Ionic and Mixed Conductivity of Solids

P. Kobeko and P. Kurchatov. Leningrad.

We distinguish two principal kinds of electrical conductivity: electrolytic, associated with the transport of matter, and electronic, without any changes whatever in the conducting medium. In pure form, electrical conductivity of the first kind is represented by aqueous solutions of salts, acids, and bases; that of the second kind, by metals. The electrical conductivity of electrolytes, as a general rule, increases with increasing temperature, whereas the electrical conductivity of metals decreases. Between these two principal groups lies a series of substances with distinctive features in their electrical conductivity; these include certain sulfides and oxides of metals with electronic conductivity, which increases, contrary to the usual case, with increasing temperature; a series of silver halide compounds with mixed ionic and electronic conductivity; and a series of insulating lattices constructed ionically and molecularly.

The study of the mechanism by which current passes through all these semiconductors and insulators is of very great importance for atomic physics and the physics of crystal structure; moreover, in this way it is often possible to elucidate the structure of one or another chemical compound. The study of the electrical properties of substances belonging to these groups began comparatively recently, but even now such clear answers have been obtained on very many questions that the mechanism of motion of an ion and an electron inside a solid has become almost as definite as the motion of an ion in a gas or of an electron in a vacuum. It may be expected that

the study of the electrical conductivity of these intermediate groups will make it possible to shed light on the still unclear mechanism of the passage of current through pure metals.

A priori one may assert that the conductivity of all substances in general can be fitted into the basic assumptions of ionic and electronic conductivity. As the first question, it is therefore necessary to determine whether the given substance conducts ionically or electronically, or whether both processes take part in the transport of electricity. The basic criterion in this case is the complete or partial fulfillment of Faraday’s law with respect to the products of electrolysis. Sometimes other regularities are also used as such criteria; among them should be counted the presence of the Hall effect in conductors with electronic conductivity and the presence of polarization in conductors with electrolytic conductivity; however, both of these latter criteria, being indirect, cannot have the decisive significance possessed by the first of those listed here.

Dielectrics with Ionic Conductivity.

Faraday’s Law.

It may be said that the field of dielectrics with ionic conductivity was opened by Warburg. He showed in 1884 that, when a current is passed through glass heated to 300°, Na is deposited at the cathode in full quantitative agreement with Faraday’s law. In these experiments the glass was taken in the form of a test tube, into which mercury was poured; the entire test tube was immersed for investigation likewise in a beaker with mercury. We shall have occasion later to dwell further on Warburg’s experiments. In 1904 Haber and Tolloczko (2) showed the applicability of Faraday’s law to BaCl₂; electrolysis was carried out between a nickel anode and a graphite cathode at 600°C.

The purest existence of ionic conductivity in a solid substance was demonstrated in silver iodide by Bruni and Scarpa (3), by Tubandt and Lorenz (4); here also belong the works of Tubandt and Lorenz

with AgCl and AgBr. The case of AgJ is of particular interest; even at room temperature its electrical conductivity is extremely high \((\sigma_{250}=2.6\ \Omega/\mathrm{cm})\), quite comparable with the electrical conductivity of aqueous solutions of sulfuric acid \((\sigma_{18}=0.7\ \Omega/\mathrm{cm})\), and nevertheless its conductivity is electrolytic in character. Let us note here that the seemingly simple question of separating ionic and electronic conductivity according to the applicability of Faraday’s law is in fact a very difficult experimental problem. Only after the work of Tubandt and his school did it become possible in many cases to determine the true character of the conductivity; moreover, it turned out that for an enormous number of previously investigated substances science had possessed incorrect information. Quite apart from reactions at the electrodes and the formation there of nonconducting layers, the chief difficulty in investigations is the growth of dendrites inside the body from the cathode side. Dendrites of this kind are also observed in liquid electrolytes (the classical example is the lead tree in a solution of lead sugar), but there dendrites are the exception, whereas in solid dielectrics they are the rule. An increase in the current strength through the specimen at constant voltage, deflections of the galvanometer—these facts are well known to everyone working in this field. In transparent crystals one can directly observe this growth of dendrites; in rock salt, for example, at a temperature of \(600\text{--}700^\circ\mathrm{C}\) and at a voltage of \(200\text{--}400\ \mathrm{V}\), one can see violet threads spreading from the cathode side, while at the same time the current strength also increases; on cooling the crystal, the thread breaks up into a countless multitude of ultramicroscopic particles, which impart to the entire specimen a yellow or bluish tint. Such proliferation of dendrites, of course, strongly distorts the results of investigations of the nature of conductivity, often giving precisely the opposite answer; it should be noted in this connection that in many substances dendrites short-circuit the plates after only a few seconds of current flow. The reasons for the rapid sprouting of dendrites in some substances and the slow sprouting in others still remain completely un-

clear. One must think that this entire phenomenon depends on the structure of the insulator; it is known that it proceeds differently in a single crystal, an alloy, and a pressed powder of one and the same substance. It is very difficult to combat the growth of dendrites, especially when in opaque substances they cannot be observed directly. Increasing the length of the specimen under investigation, and working with small voltages, is inconvenient from the experimental point of view and, moreover, does not always help.

Tubandt and Eggert (5) in 1920 gave an extremely ingenious method which, in a whole series of cases, makes it possible to circumvent this difficulty. One may say that this work opened up an entire field; only after this investigation did it become possible to speak seriously about substances with mixed electrical conductivity.

Working together with Lorenz on the electrolysis of silver halide compounds, Tubandt found that in $\alpha \mathrm{AgJ}$ bridges form very soon, only after a very thick layer of silver has been deposited on the electrode. This peculiarity of $\alpha \mathrm{AgJ}$ is preserved throughout the whole region of its existence (from $144.6^\circ$—the temperature of the transition of $\beta \mathrm{AgJ}$ into $\alpha \mathrm{AgJ}$—to $552^\circ$—the melting point of $\alpha \mathrm{AgJ}$). This remarkable property of $\alpha \mathrm{AgJ}$ makes it possible to use it as a means of “protection” against the growth of bridges in the investigation of a number of substances. The general procedure of all these experiments is as follows: cylinders $1 \ \text{cm}$ in diameter and of the same height are pressed from a very pure salt and polished at the ends; then they are placed against one another by their polished surfaces and clamped between metallic electrodes, while, in addition to the cylinders of the salt under investigation, “protective” cylinders of $\alpha \mathrm{AgJ}$ are inserted into the circuit; in the investigation, for example, of $\mathrm{PbJ}_2$, the chain of substances was the following: silver anode—$\mathrm{PbJ}_2$—$\alpha \mathrm{AgJ}$—and platinum cathode; it is clear that no bridges can form in this case.

