1, can describe all possible crystal symmetries.
3.2.2 The two La
2-x
Sr
x
CuO
4+y
crystal systems
Both the tetragonal and the ortorhombic crystal systems are important for this thesis, since
La
2−x
Sr
x
CuO
4+y
has phases in either one. The system is tetragonal at high temperatures (the
so-called high-temperature tetragonal phase, HTT), and ortorhombic at low temperatures (the
low-temperature ortorhombic phase, LTO).
‡
*
The rhombohedral crystal system is the same as the trigonal lattice system, with a few distinctions that I will not
go into detail with.
†
The extra lattice point is compared to the primitive unit cell. In theory, for the centere d lattices, it is possible to
choose a primitive unit cell with fewer lattice points – bu t choosing the centered lattice instead makes the symmetries
easier to work with.
‡
Some cuprates also have another low-temperature tetragonal phase (called LTT), but this is not the case for
La
2−x
Sr
x
CuO
4+y
.

16 3.2. Symmetries and space groups
Crystal system Unit cell lengths Unit cell angles Lattice centerings
Cubic a = b = c α = β = γ = 90
◦
P, I, F
Tetragonal a = b ≠ c α = β = γ = 90
◦
P, I
Ortorhombic a ≠ b ≠ c α = β = γ = 90
◦
P, C, I, F
Monoclinic a ≠ b ≠ c α = γ = 90
◦
≠ β > 90
◦
P, C
Triclinic a ≠ b ≠ c α ≠ β ≠ γ ≠ 90
◦
P
Hexagonal a = b ≠ c α = β = 90
◦
, γ = 120
◦
P
Rhombohedral a = b = c α ≠ β ≠ γ ≠ 90
◦
P
Table 3.1: Relations of unit cell lengths and angles for the seven different crystal systems. The lattices are
the P (primitive), C (base-centered), I (body-centered), and F (face-centered).
a
b
c
(a)
(b) (c) (d)
Figure 3.3: Illustrations of the (a) primitive (P), (b) body-centered (I), (c) face-centered (F), and (d) base-
centered (here C) lattice centerings, with the lattice vectors a, b, and c.
Tetragonal space groups
Following Tab.
3.1, it is seen that the tetragonal space groups are fairly simple, with two lattice
vectors that are the same length, a = b, and a different length last axis c. The lattice angles are
all 90
◦
, and the lattice centering is either primitive or body-centered.
Tetragonal crystals have the center of the unit cell as the center of symmetry. They have
the possibility of a four-fold rotational symmetry parallel to the c axis, as well as two-fold
rotational symmetries parallel to the a and b axes and 45
◦
to these, see Fig.
3.4(a). As illustrated
in Fig.
3.4(b), there are in total five possible mirror planes in the structure; one lying parallel
to the ab plane and four lying perpendicular to this, parallel to the a and b axes, as well as
45
◦
to these.
The seven different combinations of these symmetry operations, making up all the possible
symmetries in the tetragonal space groups, are listed in Tab. 3.2.
Ortorhombic space groups
Again looking at Tab.
3.1, it is seen that the ortorhombic space groups have only a few changes
from the tetragonal space groups. The two equal length lattice vectors are no longer the same
length, a ≠ b, and both base-centered and face-centered lattice centerings are possible.
Ortorhombic crystals have fewer symmetries than tetragonal crystals since the a and b
axes have different lengths and hence break some of the tetragonal symmetries. As shown in
Fig.
3.5(a), they have three possible two-fold rotational symmetries parallel to the three lattice
vectors – corresponding to the four-fold rotational symmetry from the tetragonal symmetry hav-
ing changed into a two-fold and the two diagonal symmetries disappearing. Further, they have
three possible mirror planes, perpendicular to each other and lying parallel to the axial planes
– while the two 45
◦
mirror planes from the tetragonal symmetry disappears, see Fig. 3.5(b).
The three different combinations of these symmetry operations, making up all the possible
symmetries in the ortorhombic space groups, are listed in Tab.
3.2.

Chapter 3. Crystallography 17
a
b = a
c
(a)
a
b = a
c
(b)
Figure 3.4: Symmetries of tetragonal crystals. (a) Rotation axes, with the four-fold rotation axis shown i n
yellow and the two-fold rotation axes in blue. (b) Mirror planes. Note that both rotation axes and mirror
planes all go through the center of the unit cell, since this is the center of symmetry.
a
b
c
(a)
a
b
c
(b)
Figure 3.5: Symmetries of ortorhombic crystals. (a) Rotation axes. (b) Mirror planes. Note that both
rotation axes and mirror planes all go through the center of the unit cell, since this is the center of
symmetry.
Crystal Crystal
system class
Description
Tetragonal 4 Four-fold rotation around c
4 Four-fold rotation around c combined with mirroring in the center of
symmetry (rotoinversion)
4/m Four-fold rotation around c combined with mirror plane perpendicular to
c
422 Four-fold rotation around c and two-fold rotations around a, b, a +b, and
a −b
4mm Four-fold rotation around c and mirror planes perpendicular to a, b, a+b,
and a −b
42m Four-fold rotation around c combined with mirroring in the center of
symmetry (rotoinversion), two-fold rotations around a and b, and mirror
planes perpendicular to a + b and a − b
4/mmm Four-fold rotation around c combined with mirror planes perpendicular
to c, a, b, a + b, and a − b
Ortorhombic 222 Two-fold rotations around a, b, and c
mm2 Mirror planes perpendicular to a and b and two-fold rotation around c
mmm Mirror planes perpendicular to a, b, and c
Table 3.2: The seven tetragonal and three ortorhombic crystal classes. The tetragonal crystal classes are
written in the order of c, a (and b because of the symmetry), and a +b (and a −b because of the symmetry),
while they are simply in the order of a, b, and c for the ortorhombic system.

18 3.2. Symmetries and space groups

19
4
Scattering
Scattering is a common name for a group of ex perime ntal methods where a beam of known
particles is used to interact with a sample. Looking at how the particles change with the interac-
tion gives information about both static and dynamic properties of the sample. Typical probing
particles to use for scattering are X-rays, neutrons, and electrons – which each have their own
strengths and weaknesses.
In this chapter, I will go through the basic theory of scattering, explain diffraction on crystals,
and then explain X-ray and neutron scattering in particular – since these are the methods used
to obtain data for this thesis. Finally, I will touch upon the differences between these two
methods, and on how to choose between them.
4.1 Definitions
Imagine an incident beam of particles, typically sent towards a sample. The flux Ψ is defined
as the number of particles passing through a unit area per second, and is used as the measure
of the intensity of the beam.
4.1.1 Scattering cross section
The cross section is a property describing how much the sample interacts with the beam, by giv-
ing how many particles end up where and with which properties after the interaction – ignoring
the incident beam. The partial differential cross section is defined as
d
2
σ
dΩ dE
f
=
particles scattered per second into a solid angle dΩ in a
given direction, with final energy between E
f
and E
f
+ dE
f
Ψ
0
dΩ dE
f
, (4.1.1)
where Ψ
0
is the initial flux. This is a measure of the efficiency of the scattering process as seen
from the given direction, where the flux of the incident beam and the size of the detection area
have been normalized away.
If the energy of the particles is not analyzed, an integral over dE
f
is performed, and the
differential cross section is obtained instead,
dσ
dΩ
=
Z
∞
0
d
2
σ
dΩ dE
f
dE
f
=
particles scattered per second into
a solid angle dΩ in a given direction
Ψ
0
dΩ
. (4.1.2)

20 4.1. Definitions
The total cross section can be found in a similar way, by also integrating over all directions,
getting
σ =
Z
all directions
dσ
dΩ
dΩ =
particles scattered per second
Ψ
0
. (4.1.3)
This expression i s the total ability of a sample to scatter the probing particles, although typically
given per atom or nucleus
*
instead of for the full sample at once. The unit of the cross section
is area, and it can be thought of as the effective area of the sample atoms or nuclei that the
particle beam interacts with in the sample.
4.1.2 Elastic scattering
The incoming wave ψ
i
at the point r is approximated by a complex plane wave of the form
ψ
i
(r) = e
ik
i
·r
, (4.1.4)
where k
i
is the incident wave vector, defined simply as the vector pointing in the direction
of the beam with length 2π/λ, with λ being the wavelength of the beam. Depending on the
probe being used, the wavelength can be calculated from the energy of the probing particles,
as indicated in Fig.
4.1.
Scattering any wave of a sample consisting on a single object which is small compared to the
wavelength – e.g. a nucleus – will result in a spherical scattered wave as illustrated in Fig.
4.2(a).
Hence, assuming a sample placed at r
′
, the scattered (final) wave ψ
f
at the point r is written
ψ
f
(r) = ψ
i
(r
′
)
−b
|r − r
′
|
e
ik
f
|r−r
′
|
, (4.1.5)
where b is the scattering length – a measure of the scattering strength of the sample. The actual
measured intensity is not the wave function directly, but instead typically the differential cross
0.1
0.5
1.0
5
10
1
5 10 50 100
X-ray photon energy E
X
[keV]
10 50 100 500
10
3
Neutron energy E
n
[meV]
100 500
10
3
5 × 10
3
10
4
Electron energy E
e
[keV]
Wavelength λ [Å]
X-rays
Neutrons
Electrons
Figure 4.1: Particle energies E versus wavelengths λ for three different scattering probes, in typical units
using the relations λ[Å] = 12.4/E
X
[keV] (or Eq. (
4.4.1)), λ[Å] = 0.28/(E
n
[eV])
1/2
(or Eqs. (4.5.1) to (4.5.2)),
and λ[Å] = 12/(E
e
[eV])
1/2
. Partly based on [
10, Fig. 1, p. 47].
*
Different probes interact with different parts of the atom, and hence see different cross sections – more on tha t
when I go through the particular probe types in Sects.
4.4 to 4.5.

Chapter 4. Scattering 21
section from Eq. (4.1.2), where the modulus squared of the wave function, |ψ(r)|
2
, gives the
number of scattered particles in the beam. Assuming elastic scattering, where no energy is
exchanged between the sample and the probing particle beam, the final scattering vector length
is the same as the ingoing, k
f
= k
i
, and it can be shown that dσ /dΩ = b
2
and σ = 4πb
2
.
If there is more than one object in the sample, interference between the scattered spherical
waves will take place, and the resulting scattered wave wi ll be a sum of the individual contri-
butions,
ψ
f
(r) =
X
j
ψ
i
(r
j
)
−b
j
|r − r
j
|
e
ik
f
|r−r
j
|
. (4.1.6)
Here it has been assumed that the individual objects in the sample only pose a small perturba-
tion to the incident beam, so that the incoming wave can be seen as having the same flux on
each of the r
j
positions.
Inserting Eq. (
4.1.4), the sum can be rewritten with a few t ricks, applying a coordinate system
with an origo far away from the sample compared to the inter-sample distances: r ≫ |r
i
− r
j
|
for any two i, j. It is then permissible to rewrite k
f
|r − r
j
| to k
f
· (r − r
j
) since the components
of the r
j
’s parallel to k
f
are much bigger than the perpendicular components. This also lets
|r − r
j
| get rewritten to r in the denominator in the final step,
ψ
f
(r) =
X
j
e
ik
i
·r
j
−b
j
|r − r
j
|
e
ik
f
·(r−r
j
)
(4.1.7)
=
X
j
−b
j
|r − r
j
|
e
i[k
i
·r
j
+k
f
·(r−r
j
)]
(4.1.8)
=
X
j
−b
j
|r − r
j
|
e
i[(k
i
−k
f
)·r
j
+k
f
·r]
(4.1.9)
= e
ik
f
·r
X
j
−b
j
|r − r
j
|
e
iq·r
j
(4.1.10)
= e
ik
f
·r
X
j
−b
j
r
e
iq·r
j
. (4.1.11)
Here, the scattering vector, as illustrated on Fig.
4.2(b), has been defined by
q = k
i
− k
f
. (4.1.12)
In the figure, the scattering angle 2θ between the incident and final beam has also been indi-
cated.
The actual measured (elastic) scattered intensity from a sample can then be shown to be
dσ
dΩ
=



X
j
b
j
e
iq·r
j



2
. (4.1.13)
4.2 Diffraction from crystals
In the case of scattering from a c rystal, the object positions r
j
can be split up using Eq. (
3.1.3)
to ∆
i
+r
j
, where the ∆’s form the basis of the atom positions in the crystal and the new r’s are
the lattice points. This means that the cross section in Eq. (
4.1.13) can be rewritten to
dσ
dΩ
= e
−2W



X
atoms i
cells j
b
i
e
iq·(∆
i
+r
j
)



2
(4.2.1)
= e
−2W



X
atoms i
b
i
e
iq·∆
i
|
{z }
F
u.c.
(q)



2



X
cells j
e
iq·r
j
|
{z }
S
N
(q)



2
, (4.2.2)

22 4.2. Diffraction from crystals
k
i
k
f
r
′
detector
(a)
θ
2θ
k
i
k
f
q
(b)
Figure 4.2: A si mple look at scattering vectors and angles. (a) A plane wave with wave vector k
i
scattering
from a single-object sample placed in r
′
, resulting in a spherical wave with wave vector k
f
in the direction
of the detector. (b) Subtracting k
i
with k
f
gives the scattering vector q.
where the extra term exp(−2W ) ≤ 1 is the site- independent Debye-Waller factor, which takes
into account that the atoms vibrate slightly around their crystal positions. The Debye-Waller
factor is close to unity for low temperatures as well as for low values of q.
4.2.1 The unit cell structure factor
The first sum in Eq. (
4.2.2) contains the unit cell structure factor (often simply referred to as
the structure factor)
F
u.c.
(q) =
X
i
b
i
e
iq·∆
i
, (4.2.3)
which only depends on the positions of the atoms in the crystal. In a typical diffraction ex-
periment, the goal is to measure |F
u.c.
(q)|
2
and then use this to calculate back to get the ∆
i
’s,
solving how the atoms are po siti oned in the unit cell.
There is one big problem with this approach though; the so called phase problem. Because
the measured value is |F
u.c.
(q)|
2
, instead of F
u.c.
(q) dire ctly, the complex phases are lost. Be-
cause of this it is not possible to do a direct calculation back to the ∆
i
’s, and several different
possible sets of ∆
i
’s can even give the same value of |F
u.c.
(q)|
2
.
A large part of crystallography is devoted to solving the phase problem, which is needed to
solve a crystal structure. The steps are typically to start out with a model structure and calculate
the resulting structure factors at given q positions, comparing to the observed structure factors
at those q. This results in a density map of the unit cell where it is possible to see if any atoms
are missing or wrong, and it is then possible to refine t he model in order to make the density
map fit better and better – resulting in the best possible model. There are several different ways
to do this, which is covered well in text books, e.g. Crystal Structure Determination by Massa [
3].
4.2.2 The Laue condition and Bragg’s law
The second sum in Eq. (4.2.2) is called the lattice or cell sum S
N
(q), and sums up all the lattice
points in the crystal, ignoring crystal boundaries. Performing this near-infinite sum, it turns out
that only when the scattering vector q coincides with a reciprocal lattice vector τ – as defined
in Eq. (
3.1.5) – will the sum element be non-zero. This rule,
q = τ, (4.2.4)
is called the Laue condition, and is illustrated in Fig. 4.3(a). Only when the incident and final
wave vectors lie in a way that they hit a reciprocal lattice point, will scattering occur.
In real space, the Laue condition corresponds to the situation illustrated in Fig.
4.3(b). An
incoming beam is scattered off a family of lattice planes with interplanar distance d at an angle

Chapter 4. Scattering 23
θ
θ
(0, 0) (1, 0)
(0, 1) (1, 1)
k
i
−k
f
q
2π
d
τ
(a)
d
θθ
k
i
k
f
d sin θ
(b)
Figure 4.3: Illustrations to help show the requirements for constructive interference. (a) In reciprocal
space, the Laue condition requires q = τ, while (b) in real space, the Bragg condition requires nλ = 2d sin θ.
θ. There will only be constructive interference if θ is such that the extra di stance traveled by
the lower beam, 2d sin θ, is an integer number n of wavelengths λ. This rule,
nλ = 2d sin θ, (4.2.5)
is called the Bragg law. For a few more details on this derivation, see Appx.
A.1, and for the
proof of the equivalence of the Laue and Bragg conditions, see Appx.
A.2.
One very simple application of Bragg’s law is a monochromator, which is an optical device used
to pick out a specific energy of a white beam.
*
A typical source used for scattering experiments
will send out many wavelengths in the same beam, and for particular experiments it is necessary
to have only one specific (and known) wavelength on the sample. A monochromator is simply a
known crystal, typically lead, silicon, or pyrolytic graphite (PG), put into the incoming beam at
an angle θ (as illustrated in Fig. 4.2(b)) where the Bragg law is met for the wanted wavelength.
The sample is then placed at and angle 2θ from the incoming beam, where the monochromatic
beam will be.
4.2.3 The structure factor for specific crystal symmetries
The Laue and Bragg conditions both say the same thing: diffraction can only take place when
the conditions are met such that S
N
(q) ≠ 0. However, the structure factor F
u.c.
(q) determines
whether or not there will actually be diffraction in the given spot, and how big it will be. If
|F
u.c.
(q)|
2
= 0, there will be no diffraction, even if the Laue and Bragg conditions are met.
Depending on the given crystal structure, there are certain reflection patterns that are always
present independent of t he actual unit cell contents – simply given by the symmetry. Some
of these for the general translational symmetry elements are summarized in Tab.
4.1. These
conditions tell when |F
u.c.
(q)|
2
≠ 0, not what the value is when it is non-zero. Adding to these,
specific space group symmetries will have their own reflection patterns – which is a great help
when initially trying to get a structural model for a crystal from a measured reflection pattern.
Below, I will go through some of the space groups used for La
2−x
Sr
x
CuO
4+y
.
I4/mmm
The tetragonal space group I4/mmm (number 139 in ITfC) is of the 4/mmm crystal class, and
La
2−x
Sr
x
CuO
4+y
is in this space group when in the tetragonal state.
*
Calling a beam “white” means that it contains a cont in u um of energies – using the naming convention known from
optical light.

24 4.2. Diffraction from crystals
Symmetry element Condition for reflection to be present
Primitive lattice P none
Base-centered lattice A k +l = 2n
B h + l = 2n
C h +k = 2n
Body-centered lattice I h + k + l = 2n
Face-centered lattice F h + k, h +l, k + l = 2n
Table 4.1: Some systematic reflection conditions due to translational symmetry elements of the lattice
symmetries. Here n is an integer.
Apart from the h + k + l = 2n condition given in Tab.
4.1, there are several other reflec-
tion conditions for scattering from crystals with the I4/mmm space group. These are given in
Tab.
4.2.
It is worth mentioning that in order to work together with the following ortorhombic space
groups, some times a larger unit cell is chosen where the lattice vectors coincide with the or-
torhombic cell, [
144]. This unit cell, F4/mmm, has the a and b lattice vectors rotated 45
◦
around the c axis and enlarged so the unit cell side is
√
2 times the I4/mmm side length. The
reflection conditions for this unit cell are also shown in Tab.
4.1.
Bmab
The ortorhombic space group Bmab is of the mmm crystal class. This is the typical space group
to use for La
2−x
Sr
x
CuO
4+y
in the ortorhombic phase, although sometimes Cmca (number 64
in ITfC) is used instead
*
. Cmca and B mab are r elated by a different choice of unit cell basis
vectors – Bmab is a variant of Cmca chosen to keep the notation closer to I4/mmm by making
the c axis the same.
The Bmab choice of basis vectors has a < b < c, and is switching the b and c axes, with a
change of sign with respect to the Cmca space group, following
{abc}
Cmca
= (100/001/0
10){abc}
Bmab
. (4.2.6)
When nothing else is mentioned, I will be using the Bmab notation rather than the Cmca no-
tation in this thesis.
Often, even when La
2−x
Sr
x
CuO
4+y
is in the tetragonal phases, the Bmab space group is still
chosen. This is a bigger unit cell with less symmetry, and will in the tetragonal case have a = b,
but when using this unit cell for both the ortorhombic and tetragonal phases, it is easier to
follow what happens through the transition.
Apart from the h + l = 2n condition given in Tab.
4.1, there are several other reflection condi-
tions for scattering from crystals with the Bmab space group. These are given in Tab.
4.2 along
with the same rules for Cmca for comparison.
Fmmm
The ortorhombic space group Fmmm (number 69 in ITfC) is also of the mmm c rystal class.
This space group is typically used for a secondary phase with disordered tilts as a reaction to
interstitial oxygen, [72].
Apart from the h + k, h + l, k + l = 2n condition given in Tab.
4.1, there are several other
reflection conditions for sc attering from crystals with the Fmmm space group. These are given
in Tab.
4.2.
*
Note that Cmca (Bmab) has actually been renamed to Cmce (Bmeb) in newer versions of ITfC because of an
ambivalence with the naming scheme. I have chosen to keep the old n am in g scheme in this thesis, to not be conflicting
with older literature.

Chapter 4. Scattering 25
I4/mmm F4/mmm Cmca Bmab Fmmm
hkl : h+k+l = 2n hkl : h + k, h + l,
k +l = 2n
hkl : h + k = 2n hkl : h + l = 2n hkl : h + k, h + l,
k +l = 2n
0kl : k +l = 2n 0kl : k, l = 2n 0kl : k = 2n 0kl : l = 2n 0kl : k, l = 2n
h0l : h + l = 2n h0l : h, l = 2n h0l : h, l = 2n hk0: h, k = 2n h0l : h, l = 2n
hk0: h +k = 2n hk0: h, k = 2n hk0: h, k = 2n h0l : h, l = 2n hk0: h, k = 2n
h00: h = 2n h00: h = 2n h00: h = 2n h00: h = 2n h00: h = 2n
0k0 : k = 2n 0k0 : k = 2n 0k0 : k = 2n 00l : l = 2n 0k0: k = 2n
00l : l = 2n 00l : l = 2n 00l : l = 2n 0k0 : k = 2n 00l : l = 2n
hhl : l = 2n hhl : h + l = 2n
hh0: h = 2n
Table 4.2: Reflection conditions for specific space groups, not showing the special conditions for specific
atom sites. Here n is an integer. Grey values are only there for completeness, since they are equivalent of
the condition above them because of th e symmetry. The conditions for I4/mmm, Cmca, and Fmmm are
directly from ITfC, while the conditions for Bmab are derived from the Cmca conditions.
4.3 Diffraction from disordered crystals
So far, I have been describing crystals as infinite perfectly repeating lattices with perfectly
placed atoms, but the crystals in nature are not always so immaculate. There are many effects
that can affect the crystal structure to be distorted in many different ways, which will all alter
the scattering from the structure. Here, I will simply focus on twinning and modulation, which
are both important for the work done in this thesis.
4.3.1 Twinning
Twinning happens when a crystal contains two or more single crystals of the same lattice and
unit cell, but with different orientations, grown together in a way that they have a real axis
(and hence a reciprocal plane) or a real plane (and hence a reciprocal axis) in common. The
relation between the single crystal twins is described by a twin element, which is basically a
symmetry element which o nly acts between the twin domains, instead of acting on each unit
cell. Twinning can happen in several different ways – e.g. by applying stress or temperature
changes to a sample and hence going through a phase transition (transformation twins) or by
the natural growing of the crystal (growth twins).
The scattering pattern from a twinned crystal will often be split into several separate reflec-
tions, as illustrated in Fig.
4.4(a) for a four-domain twinned crystal. I will go further into detail
with the specific twinning happening in La
2−x
Sr
x
CuO
4+y
in Sect.
6.3, where I will also illustrate
the twinning in real space.
4.3.2 Modulated structures
A crystal structure can be modulated, which means that a part of the structure cannot be
described with the usual or dering of the normal translational lattice, but instead show a con-
tinuous variation of atom positions, site occupations (on average, how many times the atom i s
actually there in the unit cell), or displacement factors (describing the variations of the atom
position over time) over the unit cells.
The modulation can usually be described by a suitable periodic function, and often a sine
function is used. The modulation can be divided into two diff erent categories: commensurate
and incommensurate.

26 4.4. X-rays
h
k
(a)
h
k
(b)
Figure 4.4: Examples of (hk0) reciprocal space patterns from (a ) an ortorhombic crystal with four twin
domains rotated around the c axis in relation to each other, and(b) a commensurately modulated struc-
ture, where an element repeats itself only every two unit cells along b. The dots represent the reflection
positions, not intensities, and the gridlines indicate integer h and k values.
Commensurate modulation
If the wavelength of the periodic function describing the modulation is a rational small multiple
of the basic unit cell length, the modulation is commensurate, and the resulting effect is a so-
called superstructure, where the resulting diffraction pattern has reflections between the usual
structural positions – at 1/n distances along the reciprocal lattice vectors. The crystal system
can be described with a correspondingly big supercell, made up of n unit cells completing a
full period of the modulation.
Repeating the super cell with simple translation (as is usually done for the simple unit
cell), the new – bigger – structure is no longer modulated. A simple example of this type of
modulation is shown in Fig.
4.4(b).
Incommensurate modulation
If the wavelength is, however, an irrational multiple of the unit cell, the modulation is incom-
mensurate, and a supercell cannot be chosen directly. The reflection pattern for this type of
modulation will typically contain so-called satellite reflections grouped around the main reflec-
tions in sometimes very intricate patterns.
For both types of modulation, the superstructure reflections are indexed with so called propaga-
tion vectors, which relate the superstructure reflection to the corresponding normal reflection.
For the example modulation in Fig.
4.4(b), this would be q = (0, 1/2, 0). For an incommensurate
modulation, the propagation vector can not be written with nice fractions, and will instead be
irrational.
4.4 X-rays
The most commonly used probe for scattering is X-rays – electromagnetic radiation with wave-
lengths in the region between 0.1 and 100 Å. X-rays with small wavelengths (up to 1 Å) are
typically referred to as hard X-rays, and are used for crystallography. Larger wavelength X-rays
are less useful, since they are very easily absorbed instead of being scattered.

Chapter 4. Scattering 27
The most important property of an X-ray beam is the wavelength λ – or wave vector k = 2π/λ
– which can also be expressed as the energy E by
E =
hc
λ
= ck, (4.4.1)
where h (and  = h/2π) is Planck’s constant and c is the speed of light. The relation between
these different expressions of the energy of the X-ray beam is shown in Fig.
4.5, where the
different naming conventions for hard and soft X-rays are also indicated.
gamma rays
hard X-rays
soft X-rays
ultraviolet
λ [Å]
0.1 1 10 100
E [keV]
100 10 1 0.1
k [Å
−1
]
100 10 1 0.1
Figure 4.5: Relating different energy units for X-rays. Hard X-rays and the high-energy end of the soft X-
rays are what is typically used for crystallography. Note that the wavelength axis (λ) goes in the opposite
direction of the others.
X-rays scatter from the electrons in the sample material via an electromagnetic interaction. This
means that the scattering length b in the diffraction results above – Eqs. (
4.1.5) to (4.2.3) – is
replaced with a q dependent expression,
− r
0
f
0
(q) = −r
0
Z
ρ(r)e
iq·r
dr, (4.4.2)
where r
0
= 2.82 × 10
−5
Å is the Thompson scattering length of the electron. The integral f
0
(q)
is called the atomic form factor
*
, and is the Fourier transform of the distribution of electrons
in the atom, ρ(r). For q = 0 , the atomic form factor is equal to the atomic number, Z, while
f
0
(q) tends to zero for higher q lengths.
This means that the measured scattering of X-rays from a given atom will be proportional
to Z
2
, following Eq. (4.1.13).
4.4.1 Extinction
It is sometimes necessary to take extinction into account when doing single crystal diffraction
experiments. Real crystals are not perfect, but really consist of a mosaic of small perfect blocks
with a thin distribution of orientations around some average value – called mosaicity.
If the size of the mosaic blocks are large, the risk of a scattered beam being scattered a
second time, or even multiple times, increases. These multiply scattered beams can interfere
destructively with each other, resulting in the integrated intensity from the diffraction being
reduced. This is called primary extinction.
When the X-ray beam enters the crystal, some of it diffracts from the top layers of the
crystal structure, while other parts of the beam go deeper, gradually being reflected in deeper
and deeper layers, and finally a part of the beam might transmit all the way through the sample.
The reflections from the deeper layers will naturally see a smaller flux in the incident beam,
since some of the beam has already been reflected – and hence again the integrated intensity
from the diffraction is reduced. This is called secondary extinction.
*
I am here ignoring the energy-dependent dispersion corrections f
′
and f
′′
, which should be added to f
0
to get
the total atomic form factor f (q, ω) = f
0
(q) + f
′
(ω) + if
′′
(ω) which takes resonant scattering and dissipation
into account. These are only i mporta n t at very specific energies, depen dent on the given atomic element.

28 4.5. Neutrons
Extinction happens for crystals that are “too” perfect – meaning that they consist of very few
mosaic blocks, resulting in both primary and secondary extinction being large effects. If instead
the crystal is “ideally imperfect”, the mosaic blocks have the right size and angular distribution
to let the integrated intensity from the scattering be large, but not too imprecise. Even in this
scenario, secondary extinction can still be a significant effect, and might only be eliminated by
use of a smaller sample.
4.4.2 X-ray production
X-rays used for crystallographic research are often produced in local laboratories, where X-ray
tubes with rotating anodes (typically made of copper, chromium, or molybdenum) generate
X-ray radiation from bremsstrahlung as well as fluorescent radiation specific for the anode ma-
terial (typically the K
α
and K
β
lines, which have energies between 5 and 9 keV, [
145]). However,
when a highly collimated and small, or a higher energy, beam is needed, these types of X-ray
sources are not the best option – instead the experiment is moved to a synchrotron.
Synchrotron radiation is a generic term used to describe radiation from charged particles
at relativistic speeds, forced to travel along a curved path. Typically, electrons or positrons
are kept in a storage ring, travelling around the ring in a controlled closed orbit by bending
magnets. The synchrotron radiation is produced either directly at the bending magnets, or by
insertion devices such a wigglers and undulators, which use alternating magnetic fields to force
the electrons into oscillating paths, emitting even more radiation.
An important property for X-ray sources is the so-called brilliance, which is an attempt to collect
some of the many properties the beam can have into one quantity,
Brilliance =
photons per second
mm
2
source area
|
{z }
flux
1
mrad
2
|
{z }
divergence
1
0.1% bandwidth
|
{z }
monochromacity
. (4.4.3)
The divergence is the angular spread in the beam, set by the vertical and horizontal beam slits,
while the monochromacity is the photons which fall within a bandwidth of 0.1% of the central
wavelength given by the monochromator.
Typical 3rd generation synchrotron sources have brilliances of more than ten orders of
magnitude larger than rotating anode sources at the K
α
line. A new type of pulsed X-ray source,
the free electron laser, even exceeds this with several more orders of magnitude.
4.5 Neutrons
A much less common probe to use for scattering is neutrons – the neutrally charged nuclear
particle. Neutrons are not usually found in their free form, since they are unstable outside
nuclei and decay to protons with a mean lifetime of 885.7(8) s, [
146]. However, different more
or less violent reactions can force them out of their nuclei to be used as a scattering probe.
Neutrons scatter from the nuclei in the sample material via the strong nuclear force. The
scattering length b will depend on the element and isotope of the nucleus in a non-trivial way,
and has been tabulated for most isotopes, [
147].
Just as for X-rays, the wavelength λ (or wave vector k = 2π/λ) is the main prope rty of a neutron
beam. This works by seeing the neutron – a particle – as a wave with a corresponding de Broglie
wavelength,
λ =
h
m
n
v
, (4.5.1)

Chapter 4. Scattering 29
where v is the speed of the neutron, m
n
is the neutron mass, and h is Planck’s constant. The
calculation of the corresponding energy is a little more complicated than Eq. (
4.4.1) for photons,
E =
h
2
2λ
2
m
n
=

2
k
2
2m
n
, (4.5.2)
with  = h/2π. Finally, it is also possible to give the energy of the neutron as an equivalent
temperature by
T =
E
k
B
, (4.5.3)
where k
B
is Boltzmann’s constant.
The relation between these diff erent expressions of the energy of the neutron beam is shown
in Fig.
4.6, where the different naming conventions for temperatures for the neutrons are also
indicated.
epithermal
hot
thermal
cold
ultracold
λ [Å]
0.1 1 10
E [meV]
10
4
10
3
100 10 1 0.1
k [Å
−1
]
100 10 1
v [m/s]
10
4
10
3
100
T [K]
10
5
10
4
10
3
10
2
10
1
10
0
Figure 4.6: Relating different energy units for neutrons. Typically, neutrons in the thermal region are used
for crystallography. Note that the wavelength axis (λ) goes in the opposite direction of the others.
4.5.1 Incoherent scattering
It is worth noting that neutrons interact with the nuclear spin, which results in an overall inco-
herent scattering which gives a constant non-interfering scattering of neutrons in all directions.
This scattering is usually not included in the tabulated scattering lengths, but is instead listed
as a incoherent scattering cross section for any given element.
4.5.2 Inelastic scattering
Since neutrons with typical wavelengths around inter-atomic distances in crystals at the same
time have energies comparable to elementary excitations in the same materials, they are often
used for inelastic scattering as well, where they deliver or receive energy in the scattering i nter-
action. In this way, it is possible to measure both static structure and dynamic behavior in the
same measurement, which is difficult to do with X-rays.
*
4.5.3 Neutron production
Neutrons used for crystallographic research are most often produced at one of two source
types: spallation sources or nuclear fission reactors.
*
It is possible to do inelastic X-ray scattering at very specific resonant energies, by doing X-ray Raman scattering,
or by certain triple-axis setups, [
73], but I will not go into detail with thi s here.

30 4.6. Choosing between X-rays and neutrons
The oldest type of the two – the fission reactor – relies on the chain reaction of the fission
process
235
U + n → D
1
+ D
2
+ 2-3n + Q, (4.5.4)
where the daughter nuclei D
1
and D
2
can be a range of elements, and the released energy Q is
of the order 200 MeV. The excess neutrons coming out of each reaction is on average 1.4, [
74].
The most powerful reactor neutron source in the world is located at Institut Laue Langevin
(ILL) in Grenoble, France, and yields a power of 5 8.3 MW [
153]. These sources are typically con-
tinuous, with the notable exception of the source in Dubna, Russia. Here, reflectors mounted on
a spinning wheel create a pulsed beam by making the reactor become periodically overcritical,
yielding a power of 1850 MW during the pulse, [
138].
In or der to get a really high neutron flux from a fission reactor, it is necessary to develop
cooling materials to remove the heat quickly enough to keep the fission reaction cross section
large. However, for really large fluxes, this is simply not possible – and spallation sources
have to take over. Spallation sources instead rely on 1-3 GeV proton accelerators bombarding
a target consisting of heavy neutron-rich nuclei, such as tungsten, mercury, or lead. The heavy
nuclei are destroyed in the process, releasing in the order of 10 to 12 neutrons per proton, [
75].
Spallation sources are typically pulsed, with pulses ranging from 1 µs to 3 ms, with 20 to
60 pulses per second, [
76]. An exception to this is the sem i-co ntinuous source SINQ at the Paul
Scherrer Institute (PSI) i n Villigen, Switzerland, where a continuous proton beam hits a lead
target.
For both source types the neutrons are of high energy when they are just produced, and have
to be slowed down in order to be useful for scattering. This is done in moderators consisting
of materials with a high density of hydrogen nuclei. Neutrons entering a moderator collide
with the hydrogen nuclei and lose energy, eventually exiting following a Maxwellian energy
distribution given by the temperature of the moderator.
4.6 Choosing between X-rays and neutrons
X-rays interact with the spatial distribution of electrons in the material via an electromagnetic
interaction, while neutrons interact with the atomic nuclei vi a the strong nuclear force. The
obvious large difference from this is the fact that X-rays have the same scattering length for
different isotopes, while neutrons will have different scattering lengths, see Fig.
4.7. Scattering
lengths for X-rays are simply calculated
*
using b = r
0
Z, while for neutrons it is tabulated,
e.g. b
1
H
= −3.74 fm and b
2
H
= 6.67 fm.
A selection of a few other pros and cons of the two types of probes are shown in Tab.
4.3.
While X-rays are easy and fairly cheap to produce in a small laboratory and possible to pro-
duce at enormous intensities at synchrotrons, neutrons are difficult to obtain – hardly possible
at smaller laboratories, and hence experiments have to be performed at large facilities. With
X-rays, only a small sample mass is needed because of this huge flux, whereas re latively large
samples are needed in neutron experiments. It is also worth noting that because of the dif-
ference in interaction str ength, typical X-ray experiments are very short compared to typical
neutron experiments of similar investigations. Because of incoherent scattering, neutron ex-
periments will inherently have large background signals compared to X-ray experiments, which
should also be factored in when choosi ng.
However, neutrons do not get absorbed as quickly as X-rays because of the lower interaction
rate, and the sample environment can be created out of less absorbing metals which would still
be opaque for X-rays (unless the X-ray energy is very high). Hence, neutrons are better for uses
*
The b = r
0
Z is of course only for q = 0. For other q, the atomic form factor f
0
from Eq. (
4.4.2) has been calculated
as an approximate function
P
4
n=1
a
n
exp (−b
n
q
2
/(4π)
2
) + c, where a
n
, b
n
, and c are tabulated.

