Single magnetic chirality in the magnetoelectric NdFe3(B11O3)4
ABSTRACT We have performed an extensive study of single crystals of the magnetoelectric NdFe3(B11O3)4 by means of a combination of single-crystal neutron diffraction and spherical neutron polarimetry. Our investigation did not detect significant deviations at low temperatures from space group R32 concerning the chemical structure. With respect to magnetic ordering our combined results demonstrate that in the commensurate magnetic phase below TN≈30K all three magnetic Fe moments and the magnetic Nd moment are aligned ferromagnetically in the basal hexagonal plane but align antiferromagnetically between adjacent planes. The phase transition to the low-temperature incommensurate (IC) magnetic structure observed at TIC≈13.5K appears to be continuous. By means of polarized neutron studies it could be shown that in the incommensurate magnetic phase the magnetic structure of NdFe3(B11O3)4 is transformed into a long-period antiferromagnetic helix with single chirality. Close to the commensurate-incommensurate phase transition third-order harmonics were observed, which in addition indicate the formation of magnetic solitons.
arXiv:1001.1784v2 [cond-mat.str-el] 25 Mar 2010
Single magnetic chirality in the magneto-electric NdFe3(11BO3)4
Laboratory for Neutron Scattering, Paul Scherrer Institut & ETH Zurich, CH-5232, Villigen, PSI, Switzerland and
Physik Department E21, Technische Universit¨ at M¨ unchen, D-85748 Garching, Germany∗
P. Fischer, J. Schefer, B. Roessli, and V. Pomjakushin
Laboratory for Neutron Scattering, Paul Scherrer Institut & ETH Zurich, CH-5232, Villigen, PSI, Switzerland
Forschungsneutronenquelle Heinz Maier-Leibnitz (FRM II),
Technische Universit¨ at M¨ unchen, D-85748 Garching, Germany
Institute of Physics, ASCR v.v.i, Na Slovance 2, 182 21 Praha 8, Czech Republic
G. Petrakovskii and L. Bezmaternikh
Institute of Physics SB RAS, Krasnoyarsk 660036, Russia
(Dated: March 26, 2010)
We have performed an extensive study of single-crystals of the magneto-electric NdFe3(11BO3)4
by means of a combination of single-crystal neutron diffraction and spherical neutron polarimetry.
Our investigation did not detect significant deviations at low temperatures from space group R32
concerning the chemical structure. With respect to magnetic ordering our combined results demon-
strate that in the commensurate magnetic phase below TN ≈ 30 K all three magnetic Fe moments
and the magnetic Nd moment are aligned ferromagnetically in the basal hexagonal plane but align
antiferromagnetically between adjacent planes. The phase transition to the low-temperature incom-
mensurate magnetic structure observed at TIC ≈ 13.5 K appears to be continuous. By means of
polarized neutron studies it could be shown that in the incommensurate magnetic phase the mag-
netic structure of NdFe3(11BO3)4 is transformed into a long-period antiferromagnetic helix with
single chirality. Close to the commensurate-incommensurate phase transition third-order harmonics
were observed which in addition indicate the formation of magnetic solitons.
PACS numbers: 75.85.+t, 75.50.Ee, 75.30.Kz, 75.25.-j
Over the last decades various interesting effects re-
lated to long-range spiral-like forms of magnetic order
have been revealed in multiple fields of condensed mat-
ter physics. Recent examples include multiferroic com-
pounds such as TbMnO31–3, magnetic surfaces4,5that
are interesting for applications in spintronics, e.g. the
construction of a spin field effect transistor6, magnetic in-
sulators such as Ba2CuGe2O77or itinerant magnets such
as MnSi8. Further, helimagnetism is also extensively dis-
cussed theoretically, here examples encompass the pro-
posal of a chiral universality class9, non-centrosymmetric
superconductors10and new helical Goldstone modes, so-
called helimagnons11,12.The latter have recently also
been observed experimentally13.
Spiral magnetic order that in addition exhibits chirality,
i.e. it breaks the spatial inversion symmetry, may exist in
a right- and left-handed version that are interrelated via
the inversion operation. As long as the underlying chem-
ical structure is centrosymmetric both versions are ener-
getically degenerate and right- and left-handed domains
should be observed in equal fractions14,15. However, in
case the chemical structure is non-centrosymmetric itself,
the magnetic structure is expected to show a single hand-
edness. Since 65 of the known 230 crystallographic space
groups are non-centrosymmetric (Sohncke groups)16one
would expect a great number of chiral magnetic com-
pounds with a single chirality. Yet, only a few such com-
pounds have been observed so far, with the most promi-
nent example being MnSi. MnSi exhibits many interest-
ing physical properties such as an extended non-Fermi
liquid phase17probably associated with a new kind of
metallic state18,19, a magnetic Skyrmion lattice20and the
observation of helimagnons13, that are all closely related
to its magnetic chirality21. More recently monochirality
was also observed in the non-centrosymmetric compound
Ba3NbFe3Si2O14that exhibits the interesting coexistence
of two forms of chiral magnetism, within triangles in
the basal plane and and along the c-axis as a magnetic
helix22. A further example is UPtGe that, however, dis-
plays a cycloid magnetic structure without true chirality,
but with a single turning sense of the cycloid, that was
also explained in terms of its non-centrosymmetric crys-
In this work we demonstrate that NdFe3(11BO3)4 is a
new compound that displays a chiral magnetic struc-
ture with a single chirality. Below a magnetic commen-
surate (C) to incommensurate (IC) phase transition at
TIC≈13.5 K the collinear antiferromagnetic structure of
NdFe3(11BO3)4transforms into an antiferromagnetic he-
lix that exists with a single chiral domain. Furthermore,
the observation of third-order harmonics below TICsug-
gest that the emergence of the helical magnetic structure
is accompanied by the formation of a magnetic soliton
lattice below TIC.
RM3(BO3)4 (R = Y,La-Lu, M = Al, Ga, Cr, Fe, Sc).
These borates are interesting in their own right and have
previously been studied mainly due to their special opti-
cal properties. Rare-earth ions, in general, and Nd3+, in
particular, have excellent characteristics to generate in-
frared laser action and to serve in nonlinear optics24–27.
The subfamily of ferroborates (M=Fe) is equally inter-
esting with respect to their magnetic properties due to
competing magnetic sublattices. Here GdFe3(BO3)4and
NdFe3(BO3)4are especially fascinating as both materials
show a large magneto-electric effect28,29.
The ferroborates RFe3(BO3)4 crystallize in the trigo-
nal space group R32 (group no. 155), that is they
belong to the structural type of the mineral huntite
CaMg3(BO3)430. Note that this structure is missing a
center of inversion. In our previous work31we performed
an unpolarized neutron diffraction study with both pow-
der and single crystal samples in order to investigate
the magnetic structure of NdFe3(11BO3)4. We demon-
strated that NdFe3(11BO3)4exhibits long-range antifer-
romagnetic order with the magnetic propagation vector
khex= [0,0,3/2] below TN ≈ 30 K. However, our com-
bined magnetic representational and Rietveld analysis32
yields different magnetic structures that explain the data
equally well (s. Fig. 1). All models have in common
that the magnetic moments of all three Fe sublattices
and the Nd sublattice are parallel to the hexagonal basal
plane and are coupled antiferromagetically in adjacent
planes. This is also in agreement with easy-plane type
magnetic anisotropy that was observed in several studies
of the magnetic susceptibility30,31. For a more detailed
description we refer to Refs. 31,33. In addition the study
revealed that below approximately T = 19 K the mag-
netic structure becomes incommensurate with the mag-
netic propagation vector khex,i = [0,0,3/2 + ε]. How-
ever, since the study was performed with thermal neu-
trons only, the experimental resolution did not allow for
an exact study of the propagation vector as a function of
temperature or the nature of the IC magnetic structure.