In this way Tubandt investigated a very large number of different salts: it turned out that electrolytic conductivity is observed in the hexagonal modification $\mathrm{AgJ}$—$\beta \ \mathrm{AgJ}$, $\mathrm{AgNO}_3$, $\mathrm{PbF}_2$, $\mathrm{PbCl}_2$, $\mathrm{PbBr}_2$, $\mathrm{PbJ}_2$, $\mathrm{NaNO}_3$, $\mathrm{CaCO}_3$,

KCl (6) and others. Faraday’s law was fulfilled in all these experiments with an accuracy up to 1%.

Quite recently Zeleni (7), Lukirskii, Shukarev and Trapeznikova (8) showed the electrolytic character of the conductivity of rock salt. Concluding the account of experiments on the study of the nature of ionic conductivity, one cannot fail to mention the extremely interesting work of Peters (9) on the electrolysis of lithium hydride; Faraday’s law is fulfilled here as well quite strictly, and the most remarkable thing in this investigation is that hydrogen is evolved at the anode.

All the experiments described, simple in their scheme, require preparations of great purity; the slightest impurities very strongly distort all the results.

Mobility of ions.

Let us now turn to the characterization of the properties of ionic conductivity and first of all dwell on the transport numbers, which, as is known, determine the fraction of the total current that falls to a given ion. In metals with purely electronic conductivity the transport number for the electron is equal to 1, for the ion—to 0; in liquid electrolytes the values of these numbers vary from 0.2 to 0.8.

In studying transport numbers in solid dielectrics, use is usually made of Hittorf’s method, based on determining the ratio of the mobilities of ions in aqueous solutions. Warburg and Tegetmeier had already shown that, in the electrolysis of glass, only Na ions wander. The same result was obtained by Tubandt for AgJ, AgCl, αAg₂S and αCu₂S; everywhere it turned out that the transport number for the cation is equal to 1.00. Ioffe concludes that electricity is carried purely by cations in crystals of calcite and of selenite, and Zeleni, Lukirskii, Shukarev and Trapeznikova found the same for crystals of rock salt. But in some salts the transport of electricity is due only to anions; in the electrolysis of PbCl₂ Tubandt (10) found that the transport numbers for the anion and the cation are respectively 1 and 0.

The ratio of the mobilities of ions inside an electrolytically conducting substance can be determined from the processes of diffusion of one salt into another. Obviously, the rate of diffusion is directly connected with mobility. Only the ion that moves inside the body during electrolysis diffuses. Thus, if two cylinders are pressed together—one of AgJ, the other of CuJ—and heated to 250°, then, as a result of diffusion through the contact surface, they exchange only cations and in equivalent amounts. Owing to this, the AgJ cylinder becomes lighter, and the CuJ cylinder heavier. Since the contact surface remains completely unchanged and the cylinders are easily separated from one another, the quantity of diffused Ag and Cu can be determined by direct weighing. An analogous result is obtained with Ag₂S and Cu₂S. Similar experiments were also carried out for salts that do not mix with one another, Cu₂S and CuJ, CuJ and Ag₂S. It turns out that in this case as well there is diffusion of cations, while the anions remain immobile. In this case the diffusion rate is less than that calculated from the diffusion coefficient of the two salts, which Tubandt explains by the difficulty of the transition of ions from a lattice of one type into another.

The study of the evaporation of salts enabled Schmidt to note a number of curious relationships. When salts are heated, ions fly off from their surface, and the number of ions torn away is an indicative function of the temperature. Determining the ratio \(\frac{e}{m}\) for these torn-away ions, Schmidt found that in most cases the ions that fly off are the same as those that move during the electrolysis of the salt; certain deviations, for example in the case of CuJ₂, which are observed with respect to this general rule, are due to various secondary phenomena. The cause of all these curious data for transport numbers is not yet fully clarified at present; it is undoubted, however, that these numbers are determined by the structure of the ions and, above all, by their atomic radii.

A curious example of the influence of the magnitude of the atomic radius on the mobility of the corresponding ion is provided by

electrolysis of quartz with an amalgam of lithium and sodium as the anode. If the electrolysis of a plate is carried out so that the current passes in the direction of the principal axis of the crystal, then both lithium and sodium ions can move in the lattice; but if the current passes in the perpendicular direction, then the Na ions no longer pass, and only for Li ions does the possibility of passage still remain. In quartz, in the direction of the principal axis the distances between atoms are greater than in the perpendicular direction. This, of course, also explains the different electrical conductivity of quartz in these directions.

A very interesting question concerning the absolute magnitude of the mobility of ions in heteropolar lattices has not yet been solved in any detail. Some data on this question may be found in the work of A. F. Ioffe (12) on the electrolysis of rock salt with different electrodes. When Li ions pass through NaCl, Li cannot be detected in the crystal itself, although an amount of \(0.01\,m\) in \(1\,\mathrm{cm}^3\) could already have been observed. Calculation so far shows that during the time while the current was changing and, consequently, the ions had not yet reached the opposite electrode, less than \(0.001\,m\) of substance had time to enter the crystal. Subsequently, only this amount is present in the crystal under steady current; all the remaining ions pass into the electrode and can be detected there. The mobility of Li ions in NaCl at a temperature of \(700^\circ\), according to these experiments, turns out to be equal to \(3 \cdot 10^{-4}\,\mathrm{cm}\). The absolute mobility of an ion can also be calculated from the magnitude of the jump in electrical conductivity when a body passes from the solid state into the liquid. Ioffe (13) assumed for these calculations (some investigators do not share this point of view) that at the melting temperature the degree of dissociation approaches 1, and thus the jump in electrical conductivity must be attributed exclusively to the different mobilities of the ions in the one state and in the other. If one calculates, on the basis of such considerations, the mobility of an ion for a solid at the melting temperature, then it turns out to be of the same order as the mobility determined from the rate of displacement of foreign ions in the lattice.

P. KOBEKO AND I. KURCHATOV

Dependence of Electrical Conductivity on Temperature

Electrolytic conductivity is determined, on the one hand, by the number, and on the other, by the mobility of the charge carriers. In dielectrics constructed in a purely heteropolar manner, only a very small part of the total number of ions can take part in electrical conductivity; only those ions can move whose thermal energy of motion is greater than the energy binding the ion in the lattice. An increase in temperature, by increasing the thermal energy of motion, thus changes the number of ions capable of taking part in electrical conductivity; at the same time it also changes the mobility of the ion. In the general case it does not seem possible experimentally to separate the influence of temperature on the number and on the mobility of the ions. Joffe succeeded in solving this problem only for quartz crystals. Here such a separation proved possible thanks to the existence of the so-called phenomenon of “electrical hardening” of quartz. Its conductivity remains increased after heating and subsequent cooling. It should be thought that ions torn out by thermal energy, upon cooling, do not return to their places; the number of charge carriers remains corresponding, therefore, to the maximum heating temperature, whereas their mobility, of course, corresponds to the temperature of investigation. From a comparison of the electrical conductivity at temperatures \(t_1\) and \(t_2\), then upon rapid cooling of the crystal from \(t_2\) to \(t_1\), and determination of the new electrical conductivity at \(t_1\), Joffe found that, upon heating from \(0^\circ\) to \(100^\circ\), the number of mobile ions increases by a factor of 100, and their mobility increases by a factor of 200. Neither \(\mathrm{CaCO_3}\) nor \(\mathrm{NaCl}\), substances with a simpler lattice, gave such a hardening effect; in them the electrical conductivity is always unambiguously determined by the temperature. The change of electrical conductivity with temperature, for all investigated heteropolar lattices, is expressed by two different formulas, long since known to science. One of them is:

\[ \sigma = c_1 e^{c_2 t}\ \text{or}\ \log \sigma = a + bt, \tag{1} \]

second:

\[ \sigma=C_1 e^{-\frac{C_2}{T}} \quad \text{or} \quad \log \sigma=-\frac{A}{T}+B \tag{2} \]

here \(a, b, c_1, c_2\), just like \(A_1, B_1, C_1\), and \(C_2\), are empirical constants, \(T\) is the temperature, \(\sigma\) is the conductivity. Formula (1) has no physical meaning and over almost the entire temperature interval gives the same values as formula (2). Formula (2) is based on the thermodynamic determination of the number of free ions. It is much better justified by experiment than formula (1), especially in the region of low temperatures. It is interesting to note that if the experimental results are plotted in coordinates in which the logarithms of the conductivity are laid off along the ordinate axis, and quantities reciprocal to the temperatures along the abscissa axis, then the slope of the straight lines that represent the measurements turns out to be approximately the same for all the substances investigated; the coefficient \(A\) is approximately the same for all heteropolar dielectrics.

Besides temperature, a whole series of other influences makes the lattice conducting. Such influences as light and corpuscular radiations we shall not consider; we shall dwell only on the influence of impurities on electrical conductivity. Different pieces of one and the same natural mineral conduct differently, which can be understood only on the assumption of the influence of impurities on conductivity. To this same group of phenomena one must assign Tubandt’s work with mixed crystals; the perturbing action on the lattice is also exerted by proximity to the surface of the part of the crystal lattice under consideration. As Geveshi showed (15), crystallites (pressed powders or solidified melts) conduct better than single crystals. Geveshi explains this result by the presence of a large number of surfaces inside the specimen; this causes a large number of ions moving in the electric field. According to Geveshi, disturbances of the lattice by foreign atoms are the reason why the rule of additivity is not fulfilled for mixed crystals. If, however, one takes into account the lowering of the melting temperature of the alloy and relates

values of the electrical conductivity to the corresponding temperatures, we may convince ourselves of the sufficiently good fulfillment of the rule of additivity, as was shown by Geveshi.

Polarization.

With a constant potential difference, the current passing through the crystal continuously decreases, and when the electrodes are briefly grounded a reverse current appears, which carries almost the same quantity of electricity as passed in the direct current. These phenomena were studied in especially great detail by Joffe (16), in collaboration first with Röntgen, then with Kirpicheva, and more recently with a number of persons in the Leningrad Physico-Technical Laboratory. The phenomenon of the decay of the direct current and the occurrence of the reverse current have long since been given the name of polarization phenomena. In order to elucidate the nature of polarization in crystals, it was first of all important to establish the distribution of the potential in a polarized crystal. It turned out that it is necessary to distinguish two essentially different types of polarization: in some crystals (for example, in quartz) the distortion of the field upon polarization extends over large distances from the electrodes; in others (calcite) it is concentrated at a distance of the order of \(10^{-4}\) cm from the electrode. Let us consider these two classes separately.

Type I (quartz). The measurement of the potential distribution was carried out by Joffe by the so-called probe method. The dielectric plate was surrounded by wire rings connected to electrometers; these rings and electrometers, when a voltage is applied to the crystal, will obviously be charged to the potential of the cross-section enclosed by them. In Fig. 1 the field distribution is shown at the first moment (line \(abc\)) and after 10 minutes (line \(a'b'c'\)) for quartz at a temperature of \(200^\circ\)C. Joffe assumed that the cause of the polarization here is the same as in gases, and by analogy with gaseous phenomena concluded that ions of both signs move in quartz. The potential distribution indicated above was found in a very large series of substances: in crystals of rock salt, aluminum alum, copper ...

cuprous oxide, saltpeter, fluorspar, glasses, mica, ebonite, various resins, etc. In all these cases it was possible to show that the phenomenon of current decrease upon the application of a voltage is caused by the formation of an opposing electromotive force of polarization, associated with the growth of space charge at the electrodes. If from the potential difference \(V\) applied to the material one subtracts the electromotive force of polarization \(P\), then the current \(J\) always satisfies the condition:

\[ \frac{V-P}{J}=R=\mathrm{const}. \]

Fig. 1.

The dependence of the current on the potential difference in the range from \(1\ \mathrm{V/cm}\) to \(1\cdot 10^{-6}\ \mathrm{V/cm}\), for different thicknesses of plates, agrees within the limits of observational error \((0.5\%)\) with Ohm’s law. These experiments of Joffe made it possible to explain a number of so-called “dielectric anomalies.” We shall not dwell here on all these works; we shall cite here one of the most interesting cases of the application of the theory of high-voltage polarization to such phenomena of dielectric anomalies. It is known that when any dielectric operates in an alternating field, it heats more than under the same voltage in a constant field. The amount of dis—

... of heat released in a dielectric does not obey Joule’s law; for some materials it exceeds this amount sometimes by hundreds and thousands of times and has even received the special name of Siemens heat. In the work of Sinelnikov and Walter (17) in Ioffe’s laboratory it was shown that in fact this Siemens heat is Joule heat: the phenomenon is very well explained if one takes into account that measurements of the dielectric’s resistance by investigators of dielectric losses were carried out under such conditions that the electromotive force of polarization had already risen to a value very close to the applied potential difference. Taking into account the distortion of the field in the dielectric during the passage of current, it is easy to calculate the Siemens heat, knowing \(R\) in formula (1) and the form of the function \(P=f(t)\); calculation and experiments performed by the above-named investigators showed that the heat released in a dielectric in an alternating field obeys Joule’s law to an accuracy of \(0.5\%\).

Fig. 2.

The passage of current through quartz is accompanied by a number of curious features that are not found more ...

neither in any of the dielectrics investigated up to now. We have already seen that in quartz there exists a phenomenon of “tempering”; in it there is also observed a gradual increase of the current upon successive switchings, which is shown in Fig. 2. We shall give here the explanation of this phenomenon given by Joffe: “In the absence of a field we assumed thermal dissociation and recombination, which compensate one another. The current accumulates ions of one sign near the electrodes; these ions have no possibility of recombining and are conserved as additional ions. If, having changed the direction of the current, these ions are made to move back through the entire crystal, then a large number of ions will take part in the current and the electrical conductivity will increase. When these excess ions become distributed over the whole cross-section, then at first, while they have not yet had time to recombine, the electrical conductivity must be increased.”

