MODERN THEORIES OF METALLIC ELECTRICAL CONDUCTIVITY

P. I. Lukirsky, Kiev.

1. The basic content of the theory of metallic conductivity. 2. The old, or classical, theory of metallic conductivity. 3. The physical hypotheses on which the old theory of metallic conductivity is based. 4. The fundamental equations of the old theory. 5. Attempts to reform the theory of metallic conductivity. Stark’s work. 6. W. Wien’s theory. 7. J. J. Thomson’s first theory. 8. J. J. Thomson’s second (chain) theory of conductivity. 9. Borelius’s work on the theory of metallic conductivity. 10. Vereide’s work on the theory of metallic conductivity. 11. P. Bridgman’s theory. 12. A. Joos’s equilibrium theory of metallic conductivity. 13. The influence of atomic weight on electrical conductivity. 14. Electrical conductivity and structure. 15. Results of the survey. 16. Index of literature on the theory of metallic conductivity and related questions.

The basic content of the theory of metallic conductivity.

The theory of metallic conductivity is a physical theory which, with the aid of the assumption that electrons move inside a conductor and on the basis of certain, most acceptable dynamical propositions, explains the mechanism of electric current in metallic conductors, as well as the entire complex and varied group of phenomena connected with it. This includes, first of all, a series of so-called thermoelectric phenomena: the appearance of a thermo-electromotive force in a closed circuit made of two metals whose junctions are at different temperatures, the Peltier and Thomson phenomena; then the propagation of heat in metallic conductors and the relation between the magnitude of thermal conductivity and electrical conductivity (the Wiedemann–Franz law), the dependence of electrical conductivity on temperature and pressure. Here too we are dealing with a complex group of phenomena that accompany the propagation of electric and heat currents in a magnetic field: transverse and longitudinal electrical and thermal phenomena that have received, from the names of the investigators who discovered them, the designations Hall, Ettingshausen, Nernst, Leduc–Righi, Corbino, and so on. To this same category should be assigned two phenomena recently studied—

Benedicks, indicating the role of the temperature gradient in a conductor, and the influence of the shape of the conductor in the phenomena of metallic conductivity. As a special case, we have the influence of a magnetic field on the electrical resistance of a conductor. Further, the optical properties of metals, especially for long heat rays, are closely connected with the theory of metallic conductivity. It may be said with certainty that the greater part of the physical properties of metals, especially those concerning the connection between electrical and thermal phenomena in them, bears the closest relation to the theory of metallic conductivity.

2. The old, or classical, theory of metallic conductivity.

The old electron theory of metallic conductivity, created in the period 1898–1912 thanks to the works of Riecke, Drude, J. J. Thomson, and especially H. A. Lorentz (see the bibliography at the end of the article), was the first physical theory which, proceeding from definite physical hypotheses and dynamical principles, attempted to create a theoretical scheme explaining all the phenomena enumerated above, connected by an inner unity with ideas of metallic conductivity. The further development of this theory in various directions was undertaken by: M. Reinganum, W. Wien, I. Königsberger, P. Gruner, G. Eger, Kunz, Schenck, F. Krüger, K. Baedeker, P. Debye, N. Bohr, and others (see the bibliography).

The basic dynamical scheme of this theory was borrowed entirely from the kinetic theory of gases. It would therefore be fair to call this theory of metallic conductivity kinetic.

Developed gradually and consistently to its logical limits, this theory in many respects turned out to be in contradiction with experiment. Its fate recalls the classical theory of thermal radiation. Disagreements between the results of this theory and experiment led to a radical reform of this theory and to the creation of the quantum theory. Nevertheless, there is no doubt that in the past period of the development of physics the old theory of metallic conductivity played a fruitful role: on the one hand, as a working theory that stimulated a whole series of experimental investigations which accumulated before us extensive, though not always reliable, material; on the other hand, as a test of certain assumptions, very plausible earlier and very doubtful in the light of modern physical knowledge.

