The influence of demixing on the dynamics of ionic solids: inelastic neutron scattering from AgxNa1 xCl single crystals

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INSTITUTE OF PHYSICS PUBLISHING
JOURNAL OF PHYSICS: CONDENSED MATTER
J. Phys.: Condens. Matter 15 (2003) 6415–6426 PII: S09538984(03)637793
The influence of demixing on the dynamics of ionic
solids: inelastic neutron scattering from AgxNa1−xCl
single crystals
D Caspary1, G Eckold1, P Elter1, H Gibhardt1, F G¨ uthoff1, F Demmel2,
A Hoser3,4, W Schmidt2,4and W Schweika4
1Institut f¨ ur Physikalische Chemie, Universit¨ at G¨ ottingen, Tammannstrasse 6,
D37077 G¨ ottingen, Germany
2Institut Laue–Langevin, 6 rue Jules Horowitz, BP 156, F38042 Grenoble cedex, France
3Institut f¨ ur Kristallographie, RWTH Aachen, D52056 Aachen, Germany
4Institut f¨ ur Festk¨ orperforschung, Forschungszentrum J¨ ulich, D52425 J¨ ulich, Germany
Received 21 May 2003
Published 8 September 2003
Online at stacks.iop.org/JPhysCM/15/6415
Abstract
The dynamics of mixed single crystals of AgxNa1−xCl has been investigated
by inelastic neutron scattering before and after chemical demixing. In the
homogeneousphase,theconcentrationdependenceofacousticphononsreveals
that doping of NaCl with silver chloride leads to a considerable softening
of the lattice, while the elastic properties of AgCl are almost independent
of sodium chloride additives.After quenching into the miscibility gap,
the phase separation is associated with a welldefined splitting of acoustic
phonons which provide the most direct information about the underlying
mechanism. In contrast to the local dynamical properties, the lattice structure
is essentially determined by coherency strains which hinder the relaxation of
lattice parameters. Thus diffraction and inelastic scattering yield independent
and complementary information about demixing processes in ionic solids.
1. Introduction
As shown in two previous papers [1, 2], mixed ionic systems of the silver–alkali halide type
are almost ideal systems for the study of demixing processes on a microscopic level. Simple
phase diagrams along with large ionic mobilities and the almost rigid anion sublattice make
these mixed crystals particularly suitable for kinetic investigations. Usually constituents like
AgClandNaClorAgBrandNaBrexhibitdifferentlatticeparametersand,consequently,lattice
strains play an important role during decomposition. In earlier studies [3–5], the splitting of
Braggreflectionshasbeenregardedasatooltomonitorthephaseseparation. Fromthesluggish
variation of the lattice structure as reflected by changes of the Bragg profiles or the beginning
opacity [6–8] it was concluded that demixing in silver–alkali halide systems takes place on a
rather long timescale up to weeks or months. Recent studies using diffraction in combination
09538984/03/376415+12$30.00© 2003 IOP Publishing LtdPrinted in the UK6415
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6416D Caspary et al
Table1. ElasticpropertiesofthepurecompoundsAgClandNaClatambienttemperatureaccording
to [10].
c11
c12
c44
AgCl
(ρ = 5.589 g cm−3,
a = 553.9 pm)
NaCl
(ρ = 2.164 g cm−3,
a = 563.3 pm)
Elastic constant (GPa)
Temperature coefficient (10−4K−1)
Pressure coefficient (10−12Pa−1)
59.6
−10.1
183
36.1
−3.8
121
6.22
−4.28
−81.2
12.8
−2.2
29
Elastic constant (GPa)
Temperature coefficient (10−4K−1)
Pressure coefficient (10−12Pa−1)
49.1
−7.8
239
12.8
4.7
183
withsmallangleneutronscattering[1,2],however,indicatedthatthephaseseparationappears
to bemuchfaster, withinsecondsorminutes. It was arguedthatdemixingmostprobablytakes
place within an almost rigid lattice provided by the anions. The lattice relaxation is regarded
as a subsequentand almost independentstep in the decompositionprocess. Hence, diffraction
alone seems to be unable to monitor the phase separation. Interestingly, the decomposition
can be investigated even in single crystals without destroying the coherent lattice [9]. This
observation offers the possibility for more detailed studies of the microscopic behaviour.
The time evolution of phonons, in particular, can be used as the most direct probe for local
interatomic interactions and their changes during demixing. Due to the limited free path of
phonons these provide evidence for local properties within some 50 Å.