Class II (calcite). Calcite crystals give a different distribution of potential during the passage of current. Here the entire applied potential falls across a thin layer at the cathode; when this layer is ground off (it is sufficient to grind off \(1.5—2\ \mu\)), the polarization disappears completely. As in the preceding case, Ohm’s law remains valid here up to field gradients of \(1\cdot10^{6}\ \mathrm{V/cm}\), and the current strength for any moment of time may be represented by the formula:

\[ J=\frac{V-P}{R}. \]

When negative ions (for example, \(\mathrm{NO}_3\)) enter the crystal from the cathode side, polarization is not formed. Neither temporary heating of the calcite nor the passage of current leaves any traces on its electrical conductivity.

Here we also consider it necessary to note the extraordinarily great electric strength of thin layers of calcite at the cathode. As measurements of the potential distribution with probes have shown, a layer of \(10^{-4}\ \mathrm{cm}\) can withstand a gradient of up to \(2\cdot10^{8}\ \mathrm{V/cm}\)—a voltage which exceeds by 500 times the usual strength of solid dielectrics. To such

to the “anomalously” high strength later on. The observations set forth compel Joffe to think that in calcite only negative ions, \(\mathrm{CO}_3\), and even \(\mathrm{O}\) ions move, with very low mobility. The degree of dissociation in calcite is large, which may perhaps be connected with the fact that \(\mathrm{CuCO}_3\) readily dissociates chemically into \(\mathrm{CuO}\) and \(\mathrm{CO}_2\).

Forming.

For determining the character of ion motion in a solid dielectric, the study of the electrical conductivity of substances in which thin, poorly conducting layers of electrolysis products are formed at the electrodes has proved very fruitful. The existence of such poorly conducting layers was first shown by Warburg and Tegetmeier, who studied the passage of current through glass heated to \(300^\circ\mathrm{C}\). In glass the main part of the current is caused by the displacement of \(\mathrm{Na}\) ions at the anode; in this way a layer of \(\mathrm{LiO}_2\) is formed, whose conductivity is approximately \(10^4\) times less than the conductivity of the glass. Warburg’s experimental arrangement has already been described above. When the voltage is switched on, the current through the glass decreases comparatively slowly; if the experiment is carried out at a temperature of \(300^\circ\), then after half an hour it is only \(1/100\) of its initial value. In contrast to the phenomenon of polarization, this decrease of current is irreversible: when the electrodes are short-circuited, the quantity of electricity flowing in the circuit is \(10^5\) times less than the charging quantity. When the voltage is applied again after discharge of the specimen, the current in it returns to its initial value, but falls to a small value in a time \(10^5\) times shorter than during the first passage of current; the charged quantity of electricity then becomes equal to the first discharge quantity, and upon subsequent switchings the phenomenon becomes reversible. All these regularities are, of course, self-evident: upon the first application of voltage the current flowing through the specimen forms a thin layer of \(\mathrm{LiO}_2\) at the anode and, together with this, changes the distribution of potential inside the dielectric; moreover, only a negligible fraction of the

of the total quantity of electricity that has passed through the dielectric. Upon repeated applications of voltage, the further growth of the layer proceeds very slowly, and the principal amounts of electricity passing through the dielectric are already due to redistributions of potential, so that the process becomes reversible. Warburg determined the thickness of the LiO₂ layers; it proved to be a quantity of the order of \(10^{-4}\) cm.

Fig. 3.

Since 1884, when Warburg’s work was carried out, the phenomenon of formation—as we shall call these processes of formation of poorly conducting layers in dielectrics—has not been examined in detail. In 1925–1926, in the laboratory of Ioffe, the experiments of Warburg were reconsidered by P. Kobeko, I. Kurchatov, and K. Sinelnikov, and the phenomenon of “formation” was studied further.

First of all, it was shown that “formation” is a fairly general phenomenon for an entire class of dielectrics. Phenomena entirely analogous to those observed by Warburg were found in \(Li_2B_4O_7\), \(Na_2B_4O_7\), \(K_2B_4O_7\), certain varieties of mica, AgJ, \(Ag_2S\), and other ser-

pure compounds. As in the experiments of Warburg, the formation in the listed dielectrics could be avoided (with the exception of mica; no experiments were carried out here) by using, as the anode, an amalgam of the corresponding cation. In Fig. 3 the decrease of the current with time is presented for the cases \(Li_2B_4O_7\), \(Na_2B_4O_7\), and \(K_2B_4O_7\); the current strength is plotted on the ordinate axis, time on the abscissa axis, and the corresponding dielectrics are marked on the curves. All the experiments were carried out at one and the same temperature, with plates of one and the same thickness, and with the same electrode area. As is seen from the curves, the electrical conductivity of these salts follows the order of magnitude of the mobilities of the ions Li, Na, and K, whereas the forming quantities of electricity and the magnitudes of the residual current proceed in the reverse order with respect to the mobilities of the corresponding ions.

Kobeko and Kurchatov (18), proceeding from the phenomena of formation, gave a theory of the detecting action of \(AgJ\) and \(AgS\). If columns of pressed powders of these two substances are included in a direct-current circuit, using a platinum point and a plate as electrodes, then it is easy to detect unipolarity in the current circuit. This phenomenon was discovered by Frey (19) in 1926 and explained by the different mobility of the ions Ag and S; Frey’s theory, however, as the experiments of Kobeko and Kurchatov showed, proved to be incorrect. The cause of the unipolarity lies in the fact that, owing to the different current density at the point and at the plate, the rates of growth of the nonconducting layer of J and S at the different electrodes are very different. If the point serves as the anode, then the current through the system will fall very rapidly; with the anode on the plate, the current will likewise decrease, but much more slowly, and as a result the system will detect. With such a mechanism of detection, the system will rectify only for a certain time, until the layer of nonconductor at the plate reaches a considerable thickness. Experiment showed that this is in fact observed: when the specimen is connected into an alternating-current circuit, the coefficient of unipolarity very rapidly falls to zero, while the resistance in the circuit increases by several thousand times. Probably this

the detection mechanism extends to a number of other detectors, above all to all iodide compounds, and to some sulfur compounds (some sulfur compounds conduct electronically); of course, similar phenomena are observed in all the substances listed above.

