MODERN THEORIES OF METALLIC ELECTRICAL CONDUCTIVITY1

P. G. Lapinskii, Kiev.

12. The equilibrium theory of metallic conductivity of A. Joffermann.

In 1923 A. Joffermann [^65] published an interesting work on the theory of metallic conductivity. Joffermann’s work is somewhat distinctive and, in its scheme, stands apart from most of the new works on the theory of metallic conductivity. The author, proceeding from the hypothesis of the chemical dissociation of an atom into an ion and an electron, attempts to explain the temperature variation of the electrical conductivity of both good and poor conductors. Joffermann’s work is partly related to that of Koenigsberger [^31], but has a more general character. Taking the mean free path of the free electrons to be constant, or almost constant, the author explains the change of electrical conductivity with temperature by the change in the concentration of free electrons. Like Koenigsberger, the author assumes the chemical dissociation of a neutral atom into a positive ion and a negative electron, subjecting it to the laws of chemical equilibrium.

This dissociation may be expressed conventionally by the equation:

\[ A \rightleftarrows \overset{+}{A} + \overline{\text{e}} . \tag{58} \]

The law of mass action gives:

\[ \lg \left\{ C^+ \cdot \frac{C\text{e}}{C} \right\} = k, \tag{59} \]

where \(C\), \(C^+\), \(C\text{e}\) are, respectively, the concentrations of atoms, positive ions, and electrons in equilibrium. In this case \(C^+ = C\text{e} = n\), equal to the number of free electrons per unit volume. \(C = N - n\), where \(N\) is the number of positive nuclei per unit volume. The preceding equation gives:

\[ \lg \left[ \frac{n^2}{N - n} \right] = k, \tag{60} \]

whence:

\[ n=\frac{1}{2}e^k\left(-1\pm\sqrt{1+4Ne^{-k}}\right) \tag{61} \]

or approximately:

\[ n=N^{\frac12}e^{\frac{k}{2}} \tag{62} \]

\(k\)—the equilibrium constant—is determined by the equation:

\[ \frac{\partial k}{\partial T}=\frac{q}{RT^2} \tag{63} \]

where \(q\) is the dissociation energy, \(R\) the gas constant.

If \(v_1-v_2\) is the potential difference corresponding to the electrical work of dissociation of an atom, then:

\[ q=e(v_1-v_2)=\varphi-\psi; \tag{64} \]

\(e\) is the charge of the electron. \(\psi\) is the energy of liberation of a free electron from a solid body, depending on temperature:

\[ \psi=\psi_0+\frac{3}{2}RT \tag{65} \]

(according to Richardson); \(\varphi\) is the energy of removal of a bound electron from the body. Putting

\[ q=(\varphi_0-\psi_0)-\frac{3}{2}RT, \tag{66} \]

where \(\varphi_0\) and \(\psi_0\) are constants depending on the substance, we obtain:

\[ k=-\frac{(\varphi_0-\psi_0)}{RT}-\frac{3}{2}\lg T+\mathrm{const}. \]

Then:

\[ n=AN^{\frac12}T^{-\frac34}e^{-\frac{(\varphi_0-\psi_0)}{2RT}}; \tag{67} \]

\(n\) is the number of free electrons per unit volume. \(\varphi_0-\psi_0\) is the mean energy of reaction at \(0^\circ K\). From the “classical” expression for electrical conductivity

\[ \sigma=\frac{ne^2\lambda}{2\sqrt{3mRT}} \]

it follows that:

\[ \sigma=A'N^{\frac12}T^{-\frac54}e^{-\frac{(\varphi_0-\psi_0)}{2RT}}, \tag{68} \]

where

\[ A'=\frac{Ae^2\lambda}{2\sqrt{3mk}}, \tag{69} \]

or for the specific resistance

\[ \rho = C N^{-\frac{1}{2}} T^{\frac{5}{4}} e^{\frac{\varepsilon_0-\psi_0}{2kT}} . \tag{70} \]