The purpose of this study is mainly to identify the cor-
rect magnetic structure in both the C and IC magnetic
phases. For this task we performed a detailed neutron
diffraction study with both unpolarized neutrons and
full spherical neutron polarimetry applied. Further high
resolution diffraction was carried out to study the tem-
perature dependence of the C-IC phase transition with
better resolution. The unpolarized neutron single crys-
tal diffraction data were additionally used to verify the
low-temperature chemical structure of NdFe3(11BO3)4,
the family ofborates
mostly as previous results indicated that the overall
chemical symmetry might only be R331.
Finally all our magnetic single neutron diffraction results
were evaluated by a comparative analysis with both the
standard FullProf package34and Jana200635. The latter
was only recently extended for the purpose of analysis
of magnetic neutron scattering data and emphasizes the
use of magnetic symmetries.
? ?? ???????
? ?? ??????
NdFe3(11BO3)4 are plotted with the program ATOMS36for
20 K. We note that the magnetic unit cell is doubled along
the crystallographic c-axis with respect to the chemical unit
cell. Magnetic model (M1) features three magnetic Fe mo-
ments of equal magnitude. Panels (a) and (b) show two spe-
cific cases of (M1) where the angle φNdbetween the magnetic
Fe and Nd moments is zero (M1a) or non-zero (M1b), re-
spectively. The second magnetic model (M2) shown in (c)
corresponds to magnetic moments on the Fe sublattices that
have slightly different sizes and orientations. Both models
have been previously found to explain neutron powder diffrac-
tion data on NdFe3(11BO3)4 by Fischer et al.31equally well.
Here we demonstrate that only the magnetic model (M1a)
describes all present data correctly.
Possible models for the magnetic structure of
II. EXPERIMENTAL TECHNIQUES
In this study two different samples of NdFe3(11BO3)4
were studied. A large single crystal of approximate di-
mensions 8 x 8 x 8 mm3that has been already used in
our previous work31(Sample1) was reinvestigated. Fur-
ther, a new smaller sample of the size 3 x 5 x 4mm3
was prepared for additional experiments (Sample2). The
process that was used for the preparation of the stud-
ied crystals is already described in detail in our previous
In order to reduce strong extinction effects, diffraction
measurements were carried out on the smaller Sample2
of NdFe3(11BO3)4, in a four-circle setup on the thermal
neutron diffractometer TriCS37at the continuous Swiss
spallation neutron source SINQ38and on the HEiDi hot
neutron diffractometer39, situated at the FRM II reactor
at Munich. The corresponding neutron38wavelengths
were 1.18˚ A and 0.55˚ A, respectively. The data eval-
uations were performed by means of current versions
of FullProf34and Jana200635. For the refinements the
low-temperature lattice parameters a = 9.594˚ A
c = 7.603˚ A of Ref. 31 were employed. Absorption cor-
rections were neglected. Nuclear and magnetic extinction
may be quite different, cf. e.g.40,41. We will show that
in case of NdFe3(11BO3)4such a difference is almost not
significant for hot neutrons, but turned out to be im-
portant for thermal neutrons. In contrast to FullProf,
Jana2006 permits different types of isotropic extinction
corrections. However, the corresponding differences were
not found to be essential in case of NdFe3(11BO3)4and
thus type II extinction42was used. In order to limit the
number of parameters, only isotropic temperature factors
The recent extension of the Jana2006 refinement pro-
gram allows commensurate and incommensurate mag-
netic structures to be described by four-dimensional
magnetic superspace groups analogous to occupationally
modulated chemical structures43. A more detailed dis-
cussion of these aspects of the refinement is given in the
appendix together with a comparison to the correspond-
ing FullProf results.
Further, Sample1 was investigated in a high resolution
diffraction experiment carried out on the triple-axis spec-
trometer TASP44, situated at the end position of a cold
super mirror guide of SINQ. The spectrometer was oper-
ated in its elastic mode with fixed incident and final wave
vector kf = 1.2˚ A−1. Additionally 20’ Soller collimators
were installed in the incident beam, in front of the ana-
lyzer and the detector. The second order contamination
was removed from the beam by means of a beryllium filter
that was inserted between the sample and the analyzer.
The use of a triple-axis spectrometer for diffraction ex-
periments is justified by the excellent signal-to-noise ratio
that is achieved by the use of an additional analyzer crys-
tal. This experimental setup only allows access to Bragg
reflections in a single scattering plane. For this measure-
ment the single crystal was oriented with the reciprocal
axis b∗(K) and c∗(L) within the scattering plane.
In order to perform full polarization analysis on
(SNP) option MuPAD45available at SINQ was mounted
on TASP. The neutron beam was polarized and ana-
lyzed via two polarizing supermirror benders that were
installed after the monochromator and in front of the an-
alyzer, respectively. A final wave vector kf = 1.97˚ A−1
was chosen to maximize both the intensity and the po-
larization of the neutron beam. No additional filter for
second order suppression was used because the benders
already act as such. The orientation of the crystal was
identical to the unpolarized measurements.
Only the combination of unpolarized and polarized neu-
tron diffraction measurements allowed us to find the
model for magnetic structure that gives the best agree-
ment with all data collected during the course of this
investigation. However, for the sake of clarity we will
describe the unpolarized and the SNP measurements in
separate sections, referring to the corresponding other
sections when necessary.
III.UNPOLARIZED SINGLE CRYSTAL
A. Low-temperature chemical structure
The low temperature chemical structure was verified
by measuring extended sets of nuclear neutron intensities
on HEiDi at 22 K (λ = 0.55˚ A, 287 Bragg peaks; data set
N1) and on TriCS at 6 K (λ = 1.18˚ A, 151 Bragg peaks;
data set N2). For the fits of both data sets with the space
group R32 we refined 11 parameters with two isotropic
temperature factors (one for the heavy atoms Nd and Fe
and one for the light atoms) and one isotropic extinction
parameter. The resulting structural parameters are sum-
marized in Table I together with the agreement factors.
Note, that for the data set (N1) we performed fits with
both FullProf and Jana2006. This is discussed in more
detail in the appendix.
Our findings confirm that also at low temperatures the
chemical structure of NdFe3(11BO3)4 is well described
within the space group R32.
ters derived from the two data sets are in good agree-
ment. They are close to the room-temperature val-
ues published in our previous work31.
the TriCS measurement, but in agreement with previ-
ous powder investigations31, the HEiDi single crystal re-
finement tended for the isotropic temperature factor of
Nd to negative values. Compared to the hot neutron re-
sults, the larger temperature factor values obtained from
the thermal neutron data are presumably due to the ne-
glected larger absorption in the latter case. Refinement
of occupation factors did not indicate essential deviations
The positional parame-
In contrast to
NdFe3(11BO3)4, refined from the present single crystal neu-
tron diffraction data. Results from the fits of data sets (N1)
are shown in the first and second lines were performed with
FullProf (N1F) and Jana2006 (N1J), respectively. The third
line gives the results from data set (N2) performed with Full-
prof. Standard uncertainties of the parameters are given
within parentheses. The agreement factors for the corre-
sponding fits are: (N1F) Rn,F2 = 4.5 %, Rn,F2w = 5.3 %,
Rn,F = 3.2 %, goodness of fit χ2= 6 (s. Ref. 34 for the
meaning of agreement factors); (N1J) R(all) = 3.1 %, Rw(all)
= 6.5 %, GOF(all) = 1.7, GOF(obs) = 1.8 (s. Ref. 35); (N2)
Rn,F2 = 8.0 %, Rn,F2w = 8.4 %, Rn,F = 5.6 %, goodness of
fit χ2= 258.
