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Mean Field Theory of enhanced defect formation in
AgCl and AgBr
R. Friauf
To cite this version:
JOURNAL DE PHYSIQUE Colloque C6, supplkment au no 7 , Tome 41, Juillet 1980, page C6-97
Mean Field Theory of enhanced defect formation in AgCl and AgBr
R . J. FriaufDept. of Physics, University of Kansas, Lawrence, Kansas 66045, U S A
RBsumB. - L'anomalie marquante de la conductivitk qui est observee aux temperatures hautes indique une
augmentation considerable de la formation des dkfauts prks du point de fusion T,. Puisque ni les corrections de Debye-Hiickel ni la thCorie classique diklectrique ne produisent une dipendance convenable sur la temperature, on introduit une theorie du champ moyen (MFT), dans laquelle I'energie de formation se reduit par une quantitk en proportion immediate avec la concentration des dkfauts. La MFT simple donne certainement un accroissement rapide de la concentration des dkfauts aux tempkratures hautes, mais quand on choisit la constante de force pour convenir i T,, les concentrations calculees aux temperatures plus basses sont trop petites.
Abstract. - The large conductivity anomaly observed at high temperatures indicates substantial enhancement
of defect formation near the melting point T,. Since neither Debye-Hiickel corrections nor the classical continuum theory yield a suitable temperature dependence, a mean field theory is introduced in which the formation energy is reduced by an amount directly proportional to the defect concentration. The simple MFT does gives a rapid rise in defect concentration at high T, but when the strength constant is chosen to match T, the predicted concen- trations are too small at lower temperatures.
1. Defect model. - 1 .1 NORMAL TEMPERATURE DEPENDENCE. - The ionic conductivity is dominated by cation Frenkel defects, with transport by silver ion vacancies and by two kinds of interstitialcy jumps [I]. Careful fitting of conductivity curves in an intermediate intrinsic temperature range (1500 to 280 OC for AgCl and 150° to 2200 for AgBr) gives the activation enthalpies in table I [2].
Table I. - Fornzation and migration enthalpies (eV)
for AgCl and AgBr.
Mechanism Ag Cl AgBr
- - -
Formation of Frenkel defects 1.49
+
0.02 1.16 & 0.02 Cation vacancy migration 0.31+
0.01 0.32 f 0.01 Coliinear interstitialcy 0.02 & 0.01 0.04+
0.01 Non-collinear interstitialcy 0.14+
0.01 0.28+
0.011 . 2 HIGH TEMPERATURE ANOMALY. - At higher
temperatures the measured conductivity becomes considerably larger than the extrapolated value, assuming all formation and jump enthalpies are independent of temperature. The excess appears at least 150 OC below T, and rises to more than 100
%
near T;, (see Fig. 4). Since evidence from diffusion of Na in AgCl and AgBr [3] indicates that most of the effect is due to an increase in defect concentration (rather than enhanced mobilities), we measure the anomaly by
Thus we obtain AgToT directly from experimental measurements, and then calculate the final concen- tration x, from the extrapolated value x,, without any defect interactions.
2. Defect interactions. - 2 . 1 DEBYE-HUCKEL COR- RECTIONS. - The sizable Coulomb interactions bet-
ween charged defects at large concentrations are repre- sented by the usual first order theory [4]. We find that the formation energy of a Frenkel pair is lowered by
Ag,,, which varies approximately as the square root of x,. Since AgDH cannot account for all of the observed anomaly expressed by AgToT in figure 1, we are forced to introduce some extra defect inter- action AgEX lo make up the difference. The expla-
__.-.-
0 -1-1-1-> -1-7 __.---
,
I I I I I I l l250 300 TL0C) 400 Tm
Fig. 1. - Lowering ol I'orma~ion energy for AgCI. Ag,,, is deter- mined from the measured conductivity. Ag,, is the calculated Debye-Hiickel correction based on the final defect concentration. A& would be obtained if no other defect interactions were present. Ag,, represents the difference between Ag,,, and Ag,,.
C6-98 R. J. FRIAUF
nation for AgEx will have to involve some new physical process.
2.2 CLASSICAL CONTINUUM THEORY. - Since both the dielectric constant E and lattice parameter a
also show anomalous increases near Tm, a previous treatment [I] considered the polarization energy described by the CCT of Jost [5].
In this semi-empirical approach we fix E, and a, by
setting Ag,,, = 0 at a reference temperature T o where AgEx starts to rise. Then we fit Ag,,, = AgEx
at some high temperature T , near Tm by adjusting
the e8ective cavity r a d i u s pa,. Finally we apply a self-consistent treatment with AgDH as done for the MFT below.