In the case of a multivalent atom with valence \(\nu\) we have the dissociation equation:

\[ A \rightleftarrows A' + \nu e . \tag{71} \]

Then

\[ n = (\nu N)^{\frac{1}{\nu+1}} e^{\frac{k}{\nu+1}} = A(\nu N)^{\frac{1}{\nu+1}} T^{\frac{3}{2(\nu+1)}} e^{-\frac{\varepsilon_0-\psi_0}{(\nu+1)RT}} . \tag{72} \]

For \(\sigma\) one obtains the following, rather complicated, expression:

\[ \sigma = A'(\nu N)^{\frac{1}{\nu+1}} T^{-\frac{\nu+4}{2(\nu+1)}} e^{-\frac{\varepsilon_0-\psi_0}{(\nu+1)RT}} . \tag{73} \]

Next the author establishes the dependence of the resistance on valence and temperature. Depending on the valence \(\nu\), the magnitude of the resistance \(\rho\) is found to be proportional to:

\[ \begin{aligned} \text{for } \nu &= 1:\quad \rho \sim T^{1.25} e^{\frac{\varepsilon_0-\psi_0}{2RT}} \\ \text{” } \nu &= 2:\quad \rho \sim T e^{\frac{\varepsilon_0-\psi_0}{3RT}} \\ \text{” } \nu &= 3:\quad \rho \sim T^{0.875} e^{\frac{\varepsilon_0-\psi_0}{4RT}} \\ \text{” } \nu &= 4:\quad \rho \sim T^{0.8} e^{\frac{\varepsilon_0-\psi_0}{5RT}} . \end{aligned} \tag{74} \]

And as a function of the temperature \(T\):

\[ \rho = C T^a e^{\frac{b}{T}}, \tag{75} \]

where \(C\), \(a\), \(b\) are characteristic constants of the given substance. For substances with high electrical conductivity at ordinary temperature, \(a\) is almost equal to 1; \(\frac{b}{T}\) is very small; \(e^{\frac{b}{T}}\) is close to 1. Therefore for such substances approximately:

\[ \rho = CT . \tag{76} \]

For poorly conducting substances (Se, B, and many compounds) \(\frac{b}{T}\) is large, while the factor \(T^a\) has little influence on the specific resistance. Therefore \(\rho\) decreases exponentially with temperature. For conductors of transitional type (Si, Ge, and some compounds) \(a\) is close to 1; \(b\) has an intermediate value, and \(\sigma\) has a minimum at the temperature

\[ T = \frac{2(\nu+1)(\varepsilon_0-\psi_0)}{(3\nu+4)R}. \tag{77} \]

The author gives in his work a series of values for \(a\) in the formula \(\rho \sim T^a\). For a number of metals the experimental values are constantly close to the theoretical ones. In particular, \(a\) is greater than 1.25 for the alkali metals and smaller for metals of high valence, excluding Fe and Ni. Atomic dissociation must depend: 1) on the property of the atom; 2) on the structure of the body.

Superconductivity, according to Utermann, may be caused by polymorphic transformations that change the difference \(\varphi_0-\psi_0\). For metals \(\varphi_0\) is somewhat smaller than \(\psi_0\). For poor conductors \(\varphi_0\) is considerably greater than \(\psi_0\). Utermann’s formula for poor conductors gives the same result. Only \(T^a\) is replaced by \(1+at+\beta t\). The introduction of \(\varphi_0\) and \(\psi_0\) connects the quantity \(\sigma\) with the thermoelectronic effect and the photoelectric effect, namely: \(\varphi_0\) and \(\psi_0\) give the work and the liberation energy of electrons from the given substance. For metals \(\psi_0\) is somewhat greater than \(\varphi_0\). Hence it follows, according to Utermann, that the transition of electrons is accomplished chiefly through the intra-atomic space. For compounds \(\varphi_0>\psi_0\), and the electrons, according to Utermann’s interpretation, in this case move in the free interatomic space.