Accordingto[10],theelasticconstantsofAgClandNaCldifferconsiderably,thusoffering
thepossibilityofobservingchangesinlocalinteractionsandlocalsoundvelocitiesbyinelastic
neutron scattering. The elastic properties of the pure compounds are collected in table 1. In
particular, the shear constant c44of AgCl and NaCl differs by almost a factor of 2. Hence, the
phaseseparationis expectedto beassociated witha welldefinedsplittingofphononbranches.
In this paper, we report on the spectra of transverse acoustic phonons in mixed crystals
of AgCl–NaCl before and after demixing.
miscibility gapwith a critical temperatureof about 198◦C [11]. It will be shown that phonons
unambiguously prove that the phase separation and the lattice relaxation are two different
aspects of decomposition.
Even the kinetics of the demixing process can be examined by timeresolved phonon
experiments. The corresponding results will be presented in a subsequent paper.
This system exhibits an almost symmetrical
2. Experimental details
Large (cm3) single crystals of AgxNa1−xCl were grown from the melt by the Czochralski
method in an N2 atmosphere using a ceramic crucible (FYZ Friatech) and commercially
available agents (AgCl: Acros 99.9%, NaCl: Merck >99.5%). Due to the unfavourable
shape of the liquidus–solidus curves, ingots were highly enriched with AgCl (x0≈ 0.8–0.9)
in order to obtain single crystals with intermediate silver concentrations. Either pure NaCl
crystals (Korth Kristalle GmbH) or sodiumrich fragments of preceding growth experiments
with[110]orientationwereusedasseedcrystals. Typicalgrowthratesoflessthan0.3mm h−1
yielded crystals with volumes of several cubic centimetres. A typical example is shown in
figure 1. In orderto minimize possible concentrationgradients, the crystals were annealed for
about 24–48h above400◦C. For the determinationof the actual concentration,three different
small pieces were cut from the crystals along their growth direction and analysed by atomic
absorption spectroscopy and by xray diffraction (within the homogeneous phase at 400◦C).
Page 3
The influence of demixing on the dynamics of ionic solids6417
Figure 1. Single crystal of Ag0.41Na0.59Cl as grown by the Czochralski technique.
The data prove the good homogeneity of the crystals. For the present experiments, crystals
with AgCl concentrations of x0= 0.23, 0.26 and 0.41 were used. Their quality was checked
by gammaray diffraction yielding mosaicities of less than 1◦.
Specially designed furnaces were developed which were optimized for rapid cooling
and heating. Details are described in [12]. The crystals were wrapped in silver foil in
order to guarantee the temperature homogeneity. All experiments were performed in normal
atmosphere.
The neutron scattering experiments were performed at the three axes spectrometers
UNIDAS(FRJ2, J¨ ulich),IN3andIN12(HFR,ILLGrenoble)usingdifferentcombinationsof
neutronenergy,collimation,etc. ForadditionaldiffuseelasticexperimentstheDNSinstrument
in J¨ ulich was used.
3. Results
3.1. Acoustic phonon dispersion in the homogeneous phase
As a typicalexample,figure2 showsaTA phononalongthe[100]directionat 400◦C obtained
for a mole fraction x0 = 0.41 of silver chloride.
for q = (0.2 0 0) lies in between the values of the pure components as indicated by the
arrows. The latter (νNaCl= 0.84 THz and νAgCl= 0.35 THz) are obtained by extrapolation
from room temperature data, e.g. [13, 14], using the temperature coefficients of the elastic
The phonon frequency of 0.50 THz
Page 4
6418D Caspary et al
Figure 2. Typical TA phonon spectrum of an Ag0.41Na0.59Cl mixed crystal in the homogeneous
phase at 400◦C for the wavevector q = (0.2 0 0).
Table 2. Combinations of elastic constants as obtained by the initial slopes of the acoustic phonon
branches of AgxNa1−xCl at 330◦C.
TA[100]TA[110]TA[111]LA[100]
c44
c44
c11− c12+ c44
x0
(GPa) (GPa) (GPa)(GPa)
TA[110]
c11+ c12+ 2c44
(GPa)
LA[111]
c11+ c12+ 4c44
(GPa)
c11
0.23
0.26
0.41
12.4
11.1
8.2
11.4 39.262.6 110155
constant c44(see table 1). The phononline shape of the mixed crystal is very well represented
byaLorentzian. Thelinewidthofabout0.10THzis onlyslightlylargerthantheexperimental
resolution.
