RARE GAS DIFFUSION IN α- AND β-CsCl

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RARE GAS DIFFUSION IN α- AND β-CsCl

M. Müller, M. Norgett

To cite this version:

M. Müller, M. Norgett. RARE GAS DIFFUSION IN α- AND β-CsCl. Journal de Physique Colloques,

1973, 34 (C9), pp.C9-159-C9-161. �10.1051/jphyscol:1973930�. �jpa-00215405�

JOURNAL DE PHYSIQUE

C0ll0que C9, supplhment au no 11-12, Tome 34, Novembre-DPcembre 1973, page C9-159

RARE GAS DIFFUSION IN

^a-

AND B-CsCl

Hahn-Meitner-Institut Berlin GmbH, Nuclear Chemistry a n d Reactor Division, D 1 Berlin 39 (West), Glienicker Str. 100

and M. J. N O R G E T T

Theoretical Physics Division, AERE, Harwell, Didcot, Berkshire, UK

Rbum6. -

Des calculs rendent compte de la difference des mtcanismes de diffusion des gaz rares dans les deux phases du chlorure de cbium. Dans la phase-a, les gaz peuvent migrer par

un

mkcanisme par paires de lacunes associees tandis que la diffusion est due

a

un mecanisme interstitiel dans la phase-8.

Abstract. - Calculations explain the different diffusion behaviour of inert gases in the two phases of caesium chloride. In the a-phase, the gases can migrate by a divacancy mechanism while interstitial motion occurs in the 8-phase.

Gas diffusion in NaCl structure materials has been which differ in the form of the repulsive interactions successfully explained by Born model calculations [I]. used. We have assumed the following form for the In this paper, we report similar calculations for CsCl interionic interaction

which has a characteristic ~ h a s e transformation.

In the cr phase, which has the CsCl structure, the mobility of the three gases Ar, K r and Xe is the same and the three gases have essentially identical activation energies for diffusion. I n the

/j'

phase, which has the NaCl structure, the inert gases have different mobilities and activation energies. It is also found that the mobility of Xe in both phases is reduced when the pure material is doped with Ba2+ ion and the activation energy is increased [2]. This indicates the significance of gas trapping in cation vacancies.

We wish to present calculated activation energies for gas diffusion in both phases of CsCI. Our calcu- lations are based on the Born model for ionic crystals.

The ions are bound by coulolnb forces but repel at close separations because of the overlap of the closed shells on adjacent ions. We also consider the pola- risation of the lattice in two n~odels. We either assume that the ionic polarisation rnay be represented by point dipoles (polarisable point ion (PPI) model) or use a more sophisticated shell model.

In detail, we have performed calculations with three models. In principlc, the first is the point pola- risable ion model of Murtliy and Murti [3], but we cannot reproduce their model exactly because their Paper omits certain necessary data. However, we have probably been able to substitute values so that the calculations are comp;~rable. The latticc is simulated at approximately room ternpcratusc.

Secondly, we have derived two shell models,

(ri +

^{Y j -}

I.) Cij Dij

Kj(v)

=

b exp

^{- -- - -}

-

-

.

P

r6 r8

The C i j and D i j van der Waals terms are taken from Mayer [4]. We have used radii derived by Tosi

[ 5 ] ,

neglecting any variation due to changes in crystal structure. In addition, it is necessary to increase the van der Waals interaction between second neighbours substantially if we hope to account for the observed phase transition. This point is discussed in greater detail in our earlier letter on CsI [6]. The variables b,

p

and the scaling factor for the van der Waals second neighbour interactions are estimated from the values of the lattice constant and the shear constants C, ,-C,, and C,, at approximately 0 K. This and other neces- sary data is taken from the tabulation in the paper of Mahler and Engelhardt [ 7 ] . Our two shell models difTer only in that the first (S 1) includes van der Waals interactions between nearest neighbours while they are ignored in the second model (S 2). The gas ion potentials are calculated using interpolated radii for the inert gases.

I n

both niodels, the shell para- meters are fitted to the static and high frequency dielectric constants, the transverse optic frequency and ionic polarisabilities, using low temperature values where appropriate. The shcll model parameters are collected

in

the appendix

to

this paper.

We wish to use these

potentials

for both the

^x

and

/I

phase of CsCI. Thcrc is therefore some need to inves-

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1973930

tigate their validity at different lattice spacings. We have therefore recalculated the elastic constants and their pressure derivatives using an expanded lattice corresponding to the measured interionic distance at 300 K. These can be compared with the values measured by Chang and Barsch [ 8 ] (see Table I).

The agreement is surprisingly good and gives us some confidence that we may represent the effect of tem- perature by just expanding the lattice. The potential may also be valid for the high temperature

/I

phase.

Therefore, we have carried out simulations in both phases with a lattice parameter chosen to simulate the lattice at a temperature close to the phase tran- sition (i. e. - ^{470 OC).}

Measured and calculated elastic cot~sfa~lts for cr-CsC1 at - ^{300 K}

C I 1 c12 c14 C;l Ci2

ci4

- - - - - -

E X P ~ [81 3.67 0.889 0.808 7.01 5.14 3.69 Potential S 1 3.64 0.704 0.859 6.20 5.28 3.25 Potential S 2 3.63 0.690 0.841 6.46 5.56 3.53 The elastic constants C 1 1, Clr and C44 are adiabatic values.

The pressure derivatives C ; ] , Cir and Ci4 are isothermal derivatives of the adiabatic constants.

We have used all three potentials to calculate Schottky energies. All calculations are performed with Norgett's HADES program [ 9 ] which allows the efficient relaxation of many shells about the defect.

