Influence of anharmonicity on some transport properties of AgBr

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Influence of anharmonicity on some transport properties of AgBr

P.A. Varotsos

To cite this version:

P.A. Varotsos. Influence of anharmonicity on some transport properties of AgBr. Journal de Physique, 1978, 39 (11), pp.1247-1249. �10.1051/jphys:0197800390110124700�. �jpa-00208866�

E 1247

INFLUENCE OF ANHARMONICITY

ON SOME TRANSPORT PROPERTIES OF AgBr

P. A. VAROTSOS (*)

Department ^of Physics, University ^of Athens, ^Solonos str. 104, ^Athens 144, ^Greece

(Reçu ^{le 28} avril 1978, accepté ^le 20 juillet 1978)

Résumé. ²⁰¹⁴ Des mesures récentes montrent que l’entropie ^de migration ^pour la diffusion de Na+

dans AgBr ^{est de} 5,1 kB. Cette valeur est notablement supérieure à celle trouvée dans les halogénures

alcalins pour le ^mouvement des lacunes cationiques. ^Une explication plausible ^{est, de ce} fait, proposée.

De plus, ^{on a fait un} calcul des volumes de migration pour le ^mouvement des lacunes et interstitiels de Ag+ ^dans AgBr qui ^{mène à des valeurs} comparables ^aux expériences.

Abstract. ²⁰¹⁴ Recent measurements show that the migration entropy ^for Na+ diffusion in AgBr

is 5.1 kB. This value is appreciably higher ^{than that} found in alkali halides for cation _vacancy motion.

A plausible explanation of this fact is proposed. Furthermore, ^a calculation is made of the migration

volumes for the _vacancy and interstitial motion of Ag+ ⁱⁿ AgBr which leads to values comparable

with experiment.

LE JOURNAL DE PHYSIQUE TOME 39, ^NOVEMBRE 1978,

Classification Physics ^Abstracts

66.30

Considerable interest has been recently expressed

in the study ^of transport properties ^{in silver}

halides [1-5]. ^{It has} been found that on approaching

the melting point ^their conductivity ^{shows an unusual}

increase not predicted ^{from the usual} Lidiard-Debye-

Hückel theory [6]. ^{We have} already proposed [7, 8]

that this increase is probably ^{due to the} higher ^order anharmonicity ^{of these solids at} high ^T, which reveals

a non-linear decrease of the Gibbs formation _energy

gf per cation-Frenkel defect. This proposal ^{seems to}

be in accordance with the ideas expressed recently by ^Friauf [9] and Slifkin [10]. Furthermore, by taking

the pressure variation of the elastic constants into account (clearly ^an anharmonic effect) [1l,12J, ^the

formation volume vf per Frenkel defect has been calculated [13] ^{in close} agreement ^{with the} experi-

mental results.

In the present paper ^we focus our attention on the

following problems ^{which are still} open. (i) ^The migration entropy ^{sm,v for} Na+-diffusion (through vacancies) ⁱⁿ AgBr ^has been found [4] equal ^{to 5.1} kB (kB is the Boltzmann constant). ^{This value is}

appreciably higher ^{than that} found for free cation or

anion vacancy motion in alkali halides [14]. (ii) ^The migration ^{volume vm,v} for free cation _vacancy motion in AgBr ^is [5] 5.5 ^{± 0.5} cm3/mole, ^{i.e. vm°" = 0.37 Q}

where 03A9 is the mean volume _per atom. On the other hand in alkali halides the ratio vm,v /Q ^{is around 0.5}

(*) To the memory of Antonios Varotsos.

or even more [15]. (iii) ^The migration ^{volume vm,i for}

the interstitial motion of Ag+ ⁱⁿ AgBr ^is [5] ^{~ 50} %

lower than vm, v.

1. The migration entropy ^{of Na in} AgBr. ^{- The}

Gibbs migration energy gmi’V ^for the motion of a

monovalent ion i from its normal lattice position ^{N to}

a neighbouring vacancy ^can be expressed ^as [16, 17] :

where 1 is the nearest cation-anion distance, as, aN ^are the Madelung ^{constants for the saddle} point ^{S and the}

normal site N respectively (taking ^the presence of the vacancy into account), B is the bulk modulus, F ^{is the} dielectric constant and C; ^{is a constant which is}

assumed to be temperature and pressure independent.