3. The physical hypotheses on which the old theory of metallic conductivity is based.

According to the old theory, the electric current in metallic conductors consists in the one-sided motion of electrons that have separated

from the atoms of the metallic conductor, and freely moving in the space between the atoms, under the action of the electric force of an external electric field applied to the conductor. These electrons, called free electrons, exist in the conductor even in the absence of an external electric field. In this consists one of the fundamental hypotheses of the old theory of metallic conductivity. The free electrons, taken together, have the property of a gas in thermal equilibrium with the metal. In gases the mean kinetic energy of the molecules depends only on the temperature. Therefore the thermal motion of the electrons was subjected to the same dependence:

\[ \frac{1}{2}mu^{2}=aT. \tag{1} \]

Here \(m\) is the mass of the electron, \(u\) is the velocity of translational thermal motion, and \(a=2.05\cdot 10^{-16}\). This equation (1), which is also fundamental for the old theory of metallic conductivity, contains a hypothesis that significantly and radically changed the former conceptions of the nature of heat—namely, that the carriers of heat are not only atoms and molecules, but also free electrons. From this there already follow consequences concerning the participation of electrons in heat capacity, thermal conductivity, etc. As Lorentz indicated, such electrons cause thermal radiation, which is expressed by the Rayleigh–Jeans formula and does not agree with experimental results. For short waves this formula gives a greater intensity than is obtained from experiment. The new theories of heat capacity, constructed on the basis of the quantum theory and agreeing satisfactorily with experiment, lead to the conclusion that no appreciable quantity of thermal energy can fall to the share of the electrons.

Equation (1) expresses the dynamical principle of the uniform distribution of energy over degrees of freedom, applied to the motion of electrons inside a body. In addition, it is assumed that in their motion from one collision with one or another atom to the next, the electrons are not subject to the action of any other forces except the force of the external electric field.

4. Fundamental equations of the old theory.

As a result of the indicated propositions, for the electrical conductivity \(d\) one obtains the expression:

\[ d=\frac{1}{2}\frac{ne^{2}l}{mu}=\frac{ne^{2}ul}{4aT}. \tag{2} \]

Here \(n\) is the number of free electrons in a unit volume, \(u\) is the velocity of the thermal motion of the electrons, \(e\) is the charge of the electron, \(l\) is the mean value of its free path, \(m\) is the mass of the electron, \(T\) is the absolute

temperature of the electron gas. An analogous expression is obtained for the thermal conductivity of metals \(k\):

\[ k=\frac{1}{3}nlu a, \tag{3} \]

which leads to an important consequence:

\[ \frac{k}{d}=\frac{4}{3}\left(\frac{a}{e}\right)^2 T. \tag{4} \]

Relation (4) expresses approximately one of the most important regularities established for metals, namely, the Wiedemann–Franz law: for all metals at one and the same temperature the ratio \(\frac{k}{d}\) is the same. In addition, it follows from (4) that the ratio \(\frac{k}{d}\) varies proportionally to the absolute temperature (Lorenz’s law).

5. Attempts at Reforming the Theory of Metallic Conductivity. I. Stark’s Work.

Already by 1912 it had become clear that the electronic theory of metallic conductivity had to be rebuilt on new foundations. All the works that appeared during this period of more than a decade may be regarded as attempts to find new paths toward a more satisfactory solution, consistent with experimental data, of the fundamental problem of the theory of metallic conductivity: the establishment of a detailed scheme of the motion of the current-carrying electrons, both in the form of new physical hypotheses and in the form of the application of new dynamical principles, chiefly the theory of quanta.

Already Kamerlingh-Onnes \([36]\) applied to the region of low temperatures an empirical formula according to which the electrical resistance is proportional to:

\[ \sqrt{T\frac{h\nu}{e^{\frac{h\nu}{kT}}-1}}, \tag{5} \]

where \(T\) is the absolute temperature of the metal, \(h\) is Planck’s elementary quantum of action, and \(\nu\) is the frequency of the atomic vibrations of the metal.

Stark, in 1912 \([53]\), was apparently one of the first physicists who, subjecting the old theory of metallic conductivity to criticism and casting doubt on its basic propositions, attempted radically to change its content. Stark proceeded from the hypothesis that on the surfa-

of the atom, in the form of separate grains, the electrons are located. The atom, therefore, consists of a spherical nucleus and electrons. The lines of force issuing from an electron serve to fasten atoms into a molecule, and also bind the molecule into a solid or liquid body. Starting from the electron, the lines of force terminate either at the \(+\) nucleus of its own

Fig. 1.