I: Low-temperature structural parameters of
Atom xyz B (˚ A2)
Fe 0.5506(2) 0
B2 0.4465(3) 0
0.4523(3) 0.1445(3) 0.5185(3) 0.21(3)
0.4521(3) 0.1449(3) 0.5180(4) 0.17(4)
0.4533(5) 0.1448(5) 0.5182(4) 1.14(6)
B. Magnetic commensurate-incommensurate
By means of careful cold neutron scattering experi-
ments on the triple-axis spectrometer TASP carried out
on the bigger Sample1, we investigated the temperature
dependence of the magnetic C-IC transition. Elastic Q-
scans along the reciprocal L-direction around the mag-
netic Bragg reflection (0,0,3
2) as a function of tempera-
ture were used to determine the transition temperature
as illustrated in Fig. 2(a). The propagation vector is com-
mensurate down to the temperature TIC≈ 13.5 K below
which the magnetic Bragg reflection splits into two in-
commensurate satellite peaks. Here we determined TIC
as the temperature where the maximum intensity of the
commensurate magnetic reflection is reached and then
starts to decrease as the reflection splits up into the two
incommensurate satellite peaks. Below TIC the scans
show that kz changes continuously at the C-IC phase
transition in NdFe3(11BO3)4 and can be well described
2+ ε where
ε = 1.6 · 10−3|(TIC− T)|0.58. (1)
This is demonstrated by the black solid line in Fig. 2(a).
The splitting of the magnetic Bragg reflections is only
observed along the z-direction down to T = 1.6 K as
shown by the intensity map around the (0,0,3
in the inset of Fig. 2(a). The splitting at T = 1.6 K is ε
In addition to the principal magnetic satellites also
reflections at higher order harmonics were observed as
shown in Fig. 3. Their shift with respect to khex was
found to be 3ε = 0.02. The integrated intensity of
the third order harmonics is largest near the transition
temperature TIC and decreases fast as a function of
decreasing temperature. We note that the integrated
intensity of the principle satellites remains approxi-
mately constant down the lowest temperature as would
be expected for a magnetic soliton lattice46.
Investigations performed on TriCS with thermal neu-
trons on the smaller Sample2 of NdFe3(11BO3)4clearly
show broadening of magnetic peaks at temperatures
below 12.5 K, confirming the existence C-IC magnetic
phase transition in NdFe3(11BO3)4. This is illustrated
in Fig. 2(b). However, a splitting of the (-1,0,0.5) peak
as observed before on the Sample131, could not be
reproduced under similar conditions or on the HEiDi
neutron diffractometer. This indicates a certain sample
dependence of the k-vector magnitude in the incom-
mensurate phase. At the lowest temperature kzmay be
approximated as 1.502 for Sample2.
Corresponding to the results in the preceding section,
we assumed commensurate and incommensurate mag-
netic ordering for temperatures above and below TIC, re-
spectively, for the fits of the data discussed in the follow-
ing. Moreover, we used the structural parameters from
(N1) (HEiDi results).
The spherical neutron polarimetry experiments that will
be discussed in section IV showed that only the model
(M1) for the magnetic structure is able to explain the
data correctly. Therefore, we will focus on the fits per-
formed for this model here.
for (M1) the magnetic structure can be conveniently ex-
pressed as spirals of the form
As shown in Refs. 31,33
excos(2πk · t + φj)+
(ex+ 2ey)/√3sin(2πk · t + φj)
that propagate along the hexagonal c-axis for all three
magnetic Fe moments and the magnetic Nd moment
(j = Fe,Nd) and also for both the (C) and (IC) magnetic
phases. Here exand eyare unit vectors of the hexagonal
lattice and φj describes the polar takeoff angle from the
hexagonal a axis.
In the incommensurate case we superimposed + and −
satellites in each magnetic peak, as here ±khex,iare in-
equivalent. Further, we note, that for such spirals the
magnetic neutron intensity is modulated with the factor
(1 + cos2η), where η represents the angle of the mag-
netic scattering vector to the spiral axis47. The direction
of the three parallel magnetic Fe moments, that is de-
scribed via the polar angle φFein our model, cannot be
determined from the present unpolarized neutron diffrac-
tion data. However, recent bulk magnetic measurements
of Tristan et al.48on a single crystal of NdFe3(11BO3)4
at low temperatures show easy magnetization along the
a-axis and we therefore fixed the value of the polar angle
to φFe = 0. In our previous work31we found a small
angle between magnetic moments of the Fe and Nd ions.
Hence, we performed fits in two configurations, respec-
tively, where we either kept the Nd magnetic moment
1.49 1.50 1.51
TriCS, λ= 2.1
FIG. 2: (a) The figure shows a contour map of the neutron
intensity around the reciprocal lattice position (0,0,3
ple1 in elastic Q-scans along the reciprocal L-direction as a
function of temperature determined on the triple-axis spec-
trometer TASP. The red squares are the peak positions of the
incommensurate magnetic Bragg reflections as determined by
fits of Gaussian peaks to the measured scans. Here (P) de-
notes the paramagnetic phase, (C) the commensurate mag-
netic phase and (IC) the incommensurate magnetic phase,
respectively. The inset shows a contour map of the neutron
intensity around the position (0,0,3
termined at T = 1.6 K. (b) Temperature dependencies of the
integrated neutron intensity and of the full width at half maxi-
mum (FWHM) of the magnetic Bragg peak (1,0,-1/2) of Sam-
ple2 are shown.
2) of Sam-
2) in reciprocal space de-
parallel to the Fe moments (model (M1a)) or allowed for
a refinable angle φNd between the Fe and Nd magnetic
moments (model M1(b)). Starting with a single mag-
netic domain and using the nuclear isotropic extinction
parameters, thus 2 or 3 parameters were refined ((M1a)
and (M1b), respectively). In principle up to six magnetic
orientation domains would be possible according to the
threefold- and twofold rotation axes of the paramagnetic
space group R32 in the latter case: u,v,w; −v,u−v,w;
−u + v,−u,w; u − v,−v,−w; −u,−u + v,−w; v,u,−w
in direct space notation. For the incommensurate case
only the three threefold rotations around the crystallo-
graphic c-axis are present31. However, the corresponding
FullProf calculation did not yield improved fits. This can
be understood by considering that the incommensurate
magnetic spiral described by Eq. (2) breaks the transla-
tion symmetry of the underlying chemical lattice along
the c direction. The broken translation symmetry im-
plies that any rotation of the spiral around the c-axis can
Neutron Intensity (counts/sec)
T = 7.5 K
(0, -1, L) (r.l.u.)
FIG. 3: (a) Q-scans over the (0,-1,0.5) magnetic Bragg for
different temperatures are shown. The scans were performed
with 20’ Soller collimators installed in the incident beam, in
front of the analyzer and the detector. In the panels (b) to
(d) the development of third order satellites is demonstrated.