Utermann identifies the constants introduced by him: \(\psi_0\) with the thermionic function, and \(\varphi_0\) with the photoelectric function. Physically this means that the photoelectrons must be ejected from the atoms of the substance, while the thermoelectrons are formed from “already free” electrons. Such a hypothesis makes it possible to connect both these effects closely with the phenomenon of metallic conduction. Utermann changes the usual expression for the thermionic current and gives the following, more complicated, expression, taking \(n\) as varying with \(T\):

\[ i=B''(vN)^{-\frac{1}{v}+1}T^{\left[2-\frac{3}{2(v+1)}\right]}e^{-\frac{\psi_0}{RT}+\frac{(\varphi_0-\psi_0)}{(v+1)RT}}, \tag{78} \]

which gives, for a monovalent metal:

\[ i=B''N^{\frac{1}{2}}T^{\frac{5}{4}}e^{-(\varphi_0-\psi_0)2RT}. \tag{79} \]

Utermann’s theory opens up an interesting possibility of finding, from the magnitudes of metallic conductivity, the values of \(\varphi_0\) and \(\psi_0\) and comparing them with those obtained directly. Further, important for this same purpose is the systematic study of weakly conducting bodies, especially near the minimum of their resistance and at the points of polymorphic transformations. For metallic compounds \(\varphi_0\) is considerably greater than \(\psi_0\). For example, for CuO \(\varphi_0=5\ V\), \(\psi_0=0.35\ V\). And \(\varphi_0-\psi_0=4.65\). Hence the specific resistance of CuO must be proportional to

\[ Te^{\frac{(\varphi_0-\psi_0)}{3RT}}=Te^{\frac{18000}{T}}. \]

Utermann’s theory makes it possible to establish the dependence of electrical conductivity on illumination. For poor conductors the absorption of light

must change $\varphi_0-\psi_0$ and increase electrical conductivity. In this way Utermann explained the change in the resistance of molybdenite \([65_2]\). The influence of light will be considerable in those cases where the exponential factor is large. Therefore, the electrical conductivity of metals, when they are illuminated, should change at low temperatures. The absorption of light facilitates the emission or dissociation of electrons and increases their concentration.

Utermann’s work is very valuable in that it embraces a very broad range of conductors, in particular semiconductors and compounds of metals possessing metallic conductivity. In it many questions of the theory of metallic conductivity that had earlier been left in the background are worked out in detail—for example, the influence of valency and of illumination—and a number of questions awaiting experimental resolution are posed.

13. The Influence of Atomic Weight on the Electrical Conductivity of Elements.

Comparison of the magnitude of the electrical conductivity of metals with their atomic weight led to a number of attempts to establish a regular connection between these quantities. Schimank \([55]\) indicated that for low temperatures the resistance curves of different metals $\frac{\rho_t}{\rho_{273}}$ proceed in the order of their atomic frequencies. The periodic change of electrical conductivity with the atomic weight of the elements was analyzed in the works of Benedicks \([2]\) and I. E. Grüneisen \([20]\). Benedicks characterizes each element by its atomic electrical conductivity. By this he means the product of the specific electrical conductivity of the element $K$ and its atomic volume $V$. Proceeding from theoretical considerations, Benedicks takes the expression $KV$ to be proportional to the frequency of atomic vibrations, i.e.:

\[ \frac{KV}{\nu}=\mathrm{const}=C, \tag{80} \]

where the constant $C$ characterizes the electrical conductivity of the given substance. In considering the variation of the quantity $C$ with the atomic weight of the elements, one obtains: 1) the elements of the first group have the largest value of $C$ in their semiperiod; 2) whereas among the elements of the first group K, Rb, Cs have large values of $C$, Cu, Ag, Au, which also belong to the first group, have small values. In general, the change of $C$ runs parallel to the change of the atomic volume $V$. Grüneisen also makes use of the quantity of atomic electrical conductivity, but compares it for different atoms not at one and the same temperature, but at temperatures constituting one and the same fraction, for example $\frac{1}{2}$, of a certain