Most important, however, for understanding the mechanism of ionic conductivity proved to be the study of the electrical properties of thin layers of material deposited at the electrodes. The study of these properties was carried out in the laboratory of Ioffe by Kurchatov and Sinelnikov, chiefly with thin layers of SiO\(_2\) in glass. The thickness of the layers was determined roughly by grinding off the dielectric layer from the anode side, after passing a current through the glass, and more accurately by measuring the discharge quantities of electricity. Special experiments showed that the entire potential applied to the specimen falls across the thin layer at the anode; then the thickness of this layer can easily be determined from the equality:

\[ d=\frac{\varepsilon V}{4\pi Q}, \]

where \(\varepsilon\) is the dielectric constant of the thin dielectric layer, \(V\) is the applied potential, and \(Q\) is the discharge quantity of electricity. Measurement of the potential by probes showed that, at a potential difference of 2,000 volts, in a SiO\(_2\) layer \(10^{-4}\) cm thick the currents through it still remain very small, and somewhat above this voltage they begin to increase sharply. In this SiO\(_2\) layer, fields of \(2\cdot 10^7\) V/cm are thus attained; at the same time, the maximum field that can be produced in quartz under ordinary conditions can reach only \(1\cdot 10^6\) V/cm.

Such anomalous strength is observed not only in thin layers of SiO\(_2\), but also in B\(_4\)O\(_7\) and other products of deposition on electrodes in the class of dielectrics under consideration. Measurements of the potential distribution in formed glasses showed that this strength is inherent only in very thin layers of dielectrics. If, in the same glass, the forming process is continued further, allowing the layer to grow

if it is several microns, then the limiting gradient begins to fall and, for layers of \(10\ \mu\), approaches the ordinary values, reaching only \(2\text{--}3\cdot 10^{6}\ \mathrm{V/cm}\). Measurements of this limiting gradient at different glass temperatures showed that, with increasing temperature, it decreases somewhat, remaining, however, in the temperature range from \(0^\circ\) to \(200^\circ\mathrm{C}\) still greater than the usual breakdown gradient for thick layers of the same material.

The anomalous strength of thin layers served as an impetus for the development of the idea of impact ionization in solid dielectrics. The presence of impact ionization readily makes it possible to understand the result of these experiments, if one takes into account that in thin layers the ionization process cannot develop strongly. Before setting forth the experiments that made it possible to substantiate this point of view, it is necessary to dwell on the conditions determining the breakdown gradient of a solid dielectric. When a voltage is applied to a crystal, heat is released in it; this heat is given up to the electrodes and escapes through the lateral surfaces of the specimen; as the voltage is increased, the amount of heat released will increase and, finally, at some \(V_k\) the dielectric begins to heat up, since the removal of heat is less than its generation. As soon as this happens, the heating process will bring the material avalanche-like to melting, owing to the negative temperature coefficient of resistance of ionic conductors. Proceeding from such considerations, Wagner (20), and after him Rogowski and Karman (21), theoretically developed this so-called “thermal theory” of breakdown. They arrived at the following dependence of the logarithm of the critical voltage on temperature:

\[ \log E_{kr} = \frac{A}{T} + B, \]

where \(A\) and \(B\) are coefficients depending on the thermal conductivity and other constants of the material, and \(T\) is the absolute temperature. A decrease in thickness causes strengthening of the dielectric, since in thin layers the heat removal to the electro-

dam. However, the strengthening given by this theory is very insignificant and cannot explain the large breakdown gradients of thin layers; moreover, the thermal theory proves valid for the dependence of \(E_{np}\) on \(T\) only in the range of temperatures sufficiently close to the melting temperature of the dielectric. The most complete verification of the thermal theory was carried out experimentally by N. N. Semenov, A. F. Walter, and L. D. Inge (22) for glass, rock salt, porcelain, and several other materials. The experiment showed that for glass, for example, in the temperature interval from \(50^\circ\) to \(600^\circ\), the experiment fully confirms the theory, giving not only the correct course of the curves but also the absolute values of the breakdown voltages; at temperatures from \(150^\circ\) to \(220^\circ\mathrm{C}\), however, the breakdown gradient ceases to increase, remaining almost constant.

Ionization in Solid Heteropolar Dielectrics

These deviations from the thermal theory led Ioffe, Kurchatov, and Sinelnikov (23) to propose that breakdown in the region of room temperatures has an ionization character. With an ionization mechanism of breakdown, the thickness of the dielectric has a very appreciable effect on its breakdown voltage, as is known in the case of gases. Let us consider, following the above-mentioned authors, the motion of an ion in a crystal. The electric field accelerates the ion until it reaches such a velocity that the average frictional force becomes equal to the force of the electric field \(Ee\).

Fig. 4 explains the picture of the gradual increase in the velocity of the ion. As Ohm’s law shows, the terminal velocity \(V_0\) is proportional to the field \(E\),

\[ J = neV_0 = neuE, \]

where \(n\) is the ion density, \(J\) is the current per \(1\ \mathrm{cm}^2\), and \(u\) is the mobility of the ion. The terminal value of the velocity \(V_0\) is reached over a certain distance \(\lambda_0\), which can be determined as

the ion’s acceleration distance. Some measurements have shown the authors that this \(\lambda_0\) is very small, of the order of \(10^{-5}\), \(10^{-6}\) cm.

Let now, in the field \(E\), the velocity \(V_0\) assume such a value that the relation holds:

\[ \frac{1}{2} m V_k^2 = eP, \tag{1} \]

where \(P\) is the ionization potential of the lattice; in this case, over distances of \(\lambda_0\), each ion moving in the dielectric will liberate new ions from the lattice. Let the thickness of the dielectric be \(D\); then the number of newly appearing ions, caused by the motion of one ion initially liberated by thermal motion, will be expressed by the formula:

Fig. 4.

\[ n = 2^{\frac{D}{\lambda_0}}. \]

With a uniform volume distribution of the initial ions with density \(n_0\), the average density of ions in the dielectric under impact ionization will be:

\[ n = \frac{n_0}{\log 2}\,\frac{\lambda_0}{D}\left(2^{\frac{D}{\lambda_0}} - 1\right). \tag{2} \]

If \(\frac{D}{\lambda_0}\) is a large number, then the current increases so strongly that breakdown is inevitable. A sufficient condition for breakdown

will be expressed by formula (1), and, thus, the field \(E_{\mathrm{imp}}\) is given by the relation:

\[ E_{\mathrm{imp}}=\frac{V_k}{u}=\frac{1}{u}\sqrt{\frac{2eP}{m}} . \tag{3} \]

This field does not depend on \(T\), which is also observed experimentally. The situation will be different if the thickness of the dielectric is not very large in comparison with \(\lambda_0\). If, for example, the ratio \(\dfrac{D}{\lambda_0}\) is equal to 10, then the current in the presence of impact ionization will increase by \(2^{10}\) times in comparison with the usual one, that is, it will be more than 1,000 times greater.