Figure 3 displays the lowfrequencypart of the acoustic phonon branches along the main
symmetry directions [100],[110]and [111]for a mixed crystal with x0= 0.26 at 330◦C. The
lines represent a fit using a polynomial of order 2. From the initial slopes at q = 0, the elastic
constants of the mixed crystal are obtained. The results are collected in table 2 along with
some data for two other compositions.
For the x0 = 0.26 sample, we were able to determine the whole set of three elastic
constants from six different phonon branches consistently:
c11= 58 ± 7 GPa
c12= 29 ± 6 GPa
c44= 11.3± 0.2 GPa.
For the other two concentrations, we focused our attention on the TA[100] phonon in
order to determine the concentration dependence of phonon frequencies. For q = (0.3 0 0)
the corresponding frequencies are collected in table 3.
The data of the pure compounds were obtained from the literature, e.g. [13, 14], and
are extrapolated to 330◦C. The data of the x0= 0.41 sample are determined at 400◦C and
Page 5
The influence of demixing on the dynamics of ionic solids6419
Figure 3. Dispersion of the lowenergy part of transverse (•) and longitudinal ( ) acoustic
phonon is polarized along [001].)
phonons in Ag0.26Na0.74Cl at 330◦C along the main symmetry directions. (Note that the TA[110]
Table 3. Concentration dependence of TA phonon frequencies at 330◦C and q = (0.3 0 0). The
experimental error is estimated as 0.03 THz.
x0
νTA[0.3 0 0](THz)
0
0.23
0.26
0.41
1
1.25
0.95
0.88
0.69
0.51
corrected with the help of the temperaturecoefficient of c44. Note that the frequencyshift due
to the higher temperature is only about 1%, which is smaller than the experimental error of
about0.03THz. Obviously,thereis anonlinearvariationoffrequencieswithcomposition. In
sodiumrichmixedcrystals(x < 0.5) thefrequencychangesbyalmost50%, whileit becomes
almost independent of composition in silverrich compounds.
3.2. Acoustic phonons in demixed crystals
Afterquenchingtoroomtemperatureandageingformorethansixmonths,AgxNa1−xClsingle
crystals exhibit phononspectra as shown in figure 4 for a phononwavevector of q = (0.2 0 0)
and two different initial concentrations x0. Besides an incoherent elastic peak, there are two
different and welldefined phonons which are observed in neutron energy gain (ν < 0) as
well as in neutron energy loss (ν > 0). The inelastic peak at about ±0.41 THz is due to the
TA phonon of the silverrich phase, while the phonon of the sodiumrich phase is observed at
±0.85 THz. The line widths of both phonons are only slightly larger than the experimental
Page 6
6420D Caspary et al
Figure 4. TA phonon spectra, q = (0.2 0 0), after ageing for more than six months at room
temperature with x0= 0.23 (top) and x0= 0.41 (bottom). The horizontal arrows represent the
experimental resolution.
resolution, thus indicating that chemical decomposition is almost completed. This finding is
supported by the fact that the phonon frequencies of the product phases are independent of
the initial overall concentration x0. Inspection of figure 4 shows that spectra for x0= 0.23
and 0.41 differ only in the relative phonon intensities but not in the peak position since the
concentrations of the coexisting phases (x1and x2) are the same in both cases.
The ratio of the phonon intensities is given by the volume fractions and the dynamical
structure factors of the corresponding phases. Neglecting differences in the Debye–Waller
factors and using the hightemperature approximation (hν/kBT ? 1) and the lever rule we
obtain
I1
I2
x0− x1
=x2− x0
σcoh
1
σcoh
2
?ν2
ν1
?2
(1)
Page 7
The influence of demixing on the dynamics of ionic solids6421
Figure 5. Dispersion of TA[100] phonons in demixed crystals of different overall compositions.
with
σcoh
i
= 4π(xibAg+ (1 − xi)bNa+ bCl)2
(2)
where bAg= 0.592 × 10−12cm, bNa= 0.363 × 10−12cm and bCl= 0.958 × 10−12cm are
the coherent scattering lengths of the individual elements.
If one assumes that the concentrations of the product phases are given by the binodal as
obtained by Sinistri et al [11], the two phonons correspond to x1 = 0.05 and x2 = 0.95,
respectively. Using these data in equation (1), the intensity ratio is calculated to be 0.69 and
0.26 for the sample with x0= 0.23 and 0.41, respectively, in reasonable agreement with the
data in figure 4.
