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pared with the data for the mobility of electrons in ever, the very steep temperature dependence of the the same temperature range. At room tempera- hole mobility is very strange, as one would expect ture the Hall mobility of holes of 2.0 cm.z,/v. sec. the hole to interact with the lattice in much the comparcs with the drift, mohility of electrons of 70 same way as the electron. A temperatiire dependc.m.""v. he('. from the data of Yamanaka, rt ~ d . ~tint ' effective mass would help to explain the steep The tempera1 ure dependence of the Hall mobility tempcraturc dependence. ,4t tempts presently are of the holes is approximately P4, whilc in the samc being made to measure carefully the thermoelectric temperature range the electron mobility will fit a power of these holes in order to gain more informaT-1.5 curve. Since these temperatures are well tion about the band structure. above the Debye temperature associated with the Acknowledgments.-The author gratefully aclongitudinal optical modes (approximately 200°K. knowledges the encouragement and guidance of for AgRr), one would not necessarily expect the Prof. F. C. Brown and Prof. R . J Maurer during mobility to be proportional to e+'o/T as predicted the course of this work. The fellowship provided by Lorn and I'ines38 for lower temperatures. How- by the Corning Glass Korks Foundation for 19581959 and other support from the Air Force Office (37) C. Yanirmaka, N. I t o h , a n d T. Suita, see ref. 15, p 175. of Scientific Research are sincerely appreciated (38) F. Low nnd D. Pines, P h y s . Reo., 98, 414 (1955).
IOSIC TRANSPORT PROCESSES ISTHE SILVER HALIDES BY ROBERTJ. FRIAUF University of Kansas, Lawrence, Kansas Received June 20,1862
The motion of ions in AgCl and AgBr has been studied by measurements of ionic conductivity and diffusion. The relationship between these phenomena is represented by the Einstein relation, and deviations from this relation can be interpreted in terms of correlation and displacement effects for diffusion. The presence of Frenkel defects is confirmed, and the occurrence of two kinds of interstitialcy jumps for interstitial silver ions is demonstrated. It is further estimated that the concentration of Schottky defects in AgBr cannot exceed 0.1% even at the melting point. The influence of the crystal structure and the nature of the ions on the defect structure is mentioned, and the significance of the results for the photographic process is discussed briefly.
11. Conductivity and Diffusion The silver halides are characterized by the The presence of ionic defects is revealed most presence of Frenkel defects: silver ion vacancies directly by measurements of ionic conductivity and interstitial silver ions. This feature of the and of diffusion of radioactive tracers. Both of defect structure has been recognized ever since the these phenomena occur by means of motion of depioneer work of Koch and Wagner.' Subsequent fects through the crystalline lattice : diffusion repmeasurements of ionic conductivity on samples resents a mixing of atoms by virtue of the random, doped with divalent ions have allowed the separat'e thermally induced jumping of defects, whereas evaluation of the concentration of Frenkel defects conductivity results from a slight drift velocity and of t.he mobilities of vacancies and inter- superimposed on the random motion by an aps t i t i a l ~ . * - ~The results described in this paper for plied electric field. The close relationship between AgCl and AgBr lend further confirmation to t,he the two phenomena is expressed quantitatively by existence of Frenkel defects and provide a more the microscopic Einstein re1ation.j detailed description of the ionic jumps involved in the interstitial transport process. d / p = kT/q (1) The experiments have for the most part been carried out in the intrinsic defect region, t'hat is, Here d is the microscopic diffusion coefficient, for pure crystals in the temperature range of p the mobility, and p the charge of the defect being several hundred degrees below the melting point. considered. In order to obtain a relationship between directly In this region the defect concentrations are determined by thermodynamic equilibrium, and the measurable quantities, it is necessary to introduce concentrations of vacancies and interstitials must. appropriate macroscopic quantities, namely, the be equal to maintain electric neutrality. At lower conductivity, u, and the macroscopic diffusion temperatures, below 100 to 200°, the defect con- coefficient, D centrations usually will be determined by the u = qnp, D = ( d S ) d (2) presence of impurities, but the jump mechanisms for the defects that are present should be the same where n is the concentration of defects and 11' is as a t higher temperatures unless the defect.s are the concentration of atoms of one kind. By associated with impurities. combining eq. 1 and 2 the normal form of the macro(1) E. Koch and C. Wagner, 2. p h y s i k . Chem.. B38,295 (1937). scopic Einstein relation is obtained.
I. Introduction
( 2 ) J. Teltow, .4nn. Phusik. 6, 63,71 (1949).
(3) S. W. Kurnick. J . Chem. Phys., 20, 218 (1952). (4) I . Ebert n n ' l J. T e l t o a , Ann. Physilz. 15, 268 (1963).
( 5 ) N F Mott and R W Gurney, "Electronic Processes Ciystsls " 2nd Ed , ('Inrendon Press, Ovfoirl l S i R , p ti?.
in
Ionic
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kT/Nq' (3) Since there usually are deviations from this relatioiiship, for reasons described in section 111, it is convenient to represent the experimental results in dimensionless form by introducing an experimental correlation factor .feu*
=
=
I),,,, / D o
(4
where D,,, is the measured diffusion coefficient and I), is cialculated from the conductivity by eq. X .