Fig. 2.

atom, or at the nuclei of other atoms. Metals possessing high electrical conductivity have, over large dimensions, an electric field bounded on one side by the \(+\) nucleus of the atom, and on the other by very distant electrons (see Fig. 1).

Fig. 2 gives the form of the electric field of metalloid atoms. In Fig. 3 the arrangement of positive atoms and electrons in the crystal lattice of a monovalent metal of the regular system is presented. Here the lines of force of an individual electron partly go to the nucleus of its own atom, partly to neighboring ones, and unite individual atoms into a crystal lattice. The resistance which an electron offers to its displacement in different directions is not the same. This resistance is absent when the electron is displaced in such directions in which its lines of force do not change. These directions lie in planes drawn symmetrically with respect to the \(+\) nuclei. In Fig. 3 the traces of such planes are marked by dashed lines. Along such surfaces the electron moves without the performance of work, as it would move along surfaces of equal potential. Such surfaces Stark called displacement surfaces (Schubflächen). Along such surfaces, under the action of an infinitesimally small electric force, not one electron is displaced, but together with a multitude of others. The displacement surfaces may be not only plane, but of any curvature.

Fig. 3.

Intersecting one another, they form nodal points. The number of electrons capable of moving in a monovalent metal is equal to the number of atoms and does not depend on the temperature. Under the action of an external electric force, the electron moves along those possible directions which have the smallest inclination with respect to the electric force. The electrical resistance is caused by thermal vibrations of the atomic electric fields, which act as a perturbing influence on the motion of the displaced lattice of electrons. Hence the corresponding conclusions are obtained concerning the change of resistance with temperature. According to Stark, all electrons form one indivisible solid lattice moving among the atoms. An individual electron possesses that amount of thermal energy which falls to it as a constituent part of an atom in its thermal vibrations.

For semimetallic bodies, under thermal vibrations, individual electrons may move so far away from the atom that they reach the surface of displacement and can move under the action of an electric force. Such bodies Stark called “Teilmetalle.” Whereas typical metals, according to Stark, possess at \(T = 0^\circ\) abs. an infinitely large value of conductivity, in semimetals at \(T = 0^\circ\) abs. all electrons lie outside the displacement surfaces, and their specific resistance is infinitely large. When moving near vibrating atoms, electrons experience a retarding action from the variable electric fields of the atoms. If \(n\) is the number of electrons per unit volume, then

\[ \frac{dn}{dT}=0 \tag{6} \]

for metals and

\[ \frac{dn}{dT}>0 \tag{7} \]

for semimetals. Further, if \(v\) is the mobility of the electrons in the electric field, then

\[ \frac{dv}{dT}<0 \tag{8} \]

and therefore, for metals, the change of the specific electrical conductivity \(\lambda\) with temperature is:

\[ \frac{d\lambda}{dT} = ne\frac{dv}{dT} <0 \tag{9} \]

and for semimetals:

\[ \frac{d\lambda}{dT} = ev\frac{dn}{dT} + ne\frac{dv}{dT} \tag{10} \]

Therefore there will be such a temperature \(T_1\) at which

\[ \frac{d\lambda}{dT}=0, \]

i.e. the conductivity of the semimetal reaches its greatest value. For the displacement in semimetals of an electron from its atomic nucleus to the displacement surface, work of a definite magnitude \(H\) must be performed. This work \(H\) is analogous to the work of dissociation introduced by Koenigsberger \([31]\) in his theory of the conductivity of semimetallic conductors. In metallic compounds there may also occur such a displacement of electrons from their normal state onto the surface—

…the velocity of displacement, and this is what gives rise to their metallic conductivity, which in general increases with temperature. Stark explains the change in electrical conductivity upon melting of metals and the lower conductivity in the liquid state by the fact that, in the transition to the liquid state, the crystalline lattice disappears, and the shape of the surfaces of shear of the electrons is constantly changing, becoming more curved than in the solid state.