Their shift from the commensurate position is three times
larger than for the first order peaks. The solid lines are fits
to the data with multiple Gaussian profiles.
be compensated by a translation and the spin structure
is conserved. The additional fits by means of Jana2006
gave identical results. In particular the superspace de-
scription for the magnetic structure in Jana2006 gives
a natural explanation concerning the single populated
magnetic orientation domain(s. appendix and Ref. 49).
In the commensurate case with khex= [0,0,3/2], +khex
is equivalent to −khexand this implies that the magnetic
moments in adjacent planes are arranged antiparallel.
Here the refinement according to model (M1a) yield three
in good approximation equally distributed magnetic do-
mains according to the threefold rotation axes.
At various temperatures sets of 50 magnetic and 5 nuclear
Bragg peaks were measured on HEiDi. Corresponding
characteristic refinement results obtained by both Full-
Prof and Jana2006 are given in the appendix and are
For the commensurate magnetic phase of NdFe3(11BO3)4
we may conclude φFe = φNd = 0 according to model
(M1a). We note our previously published powder neu-
tron diffraction data31may be equally well fitted with
φFe= φNd= 0 (cf. Ref. 33 additionally). Model (M1a)
is furthermore supported by the spherical neutron po-
larimetry results that will be discussed in section IV.
With respect to incommensurate magnetic ordering in
NdFe3(11BO3)4, the thermal neutron data at 6 K indi-
cate almost significance for the introduction of a non-zero
angle φNd(M1b). On the other hand, the difference be-
tween the two different HEiDi refinements is considerably
FIG. 4: Temperature dependencies of the ordered magnetic
Fe and Nd moments in NdFe3(11BO3)4 based on the eval-
uation of the present single crystal neutron diffraction data
with FullProf (F). The values found via the alternative data
refinement by means of Jana2006 (J) agree within the error-
bars (cf. appendix) apart for the data measured with thermal
neutrons on the instrument TriCS. The dashed line represents
the one expected for S = 5/2 of Fe3+(cf. Ref. 31). The linear
dashed-dotted line through the Nd moments is a guide to the
smaller, due to less important extinction corrections at
shorter neutron wavelength. Thus the simpler magnetic
structure with φFe= φNd= 0 (M1a) holds most probably
also in the incommensurate phase of NdFe3(11BO3)4(s.
appendix for details).
The resulting temperature dependencies of the ordered
magnetic moments of the Fe3+and Nd3+ions in
NdFe3(11BO3)4 are shown in Fig. 4.
between the magnetic moments derived from the 5.5 K
TriCS data and those from HEiDi at 5 K are presumably
due to the larger extinction corrections in case of ther-
mal neutrons. The magnetic moments of both Fe and Nd
seem to vary smoothly as a function of temperature, also
at the C-IC magnetic phase transition at approximately
TIC≈ 13.5 K.
IV.SPHERICAL NEUTRON POLARIMETRY
SNP measurements were performed above and below
the C-IC transition, respectively.
NdFe3(11BO3)4 an excellent Q-resolution is necessary
to observe the small splitting (ε = 0.00667 (r.l.u.) ≡
0.0055˚ A−1) of the magnetic Bragg reflections. For the
small final wave vectors kf that are required for such
a high resolution setup, the polarizing benders used at
TASP perform non-ideal with respect to transmission
and polarizing efficiency. The best tradeoff between res-
olution and transmission/polarization is reached when
TASP is operated with kf = 1.97˚ A−1. However, the
relatively moderate Q-resolution associated with kf =
1.97˚ A−1is not sufficient to perform separate SNP mea-
surements on the magnetic satellites ±khex,i and con-
sequently a superposition of intensities from both satel-
lites will be observed. Nevertheless important informa-
tion could be extracted by performing polarized Q-scans
over the magnetic satellites.
We first discuss the results obtained by conventional
SNP measurements in the commensurate phase of
NdFe3(11BO3)4 in section IVA, the findings for the
incommensurate phase are described in section sec-
In the IC phase of
A.Commensurate magnetic structure
The final polarization vector after the scattering pro-
cess at the sample is described by the Blume-Maleyev-
equations50,51that may be given in the following conve-
P′=˜ PP0+ P′′,(3)
where˜P is the polarization tensor, which describes the
rotation of the initial polarization vector P0in the scat-
tering process and P′′is the polarization created in the
scattering process at the sample. For purely magnetic
reflections, like the ones observed in NdFe3(11BO3)4,˜ P
TABLE II: Polarization matrices on all accessible magnetic Bragg reflections of NdFe3(11BO3)4 are shown for T = 20 K. The
column P0 and P′denote the direction of the initial and final polarization vector, respectively. The subscripts of P′indicate
the polarization matrices that were measured (meas) and calculated (calc) from the two distinct magnetic models models (M1a)
and (M2). The polarization tensor elements marked in bold demonstrate where model (M2) does not match the data.
+0.0 +2.0 −0.5 +x −0.861(3) +0.073(6) +0.042(6) −0.869 +0.000 +0.000−0.868+0.000 +0.000
+y −0.059(6) −0.659(4) −0.050(6) +0.000 −0.756 +0.000+0.002−0.752+0.000
+z +0.043(6) −0.050(6) +0.655(4) +0.000 +0.000 +0.756+0.002+0.000+0.752
−x +0.876(3) −0.026(6) −0.031(6) +0.869 +0.000 +0.000 +0.869 +0.000−0.000
−y +0.034(6) +0.659(4) +0.059(6) +0.000 +0.756 +0.000 +0.002+0.752+0.000
−z −0.081(6) +0.043(6) −0.655(4) +0.000 +0.000 −0.756 +0.002+0.000−0.752
+0.0 +0.0 −1.5 +x −0.872(2) +0.057(4) +0.119(4) −0.869 +0.000 +0.000−0.869 +0.000 +0.000
+y −0.010(4) +0.010(4) +0.019(4) −0.000 +0.000 −0.000 +0.000 +0.000 +0.000
+z −0.032(4) +0.020(4) −0.007(4) +0.000 −0.000 −0.000 +0.000 +0.000−0.000
−x +0.872(2) −0.058(3) −0.115(3) +0.869 +0.000 +0.000+0.869 +0.000 +0.000
−y −0.043(4) −0.018(4) −0.017(4) −0.000 −0.000 +0.000+0.000−0.000−0.000
−z −0.019(4) −0.023(4) +0.017(4) +0.000 +0.000 +0.000+0.000−0.000+0.000
+0.0 +4.0 +0.5 +x −0.853(7)+0.06(1) +0.01(1)−0.869 +0.000 +0.000−0.869 +0.000 +0.000
+y−0.05(1)−0.837(8)−0.05(1) +0.000 −0.869 +0.000−0.004−0.866+0.000
+0.03(1)−0.02(1)+0.826(8) +0.000 +0.000 +0.869−0.004+0.000 +0.866
−x +0.850(7)+0.02(1) +0.03(1) +0.869 +0.000 +0.000+0.868 +0.000 +0.000
−0.01(1) +0.807(8)+0.08(1)+0.000 +0.869 −0.000−0.004+0.866 +0.000
−0.08(1) +0.08(1)−0.851(7) +0.000 −0.000 −0.869−0.004−0.000−0.866
+0.0 −2.0 −2.5 +x −0.871(4) +0.068(7) +0.107(7) −0.869 −0.000 +0.000−0.869 −0.000+0.000
+y −0.092(7) −0.081(7) −0.012(7) +0.000 −0.184 −0.001−0.003−0.170−0.000