of the characteristic temperature \(\theta_r\) of each substance. Grüneisen gives for the value of the specific resistance of metals \(w\):

\[ w=\operatorname{const}\left(\frac{T}{\theta_r}\right)F\left(\frac{T}{\theta_r}\right), \tag{81} \]

where \(F\) is Debye’s universal function for specific heats. With the aid of this expression, knowing \(\theta_r\) for a given metal, one can calculate the atomic electrical conductivity according to Grüneisen. If we take

\[ T=\frac{1}{2}\theta_r,\quad \text{then}\quad K_{\theta_r/2}\cdot V=\operatorname{const}=C' \tag{82} \]

for the given element. If one compares the variation of the constants \(C\) and \(C\) of Benedicks and Grüneisen with atomic weight, the course of the changes is in general similar. The same question is treated in the work of Simon \([54]\). Whereas Benedicks and Grüneisen establish the value of the atomic electrical conductivity by dividing the value \(K\) of the specific conductivity by \(\frac{1}{V}\), the number of gram-atoms in \(1\ \mathrm{cm}^3\), Simon introduces the condition that the value of the electrical conductivity be referred to a gram-atom in the form of a cube. The electrical conductance of any such cube, taken perpendicular to one of its faces, is expressed by the product \(KV^{\frac{1}{3}}\). Simon proposes to regard this value as a measure of atomic conductivity; here \(K\) refers to the same “corresponding” temperatures as in Grüneisen. When comparing the values of \(KV^{\frac{1}{3}}\) according to Simon for different elements of the periodic system, it turns out that \(KV^{\frac{1}{3}}\) is very large for Na, K, Cu, Rb, Ag, Cs, Au and considerably smaller for the remaining elements, i.e. all members of the first group of the periodic system have the greatest value of atomic conductivity.

Finally, Z. Epstein \([17]\) pointed out that for elements with a polycrystalline structure the expression

\[ CK_{\tau}V^{\frac{1}{3}}, \tag{83} \]

may serve as a measure of the mean interatomic electrical conductivity. The temperatures \(\tau\) at which the electrical conductivity \(K\) is measured must, in order to obtain a comparable (adequate) measure of the electrical conductivity of the elements, be chosen in such a way that for all elements they constitute one and the same fraction of the boiling temperature of these elements at normal pressure. The values of \(K_{\tau}V^{\frac{1}{3}}\) calculated in this way give regular variations in the periodic system of elements.

14. Electrical Conductivity and Structure

Summarizing the theories considered—those of Stark, Wien, Thomson, and others—we must acknowledge that the creation of a new, more perfect theory of metallic conduction, one developed in detail and embracing the whole extensive domain of physical phenomena belonging to it, including thermoelectric, galvanomagnetic, and thermomagnetic phenomena, is a matter for the future. But from consideration of the accumulated experimental material, those paths are becoming apparent along which fruitful development of the theory may be expected. The old theories did not take sufficiently into account the structure of the conductor and the undoubted results produced by changes in this structure. We now have some material concerning the crystalline structure of solid metals, which makes it possible to divide metallic conductors into liquid, physically homogeneous ones (mercury, molten metals) and solid metals, usually consisting of conglomerates of small crystals. In the case of a chemically “pure” ^1) metal, the junction of crystals of various sizes, separated from one another by geometrical surfaces which we shall call contact surfaces, and through which the electric current must pass in going from one crystal to another. The sizes of the crystals of a given metal vary and depend on the thermal and mechanical treatment of the metal. Most cast metals and alloys treated by hot working have from 60 to 120 crystals per 1 linear inch (see Carpenter, Nature 1923, No. 2805), i.e. from 216,000 to 1,728,000 crystalline grains per 1 cm³. Often, especially for steel, the sizes of the crystalline grains are smaller. In drawn wire, usually used for electrical measurements, the sizes are still smaller. In molybdenum wire, as Sykes found, there are more than 2,000 of them per 1 cm. With an increase in the number of grains, the area of the contact surfaces correspondingly increases as well. Metals may be brought into such a state of pulverization that crystals cannot be detected. This pulverized state apparently has some resemblance to the liquid state, and is called amorphous. Such a structure, for example, is possessed by potassium at ordinary temperature, as Mac Kegan found (1922). At \(t—150^\circ\), potassium reveals a distinctly crystalline structure. Under favorable conditions metallic crystals may attain considerable size. Thus Osmond and Fremont as early as 1905 (C. R. 1905) investigated iron crystals of several cm³. Sauveur in 1912 discovered that, with careful