2.3 MEAN FIELD THEORY. - Other drastic changes near T m include a considerable decrease in some of the elastic constants and a greatly enhanced thermal diffuse scattering of X-rays. The general softening of the lattice [6] seems to foreshadow the imminent melting of the crystal. As the lattice softens more defects can form, the additional defects soften the lattice further, and before long the ultimate catastrophe occurs ! There are even a few theories in the literature relating unlimited defect formation to onset of melting [7, 81 or a superionic conducting transition [9].
Hence we propose a phenomenological defect interaction that lowers the formation energy in
direct proportion to the defect concentration
In this simplest form of the MFT the formation energy of each defect is lowered in the mean field of all other defects. The strength of the interaction is characterized by a concentration c, that is inti- mately related to T,.
Use of the MFT must also be self-consistent with the DH correction. For a trial concentration xf we first determine
calculate a new xf from eq. (I), and then iterate until xi = x f . In the MFT case the iteration converges
only if c, is not too small, which establishes the connection to T,. We obtain E , a, and xoo at T,,, by
extrapolation, and then decrease c, until the iteration process starts to diverge. With this c, the calculation converges satisfactorily for all T below T,.
3. Discussion of results. - 3 . 1 DEBYE-HUCKEL CORRECTIONS. - The Coulomb interactions shown
by AgDH in figures 1-3 are clearly significant at all
temperatures. Even at 150 O C in AgCl, for instance,
AgDH = 3.1 meV and the defect concentration is
Flg. 2. - Correcllons LO formation energy fbr AgCI. Ag,,, and Ag,, are shown for three cases : EXP, obtained from experi- mental values of om,,, ; CCT, calculated from classical continuum theory; MFT, calculated from mean field theory. AgDH shows the Debye-Hiickel correction corresponding to AgToT(EXP) ; the other two curves for AgDH(CCT) and Ag,,,(MFT) are similar and are not shown to avoid confusion.
Fig. 3. - Corrections to Ibrma~ion energy lor AgBr. Symbols are defined in figure 2.
increased by 4
%.
There is a slow rise in AgDH as the concentration increases at higher temperatures, but notice that AgDH cannot possibly account for all of the observed anomaly. Not only is the size inadequate at the higher temperatures, but also the form of the temperature dependence is inappropriate.3.2 CLASSICAL CONTINUUM THEORY. - We choose To at 2800 for AgCl and 250 OC for AgBr, and take T, at 430 OC and 400 OC, respectively, approximately 25 OC below T , in each case. These choices give p = 2.04 for AgCl and 1.12 for AgBr. In the simplest
form of the CCT we would have p = 0.5 ; in an earlier fit for AgBr we found 1.44 [l]. The rather capricious spread of values for p reveals the admittedly empirical nature of this approach.
MEAN FIELD THEORY O F ENHANCED DEFECT FORMATION I N AgCl AND AgBr C6-99 matching the experimental value at T , it levels off
too much in the interesting range below T,. Thus the general behaviour of the CCT is not very satis- factory, especially in the crucial region near T,.
3 . 3 MEAN FIELD THEORY. - The increasingly rapid
rise of AgToT near Tm strongly suggests that the conductivity anomaly is intimately associated with pre-melting behaviour of the crystal. An even more striking portrayal is shown in figure 4. The hypothe-
d /%53 d MFT
250 300 350 T ( ' c ) 400 Trn
Fig. 4. - Concentrat~on ol Frenkel delects lor AgBr. sf shows the final concentration for three cases : EXP (experimental), CCT (classical continuum theory), and MFT (mean field theory). xoo is the hypothetical extrapolated concentration of isolated defects in an ideal crystal with no defect interactions. x, represents the final concentration that would be obtained with Agg, allowing only for Debye-Hiickel corrections.
tical extrapolated value x,,, without any defect interactions, is far short of the final value xf needed to account for a,,,,. The increase from xo, to x,, "corresponding to Ag, alone, is noticeable but still
insufficient. We also note that x, from the CCT is not particularly appropriate, and see that the MFT shows some of the proper behaviour at high T.
One striking feature is that the MFT immediately relates the strength of the interaction to
T,.
For this sample of AgCl conductivity data exist up to 449.6 OC, and literature values suggest T , = 455 OC. Then we find c, = 13 106 ppm and obtain moderately good agreement with experiment in figure 2. The deter- mination of c, is astonishingly sensitive : for a decrease of only 1 ppm the iteration process already diverges ! The precise value of c, also depends strongly on the values of x,, and 8 at T,. The fitin figure 2 is rather encouraging, except that the final increase of the MFT results is slightly too rapid.