The lowenergy part of the TA phonon dispersion along [100] is shown in figure 5.
Obviously, the results of three demixed crystals with different overall concentrationscoincide
forallwavevectors. Thisfindingyieldsanotherindicationthatthechemicaldemixingisalmost
complete and all crystals consist of product phases with compositions close to the binodal.
After extrapolation to 330◦C, the phonon frequencies of these product phases can be
used along with the data of table 3 to complete the concentration dependence of νTA. All
available data are collected in figure 6 which shows that even small amounts of silver chloride
introducedintothe NaCl host lattice lead toa substantial softeningof thelattice. Interestingly,
the shape of ν(x) resembles the solidus curve as obtainedby Sinistri et al [11]. This findingis
consistent with the phenomenologicalansatz introduced by Ubbelohde [15] which relates the
shear module to the melting temperature.
This nonlinear behaviour may be discussed on the basis of elementary lattice dynamics
of the NaCl structure.
For acoustic phonons in the vicinity of the ?point, the polarization vectors u of cations
and anions are identical. Moreover, the dynamical matrix which is the Fourier transform of
the force constant matrix can be expanded in a power series of the wavevector q. The linear
range of the acoustic branch can thus be described by the sound velocity:
?2πν
κ
q
?2
=
1
M
?
?
κ?
?
l?
(u · Vκ,κ?l? · u) ·(q · rl?)2
q2
=
1
M
?
κ
?
κ?
?
l?
(u · Vκ,κ?l? · u) · z2
l?.
(3)
Page 8
6422D Caspary et al
Figure 6. Concentration dependence of the TA phonon frequency at q = (0.3 0 0). The curve is
the result of a fit to equation (6).
Vκ,κ?l? is the force constant matrix describing the interaction between particles of types κ and
κ?within different primitive cells at a distance rl?, zl? is the component of rl? along q and M
is the total mass per primitive cell. In the NaCl structure, κ is + for the cation and − for the
anion and, hence,
?2πν
4
ρa3
l?
Here, the abbreviation V±±,l? = u · V±±,l? · u is used for the projection of the force constant
matrix onto the direction of polarization, while Vc= a3/4 and ρ = (m++ m−)/Vcare the
volume of the primitive cell and the density, respectively.
In a homogeneous AgxNa1−xCl mixed crystal, silver and sodium ions are statistically
distributed among the sites of the cation sublattice. Hence, we can use the mole fraction x as
the occupation probability for the silver ions and obtain
?2πν
+ 2(1 − x)VNaCl,l? + VClCl,l?]z2
?
+ 2x
l?
?
q
?2
=
1
(m++ m−)
?
[V++,l? + 2V+−,l? + V−−,l?]z2
l?
[V++,l? + 2V+−,l? + V−−,l?]z2
l?
=
?
l?.
(4)
q
?2
=
??
l?
[x2VAgAg,l? + (1 − x)2VNaNa,l? + 2x(1 − x)VAgNa,l? + 2xVAgCl,l?
?
x2?
?
(VNaNa,l? + 2VNaCl,l? + VClCl,l?)z2
l?
(xmAg+ (1 − x)mNa+ mCl)−1
=
l?
(VAgAg,l? + VNaNa,l? − 2VAgNa,l?)z2
l?
(VAgNa,l? − VNaNa,l? + VAgCl,l? − VNaCl,l?)z2
l?
+
l?
l?
?
(xmAg+ (1 − x)mNa+ mCl)−1.
(5)
Page 9
The influence of demixing on the dynamics of ionic solids6423
If q is expressed in reciprocal lattice units (q = 2πξ/a) and zl? is given in fractions of the
lattice parameter (zl? = αl?a):
?ν
The quantity
?
describes the effect of a single silver ion substituted in NaCl, while
?
is the corresponding quantity for a single sodium ion within the AgCl matrix. If we assume
that the effectivechargesof silver and sodiumions are the same, both?VAgand ?VNadepend
on shortrange interactions only. In phenomenologicalmodels, these are usually described by
a Born–Mayer potential or, more adequately, by shell models taking into account electrical
polarization effects. For the lattice dynamical descriptionof the pure compoundsthe reader is
referred to [13] and [14].
Note that
?
depends on the cation–cation interactions only and is thus concerned with next nearest
neighbour interactions.