The relaxed region is surrounded by a dielectric continuum, usingthe method of Mott and Littleton [ l o ] to estimate the ionic displacements.

The results are as expected. The results obtained with the PPI model decrease rapidly as the relaxed region is expanded because of the discontinuity in the dielectric properties of the lattice model and the surrounding continuum. The shell model results stabilise after a few shells are relaxed and are in reasonable agreement with the measured values [ l I].

Because of this clear advantage, we have used only the shell model for our calculations of gas diffusion.

We consider two possible diffusion mechanisms.

1) Interstitial diffusion

D

=

Db exp (

- -

.

2 ) Diffusion in vacancy pair gas complexes

In addition, we must consider the partition of gas between mobile and immobile sites such as cation

vacancies. This reduces the diffusion by a factor

p.

For the two mechanisms, p has the form 1) Interstitial diffusion

2 ) Diffusion in divacancy gas complexes

where Ce and

C,

are the concentrations of divacancies and cation vacancy traps. AH: is the binding energy into traps and AH! the binding energy into vacancy pairs.

I11 substituting for the concentrations, the

C,

term contributes a factor proportional to exp(- H s / 2 k T ) , where H, is the Schottky energy, only in an intrinsic material

;

in a doped crystal, the trap concentration is constant. Thus we obtain for the activation energies.

a) Ii?t~.ilwic region.

1) Interstitial diffusion

2 ) Diffusion in divacancy gas complexes

(H: is the vacancy pair formation energy).

Scllottky etiergies calculated using d~fferent nzoclels (i) a-CsCI

No. Polarisable

of Shells point Shell models

relaxed ion model

1

2

- - -

1 1.56 eV 1.98 eV 1.77 eV

2 1.09 1.74 1.52

3 1.17 1.85 1.62

6 0.90 1.78 1.53

12

^-

1.76 1.51

15 0.78

-

Expt Schottky Energy

=

1.77 eV [ l I ] . (ii) p-CsCI

No. of shells Shell models

relaxed 1 2

- - -

I 2.57 eV 2.42 eV

2 2.55 2.39

3 2.58 2.43

4 2.51 2.36

9 2.42 2.26

RARE GAS DIFFUSION I N a- A N D 8-CsCI C9-161

b) Extrinsic region.

1)

Interstitial diffusion

AH'

⁼

AH;^,, + ^{A H ; .}

2) Diffusion in divacancy gas complexes

AH^

⁼

AH:^,,

-

AH: + ^AH; + ^H:

We have calculated all these energy values and obtained activation energies which are compared with the experimental values in table 111. For the light gas Ar and also Kr, it is not possible to distinguish the diffusion mechanism operating in u-CsCI. However, the relatively low activation energy observed for Xe diffusion can only be explained by postulating a

Experirnental a17d cnlculatec/ actioation energies (in eV)

,for

Ar, Kr mzd Xe di,!firsion itz CsCl

Intrinsic material

K-CSCI AHpx,, A H AH' A AH"

diffusion in vacancy pairs. Thus we find the same effect as we reported for caesium iodide [6]. Such a mechanism does explain the surprising fact that the activation energies for the three gases are similar.

However, in 8-CsCI, the gas diffuses preferentially by an interstitial mechanism which accounts for the different observed activation energies. In both phases, the calculated results are in qualitative and reasonable quantitative agreement with the measured values for doped samples.

APPENDIX

Shell model parameters.

-

(i) REPULSIVE

AND VAN DER

WAALS

POTENTIAL.

Model 1 Model 2

-

-

A +

⁺

(eV) 4 809.0 8 908.0

D -

-

889.0 702.5

Extrinsic material (Ba2+ doped)

P (A) ^{0.353 2 0.323 4}

a-CsC1

Ar - - 1.66-1.68 - 1.70-1.94

(ii) SPRING

CONSTANTS

(fo

AND SHELL CHARGES

( Y ) . Model 1 and 2

-

References

[I]

NORGETT, M.

J. and

LIDIARD,

A. B., Phil. Mag. 18 (1968)

1193.

[2]

FELIX,

F. W. and

MEIER,

K., P11ys. Stat. Sol. 32 (1969) K 139. (See also

FELIX,

this volume p. 149.)

[31

MURTHY,

C . S. N. and

MuRTI,

^Y. V. G. S., J. Phys. C (Solid State Physics) 4 (1971) 1108. (See also MURTHY and MURTI, this volume p. C9-337.)

t41

MAYER,

J. E., J. Chetn. Phys. 1 (1933) 270.

[51

Tosr,

M. P., Sol. Stat. Phys. 16 (1964) 1.

[61

MULLER,

M. and NORGETT, M. J., J. PIIJS. C (Solid State Physics) 5 (1972) L 256.

[7]

MAHLER,

G . and

ENGELHARDT,

P., Phys. Stat. Sol. (b) 45 (1971) 453.

[8]

CHANG,

Z. P. and

BARSCH,

G. R., Phys. Rev. Lett. 19 (1967) 1381.

[9] See for example,

LIDIARD,

A. B. and

NORGETT,

M. J., in cc Con~pr~tational Solid State Physics D, ed. F. Her- mann, N. W. Dalton and T. R. Koehler (Plenum Press) 1972, p. 385.

[lo]

MOTT,

N. F. and

LITTLETON, M.

J., Trans. Farad. Soc.

34 (1938) 485.

[l I]

HOODLESS,

I. M. and

TURNER,

R. G., PIIJJS. Stat. Sol. ( a ) 11 (1972) 689.