^1) F. Thomson [^60] gives the composition of one of the “pure” metals, namely iron:

stretching and then heating of the metal, crystals of large dimensions are obtained. G. Carpenter described a method for obtaining aluminum crystals from 0.5 to 2 cubic inches. In recent years, precise methods have already been developed for obtaining such monocrystalline metals.

It has long been known (see, e.g., Moreh, Physical States of Matter, 1912, p. 98) that, depending on the size of the crystalline grains, the physical properties of a metal change, in particular its electrical properties. Thus a plate of ordinary silver and a silver plate that has acquired a fine-grained structure from forging, when immersed in dilute acid, acquire a small potential difference (about \(1/10\) V), like two different metals. From two such varieties of silver one can make a thermoelectric pair with a thermoelectric EMF of 0.17 microvolt per \(1^\circ\) C. (For an Ag—Cu pair the thermoelectric EMF is, for example, 0.28 microvolt per \(1^\circ\) C.) Ni under the same conditions gives still greater electromotive forces than Ag. The change in specific electrical resistance as a function of the number \(n\) of crystalline grains in 1 cm of length was investigated by Thompson \([^{60}]\) for “pure” iron (99.87%), varying the number of crystals \(n\) per 1 linear cm from 10 to 690. He obtained the following results:

From this he obtained, for the specific resistance of iron containing no amorphous layers but only crystals, the following expression:

\[ \rho = 6.83 + 1.72 n \cdot 10^{-3}\ \text{microohms per }1\ \text{cm}^3. \tag{84} \]

This change in the resistance of iron, established by F. Thompson, from 7.986 to 7.152 as the sizes of the crystals change, convincingly shows that the character of the crystalline structure affects the magnitude of the electrical conductivity.

If we have a pure metal, then two cases are possible when considering its crystalline structure: 1) when the crystals are in direct contact with one another and 2) when they are separated by thin layers of amorphous metal. Both in the first case and especially

in the second case there arises the question of the role of the boundary surfaces, which we have called contact surfaces. Unfortunately, this question is at present scarcely developed, and with respect to it one can only make more or less probable suppositions. In the second case we may expect the appearance of an additional transition resistance as a result of internal thermoelectric phenomena at the contact surfaces of crystalline and amorphous metal, in accordance with Releigh’s hypothesis \[47\].

Steinberg \[57\], investigating thin layers of Ag, Cu, Fe and comparing their electrical conductivity with the conductivity of continuous masses of metal, came to the conclusion that the resistance of these metals increases with decreasing size of the crystalline grains and with increasing number of contact sites. Therefore the construction of a satisfactory theory of electrical conductivity requires due attention to questions of the influence of structure on the magnitude of electrical conductivity.

15. Results of the Review.

A review of the most important theoretical works that have appeared in the last 15 years on the explanation of the phenomena of metallic conductivity compels one to draw the following conclusions:

1. Within the metal, conditions are created that favor the free separation of electrons from the atom. Whether the work of separation is equal to 0 or to some quantity \(q\) cannot as yet be decided with sufficient definiteness. It is probable that this quantity \(q\), at least for conductors with low conductivity (semimetals, compounds), is different from zero (Uoterman). In any case, a sufficiently close approach of the atoms and the superposition of the electromagnetic fields created by them facilitates the liberation of electrons from the action of the intra-atomic forces holding them. In favor of this consideration speaks the circumstance that a decrease in the volume of a metal very often increases its electrical conductivity.