The last term on the righthand side of equation (6) is related to the elastic constant cNaCl
of the NaCl lattice via the relation
?
while for pure silver chloride the corresponding quantity is
ξ
?2
=
1
(xmAg+ (1 − x)mNa+ mCl)[−x2(?VAg+ ?VNa) + 2x?VAg+ VNaCl].
(6)
?VAg=
l?
(VAgNa,l? − VNaNa,l? + VAgCl,l? − VNaCl,l?)α2
l?
(7a)
?VNa=
l?
(VNaAg,l? − VAgAg,l? + VNaCl,l? − VAgCl,l?)α2
l?
(7b)
?VAg+ ?VNa=
l?
(2VAgNa,l? − VAgAg,l? − VNaNa,l?)α2
l?
(7c)
VNaCl=
l?
(VNaNa,l? + 2VNaCl,l? + VClCl,l?)α2
l? =1
4cNaClaNaCl
(8a)
VAgCl= VNaCl+ ?VAg− ?VNa=1
Inspection of figure 4 shows that the linear approximation holds for the TA[100] phonon
branch up to q = 0.3. Hence, it is justified to fit the concentrationdependentfrequency data
presented in figure 6 to equation (6). The results obtained for the projection of the force
constant matrix along the cubic axis are
4cAgClaAgCl.
(8b)
?VAg+ ?VNa= −1.71± 0.31 N m−1
?VAg= −1.37± 0.17 N m−1
VNaCl= 1.74 ± 0.06 N m−1.
From these data the remaining quantities
?VNa= −0.34 ± 0.35 N m−1
VAgCl = 0.71 ± 0.39 N m−1
are calculated.
Obviously, the substitution of silver ions into the NaCl host lattice has a much stronger
effect(?VAg) thandopingAgCl with sodium(?VNa). Inviewofequation(7c)it isconcluded
that the shortrangeinteractionbetween silver and sodium ions is appreciablyweaker than the
average of the Ag+–Ag+and Na+–Na+interactions.
Page 10
6424D Caspary et al
It is probably the strong polarizability of Ag+which is responsible for the pronounced
softening of NaCl. Silver ions are unique in their capability to occupy lattice sites with radii
rangingfrom0.7to1.3Å[16]. EvenPauling’sradiusofAg+(1.26Å)isclearlylargerthanthat
of Na+, the lattice parameters of silver and sodium halides exhibit the opposite relationship.
Obviously,thedeformabilityofthesilverionallowsaflexibleadaptationofitsshapeaccording
to the actual surrounding. Hence, the mechanical strength of the NaCl lattice is significantly
reduced by doping with silver. The pure AgCl lattice, on the other hand, seems to be able to
accommodateevenappreciableamountsofsodiumwithoutchangingthedynamicalbehaviour
at low frequencies.
Itisclearlydemonstratedthatboththemodificationofinteratomicforceswithcomposition
and the changes of the mass density lead to the nonlinear behaviour of the phonon frequency
and, hence, the sound velocity and elastic shear constant.
3.3. Structural properties of demixed crystals
As already shown in [1], the lattice relaxation as reflected by the splitting of Bragg reflections
takes place on a much longer timescale as compared to chemical demixing. This information
was inferred from the comparison of smallangle scattering and diffraction data. This finding
is supportedby ourpresent single crystal experiments. Figure7 shows the longitudinalprofile
ofthe(200)Braggreflectionobtainedaftermorethansixmonths. Differenttothewelldefined
phonon spectra (figure 4) which clearly reflect the two phases of different composition, the
Braggprofileconsistsofseveralbroadcomponents. Obviously,evenaftersuchalongperiodof
time, a large fraction of the crystal exhibits the lattice parameter of the homogeneousphase as
characterizedbythemainBraggcomponentnear Q = 2.016rluin the x0= 0.41sample. The
beginning contraction of the silverrich phase is reflected by the broad intensity contribution
at larger wavevectors while the expansion of the sodiumrich phase leads to the component
close to ξ = 1.985.
Notethattheexperimentalresolutionisabout0.006rlu. Moreover,SANSexperiments[1]
yield that after 8 h ageing time, the precipitates exhibit typical diameters of more than 70 nm.
Hence, grain size effects are unable to explain the broad Bragg reflections. Moreover, there
are no indications of any diffuse scattering between the Bragg peaks which might be expected
in disordered lattices.