Chapter 4. Scattering 31
H
C
O
Al
Ti
Fe
U
X-ray scattering lengths
(a)
H
C O
Al
Ti
Fe
U
Neutron scattering lengths for the average element compositions
(b)
1
H
2
H
3
H
12
C
13
C
16
O
17
O
18
O
27
Al
46
Ti
47
Ti
48
Ti
49
Ti
50
Ti
54
Fe
56
Fe
57
Fe
58
Fe
233
U
234
U
235
U
238
U
. . . and for the single isotopes seperately
Figure 4.7: Illustration of some typical (a) 100 keV X-ray and (b) thermal neut ron scattering lengths. Values
are proportional to the area diameters (different factors multiplied on X-rays and neutrons, though) and
blue areas indicate negative values. X-ray data from [
148, 149] and neutron data from [147].

32 4.6. Choosing between X-rays and neutrons
X-rays Neutrons
Distinguishing isotopes ✘ No ✔ Yes
Easy to produce
✔ Yes ✘ No
Required sample size
✔ Small ✘ Big
Measurement times
✔ Short ✘ Long
Low background
✔ Yes ✘ Perhaps (incoherent scattering)
Absorption
✘ High ✔ Low
Light elements visible
✘ No ✔ Yes
Scattering at high q
✘ Low ✔ Yes
Sees magnetism
✘ Very little ✔ Yes
Table 4.3: A few positive and negative impacts of choosing either X-rays or neutrons for crystallographic
scattering experiments.
where the sample environments are extreme, as when high pressures, large magnetic fields,
or very low temperatures are needed. Neutrons are also more interesting to use when working
with light elements, and contrast matching is possible by balancing isotope concentrations. It is
also necessary to consider that the scattering falls off towards zero for higher scattering angles
(higher q) for X-rays, while it stays independent of the angle for neutrons.
Finally, neutrons interact with magnetic structures in the sample at a comparable strength
of the interaction with the spatial structure, and can hence be used to investigate magnetic
phenomena – something which is very hard to do with X-rays.

33
5
Phase transitions
A phase transition is the transition of matter from one state to another under the influence
of for instance temperature, pressure, or applied magnetic or electric fields. To mention a few
examples, the transition could be between a solid and a liquid, a paramagnet and a ferromagnet,
or a normal metal and a superconductor.
In this chapter, I will briefly go through temperature driven phase transitions and ex plain
the importance of the critical exponents describing these.
5.1 The order parameter
A system can typically be described by a so-called order parameter µ as a measure of the
symmetry (or order) which disappears in the system at the phase transition. The m agnitude
of the order parameter will be zero when fully in the disordered phase, and non-zero when in
the ordered phase.
Examples of order parameters are the net magnetization for magnetic phase transitions and
atomic displacements for structural transitions.
For magnetic transitions, the system usually changes from a high-temperature disordered
phase with randomly oriented atomic magnetic moments into a low-temperature ordered phase
with the magnetic moments in a well defined structure.
For structural transitions, the system commonly changes from a high-temperature ordered
phase into a low-temperature phase which is a distorted version of the high-temperature phase.
This means that the symmetry is broken such that some of the symmetry operations of the
high-temperature phase are not symmetry operations of the low-temperature phase.
5.2 First order phase transitions
The order of a phase transition is traditionally given by the order of the lowest differential of
the free energy which shows a discontinuity at the transition. A first order phase transition
has a discontinuity in the first derivative of the free energy, e.g. in the volume, entropy, or
magnetization.
Such transitions are characterized by a latency where the system releases or absorbs energy
in a coexistence region, where some parts of the material have gone through the transition
while other parts have not. Only after a fixed amount of energy has been exchanged has the full
system transitioned to the new phase. A well-known example of this is water boiling, where the

34 5.3. Continuous phase transitions
water doe s not instantly turn into vapor, but instead is a turbulent mixture of liquid water and
steam bubbles at a constant temperature until enough energy (the latent heat of vaporization)
has been transferred for the whole system to change into steam.
5.3 Continuous phase transitions
A continuous phase transition,
*
also called a second order phase transition, is traditionally
considered to have a discontinuity in the second derivative of the free energy, e.g. in the com-
pressibility or heat capacity. In reality, the free energy is actually not analytical in the transition
point, which makes the name “second order” slightly deceptive.
In a simple sense, a continuous phase transition happen as follows: While still in the initial
ordered phase, small domains of the disordered final phase gradually grow already a distance
from the critical t emperature (or critical field, etc.) for the transition. At the actual phase tran-
sition the disordered domains have grown together into a large volume stretching through the
full system, covering about half of the total system volume. After the transition, the domains
that are still in the ordered phase will gradually break up into smaller and smaller domains,
each growing smaller until the system is fully disordered.
Examples of temperature dependent continuous phase transitions are many magnetic tran-
sitions and the superconducting transition.
5.3.1 The critical exponent
A continuous phase transition can be described by quantitative Landau theory (equivalent to
mean field theory), [
31, 32], where the free energy is written in polynomial expansions as pow-
ers of the relevant order parameter. When the phase transition becomes more complicated
(e.g. including long range interactions) and cannot be described by Landau theory, it can in-
stead be described by a more general renormalization group theory, [
77, 78].
With both Landau and renormalization group theory – often referred to collectively as de-
scribing Landau-like behavior – the magnitude of the order parameter µ will follow a power law
of the temperature T towards the transition point at the critical temperature T
c
. This takes
the form
µ(T ) ∝














T − T
c
T
c





β
for T ≤ T
c
(ordered phase),
0 for T ≥ T
c
(disordered phase),
(5.3.1)
where β is a so-called critical exponent.
†
The fraction (T − T
c
)/T
c
is often called the reduced
temperature t. Widely varying phase transitions follow this very general power law, which
seems surprising – but has nonetheless been verified over and over with experiments and theory
alike.
Note that the measured intensity from a scattering experiment will typically be I ∝ µ
2
, and
hence in literature it happens that β is replaced by 2β. In this thesis I will always use the
notation β to denote the critical exponent for the order parameter as written in Eq. (
5.3.1).
5.3.2 Universality classes
The value of β gives important information about the nature of the phase transition. If the
system follows Landau theory directly, β will equal 1/2 – but this is often not the case. All
*
The “continuous” in this name comes from the fact that is is continuous in the first derivative of the order
parameter, as opposed to first order phase transitions.
†
There are a number of other critical exponents defined similarly t o β, describing the behavior following changes
in magnetic field etc., but I will not go into further detail with this here. Because of this, β is sometimes alternatively
named simply the order parameter exponent, since it det er mi ne s the behavior of the or der parameter as a function of
temperature.

Chapter 5. Phase transitions 35
Landau 2D Ising 3D Ising 3D XY 3D Heisenberg
d any 2 3 3 3
D any 1 1 2 3
β 1/2 = 0.5 1/8 = 0.125 0.326 0.349 0.367
Ref. Blundell [
2] Blundell [2] Blu ndell [2] Campostrini et al. [84] Blundell [2]
Table 5.1: Examples of critical exponents for various models of system dimension d and order parameter
dimension D.
types of phase transitions can, however, be split up into categories by characterizing them
into so-called universality classes with three pieces of information: the dimensionality of the
system, d, the dimensionality of the order parameter, D, and whether the forces of the system
are long or short range.
It turns out that each universality class, independent of the smaller details of the systems,
have the same critical exponents. For example, the critical exponents for the phase transi-
tion between a gaseous and liquid phase is independent of the chemical composition of the
liquid, [
79, 80].
Most systems with d = 1 do not have continuous phase transitions, while all systems with d ≥ 4
– and many long range interactive systems – follow Landau theory exactly. The specific case of
d = 2 and D = 1 can be solved with the 2D Ising model, however most real systems have d = 3
and short range interactions, which turns out to be hard to solve exactly. Some examples of the
value of β for different model systems are given in Tab.
5.1.
Most d = 1 systems cannot have phase transitions, [
81], however a few specific examples
defy this. Examples are the “un-zipping” of DNA molecules, [
82], and the roughening of “steps”
on a crystal surface cut at a small angle to a close-packed plane, [
83].
5.3.3 Saturation of the order parameter
At low temperatures some systems tend to no longer follow the power law in Eq. (
5.3.1). Instead,
the order parameter shows a temperature-independent behavior below the so-called saturation
temperature T
s
, [
85].
This effe ct is illustrated in Fig.
5.1, where the order parameter plateaus off in the low-
temperature region instead of following the power law.
5.3.4 The critical scattering region
At temperatures close to the transition temperature T
c
, sometimes fluctuations will dominate
the system, and the order parameter will no longer follow the simple power law from Eq. (
5.3.1),
[
86]. This effect comes from the mixing of phases happening close to the transition, as ex-
plained in the beginning of this section.
This effect is also illustrated in Fig.
5.1, where the order parameter does not fall as sharply
as expected in the temperature region close to T
c
.

36 5.3. Continuous phase transitions
µ
T
T
c
T
s
µ
s
Saturated order parameter region
Landau-like behavior
Critical scattering region
Ideal power
law behavior
Figure 5.1: Illustration of a typical phase transition deviating from the ideal Landau-like power-law be-
havior. The order parameter µ saturates to µ
s
at low temperature T , starting the deviation from the
Landau-like behavior at the saturation temperature T
s
, and there is critical scattering close to the critical
temperature T
c
.

37
6
The La
2-x
Sr
x
CuO
4+y
compound
As mentioned in Chap. 2, the La
2−x
Sr
x
CuO
4
compound was first seen to be superconducting in
1987 when replacing Ba with Sr in the superconducting La
2−x
Ba
x
CuO
4
compound. It was later
found that doping the mother compound La
2
CuO
4
with O also lead to a superconductor, and
investigations has continued with co-doping with both Sr and O to La
2−x
Sr
x
CuO
4+y
.
In this chapter I will go through some of the properties of this family of compounds, men-
tioning in particular results on the structural nature.
6.1 The mother compound La
2
CuO
4
The mother compound of La
2−x
Sr
x
CuO
4+y
, the undoped material La
2
CuO
4
, is an antiferromag-
netic Mott insulator, [
87]. A Mott insulator is a type of material where the conductivity vanishes
for temperatures tending towards zero, while standard band theory would instead predict it to
be metallic, [
88].
The room-temperature crystal structure was first solved by Longo et al. [89] and later refined
further by Grande et al. [
90] to be a slightly distorted K
2
NiF
4
structure (in the ortorhombic Bmab
space group). The structure has oxygen in an octahedral coordination around the Cu
2+
ions, as
illustrated in Fig.
6.1 (where the ortorhombic unit cell is shown – this can be compared to the
tetragonal unit cell shown in Fig.
2.7(a) on p. 10). As mentioned earlier in Subsect. 2.2.2 on
cuprates, La
2
CuO
4
is a layered perovskite, formed by layering two-dimensional sheets of CuO
2
planes with LaO between the planes.
The oxygen octahedra are seen to tilt around the [1 10 ] or [1
¯
10] directions (in tetragonal
notation, meaning around a in ortorhombic notation). This distorts the structure from a simple
tetragonal structure to an ortorhombic with the same c axis, as illustrated in Fig.
6.2, where the
structure change is illustrated for a single CuO
2
layer, also showing the unit cell choices.
6.2 Doping
Hole-doping the mother compound changes it from an insulator – with one electron per Cu
2+
–
to a metal, [
91]. In practice, the doping removes electrons from the copper-oxide planes (inserts
holes), allowing electrons to move freely in the planes, and hence opening up for superconduc-
tivity.
The classic way of hole-doping La
2
CuO
4
is by substituting Sr for La, creating random sub-
stitution sites in the crystal structure. Another way to hole-dope is by intercalating oxygen,

38 6.2. Doping
c
b
a
(a)
b
c
(b)
a
c
(c)
Cu
O
La
Figure 6.1: The Bmab crystal structure of La
2
CuO
4
as solved by Grande et al. [90] (see also Tab. B.2).
The full view in 3D is shown in (a), while a rotated view from (b) the a axis and (c) the b axis show the
octahedral tilts that are only around the a axis. Figures are created using VESTA, [
163].
Cu
O
a
T
a
O
a
O
(a)
Cu
+
O above plane
−
O below plane
−
+
+
−
−
−
+
+
+
−
−
+
+
−
−
+
+
−
−
+
+
−
a
O
b
O
(b)
Figure 6.2: (a) The copper-oxide plane as seen along the c axis, showing the tetragonal (left) and ortorhom-
bic (right) unit cell conventions. (b) When crossing the tetragonal to ortorhombic phase transition, the
oxygen atoms are displaced above and below the copper plane because of the oxygen octahedral tilt, now
only described with the ortorhombic unit cell, where a
O
< b
O
.

Chapter 6. The La
2-x
Sr
x
CuO
4+y
compound 39
Tetragonal HTT
(I4/mmm)
Ortorhombic LTO
(Bmab)
Antiferromagnetic
Spin glass
Superconducting
Strontium doping x (p = x)
0.02 0.05 0.10 0.15 0.20
Temperature T [K]
0
100
200
300
400
500
(a)
Tetragonal HTT
(I4/mmm)
Ortorhombic LTO
ordered tilts
(Bmab)
Ortorhombic
disordered tilts
(Fmmm)
In-plane ordered
Antiferromagnetic
Superconducting
S
t
a
g
e
d
Stage 6 Stage 4 Stage 3 Stage 2
Oxygen doping y (p = 2y)
0.00 0.02 0.04 0.06 0.08 0.10 0.12
Temperature T [K]
0
100
200
300
400
(b)
Figure 6.3: Phase diagrams when doping La
2
CuO
4
with either (a) strontium to La
2−x
Sr
x
CuO
4
or (b) oxygen
to La
2
CuO
4+y
. The concentration p of holes per Cu atom in a CuO
2
sheet is given in parenthesis on the
two first-axes, [
92]. Adapted from [72, Fig. 1].
yielding the compound La
2
CuO
4+y
. Whereas the superconductivity in the first type of hole-
doping is strongly and non-monotonically depe ndent on the amount of Sr dopi ng as shown in
Fig.
6.3(a), the transition temperature by oxygen doping seems to be almost constant and just
above the optimal for Sr doping alone as shown in Fig.
6.3(b). This could be a consequence
of clustering and superstructures of the intercalated oxygen. In the following, I will go a little
more into detail with the different doping types.
6.2.1 Doping with strontium to La
2-x
Sr
x
CuO
4
Hole-doping the mo ther compound La
2
CuO
4
by substituting La
3+
with Sr
2+
results in a super-
conductor La
2−x
Sr
x
CuO
4
with critical temperature T
c
depending on the doping x, as found by
Dover et al. [
12]. As seen i n Fig. 6.3(a), the superconductivity appears at doping x = 0.055, [93],
and optimal doping resulting in the highest T
c
at 38 K is x = 0.15, [
94]. The superconductivity
disappears completely at x = 0.28, [
95], and the material behaves as a metal above this.
*
*
There is also a small anomalous dip in T
c
close to x = 0.125, which happens for several La-based cuprates, but
this is not of interest to this thesis.

40 6.3. Crystallographic properties
6.2.2 Doping with oxygen to La
2
CuO
4+y
Hole-doping the mother compound La
2
CuO
4
by intercalating O
2–
results in a superconductor
La
2
CuO
4+y
with critical temperature higher than with the optimal Sr doping. This was first
discovered by Grant et al. [
96], who saw T
c
≈ 40 K, although at the time they did not think that
the cause was i nterstitial oxygen. Following Fig.
6.3(b), the materi al becomes superconducting
at a doping of y ≈ 0.055 with a critical temperature T
c
≈ 32 K, while higher doping values
result in several other separate superconducting phases with criti cal temperatures as high as
T
c
= 45 K, [
72].
*
The phase diagram for La
2
CuO
4+y
, Fig. 6.3(b), is full of miscibility gaps
†
, where the oxygen
ions phase-separate into oxygen-rich and oxygen-poor regions below 290 K, [
99, 100], as well
as a wider variety of structures than that for La
2−x
Sr
x
CuO
4
. The r eason for this is the mobility
of the intercalated O
2–
ions, which phase separate into regions of oxygen-rich and oxygen-poor
regions at temperatures down to 2 0 0 K, [
100].
6.2.3 Co-doping to La
2-x
Sr
x
CuO
4+y
Doping with both Sr and O results in a compound which seemingly follows the superconductiv-
ity for the O doping, by having a critical temperature T
c
≈ 40 K independent of the Sr doping,
[
101].
Just as La
2
CuO
4+y
and La
2−x
Sr
x
CuO
4
, La
2−x
Sr
x
CuO
4+y
exhibit both antiferromagnetic and
superconducting phases, leaving a miscibility gap in the phase diagram similar to what is seen
for La
2
CuO
4+y
. There is no general consensus whether these phases appear in two distinct
structural phases or simply exist as different electronic phases in the material as argued by
Wells et al. [
100] and Mohottala et al. [102], respectively.
6.3 Crystallographic properties
Where the mother compound La
2
CuO
4
is ortorhombic (in the Bmab space group as mentioned
in Sect. 6.1) up to a temperature of around 533 K, [89, 90], doping stabilises the tetragonal
phase and decreases the ortorhombic distortion, resulting in La
2−x
Sr
x
CuO
4+y
having the phase
transition from the LTO to the HTT phase at lower temperatures, depe nding (monotonically for
Sr doping) on the doping, [
103], as can also be seen in Fig. 6.3.
As was earlier mentioned, when the compound is doped with interstitial oxygen, it will
sometimes phase separate into oxygen-rich and an oxygen-poor phases. This has lead to several
different combinations of space group choices, such as Fmmm by Radaelli et al. [
98], Bmab by
Rial et al. [
104], Fmmm in combination with Bmab by Jorgensen et al. [105] and Radaelli et al.
[
99], and several different B mab phases by Chaillout et al. [106]. Most of these investigations
used powder samples instead of single crystal samples, and it was found that any further
conclusions on the space groups would be resulting from measurements on single crystals,
[
106], however the results are still up for discussion even after these experiments. For now,
it seems like the oxygen-rich phase is either in the Bmab or Fmmm space group, while the
oxygen-poor phase is widely agreed to be in the Bmab space group, [
107]. I will go more into
detail on one possible interstitial oxygen superstructure relevant to this discussio n, staging, in
Subsect.
6.3.2.
For both Sr and O doping, some researchers further report undistorted octahedral tilts, [
104,
108], while others report distorted octahedra upon tilting, [105, 106, 109].
*
Several experiments give different doping values for different critical temperatures, [
92, 97, 98], and hence the
superconducting phases presented in Wells et al. [
72] (as in Fig. 6.3(b)) still need more refinement.
†
A mis cibi lit y gap is, in general, a region in a phase diagram where the compound exists as two or more separate
phases, splitting up into other areas of the phase diagram.

Chapter 6. The La
2-x
Sr
x
CuO
4+y
compound 41
Reflection F4/mmm Fmmm Bmab Pccn P4
2
/ncm
(hkl) HTT LTO LTLO LTT
(111) ✔ ✔ ✔ ✔ ✔
(200) ✔ ✔ ✔ ✔ ✔
(020) ✔ ✔ ✔ ✔ ✔
(002) ✔ ✔ ✔ ✔ ✔
(014) ✘ ✘ ✔ ✔ ✔
(104) ✘ ✘ ✘ ✔ ✔
(110) ✘ ✘ ✘ ✔ ✔
(100) ✘ ✘ ✘ ✘ ✔
(010) ✘ ✘ ✘ ✘ ✘
Table 6.1: Overview of a choice of reflections which are allowed and forbidden in different space groups
for La
2−x
Sr
x
CuO
4+y
. The two greyed out last columns are for related compound such as La
2−x
Ba
x
CuO
4
or La
1.875
Ba
0.125−x
Sr
x
CuO
4
, [
110]. Note that because of twinning the reflections might be seen even in the
forbidden positions, because h and k get interchanged. Table from Udby [
154, Tab. 2.1].
Other HTSC cuprates have several other crystallographic phases than the earlier mentioned
HTT and LTO phases for La
2−x
Sr
x
CuO
4+y
, and a small overview of forbidden and allowed re-
flections for some of these can be seen in Tab. 6.1, to give an idea of how to quickly distinguish
them.
It is in generally accepted that t he doped interstitial oxygen are placed in the (1/4,1/4,1/4) sites
of the ortorhombic unit cell, [100, 111], albeit sometimes with a slight displacement along c,
[
101, 104]. See also Tabs. B.3 and B.4 for a couple of structure solution examples, where the
interstitial oxygen position is given both for only oxygen doped and co-doped compounds.
6.3.1 Twinning from HTT to LTO
At high temperatures La
2−x
Sr
x
CuO
4+y
is in the tetragonal (HTT) phase (for suitable values of
the dopings x and y). Cooling the sample down will result in a transition to the ortorhombic
(LTO) phase, by tilting the CuO
6
octahedra as mentioned in Sect. 6.1. Because of this tilting, the
a and b axes change to become shorter and longer, respectively (see e.g. [
105, Fig. 2]), and the
(110) mirror plane from the tetragonal cell is lost, replaced by twin laws. As briefly mentioned
in Subsect.
4.3.1, this change in the lattice parameters can result in twinning of the crystal
structure, where several domains of different orientations occur in the same crystal, [
112].
An example of a (110) twin domain boundary is shown i n Fig.
6.4(a), where the tetragonal
and ortorhombic lattice parameters are also shown for the bottom right of the two domains,
for comparison with Fig.
6.2(b). The two twin domains result in the reciprocal plane shown in
Fig.
6.4(b), and a corresponding plane will result from the corresponding perpendicular (1
¯
10)
twin domain boundary, giving the total reciprocal plane shown in Fig.
6.4(c). Note that the dots
in Fig.
6.4(b) and (c) do not show the intensities of the reflecti ons, but simply the positions, and
that these reciprocal planes are using the tetragonal notation.
It is possible to calculate the angular distance ∆ between two twin reflections by using the
illustration in Fig.
6.5,
∆
2
= 90
◦
− 45
◦
− α, tan α =
a
b
, (6.3.1)
combined to the final result
∆ = 90
◦
− 2 arctan
a
b
. (6.3.2)
Note that the a and b here can be both in tetragonal and ortorhombic notation, since the
difference is simply a factor of
√
2 on both, cancelling out the difference.

42 6.3. Crystallographic properties
Abma
Bmab
(110)
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
−
−
−
+− −−
−
−
−+
++
−+
−
−
−
+−
−−
+−
−
−
−
−+ ++
−+
−
−
+−
−−
+−
−
−
−
−
−+
++
−+
−
+−
−−
+−
−
−
−
−
−
−+
++
−+
+−
−
−
−
−
−
a
O
b
O
a
T
b
T
Cu
+
O above plane
−
O below plane
(a)
h
k
(b)
h
k
(c)
Figure 6.4: Illustration of the twinning which happens for La
2−x
Sr
x
CuO
4+y
during the phase transition
from HTT to LTO. (a) Atom placements in the CuO
2
plane at a (110) twin domain boundary (using the
ortorhombic notation with basis vectors a
O
and b
O
), where the a and b axes are rotated 90
◦
around the c
axis. Note the oxygen close to the boundary, which are influenced by bot h sides as shown with two signs
corresponding to the influence from the left-top and right-bottom, respectively. (b) The reciprocal space
plane (hk0) (ortorhombic notation) for the twin boundary in (a). (c) The total reciprocal plane when taking
all four twins into account. Figure based on [
112, Fig. 1].
h
k
b
a
a
b
b
a
a
b
∆
(110) twin domains
(1
¯
10) twin domains
h
k
45
◦
α
∆/2
b
a
Figure 6.5: The reciprocal space plane (hk0) (tetragonal notation) as in Fig. 6.4(c), but with a larger a/b
ratio to exaggerate the splitting, showing how the angular separation ∆ is found. Note that the angle
between the a and b lines for the same point are always 90
◦
. Figure based on [
154, Fig. 4.8].

Chapter 6. The La
2-x
Sr
x
CuO
4+y
compound 43
For a typical twinning in La
2−x
Sr
x
CuO
4+y
at 10 K, where the lattice parameters are a ≈
5.32 Å and b ≈ 5.36 Å (see Tab.
7.1 in Chap. 7), this angle will be 0.4
◦
.
6.3.2 Staging
For the oxygen-rich phase (y ≥ 0.055) the LTO phase seems to have an additional superlattice
modulation along the c axis from the intercalated oxygens between neighbouring layers of LaO,
[
100]. This superstructure is consistent with a superstructure known from La
2
NiCuO
4+y
, [113],
called staging after the well-investigated stacking phenomenon in graphite, [
114].
Following Fig.
6.6, the modulation happens as follows: within the planes, the CuO
6
octahedra
are ordered like in the mother compound, and perpendicular to these planes (along c) the local
ordering is also the same. However, there are broad antiphase boundaries parallel to the CuO
2
planes consisting of layers where the CuO
6
tilts are reversed. These antiphase boundaries are
presumed to be caused by layers of interstitial oxygen in the favorable positions caused by the
strong in-plane coupling of the octahedra because of their corner sharing: if one octahedra
reverses tilt because of an interstitial oxygen, this might propagate through the entire CuO
2
plane.
The stage number n refers to the induced periodicity of n CuO
2
layers, although non-integer
staging numbers can also occur by mixing of several layerings. The staging numbers for differ-
ent dopings y in La
2
CuO
4+y
can be seen in Fig.
6.3(b). It should be noted that in reality, the
staging is a simplified picture, since data points more towards a smearing out of the bound-
aries in a sinusoidal fashion, [
72].
The staging is exactly what I am looking into in this thesis. As explained in Subsect.
4.3.2 a su-
perstructure of this type will result in scatteri ng between the integer (hkl) points in reciprocal
space belonging to the average structure. In this particular case, the staging peaks will be along
l, since this superstructure is along c. The distance from the integer (hkl) point, ∆l, will be
given by the reciprocal o f the staging number,
∆l =
1
n
, (6.3.3)
in units of l.
Note also that the staging has, so far, only been well investigated for La
2
CuO
4+y
. In this
thesis, I work with co-doped La
2−x
Sr
x
CuO
4+y
, to see how the Sr doping might affect the super-
structures. Note also that La
2−x
Sr
x
CuO
4+y
exhibits many other superstructures apart from the
staging, e.g. satellites along a
∗
, [
115]. I will show some of the overview data we have obtained
in Chap.
10.

44 6.3. Crystallographic properties
Intercalated
oxygen
In paper
plane
Out of
paper plane
tetragonal
I4/mmm
ortorhombic
Bmab
stage 6
stage 4
stage 3
stage 2
b
c
Figure 6.6: Illustration of some CuO
6
tilt structures and staging levels, as seen from the a axis (La ions are
omitted). Note that the Bmab structure is as shown for a single unit cell in Fig.
6.1(b), although here the tilt
angles are exhaggerated for clarity. Note also that the indicated b and c lengths are for the ortorhombic
unit cell (corresponding to F4/mmm for the tetragonal symmetry). The dashed octahedra are displaced
a/2 out of the paper, while the interstitial oxygen are displaced a/4 out of the paper. Figure inspired by
[72, Fig. 2], [100, Fig. 2], and [113, Fig. 2].

45
II.
Experiments

46

47
7
Measuring staging patterns
In this chapter I will go through which samples were used, how the used instruments work,
and how the staging measurements were obtained on these instruments. The data from these
measurements will be presented and analyzed in Chap.
9.
7.1 The samples
Measurements have been performed on four different La
2−x
Sr
x
CuO
4+y
single crystal samples,
with Sr doping values x = 0.00, 0.04, 0.065, and 0.09. All samples have been co-doped with O
by electrolysis in an NaOH solution over several months, making sure that they were maximally
oxygenated into the super-oxygenated regime (where the critical temperature for superconduc-
tivity is measured to an onset temperature of T
c
= 40 K).
Photos of the samples with their crystal axes indicated are shown in Fig. 7.1. Further in-
formation about the samples is shown in Tab.
7.1. Note that between the main measurements
in 2006 and the extra measurements in 200 8 the La
1.96
Sr
0.04
CuO
4+y
sample broke into several
smaller pieces such that only a smaller piece, weighing 0.018 g, was used in the latter experi-
ment. This breaking up is very common for oxygenated samples, since the oxygenation makes
the crystals more brittle.
The samples were all glued onto copper holders using GE varnish, which is good at maintaining
a hold even at very low temperatures. Both La
2
CuO
4+y
, La
1.935
Sr
0.065
CuO
4+y
, and La
1.91
Sr
0.09
-
CuO
4+y
were mounted in the (a
T
+ b
T
, c) plane,
*
while La
1.96
Sr
0.04
CuO
4+y
was mounted in the
(a, b) plane. Note, however, that most of the full reciprocal space was accessible on four-circle
setups.
7.2 The BW5 instrument
Up until October 2012 the X-ray triple-axis instrument BW5 was situated on a high-field wiggler
on one of the straight sides of the positron storage ring DORIS III at HASYLAB (DESY) in Ham-
burg, Germany, [
139]. Since then the storage ring has been replaced by the third generation
synchrotron radiation source PETRA III with a new suite of instruments, [
155, 140]. In this sec-
tion I will explain how the instrument worked (following Bouchard et al. [
116] and Zimmermann
[
168]), and how the measurements used for this thesis were produced.
*
By a
T
+ b
T
it is meant one of the ortorhombic axes, which are typically not distinguishable at ro om temperature.

48 7.2. The BW5 instrument
(a)
a
T
b
T
(b)
b
T
c
(c)
b
T
a
T
(d)
b
T
c
(e)
b
T
c
(f)
c
a
T
(g)
b
T
a
T
(h)
c
Figure 7.1: Photos of the samples with crystal axes indicated. Note that the a and b axes are in
the tetragonal notation, whereas the ortorhombic axes would be a 45
◦
rotation of these around the c
axis). (a/b) La
2
CuO
4+y
, (c/d) La
1.96
Sr
0.04
CuO
4+y
(broke into three parts before 2008 measurements),
(e/f) La
1.935
Sr
0.065
CuO
4+y
, and (g/h) La
1.91
Sr
0.09
CuO
4+y
. Photos (a-d) and (g-h) by Linda Udby. Note that
the holders used for measurements on BW5 are not the holders shown here.

Chapter 7. Measuring staging patterns 49
Doping Mass a b c Approx. size (shape)
Sample
x m [g] [Å] [Å] [Å] [mm×mm×mm]
La
2
CuO
4+y
sLSCOc_0
0.0 0.025 5.30(3)
5.326
5.37(3)
5.326
13.20(3)
13.233
0.5 ×2.5 × 3.5
(irregular disc)
La
1.96
Sr
0.04
CuO
4+y
sLSCO_0.04ABC
sLSCO_0.04B
0.04 0.068
0.018
5.32(3)
5.3583
5.38(3)
5.3583
13.19(1)
13.1963
2 ×3 × 3.9
(irregular cylinder)
La
1.935
Sr
0.065
CuO
4+y
sLSCO_0.065A
0.065 0.093 5.32(2)
5.355
5.36(3)
5.355
13.15(3)
13.135
2 ×2 × 2
(quarter cylinder)
La
1.91
Sr
0.09
CuO
4+y
sLSCO_0.09
0.09 0.417 5.33(2)
5.355
5.34(3)
5.355
13.14(2)
13.135
2.8 ×5.1 × 6.5
(half cylinder)
Table 7.1: Information about the samples used in the data collection for this thesis. Grayed out sample
names are referring to the naming used during the experiments, defined in [
154, Tab. 4.1]. Note the
La
1.96
Sr
0.04
CuO
4+y
crystal which split up into smaller pieces before 2008, where only one smaller piece
was used. Lattice constants are using the ortorhombic notation and are from low temperature neutron
measurements, [
154]. The grayed out lattice constants are the initial values given to the BW 5 instrument,
which were not further refined.
Slits
Collimators
j
Source
Beamstop
Monochromator
2θ
M
Diode
monitor
Sample
2θ
Absorber
wheel
Analyser
2θ
A
Detector
Figure 7.2: Simple illustration of the triple-axis X-ray instrument BW5, as seen from above. Angles are
exaggerated for the sake of illustration. Figure based on information from Bouchard et al. [
116, Fig. 1 and
7] and Zimmermann [
168].
7.2.1 An overview of the instrument
The BW5 instrument follows the general layout for triple-axis scattering instruments, and is
sketched in Fig.
7.2. The instrument uses a Laue transmission geometry, which is common for
triple-axis neutron instruments, but not as much for similar X-ray triple-axis instruments with
lower energies – which more often uses a direct Bragg geometry.
*
The X-ray beam enters the instrument from the wiggler through a set of collimators and
slits defining the beam shape before hitting the monochromator.
The monochromator is typically a SiGe gradient crystal used with the (111) reflection to get
an X-ray energy of around 100 keV (0.124 Å). Such a gradient crystal has a controlled homoge-
nous “gradient” of the lattice parameter over the crystal depth, which enhances the crystal
reflectivity, [
117]. The transmitted beam is caught by a beamstop, while the reflected beam
continues towards the sample position through another set of collimators and slits. Before
hitting the sample, the beam is sent through a diode monitor which is used as a measure of the
intensity of the incoming beam in data treatment.
*
For the Bragg geometry, the scattered beam is seen as coming out of the crystal surface that the incident beam
hits, whereas for Laue transmission geometry, the scattered beam is observed on the “back side” of the crystal. On
neutron triple-axis instruments, it is possible to do both because of the low intera ctio n st re ngt h of the neu tr ons .

50 7.2. The BW5 instrument
The now monochromatic beam hits the sample at the sample position at an angle of 2θ
M
to the incident beam, with one of several possible sample environments installed. For the
measurements used in this thesis, a displex cryostat was installed in order to cool down to
sub-10 K temperatures. This cryostat was mounted on top of a four-circle setup to allow for
fine tuning of the sample alignment with (χ, ψ) angles of up to ±15
◦
. Rotation of the sample
on the sample position is named ω.
The scattered X-rays are sent from the sample position through yet another set of collima-
tors and slits in the scattering angle 2θ to the incoming monochromatic beam. An absorber
wheel with different choices of exchangeable absorbers is also mounted here, making it possi-
ble to choose between different orders of magnitude in the intensity on the detector. Before
hitting the detector, however, the beam is scattered yet again on the analyser crystal in a setup
identical to the monochromator setup. The analyser improves the reciprocal space resolution
and makes sure that the detected X-rays are only the elastically scattered by only reflecti ng
energies equal to the energy picked by the monochromator. This way, unwanted energies from
e.g. fluorescence are also removed.
The detector is finally positioned at an angle of 2θ
A
to the beam from the sample, after
another set of collimators. The detector is a Ge solid state detector, which generally has a back-
ground count rate of less than 1 cts/s, and works linearly up to around 6 × 10
4
cts/s (usually a
safe margin of counts up to 4 × 10
4
cts/s is used).
Often, X-ray instruments are chosen to have the scattering plane be vertical because of polar-
ization effects, but this is not a big problem for the small Bragg angles that the 100 keV beam
gives (in the order of a few degrees), and hence the BW5 instrument was chosen to have the
scattering plane be horizontal – a common choice for neutron scattering instruments. Because
of this it is possible to install heavy sample environments on the sample position.
Also important to note is the fact that the hard X-rays used on BW5 makes it possible to
use more advanced sample environments, e.g. ILL “Orange” cryostats or big magnets which
are usually only possible to use on similar neutron instruments, [
118]. This is not normally
possible for X-ray instruments because of the strong interaction with the material (e.g. the
cryostat surface), but this interaction is weaker for higher energies. For a typical cryostat used
on BW5, the loss to absorption due to the sample environment is less than 20%, see Appx.
A.3.
7.2.2 Measurements
Measurements were done over a number of years by different people, as listed in Tab.
7.2.
The data selected for use for this thesis is only a fraction of the total data measured at the
experiments, and hence I will limit my description to how the selected data were obtained.
For all of the experiments, the gradient SiGe (1 11 ) monochromator and analyser – both with a
mosaicity of 40 arcsec – were used, set to an energy close to 100 keV. For each of the mentioned
measurements below, the cooling rate was controlled to be no faster than 2 K/min, in order to
allow the oxygen to order. Unless otherwise stated, when measurements were running the
temperature was held constant.
2006 experime nt: La
1.96
Sr
0.04
CuO
4+y
The measurements on the La
1.96
Sr
0.04
CuO
4+y
crystal were done between January 31 and Febru-
ary 2, 2006. A few measurements on the La
2
CuO
4+y
crystal were also performed, but these data
are not used in this thesis. A full beam time report has been compiled by Linda Udby, [
156],
but I will here mention a few relevant details.
The energy setting of the monochromator and analyser was set to 99.704 keV, and the beam
size before the monochromator was 1 mm × 1 mm.