2. The separation of an electron from an atom and its free existence should not be confused with the free displacement of the electron. Apparently, the spatial displacement of the electron is limited either by definite surfaces or lines (Stark), or by channels connected with the structure of the crystal lattice (J. Thomson, 2 theories; P. Bridgman). The increase of the amplitude of atomic vibrations with temperature limits the paths of possible displacements of electrons and is one of the most important causes increasing the resistance of metals (Bridgman).

3. The frequency of atomic vibrations is an important factor for the displacement of electrons (Bridgman, Wien, Vereide). However, the relation between the frequency of atomic vibrations and the magnitude of electrical conductivity has, in different authors, different interpretation and character.

1. The close connection between electrical conductivity and the laws of atomic vibrations makes the theory of metallic conductivity dependent on the theory of the structure of the solid body.

2. The question of the energy of moving electrons and of its dependence on temperature has apparently not yet been definitively resolved. In general, there is a tendency to endow the electron with a negligible amount of thermal energy in comparison with the energy of atomic vibrations (J. J. Thomson, H. Lorentz, Borelius, and others). A dependence of the electron’s energy on temperature has not been entirely rejected.

3. The free path of electrons under the action of an electric field is considerably greater than the interatomic distance (Bridgman, Borelius, Stark, J. J. Thomson, 2nd theory).

4. The enormous magnitude of the electrical conductivity observed in pure metals near \(T=0\) abs. is due above all to the absence of atomic vibrations, which facilitate the motion of electrons within the metal.

5. With increasing valence of a metal, the number of electrons participating in the transport of current changes (Vereide, Oetermann). The connection between electrical conductivity and valence has been most clearly worked out in the theory given by Oetermann.

6. The simplest mechanism of conductivity should be recognized as occurring in single-crystal pure metals. Any complication of the structure and chemical composition complicates the mechanism of the phenomena of electrical conductivity (alloys, metallic compounds).

7. A change in the structure of metals causes a change in their conductivity (F. Thompson) and in other electrical properties, a fact not sufficiently taken into account by all existing theories of metallic conductivity.

8. The study of the microstructure of metals and of the structure of their crystal lattice opens the way to the quantitative development of questions of metallic conductivity (J. J. Thomson) and to establishing their connection with the mechanism of thermal phenomena (P. Bridgman).

1. LITERATURE:

2. K. Baedeker. Phys. ZS. 11, p. 809, 1910.

3. C. Benedicks. Jahrb. d. Rad. u. El. 13, p. 351, 1916; 14, p. 471, 1917.
4. P. W. Bridgman. Phys. Rev. 9, p. 269—89, 1917; 19, p. 114—134, 1922.
   ” Proc. Americ. Acad. 52, 1917. Proc. Nation. Ac. Amer. 3, p. 10, 1917.
5. N. Bohr. Studier over Metallernes Electrontheori, Diss. Copenhagen, 1911.
6. N. Bohr u. D. Coster. ZS. für Physik. 12, p. 342, 1923.
7. T. Bialobjeski. Phil. Mag. (6) 45, p. 168, 1923.
8. A. Bramlay. Phil. Mag. (6) 46, p. 1053, 1923.
9. G. Borelius. Annal. d. Physik 57, p. 231, 1918; 57, p. 278—87, 1918; 58, p. 489, 1919.
   ” Phil. Mag. (6) 40, p. 746, 1920.
10. H. Cassel. ZS. für Physik 31, p. 477, 1925.

LITERATURE

1. J. Clay. Jahrb. d. Rad. u. El. 8, p. 383, 1911; 12, p. 209, 1915; 19, p. 59, 1922.
   Diss. Leiden, 1908; Comm. Phys. Lab. Leiden No. 107d, p. 31, 1908.