For the overall concentration x0= 0.23, which is close to the miscibility gap at ambient
temperature, the result of the continuous lattice parameter distribution is clearly seen in the
lower part of figure 7.
Obviously,thesilverrichphasecharacterizedbyits lowenergyphononsis formedwithin
an anion lattice with a larger lattice parameter than expected in equilibrium. Hence, it is
subjected to tensile stresses, while the sodiumrich phase is under compressive stress. On
relaxation of these stresses at longer times, a shift of phonon frequencies can be expected.
From the deformation?a/a and the elastic constants the residual stresses can be estimated to
be of the orderof 150MPa. Using the pressurecoefficientsofc44(table 1), changesin phonon
frequencies of some 0.01 THz might be expected at very long times.
4. Conclusion
It has been shown that demixingprocesses in ionic crystals of the silver–alkali halide type can
be characterized by inelastic neutron scattering from phonons in single crystals. The phase
separation in AgCl–NaCl mixed single crystals is associated with the splitting of acoustic
phonon branches due to the different sound velocities of the constituents. The concentration
Page 11
The influence of demixing on the dynamics of ionic solids 6425
Figure 7. Longitudinal profiles of the (200) Bragg reflection for x0= 0.41 (top) and x0= 0.23
(bottom) after more than six months ageing time.
dependence of phonon frequencies and, hence, of the sound velocities and elastic constants,
has been investigatedin detail. It could be clearly demonstratedthat not only the mass density
but also the atomic interactions vary considerably with composition. Obviously, doping of
NaCl with silver ions leads to a significant softening of the lattice, while the corresponding
dopingofAgClwith sodiumionshardlychangestheelastic properties. This effectis probably
due to the large polarizability of silver ions.
While phonons reflect most directly the changes of interatomic forces due to the phase
separation, the static properties of the crystal lattice are essentially determined by internal
coherency strains. It could be shown that the lattice relaxation is not complete even after
more than six months at ambient temperature. This finding is in agreement with the results
fromsmallanglescatteringanddiffractionfrompowdersamples[1]. Interestingly,thecrystal
qualityhardlychangesevenif theresidualstresses afterdemixingareof theorderof 150MPa.
Page 12
6426D Caspary et al
Most probably, the invariant anion sublattice acts as a rigid microscopic frame for the phase
separation and prevents the destruction of single crystals.
This observation offers the possibility to study the kinetics of the demixing process via
the time evolution of acoustic phonons. These investigations will provide direct evidence of
the underlyingmechanismandthe correspondingtimescale. Infact, stroboscopicstudies have
recently been performed. Preliminary results have been published in [16]. Details will be
presented in a subsequent paper.
References
[1] Caspary D, Eckold G, G¨ uthoff F and PyckhoutHintzen W 2001 J. Phys.: Condens. Matter 13 11521
[2] Eckold G 2001 J. Phys.: Condens. Matter 13 217
[3] Trzeciok D and N¨ olting J 1980 Z. Phys. Chem. NF 119 183
[4] Eckold G 1992 J¨ ulich Report J¨UL2675 (ISSN 03660885)
[5] Eckold G and Trzeciok D 1992 Physica B 180/181 315
[6] Suslova V N, Shmidova N I, Zavadovskaya E K and Zvinchuk R A 1970 Sov. Phys.—Dokl. 15 500
[7] Stokes R J and Li C H 1962 Acta Metall. 10 535
[8] Hendricks R W, Baro R and Newkirk J B 1964 Trans. Am. Metall. Soc. AIME 230 930
[9] Windgasse J, Eckold G and G¨ uthoff F 1997 Physica B 234–236 153
[10] Every A G and McCurdy A K 1992 Landolt–B¨ ornstein New Series Group III, vol 29a (Berlin: Springer)
[11] Sinistri C, Riccardi R, Margheritis C and Tittarelli P 1971 Z. Naturf. 27a 149
[12] Caspary D 2002 Thesis University of G¨ ottingen
[13] Schmunck R E and Winder D R 1970 J. Phys. Chem. Solids 31 131
[14] Vijayaraghavan P R, Nicklow R M, Smith H G and Wilkinson M K 1970 Phys. Rev. B 1 4819
[15] Ubbelohde A R 1978 The Molten State of Matter—Melting and Crystal Structure (New York: Wiley)
[16] Elter P, Eckold G, Caspary D, G¨ uthoff F and Hoser A 2002 Appl. Phys. A 74 S1179
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