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INSTITUTE OF PHYSICS PUBLISHING

JOURNAL OF PHYSICS: CONDENSED MATTER

J. Phys.: Condens. Matter 15 (2003) 6415–6426 PII: S0953-8984(03)63779-3

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,

D-37077 G¨ ottingen, Germany

2Institut Laue–Langevin, 6 rue Jules Horowitz, BP 156, F-38042 Grenoble cedex, France

3Institut f¨ ur Kristallographie, RWTH Aachen, D-52056 Aachen, Germany

4Institut f¨ ur Festk¨ orperforschung, Forschungszentrum J¨ ulich, D-52425 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 well-defined 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

0953-8984/03/376415+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

withsmall-angleneutronscattering[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 well-definedsplittingofphononbranches.

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 time-resolved 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 sodium-rich 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 x-ray diffraction (within the homogeneous phase at 400◦C).

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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 gamma-ray 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(FRJ-2, J¨ ulich),IN3andIN12(HFR,ILL-Grenoble)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

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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 low-frequencypart 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

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The influence of demixing on the dynamics of ionic solids6419

Figure 3. Dispersion of the low-energy 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 anon-linearvariationoffrequencieswithcomposition. In

sodium-richmixedcrystals(x < 0.5) thefrequencychangesbyalmost50%, whileit becomes

almost independent of composition in silver-rich 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 well-defined 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 silver-rich phase, while the phonon of the sodium-rich phase is observed at

±0.85 THz. The line widths of both phonons are only slightly larger than the experimental

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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 high-temperature approximation (hν/kBT ? 1) and the lever rule we

obtain

I1

I2

x0− x1

=x2− x0

σcoh

1

σcoh

2

?ν2

ν1

?2

(1)

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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 low-energy 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 non-linear 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)

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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)

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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 short-range 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 right-hand 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 concentration-dependentfrequency 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 short-rangeinteractionbetween silver and sodium ions is appreciablyweaker than the

average of the Ag+–Ag+and Na+–Na+interactions.

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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 non-linear 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 small-angle scattering and diffraction data. This finding

is supportedby ourpresent single crystal experiments. Figure7 shows the longitudinalprofile

ofthe(200)Braggreflectionobtainedaftermorethansixmonths. Differenttothewell-defined

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 silver-rich phase is reflected by the broad intensity contribution

at larger wavevectors while the expansion of the sodium-rich 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,thesilver-richphasecharacterizedbyits low-energyphononsis formedwithin

an anion lattice with a larger lattice parameter than expected in equilibrium. Hence, it is

subjected to tensile stresses, while the sodium-rich 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

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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

fromsmall-anglescatteringanddiffractionfrompowdersamples[1]. Interestingly,thecrystal

qualityhardlychangesevenif theresidualstresses afterdemixingareof theorderof 150MPa.

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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.

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