Stark’s work, although it was of a schematic character and was devoid of calculations and derivation of formulas, nevertheless introduced much that was new and original into the theory of metallic conductivity. Most important is Stark’s idea of the significance of the crystalline lattice for explaining the electrical properties of a metal. Another idea, concerning the restriction of the motion of electrons inside a metal to certain paths, later found confirmation in other theories (J. Thomson, P. Bridgman, and others).

6. Wien’s Theory.

Almost simultaneously with Stark, W. Wien \[64\] attempted to justify the theory of the electrical conductivity of metals on different grounds: first of all, he denies the applicability of the fundamental equation of the old kinetic theory of metallic conductivity (1), according to which the square of the electron velocity is proportional to the absolute temperature \(T\).

Along with this, one has to abandon the very simple derivation of the Wiedemann–Franz law. Wien retains the expression given by Drude for the magnitude of the metallic conductivity,

\[ d_1=\frac{1}{2m}e^2 n l \tag{11} \]

here \(n\) is the number of free, or conducting, electrons in \(1\ \mathrm{cm}^3\), \(l\) is their mean free path, \(u\) is the mean velocity of the electrons, and \(d_1\) is the specific electrical conductivity. With respect to \(u\), Wien assumes that this velocity does not depend on temperature and does not change its value at \(T = 0^\circ \mathrm{abs.}\); in other words, he replaces the fundamental equation (1)

\[ \frac{m u^2}{2}=aT \]

by

\[ u=\mathrm{const}. \tag{12} \]

Wien also takes the quantity \(n\) to be independent of \(T\). Only the quantity \(l\) changes with temperature, and its change alone determines the temperature dependence of the conductivity. As a consequence of Wien’s theory, it follows that

\[ \frac{d d_1}{dT}=\frac{d l}{dT} \tag{13} \]

i.e., the temperature dependence of the electrical conductivity \(d_1\) is given by the change of \(l\) with temperature. Since in a solid metal the spheres of action of the atoms overlap one another, according to Wien there is no reason to speak of the “free” motion of electrons, since in their motion they are under the action of the atoms. Assuming that the number of collisions of the electrons with the oscillating atoms of those

greater, the greater are the amplitude of the oscillations and the energy of the atoms, he finds for \(l\) the equation:

\[ \frac{1}{l_\nu}=\mathrm{const}\,\frac{\nu^2\,d\nu}{e^{h\nu/kT}-1}, \tag{14} \]

where \(\nu\) is the frequency of the atomic oscillations, and gives for the resistance \(W\) the following expression:

\[ W=C\int_{0}^{\nu_{\max}}\frac{\nu^2\,d\nu}{e^{h\nu/kT}-1}. \tag{15} \]

A detailed analysis of this expression shows that for very large \(T\) the quantity \(W_T=\mathrm{const}\, kT\), while at low temperatures \(W\) is proportional to \(T^2\). F. Gauer[^27] also arrived at the same results as Wien: the chief cause of the change in resistance is the change in the mean free path.

Wien’s theoretical investigation was of great importance in formulating the problem. From Wien’s work one can see the new questions raised by the theory of conductivity. These questions are as follows: where should one look for the influence of temperature on the electrical conductivity of metals and, in particular, how does \(T\) affect the motion of the electrons (their velocity \(u\) and mean free path \(l\)); how large is the number of free electrons \(n\) participating in the transport of the electric current, and how does their separation from the atom occur.

7. The First Theory of J. Thomson.

Somewhat differently from Stark and Wien, J. Thomson attempted to reconstruct the theory of metallic conductivity (The Nature, 1915 Dec.). Subjecting the conclusions of the old electron theory to careful consideration, he pointed out that the temperature variation of electrical conductivity and especially the phenomena of superconductivity do not agree with the foundations of the old theory of electronic conductivity. Therefore Thomson proposed a new scheme for the mechanism of conductivity, borrowed from Langevin’s theory of magnetism. Thomson assumes that each atom of a metal may be regarded as an electric bipole, analogous to a molecular magnet. Such a bipole consists of an electron and a positive charge equal to it in magnitude. Under the action of an electric force these bipoles tend to arrange themselves along the lines of electric force, just as molecular magnets arrange themselves along the lines of magnetic force. As a result, under the action of an external electric field, some of the bipoles will be arranged in the direction of the electric force, while the rest will be oriented at random in different directions. Atomic bipoles cause