+z −0.017(7) −0.006(7) +0.100(7) +0.000 +0.001 +0.184−0.003−0.000+0.170
−x +0.876(4) −0.045(7) −0.102(7) +0.869 +0.000 +0.000+0.868−0.000−0.000
−y −0.005(7) +0.085(7) +0.030(7) +0.000 +0.184 −0.001−0.003+0.170+0.000
−z −0.064(7) +0.030(7) −0.087(7) +0.000 −0.001 −0.184−0.003+0.000−0.170
+0.0 +1.0 −2.5 +x −0.860(8)+0.07(2) +0.11(2)−0.869 +0.000 +0.000−0.898+0.000−0.000
+y−0.04(2)−0.08(2)+0.01(2)+0.000 −0.055 +0.000 −0.285 −0.649
−0.04(2)−0.01(2)+0.11(2)0.000 +0.000 +0.055 −0.285
−x +0.851(8)−0.07(2)−0.10(1)+0.869 −0.000 +0.000+0.816+0.000+0.000
−0.02(2) +0.10(2) +0.02(2) +0.000 +0.055 +0.000 −0.285 +0.649
−0.03(2) +0.02(2)−0.05(2) +0.000 +0.000 −0.055 −0.285
+0.0 +1.0 +0.5 +x −0.876(3) +0.037(7) +0.084(7) −0.869 +0.000 +0.000−0.846 +0.000−0.000
+y −0.078(7) −0.485(6) −0.083(7) +0.000 −0.544 +0.000 +0.148 −0.798
+z +0.069(7) −0.062(7) +0.481(6) +0.000 +0.000 +0.544 +0.148
−x +0.880(3) −0.009(7) −0.044(7) +0.869 −0.000 −0.000+0.886+0.000+0.000
−y +0.010(7) +0.495(6) +0.079(7) +0.000 +0.544 +0.000 +0.148 +0.798
−z −0.101(7) +0.070(7) −0.478(6) +0.000 +0.000 −0.544 +0.148
and P′′are reduced to
σ˜ P =
σ = |M⊥|2+ P0x2ℑ(M∗
fined as M⊥ =
is the magnetic interaction vector de-
ˆQ × (ρ(Q) ×ˆQ), where ρ(Q) =
?ρ(r)exp(iQ · r)dr is the Fourier transform of
the magnetization density ρ(r) of the investigated sample
andˆQ is a unit vector parallel to the scattering vector
Q. The set of polarization axes is defined to have x par-
allel to Q, z perpendicular to the scattering plane and
y completing the right-handed set. We note, that the
structures that display chirality and is therefore often de-
noted as the chiral term. Finally, the measured quantity
is the polarization matrix, namely the components of the
final polarization vector after the scattering process for
all three directions of the incident beam polarization,
⊥y· M⊥z) ≡ C is only non-zero for magnetic
Pij= (Pi0˜Pji+ P′′
where i and j (i,j = x,y,z) denote the directions of the
incident and final polarization vectors, respectively.
For the commensurate magnetic phase of NdFe3(11BO3)4
we measured the polarization matrix at six magnetic
Bragg reflections and at T = 20, 25 and 30 K. The mea-
sured polarization matrices proved to be independent of
temperature (within the error bars) for T ≥ 20 K. Thus,
only the data for T = 20 K will be discussed in the fol-
lowing. The corresponding matrices are provided in table
II. From the measured polarization matrices several con-
straints on the magnetic structure can be derived:
1. On all measured Bragg peaks the elements yx and
zx are equal to zero. This implies that the magnetic
structure is not chiral at all or that it is a chiral
structure with equally populated chiral domains.
2. For the magnetic reflection (0,0,-1.5) the scatter-
ing vector Q is directed parallel to the crystallo-
graphic c direction. Hence, the magnetic interac-
tion vector only contains components in the basal
the reduction from −1 is due the polarizing ben-
ders which have a non-ideal polarization efficiency
of approximately 0.966. Thus, P′
ized whereas P′
ized. This would foremost lead to the assumption
|M⊥y|2≈ |M⊥z|2but as the elements P′
are also equal to zero it also suggests the presence
of spin domains in the basal plane.
xx= −|M⊥|2/|M⊥|2= −0.872(2). Here
xxis fully polar-
zz(−|M⊥y|2+ |M⊥z|2)/|M⊥|2are fully depolar-
3. On the magnetic reflection (0,4,0.5) the scattering
vector Q is approximately parallel to the recipro-
cal b⋆axis. As the z axis which is perpendicular
to the scattering plane lies always within the basal
hexagonal plane in the chosen scattering geometry
the y-axis is approximately parallel to the crystal-
lographic c-axis (∢(y,c) ≈ 15◦). The polarization
tensor shows P′
that |M⊥y|2≈ 0 and hence the magnetic interac-
tion vector is directed along z. Therefore the mag-
netic moments are confined in the basal plane.
We note that these constraints are satisfied by both
possible magnetic structures (M1a/b) and (M2).
order to calculate the expected polarization matrices for
both models, we used lattice constants and structural
parameters from table I. The magnitude of the magnetic
moments for the Fe and Nd ions were set to the values
as obtained from the unpolarized diffraction data. Since
SNP is generally insensitive to absolute moment sizes
in case of pure magnetic reflections they were fixed
in subsequents fits. Both models fail to explain the
observed polarization matrices when no orientation
domains were considered. Introducing the three or six
orientation domains with statistical population in the
calculation for models (M1) and (M2), respectively (cf.
appendix), resulted in polarization matrices given in
table II.Further, we considered the non-ideal polar-
ization efficiency of the used polarizing benders in the
As demonstrated in table II the magnetic model (M2)
was not able to explain the measured polarization
tensors on the two magnetic reflections (0, 1, -2.5)
and (0, 1, 0.5) as indicated by the bold entries in
table II(χ2= 9.5). The model introduces small chiral
contributions (yx and zx elements of the tensors) due
to its slightly canted spins (cf Fig. 1(b)), that are not
observed in the experiment. Performing a fit from this
starting values did not result in a better agreement
between model and data, as the fit diverged.
model (M2) can be excluded.
For the magnetic model (M1) we fixed φFe= 0 and left
φNdfree for the fits, thus corresponding to (M1b). The
fit converges to a solution with φNd = 0 (χ2= 3.8)
which corresponds to (M1a). Attempts to additionally
determine the orientation of the Fe magnetic moments
via the angle φFegave no conclusive results. The angle
φFe describes the absolute orientation of the magnetic
moments in the crystallographic ab-plane with respect
to chemical structure. Thus, the indeterminacy of φFe
is most probably related to the presence of the three
orientation domains in the hexagonal basal plane.
In summary our SNP data for the commensurate
phase is best explained by the model (M1a) when
φFe = φNd = 0 which is also in agreement with the
unpolarized neutron single crystal diffraction results
described in section IIIC. The corresponding magnetic
structure is illustrated in Fig. 1(a).
TABLE III: (a) Integrated intensities for the magnetic satel-
lite reflections in the IC phase as calculated from model
(M1a), however, with the incommensurate khex,i. For the
calculation we assumed that only one single chiral domain
is populated. (b) Measured integrated intensities for the Q-
scan in Fig. 5(a) in the different polarization channels. For
the comparison see the text.