Chapter 7. Measuring staging patterns 51
Period Instrument Participants Samples Reference
31.01.2006-
06.02.2006
BW5, DESY,
HASYLAB
L. Udby, N. H. Andersen,
Ch. Niedermayer,
M. von Zimmermann
La
1.96
Sr
0.04
CuO
4+y
,
La
2
CuO
4+y
Udby et al. [
156]
28.11.2007-
03.12.2007
BW5, DESY,
HASYLAB
L. Udby, N. H. Andersen,
S. B. Emery,
M. von Zimmermann
La
1.935
Sr
0.065
CuO
4+y
,
La
1.91
Sr
0.09
CuO
4+y
Udby et al. [
157]
03.10.2008-
10.10.2008
BW5, DESY,
HASYLAB
L. Udby, N. H. Andersen,
T. B. S. Jensen,
M. von Zimmermann
La
2
CuO
4+y
,
La
1.96
Sr
0.04
CuO
4+y
,
La
1.91
Sr
0.09
CuO
4+y
Udby et al. [
158]
21.07.2005-
04.08.2005
RITA-II,
SINQ, PSI
H. E. Mohottala,
L. Udby, B. O. Wells,
Ch. Niedermayer,
N. B. Christensen,
N. H. Andersen
La
2
CuO
4+y
,
La
1.96
Sr
0.04
CuO
4+y
,
La
1.91
Sr
0.09
CuO
4+y
Mohattala et al.
[
159]
29.10.2005-
07.11.2005
RITA-II,
SINQ, PSI
L. Udby, K. Lefmann,
N. B. Christensen,
Ch. Niedermayer
La
1.935
Sr
0.065
CuO
4+y
,
La
1.91
Sr
0.09
CuO
4+y
Udby et al. [
160]
03.10.2006-
19.10.2006
RITA-II,
SINQ, PSI
L. Udby,
N. B. Christensen,
Ch. Niedermayer
La
1.96
Sr
0.04
CuO
4+y
,
La
1.91
Sr
0.09
CuO
4+y
Udby et al. [161]
19.06.2008-
26.06.2008
RITA-II,
SINQ, PSI
L. Udby, J. í Hjøllum,
N. B. Christensen,
Ch. Niedermayer
La
2
CuO
4+y
,
La
1.96
Sr
0.04
CuO
4+y
,
La
1.91
Sr
0.09
CuO
4+y
Udby et al. [
162]
Table 7.2: An overview of staging measurement experiments. M. von Zimmermann and Ch. Niedermayer
were the instrument responsibles for BW5 and RITA-II, respectively.
The temperature dependence of the staging peaks around the Bmab position (014) was mea-
sured at 19 temperatures during cooling from 270 to 10 K, by scanning along l from 3.1 to
4.9 r.l.u. with no attenuation (overview in Fig.
D.1(b) in Appx. D). The same measurements were
done again while heating, with results consistent with the cooling measurements.
*
The peak
was not aligned before each temperature measurement, but it was later concluded that there
had been no change in the length of the b axis during cooling (only the a axis got smaller),
meaning that all the (0 14 ) scans were still properly centered along b.
Several Bmab position peaks were also measured at low temperature, varying l around (012),
(032), and (034), concluding that they had similar staging peaks to (014), [
156, Fig. 12].
2007 experiment: La
1.935
Sr
0.065
CuO
4+y
and La
1.91
Sr
0.09
CuO
4+y
The measurements on the La
1.935
Sr
0.065
CuO
4+y
and La
1.91
Sr
0.09
CuO
4+y
crystals were done be-
tween November 28 and December 3, 2007. A full beam time report has been compiled by
Linda Udby, [
157], but I will here mention a few relevant details.
The energy setting of the monochromator and analyser was set to 99.9995 keV, and the
beam size before the monochromator was 1 mm × 1 mm for the La
1.935
Sr
0.065
CuO
4+y
sample
and 2 mm × 2 mm for the La
1.91
Sr
0.09
CuO
4+y
sample. The larger beam size was because of the
large size of the La
1.91
Sr
0.09
CuO
4+y
sample, following Tab. 7.1.
For the La
1.935
Sr
0.065
CuO
4+y
sample, only six measurements along l around the Bmab position
(014) was measured, during heating up the sample from 25 to 300 K after a long list of other
measurements (overview in Fig.
D.1(c) in Appx. D). The scanned interval was for l from 3.75 to
4.25 r.l.u. after centering ω, θ and 2θ, χ, and ω again on the central peak for each temperature,
to make sure of the alignment. No attenuation was used for these measurements.
*
Because of time restraints, however, I only present the data from cooling i n this thesi s .

52 7.3. The RITA-II instrument
The (004) and (020) peaks were measured as a function of temperature as well (at smaller
temperature intervals), both with an attenuation of 12 and an alignment on ω, θ and 2θ, χ,
and ω again before the actual measurements. The temperature was allowed to change during
both measurements, but limited to a rate of 0.75 K/min such that the longest lasting of these
scans had a temperature gradient of less than 2 K. The (020) measurements ended up being
important for more than alignment purposes because they show extinction in the sample – and
hence give the transition temperature from the HTT to LTO phase more accurately than the
(014) measurements were able to. I will go more into detail on this in Chap.
9.
For the La
1.91
Sr
0.09
CuO
4+y
sample, the temperature dependence along l around (014) was mea-
sured at 19 temperatures during heating from 15 to 300 K with an attenuation of 4 (overview
in Fig.
D.1(d) in Appx. D). The scanned interval was for l from 3.95 to 4.05 r.l.u. after centering
ω, θ and 2θ, χ, and ω again o n the central peak for alignment at each temperature.
The (022) and (020) peaks were also followed as a function of temperature, with an atten-
uation of 12 for the former and 14 for the latter, and an alignment with ω, θ and 2θ, χ, and
ω before each measurement. The temperature was allowed to change during the (020) mea-
surements, but only with a rate of 0.7 K/min such that the longest lasting of the scans had a
temperature gradient of less than 1.5 K.
2008 ex pe rime nt: La
2
CuO
4+y
(and La
1.96
Sr
0.04
CuO
4+y
)
Finally, the measurements on the La
2
CuO
4+y
crystal were done between October 3 and 10, 2007.
Measurements were also done on the La
1.96
Sr
0.04
CuO
4+y
and La
1.91
Sr
0.09
CuO
4+y
crystals, but
only a few measurements for the former mentioned will be used in this thesis. A full beam time
report has been compiled by Linda Udby, [158], but I will here mention a few relevant details.
The energy setting of the monochromator and analyser was set to 99.9996 keV, and the
beam size before the monochromator was 1 mm × 1 mm.
For the La
2
CuO
4+y
sample, the temperature dependence of the staging peaks was measured
along l around the Bmab position (014) at 13 temperatures heating from 10 to 300 K (overview
in Fig.
D.1(a) in Appx. D). The measurements were done both at a long interval with l from 3.1
to 4.9 r.l.u., and a more narrow interval between 3.9 and 4.1 r.l.u. with smaller point distance
to resolve the central peak better. Before each measurement, the sample was aligned on (220)
and (004) by centering in ω, θ and 2θ, χ, and ω again, in order to fix alignment changes caused
by the temperature change.
The (020) peak was also measured as a function of temperature with no attenuation, after
centering on the peak in ω, θ and 2θ, χ, and ω again for each temperature.
For the La
1.96
Sr
0.04
CuO
4+y
sample, the (020) and (004) peaks were followed as a function of
temperature, both with an attenuation of 12 and alignment on ω, θ and 2θ, χ, and ω again
before the actual measurements.
7.3 The RITA-II instrument
The neutron triple-axis instrument RITA-II is situated on the SINQ neutron source at PSI in Villi-
gen, Switzerland. In this section I will very briefly explain how the instrument works, following
Lefmann et al. [
119] and the instrument website [141]. Several measurements sim ilar to the
BW5 measurements were performed for the same samples by Linda Udby, and details of these
experiments can be found in her thesis, [
154].

Chapter 7. Measuring staging patterns 53
Slits
Collimators
j
Source
Guides
Beamstop
Monochromator
2θ
M
Optional
Be filter
Monitor
Sample
2θ
Optional
Be/BeO filter
Analysers
2θ
A
PSD
Detector
Figure 7.3: Simple illustration of the triple-axis instrument RITA-II, as seen from above. Figure based on
information from Lefmann et al. [
119] and PSI [141].
7.3.1 An overview of the instrument
The RITA-II instrument follows a typical layout for triple-axis scattering instruments, just like
BW5, and is sketched in Fig.
7.3. The instrument is at the end of the thermal neutron guide
1RNR13, where the vertically focusing PG(002) monochromator can provide neutrons with en-
ergies between 2.5 and 20 meV. Any transmitted beam is caught by a beamstop behind the
monochromator, while the reflected beam continues at an angle 2θ
M
to the incident beam, to-
wards the sample position through a set of collimators and adjustable slits. An optional Be
filter to remove higher-order neutrons is also possible here.
The intensity of the monochromatic beam is detected by a monitor right before the sample
position. At the sample position it is possible to insert sample environments ranging from
furnaces and pressure cells to cryostats (such as the standard ILL “Orange cryostat”) and cry-
omagnets.
The scattered beam continues at the scattering angle 2θ through a second optional cooled
Be or BeO filter with built in radial collimation. It then hits the flexible nine-bladed PG(002)
analyser through a set of slits. This advanced analyser makes it possible to record data in
several different modes, e.g. a monochromatic focusing mode and a flat mode.
The position sensitive 3 He detector plate is located at the scattering angle 2θ
A
from the
analyser, after a rough radial collimator to separate the signals from the individual analyser
blades. This detector is fairly large, 500 mm × 300 mm.
7.3.2 Measurements
Measurements were done over a number of years as listed in Tab.
7.2. Since I only show results
from these experiments for comparison with the BW5 data, I will not go into detail with how
they were obtained here – this can be found elsewhere, [
154, 159–162].

54 7.3. The RITA-II instrument

55
8
Mapping out reciprocal space
In this chapter I will outline how the reciprocal space maps, givi ng overviews of several super-
structures – including staging – were measured. I will keep this short since the data has not
been fully analyzed yet and is simply here for comparison and outlook of what is still to be
done. The data will be presented in Chap. 10 with a short discussion of what was observed.
8.1 The DMC instrument
The neutron powder diffractometer DMC is situated on the SINQ neutron source at PSI in Vil-
ligen, Switzerland. In this section I will briefly explain how the instrument works, following
Schefer et al. [
120] and the i nstrument website [142], and describe how it was used to measure
reciprocal space maps for several of our samples.
8.1.1 An overview of the instrument
The DMC instrument follows a simple two-axis scattering layout, as sketched in Fig.
8.1. The
instrument is situated at the cold neutron guide RNR12, where the vertically focusing PG(002)
monochromator
*
can provide neutrons with energies between 3 and 15 meV (between 2.3 and
5.1 Å). The transmitted beam is sent on through the guide system to another instrument, while
the reflected beam continues at an angle 2θ
M
to the incident beam, towards the sample position
through several set of slits and an optional collimator.
The intensity of the monochromatic beam is measured by a monitor in the beam before the
sample position. It is possible to install a variety of sample environments in the sample area,
amongst them the standard ILL “Orange” cryostat (on top of a three-circle setup to allow for
fine tuning of the sample alignment with (χ, ψ) angles of up to ±15
◦
), which we used for all
our reciprocal space mapping measurements.
In front of the detector is an oscillating radial collimator, covering the full angular span of
the detector. The detector itself is interesting, since it uses 400 linear position-sensitive BF
3
counters in a large “banana” array spanning 79.8
◦
at a radius of 1.5 m. The angular distance
between the detectors is 0.2
◦
, and it is normal to measure once, rotate the detector 0.1
◦
in the
horizontal plane, and measure again, in order to get a better angular resolution.
*
It is also possible to instead use a Ge crystal, scattering on th e (311), (511), or (400) planes.

56 8.1. The DMC instrument
Slits
Collimators
j
Source
Guides
Guides
Monochromator
2θ
M
Monitor
Sample
2θ
Radial
collimator
Radial 80
◦
detector
Figure 8.1: Simple illustration of the powder diffraction instrument DMC, as seen from above. Figure
based on information from Schefer et al. [
120] and PSI [142].
Period Instrument Participants Samples
12.11.2012-
17.11.2012
DMC, SINQ, PSI P. J. Ray, L. Udby,
K. Lefmann, M. Frontzek
La
2
CuO
4+y
05.07.2013-
07.07.2013
DMC, SINQ, PSI L. Udby, U. B. Hansen,
M. Frontzek
La
2
CuO
4+y
,
La
1.94
Sr
0.06
CuO
4+y
24.11.2013-
28.11.2013
DMC, SINQ, PSI P. J. Ray,
M. Frontzek
La
1.94
Sr
0.06
CuO
4+y
,
La
1.91
Sr
0.09
CuO
4+y
Table 8.1: An overview of reciprocal space mapping experiments. M. Frontzek was the instrument respon-
sible for DMC.
8.1.2 Measurements
Measurements
*
have been done over a number of experiments, as outlined in Tab. 8.1. The
samples are the same, or other c rystals of the same composition, as the samples used at BW5
as presented in Chap.
7. Crystals were mounted in the (a, b) and the (b, c) planes on aluminium
holders, using aluminium thread and GE varnish.
†
In order to map out slices of reciprocal space, a sample is placed in the cryostat, and counts
are then recorded in the channels of the detector as a function of the horizontal rotation of
the crystal, ω. In this way a (2θ, ω) map is collected, which can then be transformed into
the recipro cal lattice directions of the given crystal or ientation. Measurements were done for a
low temperature (either 2, 5 , or 10 K), a medium temperature of 100 K, and room tem perature
(300 K), for both the (a, b) and (b, c) alignments, for most of the samples. A full overview of
the measurements is presented in Appx.
E.
*
The used samples were not all the same samples as the samples measured on at BW5 and RITA-II, but the compo-
sitions are the same, so I will not go into further details with this here, since I do not perform any deeper analysis of
the reciprocal space map results.
†
Araldite glue covered with a small amount of Gd
2
O
3
was u se d for the La
2
CuO
4+y
sample at the November 2012
experiment, which resulted in a features in the background.

57
III.
Results and analysis

58

59
9
Staging measurements
In this chapter I will go through the measurements that were done on BW5, analysing them to
see the phase transitions. I will also show some of the complementary neutron results taken
on RITA-II from Udby [
154].
Unless otherwise stated, peak fits have been done with the iFit data analysis framework,
[
164], while power law fits are done with the spec1d software package, [165], both for MATLAB,
[
166].
9.1 Initial treatment of BW5 staging data
The raw (014) l-scan data for all four samples at all the measured temperatures can be seen
in Appx. D, where they are shown plotted individually (with an o verview in Fig. D.1 and the
individual plots in Figs.
D.2 to D.13).
For the La
2
CuO
4+y
sample the measurements were split up into a wide scan and a narrower
central scan with a closer density of measurements. In order to treat these scans as one, they
have been combined into a single dataset for each temperature.
*
All data has been normalized
to the monitor value. The only other initial treatment of the raw data was to remove a few
erroneous points where the detector had zero counts as an obvious error. For the La
1.96
Sr
0.04
-
CuO
4+y
sample at T = 70 K there was one of these points, while for the La
2
CuO
4+y
sample
there were two points at T = 1 0 K, one point at T = 30 K, and one point at T = 290 K. These
removed points are marked with red in the raw data in Figs.
D.2 to D.13.
The data for each sample has been plotted together as an overview in Fig.
9.1, and on a
logarithmic scale in Fig.
9.2. To be able to compare the data more directly, they have also been
plotted in this way, but with the same l axis limits, in Figs.
D.14 and D.15.
9.1.1 Correcting for changing lattice parameter
With the exception of the La
2
CuO
4+y
measurements, each l scan at each temperature was
aligned on the central (014) peak in order to scan at the correct position in reciprocal space.
For the La
2
CuO
4+y
sample, it was concluded that even though the alignment was not done, the
position was still correct, [15 8 ]. These alignments did, however, not change the UB matrix of
the instrument, and hence the positions in reciprocal space written down in the data files did
not have the correct values. The position of the (014) peak should always be at l = 4 in the
*
Combining the data is done after normalizing to the monitor value, using the combine() command from the
spec1d software package for MATLAB, [
165, 166].

60 9.1. Initial treatment of BW5 staging data
3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8
0
20
40
60
80
100
Intensity I per monitor [a.u.]
Temperature T [K]
50
100
150
200
250
300
(a)
Intensity I per m onitor [a.u.]
Temperature T [K]
3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8
0
5
10
15
20
Intensity I per monitor [a.u.]
Temperature T [K]
0
50
100
150
200
250
(b)
Intensity I per m onitor [a.u.]
Temperature T [K]
3.8 3.9 4 4.1 4.2 4.3
0
50
100
150
Intensity I per monitor [a.u.]
Temperature T [K]
0
50
100
150
200
250
300
(c)
Intensity I per m onitor [a.u.]
Temperature T [K]
3.98 4 4.02 4.04 4.06 4.08
0
50
100
150
200
l through (014) [r.l.u.]
Intensity I per monitor [a.u.]
Temperature T [K]
0
50
100
150
200
250
300
(d)
l through (014) [r.l.u.]
Intensity I per m onitor [a.u.]
Temperature T [K]
Figure 9.1: Measurement on the (a) La
2
CuO
4+y
, (b) La
1.96
Sr
0.04
CuO
4+y
, (c) La
1.935
Sr
0.065
CuO
4+y
, and
(d) La
1.91
Sr
0.09
CuO
4+y
crystals along l over the (014) peak on BW5 before any correction of the instrument
lattice parameter. The curve color shows the temperature, with blue being colder and red being warmer.
Note the different l axis intervals on the graphs – the plots are shown all with the same interval in figure
Fig. D.14.

Chapter 9. Staging measurements 61
3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8
10
0
10
1
10
2
Intensity I per monitor [a.u.]
Temperature T [K]
50
100
150
200
250
300
(a)
Intensity I per m onitor [a.u.]
Temperature T [K]
3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8
10
−1
10
0
10
1
Intensity I per monitor [a.u.]
Temperature T [K]
0
50
100
150
200
250
(b)
Intensity I per m onitor [a.u.]
Temperature T [K]
3.8 3.9 4 4.1 4.2 4.3
10
0
10
1
10
2
Intensity I per monitor [a.u.]
Temperature T [K]
0
50
100
150
200
250
300
(c)
Intensity I per m onitor [a.u.]
Temperature T [K]
3.98 4 4.02 4.04 4.06 4.08
10
0
10
2
l through (014) [r.l.u.]
Intensity I per monitor [a.u.]
Temperature T [K]
0
50
100
150
200
250
300
(d)
l through (014) [r.l.u.]
Intensity I per m onitor [a.u.]
Temperature T [K]
Figure 9.2: As figure Fig. 9.1, but with the intensity on a logarithmic scale. (a) La
2
CuO
4+y
, (b) La
1.96
Sr
0.04
-
CuO
4+y
, (c) La
1.935
Sr
0.065
CuO
4+y
, and (d) La
1.91
Sr
0.09
CuO
4+y
. These plots are shown all with the same l
axis in figure Fig.
D.15.

62 9.1. Initial treatment of BW5 staging data
3.96 3.98 4 4.02 4.04
0
20
40
60
80
100
Intensity I per monitor [a.u.]
(a)
Intensity I per m onitor [a.u.]
3.6 3.8 4 4.2 4.4
0
5
10
15
20
Intensity I per monitor [a.u.]
(b)
3.8 3.9 4 4.1 4.2
0
20
40
60
80
100
120
140
160
180
l through (014) [r.l.u.]
Intensity I per monitor [a.u.]
(c)
l through (014) [r.l.u.]
Intensity I per m onitor [a.u.]
3.98 4 4.02 4.04 4.06
0
20
40
60
80
100
120
140
160
180
l through (014) [r.l.u.]
Intensity I per monitor [a.u.]
(d)
l through (014) [r.l.u.]
Figure 9.3: Examples of fits done to find the correction on the l axis. (a) La
2
CuO
4+y
at 100 K, (b) La
1.96
-
Sr
0.04
CuO
4+y
at 70 K, (c) La
1.935
Sr
0.065
CuO
4+y
at 75 K, and (d) La
1.91
Sr
0.09
CuO
4+y
at 100 K. The green
points are the original data, with the total fitted curve in blue (with the single peaks shown as dashed
lines). The gray data is scaled on the l axis to get the fitted curve (also shown in gray) centered around
l = 4. Note that the error bars are smaller than the points.
measurements, but this is not the case in the data files. The scans are centered around the im-
mediate central point of the scan at the current temperature, but the l value of the data has to
be corrected: the reciprocal lattice points are a bit off at lower temperatures, where the lattice
parameters have changed slightly.
To corre ct for this, the position of the central peak of the data in each scan has been found
with a fit. This value has then been used as a factor on the l axis in orde r to center the data
correctly – as if the lattice parameter had actually been recalculated at each temperature. Ex-
amples of these fits for one temperature for each sample have been shown in Fig.
9.3, while an
overview of the fitted positions for all of the datasets is shown in Fig.
9.4.
For La
2
CuO
4+y
the fitting function was simply a Gaussian
*
on a restricted narrow central cut-
out of the scan, between l = 3.95 and 4.05 r.l.u., see Fig.
9.3(a). The fitting function had to be
a little more complicated for the three other samples, where the full measured intervals were
taken into account.
*
For an overview of the functions used in this thesi s, see Appx.
C.

Chapter 9. Staging measurements 63
For La
1.96
Sr
0.04
CuO
4+y
the fitting function was two Lorentzians around a common center,
with a Lorentzian with a free center between these (see Fig.
9.3(b)). Since the inner central peak
is very small (especially for higher temperatures where it almost disappears), and hence very
difficult to fit precisely, the mid-point of the two outer peaks was used for the correction of
the l axis.
For La
1.935
Sr
0.065
CuO
4+y
the fitting function was a central Lorentzian with two more Lorent-
zians in the shoulders around another free common center (see Fig.
9.3(c)). The position of
the first mentioned was used for the correction of the l axis. One exception, however, was the
T = 300 K dataset, which does not show a peak. I have tried fitting it with a simple Lorentzian
on a sloping background, but there is no trace of a peak, and hence the l axis for this temper-
ature has simply been scaled to a linearly extrapolated point from the central positions of the
three lower closest tem peratures.
For La
1.91
Sr
0.09
CuO
4+y
the fitting function was like the function for La
1.935
Sr
0.065
CuO
4+y
,
except that the central peak was fitted with a Gaussian instead of a Lorentzian (see Fig.
9.3(d)).
From the peak positions it is possible to calculate the length of the sample axis c from the
one given to the instrument initially (following Tab.
7.1). Naming the instrument value for the
lattice parameter c
instr
and the real lattice parameter c
real
,
l
center
2π
c
instr
= 4
2π
c
real
⇒ c
real
=
4c
instr
l
center
, (9.1.1)
where l
center
is the center position found for the dataset as shown in Fig.
9.4. The found values
are shown in Fig.
9.5. The low t emperature values in this figure can be compared with the
values from Tab.
7.1 as shown in Tab. 9.1.
The La
2
CuO
4+y
value matches easily within the uncertainty, but for the other samples the
neutron values are significantly larger, although the general trend of higher Sr doping mean-
ing a shorter lattice parameter is still seen. This discrepancy of the higher Sr doped lattice
parameters between the neutron results and these will be discussed further in Sect. 11.1.
The full datasets after the correction are shown in Figs.
9.6 and 9.7, to compare with the uncor-
rected full datasets in Figs.
9.1 and 9.2. As was done with the uncorrected data, the corrected
data has also been plotted with the same l axis limits in Fig.
D.16 and Fig. D.17.
For comparison, the central regions of the data is shown before and after the scaling of the
l axis has been done in Figs.
D.18 to D.21. The data after the scaling is now clearly centered
around l = 4 as was expected.
Lattice parameter Lattice parameter Temperature T for X-ray
Sample
c from neutrons [Å] c from X-rays [Å] lattice parameter value [K]
La
2
CuO
4+y
13.20(3) 13.2038(11) 10.279(10)
La
1.96
Sr
0.04
CuO
4+y
13.19(1) 13.1367(7) 9.331(19)
La
1.935
Sr
0.065
CuO
4+y
13.15(3) 13.0857(2) 25.001(4)
La
1.91
Sr
0.09
CuO
4+y
13.14(2) 13.0346(1) 15.003(15)
Table 9.1: A comparison of lattice parameter lengths c for the four samples from neutron experiments,
[
154] (also given in T ab. 7.1), with the found lattice parameters for the lowest temperature as shown in
Fig.
9.5, using Eq. (9.1.1). Note that the X-ray lattice parameter for La
1.96
Sr
0.04
CuO
4+y
is found for the
mid-point between the large staging peaks – not the center of the smaller central peak.

64 9.1. Initial treatment of BW5 staging data
0 50 100 150 200 250 300
3.998
4
4.002
4.004
4.006
4.008
Peak position [r.l.u.]
(a)
Central position [r.l.u.]
0 50 100 150 200 250
4.002
4.004
4.006
4.008
4.01
4.012
4.014
4.016
4.018
Peak position [r.l.u.]
(b)
0 50 100 150 200 250 300
4
4.005
4.01
4.015
extrapolated
Temperature T [K]
Peak position [r.l.u.]
(c)
Temperature T [K]
Central position [r.l.u.]
0 50 100 150 200 250 300
4.014
4.016
4.018
4.02
4.022
4.024
4.026
4.028
4.03
Temperature T [K]
Peak position [r.l.u.]
(d)
Temperature T [K]
Figure 9.4: Overview of the peak center positions before the l axis correction. (a) La
2
CuO
4+y
, (b) La
1.96
-
Sr
0.04
CuO
4+y
, (c) La
1.935
Sr
0.065
CuO
4+y
, and (d) La
1.91
Sr
0.09
CuO
4+y
. Note that the T = 300 K point for
La
1.935
Sr
0.065
CuO
4+y
is an extrapolation from the other points, since no peak was observed.
0 50 100 150 200 250 300
13.05
13.1
13.15
13.2
Temperature T [K]
c axis length [
˚
A]
Temperature T [K]
c axis length [Å]
Figure 9.5: Found lattice parameter lengths c from the l axis corrections on BW5 data following Fig. 9.4.
Green points are La
2
CuO
4+y
, blue points are La
1.96
Sr
0.04
CuO
4+y
, yellow points are La
1.935
Sr
0.065
CuO
4+y
,
and red points are La
1.91
Sr
0.09
CuO
4+y
.

Chapter 9. Staging measurements 65
3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8
0
20
40
60
80
100
Intensity I per monitor [a.u.]
Temperature T [K]
50
100
150
200
250
300
(a)
Intensity I per m onitor [a.u.]
Temperature T [K]
3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8
0
5
10
15
20
Intensity I per monitor [a.u.]
Temperature T [K]
0
50
100
150
200
250
(b)
Intensity I per m onitor [a.u.]
Temperature T [K]
3.7 3.8 3.9 4 4.1 4.2
0
50
100
150
Intensity I per monitor [a.u.]
Temperature T [K]
0
50
100
150
200
250
300
(c)
Intensity I per m onitor [a.u.]
Temperature T [K]
3.95 3.96 3.97 3.98 3.99 4 4.01 4.02 4.03 4.04 4.05
0
50
100
150
200
l through (014) [r.l.u.]
Intensity I per monitor [a.u.]
Temperature T [K]
0
50
100
150
200
250
300
(d)
l through (014) [r.l.u.]
Intensity I per m onitor [a.u.]
Temperature T [K]
Figure 9.6: Measurement on the (a) La
2
CuO
4+y
, (b) La
1.96
Sr
0.04
CuO
4+y
, (c) La
1.935
Sr
0.065
CuO
4+y
, and
(d) La
1.91
Sr
0.09
CuO
4+y
crystals along l over the (014) peak on BW5 after correction of the first-axis, see
also Figs.
D.18 to D.21. The curve color shows the temperature, with blue being colder and red being
warmer.

66 9.1. Initial treatment of BW5 staging data
3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8
10
0
10
1
10
2
Intensity I per monitor [a.u.]
Temperature T [K]
50
100
150
200
250
300
(a)
Intensity I per m onitor [a.u.]
Temperature T [K]
3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8
10
−1
10
0
10
1
Intensity I per monitor [a.u.]
Temperature T [K]
0
50
100
150
200
250
(b)
Intensity I per m onitor [a.u.]
Temperature T [K]
3.7 3.8 3.9 4 4.1 4.2
10
0
10
1
10
2
Intensity I per monitor [a.u.]
Temperature T [K]
0
50
100
150
200
250
300
(c)
Intensity I per m onitor [a.u.]
Temperature T [K]
3.95 3.96 3.97 3.98 3.99 4 4.01 4.02 4.03 4.04 4.05
10
0
10
2
l through (014) [r.l.u.]
Intensity I per monitor [a.u.]
Temperature T [K]
0
50
100
150
200
250
300
(d)
l through (014) [r.l.u.]
Intensity I per m onitor [a.u.]
Temperature T [K]
Figure 9.7: As figure Fig. 9.6, but with the intensity on a logarithmic scale. Measurement on the
(a) La
2
CuO
4+y
, (b) La
1.96
Sr
0.04
CuO
4+y
, (c) La
1.935
Sr
0.065
CuO
4+y
, and (d) La
1.91
Sr
0.09
CuO
4+y
crystals along
l over the (014) peak on BW5 after correction of the first-axis.

Chapter 9. Staging measurements 67
9.2 Fitting to find integrated intensities for BW5 staging data
After the correction of the staging data, it is now possible to fit the scans to obtain the total
intensities of the contained peaks. Since the difficulty level of the data treatment was very
different for the four samples, I will start with La
1.96
Sr
0.04
CuO
4+y
which was the most straight-
forward. Then I will go over the data treatment for the La
1.935
Sr
0.065
CuO
4+y
and La
1.91
Sr
0.09
Cu-
O
4+y
samples, and finally for the La
2
CuO
4+y
sample, which was the most complicated.
In the following subsections the fitting functions are chosen empirically, based on which
peak shapes gives a better fit. The desired end result is the integrated intensities of the reflec-
tions – not the peak shapes or resolution – and hence Lorentzians and Gaussians are used. It
could have also been interesting to use Voigts (explained in further detail in Appx.
C) for the
fitting, however with the large number of fitting parameters already used this is an unwanted
complication of the data analysis.
9.2.1 Fitting La
1.96
Sr
0.04
CuO
4+y
peaks
The measured peaks for the La
1.96
Sr
0.04
CuO
4+y
sample were shown in Fig. 9.6(b) and on a loga-
rithmic scale in Fig.
9.7(b). In order to find the integrated intensity for each of the elements in
the spectra, each full scan has been fitted with a constant background and three Lorentzians:
one small central peak for the Bmab peak, and one larger peak on either side for the staging.
The mid-poi nt of the staging peaks and their distance to this are used as fitting parameters, in-
stead of fitting the location of the two individual peaks directly. Examples of fits for the T = 20
and 230 K datasets are shown in Fig.
9.8.
The position of the central Bmab peak has been held fixed to the T = 60 K value for temper-
atures above this, since the peak intensity becomes very small at higher temperatures, making
3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8
0
5
10
15
20
Intensity I per monitor [a.u.
(a)
Intensity I per m onitor [a.u.]
3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8
0
0.5
1
1.5
2
2.5
l through (014) [r.l.u.]
Intensity I per monitor [a.u.
(b)
l through (014) [r.l.u.]
Intensity I per m onitor [a.u.]
Figure 9.8: Examples of fits done for the La
1.96
Sr
0.04
CuO
4+y
BW5 data, here for the (a) T = 20 K and
(b) T = 230 K measurements. The individual peaks of the fit (plus the background) are shown as dashed
lines, while the full fit is the solid blue line. For points without visible errorbars, the errors are smaller
than the marker.