around themselves intense electric forces, which tend to pull an electron from one atom to another. The difference between an insulator and a metal, according to Thomson, consists in the fact that the electrons in the atoms of an insulator are able to offer sufficient resistance to this force and remain in the atoms. In metals, however, they are more easily torn away, yielding to the force of attraction, and thus pass along a chain from one atom to another. The force that pulls the electrons is caused by neighboring atoms and therefore does not depend on the external electric force. If along one chain \(p\) electrons pass per second, and the number of such chains per \(1\ \mathrm{cm}^2\) is \(N\), then the current density is \(Npe=i\) (16), where \(e\) is the charge of the electron. Further, if in a unit volume in the direction of the electric force there are \(I\) dipoles, and \(d\) is the distance between the centers of neighboring dipoles, then per unit length of the chain their number will be: \(\frac{1}{d}\)

\[ I=\frac{N}{d}\quad \text{and}\quad i=Idpe. \tag{17} \]

Further Thomson, by analogy with the well-known formulas of Langevin’s theory of magnetism, obtains the expression \(I\), corresponding to the given external electric force \(X_0\):

\[ I=-\frac{\dfrac{Nm^2}{3}X_0\dfrac{1}{RT}}{1-\dfrac{Nm^2k}{3RT}}. \tag{18} \]

Here \(T\) is the absolute temperature, \(R\) the gas constant, \(m\) the electric moment of the dipole, and \(kI\) the internal electric field arising from the action on the given dipole of neighboring atoms. Or, denoting by \(T_0\) the temperature at which \(\dfrac{Nm^2k}{3RT}=1\), we obtain:

\[ I=\frac{1}{k}\frac{X_0T_0}{T-T_0}\quad \text{or}\quad i=\frac{dpe}{k}\frac{X_0T_0}{T-T_0}. \tag{19} \]

Thus, the specific conductivity of the metal is

\[ \sigma=\frac{dpe}{k}\frac{T_0}{T-T_0}. \tag{20} \]

At \(T=T_0\) it becomes infinite. The role of the internal force \(kI\) deserves attention. First of all, this force is of a purely electrical character. Further, from the equation:

\[ \frac{kI}{X_0}=\frac{T_0}{T-T_0}=F(T) \tag{21} \]

it is seen that this internal electric field is always proportional to the external field and reaches a particularly large value at \(T=T_0\), when it is considerably greater than the external field, since the chains of atomic dipoles are held together chiefly by interatomic forces.

Therefore, according to Thomson, the role of the external electric force in metallic conduction consists in polarizing the metal, i.e., in forming a series of chains. Hence Joule heat corresponds to the work of forming the chains, i.e., of overcoming the forces that hinder the orientation of the dipoles. When the chains have been formed, the electric current is transmitted along them under the action of atomic forces alone, i.e., adiabatically. Therefore, if with the removal of the electric force the polarization remains, then the current too must remain. Thomson’s theory departs even more than Wien’s theory from the former schemes on which the old theory of conductivity was based. It introduces, as an important physical quantity, the intermolecular electric field \(E_1=\dfrac{x_0T_0}{T-T_0}\).

Unfortunately, this scheme for the transmission of electric current says nothing about other phenomena connected with electrical conductivity: thermoelectric phenomena, thermal conductivity, the influence of a magnetic field on resistance, etc.

8. The Second (Chain) Theory of Conductivity of J. Thomson.

In 1922 J. Thomson published \([^{59}_8]\) a second theory of metallic conductivity, closer in its dynamical foundations to the old kinetic electronic theory of metallic conductivity. The role which individual electrons played in the old electronic theory is, in Thomson’s new theory, performed by electron chains. According to Thomson’s conception, the lattice of a solid metal is built of atoms and electrons, which are arranged alternately. An electron is not bound to any individual atom more than to the others. When an electron is displaced from its position of equilibrium, vibrations of very high frequency arise, corresponding to visible or ultraviolet rays. Such electrons become firmly connected with one another and form a “solid” chain. This chain, when displaced from its position of equilibrium, can execute oscillations whose period is very large, so that even at low temperatures such a chain receives an amount of energy corresponding to one degree of freedom under a uniform distribution. The chain can also execute translational motion along the lines of the crystal lattice, as a single solid body, carrying electricity and thermal energy inside the metal. On the basis of these ideas J. Thomson explains the change of resistance with temperature, superconductivity, and the Wiedemann–Franz law. If the amount of kinetic energy of an electron at temperature \(T\) in the state of thermal equilibrium is denoted by \(3kT\), then the energy of the whole chain, however large the number \(n\) of electrons forming it may be, will, according to Thomson, be equal to \(kT\); whence each electron receives only \(\dfrac{kT}{n}\), a quantity very small for consider-