934(31) 946(31) 65(8)
integrated over ±khex,i
B.Chirality in the incommensurate magnetic
Fig. 5(a) shows polarized constant-energy-scans along
the L-direction performed around the reciprocal space
position (0, 0, -1.5) at T = 1.5 K in the IC-phase. Four
different polarization channels were measured, namely
Ixx, Ix−x, I−xx and I−x−x. The scans for the chan-
Neutron Intensity (counts/sec)
(0, 0, L)(r.l.u)
σ(P= +ex)−σ(P=−ex) (counts/sec)
FIG. 5: (a) Q-scans in the polarized mode over the magnetic
Bragg satellites (0,0,0)-khex,iand 0,0,-3)+khex,iat T = 1.5 K
are shown. (b) The difference for intensities with the initial
polarization vector directed parallel and antiparallel to Q for
the same Q-scan are shown. This difference is directly propor-
tional to the chiral contribution to the scattering cross-section
as shown in Eq. (9). We note, that the installed supermirror
benders simultaneously act as 100’ collimators in front of and
behind the sample, respectively.
nels Ix−xand I−xxappear to be slightly shifted towards
the positions of the magnetic satellite reflections (0,0,-
3)+khex,i (Ix−x) and (0,0,0)-khex,i(I−xx), respectively,
in agreement with the observations made in the high res-
olution measurement in section IIIB.
The effect is more significant in the sum of the four terms
∆I = (Ixx+ Ix−x) − (I−xx+ I−x−x)
= σ(P0x= +1) − σ(P0x= −1),
that is equivalent of measuring the difference of the two
polarized neutron cross-sections with the initial polariza-
tion vector P0directed parallel and antiparallel to the x
direction (thus parallel or antiparallel to Q). The result-
ing scan is shown in Fig. 5(b).
By means of Eqns. (6) and (8) we further obtain the re-
∆I = (|M⊥|2+ C) − (|M⊥|2− C)
demonstrating that ∆I is proportional to the chiral term.
Thus, by carrying out the polarized scans we were able to
separate the resolution limited peak at (0,0,-1.5) in two
satellites (0,0,0)-khex,i(I−xx) and (0,0,-3)+khex,i(Ix−x).
Since two distinct extrema with opposite signs for the
two satellites can be distinguished the collinear antifer-
romagnetic structure of NdFe3(11BO3)4observed in the
commensurate phase appears to transform to an antifer-
romagnetic long-period spiral below TIC. In addition the
non-zero chiral term also suggests that at least unequally
populated chiral domains are present in NdFe3(11BO3)4.
In order to investigate this in more detail we performed
calculations based on the magnetic model (M1a), how-
ever together with the incommensurate propagation vec-
tor khex,ithat leads to a small rotation of the magnetic
moments between neighboring layers. We calculated the
integrated intensities for the four measured polarization
channels Ixx, Ix−x, I−xx and I−x−x for both satellites
(0,0,0)-khex,i and (0,0,-3)+khex,i, where we used ε =
0.0667 in khex,i= [0, 0,3
2+ ε] as deduced from our high
resolution diffraction data. The calculated integrated in-
tensities are given in table III(a) for the case of only a
single chirality domain being populated. In addition we
assumed only one orientation domain, similar as for the
unpolarized results. For comparison we give the mea-
sured integrated intensities from the scans for the four
polarization channels in table III(b). Due to the limited
resolution of the setup the measured values for the in-
dividual polarization channels are integrated over both
peaks and cannot be compared directly to the calcula-
tions. This is especially true for the difference of the
intensities. However, two reasonable assumptions can be
made to allow for a comparison:
(i) the measured integrated intensity in the channel
‘−xx’ is only due to the peak at (0,0,0)-khex,i
whereas the channel ‘x−x’ is only due to the peak
at (0,0,-3)+khex,i. This assumption is justified as
the measured points of each respective polarization
channel (red squares and blue circles in Fig. 5(a))
are shifted towards the direction of the correspond-
(ii) the channels ‘xx’ and ‘−x − x’ are equally con-
tributed by both satellite reflections as they are
centered on top of each other and between the other
two channels (black circles and green triangles in
Based on these assumptions we are able to calculate the
xx/2) = 28(4) for the intensities measured
on the magnetic Bragg reflection (0,0,-3)+khex,i. Note
that the division by 2 is due to assumption (ii). The same
ratio for the calculated intensities for this peak amounts
xx= 24. For the satellite (0,0,0)-khex,iwe ob-
tain the measured ratio Im
calculated ratio Ic
−x−x= 24. This indicates that
our model of a long-period antiferromagnetic helix propa-
gating along the hexagonal c-axis with the magnetic mo-
ments parallel to the hexagonal basal plane and single
−x−x/2) = 28(4) and the
chirality domain is in good agreement with our data.
In addition we verified our assumptions by performing
a convolution of the calculated integrated intensities for
each of the four polarization channels (cf. table III) with
the four-dimensional resolution function of the spectrom-
eter in a simulation. The results of the simulations are
the solid lines in Fig. 5(a). To match the intensity a
single scale factor four all four polarization channels was
introduced in the calculation. Similar as for the previous
calculations the simulation was performed with only one
of the two chiral domains being populated, and a sin-
gle orientation domain. The solid line in Fig. 5(b) was
obtained by summating the individual curves for each
polarization channel with respect to Eq. (8). The pecu-
liar shape of difference curve is due to the slightly asym-
metric shape of the magnetic satellite peaks. After the
asymmetric peak shape was taken account for within the
simulation good agreement between simulation and ex-
perimental data was achieved.
The present single-crystal neutron diffraction investi-
gations did not detect significant deviations from space
group R32 concerning the chemical structure of mul-
tiferroic NdFe3(11BO3)4 at low temperatures.
respect to magnetic ordering this study shows that
only the magnetic model (M1a) is in agreement with
our polarized neutron data on the noncentrosymmetric
However, in contrast to the previous neutron powder
diffraction study of NdFe3(11BO3)431, we may conclude
from our combined investigation with unpolarized and
polarized neutrons that in addition, ferromagnetic align-
ment of the magnetic Fe and Nd sublattices holds. This
implies low magnetic symmetry such as R12 (monoclinic
C2) in the commensurate phase. Our conclusion is fur-
ther supported by measurements of the magnetic suscep-
tibility by Tristan et al.48that yield an easy magnetiza-
tion along the a-axis. In addition, we should emphasize
that not only Fe3+, but also the Nd3+ions show antifer-
romagnetic long-range order below the N´ eel temperature
in case of neodymium ferroborate.
Concerning the incommensurate phase the polarized Q-
scans over the positions of the magnetic peaks could be
well explained via the magnetic model (M1a) that was
found for the commensurate phase simply by introducing
the incommensurate propagation vector khex,i. This sug-
gests that the magnetic structure transforms into a long-
period antiferromagnetic helix that propagates along the
hexagonal c-axis with the magnetic moments perpen-
dicular to it. The incommensurate magnetic propaga-
tion vector khex,i= [0, 0,3
2+ ε] is therefore associated
with a rotation of the magnetic moments about 180◦+
γ around the c-axis between adjacent hexagonal planes
that are interrelated via trigonal translations. The mea-
sured value of the splitting ε = 0.00667 corresponds to
γ ≈ 0.8◦and the full period of the helix amounts to ap-
proximately 1140˚ A. The mere observation of a chiral
contribution by means of polarization analysis signifies
unequally populated chirality domains. Our data further
suggest that only one of the two chirality domains is pop-
ulated. The antiferromagnetic helix in the IC phase of
NdFe3(11BO3)4therefore exists with an unique handed-
ness. A single chirality domain is in principle not ex-
pected, since left- and right-handed spirals are energeti-
cally degenerate, however in the case of NdFe3(11BO3)4
this might be related to the fact that the chemical struc-
ture is non-centrosymmetric. This is similar to the mag-
netic spirals in MnSi or UPtGe52, and the more recent
example Ba3NbFe3Si2O1422that all three possess no in-
version symmetry. In particular the Jana2006 analysis
has shown that in addition the incommensurate magnetic
structure exists in a single magnetic orientation domain
with full superspace group symmetry R32(00γ)t0 and in
full agreement with the polarized results.