68 9.2. Fitting to find integrated intensities for BW5 staging data
0 50 100 150 200 250
3.8
3.9
4
4.1
4.2
Temperature T [K]
Staging peak positions [r.
Temperature T [K]
Peak positions [r.l.u.]
Figure 9.9: Positions of fitted peaks for the La
1.96
Sr
0.04
CuO
4+y
sample. The blue points show the position
of the central Bmab peak while the green points show the mid-point of the staging peaks, which are in turn
positioned at the yellow and red points. The points with crossmarks indicate that the position was held
fixed to the T = 60 K value. For points without visible errorbars, the errors are smaller than the marker.
the fit difficult. In the same way, the width of the Bmab peak is held fixed to the T = 90 K value
for temperatures above this.
An overview of the positions of the fitted peaks is shown in Fig.
9.9. Note how the Bmab
peak (shown in blue) is always at a higher value of l compared to the mid-poi nt of the staging
peaks (shown in green). This distance is δl = 0.014(2) r.l.u., as determined from the fits at
temperatures below 60 K, above which the Bmab position is held fixed in the fit (although the
distance seems to stay the same all the way to the highest measured temperature, assuming that
the Bmab peak is correctly positioned at the fixed value). This indicates that the superstructure
is not a superstructure for the Bmab reflection, but instead possibly for a structural Fmmm
phase (for which the central position is not an allowed reflection following Tab.
6.1) with a
slightly longer lattice parameter c. I will go more into detail with this in Subsect.
11.1.1.
The distance between the staging peaks (their positions shown as yellow and red points)
and their mid-point (green points) for temperatures below 200 K has an average value ∆l =
0.193(8) r.l.u., corresponding to a staging number of just above n = 5 following Eq. (
6.3.3).
Above 200 K, the distance slowly increases all the way up to ∆l = 0.227(10) r.l.u. for T = 270 K,
corresponding to a staging number n of between 4 and 5.
The integrated intensities for the fitted peaks are calculated as explained in Appx.
C and shown
in Fig.
9.10. I will work further with these intensities in Sect. 9.4, where I find transition tem-
peratures and critical exponents for all the samples together. As an initial observation, the
staging peaks follow power laws nicely at T > 150 K (below which saturation takes a hold) – as
is expected for a continuous phase transition, following Chap.
5 – while the central Bmab peak
seems to fall very slowly, and might have a large amount of critical scattering. The transition
temperature for the Bmab peak also seems much lower than for the staging peaks.
The widths of the fitted peaks are shown in Fig.
9.11. Since the intensity of the central Bmab
peak was so small, and the width was hence kept fixed for higher temperatures, it is difficult to
conclude anything for this parameter. For the staging peaks, however, the width seems fairly
constant at low temperatures, but rising significantly when getting close to the phase transition.
9.2.2 Fitting La
1.935
Sr
0.065
CuO
4+y
peaks
The measured peaks for the La
1.935
Sr
0.065
CuO
4+y
sample were shown in Fig.
9.6(c) and on a
logarithmic scale in Fig.
9.7(c). To find the integrated intensities for each of the elements in the
spectra, each full scan has been fitted with a constant background plus three Lorentzians: a
large central peak for the Bmab peak, and one small peak on either side for the staging. As for
the La
1.96
Sr
0.04
CuO
4+y
sample, the mid-point of the staging peaks and their distances to this

Chapter 9. Staging measurements 69
0 50 100 150 200 250
0
0.1
0.2
0.3
Temperature T [K]
Intensity I [a.u.]
(a)
Temperature T [K]
Integrated intensity I [a.u.]
0 50 100 150 200 250
0
0.5
1
1.5
2
Temperature T [K]
(b)
Temperature T [K]
Figure 9.10: Integrated peak intensities of fitted peaks for the La
1.96
Sr
0.04
CuO
4+y
sample for (a) the
central Bmab peak and (b) the staging peaks: red points for the low l and yellow for the high l peak (as in
Fig.
9.9).
0 50 100 150 200 250
0.055
0.06
0.065
0.07
0.075
Temperature T [K]
width [r.l.u.]
(a)
Temperature T [K]
Peak width (HWHM) [r.l.u.]
0 50 100 150 200 250
0.05
0.1
0.15
Temperature T [K]
(b)
Temperature T [K]
Figure 9.11: Widths of fitted peaks for the La
1.96
Sr
0.04
CuO
4+y
sample for (a) the central Bmab peak (held
fixed at the T = 90 K value at temperatures higher than this, marked with crossmarks) and (b) the staging
peaks: red points for the low l and yellow for the high l peak (as in Fig.
9.9).
are used as fitting parameters. Examples of these fits for the T = 25 and 175 K datasets are
shown in Fig.
9.12. The T = 300 K data did not exhibit signs of a peak even when tried fitted
with a single peak on a sloping background.
The choice of fitting two side-peaks in the shoulders of the central Bmab peak – instead of a
wide bottom peak below it – was simply because the former gave a slightly better fit. Fitting with
a single Lorentzian, two Lorentzians, two Gaussians, or a Lorentzian and a Gaussian, all gave
worse goodness-of-fit values for the fits than fitting with a central Lorentzian and a Lorentzian
in either shoulder. This result could also be seen by eye, noticing how one or two peaks only
could not account for both the width and the tails of the total signal. As an example, for the
T = 25 K dataset the following correlations (a value between 0 and 1 given for an iFit fit, with
higher values corresponding to a better fit) were found: 0.868 for one Lorentzian, 0.917 for two
Lorentzians, 0.979 for two Gaussians, 0.921 for a Gaussian and a Lorentzian, and 0.995 for a
Lorentzian with a Lorentzian on each side. With more time, it would be interesting to also do a
full analysis with the latter mentioned fit function, especially if i t is possible to get more data
than the only six scans (one of them with no peak) that have currently been obtained from BW5.
The widths of the staging peaks were held fixed to the T = 25 K value for temperatures
above this, since these peaks are so small in intensity that the fit is difficult. In the same
way, the mid-point and distance to the mid-point for the staging peaks were kept fixed to the
T = 175 K value for the 225 K scan.
An overview of the positions of the fitted peaks is shown in Fig.
9.13. The position of the Bmab
peak and the mid-point of the staging peaks – shown in blue and green, respectively – are very

70 9.2. Fitting to find integrated intensities for BW5 staging data
3.7 3.8 3.9 4 4.1 4.2
0
50
100
150
200
Intensity I per monitor [a.u.
(a)
Intensity I per m onitor [a.u.]
3.8 3.85 3.9 3.95 4 4.05 4.1 4.15 4.2 4.25
0
20
40
60
80
100
120
l through (014) [r.l.u.]
Intensity I per monitor [a.u.
(b)
l through (014) [r.l.u.]
Intensity I per m onitor [a.u.]
Figure 9.12: Examples of fits done for the La
1.935
Sr
0.065
CuO
4+y
BW5 data, here for the (a) T = 25 K and
(b) T = 175 K measurements. The individual peaks of the fit are shown as dashed lines (plus the constant
background), while the full fit is the solid blue line. For points without visible errorbars, the errors are
smaller than the marker.
0 50 100 150 200
3.95
4
4.05
Temperature T [K]
Staging peak positions [r.
Temperature T [K]
Peak positions [r.l.u.]
Figure 9.13: Positions of fitted peaks for the La
1.935
Sr
0.065
CuO
4+y
sample. The blue points show the
position of the central Bmab peak while the green points show the mid-point of the staging peaks, which
are in turn positioned at the yellow and red points. Note that no points are shown for the T = 300 K data
since no peak was seen here. The crossmarks indicate parameters that were fixed to the value from the
neighboring scan with lower temperature. For point s without visible errorbars, the errors are smaller than
the marker.
close to each other, with an average distance of only δl = 0.0017(11) r.l.u. (not including the
T = 225 K data, where the mid-point and distance to this for the staging peaks was fixed at
the 175 K values). As opposed to the L a
1.96
Sr
0.04
CuO
4+y
sample, though, here the Bmab peak
is at lower values of l compared to the mid-point of the staging peaks. With this very small δl,
and the large uncertainty on the value, there is not much incentive to say that two structural
phases, one Bmab and one Fmmm as was concluded for the La
1.96
Sr
0.04
CuO
4+y
sample, are at
play for this sample.
The distance between the staging peaks (their positions shown in yellow and red points)
and their mid-point (green points) for the lowest temperature of 25 K was found to be ∆l =

Chapter 9. Staging measurements 71
0 50 100 150 200 250 300
0
2
4
6
8
Temperature T [K]
Intensity I [a.u.]
(a)
Temperature T [K]
Integrated intensity I [a.u.]
0 50 100 150 200 250 300
0
0.5
1
1.5
Temperature T [K]
(b)
Temperature T [K]
Figure 9.14: Integrated peak intensities of fitted peaks for the La
1.935
Sr
0.065
CuO
4+y
sample for (a) the
central Bmab peak and (b) the staging peaks: red points for the low l and yellow for the high l peak (as in
Fig.
9.13). For points without visible errorbars, the errors are smaller than the marker.
0 50 100 150 200
0.008
0.009
0.01
0.011
0.012
0.013
Temperature T [K]
width [r.l.u.]
(a)
Temperature T [K]
Peak width (HWHM) [r.l.u.]
0 50 100 150 200
0.04
0.045
0.05
0.055
0.06
0.065
0.07
Temperature T [K]
(b)
Temperature T [K]
Figure 9.15: Widths of fitted peaks for the La
1.350
Sr
0.650
CuO
4+y
sample for (a) the central Bmab peak
and (b) the staging peaks: red points for the low l and yellow for the high l peak (as in Fig.
9.13). Both
staging peak widths were held fixed at the T = 25 K value at temperatures higher than this (marked with
crossmarks). For points without visible errorbars, the errors are smaller than the marker.
0.0755(14) r.l.u., corresponding to a staging number n larger than 13. For higher temperatures
the distance grows slightly, to ∆l = 0.0847(11) r.l.u. for T = 175 K, corresponding to n between
11 and 12.
The integrated intensities for the fitted peaks are shown in Fig.
9.14. I will work further with
these intensities in Sect.
9.4, however at a first glance they follow power laws well.
The widths of the fitted peaks are shown in Fig.
9.15. Since the intensities of the staging
peaks were so small, and their widths were hence kept fixed for higher temperatures, it is
difficult to conclude anything for these parameters. For the central Bmab peak, however, the
width is seen to decrease slowly before the phase transition, close to which it falls significantly.
Since the value for the staging number is fairly high, n = 13, it i s interesting to take a look
at correlation lengths for this sample. The widths found for the staging peaks were shown
in Fig.
9.15(b) as half width half maximum (HWHM) values of around HWHM = 0.055 r.l.u.. A
typical domain size D in a sample, connected to the measured reflection, is
D =
2π
FWHM
=
π
HWHM
, (9.2.1)
where FWHM is the full width at half maximum. Using the value for the lattice constant given
in Tab.
9.1 to get HWHM = 0.055 · 2π/c ≈ 0.053 Å
−1
, it is then given that D ≈ 119 Å, which
corresponds to 119 Å/c ≈ 9 unit cells.
Since the staging number n only counts half a unit cell (as shown in Fig. 6.6 with e.g. stage 6
having three unit cells per level), this means that the correlation length in the sample is longer

72 9.2. Fitting to find integrated intensities for BW5 staging data
than the staging number needs to order the oxygen, and it is concluded that staging is indeed
possible with such a high value.
9.2.3 Fitting La
1.91
Sr
0.09
CuO
4+y
peaks
The measured peaks for the La
1.91
Sr
0.09
CuO
4+y
sample were shown in Fig.
9.6(d) and on a loga-
rithmic scale in Fig.
9.7(d). In order to find the integrated intensities for each of the elements in
the spectra, each full scan has been fitted with a large central Gaussian for the Bmab peak, and
one small Lorentzian peak on either side for the staging. Again, the mid-point of the staging
peaks and their distances to this are used as fitting parameters. Examples of these fits for the
T = 60 and 220 K datasets are shown in Fig. 9.16. The mid-point between the staging peaks
was kept fixed to the T = 200 K value for temperatures higher than this because of the low
intensity of the staging peaks at these temperatures.
3.96 3.97 3.98 3.99 4 4.01 4.02 4.03 4.04 4.05
0
50
100
150
200
through (014) [r.l.u.]
Intensity I per monitor [a.u.
(a)
Intensity I per m onitor [a.u.]
3.96 3.97 3.98 3.99 4 4.01 4.02 4.03 4.04 4.05
0
20
40
60
80
100
120
l through (014) [r.l.u.]
Intensity I per monitor [a.u.
(b)
l through (014) [r.l.u.]
Intensity I per m onitor [a.u.]
Figure 9.16: Examples of fits done for the La
1.91
Sr
0.09
CuO
4+y
BW5 data, here for the (a) T = 60 K and
(b) T = 220 K measurements. The individual peaks of the fit are shown as dashed lines (plus the constant
background), while the full fit is the solid blue line. For points without visible errorbars, the errors are
smaller than the marker.
The fitting function was chosen with a central Gaussian because of the shape of the main
peak, with two Lorentzians in the shoulders of this peak to account for the asymmetry seen in
the measured data. Several other fitting functions were used before the final chosen function,
however a central Gaussian with Lorentzians in the shoulders gave the best results. Note also
the strong asymmetry in the staging peaks in the shoulders, with the high l peak being much
larger than the low l peak.
An overview of the positions of the fitted peaks is shown in Fig.
9.17. The position of the Bmab
peak and the mid-point of the staging peaks – shown in blue and gree n, respectively – are, as
for the La
1.935
Sr
0.065
CuO
4+y
sample, very close to each other, with an average distance of only
δl = 0.000 39(16) r.l.u. for temperatures below 200 K. The Bmab peak is at lower values of l
compared to the mid-point of the staging peaks. As was mentioned for the La
1.935
Sr
0.065
CuO
4+y

Chapter 9. Staging measurements 73
0 50 100 150 200 250 300
3.99
3.995
4
4.005
4.01
Temperature T [K]
Staging peak positions [r.
Temperature T [K]
Peak positions [r.l.u.]
Figure 9.17: Positions of fitted peaks for the La
1.91
Sr
0.09
CuO
4+y
sample. The blue points sh ow the posi-
tion of the central Bmab peak while the green points show the mid-point of the staging peaks, which are
in turn positi oned at the yellow and red points. The crossmark indicates a parameter that was fixed to the
value from a lower temperature scan. For points without visible errorbars, the errors are smaller than the
marker.
0 50 100 150 200 250 300
0
0.5
1
1.5
2
2.5
Temperature T [K]
Intensity I [a.u.]
(a)
Temperature T [K]
Integrated intensity I [a.u.]
0 50 100 150 200 250 300
0
0.1
0.2
0.3
0.4
Temperature T [K]
(b)
Temperature T [K]
Figure 9.18: Integrated peak intensities of fitted peaks for the La
1.91
Sr
0.09
CuO
4+y
sample for (a) the
central Bmab peak and (b) the staging peaks: red points for the low l and yellow for the high l peak (as in
Fig.
9.17).
sample, this very small δl value with a large associated uncertainty, leads to the conclusion that
there is only a Bmab phase in this sample, as opposed to two structural phases.
The distance between the staging peaks (their position shown in yellow and red points) and
their mid-point (green points) for temperatures below 200 K is on average ∆l = 0.011(3) r.l.u.,
which would correspond to a staging number of just above 89. For higher temperatures this
distance becomes even smaller, all the way down to ∆l = 0.007 19(2) r.l.u., corresponding to
an n larger than 1 39 , for T = 300 K. These values for n are both absurdly high, and most
likely means that the fitted shoulder peaks are not actually from staging, but instead stems
from random disorder.
The integrated intensities for the fitted peaks are shown in Fig. 9.18. I will work further with
these intensities in Sect.
9.4, although at a first glance none of them follow a simple power law
very well. They do, however, all fall with temperature: the Bmab peak might show two phase
transitions instead of one – with a jump in the intensity around 70 K; while the staging peaks
fall eratically, most likely caused by noise because of the very low intensities.
The widths of the fitted peaks are shown in Fig.
9.19. For the central Bmab peak the width
generally falls with rising temperature. The staging peaks also have this general trend, although
the high l peak width has a jump in width around T = 220 K.
Just like for the L a
1.935
Sr
0.065
CuO
4+y
sample, the value for the staging number for this sample
is high, n = 89 and higher. It is again interesting t o look at the the correlation length for
this sample. The widths found for the staging peaks were shown in Fig. 9.19(b) with values of

74 9.2. Fitting to find integrated intensities for BW5 staging data
0 50 100 150 200 250 300
0.007
0.008
0.009
0.01
0.011
0.012
Temperature T [K]
width [r.l.u.]
(a)
Temperature T [K]
Peak width (HWHM) [r.l.u.]
0 50 100 150 200 250 300
0.002
0.003
0.004
0.005
Temperature T [K]
(b)
Temperature T [K]
Figure 9.19: Widths of fitted peaks for the La
1.91
Sr
0.09
CuO
4+y
sample for (a) the central Bmab peak and
(b) the staging peaks: red points for the low l and yellow for the high l peak (as in Fig.
9.17).
around HWHM = 0.0045 r.l.u. = 0.0045 ·2π/c ≈ 0.0022 Å
−1
for low temper atures (with c from
Tab. 9.1). A typical domain size i n the sample will then be given by Eq. (9.2.1) as D ≈ 14 48 Å,
which correspo nds to 1448 Å/c ≈ 111 unit cells.
This means that a staging number of n = 89 (needing interactions reaching out n/2 =
44.5 unit cells) is indeed possible within the domain sizes, as it was concluded similarly for the
La
1.935
Sr
0.065
CuO
4+y
sample. Even the very large staging number at higher temperatures, up
towards 139, is not going against the domain sizes.
9.2.4 Fitting La
2
CuO
4+y
peaks
The measured peaks for the La
2
CuO
4+y
sample were shown in Fig.
9.6(a) and on a logarithmic
scale in Fig.
9.7(a). To find the integrated intensities for each of the elements in the spectra,
each full scan has been fitted with a constant background plus a c entral Gaussian with a smaller
Lorentzian at the same center – the Bmab peak – and five pairs of Lorentzians – representing
staging – fitted with a common mid-point and the same distance to this mid-po int in the pairs.
Examples of these fits are shown for the T = 10, 161, and 220 K datasets in Fig. 9.20.
The choice of fitting function was decided on the basis of the T = 10 K data, which show the
fitted peaks most clearly. Several parameters were held fixed at higher temperatures: from 30 K
and upwards, the distance from the mid-point to the innermost staging peaks at ∆l ≈ 0.13 r.l.u.
(plotted with left-pointing triangles in the coming figures) was kept fixed to the T = 10 K pa-
rameter; from 60 K and upwards, the widths of the outermost staging peaks at ∆l ≈ 0.6 r.l.u.
(up-pointing triangles) were kept fixed to the T = 50 K parameters; from 190 K and upwards,
the widths of the outer shoulder peaks on the main staging peaks (∆l ≈ 0.33 r.l.u., squares)
were kept fixed at the T = 161 K parameters; from 220 K and upwards, the widths of the in-
nermost staging peaks (∆l ≈ 0.13 r.l.u., left-pointing triangles) and the distance between the
mid-point and the three outermost staging peaks (∆l ≈ 0.33 r.l.u., ∆l ≈ 0.5 r.l.u., ∆l ≈ 0.6 r.l.u.,
or squares, diamonds, up-pointing triangles) were kept fixed at the T = 190 K parameters; from
250 K and upwards, the widths of the ∆l ≈ 0.5 r.l.u. staging peaks (diamonds) were kept fixed
at the T = 220 K parameters; for 300 K the widths, the mid-point of all the staging peaks, and
the distance from the main staging peaks (∆l ≈ 0.2 r.l.u., circles) to the mid-point were kept
fixed to the T = 290 K parameters.
An overview of the positions of all the fitted peaks is shown in Fig.
9.21. The positions of the
Bmab peak and the mid-point of the staging peaks – shown i n blue and green, respectively – are
fairly close, with an average distance of δl = 0.0094(7) r.l.u. for temperatures below 250 K. For
temperatures above this, the distance is first seen to get smaller, and then slightly larger. This

Chapter 9. Staging measurements 75
3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8
0
5
10
15
20
25
Intensity I per monitor [a.u.]
2345
3.95 4 4.05
0
20
40
60
80
100
(a)
Intensity I per m onitor [a.u.]
3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8
0
5
10
15
20
25
Intensity I per monitor [a.u.]
2345
3.95 4 4.05
0
20
40
60
80
(b)
Intensity I per m onitor [a.u.]
3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8
0
5
10
15
20
25
l through (014) [r.l.u.]
Intensity I per monitor [a.u.
2345
3.95 4 4.05
0
20
40
60
80
(c)
l through (014) [r.l.u.]
Intensity I per m onitor [a.u.]
Figure 9.20: Examples of fits done for the La
2
CuO
4+y
BW5 data, here for the (a) T = 10 K, (b) T = 161 K,
and (c) T = 220 K measurements. The individual peaks of the fit are shown as dashed lines (plus the
constant background), while the full fit is the solid blue line. The red lines in the background mark integer
staging positions on the high l side up to n = 10, with n indicated in the top for the smaller values. For
points without visible errorbars, the errors are smaller than the marker. T he insets show zoom-ins of the
central peak in its full height.

76 9.2. Fitting to find integrated intensities for BW5 staging data
0 50 100 150 200 250 300
3.4
3.6
3.8
4
4.2
4.4
4.6
Temperature T [K]
Staging peak positions [r.l.u.]
Temperature T [K]
Peak positions [r.l.u.]
Figure 9.21: Positions of fitted peaks for the La
2
CuO
4+y
sample. The blue points show the position of
the central Bmab peak while the green points show the mid-point of the staging peaks. The staging peaks
are positioned at the other colored points, with the circle points being the large main staging peaks. For
points without visible errorbars, the errors are smaller than the mark er. Crossmarks over points in dicate
that the distance from the point to the mid-point were fixed to the value for the temperature lower than
where the crossmarks start.
value for δl is small, but significantly larger than zero with the associated uncertainty, indicat-
ing that the superstructures are not connected to the Bmab reflection, but instead possibly a
structural Fmmm phase, as was concluded for the La
1.96
Sr
0.04
CuO
4+y
sample.
The distance between the staging peaks (their positions shown in yellow and red, with
different symbols used for the different pairs), and their mid-point (green points) are as ex-
plained in the following: For the innermost staging peaks (left-pointing triangles, only fitted
for 10 K) ∆l = 0.13(7) r.l.u., corresponding to a staging number n between 7 and 8. The un-
certainty on this value is large because of difficulties in fitting the peak resulting from the
very low intensities. The largest staging peaks (circles) have ∆l = 0.2159(16) r.l.u. for tem-
peratures below T = 170 K, above which they are seen to move closer to the center, to about
∆l = 0.1793(4 ) r.l.u. at the smallest value at T = 220 K, corresponding to n just below 5
and between 5 and 6, respectively. The small peaks in the shoulder of the large staging peak
(squares) have ∆l = 0.327 (13 ) r.l.u. for temperatures below T = 210 K, above which the dis-
tance to the mid-point is kept fixed. This corresponds to a staging number of n = 3. The next
peaks (diamonds) have ∆l = 0.500(6) r.l.u. for temperatures below T = 210 K, above which the
distance to the mid-point is kept fixed. This corresponds to a staging number of n = 2. The
outermost peaks (up-pointing triangles) ∆l = 0.597(5) r.l.u. for temperatures below T = 210 K,
above which the distance to the mid-point is kept fixed. This corresponds to a staging number
n between 1 and 2, which might indicate that these outermost peaks are not actually a super-
structure of (014), but rather the surrounding integer (hkl) positions, here (013) and (015).
The integrated intensities for all of the fitted peaks are shown in Fig.
9.22. I will work further
with some of these in Sect.
9.4, but I will go through the results here in a descriptive way.
As an initial observation the central Bmab peak (shown both split up in the Gaussian and
Lorentzian and as the total in Fig.
9.22(a)) does not follow a power law, but simply falls off

Chapter 9. Staging measurements 77
0 50 100 150 200 250 300
0
0.5
1
1.5
2
2.5
Intensity I [a.u.]
(a)
Integrated intensity I [a.u.]
0 50 100 150 200 250 300
0
0.5
1
1.5
2
2.5
(b)
0 50 100 150 200 250 300
0
0.05
0.1
0.15
0.2
0.25
Intensity I [a.u.]
(c)
Integrated intensity I [a.u.]
0 50 100 150 200 250 300
0
0.05
0.1
0.15
(d)
0 50 100 150 200 250 300
0
0.01
0.02
0.03
0.04
Temperature T [K]
Intensity I [a.u.]
(e)
Temperature T [K]
Integrated intensity I [a.u.]
0 50 100 150 200 250 300
0
0.1
0.2
0.3
Temperature T [K]
(f)
Temperature T [K]
Figure 9.22: Integrated peak intensities of fitted peaks for the La
2
CuO
4+y
sample. (a) The central Bmab
peak: empty diamonds for the Lorentzian tail, empty squares for the Gaussian main part of the peak, and
filled circles for the sum of the two. (b-f) The staging peaks: red points for the low l and yellow for the high
l peaks, with different symbols depending on the set of peaks following the symbols and colors of Fig.
9.21:
the circles in (b) are the two large main staging peaks, the squares in (c) are the two ∆l ≈ 0.33 r.l.u. staging
peaks, the diamonds in (d) are the two ∆l ≈ 0.5 r.l.u. staging peaks, the up-pointing triangles in (e) are the
two ∆l ≈ 0.6 r.l.u. staging peaks, and the left-pointing triangles in (f) are the two ∆l ≈ 0.13 r.l.u. staging
peaks. The smaller staging peaks are sh own both in (b) for an easy comparison, and in their individual
plots in (c-f).

78 9.2. Fitting to find integrated intensities for BW5 staging data
slowly with rising temperature. This is expected, since the HTT to LTO phase transition is
above room temperature for the La
2
CuO
4+y
sample.
The main staging peaks, shown with circle points in Fig.
9.22(b), seem to follow a power
law, although the sudden fall at T = 190 K might be problematic. This fall could stem from a
slight change in the peak shape that has not been resolved well enough, or maybe from these
staging peaks actually consisting of a combination of n = 4 and 5. Note, however, that this fall
in intensity coincides with the rise in intensity seen on the inner-most staging peak, as seen
in Fig.
9.22(f).
The four smaller staging peaks are harder to conclude anything from. The data in Fig.
9.22(c)
and (d) show a phase transition, although it is not very Landau-like. The intensities in Fig.
9.22(e)
and (f) are so noisy that I cannot conclude much from them.
Finally, the widths of the fitted peaks are shown in Fig.
9.23. For the Bmab peak the width is
shown for each of the two parts of the peak, and seems fairly constant. There is, however, a rise
in the width for the Gaussian part in Fig.
9.23(b), which might indicate that a phase transition
is close.
For the main staging peaks, in Fig.
9.23(c), the width seems to fall, but suddenly rise for the
high l peak (yellow points). This rise might not mean much, though, since the intensity of the
peak is very low at these temperatures. For the other staging peaks, the widths are kept fixed
for many of the higher temperatures, although a small fall in width with temperature is seen in
Fig.
9.23(d) with the opposite happening in Fig. 9.23(e).

Chapter 9. Staging measurements 79
0 50 100 150 200 250 300
0.018
0.0182
0.0184
0.0186
0.0188
Temperature T [K]
width [r.l.u.]
(a)
Temperature T [K]
Peak width (HWHM) [r.l.u.]
0 50 100 150 200 250 300
0
0.02
0.04
0.06
0.08
Temperature T [K]
(b)
Temperature T [K]
0 50 100 150 200 250 300
0.01
0.02
0.03
0.04
0.05
Temperature T [K]
width [r.l.u.]
(c)
Temperature T [K]
Peak width (HWHM) [r.l.u.]
0 50 100 150 200 250 300
0
0.005
0.01
0.015
0.02
0.025
0.03
Temperature T [K]
width [r.l.u.]
(d)
Temperature T [K]
Peak width (HWHM) [r.l.u.]
0 50 100 150 200 250 300
0
0.005
0.01
0.015
0.02
0.025
Temperature T [K]
(e)
Temperature T [K]
0 50 100 150 200 250 300
0.005
0.01
0.015
0.02
0.025
0.03
Temperature T [K]
width [r.l.u.]
(f)
Temperature T [K]
Peak width (HWHM) [r.l.u.]
0 50 100 150 200 250 300
0
0.01
0.02
0.03
0.04
0.05
Temperature T [K]
(g)
Temperature T [K]
Figure 9.23: Widths of fitted peaks for the La
2
CuO
4+y
sample. (a) Th e Lorentzian tail and (b) the Gaussian
main part of the central Bmab peak (following the symbols from Fig.
9.22(a)). (c) The main staging peaks.
(d-g) The smaller staging peaks. All the staging peaks follow the symbols and colors from Fig.
9.21, with
red points for th e low l and yellow for the high l peaks: the circles in (c) are the two large main staging
peaks, the squares in (d) are the two ∆l ≈ 0.33 r.l.u. staging peaks, the diamonds in (e) are the two
∆l ≈ 0.5 r.l.u. s taging peaks, the up-pointing triangles in (f) are the two ∆l ≈ 0.6 r.l.u. staging peaks, and
the left-pointing triangles in (g) are the two ∆l ≈ 0.13 r.l.u. staging peaks. Crossmarks over points indicate
that the width was fixed to the value of the temperature lower than where the crossmarks start.

80 9.3. Alternative ways to find the transition temperature
9.3 Alternative ways to find the transition temperature
In order to analyze the phase transitions and get out the transition temperature and critical
exponent, data will be fitted with power laws as suggested in Chap. 5. However, some of the
found temperature dependent integrated intensity datasets have very few useable points, or
behave in a noisy way. In order to find the critical exponent for the phase transition for these
it is helpful to determine the transition temperature first – in an alternative way than simply
fitting with a power law to get both parameters out simultaneously. There are several ways to
do this, but below I will explain the two methods I have used for the data analysis in this thesis,
and then continue to fitting with power laws in Sect.
9.4.
9.3.1 Taking extinction and twinning at the phase transition into account
When going from the high temperature tetragonal (HTT) phase to the low temperature or-
torhombic (LTO) phase, several new reflections show up in the data. These reflections were
forbidden in the HTT phase, and going through the transition, more of the crystal is hence
“seen”, simply from the change in the structure factor. This results in a lowering in intensity of
the reflections that are allowed in both phases (but split up in the LTO phase), with a very sharp
transition. On the other hand, twinning also happens at this transition, causing the sample to
be divided into several domains, which will relieve extinction just as mosaicity would – also
happening as a very sharp transition, but with the effect of increasing the intensity, [
112, 121].
This sharp transition should be at the same temperature as the central Bmab peak of the
(014) spectra show up, because this is where the cryst al symmetry changes from HTT to LTO,
and hence where the point (014) becomes an allowed reflection in reciprocal space (since it
was forbidden for the tetragonal crystal symmetry following Tab. 6.1). As has earlier been
mentioned, the staging peaks are likely a superstructure of another structural phase (Fmmm),
and can show up at a different temperature than this. Hence, this method can only be used to
find the transition temperature for the central peak, not the staging peaks.
Several measurements of fundamental peaks such as (020), (004), and (022 ) were taken as a
function of temperature during the different experiments. In order to check for extinction
effects for the samples, the (020) peaks were investigated for each of these. The raw data
*
is
shown in Fig.
9.24.
The La
2
CuO
4+y
sample did not show any significant change in intensity over temperature,
however this is expected since the transition temperature from the HTT to the LTO phase is
above room temperature for this doping.
It is also as expected that extinction is a small effect for the La
1.96
Sr
0.04
CuO
4+y
sample, since
this sample is very small – and the data does indeed not change much over temperature.
The data for the La
1.935
Sr
0.065
CuO
4+y
sample shows a clear change when the temperatures
get low, with the peak almost halving in intensity.
Extinction was expected to be a significant effect for the La
1.91
Sr
0.09
CuO
4+y
sample, but as
is seen i n Fig.
9.24(d), the temperature dependence is more complicated, with the peak splitting
up at lower temperatures, possibly because of twinning (which is expected to split the peak
with about 0.1
◦
, following the calculations shown in Appx.
A.4).
To find the transition temperature for the HTT to LTO transition in La
1.935
Sr
0.065
CuO
4+y
from
the data in Fig.
9.24(c), each tempe rature scan has been fitted with a Gaussian plus a Lorentzian
fixed at the same center. The integrated intensity of the sum of these two peaks as a function of
temperature is shown in Fig.
9.25(a). The effect is clearly a sm aller intensity on the fundamental
peak when getting below the transition temperature, which either means that the extinction
*
The raw data for the measured (004) and (022) peaks for some of the samples can be seen in Fig.
D.22, but none
of these showed any significant extinction effects.

Chapter 9. Staging measurements 81
1.97 1.98 1.99 2 2.01 2.02 2.03
0
50
100
150
200
250
300
350
Intensity I per monitor [a.u.]
Temperature T [K]
50
100
150
200
250
(a)
Intensity I per m onitor [a.u.]
Temperature T [K]
0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2
0
20
40
60
80
100
120
Intensity I per monitor [a.u.]
Temperature T [K]
0
50
100
150
200
250
(b)
Intensity I per m onitor [a.u.]
Temperature T [K]
0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7
0
20
40
60
80
100
Intensity I per monitor [a.u.]
Temperature T [K]
0
50
100
150
200
250
(c)
Intensity I per m onitor [a.u.]
Temperature T [K]
1.1 1.2 1.3 1.4 1.5 1.6
0
10
20
30
40
50
ω through (020) [deg]
Intensity I per monitor [a.u.]
Temperature T [K]
0
50
100
150
200
250
(d)
Sample angle ω through the (020) reflection [deg]
Intensity I per m onitor [a.u.]
Temperature T [K]
Figure 9.24: Measurement along ω over the (020) peak on BW5 for the (a) La
2
CuO
4+y
, (b) La
1.96
Sr
0.04
Cu-
O
4+y
, (c) La
1.935
Sr
0.065
CuO
4+y
, and (d) La
1.91
Sr
0.09
CuO
4+y
crystals.

82 9.3. Alternative ways to find the transition temperature
100 150 200 250
8
10
12
14
16
Temperature T [K]
Area of peak [a.u.]
(a)
Temperature T [K]
Integrated intensity I [a.u.]
100 150 200 250
4
6
8
10
12
Temperature T [K]
Constant minus area of peak [a.u
(b)
Temperature T [K]
Constant minus intensity I [a.u.]
Figure 9.25: Integrated peak intensities for the La
1.935
Sr
0.065
CuO
4+y
(020) peak scans shown in Fig. 9.24(c).
(a) The intensit ies directly from the fits. (b) The intensities subtracted from an arbitraty constant in order
to fit with a simple power law shown in blue. The transition temperature was found to be T
HTT→LTO
=
216.3(5) K. Crossmarks indicate points that are not used for the fit.
50 100 150 200 250
1
1.5
2
2.5
3
Temperature T [K]
Area of peak [a.u.]
Temperature T [K]
Integrated intensity I [a.u.]
Figure 9.26: Integrated (020) peak intensities for the La
1.91
Sr
0.09
CuO
4+y
sample, obtained with a double
Gaussian fit of the peak in Fig.
9.24(d).
dominates, and the twinning – in this case – is not a large enough effect to counteract this, or
simply that the structure factor changes significantly between the HTT and LTO phases.
In order to fit the intensities to a standard power law (as defined in Appx.
C), these data
are subtracted from an arbitrarily chosen constant, resulting in the fit shown in Fig.
9.25(b).
This fit gave a transition temperature of T
HTT→LTO
= 216.3(5) K, which will be used later when
working with the (014) peak data. Note however that the statistical uncertainty on this transition
temperature, as found by fitting with spec1d, is very low and should probably be a little larger
due to the model choice: the transition temperature is mostly determined by the one datapoint
exactly at this temperature in Fig.
9.25(b).
The (020) peaks for the La
1.91
Sr
0.09
CuO
4+y
sample in Fig. 9.24(d) were fitted in order to get the
integrated intensities as well. Two Gaussians were used in order to take the peak splitting into
account. The integrated intensities as a function of temperature can be seen in Fig.
9.26.
The results – significantly so at high temperatures – have large errorbars because of the poor
resolution of the (020) peak measurements. However two things can be seen as general features:
at temperatures lower than T ≈ 80 K the intensity rises, coinciding with the temperature where
the peaks visibly split up into two peaks instead of overlapping; and at temperatures above
T ≈ 220 K the intensity also rises. The low temperature effect is the opposite of what is seen in
Fig.
9.25 for La
1.935
Sr
0.065
CuO
4+y
, meaning that the twinning relief of the extinction dominates

Chapter 9. Staging measurements 83
here. The higher temperature effect is just as was seen for La
1.935
Sr
0.065
CuO
4+y
, where the
extinction dominates.
The effect at low temperature i s much like was seen on the integrated intensities for the
Bmab (014) peak for the La
1.935
Sr
0.065
CuO
4+y
sample in Fig.
9.18(a), indicating that a small
second phase transition might indeed happen at lower temperatures. By eye, the transition
temperature for the higher temperature effect in Fig.
9.26, at around 220 K, also matches the
behavior of the Bmab (014) peak.
9.3.2 Slope ana lysis of the integrated intensities
One measure of the position of the transition temperature is where the slope of the integrated
intensities as a function of temperature is the highest (of course assuming that a phase tran-
sition is seen at all), [
122]. In order to do this analysis, the integrated intensity data has been
interpolated linearly, and then this data has been smoothed with a low-pass Butterworth filter
*
in order to be able to find the gradient in a meaningful way. The minimum of the gradient of
this can then be found, and hence indicate the transition temperature.
This analysis has been done for all of the found integrated intensities, although it is only
used directly in the further analysis for the main staging peaks of the La
2
CuO
4+y
sample (shown
in Fig.
9.27(a) and (b), giving T
staging
= 175(5) K for both), the central Bmab peak for the La
1.96
-
Sr
0.04
CuO
4+y
sample (shown in Fig.
9.27(c), giving T
HTT→LTO
= 57(5) K), and the central Bmab
peak for the La
1.91
Sr
0.09
CuO
4+y
sample (shown in Fig.
9.27(d), giving T
HTT→LTO
= 247(5) K). The
errors on these numbers have been estimated manually, since the used method was not very
well developed.
Apart from these directly used results, it is also interesting to compare the slope analysis with
the power fit analysis done on the rest of the integrated intensities. The analysis done for the
∆l ≈ 0.33 r.l.u. and ∆l ≈ 0.5 r.l.u. staging peaks for the La
2
CuO
4+y
sample is shown in Fig.
9.28.
For the ∆l ≈ 0.33 r.l.u. staging peaks in Fig.
9.28(a) and (b) the slope analysis results in very
low values because of the somewhat erratic behavior of the data. However, they both seem to
have a second minimum around T
staging
≈ 175 K.
For the ∆l ≈ 0.5 r.l.u. staging peaks in Fig.
9.28(c) and (d) the slope analysis give meaningful
results with T
staging
= 121(5) K for the low l peak and T
staging
= 174(5) K, although it could
be argued if they should both be around the 174 K value with the very wide minimum seen in
the gradient of the low l results: I choose to set the low l transition temperature manually to
175(5) K because of this.
The remaining slope analyses are shown in Fig.
D.23, and are mostly for comparison with the
transition temperatures found directly from the tem perature series for the (014) peak scans
later.
For La
1.96
Sr
0.04
CuO
4+y
the transition temperatures were found to be T
staging
≈ 212(5) K for
the staging peak at low l and T
staging
≈ 214(5) K at high l with this method.
For La
1.935
Sr
0.065
CuO
4+y
, the method gave a transition temperature T
HTT→LTO
≈ 202(5) K for
the central Bmab peak, and T
staging
≈ 202(5) K for both the staging peak at low and high l.
The results for the staging peaks for the La
1.91
Sr
0.09
CuO
4+y
sample were both inconclusive,
with a best guess that the transition temperatures are around 250 K or higher for both peaks.
*
This is done using the functions butter() and filtfilt() for MATLAB, [
166], with settings such that the
smoothed data – evaluated by eye – seemed right.