111. Correlation and Displacement Effects Deviations from the macroscopic Einstein relationship usually are traceable to a modification of the expression for the macroscopic diffusion coefficient in eci. 2. The existence of correlation effects for vacancy diffusion was first recognized by Bardeen and Herring6 They pointed out that whereas a vacancy moves through the lattice in a genuine random walk sequence, the motion of a tracer atom has correlations between the directions of successive jumps. Suppose, for instance, that a vacancy has just caused a tracer atom to jump to the right; since the vacancy is now to the left of the tracer atom, a return jump is more likely than any other. Hence the diffusion coefficient measured by means of tracer atoms is somewhat less than would be el pected from the cond~ctivity.~ The various possible jump mechanisms for an interstitial ion in a silver halide are shown in Fig. 1. For the direct jump there is no correlation effect a t all. For the remaining interstitialcy iumps, in which the interstitial ion knocks a neighboring silver ion out of a lattice site and then occupies the vacated site,2,*there are displacement as well as correlation effects. For the collinear interstitialcy jump, for instance, the displacement of charge is twice as large at, the displacement of either of the ions involved in the jump, and hence the relationship between conductivity and diffusion is affected. The experimental results indicate that both v 1 and u2 jumps occur ( v g and v4 jumps presumably have too large an activation energy), and these will be designated simply as collinear (vl) and noncollinear ( v2) in the following discussion. The actual path of an interstitial defect consists of a random mixture of the two kinds of jumps, with a frequency ratio depending on the temperature, and the correlation and displacement factors depend on this ratio. T alues of correlation, displacement, and over-all combined correlation factors are given in Table I. The theoretical evaluation of displacement factors is rather simple. The calculation of correlation factors is more complicated, but it can be reduced to a problem in diffusion of probability, and values have been obtained for all of the processes of in(6) J. Bardeen a n d L Herring, in "Imperfections i n Nearly Perfect Crystals," W. Shockley, E d , J o h n Wiley & Sons, Inc , S e w York. N Y , 1952, p. 261 ( 7 ) A more completi. description. a n d a mathematical fornlulatlon, of correlation a n d displacement effects 1s glren, for instance In R J. r n a u f , J. -4ppZ. P h y s , 33, 492 (1962) (8) F Seitz, Acta Cryst 3, 358 (1930) (9) K Compaan a n d Y I-Iaien ?'ran8 F a i a d a ~ Sor , 62, 786 (1976). 64, 1408 (195x1
Fig. 1.-Interstitial jump mechanisms in AgCl and AgBr. Direct jump, vo; interstitialcy jumps: collinear, v i ; noncollinear forward, v r ; non-collinear bnckn-ards, va; place exchange, v4.
TABLE I FACTORS SILVER HALIDES(sac1 T Y P E STRUCTURE)'
THEORETICAL CORRELATIOS .4ND DISPLACEMEST FOR
I~IPWSIOSI X
THE
Mechanism
Vacancy Direct interstitial ( y o ) Collinear interstitialcy (v1) Non-collinear interstitialcy (ut)
Correlation factor
0.78146 1
Dieplarement factos
1 1 '/a
3/4
Mixed interstitialcy ( v 1 = Y ? ) 3/& a Values are taken from ref. 9.
Over-all correlation factor
0.78146 1 0.3333
32/33 0.7273 0.8782 0,52G!)
terest in the silver halides. The important feature for our present purposes is that these values are det'ermined by the geometry of the structure and of the mechanism for defect motion. Hence by comparing experimentally determined correlation factors from eq. 4 to a table of theoretical values such as Table I, it is possible to identify the pafticular kind of defect mot'ion in considerable detail. IV. Interpretation of Experimental Results The rather similar behavior of AgCl and AgRr is shown by the experimental results in Fig. 2 and 3. In each case the halogen diffusion is very small, indicating that any contribution by vacancy pair diffusion to the motion of silver ions is negligible; (10) a n d R. (11) (12) (1958). (13)
W. D. Compton, Phys. Rev., 101, 1209 (1956): \Y,D . Compton J. Maurer. J. Phys. Chem. Solids, 1, 191 (1956). R. J. Friauf, Phys. Rev.. 105, 843 (1957). A . S. Miller a n d K. J. Rlaurer, J . Phps. Chem. Solid.?, 4, IQG D. Tannhauser, ibid., 6, 224 (1958).
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Fig. 3.-Diffusion coefficients for AgBr. Ionic conductivity and silver diffusion 0 are from ref. 11, silver diffusion A
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ascribed to the large polarizability of the silver For TlCl,7922CsBr,28and however, all of which -----Activation energies, e.v.have the CsCl structure, it appears that Schottky AgCl AgCl AgBr Process (exptl.) (theory) (exptl.) defects predominate with the anion more mobile in all cases. Thus even the larger polarizability Formation of Frenkel defects 1.6gb 1.76" 1.2P and smaller ionic radius of T1+ compared to Cs+ Vacancy jump 0.33h