For the AgBr sample conductivity data exist up to 415.9 OC. We make the interaction as strong as possible by choosing Tm = 417 OC (lower than the
Fig. 5 . - Dependence ol' AgEx on deleci concentration for AgCl and AgBr. The temperature is indicated at several points on each of the curves. For the MFT the melting temperatures are taken as 455 OC for AgCl and 417 OC for AgBr.
usually quoted range of 422 to 428 OC). Then we obtain
c, = 13 856 ppm and the results in figures 3 and 4, but still find an inadequate number of defects at the highest observation T. We just can't make the MFT interactions strong enough to match the observed
a,,,, ; if we did, the crystal would already be melted !
Another interesting feature is displayed by plotting AgEx vs. x, in figure 5. We see a linear relationship for the MFT (as our theory demands) and observe an approximately square-root dependence for the CCT. The experimental behaviour lies in between, with a dependence just slightly less than linear for AgCl, but the situation is less satisfactory for AgBr. These curves also show how the anomaly is stretched our for the final few degrees below T, ; the effect is very noticeable in AgCl between 4500 and 455 OC. The shortfall of the MFT in AgBr is also demonstrated by the compression of the calculated curve relative to the experimental results.
C6-100 R. J. FRIAUF
In conclusion we feel that the MFT is definitely a step in the right direction. This approach simulates the very rapid rise near T , and qualitatively has just the right feel ! But ramifications beyond the simple linear MFT are needed, especially for AgBr. The more elegant theoretical approaches 17, 91 do introduce more than linear dependence between AgEx and xf and hence offer hope of improved agreement. From
the experimental point of view, figure 5 shows the need for additional measurements in the crucial range right up to the melting point, where the most interesting effects occur and the test of the theory is most distinctive. Because of the sensitive role of
Tm in determining the strength of the MFT inter- actions, a much better in situ measurement of T m is also important.
DISCUSSION
Question. - F. BBNIERE. Question. - M . J . GILLAN.
You recall that Debye-Huckel theory was applied by Prof. Lidiard to the ionic crystals from the liquid solutions. DH is valid for concentrations
which is equivalent to 3 x lop6 in mole fraction. For higher concentrations Onsager and Bjerum's theories are applied. Would those improve the analysis of AgCl when C = 1
%
in mole fraction.Reply. - R. J . FRIAUF.
The most extensive application of improvements to the Debye-Huckel theory for ionic crystals, of which I am aware, is given by the calculations of Sevenich and Kliewer for AgC1, based on the theory of Allnatt and Cohen. Even with these improvements, their results look very similar to the linear Debye-Hiickel calculations when plotted as Ag,-vs. T. Therefore I conclude that there are not likely to be any substantial changes even when extended forms of the theory are introduced. In any case, 1 am convinced that even the linear DHL theory is much better than nothing at all.
I believe the catastrophe you find in the vacancy concentration is the same as the one which occurs in Frenkel's old theory of melting. Ideas similar to yours have recently been proposed to account for the behaviour of the defect concentration in superionic fluorites (Gillan, Richardson, March and Tosi, to be published).
Reply. - R. J . FRIAUF.
Thank you for the reminder. I have undoubtedly been influenced by reading Frenkel's remarkably provocative book on Kinetic Theory ofLzquids. I agree that the behaviour in AgCl and AgBr seems to be very similar to that of the fluorites, and will be most interested to learn of your approach.
Question. - W . HAYES.
Do you feel that your studies of AgCl and AgBr have relevance to the superionic AgI ?
Reply. - R. J . FRIAUF.
Yes, it almost seems as if the qhloride and bromide are striving toward a superionic phase transformation, but that the process is interrupted by melting. The behaviour is perhaps even more similar to the fluorites, because of the large rise in conductivity, but without any indication of a first order transition.
References
[I] FRIAUF, R. J., J. Physique 38 (1977) 1077. [6] TELTOW, F., Ann. Phys. 5 (1950) 63, MULLER, P., Phys. S t a t u ~ [2] KAO, K.-J., Thesis, University of Kansas (1978). Solidi 21 (1967) 693.
[3] BATRA, A. P. and SLIFKIN, L. M., Phys. Rev. B 12 (1975) 3473. [7] MATYAS, A., Czech. J. Phys. 4 (1954) 14. [4] LIDIARD, A. B., in Encyclopedia of Physics, ed. S. Flugge (Sprin- [8] O'REILLY, D. E., Phys. Rev. A 15 (1977) 1198.
ger-Verlag, Berlin) 1957, Vol. XX, p. 246. [9] WELCH, D. 0 . and DIENES, G. J., J. Electron. Muter. 4 (1975) 973. [5] JOST, W., Diffusion in Solids, Liquids, and Gases (Academic