84 9.3. Alternative ways to find the transition temperature
0
1
2
3
175(5) K
0 50 100 150 200 250 300
(a)
Temperature T [K]
Intensity I [a.u.]
Gradient
0
0.5
1
1.5
2
2.5
175(5) K
0 50 100 150 200 250 300
(b)
Temperature T [K]
Intensity I [a.u.]
Gradient
0
0.1
0.2
0.3
0.4
57(5) K
0 50 100 150 200 250 300
(c)
Temperature T [K]
Intensity I [a.u.]
Gradient
0
1
2
3
247(5) K
0 50 100 150 200 250 300
(d)
Temperature T [K]
Intensity I [a.u.]
Gradient
Figure 9.27: Slope analysis of integrated intensities of (a) the low l and (b) high l main staging peaks
(∆l ≈ 0.2 r.l.u.) of the La
2
CuO
4+y
sample, (c) the central Bmab peak of the La
1.96
Sr
0.04
CuO
4+y
sample, and
(d) the central Bmab peak of the La
1.91
Sr
0.09
CuO
4+y
sample. The integrated intensity points are plotted
directly, while linearly interpolated points are shown as a colored line underneath. A smoothing filter has
been applied to the data points (grey) and the interpolated points (black), resulting in the gradients shown
in the bottom parts of each plot. The gradient minimum for the smooth data is marked by a green line.
0
0.1
0.2
0.3
19(5) K
0 50 100 150 200 250 300
(a)
Temperature T [K]
Intensity I [a.u.]
Gradient
0
0.05
0.1
0.15
0.2
18(5) K
0 50 100 150 200 250 300
(b)
Temperature T [K]
Intensity I [a.u.]
Gradient
0
0.05
0.1
0.15
121(5) K
0 50 100 150 200 250 300
(c)
Temperature T [K]
Intensity I [a.u.]
Gradient
0
0.05
0.1
0.15
0.2
174(5) K
0 50 100 150 200 250 300
(d)
Temperature T [K]
Intensity I [a.u.]
Gradient
Figure 9.28: Additional slope analysis of the integrated intensity of some of the smaller staging peaks for
La
2
CuO
4+y
. (a) The low l and (b) high l ∆l ≈ 0.33 r.l.u. staging and (c) the low l and (d) high l ∆l ≈ 0.5 r.l.u.
staging. The method and colorin g of the different elements are as explained in the caption of Fig. 9.27.

Chapter 9. Staging measurements 85
9.4 Analysing the Bmab and staging phase transitions
The integrated peak intensities were shown for the four samples in Figs.
9.10, 9.14, 9.18
and 9.22. As explained in Chap. 5, these intensities should be fitted with a power law to find
the transition temperature and critical exponent for the phase transitions. Saturation of the
order parameter and critical scattering will have to be taken into account by only fitting some
temperature regions of the data.
9.4.1 Choosing which part of the data to fit
As was described in Subsect.
5.3.3 the intensities tend to saturate for very low temperatures,
which is especially seen for the La
2
CuO
4+y
and La
1.96
Sr
0.04
CuO
4+y
(staging) results here. The
fits should be cut off such that these temperatures are not used. As a rule of thumb if the
transition temperature is already known, the cut-off happens at T equal to a third of this, [
85,
Fig. 1]. If the transition temperature has already been found by alternative methods, such as
looking at the (020) peak or the slope analysis, this value is used to determine the cut-off, and
directly for the fit instead of fitting this parameter. If the transition temperature is not already
known, this is simply a matter of trying out different cut-offs and evaluating which one gives
the better and more realistic fit, combined with looking at the data by eye to evaluate if the
results are meaningful.
The critical scattering happening close to the transition temperature, resulting in the inten-
sity not falling as fast as was expected as explained in Subsect.
5.3.4, also has to be taken into
account. This is done by omitting points above and close to the transition temperature, in order
to ignore the smoothing of the transition and only fit the data unaffected by critical scattering.
Several differe nt cut-offs were tried for each fit, and the ones giving the best fits were chosen.
9.4.2 Fit results for the phase transitions
The final fitted integrated intensities for each of the peaks in the (014) scans are shown in
Fig.
9.29, with overlapping axes for the staging peaks (left axis) and central Bmab peak (right
axis shown in blue) for easier comparison. They are all fitted with power laws (as defined in
Appx.
C) as explained in the following. The results are also shown plotted on double logarithmic
axes in Fig.
D.24.
The La
2
CuO
4+y
staging peak transition tem peratures were held fixed to the values found with
the slope analysis, as explained in Subsect. 9.3.2 and shown in Fig. 9.27(a) and (b). The Bmab
peak for this sample was not fitted, since no clear phase transition is seen.
The La
1.96
Sr
0.04
CuO
4+y
Bmab peak transition temperature was also held fixed to the value
found with the slope analysis, as shown in Fig. 9.27(c). This was especially necessary for this
peak, since the phase transition is so wide, possibly because of a large contribution from critical
scattering.
The La
1.935
Sr
0.065
CuO
4+y
Bmab peak transition temperature was held fixed at the value
found with the (020) peak data, as explained in Subsect.
9.3.1 and shown in Fig. 9.25.
Finally, the La
1.91
Sr
0.09
CuO
4+y
Bmab peak transition temperature was also held fixed to the
value found with the slope analysis, as shown in Fig.
9.27(d).
The fits done for the two main staging peaks for the La
2
CuO
4+y
sample are slightly inaccurate,
mainly because of the very few points that were useable for the fit. The critical exponents ended
up being 2β = 0.06(4) and 0.09(5) for the low and high l peaks, respectively, but with reduced
χ
2
values of 55 and 183, these values are not to be trusted.
However, looking at the double logarithmic plots in Fig.
D.24(a) and (b), where the fitted
points should lie on a straight line to follow t he power law behavior, i t seems that this is how

86 9.4. Analysing the Bmab and staging phase transitions
0 100 200 300
0
0.5
1
1.5
2
2.5
3
Temperature T [K]
Area I of staging peak [a.u.]
0
0.5
1
1.5
2
2.5
Area of middle peak [a.u.]
(a)
Temperature T [K]
Staging integrated intensity I [a.u.]
Bmab integrated intensity I [a.u.]
0 100 200 300
0
0.5
1
1.5
2
2.5
Temperature T [K]
Area I of staging peak [a.u.]
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Area of middle peak [a.u.]
(b)
Temperature T [K]
Staging integrated intensity I [a.u.]
Bmab integrated intensity I [a.u.]
0 100 200 300
0
0.5
1
1.5
2
Temperature T [K]
Area I of staging peak [a.u.]
0
2
4
6
8
10
Area of middle peak [a.u.]
(c)
Temperature T [K]
Staging integrated intensity I [a.u.]
Bmab integrated intensity I [a.u.]
0 100 200 300
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
Temperature T [K]
Area I of staging peak [a.u.]
0
0.5
1
1.5
2
2.5
3
Area of middle peak [a.u.]
(d)
Temperature T [K]
Staging integrated intensity I [a.u.]
Bmab integrated intensity I [a.u.]
Figure 9.29: Integrated int ensiti es fitted w ith power laws, with the Bmab peaks (show in blue) and staging
peaks (shown in red and yellow) on different axes to be able to compare them more easily. Crossmarks indi-
cate points that are not used for the fit, either because of saturation (low T ) or critical scattering (T close to
transition temperature). (a) La
2
CuO
4+y
, (b) La
1.96
Sr
0.04
CuO
4+y
, (c) La
1.935
Sr
0.065
CuO
4+y
, (d) La
1.91
Sr
0.09
-
CuO
4+y
.
good the fit can get with the low number of points available. The points omitted from the fit
due to saturation also fall underneath the straight line, as expected.
The fits done for the staging peaks for the La
1.96
Sr
0.04
CuO
4+y
sample were much better, with
enough datapoints in the relevant temperature regions to get fits with reduced χ
2
values of 2.2
and 0.7, critical exponents 2β of 0.36(2) and 0.312(14), for the low and high l peaks, respectively.
The transition temperatures T
staging
were found to be 216.2(1.1) and 215.3(1.0) K, respectively,
compared to the values from the slope analysis of 212(5) and 214(5) K.
The central Bmab peak was only fitted for narrow temperature intervals at low and high
temperature, and gave a critical exponent of 2β = 0.12(4) with a reduced χ
2
of 1.3. Note that
these very low reduced χ
2
values might indicate an over-estimation of the uncertainties of the
datapoints, originating from the fitting routine used to fit the peak shapes – this is also true
for the following results.
Looking at the double logarithmic plots in Fig.
D.24(c) to (e), it is easily seen that these fits
are well determined, with the points lying on a straight line, and the saturated point falling
below this.

Chapter 9. Staging measurements 87
The fits done for the staging peaks for the La
1.935
Sr
0.065
CuO
4+y
sample resulted in reduced χ
2
values of 0.04 and 0.23, critical exponents 2β of 0.48(18) and 0.48(15), and transition tempera-
tures T
staging
of 225.3(9) and 225(5) K for the low and high l peak, respectively. The transition
temperatures can be compared to the ones found with the slope analysis, both at 202(5 ) K. The
fit of the central Bmab peak gave a critical exponent of 2β = 0.41(3) with a reduced χ
2
of 0.02.
Again looking at the double logarithmic plots, now in Fig.
D.24(f) to (h), the fits are well
determined with the few available points.
The fits done for the staging peaks for the La
1.91
Sr
0.09
CuO
4+y
sample were fairly inaccurate
because o f the very noisy data, resulting in fits with reduced χ
2
values of 58 and 77, critical
exponents 2β of 0.52(6) and 0.50(8), and transition temperatures T
staging
of 279(4) and 300(8) K
for the low and high l peak, respectively. The fit of the central Bmab peak gave a cri tical
exponent of 2β = 0.332(16) with a reduced χ
2
of 3, and a much better agreement with data.
Looking at the double logarithmic plots for these fits, shown in Fig.
D.24(i) to (k), the central
Bmab peak fit is well determined as expected.
*
The precision of the staging peak fits are clearly
limited by the noisy data: the fits do, however, agree overall with a straight line.
A final overview of all of the results mentio ned so far in this chapter is shown in Tab.
9.2, which
also refers to the individual figures where the results were found.
Note how the transition temperature T
staging
for each of the staging peak sets (low and high
l) agree well. In the same way the staging peaks for all the different staging numbers agree
well for the same sample, as seen for the La
2
CuO
4+y
sample. In most cases, the direct fit and
alternative method (slope analysis or (020) peak extinction analysis, when available) also agree,
with exception of the La
1.935
Sr
0.065
CuO
4+y
staging peaks, where the slope analysis gave lower
values. This could be interesting to investigate further, by measuring for more temperatures.
The critical exponents for the staging peaks are seen to increase for increasing Sr doping,
while the critical exponents for the central Bmab peaks seem a little more erratic.
*
Possibly with the hint of a second phase transition at low temperatures as was also mentioned earlier, indicated
in the double logarithmic plot as low t em pera tu re points being above the lin e instead of below it (as would be expected
for the saturation seen in the other samples).

88 9.4. Analysing the Bmab and staging phase transitions
Staging distance Staging Transition Critical Result from
Sample and peak
∆l [r.l.u.] number n temperature [K]
Result from
exponent 2β direct fit in
La
2
CuO
4+y
(δl = 0.0094(7) r.l.u., Bmab higher in l – significant difference)
Central Bmab > 300 Direct fit: Fig. 9.22(a)
Low l staging 0.2159(16) 4.6 175(5) Slope analysis: Fig. 9.27(a) 0.06(4) Fig. 9.29(a)
High l staging — — 175(5) Slope analysis: Fig. 9.27(b) 0.09(5) Fig. 9.29(a)
Low l staging 0.327(13) 3.1 ∼175 Slope analysis: Fig. 9.28(a)
High l staging — — ∼175 Slope analysis: Fig. 9.28(b)
Low l staging 0.500(6) 2.0 175(5) Slope analysis: Fig. 9.28(c)
High l staging — — 174(5) Slope analysis: Fig. 9.28(d)
La
1.96
Sr
0.04
CuO
4+y
(δl = 0.014(2) r.l.u., Bmab higher in l – significant difference)
Central Bmab 57(5) (atypical) Slope analysis: Fig. 9.27(c) 0.12(4) Fig. 9.29(b)
Low l staging 0.193(8) 5.1 216.2(1.1) Direct fit: Fig. 9.29(b) 0.36(2) Fig. 9.29(b)
212(5) Slope analysis: Fig.
D.23(b)
High l staging — — 215.3(1.0) Direct fit: Fig. 9.29(b) 0.312(14) Fig. 9.29(b)
214(5) Slope analysis: Fig.
D.23(c)
La
1.935
Sr
0.065
CuO
4+y
(δl = 0.0017(11) r.l.u., Bmab lower in l – no significant difference)
Central Bmab 216.3(5) (020) analysis: Fig. 9.25(b) 0.41(3) Fig. 9.29(c)
202(5) Slope analysis: Fig.
D.23(d)
Low l staging 0.0755(14) 13.2 225.3(9) Direct fit: Fig. 9.29(c) 0.48(18) Fig. 9.29(c)
202(5) Slope analysis: Fig.
D.23(e)
High l staging — — 225(5) Direct fit: Fig. 9.29(c) 0.48(15) Fig. 9.29(c)
202(5) Slope analysis: Fig.
D.23(f)
La
1.91
Sr
0.09
CuO
4+y
(δl = 0.000 39(16) r.l.u., Bmab lower in l – no significant difference)
Central Bmab 247(5) Slope analysis: Fig. 9.27(d) 0.332(16) Fig. 9.29(d)
Low l staging 0.011(3) 89.0 279(4) Direct fit: Fig. 9.29(d) 0.52(6) Fig. 9.29(d)
High l staging — — 300(8) Direct fit: Fig. 9.29(d) 0.50(8) Fig. 9.29(d)
Table 9.2: Overview of the transition temperatures, critical exponents, and (low temperature) staging in-
formation found for the different peaks from the four samples on BW5, with references to figures showing
how the results were found for the two first mentioned. Note that the staging distance ∆l (and hence
staging number n) is the same for the low and high l peaks for one set of staging peaks, and is hence only
shown for one of them. The distance between the central Bmab peaks and the mid-points of the staging
peaks, δl have also been shown. Note the La
2
CuO
4+y
Bmab peak transition, which was very atypical with
a large amount of critical scattering and a very low transition temperature.

Chapter 9. Staging measurements 89
9.5 Results from RITA -II measurements
Measurements on the neutron i nstrument RITA-II were done for all four samples as a function
of temperature, just like on BW5.
The results along with some examples of fitted data for a few temperatures are shown in
Fig. 9.30(a) for La
2
CuO
4+y
, Fig. 9.30(b) for La
1.96
Sr
0.04
CuO
4+y
, Fig. 9.31(a) for La
1.935
Sr
0.065
Cu-
O
4+y
, and Fig.
9.31(b) for La
1.91
Sr
0.09
CuO
4+y
. The initial analysis resulting in these figures was
done by Linda Udby, [
154].
The temperature series of the peak areas (integrated peak intensities) from the insets of the
figures can be analysed in the same way as I did for the BW5 data, but I will here simply do
a couple of observations for comparison with the results from BW5 – leaving the full analysis
for the future.
For La
2
CuO
4+y
it is seen that the central Bmab (014) peak does not change much in integrated
intensity (black points in the third inset of Fig.
9.30(a)). This agrees well with the X-ray data,
where the Bmab peak did not show a phase transition either.
The staging peaks are positioned between ∆l = 0.25 and 0.3 from l = 4 (colored points in
the first inset of Fig.
9.30(a)), corresponding to staging number n = 4 or a little lower. They
have a slight tendency to a larger distance for higher temperatures. The larger ∆l on RITA-II
compared to the one on BW5 could be explained by the smaller peaks at ∆l ≈ 0.33 r.l.u. being
explicitly fitted as a different set of staging reflections, which they were not for the RITA-II
data because they were not directly seen – either because of the low flux or possibly a wider
resolution. In the same way, none of the smaller staging peaks seen in the X-ray data were seen
in the neutron data – it should, however, be mentioned that these are seen wit h neutrons in the
reciprocal space maps presented in Chap.
10 for two different La
2
CuO
4+y
samples.
The staging seems to have a transition temperature of T = 300(10) K (colored points in
the third inset of Fig. 9.30(a)), which does not seem to agree with the X-ray data – although
the fit for the X-ray staging data for this sample had a very large uncertainty since only a few
datapoints could be used close to the transition temperature.
For La
1.96
Sr
0.04
CuO
4+y
the central Bmab (014) peak falls off at a fairly low temperature around
T = 110(25) K (black points in the third inset of Fig.
9.30(b)), although the interval with no data
around 100 K limits the precision of this conclusion. The transition temperature was found to
be 57(5) K with slope analysis on the X-ray data – and hence at least both neutron and X-ray data
agree that the transition temperature is much lower than the one found for the corresponding
staging peaks. Note also that the central peak is significantly larger for these neutron data from
RITA-II compared to the X-ray data from BW5, which will be discussed further in Chap.
11.
The staging peaks are posi tioned close to ∆l = 0.2 from l = 4 (colored points in the first
inset of Fig.
9.30(b)) corresponding to staging number n = 5 , again with a tendency for larger
distance at higher temperatures – slightly more so than was seen for L a
2
CuO
4+y
. This agrees
well with the X-ray data. The staging peaks seem to have a transition temperature of around
T = 245(15) K (colored points in the third inset of Fig.
9.30(b)), seemingly around 20 to 30 K
higher than what was observed for the X-ray data. The conclusion for the neutron data does
not take critical scattering into account, however, which could lower the transition temperature
somewhat.
For La
1.935
Sr
0.065
CuO
4+y
the central Bmab (014) peak seems to fall off at a temperature of
around T = 220(20) K (black points in the third inset of Fig.
9.31(a)). This agrees well with
the X-ray data.
The underlying broader feature was fitted with a single wide Gaussian for these data, and
this seems to fall off at a slightly lower temperature, around T = 210(20) K (colored points in

90 9.5. Results from RITA-II measurements
3.6 3.8 4 4.2 4.4 4.6
0
0.2
0.4
0.6
0.8
1
l through (014) [r.l.u.]
Intensity I [cts/s]
• T = 10 K
• T = 200 K
• T = 300 K
0 100 200 300
3.6
3.8
4
4.2
4.4
Position l [r.l.u.]
T [K]
0 100 200 300
0
0.01
0.02
0.03
0.04
0.05
FWHM [Å
−1
]
T [K]
0 100 200 300
0
0.2
0.4
0.6
0.8
1
1.2
Area A [cts/s]
T [K]
(a)
3.6 3.8 4 4.2 4.4 4.6
0
0.5
1
1.5
2
2.5
3
3.5
4
l through (014) [r.l.u.]
Intensity I [cts/s]
• T = 10 K
• T = 150 K
• T = 250 K
0 100 200 300
3.6
3.8
4
4.2
4.4
Position l [r.l.u.]
T [K]
0 100 200 300
0
0.02
0.04
0.06
0.08
0.1
FWHM [Å
−1
]
T [K]
0 100 200 300
0
2
4
6
8
10
Area A [cts/s]
T [K]
(b)
Figure 9.30: Results from RITA-II for La
2
CuO
4+y
and La
1.96
Sr
0.04
CuO
4+y
. (a) La
2
CuO
4+y
fitted with a
central Gaussian and two Voigts. (b) La
1.96
Sr
0.04
CuO
4+y
fitted with three Voigts. The fitted positions,
widths and areas (integrated intensities) are shown in the inset plots (red circles are the right staging
peaks, left-pointing triangles are the left staging peaks, black circles are the central peaks). Adapted from
Udby [
154, Fig. 4.10 and 4.12].

Chapter 9. Staging measurements 91
3.6 3.8 4 4. 2 4.4 4.6
0
5
10
15
20
25
30
35
40
l through (014) [r.l.u.]
Intensity I [cts/s]
• T = 2 K
• T = 180 K
• T = 235 K
0 100 200 300
3.6
3.8
4
4.2
4.4
Position l [r.l.u.]
T [K]
0 100 200 300
0
0.05
0.1
0.15
0.2
FWHM [Å
−1
]
T [K]
0 100 200 300
0
10
20
30
40
50
Area A [cts/s]
T [K]
(a)
3.7 3.8 3.9 4 4.1 4.2 4.3 4.4 4.5
0
200
400
600
800
1000
1200
1400
1600
1800
2000
l through (014) [r.l.u.]
Intensity I [cts/s]
• T = 2 K
• T = 193 K
• T = 307 K
0 100 200 300
3.6
3.8
4
4.2
4.4
Position l [r.l.u.]
T [K]
0 100 200 300
0
0.01
0.02
0.03
0.04
0.05
FWHM [Å
−1
]
T [K]
0 100 200 300
0
200
400
600
800
1000
1200
Area A [cts/s]
T [K]
(b)
Figure 9.31: Results from RITA-II for La
1.935
Sr
0.065
CuO
4+y
and La
1.91
Sr
0.09
CuO
4+y
. (a) La
1.935
Sr
0.065
Cu-
O
4+y
fitted with a Gaussian and a Voigt. (b) La
1.91
Sr
0.09
CuO
4+y
fitted with a Gaussian. The fitted positions,
widths and areas (integrated intensities) are shown in the inset plots (left-pointing triangles are the wide
bottom peaks for the La
1.935
Sr
0.065
CuO
4+y
sample). Adapted from Udby [
154, Fig. 4.14 and 4.16].

92 9.5. Results from RITA-II measurements
the third inset of Fig. 9.31(a)), agreeing with the transition temperatures found for the staging
peaks in the X-ray data.
For La
1.91
Sr
0.09
CuO
4+y
there was no visible staging in the neutron data, and the Bmab (014)
peak falls off at around T = 295(15) K (black points in the third inset of Fig.
9.31(b)). The found
transition temperature does not fit perfectly with the X-ray data results for the central Bmab,
which had T
HTT→LTO
= 247(5) K from the slope analysis. However the transition temperature
for the staging peaks in these data do get close to room temperature, and hence might explain
the discrepancy since the staging peaks are not resolved in the neutron data, and hence might
simply be seen as an extra intensity in the central peak.
An overview of all of the neutron data results mentioned above is shown in Tab.
9.3, for later
reference and for an easier comparison with the X-ray results in Tab.
9.2.
Staging distance Staging Transition
Sample and peak
∆l [r.l.u.] number n temperature [K]
Result from
La
2
CuO
4+y
Central Bmab > 300 Fig. 9.30(a)
Low l staging 0.25-0.3 4 or lower 300(10) Fig. 9.30(a)
High l staging — — 300(10) Fig. 9.30(a)
La
1.96
Sr
0.04
CuO
4+y
Central Bmab 110(25) Fig. 9.30(b)
Low l staging 0.2 5 245(15) Fig. 9.30(b)
High l staging — — 245(15) Fig. 9.30(b)
La
1.935
Sr
0.065
CuO
4+y
Central Bmab 220(20) Fig. 9.31(a)
Underlying peak 210(20) Fig. 9.31(a)
La
1.91
Sr
0.09
CuO
4+y
Central Bmab 295(15) Fig. 9.31(b)
Table 9.3: Overview of the transition temperatures and staging information found for the different peaks
from the four samples on RITA-II, with references to figures showing the results.

93
10
Reciprocal space maps
10.1 Raw data from DMC
DMC is a neutron powder diffraction instrument, so the typical datafiles come out as inten-
sity values with their respective 2θ values for each detector tube position. In order to get an
overview from a single crystal dataset out of all these, it is necessary to first combine many of
these datafiles, and then later transform this to a reciprocal space map.
10.1.1 Combining many datasets
The large number of 2θ spectra from each single measurement at each cr ystal rotation ω and
detector position is combined into one single (2θ, ω) dataset using the instrument software
csc, which also corrects for the detector efficiency by using a standard vanadium normalization
scan. An example of such a dataset is shown in Fig.
10.1 for the La
2
CuO
4+y
sample at T = 10 K.
Note the intense lines at constant 2θ values close to 63◦ and 74
◦
: these are powder lines
from Al in the beam from the sample holder and sample environment. The lines have a small
change in intensity for different ω, which is because of asymmetries in the sample holder.
The background also tend to change slightly with ω, differently from sample to sample
depending on the sample holder and mounting. For the La
2
CuO
4+y
sample, some of the sample
holder was covered with Gd
2
O
3
to mask glue (as mentioned in Chap.
8), and this stops a small
part of the incoherent scattering and makes this effect even more noticeable. On the other hand,
large amounts of glue on the holder can give a larger background due to the extra incoherent
scattering from the hydrogen in the glue.
The streaks going through the more intense peaks along 2θ are faults in the m ulti-detector,
where the tubes “saturate” when going through a Bragg peak. When this happens the neutrons
diffuse and get caught up in the neighboring tubes.
*
10.2 Transformed data as reciprocal planes
When the raw data has been collected and combined into (2θ, ω) datasets as shown in Fig.
10.1,
it is possible to transform it into rec iproc al space planes using the software TVtueb, [
167].
Here, two peaks are marked as known – say, the (2θ, ω) ≈ (55
◦
, −245
◦
) peak as (020) and the
*
Technically this is not an actual detector saturation since all neutrons are still detected, albeit be it in the wrong
detector tube. This effect is also called bleeding. In the reciprocal space maps I show in this thesis, the bleeding is
especially visible because the colormaps used for plotting are chosen to show small effect, whereas the large Bragg
reflections are cut off by ending the colormap long befor e their maximum intensity.

94 10.2. Transformed data as reciprocal planes
2θ [deg]
ω [deg]
20 30 40 50 60 70 80 90
−240
−220
−200
−180
−160
−140
−120
−100
−80
Intensity [arb.unit]
0
20
40
60
80
100
120
140
160
Detector angle 2θ [deg]
Sample rotation ω [deg]
Intensity I [a.u.]
Figure 10.1: Example of a measured (2θ, ω) plane for La
2
CuO
4+y
at T = 10 K, created by combining
many separate “powder spectra” from DMC with the csc program.
k [r.l.u.]
l [r.l.u.]
−2 −1 0 1 2 3
−7
−6
−5
−4
−3
−2
−1
0
1
2
3
Intensity [arb.unit]
0
20
40
60
80
100
120
140
160
k [r.l.u.]
l [r.l.u.]
Intensity I [a.u.]
Figure 10.2: Example of a reciprocal space map for La
2
CuO
4+y
at T = 10 K, created by transforming the
data in Fig.
10.1 with the TVtueb program.
(2θ, ω) ≈ (67
◦
, −148
◦
) peak as (006) – and the program then calculates the transformation
from the position of these. In this way, the data from Fig. 10.1 turns into a reciprocal space
map as shown in Fig.
10.2.
Note how the powder lines now turn into rings – as is fully expected – and the detector
saturation streaks turn into curves.
With TVtueb it is also possible to take symmetries into account, meaning that it is possible to
use a two-fold rotational symmetry on the plane shown in Fig.
10.2 in order to get a clearer
picture of the plane. This results in the plot shown in Fig.
10.3(a). The same treatment has been
done on several other datasets for three crystals, although I will only shown the low temperature
reciprocal space maps here, in Figs. 10.3 to 10.5 for La
2
CuO
4+y
, La
1.94
Sr
0.06
CuO
4+y
, and La
1.91
-
Sr
0.09
CuO
4+y
, respectively. An overview of all of the measurements can be seen in Appx. E.

Chapter 10. Reciprocal space maps 95
k [r.l.u.]
l [r.l.u.]
−3 −2 −1 0 1 2 3
−6
−4
−2
0
2
4
6
Intensity [arb.unit]
0
20
40
60
80
100
120
140
160
k [r.l.u.]
l [r.l.u.]
Intensity I [a.u.]
(a)
h [r.l.u.]
k [r.l.u.]
−3 −2 −1 0 1 2 3
−3
−2
−1
0
1
2
3
Intensity [arb.unit]
0
20
40
60
80
100
120
140
160
h [r.l.u.]
k [r.l.u.]
Intensity I [a.u.]
(b)
Figure 10.3: Symmetrized reciprocal space maps measured at DMC at T = 10 K for the La
2
CuO
4+y
sample
in two different settings; (a) (0kl) plane and (b) (hk0) plane. Measurements at several other temperatures
can be found in Appx.
E.

96 10.2. Transformed data as reciprocal planes
k [r.l.u.]
l [r.l.u.]
−3 −2 −1 0 1 2 3
−6
−4
−2
0
2
4
6
Intensity [arb.unit]
0
5
10
15
20
25
30
35
40
45
50
k [r.l.u.]
l [r.l.u.]
Intensity I [a.u.]
(a)
h [r.l.u.]
k [r.l.u.]
−3 −2 −1 0 1 2 3
−3
−2
−1
0
1
2
3
Intensity [arb.unit]
0
5
10
15
20
25
30
35
40
45
50
h [r.l.u.]
k [r.l.u.]
Intensity I [a.u.]
(b)
Figure 10.4: Symmetrized reciprocal space maps measured at DMC at low temperatures for the La
1.94
-
Sr
0.06
CuO
4+y
sample in two different settings; (a) (0kl) plane with temperatures between 100 and 1.5 K
(measured while cooling) and (b) (hk0) plane with T = 2 K. Measurements at several other temperatures
can be found in Appx.
E.

Chapter 10. Reciprocal space maps 97
k [r.l.u.]
l [r.l.u.]
−3 −2 −1 0 1 2 3
−6
−4
−2
0
2
4
6
Intensity [arb.unit]
0
10
20
30
40
50
60
70
80
90
k [r.l.u.]
l [r.l.u.]
Intensity I [a.u.]
(a)
h [r.l.u.]
k [r.l.u.]
−3 −2 −1 0 1 2 3
−3
−2
−1
0
1
2
3
Intensity [arb.unit]
0
10
20
30
40
50
60
70
80
90
h [r.l.u.]
k [r.l.u.]
Intensity I [a.u.]
(b)
Figure 10.5: Symmetrized reciprocal space maps measured at DMC at T = 100 K for the La
1.91
Sr
0.09
Cu-
O
4+y
sample in two different settings; (a) (0kl) plane and (b) (hk0) plane. Measurements at several other
temperatures can be found in Appx.
E.

98 10.3. Observations in the reciprocal space maps
10.3 Observations in the reciprocal space maps
Looking at the r ecipro cal space maps in Figs.
10.3 to 10.5, one thing is apparent: the number
of superstructural peaks – especially for the La
2
CuO
4+y
sample – is staggering. I will try to give
an overview of the reflections in the reciprocal space maps in the following.
10.3.1 Staging
The staging is very visible for the La
2
CuO
4+y
sample in Fig.
10.3(a) at positions with even l and
odd k, where sets of three or five peaks (depending on the central peak location) are clearly
seen along l, possibly with a diffuse background or several smaller peaks hidden underneath.
For the La
1.94
Sr
0.06
CuO
4+y
sample in Fig.
10.4(a) the positions with even l and odd k seem
to have more of a diffuse background, with no resolved superstructure peaks.
For the La
1.91
Sr
0.09
CuO
4+y
sample in Fig.
10.5(a), there is only a hint of a diffuse background
on these positions.
10.3.2 Other superstructures and forbidden reflections
Table
4.2 (on p. 25) give the allowed reflections for the Bmab and Fmmm space groups. For
both Bmab and Fmmm, a (hk0) plane requires h and k be even for reflection, while they are
different in the (0kl) plane: for Bmab it is simply required that l is even (and k is even if l = 0);
while for Fmmm i t is required that both k and l be even on the whole plane.
For La
2
CuO
4+y
, the (0kl) plane, apart from the clear staging peaks around the Bmab allowed
positions at odd k and even l, there are a few similar superstructure peak “chains” along k at
l = ±1 around k = ±2.5 and l = ±5 around k = ±1 ± 0.5. Single superstructure peaks at odd l
for even k (including 0) are also seen. Finally, complicated superstructures appear in “crosses”
around the fundamental reflection positions with both k and l even.
In the (hk0) plane there is also a multitude of superstructural peaks, especially concentrated
around the fundamental (200), (220), and (020) peaks. The crystal may have been slightly out
of alignment for the scan, however, since the (220) peak seems to be very faint and might be
out-of-plane, as opposed to the (2
¯
20) which ought to have the same intensity because of the
symmetry.
For La
1.94
Sr
0.06
CuO
4+y
, the (0kl) plane contains peaks at integer positions: the positions with
both k and l even are allowed in both Bmab and Fmmm, while the k odd and l even positions
are only allowed in Bmab. The small peaks with both k and l odd are not allowed in either
space group, and likely related to a commensurately modulated superstructure. Large diffuse
backgrounds behind the fundamental peaks with both k and l even are also seen, especially
for non-zero k.
In the (hk0) plane there is a hint of a forbidden peak at k = 3 for l = 0. Incommensurate
peaks are also observed close to the fundamental peaks at (200), (22 0), and (020), although
several of these are very smeared.
For La
1.91
Sr
0.09
CuO
4+y
, the (0kl) plane seemingly contains only peaks at integer positions. The
positions with both k and l even are allowed in both Bmab and Fmmm, while the k odd and l
even positions are only allowed in Bmab. The small peaks with both k and l odd are not allowed
in either space group, and must be related to a superstructure as for La
1.94
Sr
0.06
CuO
4+y
.
In the (hk0) plane there are small visible peaks at (210) and (120) (as well as the correspond-
ing peaks in the other quadrants): these are disallowed for both Bmab and Fmmm. A few small
incommensurate peaks are also seen close to the fundamental peaks at (200), (220), and (020).