The determined saturation value of the magnetic Fe mo-
ment of approximately 4.2 µB (see Fig. 4) is less than
5 µB which would be expected for a free Fe3+ ion, in
contrast to our previous powder results31. Moreover, the
ferromagnetic alignment of the Fe and Nd magnetic mo-
ments implies considerably smaller magnitudes for Nd
than derived from the powder diffraction data. The latter
are not affected by extinction. The difference may be to
a certain extent due to the extinction effects, as the even
smaller saturation value of Fe in case of thermal neutrons
is caused by the considerably larger extinction, compared
to hot neutrons. On the other hand, for TbFe3(11BO3)4
at 2 K, Ritter at al. determined by means of powder neu-
tron diffraction an antiparallel alignment of the Fe and
Tb magnetic moments with ordered magnitudes µFe =
4.39(4) µB and µTb= 8.53(5) µB53. Despite the bond-
valence result 3+ for Fe derived by these authors, the
ordered Fe magnetic moment is also reduced. Hence, an-
other possible reason could be frustration effects, yielding
partially disordered magnetic moments.
Furthermore, the observation of third order harmonics of
the magnetic satellites at the positions (0, 0, 3/2 ± 3ε) in
the incommensurate phase additionally suggest the for-
mation of a magnetic soliton lattice in NdFe3(11BO3)446.
A soliton is the appearance of localized or topologi-
cal defects in periodic structures due to the presence
of non-linear forces.Such non-linear forces can be
due to an external magnetic field that interacts with
the magnetic moments or due to magnetic anisotropy
as shown theoretically by Izyumov and Laptev54. In-
deed NdFe3(11BO3)4exhibits magnetic anisotropy in the
hexagonal basal plane as demonstrated by the results of
Tristan et al.48. The calculation of Izyumov and Laptev
is based on a magnetic helix that forms due to the the
presence of the Dzyaloshinsky-Moriya interaction (DMI).
Until now no explicit statement about the existence of
the DMI in NdFe3(11BO3)4has been made. However, as
NdFe3(11BO3)4 is non-centrosymmetric the presence of
the DMI is allowed from symmetry. Therefore, the for-
mation of the observed magnetic helix is possibly driven
by the DMI.
In Figs. 3 (b) to (d) we see that the intensities of
the second order satellites are highest for temperatures
T ? 13.5 K and the distortions of the incommensurate
periodic structures seem to be largest near to the C-
IC phase transition. We therefore assume that at TIC
the interaction that favors a magnetic order that is in-
commensurate with respect to the underlying crystal lat-
tice becomes non-negligible and leads to non-linear forces
onto the magnetic subsystem slightly below TIC as it
still wants to remain in its commensurate magnetic order.
Within a small temperature regime below TICthe mag-
netic structure consequently is not yet completely incom-
mensurate but can be rather viewed as a distorted com-
mensurate magnetic structure with domain walls. Al-
ternating periods of commensurate parts and domains
walls then lead to the observed third order harmonics.
The observation of a magnetic soliton lattice without the
application of external forces like magnetic fields or me-
chanical stress are rather unlikely and to the best of our
knowledge the only other compound for which a magnetic
soliton lattice was reported without the application of an
external magnetic field is CuB2O446. The observed tem-
perature dependence of the propagation vector is contin-
uous and described by Eq. (1). This behavior is close to
k(T) ∝ |(TIC− T)|0.48reported in Ref. 46. In addition,
similar to CuB2O4 the commensurate phase is realized
when the temperature is increased, which is in contra-
diction to the prediction of the theory54. For CuB2O4it
was proposed that the difference to the theory can be ex-
plained by assuming that the change of the propagation
vector is not due to a temperature dependent magnetic
anisotropy as in Ref. 54 but rather due to the magnitude
of the DMI that decreases as a function of increasing
temperature55. We assume that is is similarly true for
NdFe3(11BO3)4. In summary our experimental results
are well described by the assumption of a magnetic soli-
The origin of the less pronounced incommensurability in
case of the small crystal studied (Sample2) in the present
work, compared to the larger one (Sample1) that was also
used in Ref. 31, is not yet clear and should be clarified
by future systematic investigations of possible sample de-
Our neutron diffraction results show that the long-
range magnetic order of multiferroic NdFe3(11BO3)4ob-
served below TN ≈ 30 K consists of antiferromagnetic
stacking along the c-axis, where the magnetic moments
of all three Fe3+sublattices and the Nd3+sublattice
are aligned ferromagnetically and parallel to the hexag-
onal basal plane, corresponding to model (M1a). Below
TIC ≈ 13.5 K the magnetic structure turns into an in-
commensurate antiferromagnetic helix propagating along
the c-axis with a period of approximately 1140˚ A.
Our polarized neutron diffraction data further suggests
that the helix is monochiral, i.e. only one of the two
possible chiral domains is fully populated. The single
magnetic chirality in neodymium ferroborate can be ex-
plained in terms of its non-centrosymmetric chemical
structure, similar as for MnSi52, and the more recent ex-
ample Ba3NbFe3Si2O1422. To the best of our knowledge
the former two materials are the only examples apart
from NdFe3(11BO3)4that show this peculiar property.
In the case of NdFe3(11BO3)4 the commensurate-
incommensurate magnetic phase transition is possibly ac-
companied by the formation of a magnetic soliton lattice,
as indicated by the observation of third order harmonics
of the magnetic Bragg peaks. This further suggests that
the magnetic helix in NdFe3(11BO3)4may be driven by
the Dzyaloshinskii-Moriya interaction, which would be
allowed by symmetry.
In conclusion, we identified the new monochiral com-
pound NdFe3(11BO3)4that provides us with a new model
system to investigate the interesting properties of mag-
netic chirality in condensed matter.
Partially this work has been performed at the Swiss
spallation neutron source SINQ38(instruments TriCS37,
TASP44, MuPAD45) and another part at the single crys-
tal neutron diffractometer HEiDi39for hot neutrons, sit-
uated at the Forschungsneutronenquelle Heinz Maier-
Leibnitz (FRM II). Further we are thankful to Seve-
rian Gvasaliya for providing experimental support for the
measurements performed on TASP. MJ is grateful to Ben
Taylor and Dominik Bauer for useful discussions.
Appendix A: Characteristic results from FullProf
Here we want to discuss the results of our magnetic
refinements for NdFe3(11BO3)4by means of FullProf in
more detail. We have chosen several characteristic data
sets that highlight the results of the fits provided in ta-
First we will discuss the data sets measured on HEiDi
with hot neutrons (λ = 0.55˚ A). For the commensurate
magnetic phase of NdFe3(11BO3)4we performed fits with
φNdfixed to zero and free, respectively. The correspond-
ing sets in table IV are (F1a) and (F1b) at T = 22.5 K,
respectively. The refinements were carried out with the
nuclear scale factor and extinction parameter 2.9(1) that
was determined by the fit (N1) (cf. table I). The deter-
mined angle φNdfor (F1b) is of similar magnitude as the
value 76(3)◦of the previous powder neutron diffraction
results31. However, according to Hamilton’s signficance
test56, one should consider the ratio 1.01 of the weighted
R-factors. Then R1,79,0.25≈1.01 from table 1 in Ref. 56
TABLE IV: Magnetic refinements of NdFe3(11BO3)4 by means of FullProf34. Here the three character encoding (column set
in the table) of the different fit runs is as follows. The first letter ’F’ signifies that the fit was performed with FullProf (cf.
