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Diffusion and Electrolytic Conduction in Crystals (Ionic Semiconductors)
1.A survey of the field of electrolytic conduction is given by A. Joffé, The Physics of Crystals, New York, 1928;
1.G. v. Hevesy, Handbuch d. Physik XIII;
1.C. Tubandt, Handbuch d. Exp. Physik XIII, 1 (1932);
1.W. Jost, Müller‐Pouillets Lehrbuch d. Physik IV, 4. In print.
2.First observed experimentally by T. E. Phipps, W. D. Lansing, and T. G. Cooke, J. Am. Chem. Soc. 48, 112 (1926).
3.A. Smekal, Phys. Zeits. 26, 707 (1925).
4.Discussion: see W. Jost, Zeits. f. Physik. Chemie B6, 88, 210 (1929);
4.W. Jost, B7, 234 (1930);
4.A. Joffé, Zeits. f. Physik 62, 730 (1930).
5.W. Jost, Zeits. f. Physik. Chemie B16, 129 (1932).
6.I. Frenkel, Zeits. f. Physik 35, 652 (1926).
6.Of other contributions to this field we mention: H. Braune, Zeits. f. Physik. Chemie 110, 147 (1924);
6.W. Braunbek, Zeits. f. Physik 44, 684 (1927).
6.S. Dushman and I. Langmuir, Phys. Rev. 20, 113 (1922);
6.S. Dushman and I. Langmuir, 22, 357 (1923). , Phys. Rev.
6.A. van Liempt, Rec. Trav. Chim. 51, 114 (1932).
7.It is remarkable, that in the theory of electronic semiconductors the same expression is obtained. See A. H. Wilson, Proc. Roy. Soc. A133, 458 (1931);
7.A. H. Wilson, 134, 277 (1931).
7.R. Peierls, Ergebnisse der Exakten Naturwissenschaften 11, 317 (1932).
8.C. Wagner, Zeits. f. Physik. Chemie B11, 139 (1930).
9.W. Braunbek, Zeits. f. Physik 44, 684 (1927).
10.Work functions, so called “Platzwechsel‐energies” have been calculated before, by Rogowski (Arch. f. Elektrotechnik 18, 123 (1927); and by Braunbek.6 But both of them derived them for the displacement of the lattice of the positive ions as whole towards the negative ones. Though this calculation for Rogowski’s purpose was justified—discussion of the electrical breakdown—its results cannot be used for the process of electrolytical conduction and diffusion which we consider.
11.K. Fajans, Fortschr. d. Physikal. Wissensch. (russ.) V, 294 (1926).
12.A. Reis, Zeits. f. Physik 44, 353 (1927).
13.M. Born (Zeits. f. Physik 79, 62 (1932)) has lately discussed in a similar case the “hole energy,” in connection with the discrepancy between the theoretical value for the energy of the photographic primary process and the empirical value as determined by Pohl and coworkers. Although in this process the polarization energy in the neighborhood of two neutralized atoms is not quite negligible, it is not mentioned by Born. However, in this case the polarization energy is of a smaller order of magnitude than in ours, because the assymmetrical part of the field, produced by neutralizing two neighboring ions can be represented by a dipole field which decreases much more rapidly with the distance than the assymmetrical Coulomb fields around a hole and around a displaced ion at a distant point in the interlattice space.
13.See, however, W. Klemm, Zeits. f. Physik 82, 529 (1933).
14.In the case of the silver halides ε is
15.Unfortunately the nature of the conductivity of silver sulphide is still under discussion (W. Jost u. H. Rüter, Zeits. f. Physik. Chemie B21, 48 (1933).
15.C. Wagner, Zeits. f. Physik. Chemie B21, 25 (1933).
15.We can, however, avoid any of these difficulties by basing our discussion on the rate of diffusion which has been measured for copper ions in small concentrations in silver sulphide (H. Braune, Zeits. f. Physik. Chemie 112, 270 (1924);
15.C. Tubandt, H. Reinhold, Z. Anorg. Chem. 177, 253 (1928);
15.W. Jost u. H. Rüter, Zeits. f. Physik. Chemie B21, 48 (1933).
15.From Braunes measurements we take the diffusion constant . This velocity is as high or higher than that of salts in aqueous solutions !
16.Holes can also be produced by removing ions to surfaces. But as long as only ions of one type are removed this would give an amount, depending on the size of the surface. Such a process, though it might occur, could not be identified with the generally observed conductivity.
17.This approximation for the repulsive potentials may be not very good. But the inaccuracy certainly lies within the limits of other approximations we have made, and a more accurate treatment would be worth while only if a much better approximation could be obtained. This would require that besides considering van der Waals forces (M. Born and J. E. Meyer, Zeits. f. Physik 75, 1 (1932)) one would have to abandon the simple expression for the repulsive forces consisting of an inverse power or an exponential function of the distance. Both assumptions are too rough for such a close approach of the ions though they allow one to calculate good values for the equilibrium energy of the lattice. And furthermore one would have to consider the displacement of ions in the neighborhood both of holes and of migrating ions. Moreover the polarizability may not be assumed to be constant.
17.See also J. E. Meyer, J. Chem. Phys. 1, 327 (1933).
18.Van Vleck, Electric and Magnetic Susceptibilities, Oxford, 1932.
19.By using the expression for a continuous dielectric, as we did above, we should obtain for the polarization energy if we use Goldschmidt’s data for the radius of the Ag‐ion, and for the dielectric constant (this corresponds about to the value for AgCl; the dielectric constant of has not been measured, but cannot be smaller than this value).
20.In the case, where the work function is almost zero, we may assume for the present approximation that a certain fraction of ions is removed from regular places, depending only very little on temperature, whereas it would require much higher energy to remove more ions. It is noteworthy that this result is equivalent to a picture given by G. C. Schmidt (Ann. d. Physik 80, 605 (1926)). He assumed for the explanation of the experimentally found unipolar conductivity, that the crystal is built up of complex molecules, from which only one ion can dissociate with little energy.
21.E. Madelung, Phys. Zeits. 19, 524 (1918).
22.We have not yet discussed the mechanism of transfer of an ion between normal positions and abnormal ones in the interlattice space. It may happen that the potential barrier, which an ion must cross is greater than the energy difference between its initial and final positions. Apparently this additional potential barrier is unimportant for electrolytic conduction, which depends only on the equilibrium concentration of ions in the interlattice space. However, for diffusion this barrier is quite important, because in the process of concentration equalization the exchange of ions in normal positions and in the interlattice space has to be considered. It is necessary, that the microscopic equalization occurs in a time interval very small compared to the time required for the propagation of ions over observable distances. For a detailed discussion one may refer to the treatment of an analogous problem given in a previous article (W. Jost, Zeits. f. Physik. Chemie B9, 73 (1930)).
23.Using the formula for a continuous dielectric to obtain approximate results, one would have to assume for in case of a higher value than in the case of For not the diameter of the cation is of importance, but the distance to the nearest surrounding ions.
24.See also: W. Jost, Zeits. f. Physik. Chemie B16, 129 (1932).
25.Compare A. v. Hippel, Zeits. f. Physik 75, 145 (1932), who discusses the same question for electrons in the case of the electric breakdown of insulators.
26.W. Schottky, H. Ulich u. C. Wagner, Thermodynamik S, 380 (1929);
26.W. Schottky and C. Wagner, Zeits. f. Physik. Chemie B11, 163 (1930);
26.C. Wagner, Zeits. f. Physik. Chemie. Bodenstein‐Festband, S, 177 (1931).
27.This means: extending the crystal beyond its original limits.
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