99
IV.
Discussion of results

100

101
11
Discussions
In this chapter I will discuss the results presented in the last two chapters, as well as compare
these with results from literature. I will discuss the lattice parameters found for the low temper-
ature X-ray measurements and compare the doping dependence on these with earlier results. I
will then comment on the differences in peak intensities between the RITA-II neutron data and
the BW5 X-ray results. After this, I will compare the found transition temperatures and critical
exponents with similar earlier results, arguing what these results might mean in the context of
a few known theoretical models. Finally, I will comment on staging as it has been presented in
literature up till now, and discuss a few ways the data analysis could be improved.
11.1 Lattice parameters
The lattice parameters given to the BW5 instrument during the experiments (shown in Tab.
7.1)
were mostly off from the real lattice parameters observed while scanning. The calculated c axis
lengths found by calibrating the l axis to have data correctly centered around l = 4 was shown
in Fig.
9.5, with the low temperature values listed in Tab. 9.1. The low temperature values for
the lattice constants found from neutron experiments (also shown in Tab.
9.1) did not agree
well with the X-ray results, except for the La
2
CuO
4+y
sample. The discrepancy – between 0 .03
and 0.8% difference in length, with the larger values for the samples with higher Sr content
– may be explained by the fact that the neutron measurements to get the lattice parameters
were done quickly on the (002) peaks (or similar), according to Udby [154], and might not be
as precise as their uncertainties would indicate. What both sets of c lengths could agree on,
however, was that they decrease for larger Sr doping.
In the following, I will conclude a little further on the differing lattice constants, as well as
calculate the lattice length difference between the seemingly different (at least for some of the
samples) structural phases Bmab and Fmmm. Finally, I will compare with a few results from
literature.
11.1.1 Two different structural phases
The distances in reciprocal space between the Bmab central peak and the mid-point for the
staging peaks in the Fmmm phase, δl, were found in Sect. 9.2 and the low temperature results
were listed in Tab.
9.2. These distances can be recalculated into units of Å
−1
by using the
approximate low temperature lattice parameters for the samples from Tab.
9.1, and
δl [Å
−1
] = δl [r.l.u.]
2π
c [Å]
. (11.1.1)

102 11.1. Lattice parameters
Sample c [Å] δl [r.l.u.] δl [Å
−1
] c
Bmab
[Å] c
Fmmm
[Å]
La
2
CuO
4+y
13.2038(11) 0.0094(7) 0.0045(5) 13.2038(11) 13.235(4)
La
1.96
Sr
0.04
CuO
4+y
13.1367(7) 0.014(2) 0.0067(3) 13.091(3)
13.1367(7)
La
1.935
Sr
0.065
CuO
4+y
13.0857(2) 0.0017(9) 0.000 82(10) 13.0857(2) 13.0801(11)
La
1.91
Sr
0.09
CuO
4+y
13.0346(1) 0.000 39(16) 0.000 19(5)
13.0346(1) 13.0333(3)
Table 11.1: Lattice constant length c from the l axis calibration used together with the distance δl between
the Bmab peak position and the staging mid-point to find the lattice constant for both the Bmab phase
and the Fmmm phase at low temperatures. The greyed out c
Bmab
and c
Fmmm
lattice constants are the
ones that were found directly (same value as the second column), while t he others are calculated with the
values in the table using Eq. (11.1.2) as explained in the text.
The results from this calculation are shown in Tab.
11.1.
The two different lattice parameters for the two individual structural phases are also shown
in this table – calculated using the one known lattice parameter and the distance between the
two in reciprocal space. For L a
2
CuO
4+y
this was done by
c
Fmmm
=
2π
c
∗
Fmmm
=
2π
c
∗
Bmab
− δl/4
=
2π
(2π)/c
Bmab
− δl/4
, (11.1.2)
where the δl is divided by four since the δl value was found at the l = 4 position. For La
1.96
-
Sr
0.04
CuO
4+y
the sign in the denominator is flipped to a plus sign and Fmmm and Bmab switch
places (the directly found lattice constant was the one belonging to Fmmm because the mid-
point of the staging peaks – as opposed to the Bmab peak – was used for the l axis calibration
for this sample). For La
1.935
Sr
0.065
CuO
4+y
and La
1.91
Sr
0.09
CuO
4+y
, the sign is flipped because
the Bmab peak was lower in l instead of higher (as also indicated in Tab. 9.2).
As was already mentioned in Sect.
9.2, the differences in apparent lattice parameter for the
central Bmab peak, c
Bmab
, and the mid-point of the staging peaks, c
Fmmm
, only seem significant
for the two first samples, La
2
CuO
4+y
and La
1.96
Sr
0.04
CuO
4+y
, with a difference of 0.24 and 0.35%,
respectively – whereas the two parameters are very close (and even have the opposite relation)
for La
1.935
Sr
0.065
CuO
4+y
and La
1.91
Sr
0.09
CuO
4+y
, with differences of 0.04 and 0.01%. It is also
only these two first mentioned samples that show the staging peaks to be a clear staging with
small values of the staging number n, while the La
1.935
Sr
0.065
CuO
4+y
and La
1.91
Sr
0.09
CuO
4+y
samples only had very low intensity staging peaks with higher values of n.
As a final comment it should be noted that the splitting of the structure into two phases with
different lengths of the crystal axis c has been seen before for at least La
2
CuO
4+y
, [
100]. It
was found that the Bmab c axis had a low temperature length of around 13.13 Å, with the axis
connected to the staging phase being instead around 13.18 Å, [
100, Fig. 5]. This difference in the
axis length of about 0.05 Å show a similar relation to what was found for the BW5 X-ray data,
although the absolute lengths are lower, with the Bmab value matching other low temperature
results from literature well, e.g. Radaelli et al. [
108].
11.1.2 Lattice length as a function of doping
As mentioned, it was found that the lattice parameter length c becomes shorter for increasing
Sr doping in co-doped samples, both for the Bmab and Fmmm phase. However, interestingly,
Radaelli et al. [108] found that the opposite happens when only doping with Sr in oxygen-
stoichiometric samples: higher Sr doping meant a longer c axis, as shown in Fig.
11.1 with
the gray-scale points. The results from this thesis have been marked in this figure as well
(blue diamonds and green triangles for Bmab and Fmmm, respectively) along with the earlier
neutron experiments (yellow squares).
Since the behavior of the c axis length (for both structural phases) i s directly opposite to
what was found for only Sr doped samples, the O doping must affect the crystal structure along

Chapter 11. Discussions 103
0.00 0.05 0.10 0.15 0.20 0.25
13.00
13.05
13.10
13.15
13.20
13.25
13.30
Strontium doping x
Lattice parameter c [Å]
Figure 11.1: Lattice constant c as a function of Sr doping (no oxygenation) for T = 10 K (dark gray circles),
70 K (black circles), and 295 K (light gray circles) foun d with neutron and X-ray powder diffraction by
Radaelli et al. [
108] and for T = 300 K (light gray squares) found with neutron powder diffraction by
Rial et al. [
101]. The purple and red squares are measurements on co-doped powder samples at T = 10
and 300 K, respectively, by Rial et al. [
101]. The yellow squares mark the c axis lengths for the four
single-crystal samples used in this thesis, found at low temperature with neutrons by Udby [
154]. The
blue diamonds mark the c axis lengths found for the low temperature BW5 data Bmab peaks (see Tab.
9.1),
while the green triangles mark the corresponding Fmmm values. For points without visible errorbars, the
errors are smaller than the marker. Lines through the points are guides to the eye.
c in a way that stops the unit cell from stretching out when adding Sr. This was exactly found
by Rial et al. [
101], who used neutron powder diffraction to investigate a set of samples with
several different Sr dopings, before and after superoxygenating them (their structure results
are listed in Tab.
B.4). They indeed saw the c axis length increasing for increasing Sr content
before oxygenation (their investigations were only done at room temperature). These results
are also shown in Fig.
11.1, as the light gray squares that match well with the Radaelli et al. high
temperature data. After oxygenation however, their results were as shown with purple and red
squares in the figure, for T = 10 and 300 K, respectively. At high temperature, they saw that
the c axis length increased less with x for the oxygenated samples, while it was decreasing with
x at low temperatures, as we see i n our data (although not with as high a slope as seen in the
present results).
Replacing some of the small
*
La
3+
ions with larger Sr
2+
ions, electrons are removed from the
CuO
2
planes and reliefs compressive stress, while tensile stress i s released within the (La,Sr)O
planes by making the distances La−O and Cu−O in the structure more similar, [
123]. Similarly,
adding O
2–
into interstitial positions again reduces the tensile stress in the (La,Sr)O planes and
“steals” electrons from the CuO
2
planes, with the same result.
Both of these effects individually c ause the unit cell axis length c to increase for Sr-only
doped samples ([
108], shown in Fig. 11.1 and listed in Tab. B.4), and for O-only doped samples
([
98, 100, 104], some results listed in Tab. B.3). In many of these publications the unit cell
axis length a instead becomes smaller and b remains constant with O doping, while a seems
constant and b decreases towards the value of a with Sr doping.
*
Looking up the crystal ionic radii for the constituents of the samples, it is seen that the size of La
3+
is 1.17 Å,
while the size of Sr
2+
is slightly larger at 1.32 Å, [
150].

104 11.2. Comparing measured intensities for X-ray and neutron data
The present results indicate that when co-doping, these two effects somehow work together
to make c decreasing for increasing Sr doping, with the exact value of the O-doping unfortu-
nately unknown. The behavior of the a and b axes were not investigates, however. Rial et al.
[
101] saw the lattice constant a become smaller after oxygenation for all Sr do pings, so it could
be interesting to measure values for the other lattice constants more precisely for the present
work as well. This would give a better understanding of how these behave with temperature
for the different single-crystal samples. Further, it would be very relevant to find the absolute
value of the O doping, y, in order to compare more precisely with the results from literature.
11.2 Comparing measured intensities for X-ray and neutron data
The integrated intensities found for the X-ray data from BW5 and the neutron data from RITA-II,
as presented in Sects. 9.2 and 9.5, have certain differences in the relations between the central
and staging peak values. I will here mention these differences, and discuss what might cause
them.
For La
2
CuO
4+y
seen with neutrons, the integrated intensity of the central peak is a little smaller
than the values for the staging peaks, compared to what is seen in the X-ray data. For the
neutron data the integrated intensity of the central peak (at low temperatures) is about the
same size as the low l staging peak value (black circles and colored triangles in the third inset of
Fig.
9.30(a), respectively), both around 0.6 cts/s, whereas the high l staging peak is significantly
higher than this (colored circles in the third inset of Fig.
9.30(a)), at around 1.1 cts/s. For the
X-ray data, the low temperature integrated intensities for the two main staging peaks and the
central peak are all comparable, as seen in Fig.
9.22(a) and (b), all around 2.3 a.u.
*
This small difference might simply be explained by several staging peaks overlapping in the
neutron data (due to a larger resolution function), and hence being counted in the main staging
peaks, explaining why the high l neutron staging peak is larger (although the low l peak is not,
which might go against this explanation).
For La
1.96
Sr
0.04
CuO
4+y
seen with neutrons, the integrated intensity of the central peak is signifi-
cantly larger than the values for the staging peaks, compared to what is seen with X-rays, where
the central peak is very small. For the neutron data the integrated intensity of the central peak
(low temperatures) is about 4.5 cts/s (black circles in the third inset of Fig.
9.30(b)), whereas
the staging peaks lie between 5.5 and 8 cts/s (colored circles and triangles in the third inset
of Fig.
9.30(b)). For the X-ray data, however, the low temperature integrated intensity of the
central peak lies only around 0.38 a.u. compared to the 2.0 to 2.2 a.u. values for the staging
peaks (Fig.
9.10(a) and (b)).
One explanation for the large difference in the central peak to staging ratio as seen with neu-
trons and X-rays could be that the peak is magnetic in origin: magnetic order is easily seen with
neutrons, while the signal from this is orders of magnitude lower when using X-rays. Magnetism
in La
2−x
Sr
x
CuO
4+y
tends to disappear very quickly with both x and y, however (as shown in
Fig.
6.3), and so far only incommensurate magnetic order has been seen for these compounds
at higher dopings, [
124]. That is, a magnetic peak at integer (hkl) would be something very
unexpected. A second, much simpler, explanation could be the structure factor in the central
peak being very dependent on the oxygen scattering length (more so than the staging peaks), in
which case the neutrons will see the signal as larger since the oxygen scattering length is larger
for neutrons than for X-rays compared to the scattering lengths of the other elements in the
compound. A way to investigate this further could be to use polarized neutrons, which make it
possible to obtain additional information about magnetic structures in the material.
*
Note that the X-ray data integra te d intensities are all given with arbitrary units (a.u.), although they are all found
in the same way, and normalized to the monitor – so the values are still comparable fo r the same sample.

Chapter 11. Discussions 105
For La
1.935
Sr
0.065
CuO
4+y
seen with neutrons, the central peak was simply fitted with one narrow
and one underlying wider peak, which makes it more difficult to compare intensities to the X-ray
data, where the wider peak was instead split up into two staging peaks, one in each shoulder of
the central peak. For the neutron data the narrow central peak, thought of as the Bmab peak,
has a low te mperature integrated intensity of around 20 cts/s (black circles in the third inset
of Fig.
9.31(a)), whereas the wider peak, thought of as the underlying disorder, has the value
46 cts/s (colored triangles in the third inset of Fig.
9.31(a)). For the X-ray data, on the other
hand, the central Bmab peak has the largest integrated intensity, at 8.3 a.u., whereas the two
staging peaks each have values of between 1.0 and 1.7 a.u. (Fig.
9.14(a) and (b)).
The difference in method of analysis on the data from the two methods makes the compar-
ison a little more difficult. It would seem that part of the intensity from the central peak seen
with X-rays might be attributed to the underlying peak in the neutron data. In general, though,
the peak shapes seem fairly similar. It could be interesting to have both full datasets and do
the analysis in the same way for both of them, making a more meaningful comparison possible.
For La
1.91
Sr
0.09
CuO
4+y
seen with neutrons, only a simple central peak was observed, whereas
a strong asymmetry interpreted as staging was seen in the X-ray data. This can very well be
explained by the difference in resolution with the two different methods – with the neutron data
showing a much wider peak: around 0.015 Å
−1
for the neutron data (second inset of
9.31(b))
versus around 0.0045 Å
−1
for the X-ray data (as calculated in Subsect.
9.2.3). The staging peaks
are simply not resolvable in the neutron data with this r esolution.
11.2.1 About absolute scale intensities
In order to be able t o compare the measured integrated intensities for each of the measured
reflections with calculated structure factors for the different techniques, it is necessary to nor-
malize the values to an absolute scale.
With neutron scattering, it is fairly straight-forward to normalize the intensity data using
the mass of the samples, since the incoming beam tends to be large enough to cover the full
sample – and at the same time has an interaction small enough to actually see the bulk of the
material. For X-rays, however, the beam is very small, and hence all of the sample is not neces-
sarily illuminated, on top of the fact that the interaction is much larger: a typical attenuation
length for La
2−x
Sr
x
CuO
4+y
is λ = 0.85 mm for hard X-rays, following the calculations done in
Appx.
A.5. Most usual X-ray samples are fairly small, in t he order of µm, but the samples used
on BW5 had sizes of the order of mm, as listed in Tab.
7.1, which just makes the normalization
even more difficult.
It is possible to instead normalize the measured X-ray intensities by normalizing to the
size of a fundamental reflection, e.g. (200), at the same temperature instead. This might be
interesting to do for the results in this thesis, in the case where it is possible to find a reason-
able number of fundamental peak measurements with the same attenuation as the reflections I
want to normalize. This does, however, also assume that the extinction in the sample is negli-
gible, which could require measurements be done at specific fundamental reflections that have
smaller structure factors instead (since extinction is a larger eff ect for larger reflections, follow-
ing the description in Subsect.
4.4.1). If this is not possible, it would be necessary to correct for
the extinction using the sample geometries – or, in the worst case scenario, instead go back and
repeat measurements on smaller samples. A second effect that might make the normalization
difficult is if the samples have very imhomgeneous shapes, which is mostly the case, as seen
in Fig. 7.1. In this case, absorption correction of the data will be almost impossible, and repeat
measurements on samples chosen for their shape will have to be done.

106 11.3. Discussion of phase transition results
11.3 Discussion of phase transition results
A full overview of the found transition temperatures and critical exponents for the BW5 ex-
periments was shown in Tab. 9.2 and the observed transition temperatures for the RITA-II
experiments were shown in Tab.
9.3. Some results from literature are shown in Tab. 11.2, while
the theoretically calculated values were shown in Tab.
5.1 (note that the critical exponents are
given as β, not 2β, in the table with theoretical values). Some of all these values are compared
in Fig.
11.2.
11.3.1 Critical exponents
Starting with the critical exponents shown in Fig.
11.2(a) – which were found for most of the
reflections analyzed from the BW5 X-ray data – an actual tendency with Sr doping is not visible
for the central Bmab reflections, while there is a clear tendency for β to increase with increasing
Sr doping for the staging reflections.
There are several publications with investigations of the Bmab peak on relevant compounds,
as shown in the upper part of Tab.
11.2, with the points shown as green triangles in Fig. 11.2(a).
From comparing with these, it is slightly difficult to conclude whether the BW5 X-ray results
follow the same behavior as the literature examples, mainly because of the fact that it was
not possible to find literature on compounds with Sr dopings comparable to the current work
– with or without O doping – that state the critical exponents for their results. The present
results seem consistently lower in value, and while both Böni et al. [
125] and Braden et al. [126]
concluded that their systems could be modeled by the 3D XY model, the present results more
point in the direction of a 2D model like the Ising model marked in the figure with the lowest
green line.
The staging in the isostructural compound La
2
NiO
4+y
(shown in the lower part of Tab.
11.2)
seems to have the same tendency as the staging results from the BW5 X-ray measurements, with
a higher doping meaning a higher value of β, even though the actual critical exponents were
not me ntioned directly in the published work. Note, though, that the dopings investigated by
Tranquada et al. are only with oxygen, so the applicability of this comparison is limited.
However, a comparison between the phase transition seen for the staging in La
2
NiO
4.06
with
n between 3 and 4, shown in Fig.
11.3, with the phase transition seen for the main staging
peaks in the La
2
CuO
4+y
sample with n between 4 and 5, shown in Fig.
9.22(b), is especially
noteworthy. Both were very rapid, almost approaching first order – something which is difficult
to explain with simple models for continuous phase transitions. Evidence for the oxygen order-
ing in La
2
NiO
4.105
being first-order (with the kinetics of the oxygen ordering being understood
in terms of nucleation and growth) was actually found, [
128], with the shape of the measured
transition being dependent on the cooling conditions – indicating that something similar could
be happening for the La
2
CuO
4+y
sample, which could be very interesting to investigate further.
In general, it is observed that the critical exponent for both the Bmab and staging reflections
vary widely, from very low values for the staging peaks in La
2
CuO
4+y
to values all the way up
to 2β ≈ 0.5 for the staging peaks in La
1.91
Sr
0.09
CuO
4+y
.
Very low values of β could indicate a close to one-dimensional system
*
or a close to first-
order transition, while the higher values – especially seen for the staging – could point in the
direction of the conclusions made by Böni and Braden et al., with describing the long-range
order in the 3D XY model. There is no ground to rule out the other two shown 3D models,
the Ising and Heisenberg models, here, however, which both have critical exponents very close
to the 3D XY model. This indicates having to use very different models to describe the long
range order in the different samples, while a broader scoped model able to describe all the
*
Note however, as was mentioned in Chap.
5, that many pure 1D systems do not show phase tr an si ti on s – e.g. the
1D Ising model, [129].

Chapter 11. Discussions 107
Critical Critical
Sample and peak
temperature [K] exponent 2β
Reference
Central Bmab peak results T
HTT→LTO
La
2
Cu
0.95
Li
0.05
O
4−δ
423(1) 0.55(8) Böni et al. [125]
La
2
CuO
4
507(1) 0.56(2) Braden et al. [
126]
La
1.97
Sr
0.13
Cu
0.95
Li
0.05
O
4−δ
351(1) 0.58(8) Böni et al. [
125]
La
1.87
Sr
0.13
CuO
4+y
196.5(5) 0.71(2) Braden et al. [
126]
Staging results T
staging
La
2
NiO
4.06
, n = 3-4 235(5) very small Tranquada et al. [113]
La
2
NiO
4.074
, n = 2 280 Tranquada et al. [
113]
La
2
NiO
4.074
, n = 3 245(10) Tranquada et al. [
113]
La
2
NiO
4.105
, n ≈ 2 ∼290
large Tranquada et al. [113]
Table 11.2: Observed critical temperatures and critical exponents from literature. No staging results are
available in literature for staging peaks in La
2−x
Sr
x
CuO
4+y
, but the isostructural compound La
2
NiO
4+δ
is a common comparison.
2D Ising
3D Ising
3D XY
3D Heisenberg
0
0.25
0.5
0.75
Critical exponent 2β [Å]
( )
(a)
0 0.02 0.04 0.06 0.08 0.10 0.12
0
100
200
300
400
500
Strontium doping x
Transition temperature [K]
oxygen stoichiometric
( )
( )
(b)
Figure 11.2: (a) Critical exponents 2β and (b) transition temperatures found for the BW5 X-ray measure-
ments (Tab.
9.2), with blue circles m arking the central Bmab peak and red and yellow circles marking the
low and high l staging peaks, respectively. Note the x = 0.00 Bmab transition temperature, which was only
determined to be above T
HTT→LTO
= 300 K, marked with an arrow. The transi tion temperatures for the
RITA-II neutron measurements (Tab.
9.3) are shown with squares, with colors corresponding to the X-ray
points. There is a general tendency for increasing 2β with increasing x for the staging reflections, but no
clear tendency seen for the Bmab reflections. In the same way, a tendency for increasing T
staging
with x
is observed, while T
HTT→LTO
is seen to have a clear minimum at x = 0.04, both for neutrons and X-rays.
Results from literature (only for the Bmab reflection, top part of Tab. 11.2) are marked as green triangles
(Li doped samples are shown with parentheses). The critical exponents are compared to theoretical values
from Tab.
5.1 (green lines). For comparison, the T
HTT→LTO
for only Sr-doped found in e.g. Chang [127] are
also shown (green line, some values written in Tab.
B.1). For points without visible errorbars, the errors
are smaller than the marker.

108 11.3. Discussion of phase transition results
200 220 240 260 280 300
0
2
4
6
8
10
Temperature T [K]
Peak intensity I [10
3
cts/min]
(0,3,1.72)
(0,3,2)
(0,3,1.8)
÷10
Figure 11.3: Temperature dependence of the scattered intensity for three different peaks measured along
l for a La
2
NiO
4.06
single crystal with neutron scattering. Especially the (0,3,1.72) peak (blue squares) is
relevant for comparison with the La
2
CuO
4+y
results in this thesis, with the very rapid change in intensity,
just as what was seen for the main staging peaks in the BW5 X-ray data (Fig.
9.22(b)). Figure adapted from
[
113, Fig. 13].
critical exponents at the same time would be favorable. As a comment to this, the electronic
phases and conduction in La
2
CuO
4+y
and La
2−x
Sr
x
CuO
4
have been investigated in the context
of percolation theory, which describe networks of conductive regions in the materials, [
130–
132]. Percolation theory might be interesting to look more into for the structural phases as well,
since this might be able to describe the apparent di fferent dimensionalities of the observed
order in the samples.
11.3.2 Transition temperatures
The transition temperatures were shown in Fig.
11.2(b), both for the BW5 X-ray measurements
(circles) and the RITA-II neutron measurements (squares). The blue points mark the transition
temperatures for the central Bmab peaks, and these are seen to have a minimum around the
La
1.96
Sr
0.04
CuO
4+y
sample. The red and blue points mark the low and high l staging peaks, and
the general tendency for both of these seem to be a slightly higher transition temperature with
increasing Sr doping.
For the Bmab peaks, the transition temperature T
HTT→LTO
is already well known for the Sr-
only doped compounds, for which the values are shown in Fig.
11.2(b) as a green line (see also
Fig.
6.3(a)). It is expected that the transition temperature will decrease for co-doped samples
after oxygenation, as it is seen in Fig.
6.3(b) showing the doping dependence of the O-only doped
compounds. The values listed in the top part of Tab.
11.2 from literature do not seem to be in
disagreement with the present results, although (again) this comparison is difficult because of
the limited number of values found in literature. The minimum at x = 0.04 is interesting, but
yet unexplained: the very stretched out phase transition seen for the La
1.96
Sr
0.04
CuO
4+y
sample,
shown with blue in Fig. 9.29(b), could be investigated further by taking the critical scattering
from the sample into account. Note that the minimum is seen in both the X-ray and neutron
data – the small intensity seen with X-rays, and the connected problems with data analysis to
this, cannot explain the minimum alone, since the large signal seen with neutrons show the
same tendency.
For the staging peaks, the transition temperatures T
staging
are seen to increase with increas-
ing Sr doping, as mentioned. Again it is interesting to compare with the staging seen in the

Chapter 11. Discussions 109
isostructural compound La
2
NiO
4+y
, with results from Tranquada et al. [113] shown in the bot-
tom part of Tab.
11.2. The doping is only O-doping, but a slight tendency for larger T
staging
for
larger y is also seen for these compounds.
11.4 Further comments on staging in literature
There has not been any re search on the staging in the co-doped compound La
2−x
Sr
x
CuO
4+y
outside our research group. However, staging in the oxygen-doped compound, La
2
CuO
4+y
, was
investigated vigorously by Wells et al. [
100]. They observed stagings between 2 and 6.6 for their
samples depending on the ox ygenation treatment. This agrees well with the different staging
numbers found in the results for the La
2
CuO
4+y
sample presented in this thesis, which were
between 2 and 6, with a very small intensity of a possible staging peak all the way up to n
between 8 and 9.
11.4.1 Phase transition behavior for varying staging number
Apart from the results from Wells et al., the main results in literature on staging stems from the
isostructural compound La
2
NiO
4+y
, as has also been mentioned earlier. Tranquada et al. [113]
investigated this and characterized the staging for different oxygen dopings. Some results of
these investigations were shown in Tab.
11.2, where the staging numbers n observed were also
listed. It was, in rough terms, seen that lower staging numbers had higher transition temper-
atures: the n = 2 and close to 2 stagings seen in La
2
NiO
4.074
and La
2
NiO
4.105
had transition
temperatures of 280 and around 290 K, respectively, while the higher n = 3 and between 3 and
4 stagings instead had transition temperatures at around 235 and 245 K.
The results in this thesis go directly against this, with the n = 2, 3.1, and 4.6 staging peaks
for La
2
CuO
4+y
having the same transition temperature T
staging
of around 175 K (see Tab.
9.2);
the n = 5.1 staging peaks for La
1.96
Sr
0.04
CuO
4+y
having T
staging
= 215-216 K; the n = 13.2
staging peaks for La
1.935
Sr
0.065
CuO
4+y
having T
staging
= 225 K; and finally the n = 89 (and larger)
staging peaks for La
1.91
Sr
0.09
CuO
4+y
having T
staging
= 2 79 -30 0 K. That is, a rising transition
temperature for increasing staging number, although not within the same sample, where the
transition temperature stays constant (also opposed to the results from Tranquada et al.).
From the Tranquada et al. results shown in Tab.
11.2, it was also visible how a larger staging
number n tended to have a smaller critical exponent 2β. This is also the direct opposite be-
havior to what is seen in the present work, where increasing n show a clear tendency to an
increasing β, although it is difficult to finally conclude on the smaller staging reflections for
the La
2
CuO
4+y
sample – which seems to be the sample that can easiest be compared with the
Tranquada et al. results on La
2
NiO
4+y
.
In the end, based on the present data, it is difficult to finally conclude what determines the
transition temperature and critical exponents for the different staging numbers – there might
not even be a simple relation. Further investigating this with more doping values and more
temperature measurements in the relevant temperature regions could be interesting.
11.5 Other superstructures
Staging was also observed in the r ecipr ocal space maps presented in Chap.
10, especially well
for the La
2
CuO
4+y
sample. Several other superstructures were also observed, amongst them
commensurate peaks indicating that the samples indeed do have at least a fraction of the
sample in the Bmab phase instead of the Fmmm or the tetragonal I4/mmm phases.
For La
2
CuO
4+y
, Fig.
10.3(a), small peaks similar to staging superstructures were seen along
k, which might indicate a corresponding (oxygen) ordering along b (or a such that the signal is

110 11.6. Peak widths and shapes
simply seen due to twinning) in the crystal structure, as was also observed by Wochner et al.
[
115] in La
2
CuO
4+y
with y ≈ 0.015. This could of course be interesting to investigate further
in our samples, especially if extending the investigations to the co-doped samples.
Fairly recently, Fratini et al. [
133], Poccia et al. [134 ] investigated some of the other observed
superstructures for La
2
CuO
4+y
, specifically parts of the “crosses” also observed in the present
work. They concluded that the oxygen ordering split into several different real-space regi ons in
the samples, connected in fractal patterns – as opposed to isolated domains – in samples show-
ing optimal superconductivity. This again points in the direction of investigating the present
results in connection to percolation theory (as was also mentioned in Sect.
11.3).
11.6 Peak widths and shapes
Through the data analysis, the peak shapes have been fitted with the empirically best matching
peak shapes, be it Gaussian, Lorentzian, or a superposition of these. When a peak is very wide,
and it fits best with a Gaussian, this might come from the mosaicity in the sample determining
the width. This was especially the case with the central peaks for the La
2
CuO
4+y
and La
1.91
-
Sr
0.09
CuO
4+y
samples (the former though with a low Lorentzian tail).
As was already mentioned once in Sect. 9.2, it could be interesting to fit all of the reflections
with Voigts instead of Gaussians and Lorentzians. In t his way, it might be possible to follow
the peak shapes slightly more accurately, although with the cost of having to fit with more
parameters. If the resolution of the instrument is understood, this could result in a better
understanding of the mosaicity of the samples.
Another advantage of looking mo re into the peak widths would be to try to get out a second
critical exponent, ν, which is related to the correlation lengths in the sample, and hence related
to the inverse of the reflection widths. This could make it possible to easier distinguish between
which types of theore tical models might explain the observed phase transitions for the different
types of reflections in the samples.

111
12
Conclusions
In this master’s thesis the superoxygenated La
2−x
Sr
x
CuO
4+y
compound has been investigated
as single crystals with four different strontium dopings x = 0.00, 0.04, 0.065, and 0.09, by
following the temperature dependence of a specific superstructure. This superstructure is be-
lieved to be related to the periodicity of intercalated oxygen along the long axis of the crystal
structure, in a similar manner to what has been heavily investigated in the isostructural com-
pound La
2
NiO
4+y
, where the superstructure is known as staging.
The primary focus of this thesis was on analyzing X-ray data from a total of four different
experiments at the BW5 instrument at DESY, Hamburg, Germany. Is was found that the crystal
structure of La
2−x
Sr
x
CuO
4+y
does indeed show staging, also for the co-doped samples: samples
with higher strontium content showed higher staging numbers with smaller signals than what
has previously been observed for the only oxygen-doped compound.
A difference was observed between the c axis length for the structural Bmab phase and the
corresponding c axis length for the staging peak mid-poi nts, interpreted as belonging to an
oxygen-rich Fmmm phase. These two structural phases were assumed to divide the samples
into different phase fractions. The difference in lattice axes was the m ost significant for the
x = 0.00 and 0.04 samples, with a difference of 0.24% and 0.35%, respectively, while the x =
0.065 and 0.09 samples showed only a difference of 0.04% and 0.01%.
Both the staging and the Bmab phases were investigated as a function of temperature by fit-
ting measured reflections with empirically found Gaussian and Lorentzian peak shapes. The
observed phase transitions were investigated by fitting power laws to the found integrated
intensities, several times in combination with alternative analysis for finding the transition
temperature.
It was found that the critical exponents for the central integer position Bmab reflections
were in general lower than results from lit erature, and they did not show a clear tendency with
doping for the available data. For the staging reflections, however, the critical exponents for the
observed phase transitions showed a clear increase following strontium doping x. This seemed
to be consistent with earlier results on La
2
NiO
4+y
. The x = 0.00 sample showed a very sharp
transition – consistent with results on La
2
NiO
4.105
, where the oxygen ordering was found to be
first-order, with the shape of the integrated intensity curve being strongly dependent on the
cooling conditions.
The transition temperatures found for the Bmab reflections showed a minimum for the
x = 0.04 sample – possibly explained by a very stretched out phase transition with a large

112 12.1. Outlook
contribution from critical scattering – and were all consistently lower than the corresponding
oxygen-stoichiometric values for samples with the same strontium content. This lowering with
oxygenation was consistent with earlier results. The transition temperatures for the staging re-
flections showed a slow increase with strontium doping, which was also co nsistent with results
from La
2
NiO
4+y
.
Both critical exponents and transition temperatures for the staging reflections, seen as a
function of the staging number, seemed to be going against earlier results, or simply not show
a direct relation. However, considering how few results are found in both literature and the
present work, this will have to be investigated further before a clear conclusion can be made.
Comparisons with earlier analyzed neutron data from the RITA-II instrument (at SINQ, Paul
Scherrer Institute, Villigen, Switzerland), as well as newly measured reciprocal space maps also
done with neutrons on the DMC instrument (also at SINQ), concluded that the staging is also
visible for neutrons. The general behavior of the samples was reproduced between the tech-
niques. The coarser resolution on the neutron instruments did, however, make it difficult to
resolve staging for the samples with higher strontium content.
Finally, it was concluded that several other superstructures apart from the staging could
be interesting to investigate in further detail. Analyzing the present results as well as further
measurements in the context of percolation theory (or other networking mechanisms) and the
oxygenated structure being divided into fractal patterns could possibly provide a morphologic
explanation for the widely varying phase transitions over the range of doping values. In an
average picture, this could also give an understanding of the volume fractions for the differ-
ent observed superstructures as well as of the superconducting domains. Understanding the
morphology and composition of the structure in detail in this way would, hopefully, result in a
better understanding of the superconductivity in the compound as a whole.
12.1 Outlook
One of the main challenges of this thesis was the limitation of the analyzed data. More samples
with different doping values, a better knowledge of the absolute value of the oxygen content
of the samples, as well as further measurements at smaller temperature steps and on different
reflections, could make the conclusions on the results more clear – especially in the context
of earlier work.
Some preparations have already been made in order to do a full analysis of many more
superstructures in especially the x = 0.00 compound, where detailed data have been obtained
for several different temperatures at the neutron time-of-flight single-crystal diffractometer
SXD at ISIS in Harwell, Oxfordshire, England. These data will hopefully result in a full structural
model of the 3D oxygen ordering in the compound, which has been sorely missed for a long
time.
To this day the mechanisms behind the high-temperature superconductors have not been un-
derstood fully. Because of this it is important to investigate even the earliest discovered of
these materials with varying experimental techniques, building up a large puzzle that can hope-
fully – one day – be solved. With this work, I have added a small piece to the collected work,
and I hope that I will be able to add many more in the future.

113
.
Appendixes

114

115
A
Calculations
In this appendix I will show calculations that are relevant for this thesis, but which I deemed to
not be important enough to have in the main text.
A.1 Derivation of the Bragg condition for scattering
To de rive the Bragg condition for scattering, consider a crystal made o ut of parallel planes of
ions with a interplanar distance d, as shown in Fig.
A.1.
To get constructive interference of a beam scattered by two consecutive layers, the differ-
ence in distance traveled by either beam must be an integer number, n, of wavelengths λ. This
assumes elastic scattering so λ = λ
i
= λ
f
(equivalent to k = |k
i
| = |k
f
|). Looking at the figure,
the extra distance traveled by the lower of the two shown beams is 2d sin θ, and hence the
condition for constructive interference is
nλ = 2d sin θ, (A.1.1)
also known as Bragg’s law. The Laue condition further states that q = τ, such that q must equal
τ = 2πn/d. Further applying that k = 2π/λ, Bragg’s law can be rewritten to
q = 2k sin θ, (A.1.2)
which is very useful when working with reciprocal space instead of real space.
d
θθ
|k
i
| =
2π
λ
i
|k
f
| =
2π
λ
f
d sin θ
d sin θ
Figure A.1: Illustration to derive the Bragg law, showing the ingoing and outgoing beams being scattered
elastically at an angle of 2θ on crystal planes with interplanar distance d.

116 A.2. Proof that the Bragg and Laue conditions are equivalent
A.2 Proof that the Bragg and Laue conditions are equivalent
Suppose the incident and scattered wave vectors k
i
and k
f
satisfy the Laue condition, such that
q = k
i
− k
f
is a reciprocal lattice vector τ. The scattering vector must hence be a normal vector
to the scattering planes defined by τ.
Assuming elastic scattering, |k
i
| = |k
f
|, it follows that k
i
and k
f
must make the same angle
θ with the plane perpendicular to q. Hence, as seen on Fig.
A.2, the scattering can be seen as
a Bragg reflection from the family of real space lattice planes perpendicular to the reciprocal
lattice vector τ, with Bragg angle θ.
From the figure, it is geometrically visible that
|q| = 2k sin θ, (A.2.1)
and hence we come directly back to Bragg’s law in the form of Eq. (
A.1.2).
θ
θ
−k
f
k
i
k
f
q = τ
Figure A.2: Illustration to show how q = k
i
− k
f
being perpendicular to a lattice plane means that k
i
and
k
f
must have the same angle θ to the plane for elastic scattering with |k
i
| = |k
f
|.
A similar way to see the equivalence of the Laue and Bragg conditions is to look at the vectors
in reciprocal space as in Fig. A.3. Again assuming elastic scattering |k
i
| = |k
f
|, and defining the
angle between k
i
and k
f
to be 2θ, it is simple to see trigonometrically that
sin θ =
q/2
k
, (A.2.2)
which once more leads to Bragg’s law in the form of Eq. (
A.1.2).
θ
θ
k
i
−k
f
q
2π
d
τ
Figure A.3: Illustration to show how the length of q can be expressed as a function of k and θ for elastic
scattering |k
i
| = |k
f
|.