Usually, even immediately before breakdown, the currents through the specimen are of the order of \(10^{-11}\,\mathrm{A}\), and increasing them to \(10^{-8}\) is not dangerous for the dielectric. As a condition for the breakdown of thin layers, Joffe proposes the condition

\[ n_0 \lambda^{\frac{D}{\lambda}}=\mathrm{const.}, \tag{4} \]

i.e. he requires constancy of the strength of the breakdown current. As a first approximation, (4) may be replaced by

\[ \frac{\log n_0}{\log 2}+\frac{D}{\lambda}=\mathrm{const.} \tag{5} \]

The quantity \(\lambda\) in this formula is not identical with the acceleration distance \(\lambda_0\). Here \(\lambda\) is the distance over which, in any field, the ion reaches the velocity \(V_k\); \(\lambda_0\) is, therefore, the greatest of all \(\lambda\).

As a first approximation one may neglect the loss of energy by the ion during acceleration; then the energy of the ion is determined by the potential difference at the ends of the path which it has traversed. Condition (5) becomes

\[ \frac{\log n_0}{\log 2}+\frac{D}{X} = \frac{\log n_0}{\log 2}+\frac{V}{P} =\mathrm{const.}, \tag{6} \]

since

\[ \lambda=\frac{P}{E}=\frac{PD}{V}. \]

The condition (6) at constant temperature is expressed by a very simple and extraordinarily interesting equality:

\[ V=\mathrm{const.};\ ED=\mathrm{const.}, \tag{7} \]

which gives an answer to the question of the breakdown voltage of thin layers. As for the number of ions produced by ionizations, under these simple assumptions it is expressed by the formula:

\[ n=\frac{n_0}{\log 2}\frac{P}{V}\left(2^{\frac{V}{P}}-1\right). \tag{8} \]

Such, in general outline, is this theory. The phenomena of dielectric strengthening that are required here were already partly described by us in setting forth the forming processes; however, those experiments gave very little for the theory of ionization phenomena; even the basic dependence \(ED=\mathrm{const.}\) could not be established there, since the thickness \(D\) of the formed layer was increasing all the time. The principal material for this theory is provided by the experimental data of the above-mentioned authors on the study of the electrical properties of thin layers of glass, mica, and a number of other dielectrics. It is not without some interest to give a brief description of the experimental method, while noting in advance that the thickness of the layers studied in some experiments was only \(2\cdot 10^{-6}\) cm. Thin layers of mica were obtained by successive splitting of thick specimens with tweezers, or better with a needle; thin layers of glass were blown from thin-walled tubes of small diameter. For breakdown and for measuring the current strengths, these thin plates were placed on a paraffin cube, the mercury in a tube serving as one of the electrodes; the other electrode was a drop of mercury, which was carefully poured on with a capillary pipette. The results for the dependence of the breakdown gradient on thickness are presented in Figs. 5 and 6, where the data are compared for breakdowns with different electrodes: mercury (\(\circ\)), water (\(\times\)), and soot electrodes (\(\bullet\)). As Fig. 5 shows, in the range of thicknesses from \(1.5\cdot 10^{-4}\) cm to \(1\cdot 10^{-5}\) cm, the breakdown potential does not depend

from the thickness, as is also required by formula (7). The curve in Fig. 5 is a hyperbola in accordance with the formula \(ED=\mathrm{const}\). This becomes especially noticeable if the data of Fig. 5 are replotted

Fig. 5.

in logarithmic coordinates. The formula \(ED=\mathrm{const}\) then gives

\[ \log E=-\log D+\mathrm{const}, \]

a straight line with an angle of inclination to the abscissa axis of \(135^\circ\); as can be seen from Fig. 6, such a course is fully confirmed by experiment. In the range of thicknesses from \(1\cdot 10^{-5}\,\mathrm{cm}\) to \(3\cdot 10^{-6}\,\mathrm{cm}\), deviations from the ionization theory begin: the breakdown gradient ceases to increase, reaching a limiting value of \(1.5\cdot 10^{8}\ \dfrac{\mathrm{volts}}{\mathrm{cm}}\) for glass.

Fig. 6.

This result was explained by the rupture of the lattice by the forces of the electric field; a field of \(1.5\cdot 10^{8}\ \dfrac{\mathrm{volts}}{\mathrm{cm}}\) thus gives the true electric strength of the substance. In 1926 Borman was

a calculation was made of the limiting electric field for rock salt; it turned out that there this field reaches a value of 70 million volts, while the strength measured experimentally, 150 million volts, thus gives an interesting confirmation of the theory of ionic lattices. The experiments with various electrodes, of which mention was made above, were carried out in order to exclude the explanation of the constancy of the breakdown gradient in this range of thicknesses by destruction of the dielectric by an electron current.

Fig. 7.

Fields of the order of \(1.5 \cdot 10^8 \dfrac{\text{volts}}{\text{cm}}\) can tear electrons out of the electrodes; thus a large current may arise, which will bring the dielectric to catastrophe; since the surface fields by which the onset of the tearing-out of electrons from the electrode is determined are very different for mercury, water, and carbon, and since the experiment gave one and the same limiting gradient in all cases, one must settle on the hypothesis of rupture of the lattice by the forces of the electric field.

For the theoretical calculation, the data on the dependence of the current strength in plates of different

thickness on the gradient. In Fig. 7 are presented the results of experiments on glass. Along the ordinate is plotted here the current, along the abscissa—the gradient; the curves correspond to different thicknesses. These curves agree well with formula (8), provided that the currents in plates of different thickness are compared only at one and the same gradient. As a function of the gradient, however, the experimental data give changes in the quantity \(P\); it decreases as the gradient increases. This result is a consequence of the assumption that the ions move without loss of energy inside the dielectric. The authors developed a more rigorous theory of all the ionization phenomena, which already embraced all the experimental material. Without presenting the theory itself here, we shall confine ourselves to setting forth the results obtained. It turned out that the ionization potential for glass, the quantity \(P\) in formulas (6) and (8), reaches a value of 10 volts; the length \(\lambda\)—the distance between separate ionizations—varies from \(10^{-5}\ \text{cm}\) to \(10^{-7}\), depending on the gradient of the electric field.

Concluding the chapter on impact ionization, let us mention that this property is a very general one for dielectrics with ionic conductivity; it was observed in a mixture of wax with paraffin, in rosin and other resins, whose specific resistance is higher than \(10^{14}\ \Omega/\text{cm}\).