Jana2006 refinements in table V), the number at second position indicates which data set has been used, the characters at the
third position characterize constraints that were applied. Here a and b indicate that the polar angle φNdwas fixed to zero and
left free for the fits, respectively (cf. cases M1a and M1b in section IIIC).
Instrument T (K) Phase # refl.SetRm,F2(%) Rm,F2w(%) Rm,F(%) χ2
µ(Fe) (µB) µ(Nd) (µB) φNd(◦) # domains
HEiDi 15C 50F2a
TriCS5.5 IC152 F4a
1φNd was fixed at the given value different from φNd = 0.
indicates only a marginal significance level of 0.25 that
(F1b) is the correct result. Further refinements at lower
temperatures support this as will be demonstrated in the
following. Further, variation of also the extinction pa-
rameter yields the value 3.0(8) and χ2= 1.07. Therefore
in case of NdFe3(11BO3)4the nuclear and magnetic ex-
tinction parameters agree within error limits. Thus we
use the former also for the refinements of the subsequent
’small’ HEiDi data sets with 5 nuclear and 50 magnetic
peaks. The former were used to obtain the scale factor.
At T = 15 K we performed three different sets of fits
where the angle φNdwas consequently fixed to 0, 15 and
30◦(cf. sets (F2a,F2a’,F2a”) in table IV). There is ap-
parently a very flat minimum centered at φNd= 0 which
proves the assumption φNd= 0 to be correct.
The same is valid for the incommensurate magnetic phase
of NdFe3(11BO3)4. Fits of the data at 5 K show bet-
ter agreement factor when φNdis fixed to zero (cf. sets
(F3a+b) in table IV).
Finally, the results obtained by fits of the TriCS
data measured with thermal neutrons (λ = 1.18˚ A) at
T = 5.5 K gives the following results. The nuclear fit (N2)
yielded the scale factor 40(1) and a considerably larger
extinction parameter 18(1), compared to the hot neutron
HEiDi data. Best fits of the magnetic neutron intensities
had been obtained with temperature parameter B = 0 of
the metal atoms. The positional x-parameter of Fe had
been taken from (N2), table I. Good fits were only ob-
tained by refining also the extinction parameter, resulting
in the considerably smaller magnetic value 2.3(2), com-
pared to the almost an order larger nuclear value. The
results of the refinement of the magnetic intensities is
given in table IV sets (F4a) and (F4b), corresponding
again to fixed and free polar angle φNd, respectively. For
(F4b) the extinction parameter was fixed to the value
from (F4a). The latter does not change essentially, if it
is varied too. Although refinement (F4a) seems with an
additional parameter somewhat better, we think that it
is hardly significant with respect to the comparable hot
neutron 5 K results, which in principle are considerably
less affected by extinction.
Appendix B: Characteristic results from Jana2006
In the following we will discuss our additional refine-
ments that were performed with Jana200635. Recently
the option to refine magnetic structures has been imple-
mented. It should be emphasized that magnetic struc-
tures have to be defined in a different way in Jana2006
compared to FullProf. In Jana2006 commensurate and
incommensurate magnetic structures are described with
superspace groups43similar to occupationally modulated
chemical structures.The modulation function of the
magnetic axial vector configuration is represented by
means of a Fourier expansion according to the observed
k-vectors, similar to Ref. 47. Moreover, the irreducible
representations of the chemical structure with their lit-
tle group Gkand the associated magnetic basis functions
as well as time inversion are taken into account. The
atomic magnetic moments are described in a polar coor-
The incommensurate and commensurate magnetic struc-
tures of NdFe3(11BO3)4 correspond to a complex one-
dimensional and to a two-dimensional representation,
respectively.Keeping the chemical structure accord-
ing to space group R32, the former is described in by
means of a four-dimensional superspace approach with
cosine and sine waves associated with the k-vectors
khex,i = [0,0,±1.502].
the magnetic Fe and Nd moments in the ’easy’ (a,b)-
plane of NdFe3(11BO3)4, the magnetic superspace group
R32(00γ)t0 was used. With the Nd moments deviating
For ferromagnetic coupling of
TABLE V: Characteristic magnetic refinements by means of Jana200635. Here the three character encoding (column set in
the table) of the different fit runs is as for table IV). Thus, the first character ’J’ indicates that the fits were performed with
Jana2006. E.g. (J3x) means that the data is identical to the data set for the case (F3x) shown in table IV.
Instrument T (K) Set superspace group Rm(all) Rmw(all) GOF(all) GOF(obs) µ(Fe) (µB) µ(Nd) (µB) φNd(◦) domains
J1b14.112.11.02 1.133.044(3) 1.29(2) 73.7(2)6
J2b6.310.0 1.62 1.65 3.782(3)1.27(3) 65.2(5)6
from φFe = 0 in the (a,b)-plane, magnetic superspace
group R3(00γ)t holds. Here the symmetry implies one
or two magnetic domains, respectively. In the latter case
they were found to be statistically populated.
In the magnetic commensurate case with khex =
[0,0,3/2], being equivalent to the two-fold superstruc-
ture, only a cosine wave with magnetic superspace group
R12(α,α,γ)00 (α=β=0;γ=3/2) is appropriate for ferro-
magnetic alignment of the magnetic Nd and Fe moments
in the (a,b)-plane. For magnetic monoclinic superspace
group R1(α,β,γ)t (α=β=0;γ=3/2) the magnetic Nd and
Fe moments are not oriented parallel in the (a,b)-plane.
The two cases imply three or six magnetic domains, re-
spectively. They were found to be statistically populated.
As an example we discussed the results of combined re-
finement of 5 nuclear (to obtain the scale factor) and 50
magnetic peaks at 15 K (HEiDi measurements) that are
given in table V. Here we used the extinction parameter
RhiIso = 0.109(8) and the atom positions from table I.
(J2a) and (J2b) correspond to fixed and free polar angle
φNd. The resulting R-factors of the refinement for this
temperature show obviously no significance for an addi-
tional parameter φNdand again indicate that the ferro-
magnetic alignment of the Fe and Nd magnetic moments
For the data at 5 K a non-zero angle φNdshows a small
improvement of the refinement (J3b), however, in view of
the additional refinement parameter φNd, one may ques-
tion whether the small improvement is significant. Based
on Hamilton’s significance test56, the Rmw-ratio 8.2/7.9
= 1.04 should be considered. In cases (J3a) and (J3b) we
have 2 and three parameters, respectively and 55 reflec-
tions, i.e R1,52,0.05≈ 1.04 from table 1 in Ref. 56 indicates
the correctness of φNd=0 (J3a) at a significance level of
On the other hand (J4b) appears to be better than (J4a).
Presumably this is related to the considerably larger ex-
tinction effects associated with thermal neutrons com-
pared to hot neutrons. Apart from this temperature, the
Jana2006 data evaluation yields values for the magnetic
moments that agree within the error bars with the values
shown in Fig. 4 which is based on the FullProf evaluation.
It should be noted that the superspace group description
yields a natural explanation for the number of magnetic
domains49. The latter were found to be statistically oc-
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