2. C. Crommelin. Phys. ZS. 21, p. 274, p. 300, p. 331, 1920. Jahrb. d. Rad. u. El. 19, p. 33, 1922.
   K. Crommelin. Uspekhi Fiz. Nauk 5, issue II, pp. 161–76, 1925.

3. P. Drude. Annal. d. Physik. 1, p. 566, 1900; 3, p. 369, p. 514, p. 589, 1900; 7, p. 687, 1902.
   ” Phys. ZS. 1, p. 161, 1900; Intern. Electr. Congr. St. Louis 1, p. 317, 1904; Jons, Electrons, Corpuscles, p. 162, 1905.

4. P. Debye. Annal. d. Phys. 33, p. 441, 1910.

5. J. Dewar and I. Fleming. Phil. Mag. (5) 36, p. 271, 1893; 40, p. 303, 1895; Proc. Roy. Soc. London, 60, p. 72, 1896; 60, p. 425, 1897; Electrician 37, p. 267, 1896.

6. Evan Everdingen. Arch. Néerl. (2) 6, p. 294, 1901; Comm. Phys. Lab. Leiden. 72, p. 1, 1901.

7. A. Einstein. Het Natuurk Lab. Leiden. Jubelband H. K. Onnes. Leiden. 1922, p. 429.

8. Z. Epstein. ZS. für Physik 32, p. 620, 1925; Zhurn. R. F. Kh. O. 57, p. 341, 1925.

9. J. Frenkel. ZS. für Physik 29, p. 214, 1924.

10. R. Gans. Annal. d. Phys. 20, p. 293, 1906.

11. W. Giese. Wied. Annal. 37, p. 576, 1889.

12. P. Gruner. Verh. D. Phys. Ges. (6), p. 509, 1908; Phys. ZS. 10, p. 48, 1909.

13. E. Grüneisen. Ber. D. Phys. Ges. 20, 1918; 15, pp. 186–200, 1913; Annal. d. Phys. 39, p. 276, 1912; Phys. ZS. 19, p. 382, 1918.

14. L. Holborn. Annal. d. Phys. 6, p. 242, 1901.

15. L. Holborn u. Wien. Wied. Annal. 59, p. 211, 1896.

16. Höjendahl. Phil. Mag. (6) 48, p. 349, 1924.

17. F. Haber. Ber. d. D. Chem. Ges. p. 506, 1919.

18. F. Hauer. Annal. d. Phys. 51, p. 215, 1916.

19. E. H. Hall. Proc. Americ. Acad. 50, p. 67, 1914; Proc. Nat. Acad. Sc. 3, p. 163, 1917.

20. G. Jäger. Wien. Ber. 117, p. 843, 869, 1908.

21. J. Jeans. Phil. Mag. (6), 17, p. 773, 1909.

22. I. Königsberger. Phys. ZS. 8, p. 237, 1907; Jahrb. d. Rad. u. El. 4, p. 158, 1907; Annal. d. Phys. 32, p. 179, 1910.

23. W. H. Keesom. Phys. ZS. 14, p. 670, 1913; Versl. Akad. Amsterdam. Mai, 19 Suppl. No. 306.

24. J. Kunz. Phil. Mag. (6) 16, p. 764, 1908.

25. Lord Kelvin. Phil. Mag. (6), 3, p. 257, 1902.

26. E. Kretschmann. Annal. d. Phys. 74, p. 405, 448, 1924.

27. H. Kamerlingh-Onnes. Comm. Phys. Lab. Leiden. No. 99, No. 119, 120b, 122b, 124c, 133a, bc, d, 139f, 140b, c, 141b; Suppl. No. 29, 34b, 35; 4 Conseil de Physique Solvay 1924. Suppl. No. 50a.