ABSOLUTE VALUE OF THE ELECTRICAL CONDUCTIVITY

It may be said that theoretical calculations in this field do not exist up to the present time, if one requires of these calculations a quite definite answer to the question of the absolute value of the electrical conductivity. Of all the attempts to compute this quantity, the most successful should be considered the quite recent work of Braunbek (24). Considering theoretically the motion of an ion in crystals of rock salt in connection with melting phenomena, this scholar arrived theoretically at the same values of the electrical conductivity and of the temperature coefficient as those measured by Ioffe in experiment. This attempt by Braunbek is extremely

important in the sense that all theories which attribute the electrical conductivity of a dielectric to exclusively local destructions of the lattice lose their right to exist. Such theories include those of Smekal (25) and Rogovskii (26). According to Smekal, within any single crystal there exists an extremely large number of very small cracks, with dimensions of \(10^{-6}\), \(10^{-7}\) cm. On the internal cleavage surfaces of such cracks the binding energy of an ion in the lattice is less than inside the crystal; therefore dissociation proceeds easily here, and the entire current through the crystal is due to the motion of such ions. Rogovskii holds the same views, differing from Smekal only as to the nature of the carriers of electricity. Smekal believes that the passage of current is caused by the motion of ions dissociated on the surfaces of the cracks, whereas Rogovskii attributes it to the motion of ion pairs inside the cracks. Both theories arose as a result of attempts to find a way out of the very large values of the electrical conductivity of dielectrics, on the one hand (Smekal), and, on the other, of the small values of dielectric strength at low temperatures (Rogovskii).

As for the second consideration, under certain experimental conditions, as we know, in thin layers of dielectrics at low temperatures it is possible experimentally to obtain the limiting theoretical gradients; Smekal’s considerations, however, may be thought to have become superfluous after Braunbek’s work. Let us now turn to a review of the experimental data concerning the absolute magnitude of the electrical conductivity of this type of dielectric. Here we encounter uncertainties: for the conductivity of poor conductors we know only the order of magnitude, and, as regards the electrical conductivity of semiconductors, the data of different authors differ greatly. The only material beyond reproach may be considered to be the measurements of Joffe and Kirpicheva on crystals obtained by the method of successive recrystallizations. These measurements established, first of all, the constancy of the magnitude of the electrical conductivity for crystals of a given substance and opened the way to further study of the absolute electrical conductivity of a dielectric. This small theoretical and

the experimental material on the question of the magnitude of the electrical conductivity shows that these aspects of the question are still very far from being resolved and, chiefly, with regard to its theoretical part: the complexity of the question for the time being precludes the possibility of a mathematical solution. Much more thoroughly studied is the domain of general relations between electrical conductivity and a number of other quantities. These relations sometimes make it possible to understand the course of electrical conductivity in a series of particular chemical compounds. Here the works of G. Hevesy and Blitz (27) are of greatest interest. With very rare exceptions \((\alpha \mathrm{AgJ})\), the transition of a substance from the solid state to the liquid is accompanied by a jump in electrical conductivity. This jump is caused by the difference in the fixation of ions in the solid and liquid states. Different substances give different magnitudes of the jump, and one may think that this jump characterizes the degree of perfection of the lattice, the degree of strength of the ion’s bonds within the solid body.

As a rough estimate of the imperfection of the lattice in this sense Hevesy proposed taking the coefficient

\[ a=\frac{\sigma_{\text{solid}}}{\sigma_{\text{liquid}}}. \]

No connection of the coefficient \(a\) with the coefficient of expansion of the dielectric, the lattice constant, or the magnitude of the ions could be found, but Hevesy noted a curious parallelism between \(a\) and the quantity \(E\)—the energy of transition from \(\overset{+}{X}\overset{-}{Y}\) to \(XY\). In the case of the transition \(\overset{+}{K}\overset{-}{Cl}\) to \(\mathrm{KCl}\), this quantity \(E\) is equal to 3 calories; the electron affinity of chlorine is 96 calories, the ionization work of potassium is 99 calories; for \(\mathrm{AgJ}\) this energy turns out to be of an entirely different order. It is equal here to

\[ (-59+328)=269 \]

calories. In the table the quantities \(a\) and \(E\) are compared. This parallelism allows one to think that within the lattice thermal motion may cause a transition of the ionic structure of the compound into the molecular one, with these “neutral” centers serving as centers of dissociation of the lattice. It is interesting to note that in \(\mathrm{AgJ}\), where the quantity \(E\) reaches a very large value and where we can, thus, suppose a high frequency of transitions and strong-

… destruction of the lattice; mechanical tests reveal a very high elasticity of the material.

Geveshi further notes the different behavior of thermal conductivity and electrical conductivity in solid ionic conductors. The former is due to elastic waves in the lattice; these elastic waves are scattered little, only in the case of “sufficiently pure” lattices; the thermal conductivity of a dielectric is the worse, the farther the lattice is from the ideal structure. For the processes of electrical conductivity, on the contrary, destruction of the

TABLE

lattice is a favorable factor; owing to imperfection, dissociation in it increases. The opposite course of thermal conductivity and electrical conductivity with temperature illustrates this point of view. In RbF and CsF—very “pure lattices” with small $E$—we have very good conductors of heat and poor conductors of electricity; in AgJ, AgCl, and AgBr$_2$, on the contrary, large $E$ values are combined with good insulating properties with respect to thermal processes and poor ones with respect to electrical processes. Geveshi obtained a very interesting development and confirmation of this idea in Tubandt’s recent experiments, which will be set forth in the chapter on semiconductors with mixed conductivity.

Mixed Electrical Conductivity.

The question, studied over the course of many years, of whether, alongside ionic electrical conductivity in solid salts, electronic conductivity also exists, first received its resolution in the work of Tubandt with Ag₂S (28). This salt exists in two modifications, α and β. The first is stable above 179°, the second at lower temperatures. The transition from one modification to the other takes place spontaneously under the influence of temperature. Mixed electrical conductivity is possessed by β Ag₂S. Tubandt investigated the applicability of Faraday’s law in this salt according to the usual scheme, using AgJ as a protective cylinder at the cathode, according to the scheme:

Pt — cathode — α AgJ — β Ag₂S — β Ag₂S — β Ag₂S — Ag — anode;

in this way, however, it was possible to work only in a very narrow temperature interval, between 179° (the transition point of α Ag₂S into β Ag₂S) and 145° (the transition point of α AgJ into β AgJ, which has no protective action). Tubandt nevertheless showed that at 170° 19.23% of the entire current is carried electronically, and 17.1% at 150°. The work on β Ag₂S was completed only in 1927, when Tubandt used, instead of the protective cylinder of AgS, an aqueous solution of KNO₃. A disk of gauze was applied to a glass tube with a flat bottom, in which there was a small aperture, and 4 cylinders of Ag₂S were pressed against it. The tube was lowered into a solution of KNO₃, which served as the cathode, while the anode was a silver plate. By special experiments it was ascertained that Ag₂S does not dissolve and does not swell in the liquid at the cathode. It turned out that after 24 hours of contact with the solution (without passage of current) the weight of the Ag₂S cylinders remained entirely unchanged. When current was passed, no dendrites formed. Thanks to this method, Tubandt succeeded in investigating the electrical conductivity of α Ag₂S below 100° and in establishing the following dependence of it on temperature:

Thus, for each temperature there exists a definite equilibrium between the amount of electricity transferred in one way and in the other. With increasing temperature, the equilibrium shifts toward an increase in metallic electrical conductivity. The inverse dependence was indicated by Tubandt (29) for $\gamma\mathrm{CuJ}$, which also possesses mixed electrical conductivity. Studying this substance from $0^\circ$ to $402^\circ$ (the transition point to $\beta\mathrm{CuJ}$), Tubandt found the following distribution between ionic and metallic conductivity:

As we see, metallic conductivity decreases with increasing temperature, and below $200^\circ$ CuJ conducts purely electronically; the distribution of conductivity as a function of temperature is completely opposite in $\gamma\mathrm{CuJ}$ and $\beta\mathrm{Ag}_2\mathrm{S}$, the latter at low temperatures being a purely ionic conductor. Besides CuJ and $\mathrm{Ag}_2\mathrm{S}$, $\mathrm{Cu}_2\mathrm{S}$, CuCl, and $\mathrm{CuBr}_2$ behave analogously. The presence in one substance of electrical conductivity of two kinds is explained by Tubandt as due to the simultaneous presence of atomic and ionic lattices. Under the influence of temperature one lattice may pass into the other, and together with this the character of the electrical conductivity also changes. The transition of an ionic lattice into an atomic one with increasing temperature, which we had in $\mathrm{Ag}_2\mathrm{S}$, had been observed earlier as well. In an atomic lattice the valence electrons are in a state analogous to that in metals and in purely electronically conducting substances: oxides and sulfides of heavy metals. This transition of lattices should take place the more easily, the smaller the electron affinity of the anion and the greater the ionization work of the metal.

If one recalls Geveshi’s general scheme for such transitions, it is interesting to note that in substances with mixed electrical conductivity there is usually no jump in electrical conductivity at the melting temperature. As can be seen from

Fig. 8, in CuJ the conductivity falls only slightly upon solidification, but strong changes in this quantity are observed upon transition into the region of mixed electrical conductivity. In conclusion it should be noted that in all these salts with mixed electrical conductivity, metallic conductivity is characteristic of modifications stable at low temperatures; for high-temperature modifications, on the contrary, electrolytic conductivity is characteristic.

Fig. 8.

In conclusion let us dwell on the mechanism of electrical conductivity of semiconductors. Before Tubandt the generally accepted theory was that of Königsberger. According to Königsberger, the number of electrons increases together with the temperature, owing to the dissociation of (excited) atoms, while the mean free path decreases with increasing temperature. As a result the resistance of a semiconductor may be represented by the formula:

\[ W = W_0(a + bt + ct^2)e^{-\frac{Q}{T}} . \tag{9} \]

Thus the resistance at first falls on heating (the first factor, the term \(e^{-\frac{Q}{T}}\), plays the determining role) and then begins to increase, owing to the decrease in the magnitude of the mean free path. Although formula (9) agrees rather well with experiment, at the present time it is difficult

to preserve it. The experimental material on which Königsberger based his work is not free from objections: we know that in the salts \(Ag_2S\) and \(Cu_2S\), which, among others, figure in his work, the conductivity is electronic only to an insignificant degree, while the main part of the current is carried by ions. Utermann’s data for \(MoS_2\) and \(Sb_2S_3\), which Königsberger also used, are unreliable, since Utermann took no measures to eliminate bridges.

In many cases the increase in conductivity with heating can be explained while retaining the assumption of an unchanged number of electrons participating in electrical conduction at different temperatures; it is necessary only to take into account the conditions for the passage of electrons from crystal to crystal in an aggregate. Phenomena at the boundaries of individual crystals can play a substantial role, as can be clearly seen in the experiments of Ryskevitch, who investigated the electrical conductivity of a graphite crystal (\(4 \times 1 \times 0.04\) mm). The conductivity of graphite proved to be \(2\frac{1}{2}\) times greater than that of silver; the temperature coefficient was \(0.004\), as in all other metals, and had the same sign. Besides graphite, silicon and titanium should be transferred from the class of semiconductors to the category of ordinary metals. The very considerable influence of surface conditions in \(Si\) on the passage of electrons affects the detecting capacity of this metal (32).

The mechanism of electrical conductivity of the remaining semiconductors still remains unresolved up to the present time; one may hope that in the near future science will obtain here as well a definite answer on the basis of the work of Tubandt and Ryskevitch.

LITERATURE.

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10) Tubandt. ZS. f. anorg. Chem. 115, 105. 1920.

11) Schmidt. Ann. d. Phys. 75, 337. 1924; 80, 588. 1926.

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13) Ioffe. Ibid.

14) Ioffe. Proceedings of the Polytechnic Institute in Petrograd. 21, 62. 1915. Ann. d. Phys. 72, 461. 1923.

15) Hevesy. ZS. f. Phys. 10, 80. 1822; 36, 481. 1926.

16) Ioffe. Ann. d. Phys. 72, 461. 1923; 20, 919. 1906; 41, 449. 1913; 61, 1. 1921.

17) K. Sinelnikov and A. Walter. ZS. f. Phys. 40, 786. 1927.

18) P. Kobeko and I. Kurchatov. Proceedings of L. F. T. L. 1928.

19) Trey. Phys. ZS. 23, 849. 1925.

20) Wagner. J. A. I. E. E. 1922.

21) Rogovsky and Karnan. Arch. f. Elektrotechn. 13, 153. 1924. Ibid. 13, 174. 1924.

22) Semenov, Walter and Inge. ZS. f. Phys. 32, 273. 1925. Elektrichestvo No. 9. 1927.

23) Ioffe, Kurchatov and Sinelnikov. Reports of the Academy of Sciences 65. 1927. Publications from the Massachusetts Institute of Technology II, 46, 133. 1927.

24) Braunbek. ZS. f. Phys. 44, 684. 1927.

25) Smekal. ZS. f. Phys. 36, 238. 1926; Wien. Anz. 159. 1923; Phys. ZS. 26, 707. 1925; Verh. d. D. Phys. Ges. 630, 52. 1925.

26) Rogowsky. Arch. f. Elektrochem. 18, 123. 1927.

27) Hevesy and Blitz. ZS. f. anorg. Chem. 133, 306. 1924.

28) Tubandt and Reinhold. ZS. f. anorg. Chem. 160, 22. 1927.

29) Tubandt, Rintroff and W. Jost. ZS. f. anorg. Chem. 165, 195. 1927.

30) Waterman. Phys. Rev. 21, 54. 1923.

31) Ryschkevitch. ZS. f. Elektrochem. 29, 474. 1923.

32) Streintz and Wesely. Phys. ZS. 21, 42, 316, 367. 1920.