The value of the constant Ci ^{can be} calculated from the boundary condition that gî v ^becomes equal ^to

the migration enthalpy hmvôi ^{at absolute zero. In the}

case of AgBr ^{at T = 0 K we have} [18-20,16]

For Na in AgBr ^{we use the} approximation

and therefore a direct application of eq. (1) ^{at T =} 0 K leads to CNa ^{= 0.077. The} corresponding migration

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0197800390110124700

1248

entropy sNa ^{can now be} calculated from direct diffe- rentiation of _eq. (1) ^with respect ^to temperature :

For the application ^of eq. (2) ^{in the} region ^160-

180 °C we have used : (a) ^the temperature ^variation of the elastic constants and expansivity ^values report- ed by Tannhauser, ^{Bruner and Lawson} [19] ^and (b)

the temperature variation ofs measured by ^Smith [20].

The values obtained for D(B03A9)DT ^and â(El )/aT

are 5.8 x 10- 3 eV/K ^{and 7.06 x 10- 2} Â/K respecti- vely. Therefore the first term of the right side is found

to be ~ 0.9 kB, whereas the second one is appreciably higher ^{at 4.8} ko. ^{Therefore the} calculated value is

s’Nà’ ;:~t ^5.7 kB which is in satisfactory agreement ^with

experiment [4]. This calculation shows that the major

contribution to s-N-1 ^comes mainly ^{from the} rapid

decrease of elastic constants with the température ; it finally ^{means that the} migration entropy ^{for Na-} motion in AgBr ^is appreciably larger ^{than that found}

in alkali halides due to the higher anharmonicity ^of AgBr.

2. The migration volume for a free cation _vacancy in AgBr. ^{- The} parameter ^{vm,v can now} be obtained

by ^a direct dBiflerentiation of _eq. (1) ^with respect ^to pressure. The calculation of this derivative demands the knowledge of D(In s)/ôP ^which unfortunately ^{is not}

available to us. In alkali halides, experiments ^show

that [21-23] D(In E)IDP =- _ 10-2 kbar-1. By ^assum- ing that this value is also appropriate ^for AgBr ^we

estimate that the _pressure derivative of the first term

of the right ^{side of} eq. (1) ^is considerably ^{smaller than} Ci à(BQ)/ôP ^{at constant T. Therefore we can} approxi- mately ^{set :}

where now the constant Ci corresponds ^{to the} migra-

tion of Ag+ ^{to a} neighbouring vacancy. The value of

C¡g ^{can be extracted from the} application of eq. (1)

at T ⁼ 0 K in a manner similar to that indicated above for Na+-motion :

By bearing ^{in mind that the} migration enthalpy ^for

the (vacancy) Ag+-motion is 0.33 eV [1, 5], eq. (4) gives : CÂg ‘~ ^0.037.

Loje ^{and Schuelle} [18] ^found that the pressure derivative of the (isothermal) bulk modulus B at R.T is 7.49. The same authors, ^{at T = 195} K, gave the value

ôBs/ôP ^{= 6.26,} where Bs ^denotes the adiabatic bulk modulus. It is reasonable to assume that

(DBî ,DP)~ (ôB/ôP)

for T ⁼ 195 K ; ^{therefore a linear} extrapolation up ^to 500 K gives aB/op ~ ^9.8.

By inserting the above values into _eq. (3) ^we finally get : v Âg mv ~ ^{0.33 03A9 ;} this result is comparable ^{to the}

value 0.37 Q + 10 % ^obtained experimentally ^[5].

3. The migration ^{volume for the} interstitial motion of Ag+. - ^Similar considerations should be applied

for the interstitial motion of Ag+. By using eq. (4)

and the _average migration enthalpy [5] hmiÂg‘ ^{= 0.20 eV,}

one obtains Clg ⁼ 0.022. Therefore the application

of _eq. (3) ^{leads to :} vÂg N ^{0.2 03A9 which has to be}

compared ^{with the} experimental ^value [5] ^{of 0.25 Q.}

4. Concluding ^{remarks. - It is an} undisputed ^fact

that a calculation of sm,vNa Na vÂg, Ag v/§flo Ag ^{on a} microscopic

basis still remains an extremely ^difficult problem. ^{It is}

therefore fortunate that the macroscopic ^model proposed ^here gives ^a satisfactory agreement ^{with the} experimental ^{results. It is} worthwhile to emphasize again that the basis of the present model is the strong

anharmonicity ^of AgBr. ^{If one} recalls that the tempe-

rature variation of e is due mainly ^to the volume

dependence ^{of the} microscopic ^{force constants} [24],

we can state that the present paper ^- being ^{in basic}

agreement ^with [8] suggests ^that anharmonic effects

play ^an important ^{role in the} transport properties of AgBr.

References

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