Appendix A. Calculations 117
A.3 Loss of flux due to typical sample environments on BW5
A typical cooling sample environment on the BW5 instrument, such as a displex cryostat or an
ILL “Orange” cryostat, has in total 3 mm aluminium in the beam, divided into two times 1 mm
vacuum shielding and two times 0.5 mm thermal radiation shielding, [
168].
Assuming an X-ray energy of 100 keV, it is possible to look up the absorption coefficient to
be µ = 0.458 877 08 cm
−1
, [
151]. Through the total length of aluminium in the beam, it is then
possible to calculate the remaining flux by
Ψ (z = 3 mm) = Ψ
0
exp (−µz) = Ψ
0
exp (−0.458 877 08 cm
−1
· 0.3 cm) = 0.87139 Ψ
0
. (A.3.1)
In Fig.
A.4 this calculation has been done for several more thicknesses and X-ray energies, as
an overview.
0 10 20 30 40 50 60 70 80 90 100
0
10
20
30
40
50
60
70
80
90
100
X-ray energy E [keV]
Remaining flux Ψ (z) [%]
z = 1 mm
z = 3 mm
z = 7 mm
z = 15 mm
Figure A.4: Remaining flux after sending varying energies of X-rays through several different thicknesses
of aluminium. The curves have been calculated as in Eq. (
A.3.1) using absorption coefficients from Bandy-
opadhyay et al. [
151].
A.4 Typical twinning separation
Using Eq. (6.3.2), it is possible to calculate typical angular separations between measured peaks
in the hk plane. The low temperature lattice constants were given in Tab.
7.1, and from t his I get
La
2
CuO
4+y
: ∆ = 90
◦
− 2 tan
−1

a
b

= 90
◦
− 2 tan
−1
5.30 Å
5.37 Å
!
= 0.75
◦
, (A.4.1)
La
1.96
Sr
0.04
CuO
4+y
: ∆ = 90
◦
− 2 tan
−1

a
b

= 90
◦
− 2 tan
−1
5.32 Å
5.38 Å
!
= 0.64
◦
, (A.4.2)
La
1.935
Sr
0.065
CuO
4+y
: ∆ = 90
◦
− 2 tan
−1

a
b

= 90
◦
− 2 tan
−1
5.32 Å
5.36 Å
!
= 0.43
◦
, (A.4.3)
La
1.91
Sr
0.09
CuO
4+y
: ∆ = 90
◦
− 2 tan
−1

a
b

= 90
◦
− 2 tan
−1
5.33 Å
5.34 Å
!
= 0.11
◦
. (A.4.4)

118 A.5. X-ray attenuation lengths
A.5 X-ray attenuation lengths
In order to find the X-ray attenuation coefficient for a sample material, it is necessary to first
know the density of the material. For La
2−x
Sr
x
CuO
4+y
, this is calculated by
ρ = 4
(2 − x)m
La
+ xm
Sr
+ m
Cu
+ (4 + y)m
O
V
0
, (A.5.1)
where V
0
= abc is the unit cell volume, and the 4 comes from the fact that there are four chem-
ical units in each crystallographic unit cell. The crystal lattice parameters for the four samples
can be found in Tab.
7.1, and the values for the atomic masses are m
La
= 138.9055 amu,
m
Sr
= 87.62 amu, m
Cu
= 63.546 amu, and m
O
= 15.9994 amu. Setting y = 0.055 for simplic-
ity, this results in
La
2
CuO
4+y
: ρ = 7.182 g/cm
3
, (A.5.2)
La
1.96
Sr
0.04
CuO
4+y
: ρ = 7.112 g/cm
3
, (A.5.3)
La
1.935
Sr
0.065
CuO
4+y
: ρ = 7.137 g/cm
3
, (A.5.4)
La
1.91
Sr
0.09
CuO
4+y
: ρ = 7.133 g/cm
3
. (A.5.5)
With the densities known, it is possible to look up the X-ray attenuation length λ for the material
using the NIST database X-ray Form Factor, Attenuation, and Scattering Tables, [
152]. The result
for a lookup with density 7.1 g/cm
3
is plotted in Fig.
A.5 both on a linear and logarithmic
scale for a large range of energies. For an energy of E ≈ 100 keV, the attenuation length is
λ ≈ 0.85 mm.
10
−3
10
−2
10
−1
10
0
Attenuation length λ [mm]
10 20 30 40 50 60 70 80 90 100 110
0.2
0.4
0.6
0.8
1
X-ray photon energy [keV]
Attenuation length λ [mm]
X-ray energy E [keV]
Attenuation length λ [mm]
Attenuation length λ (log scale) [mm]
Figure A.5: X-ray attenuation lengths plotted linearly (left axis, red points) and logarithmic (right axis,
blue points). Data from table lookup, [
152]. Lines are a guide to the eye.

119
B
Additional crystallographic information
B.1 Overview of space group investigations
An overview of a few of t he more extensive studies on the space group/space groups of differ-
ently doped La
2−x
Sr
x
CuO
4+y
is shown in Tab.
B.1.
B.2 Examples of crystal refinement results
There are many refinement results of the average structure for different Sr and O dopings of
La
2−x
Sr
x
CuO
4+y
in literature, both using powder and single crystal samples with neutrons and
X-rays. I have tried to collect a few of these results in the following t ables.
Grande et al. [
90] were the first to solve the un-doped structure, using X-rays on powder
samples to solve the average structure at room temperature (T = 300 K) for La
2
CuO
4
as shown
in Tab.
B.2. These data were also visualized in Fig. 6.1.
Rial et al. [
104] used neutrons on powder samples to solve the average structure at room
temperature for L a
2
CuO
4
and La
2
CuO
4+y
, as shown in Tab.
B.3. Note that the un-doped struc-
ture matches well with the old results from Tab.
B.2, even with the O(2) 8f position, which for
the y value is simply chosen below the unit cell instead of in the top (the large value in Tab.
B.2
versus the negative value in Tab. B.3).
Rial et al. [
101] also used neutrons on powder samples to solve the average structure at
room temperature for a set of La
2−x
Sr
x
CuO
4
samples, then oxygenated them and solved the
structure similarly for both room temperature and low temperature, as shown in Tab.
B.4.

120 B.2. Examples of crystal refinement results
Sr doping x O doping y Fmmm Bmab I4/mmm Ref.
Un-doped
0 0 T < 533 K 533 K < T [89, 90]
Only strontium doped
0.025 0 T Ú 475 K 475 K Û T [127]
0.05 0 T Ú 420 K 420 K Û T [
127]
0.105 0 T Ú 295 K 295 K Û T [
127]
0.12 0 T Ú 255 K 255 K Û T [
127]
0.145 0 T Ú 195 K 195 K Û T [
127]
0.17 0 T Ú 140 K 140 K Û T [
127]
Only oxygen doped
0 0.018 T < 315 K T < 520 K 520 K < T [99]
0 0.03 T < 320 K T < 430 K 430 K < T [
105]
0 0.032 T < 415 K
a = b for 380 K < T
T < 415 K 415 K < T [99]
0 0.044 T < 340 K T < 275 K 340 K < T [
99]
Co-doped
Several oxygen annealed samples with 0 ≥ (x, y) ≥ 0.03 showed phase separation into oxygen-rich
and oxygen-stoichiometric domains, possibly Fmmm and Bmab (although not solved directly):
0.004 0.03 T Ú 280 K T Ú 520 K 520 K Û T [
135]
0.016 0.03 T Ú 130 K T Ú 500 K 500 K Û T [
135]
0.03 0.03 T Ú 460 K 460 K Û T [
135]
Table B.1: Overview of space group results from investigations where refinements were done for a large
number of temperatures to obtain accurate transition temperatures. Many more experiments have been
done exclusively at, say, 10-20 K and then again at 280-320 K. There is a distinct lack of investigations for
La
2
CuO
4+y
samples with large y, while the investigations on co-doped materials mentioned here (shown
in grey, with the values being very roughly read from a graph) did not actually solve the structure, but
simply conclude on phase separation.
Atom Site x [a] y [b] z [c] Occ
La 8f (m...) 0.00 0.007 0.362 2
Cu 4a (2/m..) 0.00 0.000 0.000 1
O(1) 8e (..2) 0.25 0.25 0.007 2
O(2) 8f (m..) 0.00 0.969 0.187 2
Table B.2: Room temperature (T = 300 K) structure from Grande et al. [90, Tab. 1] for La
2
CuO
4
in space
group Bmab (changed from their used space group Abma, where the a and b directions are exchanged)
with a = 5.370 Å, b = 5.406 Å, c = 13.15 Å. Occ is the occupation factor, which is simply here for com-
parison with other results (the occupation was not changed from the chemical content in their structure
solution).

Appendix B. Additional crystallographic information 121
Atom Site x [a] y [b] z [c] Occ B [Å
2
]
La
2
CuO
4
with a = 5.3562(2) Å, b = 5.4024(2) Å, c = 13.1494(6) Å
La 8f (m..) 0 0.0060(6) 0.3610(1) 2 0.33(3)
Cu 4a (2/m..) 0 0 0 1 0.12(3)
O(1) 8e (..2) 0.25 0.25 0.0081(3) 1.99(2) 0.43(5)
O(2) 8f (m..) 0 −0.0332(6) 0.1839(3) 2.01(2) 0.98(6)
La
2
CuO
4+y
with a = 5.3427(3) Å, b = 5.4040(3) Å, c = 13.2124(7) Å
La 8f (m...) 0 −0.002(1) 0.3597(2) 2 0.49(4)
Cu 4a (2/m..) 0 0 0 1 0.20(7)
O(1) 8e (..2) 0.25 0.25 0.0043(6) 1.99(3) 0.60(7)
O(2) 8f (m..) 0 −0.019(1) 0.1825(4) 1.69(1) 0.93(7)
O(3) 16g (1) 0.029(7) 0.093(5) 0.179(3) 0.31(1) 0.93(7)
O(4) 8e (..2) 0.25 0.25 0.238(6) 0.105(14) 0.93(7)
Table B.3: Room temperature (T = 300 K) structures from Rial et al. [104, Tab. 1] for La
2
CuO
4
and
La
2
CuO
4+y
in space group Bmab. Occ is the occupation factor, while B is the equivalent isotropic thermal
parameter.

122 B.2. Examples of crystal refinement results
Sr doping x a [Å] b [Å] c [Å] y (La/Sr) z (La/Sr) z (O(1)) y (O(2)) z (O(2)) x (O(3)) y (O(3)) z (O(3)) z (O(4))
La
2−x
Sr
x
CuO
4
at T = 300 K
0.05 5.3541(4) 5.3866(4) 13.1846(8) 0.0047(9) 0.3612(2) 0.0067(5) −0.0279(8) 0.1823(3)
0.09 5.3510(5) 5.3633(5) 13.2133(8) 0.004(1) 0.3608(2) 0.0051(6) −0.021(1) 0.1817(3)
0.12 5.3539(2) 13.2249(7) 0 0.3610(1) 0 0 0.1813(3)
0.125 5.3518(2) 13.2287(7) 0 0.3609(1) 0 0 0.1814(3)
0.13 5.3502(2) 13.2333(7) 0 0.3609(1) 0 0 0.1813(3)
0.14 5.3493(2) 13.2368(7) 0 0.3609(1) 0 0 0.1815(3)
La
2−x
Sr
x
CuO
4+y
at T = 300 K
0.05 5.3545(4) 5.3856(5) 13.232(1) −0.001(2) 0.3602(2) −0.003(1) −0.016(2) 0.1815(5) 0.02(1) 0.089(5) 0.177(3) 0.218(5)
0.09 5.3468(5) 5.3663(5) 13.2378(9) 0.003(2) 0.3603(2) −0.002(1) −0.021(2) 0.1813(6) 0.02(1) 0.078(7) 0.179(4) 0.216(5)
0.12 5.3496(3) 13.2377(9) 0 0.3602(2) 0 0 0.1812(6) 0.056(5) 0.056(5) 0.181(4) 0.226(7)
0.125 5.3499(3) 13.2397(9) 0 0.3605(2) 0 0 0.1809(6) 0.056(5) 0.056(5) 0.181(4) 0.217(7)
0.13 5.3486(3) 13.2370(8) 0 0.3602(2) 0 0 0.1812(6) 0.055(6) 0.055(6) 0.182(5) 0.222(8)
0.14 5.3476(3) 13.2405(8) 0 0.3605(2) 0 0 0.1813(6) 0.052(3) 0.052(3) 0.182(4) 0.216(7)
La
2−x
Sr
x
CuO
4+y
at T = 10 K
0.05 5.3312(4) 5.3793(5) 13.201(1) 0.005(2) 0.3603(2) 0.004(1) −0.014(3) 0.1819(6) −0.01(1) 0.087(6) 0.179(3) 0.224(5)
0.09 5.3333(4) 5.3548(5) 13.1976(9) 0.009(1) 0.3604(2) 0.003(1) −0.016(2) 0.1814(6) −0.02(1) 0.077(7) 0.179(3) 0.219(5)
0.12 5.3274(5) 5.3505(5) 13.1968(9) 0.009(1) 0.3602(2) −0.0047(8) −0.022(2) 0.1818(5) 0.033(9) 0.074(9) 0.179(4) 0.225(7)
0.125 5.3301(4) 5.3482(5) 13.1947(8) 0.007(1) 0.3604(2) −0.0033(9) −0.026(2) 0.1818(7) 0.04(1) 0.03(1) 0.179(6) 0.213(7)
0.14 5.3245(4) 5.3473(9) 13.1935(8) 0.009(1) 0.3604(2) −0.0033(9) −0.025(2) 0.1819(6) 0.04(1) 0.05(1) 0.181(6) 0.216(7)
Table B.4: Room temperature and low temperature structures from Rial et al. [101, Tab. 2, 3, and 4] for
La
2−x
Sr
x
CuO
4
and La
2−x
Sr
x
CuO
4+y
in space group Bmab and F4/mmm (when a = b). The site positions
refer to the same naming convention as used in Tab.
B.3, with the site-forced 0 and 0.25 positions omitted
here.

123
C
Mathematical functions
C.1 Gaussian
The Gaussian function is the classical bell shaped function with amplitude I
0
, center µ, and
width σ ,
I(x) = I
0
exp
−
1
2

x − µ
σ

2
!
, (C.1.1)
as shown in Fig.
C.1(a). The area of the curve is found by
A = I
0
Z
∞
−∞
exp
−
1
2

x − µ
σ

2
!
dx (C.1.2)
= I
0
Z
∞
−∞
exp
−
1
2

y
σ

2
!
dy (C.1.3)
=
√
2I
0
σ
Z
∞
−∞
exp (−z
2
) dz (C.1.4)
=
√
2πI
0
σ , (C.1.5)
where the variable of integration was first changed from x to y = x −µ and then to z = y/
√
2σ ,
and finally recognizing the Gaussian integral identity of
√
π.
Using the law of combinations of errors [
11, Eq. (4.14)] (assuming uncorrelated uncertainties),
it is possible to find the error on the area, δA, following the error on the amplitude and width,
δI
0
and δσ ,
δA
2
=
dA
dI
0
!
2
δI
0
2
+

dA
dσ

2
δσ
2
(C.1.6)
= 2πσ
2
δI
0
2
+ 2πI
2
0
δσ
2
(C.1.7)
= A
2
δI
0
2
I
2
0
+
δσ
2
σ
2
!
, (C.1.8)
inserting the expression for the area from Eq. (
C.1.5) in the last line.
The parameter σ controls the width of the peak, but not in a simple way: About 68.2% of
the area of the peak lie within µ ± σ .
Often it is preferable to know the full width at half maximum (FWHM), or half width at half
maximum (HWHM), instead. For a Gaussian curve this can be found by letting x = w at half the

124 C.2. Lorentzian
(a)
0
I
0
/2
I
0
Function value I
FWHM = 2
√
2 ln 2σ
I
0
A =
√
2πI
0
σ
(b)
µ
0
I
0
/2
I
0
Position x
Function value I
FWHM = 2Γ
I
0
A = πI
0
Γ
Figure C.1: A Gaussian (a) and a Lorentzian (b) normalized to have the same height and area. Note how
much more of the area is hidden in the long tails in the Lorentzian.
maximum height, keeping the center of the curve in zero (µ = 0), which means that
1
2
I
0
= I
0
exp
−
1
2

w
σ

2
!
⇒ w =
p
2 ln 2σ , (C.1.9)
so FWHM = 2w = 2
√
2 ln 2σ ≈ 2.355σ and HWHM = w =
√
2 ln 2σ ≈ 1.177σ .
C.2 Lorentzian
The Lorentzian function is another typical peak shape, which has larger tails than the Gaussian,
and a sharper middle part. It is given by the amplitude I
0
, center µ, and width Γ ,
I(x) = I
0
Γ
2
Γ
2
+ (x − µ)
2
, (C.2.1)
as shown in Fig.
C.1(b). The area of this curve is found by
A = I
0
Γ
2
Z
∞
−∞
1
Γ
2
+ (x − µ)
2
dx (C.2.2)
= I
0
Γ
2
Z
∞
−∞
1
Γ
2
+ y
2
dy (C.2.3)
= I
0
Γ
2

1
Γ
tan
−1

y
Γ

∞
−∞
(C.2.4)
= I
0
Γ

π
2
+
π
2

(C.2.5)
= πI
0
Γ , (C.2.6)
where the variable of integration was changed from x to y = x − µ in the first step.
The error on the area, δA, is found with the law of combinations of errors from the errors
on the amplitude and width, δI
0
and δΓ ,
δA
2
=
dA
dI
0
!
2
δI
0
2
+

dA
dΓ

2
δΓ
2
(C.2.7)
= π
2
Γ
2
δI
0
2
+ π
2
I
2
0
δΓ
2
(C.2.8)

Appendix C. Mathematical functions 125
= A
2
δI
0
2
I
2
0
+
δΓ
2
Γ
2
!
, (C.2.9)
inserting the expression for the area from Eq. (
C.2.6) in the last line.
Setting µ = 0, it is easy to see that the width Γ is defined simply by being the half width at
half maximum, so HWHM = Γ and FWHM = 2Γ for the Lorentzian.
C.3 Voigt
The Voigt is a third peak shape, which I could have used for some of my data analysis, but
chose not to because of it having more parameters. The Voigt is a convolution of a Gaussian
and a Lorentzian peak shape,
I(x) = I
0
Z
∞
−∞
G(x
′
, σ ) L(x − x
′
, Γ ) dx
′
, (C.3.1)
where G(x
′
, σ ) is a Gaussian with center µ
G
and width σ as defined in Sect.
C.1, and L(x −x
′
, Γ )
is a Lorentzian with center µ
L
and width Γ as defined in Sect.
C.2, giving the center of the Voigt
as µ = µ
G
+ µ
L
.
Often a pseudo-Voigt is defined as a linear combination of a Gaussian and a Lorentzian at
the same center µ instead, as
I(x) = ηL(x, Γ ) +(1 − η)G(x, σ) (C.3.2)
with 0 < η < 1. This function is simpler to use for fitting peak shapes, compared to the full
convolution necessary for each fit step if using the original definition.
The Voigt is often used for peak shape analysis, where the peak shape is a combination of a
Gaussian and a Lorentzian because of e.g. resolution effects, [
136].
C.4 Power law
The classical power law used to fit transition curves as a function of temperature T , is given
with the following expression.
f (T) =





I
0
|T − T
c
|
β
+ I
b
for T ≤ T
c
,
I
b
for T ≥ T
c
,
(C.4.1)
where I
0
is the amplitude, T
c
is the critical temperature for the transition, β is the critical
exponent, and I
b
is the background.

126 C.4. Power law

127
D
Additional staging figures
D.1 All raw BW5 staging measurements
As explained in Chap.
7, measurements over t he (014) peak along l were done on the BW5
instrument for four different samples, La
2
CuO
4+y
, La
1.96
Sr
0.04
CuO
4+y
, La
1.935
Sr
0.065
CuO
4+y
, and
La
1.91
Sr
0.09
CuO
4+y
(see Tab.
7.2 on p. 51 for an overview of the experiments). An overview of
the different datafiles – showing the temperature and the l interval they were measured for, is
shown in Fig.
D.1.
The raw data from the measurements are shown in Figs.
D.2 to D.13, first showing the plots
on a linear scale,
◮ Figure D.2; La
2
CuO
4+y
,
◮ Figure D.3 and D.4; La
1.96
Sr
0.04
CuO
4+y
,
◮ Figure D.5; La
1.935
Sr
0.065
CuO
4+y
,
◮ Figure D.6 and D.7; La
1.91
Sr
0.09
CuO
4+y
,
and then on a logarithmic scale,
◮ Figure D.8; La
2
CuO
4+y
,
◮ Figure D.9 and D.10; L a
1.96
Sr
0.04
CuO
4+y
,
◮ Figure D.11; La
1.935
Sr
0.065
CuO
4+y
,
◮ Figure D.12 and D.13; La
1.91
Sr
0.09
CuO
4+y
.
A few overview figures have been plotted with the same l axes above each other, to make
comparison of peak sizes easier. For the uncorrected data in Fig.
9.1 (p. 60) and the logarithmic
scale plot in Fig.
9.2 (p. 61), these are Fig. D.14 and Fig. D.15. For the corrected data in Fig. 9.6
(p. 65) and the logarithmic scale plot in Fig. 9.7 (p. 66), these are Fig. D.1 6 and Fig. D.17.

128 D.1. All raw BW5 staging measurements
3 3.5 4 4.5 5
0
50
100
150
200
250
300
425 10.273 ±0.007 K426 10.290 ±0.007 K
442 29.986 ±0.006 K443 29.987 ±0.005 K
465 49.994 ±0.010 K466 49.995 ±0.014 K
476 59.990 ±0.013 K477 59.99 ±0.02 K
488 69.989 ±0.014 K489 69.99 ±0.02 K
499 99.992 ±0.008 K
501 103.666 ±0.006 K
532 161.224 ±0.005 K533 161.230 ±0.003 K
544 189.995 ±0.005 K545 189.996 ±0.006 K
555 219.996 ±0.005 K556 219.996 ±0.005 K
567 249.996 ±0.005 K568 249.997 ±0.005 K
579 279.997 ±0.005 K580 279.998 ±0.005 K
590 289.998 ±0.005 K591 289.998 ±0.004 K
602 299.997 ±0.005 K603 299.998 ±0.005 K
Temperature T [K]
(a)
Temperature T [K]
3 3.5 4 4.5 5
0
50
100
150
200
250
300
651 9.33 ±0.02 K
646 20.000 ±0.009 K
642 30.000 ±0.002 K
638 34.999 ±0.002 K
634 39.9997 ±0.0015 K
630 44.9995 ±0.0013 K
626 50.8 ±1.2 K
622 60.000 ±0.002 K
618 69.999 ±0.002 K
614 89.999 ±0.002 K
610 110 K
606 129.9999 ±0.0007 K
602 149.9999 ±0.0007 K
598 170 K
594 189.9998 ±0.0015 K
590 209.9998 ±0.0015 K
586 230 K
582 250.000 ±0.002 K
577 269.997 ±0.010 K
(b)
3 3.5 4 4.5 5
0
50
100
150
200
250
300
287 25.001 ±0.004 K
298 74.996 ±0.002 K
309 125.000 ±0.002 K
320 175 K
353 225 K
364 300 K
l through (014) [r.l.u.]
Temperature T [K]
(c)
l through (014) [r.l.u.]
Temperature T [K]
3 3.5 4 4.5 5
0
50
100
150
200
250
300
506 15.00 ±0.02 K
517 20.000 ±0.007 K
539 39.9972 ±0.0015 K
550 49.998 ±0.002 K
561 59.996 ±0.002 K
572 69.996 ±0.003 K
583 79.996 ±0.002 K
594 89.997 ±0.003 K
605 99.997 ±0.002 K
616 119.998 ±0.004 K
627 139.9998 ±0.0014 K
654 160 K
665 180 K
676 200 K
687 220 K
698 240 K
709 260 K
720 280 K
731 300 K
l through (014) [r.l.u.]
(d)
l through (014) [r.l.u.]
Figure D.1: Overview of the BW5 staging scans on the (a) La
2
CuO
4+y
, (b) La
1.96
Sr
0.04
CuO
4+y
, (c) La
1.935
-
Sr
0.065
CuO
4+y
, and (d) La
1.91
Sr
0.09
CuO
4+y
samples, with file numbers and corresponding average
temperatures. The errors given on the temperatures are the standard deviation values.

Appendix D. Additional staging figures 129
3.5 4 4.5
0
20
40
60
80
100
Intensity I per monitor [a.u.]
Datafile numbers 4 25 + 426
T = 10.279 0 ± 0.0 00 6 K
3.5 4 4.5
0
20
40
60
80
100
Datafile numbers 4 42 + 443
T = 29.986 2 ± 0.0 00 3 K
3.5 4 4.5
0
20
40
60
80
100
Datafile numbers 4 65 + 466
T = 49.994 2 ± 0.0 00 7 K
3.5 4 4.5
0
20
40
60
80
100
Intensity I per monitor [a.u.]
Datafile numbers 4 76 + 477
T = 59.990 1 ± 0.0 00 9 K
3.5 4 4.5
0
20
40
60
80
100
Datafile numbers 4 88 + 489
T = 69.989 8 ± 0.0 00 9 K
3.5 4 4.5
0
20
40
60
80
100
Datafile numbers 4 99 + 501
T = 101 .3 1 ± 0 .1 1 K
3.5 4 4.5
0
20
40
60
80
100
Intensity I per monitor [a.u.]
Datafile numbers 5 32 + 533
T = 161 .2 25 8 ± 0.0003 K
3.5 4 4.5
0
20
40
60
80
100
Datafile numbers 5 44 + 545
T = 189 .9 95 5 ± 0.00 03 K
3.5 4 4.5
0
20
40
60
80
100
Datafile numbers 5 55 + 556
T = 219 .9 96 2 ± 0.00 03 K
3.5 4 4.5
0
20
40
60
80
100
Intensity I per monitor [a.u.]
Datafile numbers 5 67 + 568
T = 249 .9 96 5 ± 0.0003 K
3.5 4 4.5
0
20
40
60
80
100
l through (014) [r.l.u.]
Datafile numbers 5 79 + 580
T = 279 .9 97 0 ± 0.00 03 K
3.5 4 4.5
0
20
40
60
80
100
l through (014) [r.l.u.]
Datafile numbers 5 90 + 591
T = 289 .9 97 7 ± 0.00 03 K
3.5 4 4.5
0
20
40
60
80
100
l through (014) [r.l.u.]
Intensity I per monitor [a.u.]
Datafile numbers 6 02 + 603
T = 299 .9 97 8 ± 0.0003 K
Figure D.2: Raw BW5 staging scans for the La
2
CuO
4+y
sample (see Fig. D.1(a) for an overall vi ew). The
narrow scans around l = 4, as seen on Fig.
D.1(a), have been combined with the wider scans at the same
temperature, to form one single dataset. These are shown on a logarithmic scale in Fig.
D.8.

130 D.1. All raw BW5 staging measurements
3.5 4 4.5
0
5
10
15
20
Intensity I per monitor [a.u.]
Datafile number 6 51
T = 9.3 30 5 ± 0 .0 01 4 K
3.5 4 4.5
0
5
10
15
20
Datafile number 6 46
T = 20.000 2 ± 0.0 00 7 K
3.5 4 4.5
0
5
10
15
20
Datafile number 6 42
T = 29.999 6 ± 0.0 00 2 K
3.5 4 4.5
0
5
10
15
20
Intensity I per monitor [a.u.]
Datafile number 6 38
T = 34.999 40 ± 0 .00014 K
3.5 4 4.5
0
5
10
15
20
Datafile number 6 34
T = 39.999 66 ± 0 .00011 K
3.5 4 4.5
0
5
10
15
20
Datafile number 6 30
T = 44.999 52 ± 0 .00010 K
3.5 4 4.5
0
5
10
15
20
Intensity I per monitor [a.u.]
Datafile number 6 26
T = 50.79 ± 0.0 9 K
3.5 4 4.5
0
5
10
15
20
Datafile number 6 22
T = 59.999 51 ± 0 .00012 K
3.5 4 4.5
0
5
10
15
20
Datafile number 6 18
T = 69.999 19 ± 0 .00013 K
3.5 4 4.5
0
5
10
15
20
Intensity I per monitor [a.u.]
Datafile number 6 14
T = 89.999 17 ± 0 .00013 K
3.5 4 4.5
0
5
10
15
20
Datafile number 6 10
T = 110 K
3.5 4 4.5
0
5
10
15
20
Datafile number 6 06
T = 129 .9 99 94 ± 0.0000 6 K
3.5 4 4.5
0
5
10
15
20
l through (014) [r.l.u.]
Intensity I per monitor [a.u.]
Datafile number 6 02
T = 149 .9 99 94 ± 0.0000 6 K
3.5 4 4.5
0
5
10
15
20
l through (014) [r.l.u.]
Datafile number 5 98
T = 170 K
3.5 4 4.5
0
5
10
15
20
l through (014) [r.l.u.]
Datafile number 5 94
T = 189 .9 99 78 ± 0.0001 1 K
Figure D.3: Raw BW5 staging scans for the La
1.96
Sr
0.04
CuO
4+y
sample (see Fig. D.1(b) for an overall view).
Continued in Fig.
D.4. These are shown on a logarithmic scale in Fig. D.9.

Appendix D. Additional staging figures 131
3.5 4 4.5
0
5
10
15
20
Intensity I per monitor [a.u.]
Datafile number 5 90
T = 209 .9 99 78 ± 0.0001 1 K
3.5 4 4.5
0
5
10
15
20
l through (014) [r.l.u.]
Datafile number 5 86
T = 230 K
3.5 4 4.5
0
5
10
15
20
l through (014) [r.l.u.]
Datafile number 5 82
T = 249 .9 99 61 ± 0.0001 4 K
3.5 4 4.5
0
5
10
15
20
l through (014) [r.l.u.]
Intensity I per monitor [a.u.]
Datafile number 5 77
T = 269 .9 97 5 ± 0.0007 K
Figure D.4: Raw BW5 staging scans for t he La
1.96
Sr
0.04
CuO
4+y
sample (see Fig. D.1(b) for an overall view).
Continued from Fig.
D.3. These are shown on a logarithmic scale in Fig. D.10.
3.8 3.9 4 4.1 4.2 4.3
0
50
100
150
Intensity I per monitor [a.u.]
Datafile number 2 87
T = 25.001 3 ± 0.0 00 4 K
3.8 3.9 4 4.1 4.2
0
50
100
150
Datafile number 2 98
T = 74.996 1 ± 0.0 00 2 K
3.8 3.9 4 4.1 4.2
0
50
100
150
Datafile number 3 09
T = 124 .9 99 7 ± 0.00 02 K
3.8 3.9 4 4.1 4.2
0
50
100
150
l through (014) [r.l.u.]
Intensity I per monitor [a.u.]
Datafile number 3 20
T = 175 K
3.8 3.9 4 4.1 4.2
0
50
100
150
l through (014) [r.l.u.]
Datafile number 3 53
T = 225 K
3.8 3.9 4 4.1 4.2
0
50
100
150
l through (014) [r.l.u.]
Datafile number 3 64
T = 300 K
Figure D.5: Raw BW5 staging scans for the La
1.935
Sr
0.065
CuO
4+y
sample (see Fig. D.1(c) for an overall
view). These are shown on a logarithmic scale in Fig.
D.11.

132 D.1. All raw BW5 staging measurements
3.98 4 4.02 4.04 4.06
0
50
100
150
200
Intensity I per monitor [a.u.]
Datafile number 5 06
T = 15.003 ± 0.00 2 K
4 4.02 4.04 4.06 4.08
0
50
100
150
200
Datafile number 5 17
T = 19.999 7 ± 0.0 01 1 K
3.98 4 4.02 4.04 4.06
0
50
100
150
200
Datafile number 5 39
T = 39.997 2 ± 0.0 00 2 K
4 4.02 4.04 4.06 4.08
0
50
100
150
200
Intensity I per monitor [a.u.]
Datafile number 5 50
T = 49.998 0 ± 0.0 00 2 K
3.98 4 4.02 4.04 4.06
0
50
100
150
200
Datafile number 5 61
T = 59.996 3 ± 0.0 00 2 K
3.98 4 4.02 4.04 4.06
0
50
100
150
200
Datafile number 5 72
T = 69.996 4 ± 0.0 00 4 K
3.98 4 4.02 4.04 4.06
0
50
100
150
200
Intensity I per monitor [a.u.]
Datafile number 5 83
T = 79.996 2 ± 0.0 00 3 K
3.98 4 4.02 4.04 4.06
0
50
100
150
200
Datafile number 5 94
T = 89.996 6 ± 0.0 00 4 K
3.98 4 4.02 4.04 4.06
0
50
100
150
200
Datafile number 6 05
T = 99.997 1 ± 0.0 00 3 K
3.98 4 4.02 4.04 4.06
0
50
100
150
200
Intensity I per monitor [a.u.]
Datafile number 6 16
T = 119 .9 98 4 ± 0.0005 K
3.98 4 4.02 4.04 4.06
0
50
100
150
200
Datafile number 6 27
T = 139 .9 99 8 ± 0.00 02 K
3.98 4 4.02 4.04 4.06
0
50
100
150
200
Datafile number 6 54
T = 160 K
3.98 4 4.02 4.04 4.06
0
50
100
150
200
l through (014) [r.l.u.]
Intensity I per monitor [a.u.]
Datafile number 6 65
T = 180 K
3.98 4 4.02 4.04 4.06
0
50
100
150
200
l through (014) [r.l.u.]
Datafile number 6 76
T = 200 K
3.98 4 4.02 4.04 4.06
0
50
100
150
200
l through (014) [r.l.u.]
Datafile number 6 87
T = 220 K
Figure D.6: Raw BW5 staging scans for the La
1.91
Sr
0.09
CuO
4+y
sample (see Fig. D.1(d) for an overall view).
Continued in Fig.
D.7. These are shown on a logarithmic scale in Fig. D.12.

Appendix D. Additional staging figures 133
3.98 4 4.02 4.04 4.06
0
50
100
150
200
Intensity I per monitor [a.u.]
Datafile number 6 98
T = 240 K
3.98 4 4.02 4.04 4.06
0
50
100
150
200
l through (014) [r.l.u.]
Datafile number 7 09
T = 260 K
3.98 4 4.02 4.04 4.06
0
50
100
150
200
l through (014) [r.l.u.]
Datafile number 7 20
T = 280 K
3.98 4 4.02 4.04 4.06
0
50
100
150
200
l through (014) [r.l.u.]
Intensity I per monitor [a.u.]
Datafile number 7 31
T = 300 K
Figure D.7: Raw BW5 staging scans for the La
1.91
Sr
0.09
CuO
4+y
sample (see Fig. D.1(d) for an overall view).
Continued from Fig.
D.6. These are shown on a logarithmic scale in Fig. D.13.

134 D.1. All raw BW5 staging measurements
3.5 4 4.5
10
0
10
1
10
2
Intensity I per monitor [a.u.]
Datafile numbers 4 25 + 426
T = 10.279 0 ± 0.0 00 6 K
3.5 4 4.5
10
0
10
1
10
2
Datafile numbers 4 42 + 443
T = 29.986 2 ± 0.0 00 3 K
3.5 4 4.5
10
0
10
1
10
2
Datafile numbers 4 65 + 466
T = 49.994 2 ± 0.0 00 7 K
3.5 4 4.5
10
0
10
1
10
2
Intensity I per monitor [a.u.]
Datafile numbers 4 76 + 477
T = 59.990 1 ± 0.0 00 9 K
3.5 4 4.5
10
0
10
1
10
2
Datafile numbers 4 88 + 489
T = 69.989 8 ± 0.0 00 9 K
3.5 4 4.5
10
0
10
1
10
2
Datafile numbers 4 99 + 501
T = 101 .3 1 ± 0 .1 1 K
3.5 4 4.5
10
0
10
1
10
2
Intensity I per monitor [a.u.]
Datafile numbers 5 32 + 533
T = 161 .2 25 8 ± 0.0003 K
3.5 4 4.5
10
0
10
1
10
2
Datafile numbers 5 44 + 545
T = 189 .9 95 5 ± 0.00 03 K
3.5 4 4.5
10
0
10
1
10
2
Datafile numbers 5 55 + 556
T = 219 .9 96 2 ± 0.00 03 K
3.5 4 4.5
10
0
10
1
10
2
Intensity I per monitor [a.u.]
Datafile numbers 5 67 + 568
T = 249 .9 96 5 ± 0.0003 K
3.5 4 4.5
10
0
10
1
10
2
l through (014) [r.l.u.]
Datafile numbers 5 79 + 580
T = 279 .9 97 0 ± 0.00 03 K
3.5 4 4.5
10
0
10
1
10
2
l through (014) [r.l.u.]
Datafile numbers 5 90 + 591
T = 289 .9 97 7 ± 0.00 03 K
3.5 4 4.5
10
0
10
1
10
2
l through (014) [r.l.u.]
Intensity I per monitor [a.u.]
Datafile numbers 6 02 + 603
T = 299 .9 97 8 ± 0.0003 K
Figure D.8: Raw BW5 staging scans for the La
2
CuO
4+y
sample (see Fig. D.1(a) for an overall view). These
are shown on a linear scale in Fig. D.2.