28. H. Kamerlingh-Onnes u. G. Holst. Comm. Leid. No. 141a, 142a.

29. H. Kamerlingh-Onnes u. W. Tüyn. Comm. Leid. No. 160, 167a.

30. H. Kamerlingh-Onnes u. I. Clay Comm. Leid. 95d, 99c, 107c, 197; Suppl. No. 17.

31. G. H. Livens. Phil. Mag. (6) 29, p. 173, 425, 1915; 30, p. 105, 112, 287, 434, 549, 1915.

32. F. A. Lindemann. Sitz. Ber. Berlin 1911, p. 316.

33. H A. Lorentz. Versl. Akad. Amsterdam. 11, p. 729, 1903; Proc. Roy. Acad. Amstrd. 5, p. 608, 1903; 7, p. 438, 585, 684; Versl. Akad. Amsterd. 13, p. 493, 1904; p. 565, 710, 1905; Ergebnisse und Probleme d. Elektronentheorie, p. 43, 1905.

34. L. Lorentz. Wied. Annal. 13, p. 422, 582, 1881.

35. Mc Lennan. Natur 112, p. 135, 1923.

36. W. Meissner. Naturwiss. 1925, p. 695. Jahrb. d. Rad. u. El. 17, p. 229, 1920; Phys. ZS, p. 689, 1925.

37. W. Nernst. Sitz. Ber. Berlin 1911, p. 311.

1. Lord Rayleigh. Sc. Papers 4, p. 232.

2. E. Riecke. Wied. Annal. 66, p. 353, 545, 1190; 1898; Gött. Nachr. 1898, p. 48, 137; Annal. d. Phys. (4), 4, p. 378, 1901; Phys. ZS. 2, p. 217, 1901; 3, p. 182, 1902; 10, p. 508, 1909; Jahrb. d. Rad. u. El. 3, p. 24, 1906.

3. O. W. Richardson. Phil. Mag. (6) 23, p. 594, 1912; 30, p. 295, 1915.

4. M. Reinganum. Annal. d. Phys. 2, p. 398, 1900; Phys. ZS. 10, p. 355, 645, 1909; 11, p. 673, 1910.

5. F. B. Silsbee. Journ. Wash. Acad. Sc. J. 6, p. 597, 1916; Bul. Bureau Standards 14, p. 301, 1918, No. 307. 1917.

6. A. Schuster. Phil. Mag. (6) 7, p. 151, 1904.

7. I. Stark. Jahrb. d. Rad. u. El. 9, p. 188, 1912.

8. F. Simon. ZS. für Physik 27, p. 157, 1924; ZS. f. Phys. Chem. 109, p. 136, 1924.

9. H. Schimank. Annal. d. Phys. 45, p. 706, 1914.

10. R. Schenk. Phys. ZS. 8, p. 239, 1907; Annal. d. Phys. 32, p. 261, 1910.

11. Steinberg. Phys. Rev. 21, p. 22, 1923.

12. Scaupy. Phys. ZS. 1920, p. 597.

13. J. J. Thomson. Rapp. Congr. Intrn. Paris, 1900, 3, p. 138; Journ. inst. Electr. Eng. 38, p. 455, 1907; Electrician. 58, p. 839, 1907; Eclair. Electr. 51, p. 21, 94, 129, 1907; Inst. of Metals J. 14, p. 3, 1915; Natur. 96, p. 494, 1915 (Dec.); Phil. Mag. (6) 30, p. 192, 1915; 44, p. 658, 1922.

14. F. Thompson. Phil. Mag. (6) 31, p. 357, 1916.

15. W. Tuyn. Doktor. dissertation. Leiden, 1924.

16. M. Travers and Gwyer. Proc. Roy. Soc. 74, p. 74, 1915.

17. T. Vereide. Annal. d. Phys. 55, p. 589, 1918.

18. W. Wien. Bericht. Berl. Akad. 1913, p. 184; Vorles. über Probl. Theor. Phys p. 29—48, 1913.

19. A. Watermann. Phys. Rev. 22, p. 259, 1923; 21, p. 514, 1923.

1. Conclusion. See issue 1, p. 47. ↩