Appendix D. Additional staging figures 135
3.5 4 4.5
10
0
10
1
Intensity I per monitor [a.u.]
Datafile number 6 51
T = 9.3 30 5 ± 0 .0 01 4 K
3.5 4 4.5
10
0
10
1
Datafile number 6 46
T = 20.000 2 ± 0.0 00 7 K
3.5 4 4.5
10
0
10
1
Datafile number 6 42
T = 29.999 6 ± 0.0 00 2 K
3.5 4 4.5
10
0
10
1
Intensity I per monitor [a.u.]
Datafile number 6 38
T = 34.999 40 ± 0 .00014 K
3.5 4 4.5
10
0
10
1
Datafile number 6 34
T = 39.999 66 ± 0 .00011 K
3.5 4 4.5
10
0
10
1
Datafile number 6 30
T = 44.999 52 ± 0 .00010 K
3.5 4 4.5
10
0
10
1
Intensity I per monitor [a.u.]
Datafile number 6 26
T = 50.79 ± 0.0 9 K
3.5 4 4.5
10
0
10
1
Datafile number 6 22
T = 59.999 51 ± 0 .00012 K
3.5 4 4.5
10
0
10
1
Datafile number 6 18
T = 69.999 19 ± 0 .00013 K
3.5 4 4.5
10
0
10
1
Intensity I per monitor [a.u.]
Datafile number 6 14
T = 89.999 17 ± 0 .00013 K
3.5 4 4.5
10
0
10
1
Datafile number 6 10
T = 110 K
3.5 4 4.5
10
0
10
1
Datafile number 6 06
T = 129 .9 99 94 ± 0.0000 6 K
3.5 4 4.5
10
0
10
1
l through (014) [r.l.u.]
Intensity I per monitor [a.u.]
Datafile number 6 02
T = 149 .9 99 94 ± 0.0000 6 K
3.5 4 4.5
10
0
10
1
l through (014) [r.l.u.]
Datafile number 5 98
T = 170 K
3.5 4 4.5
10
0
10
1
l through (014) [r.l.u.]
Datafile number 5 94
T = 189 .9 99 78 ± 0.0001 1 K
Figure D.9: Raw BW5 staging scans for t he La
1.96
Sr
0.04
CuO
4+y
sample (see Fig. D.1(b) for an overall view).
Continued in Fig.
D.4. These are shown on a linear scale in Fig. D.3.

136 D.1. All raw BW5 staging measurements
3.5 4 4.5
10
0
10
1
Intensity I per monitor [a.u.]
Datafile number 5 90
T = 209 .9 99 78 ± 0.0001 1 K
3.5 4 4.5
10
0
10
1
l through (014) [r.l.u.]
Datafile number 5 86
T = 230 K
3.5 4 4.5
10
0
10
1
l through (014) [r.l.u.]
Datafile number 5 82
T = 249 .9 99 61 ± 0.0001 4 K
3.5 4 4.5
10
0
10
1
l through (014) [r.l.u.]
Intensity I per monitor [a.u.]
Datafile number 5 77
T = 269 .9 97 5 ± 0.0007 K
Figure D.10: Raw BW5 staging scans for La
1.96
Sr
0.04
CuO
4+y
(see Fig. D.1(b) for an overall view). Continued
from Fig.
D.3. These are shown on a linear scale in Fig. D.4.
3.8 3.9 4 4.1 4.2 4.3
10
0
10
1
10
2
Intensity I per monitor [a.u.]
Datafile number 2 87
T = 25.001 3 ± 0.0 00 4 K
3.8 3.9 4 4.1 4.2
10
0
10
1
10
2
Datafile number 2 98
T = 74.996 1 ± 0.0 00 2 K
3.8 3.9 4 4.1 4.2
10
0
10
1
10
2
Datafile number 3 09
T = 124 .9 99 7 ± 0.00 02 K
3.8 3.9 4 4.1 4.2
10
0
10
1
10
2
l through (014) [r.l.u.]
Intensity I per monitor [a.u.]
Datafile number 3 20
T = 175 K
3.8 3.9 4 4.1 4.2
10
0
10
1
10
2
l through (014) [r.l.u.]
Datafile number 3 53
T = 225 K
3.8 3.9 4 4.1 4.2
10
0
10
1
10
2
l through (014) [r.l.u.]
Datafile number 3 64
T = 300 K
Figure D.11: Raw BW5 staging scans for the La
1.935
Sr
0.065
CuO
4+y
sample (see Fig. D.1(c) for an overall
view). These are shown on a linear scale in Fig.
D.5.

Appendix D. Additional staging figures 137
3.98 4 4.02 4.04 4.06
10
0
10
1
10
2
Intensity I per monitor [a.u.]
Datafile number 5 06
T = 15.003 ± 0.00 2 K
4 4.02 4.04 4.06 4.08
10
0
10
1
10
2
Datafile number 5 17
T = 19.999 7 ± 0.0 01 1 K
3.98 4 4.02 4.04 4.06
10
0
10
1
10
2
Datafile number 5 39
T = 39.997 2 ± 0.0 00 2 K
4 4.02 4.04 4.06 4.08
10
0
10
1
10
2
Intensity I per monitor [a.u.]
Datafile number 5 50
T = 49.998 0 ± 0.0 00 2 K
3.98 4 4.02 4.04 4.06
10
0
10
1
10
2
Datafile number 5 61
T = 59.996 3 ± 0.0 00 2 K
3.98 4 4.02 4.04 4.06
10
0
10
1
10
2
Datafile number 5 72
T = 69.996 4 ± 0.0 00 4 K
3.98 4 4.02 4.04 4.06
10
0
10
1
10
2
Intensity I per monitor [a.u.]
Datafile number 5 83
T = 79.996 2 ± 0.0 00 3 K
3.98 4 4.02 4.04 4.06
10
0
10
1
10
2
Datafile number 5 94
T = 89.996 6 ± 0.0 00 4 K
3.98 4 4.02 4.04 4.06
10
0
10
1
10
2
Datafile number 6 05
T = 99.997 1 ± 0.0 00 3 K
3.98 4 4.02 4.04 4.06
10
0
10
1
10
2
Intensity I per monitor [a.u.]
Datafile number 6 16
T = 119 .9 98 4 ± 0.0005 K
3.98 4 4.02 4.04 4.06
10
0
10
1
10
2
Datafile number 6 27
T = 139 .9 99 8 ± 0.00 02 K
3.98 4 4.02 4.04 4.06
10
0
10
1
10
2
Datafile number 6 54
T = 160 K
3.98 4 4.02 4.04 4.06
10
0
10
1
10
2
l through (014) [r.l.u.]
Intensity I per monitor [a.u.]
Datafile number 6 65
T = 180 K
3.98 4 4.02 4.04 4.06
10
0
10
1
10
2
l through (014) [r.l.u.]
Datafile number 6 76
T = 200 K
3.98 4 4.02 4.04 4.06
10
0
10
1
10
2
l through (014) [r.l.u.]
Datafile number 6 87
T = 220 K
Figure D.12: Raw BW5 staging scans for the La
1.91
Sr
0.09
CuO
4+y
sample (see Fig. D.1(d) for an overall view).
Continued in Fig.
D.7. These are shown on a linear scale in Fig. D.6.

138 D.1. All raw BW5 staging measurements
3.98 4 4.02 4.04 4.06
10
0
10
1
10
2
Intensity I per monitor [a.u.]
Datafile number 6 98
T = 240 K
3.98 4 4.02 4.04 4.06
10
0
10
1
10
2
l through (014) [r.l.u.]
Datafile number 7 09
T = 260 K
3.98 4 4.02 4.04 4.06
10
0
10
1
10
2
l through (014) [r.l.u.]
Datafile number 7 20
T = 280 K
3.98 4 4.02 4.04 4.06
10
0
10
1
10
2
l through (014) [r.l.u.]
Intensity I per monitor [a.u.]
Datafile number 7 31
T = 300 K
Figure D.13: Raw BW5 staging scans for the La
1.91
Sr
0.09
CuO
4+y
sample (see Fig. D.1(d) for an overall view).
Continued from Fig.
D.6. These are shown on a linear scale in Fig. D.7.

Appendix D. Additional staging figures 139
3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8
0
20
40
60
80
100
Intensity I per monitor [a.u.]
Temperature T [K]
50
100
150
200
250
300
(a)
Intensity I per m onitor [a.u.]
Temperature T [K]
3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8
0
5
10
15
20
Intensity I per monitor [a.u.]
Temperature T [K]
0
50
100
150
200
250
(b)
Intensity I per m onitor [a.u.]
Temperature T [K]
3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8
0
50
100
150
Intensity I per monitor [a.u.]
Temperature T [K]
0
50
100
150
200
250
300
(c)
Intensity I per m onitor [a.u.]
Temperature T [K]
3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8
0
50
100
150
200
l through (014) [r.l.u.]
Intensity I per monitor [a.u.]
Temperature T [K]
0
50
100
150
200
250
300
(d)
l through (014) [r.l.u.]
Intensity I per m onitor [a.u.]
Temperature T [K]
Figure D.14: Measurement on the (a) La
2
CuO
4+y
, (b) La
1.96
Sr
0.04
CuO
4+y
, (c) La
1.935
Sr
0.065
CuO
4+y
, and
(d) La
1.91
Sr
0.09
CuO
4+y
crystals along l over the (014) peak on BW5 before any correction of the first-axis.
Here the data are plotted such that the l axes are the same, to compare better – a more zoomed in version
can be seen in Fig.
9.1 (p. 60).

140 D.1. All raw BW5 staging measurements
3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8
10
0
10
1
10
2
Intensity I per monitor [a.u.]
Temperature T [K]
50
100
150
200
250
300
(a)
Intensity I per m onitor [a.u.]
Temperature T [K]
3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8
10
−1
10
0
10
1
Intensity I per monitor [a.u.]
Temperature T [K]
0
50
100
150
200
250
(b)
Intensity I per m onitor [a.u.]
Temperature T [K]
3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8
10
0
10
1
10
2
Intensity I per monitor [a.u.]
Temperature T [K]
0
50
100
150
200
250
300
(c)
Intensity I per m onitor [a.u.]
Temperature T [K]
3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8
10
0
10
2
l through (014) [r.l.u.]
Intensity I per monitor [a.u.]
Temperature T [K]
0
50
100
150
200
250
300
(d)
l through (014) [r.l.u.]
Intensity I per m onitor [a.u.]
Temperature T [K]
Figure D.15: As figure Fig. D.14, but with the intensity on a logarithmic scale. (a) La
2
CuO
4+y
, (b) La
1.96
-
Sr
0.04
CuO
4+y
, (c) La
1.935
Sr
0.065
CuO
4+y
, and (d) La
1.91
Sr
0.09
CuO
4+y
. Here the data are plotted such that
the l axes are the same, to compare better – a more zoomed in version can be seen in Fig.
9.2 (p. 61).

Appendix D. Additional staging figures 141
3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8
0
20
40
60
80
100
Intensity I per monitor [a.u.]
Temperature T [K]
50
100
150
200
250
300
(a)
Intensity I per m onitor [a.u.]
Temperature T [K]
3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8
0
5
10
15
20
Intensity I per monitor [a.u.]
Temperature T [K]
0
50
100
150
200
250
(b)
Intensity I per m onitor [a.u.]
Temperature T [K]
3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8
0
50
100
150
Intensity I per monitor [a.u.]
Temperature T [K]
0
50
100
150
200
250
300
(c)
Intensity I per m onitor [a.u.]
Temperature T [K]
3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8
0
50
100
150
200
l through (014) [r.l.u.]
Intensity I per monitor [a.u.]
Temperature T [K]
0
50
100
150
200
250
300
(d)
l through (014) [r.l.u.]
Intensity I per m onitor [a.u.]
Temperature T [K]
Figure D.16: Measurement on the (a) La
2
CuO
4+y
, (b) La
1.96
Sr
0.04
CuO
4+y
, (c) La
1.935
Sr
0.065
CuO
4+y
, and
(d) La
1.91
Sr
0.09
CuO
4+y
crystals along l over the (014) peak on BW5 after correction of the first-axis. Here
the data are plotted such that the l axes are the same, to compare better – a more zoomed in version can
be seen in Fig.
9.6 (p. 65).

142 D.1. All raw BW5 staging measurements
3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8
10
0
10
1
10
2
Intensity I per monitor [a.u.]
Temperature T [K]
50
100
150
200
250
300
(a)
Intensity I per m onitor [a.u.]
Temperature T [K]
3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8
10
−1
10
0
10
1
Intensity I per monitor [a.u.]
Temperature T [K]
0
50
100
150
200
250
(b)
Intensity I per m onitor [a.u.]
Temperature T [K]
3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8
10
0
10
1
10
2
Intensity I per monitor [a.u.]
Temperature T [K]
0
50
100
150
200
250
300
(c)
Intensity I per m onitor [a.u.]
Temperature T [K]
3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8
10
0
10
2
l through (014) [r.l.u.]
Intensity I per monitor [a.u.]
Temperature T [K]
0
50
100
150
200
250
300
(d)
l through (014) [r.l.u.]
Intensity I per m onitor [a.u.]
Temperature T [K]
Figure D.17: As figure Fig. D.16, but with the intensity on a logarithmic scale. (a) La
2
CuO
4+y
, (b) La
1.96
-
Sr
0.04
CuO
4+y
, (c) La
1.935
Sr
0.065
CuO
4+y
, and (d) La
1.91
Sr
0.09
CuO
4+y
. Here the data are plotted such that
the l axes are the same, to compare better – a more zoomed in version can be seen in Fig.
9.7 (p. 66).

Appendix D. Additional staging figures 143
D.2 Extra BW5 data analysis figures
In Figs.
D.18 to D.21 the central regio ns of the data is shown before and after the scaling of
the l axis, to see the change in the data as clearly as possible. These figures were referred to
in Subsect.
9.1.1.
In Fig.
D.22, extra measurements over the (004) and (022) peaks for the La
1.96
Sr
0.04
CuO
4+y
,
La
1.935
Sr
0.065
CuO
4+y
, and La
1.91
Sr
0.09
CuO
4+y
samples are shown. These figures were referred
to in Subsect.
9.3.1.
In Fig.
D.23, the slope analysis of integrated intensities not shown in Figs. 9.27 and 9.28
(p. 84) are shown. This figure was referred t o in Subsect. 9.3.2.
In Fig.
D.24 the fitted points, along with the fitted curve, from Fig. 9.29 (p. 86) are shown
individually in double logarithmic plots, to give an overview of the validity of the fits.

144 D.2. Extra BW5 data analysis figures
3.97 3.98 3.99 4 4.01 4.02 4.03
20
40
60
80
100
Intensity I per monitor [a.u.]
Temperature T [K]
50
100
150
200
250
300
Intensity I per m onitor [a.u.]
Temperature T [K]
3.97 3.98 3.99 4 4.01 4.02 4.03
20
40
60
80
100
l through (014) [r.l.u.]
Intensity I per monitor [a.u.]
Temperature T [K]
50
100
150
200
250
300
l through (014) [r.l.u.]
Intensity I per m onitor [a.u.]
Temperature T [K]
Figure D.18: Measurement on the La
2
CuO
4+y
crystal along l over the (014) peak on BW5 (top) before and
(bottom) after correction of the first-axis.
3.7 3.8 3.9 4 4.1 4.2
5
10
15
20
Intensity I per monitor [a.u.]
Temperature T [K]
0
50
100
150
200
250
Intensity I per m onitor [a.u.]
Temperature T [K]
3.7 3.8 3.9 4 4.1 4.2
5
10
15
20
l through (014) [r.l.u.]
Intensity I per monitor [a.u.]
Temperature T [K]
0
50
100
150
200
250
l through (014) [r.l.u.]
Intensity I per m onitor [a.u.]
Temperature T [K]
Figure D.19: Measurement on the La
1.96
Sr
0.04
CuO
4+y
crystal along l over the (014) peak on BW5 (top)
before and (bottom) after correction of the first-axis.

Appendix D. Additional staging figures 145
3.85 3.9 3.95 4 4.05 4.1 4.15
50
100
150
Intensity I per monitor [a.u.]
Temperature T [K]
0
50
100
150
200
250
300
Intensity I per m onitor [a.u.]
Temperature T [K]
3.85 3.9 3.95 4 4.05 4.1 4.15
50
100
150
l through (014) [r.l.u.]
Intensity I per monitor [a.u.]
Temperature T [K]
0
50
100
150
200
250
300
l through (014) [r.l.u.]
Intensity I per m onitor [a.u.]
Temperature T [K]
Figure D.20: Measurement on the La
1.935
Sr
0.065
CuO
4+y
crystal along l over the (014) peak on BW5 (top)
before and (bottom) after correction of the first-axis.
3.94 3.96 3.98 4 4.02 4.04 4.06
50
100
150
200
Intensity I per monitor [a.u.]
Temperature T [K]
0
50
100
150
200
250
300
Intensity I per m onitor [a.u.]
Temperature T [K]
3.94 3.96 3.98 4 4.02 4.04 4.06
50
100
150
200
l through (014) [r.l.u.]
Intensity I per monitor [a.u.]
Temperature T [K]
0
50
100
150
200
250
300
l through (014) [r.l.u.]
Intensity I per m onitor [a.u.]
Temperature T [K]
Figure D.21: Measurement on the La
1.91
Sr
0.09
CuO
4+y
crystal along l over the (014) peak on BW5 (top)
before and (bottom) after correction of the first-axis.

146 D.2. Extra BW5 data analysis figures
0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5
0
50
100
150
ω through (004) [deg]
Intensity I per monitor [a.u.]
Temperature T [K]
0
50
100
150
200
250
(a)
Sample angle ω through (004) [deg]
Intensity I per m onitor [a.u.]
Temperature T [K]
0.7 0.8 0.9 1 1.1 1.2 1.3 1.4
0
5
10
15
20
25
30
35
ω through (004) [deg]
Intensity I per monitor [a.u.]
Temperature T [K]
50
100
150
200
250
(b)
Sample angle ω through (004) [deg]
Intensity I per m onitor [a.u.]
Temperature T [K]
1.65 1.7 1.75 1.8 1.85 1.9 1.95 2 2.05 2.1
0
50
100
150
ω through (022) [deg]
Intensity I per monitor [a.u.]
Temperature T [K]
50
100
150
200
250
300
(c)
Sample angle ω through (022) [deg]
Intensity I per m onitor [a.u.]
Temperature T [K]
Figure D.22: Measurement of ω over a peak on BW5. (a) The La
1.96
Sr
0.04
CuO
4+y
(004) peak, (b) the
La
1.935
Sr
0.065
CuO
4+y
(004) peak, and (c) the La
1.91
Sr
0.09
CuO
4+y
(022) peak.

Appendix D. Additional staging figures 147
0
1
2
175(5) K
0 50 100 150 200 250 300
(a)
Temperature T [K]
Intensity I [a.u.]
Gradient
0
0.5
1
1.5
2
2.5
212(5) K
0 50 100 150 200 250 300
(b)
Temperature T [K]
Intensity I [a.u.]
Gradient
0
0.5
1
1.5
2
2.5
214(5) K
0 50 100 150 200 250 300
(c)
Temperature T [K]
Intensity I [a.u.]
Gradient
0
2
4
6
8
202(5) K
0 50 100 150 200 250 300
(d)
Temperature T [K]
Intensity I [a.u.]
Gradient
0
0.5
1
1.5
202(5) K
0 50 100 150 200 250 300
(e)
Temperature T [K]
Intensity I [a.u.]
Gradient
0
0.5
1
202(5) K
0 50 100 150 200 250 300
(f)
Temperature T [K]
Intensity I [a.u.]
Gradient
0
0.1
0.2
0.3
17(5) K
0 50 100 150 200 250 300
(g)
Temperature T [K]
Intensity I [a.u.]
Gradient
0
0.1
0.2
0.3
0.4
77(5) K
0 50 100 150 200 250 300
(h)
Temperature T [K]
Intensity I [a.u.]
Gradient
Figure D.23: Additional slope analysis done on the integrated intensities for the following peaks:
(a) La
2
CuO
4+y
central Bmab peak (does not show a phase transition). (b) La
1.96
Sr
0.04
CuO
4+y
low l staging
peak. (c) La
1.96
Sr
0.04
CuO
4+y
high l staging peak. (d) La
1.935
Sr
0.065
CuO
4+y
central Bmab peak. (e) La
1.935
-
Sr
0.065
CuO
4+y
low l staging peak. (f) La
1.935
Sr
0.065
CuO
4+y
high l staging peak. (g) La
1.91
Sr
0.09
CuO
4+y
low
l staging peak (result should probably be around room temperature, if analysing by eye). (h) La
1.91
Sr
0.09
-
CuO
4+y
high l staging peak (around room temperature as for the low l). The integrated intensity points
are plotted directly, while the linearly interpolated points are shown as a colored line underneath these. A
smoothing filter has been applied to both the actual data (grey) and the interpolated data (black), resulting
in the gradients shown in the bottom parts of each plot. The gradient minimum for th e sm oothed data is
marked with a green line. This is in addition to the graphs in Figs.
9.27 and 9.28 (p. 84).

148 D.2. Extra BW5 data analysis figures
−2 −1 0
0.7
0.75
0.8
0.85
0.9
0.95
log(I)
(a)
Intensity log (I [a.u.])
−2 −1 0
0.55
0.6
0.65
0.7
0.75
0.8
(b)
−2 −1.5 −1 −0.5
−1.25
−1.2
−1.15
−1.1
−1.05
−1
−0.95
log(I)
(c)
Intensity log (I [a.u.])
−3 −2 −1 0
−0.2
0
0.2
0.4
0.6
0.8
(d)
−3 −2 −1 0
0
0.2
0.4
0.6
(e)
−1.5 −1 −0.5
1.6
1.7
1.8
1.9
2
2.1
log(I)
(f)
Intensity log (I [a.u.])
−6 −4 −2 0
−3
−2.5
−2
−1.5
−1
−0.5
0
0.5
(g)
−1.5 −1 −0.5
−0.6
−0.4
−0.2
0
(h)
−3 −2 −1 0
−0.2
0
0.2
0.4
0.6
0.8
log(t)
log(I)
(i)
Red. temperature log (t [K])
Intensity log (I [a.u.])
−3 −2 −1 0
−2.6
−2.4
−2.2
−2
−1.8
−1.6
−1.4
−1.2
log(t)
(j)
Red. temperature log (t [K])
−2 −1 0
−1.8
−1.6
−1.4
−1.2
−1
log(t)
(k)
Red. temperature log (t [K])
Figure D.24: Double logarithmic plots of the integrated intensities versus the reduced temperature t =
(T
c
−T)/T
c
for temperature points below the found critical temperature T
c
from the individual fits shown
in Fig.
9.29 (p. 86) (the shown lines are the fitted curves from this figure as well). Blue point s are for
the central Bmab peak while red and yellow points are the low and high l staging peaks, respectively,
for (a-b) La
2
CuO
4+y
, (c-e) La
1.96
Sr
0.04
CuO
4+y
, (f-h) La
1.935
Sr
0.065
CuO
4+y
, and (i-k) La
1.91
Sr
0.09
CuO
4+y
.
Crossmarks indicate points that were not used for the fit. Note how saturation causes the points at high
t to fall below the fitted line, as expected – with exception of (i), where th ere are signs of a second phase
transition.

149
E
Additional reciprocal space figures
E.1 All DMC maps
As explained in Chap.
8, measurements were done on the DMC instrument for four different
samples, La
2
CuO
4+y
, La
1.94
Sr
0.06
CuO
4+y
, and La
1.91
Sr
0.09
CuO
4+y
(see Tab.
8.1 on p. 56 for an
overview of the experiments). In this appendix I show all of these data. They are as follows:
◮ For La
2
CuO
4+y
• (0kl) plane at T = 10, 100, and 300 K in Figs. E.1 to E.3
• (hk0) plane at T = 10 and 300 K in Figs. E.4 and E.5
• (0kl) plane for different sample with same doping at T = 5 K in Fig. E.6
◮
For La
1.94
Sr
0.06
CuO
4+y
• (0kl) plane with temperature between T = 100 and 1.5 K (cooling during measure-
ment) and at T = 100 and 300 K in Figs.
E.7 to E.9
•
(hk0) plane at T = 2 and 300 K in Figs. E.10 and E.11
◮
For La
1.91
Sr
0.09
CuO
4+y
• (0kl) plane at T = 100 and 300 K in Figs. E.12 and E.13
• (hk0) plane at T = 100 K in Fig. E.14
All the data have been symmetrized to make it easier to look at the reciprocal space patterns.
The actual non-symmetrized data are shown in small insets of each figure. The process of going
from measured 2θ spectra to the reciprocal space planes has been explained in Chap.
10.

150 E.1. All DMC maps
k [r.l.u.]
l [r.l.u.]
−3 −2 −1 0 1 2 3
−6
−4
−2
0
2
4
6
Intensity [arb.unit]
0
20
40
60
80
100
120
140
160
k [r.l.u.]
l [r.l.u.]
Intensity I [a.u.]
Non-symmetrized data
Figure E.1: Reciprocal kl plane for La
2
CuO
4+y
at T = 10 K with lattice constants b = 5.3567 Å and
c = 13.1953 Å. Data taken during November 2012. Measurement time was ∼10 hr and 10 min, the
instrument wavelength was λ = 2.45 Å, and the monitor preset was 20. Measurements were done at
angles ω ∈ [70.1°; 249.9°], ∆ω = 0.2° (detector at 2θ = 13.0°), and ω ∈ [70.0°; 249.8°], ∆ω = 0.2° (detector
at 2θ = 13.1°). The last of the two sample angles was stopped early, and hence the full space is not filled –
as opposed to in all of the the following plots.
k [r.l.u.]
l [r.l.u.]
−3 −2 −1 0 1 2 3
−6
−4
−2
0
2
4
6
Intensity [arb.unit]
0
20
40
60
80
100
120
140
160
k [r.l.u.]
l [r.l.u.]
Intensity I [a.u.]
Non-symmetrized data
Figure E.2: Reciprocal kl plane for La
2
CuO
4+y
at T = 100 K with lattice constants b = 5.358 Å and
c = 13.1997 Å. Data taken during November 2012. Measurement time was ∼10 hr and 50 min, the
instrument wavelength was λ = 2.45 Å, and the monitor preset was 20. Measurements were done at
angles ω ∈ [70.1°; 249.9°], ∆ω = 0.2° (detector at 2θ = 13.0°).

Appendix E. Additional reciprocal space figures 151
k [r.l.u.]
l [r.l.u.]
−3 −2 −1 0 1 2 3
−6
−4
−2
0
2
4
6
Intensity [arb.unit]
0
20
40
60
80
100
120
140
160
k [r.l.u.]
l [r.l.u.]
Intensity I [a.u.]
Non-symmetrized data
Figure E.3: Reciprocal kl plane for La
2
CuO
4+y
at T = 300 K with lattice constants b = 5.3771 Å and
c = 13.2352 Å. Data taken during November 2012. Measurement time was ∼12 hr and 50 min, the
instrument wavelength w as λ = 2.45 Å, and the monitor preset was 20. Measurements were done at
angles ω ∈ [70.1°; 249.9°], ∆ω = 0.2° (detector at 2θ = 13.0°).
h [r.l.u.]
k [r.l.u.]
−3 −2 −1 0 1 2 3
−3
−2
−1
0
1
2
3
Intensity [arb.unit]
0
20
40
60
80
100
120
140
160
h [r.l.u.]
k [r.l.u.]
Intensity I [a.u.]
Non-symmetrized data
Figure E.4: Reciprocal hk plane for La
2
CuO
4+y
at T = 10 K with lattice constants a = 5.3417 Å and b =
5.3442 Å. Data taken during November 2012. Measurement time was ∼9 hr and 30 min, the instrument
wavelength was λ = 2.45 Å, and the monitor preset was 20. Measurements were done at angles ω ∈
[90.0°; 270°], ∆ω = 0.2° (detector at 2θ = 13.0°).

152 E.1. All DMC maps
h [r.l.u.]
k [r.l.u.]
−3 −2 −1 0 1 2 3
−3
−2
−1
0
1
2
3
Intensity [arb.unit]
0
20
40
60
80
100
120
140
160
h [r.l.u.]
k [r.l.u.]
Intensity I [a.u.]
Non-symmetrized data
Figure E.5: Reciprocal hk plane for La
2
CuO
4+y
at T = 300 K with lattice constants a = 5.3678 Å and b =
5.3682 Å. Data taken during November 2012. Measurement time was ∼9 hr, the instrument wavelength
was λ = 2.45 Å, and the monitor preset was 20. Measurements were done at angles ω ∈ [90.0°; 270°],
∆ω = 0.2° (detector at 2θ = 13.0°).
k [r.l.u.]
l [r.l.u.]
−3 −2 −1 0 1 2 3
−6
−4
−2
0
2
4
6
Intensity [arb.unit]
0
5
10
15
20
25
30
35
40
45
50
k [r.l.u.]
l [r.l.u.]
Intensity I [a.u.]
Non-symmetrized data
Figure E.6: Reciprocal kl plane for La
2
CuO
4+y
(other sample) at T = 5 K with lattice constants b =
5.3107 Å and c = 13.1171 Å. Data taken during July 2013. Measurement time was ∼22 hr and 8 min,
the instrument wavelength was λ = 2.45 Å, and the monitor preset was 1.0. Measurements were done at
angles ω ∈ [0°; 180°], ∆ω = 0.2° (detector at 2θ = 13°). I am unsure about sample rotations, since it was
not written in a logbook.

Appendix E. Additional reciprocal space figures 153
k [r.l.u.]
l [r.l.u.]
−3 −2 −1 0 1 2 3
−6
−4
−2
0
2
4
6
Intensity [arb.unit]
0
5
10
15
20
25
30
35
40
45
50
k [r.l.u.]
l [r.l.u.]
Intensity I [a.u.]
Non-symmetrized data
Figure E.7: Reciprocal kl plane for La
1.94
Sr
0.06
CuO
4+y
at T = 1.5-100 K with lattice constants b = 5.3171 Å
and c = 13.1268 Å. Data taken during July 2013. Measurement time was ∼12 hr and 50 min, the instru-
ment wavelength was λ = 2.45 Å, and the monitor preset was 1.0. Measurements were done at angles
ω ∈ [180.0°; 360.1°], ∆ω = 0.2° (detector at 2θ = 13.0°), and ω ∈ [180.1°; 360.0°], ∆ω = 0.2° (detector at
2θ = 13.1°). I am unsure about sample rotations, since it was not written in a logbook.
k [r.l.u.]
l [r.l.u.]
−3 −2 −1 0 1 2 3
−6
−4
−2
0
2
4
6
Intensity [arb.unit]
0
50
100
150
200
250
k [r.l.u.]
l [r.l.u.]
Intensity I [a.u.]
Non-symmetrized data
Figure E.8: Reciprocal kl plane for La
1.94
Sr
0.06
CuO
4+y
at T = 100 K with lattice constants b = 5.3413 Å
and c = 13.1533 Å. Data taken during November 2013. Measurement time was ∼9 hr and 20 min , the
instrument wavelength was λ = 2.45 Å, and the monitor preset was 0.79 (1.58 initially). Measurements
were done at angles ω ∈ [90.0°; 270.1°], ∆ω = 0.2° (detector at 2θ = 10.0°), and ω ∈ [90.1°; 270.2°],
∆ω = 0.2° (detector at 2θ = 10.1°). Statistics was lowered from 1.58 to 0.79 part way through.

154 E.1. All DMC maps
k [r.l.u.]
l [r.l.u.]
−3 −2 −1 0 1 2 3
−6
−4
−2
0
2
4
6
Intensity [arb.unit]
0
50
100
150
200
250
k [r.l.u.]
l [r.l.u.]
Intensity I [a.u.]
Non-symmetrized data
Figure E.9: Reciprocal kl plane for La
1.94
Sr
0.06
CuO
4+y
at T = 300 K with lattice constants b = 5.3501 Å
and c = 13.1889 Å. Data taken during November 2013. Measurement time was ∼16 hr and 55 min, the
instrument wavelength was λ = 2.45 Å, and the monitor preset was 1.58. Measurements were done at
angles ω ∈ [90.0°; 270.1°], ∆ω = 0.2° (detector at 2θ = 10.0°), and ω ∈ [90.1°; 270.2°], ∆ω = 0.2° (detector
at 2θ = 10.1°).
h [r.l.u.]
k [r.l.u.]
−3 −2 −1 0 1 2 3
−3
−2
−1
0
1
2
3
Intensity [arb.unit]
0
5
10
15
20
25
30
35
40
45
50
h [r.l.u.]
k [r.l.u.]
Intensity I [a.u.]
Non-symmetrized data
Figure E.10: Reciprocal hk plane for La
1.94
Sr
0.06
CuO
4+y
at T = 2 K with lattice constants a = 5.3349 Å and
b = 5.3291 Å. Data taken during July 2013. Measurement time was ∼11 hr and 50 min, the instrument
wavelength was λ = 2.45 Å, and the monitor preset was 1.12. Measurements were done at angles ω ∈
[180.0°; 360.1°], ∆ω = 0.2° (detector at 2θ = 13.0°), and ω ∈ [180.1°; 360.0°], ∆ω = 0.2° (detector at
2θ = 13.1°).

Appendix E. Additional reciprocal space figures 155
h [r.l.u.]
k [r.l.u.]
−3 −2 −1 0 1 2 3
−3
−2
−1
0
1
2
3
Intensity [arb.unit]
0
5
10
15
20
25
30
35
40
45
50
h [r.l.u.]
k [r.l.u.]
Intensity I [a.u.]
Non-symmetrized data
Figure E.11: Reciprocal hk plane for La
1.94
Sr
0.06
CuO
4+y
at T = 300 K with lattice constants a = 5.3413 Å
and b = 5.3361 Å. Data taken during July 2013. Measurement time was ∼7 hr and 30 min, the instrument
wavelength was λ = 2.45 Å, and the monitor preset was 1.12. Measurements were done at angles ω ∈
[180.0°; 360.1°], ∆ω = 0.2° (detector at 2θ = 13.0°), and ω ∈ [180.1°; 360.0°], ∆ω = 0.2° (detector at
2θ = 13.1°). Stopped quite shortly into the second scan part with the detector moved 0.1°.
k [r.l.u.]
l [r.l.u.]
−3 −2 −1 0 1 2 3
−6
−4
−2
0
2
4
6
Intensity [arb.unit]
0
10
20
30
40
50
60
70
80
90
k [r.l.u.]
l [r.l.u.]
Intensity I [a.u.]
Non-symmetrized data
Figure E.12: Reciprocal kl plane for La
1.91
Sr
0.09
CuO
4+y
at T = 100 K with lattice constants b = 5.3288 Å
and c = 13.138 Å. Data taken during November 2013. Measurement time was ∼33 hr and 15 min , the
instrument wavelength was λ = 2.45 Å, and the monitor preset was 0.79. Measurements were done at
angles ω ∈ [90.0°; 270.1°], ∆ω = 0.2° (detector at 2θ = 13.0°), and ω ∈ [90.1°; 270.2°], ∆ω = 0.2° (detector
at 2θ = 13.1°). The time is longer than usual because the neutron beam was down for part of it.

156 E.1. All DMC maps
k [r.l.u.]
l [r.l.u.]
−3 −2 −1 0 1 2 3
−6
−4
−2
0
2
4
6
Intensity [arb.unit]
0
10
20
30
40
50
60
70
80
90
k [r.l.u.]
l [r.l.u.]
Intensity I [a.u.]
Non-symmetrized data
Figure E.13: Reciprocal kl plane for La
1.91
Sr
0.09
CuO
4+y
at T = 300 K with lattice constants b = 5.3322 Å
and c = 13.183 Å. Data taken during November 2013. Measurement time was ∼38 hr and 47 min, the
instrument wavelength was λ = 2.45 Å, and the monitor preset was 0.79. Measurements were done at
angles ω ∈ [90.0°; 270.1°], ∆ω = 0.2° (detector at 2θ = 13.0°), and ω ∈ [90.1°; 270.2°], ∆ω = 0.2° (detector
at 2θ = 13.1°).
h [r.l.u.]
k [r.l.u.]
−3 −2 −1 0 1 2 3
−3
−2
−1
0
1
2
3
Intensity [arb.unit]
0
10
20
30
40
50
60
70
80
90
h [r.l.u.]
k [r.l.u.]
Intensity I [a.u.]
Non-symmetrized data
Figure E.14: Reciprocal hk plane for La
1.91
Sr
0.09
CuO
4+y
at T = 100 K with lattice constants a = 5.3375 Å
and b = 5.3264 Å. Data taken during November 2013. The instrument wavelength was λ = 2.45 Å, and
the monitor preset was 0.79. Measurements were done at angles ω ∈ [90.0°; 270.1°], ∆ω = 0.2° (detector
at 2θ = 13.0°), and ω ∈ [90.1°; 270.2°], ∆ω = 0.2° (detector at 2θ = 13.1°).

I
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