... \(n\). Thomson explains the formation of such chains inside the metal in the following way: the alternating electric field of black radiation occupies a volume sufficiently large in comparison with the distance between two neighboring electrons. Therefore the electric field of black radiation links the electrons into one chain. The oscillation of such a chain produces a reverse reactive wave, weakening the intensity of the black radiation. When there is no external electric field applied to the metal, the chains move in all directions, forming local currents whose distribution can be changed by a magnetic field.

If by \(t\) we denote the time between two collisions of an electron with atoms, by \(X\) the magnitude of the electric force, and by \(e\) the charge of the electron, then \(v\)—the magnitude of its velocity—will, according to the old theory, be

\[ v=\frac{1}{2}\frac{Xet}{m} \quad \text{or} \quad v=\frac{1}{2}\frac{Xel}{mv}; \]

here \(m\) is the mass of the electron, \(l\) its free path.

According to Thomson’s chain theory, the kinetic energy of a chain of \(n\) electrons at temperature \(T\) is equal to

\[ \frac{1}{2}nmv^{2}=\frac{1}{2}kT. \tag{22} \]

The chains move in the interval between atoms, experiencing near an atom their strongest action, partly resembling a collision in the old theory. If \(2c\) is the distance between the centers of neighboring atoms, and \(v\) is the velocity of motion of the chain, then the time of passage of an electron from atom to atom is \(\frac{2c}{v}\). During this time the electric force \(X\) increases the velocity of motion of the chain by \(Xe\frac{2c}{mv}\). But part of the kinetic energy is lost as the chain passes near an atom. If the magnitude of the friction experienced by the electron is \(\eta\), then its velocity under the action of the electric force \(X\) will be

\[ u=\frac{Xe}{m\eta} \quad \text{and} \quad u=\frac{Xencv}{kT} = \frac{eXlv}{2kT} \quad (l\text{ is the length of the chain}). \tag{23} \]

If the number of chains having the direction \(X\) in unit volume is \(q\), then the current density will be:

\[ qn\,\frac{Xelv}{2kT} = ql\,\frac{Xe^{2}nv}{2kT}. \tag{24} \]

Therefore the magnitude of the electrical conductivity \(z\) is expressed as follows:

\[ z = q\,\frac{e^{2}lnv}{2kT} = q\,\frac{e^{2}l^{2}fp}{2kT}, \tag{25} \]

where \(p\) is the number of electrons per unit volume, and \(f\) is their friction during the motion of the chains. As was indicated above, the chains not only determine the electrical conductivity of the metal, but are also centers of emission and absorption of rays in the interatomic space. Analogously to Wien’s displacement law, \(\lambda_{\max}T=\mathrm{const}\), the length of a chain is inversely proportional to \(T\).

Further, Thomson gives the relations: \(\omega l=\dfrac{\beta}{T}\) (26); here \(\beta\) is a constant, and

\[ v=\left(\frac{Re}{\xi m}\right)^{\frac12}T \tag{27} \]

whence it follows that \(ncv\) does not depend on the temperature. Therefore \(p\) is constant and \(\xi\) is inversely proportional to \(T\). Thomson explains the phenomenon of superconductivity of metals as follows: when a chain collides with atoms, part of the energy of the chain passes to the atom. When the period of oscillation of the chain increases, the amount of energy transferred to the atom decreases. With slow motion of the electrons and a prolonged impact, the atom receives a very small part of the energy: according to Jeans, \(e^{-2c\nu}\), where \(c\) is the duration of the impact and \(\nu\) is the frequency of the atomic oscillations. In this case the velocity of the electron chains, and therefore also the magnitude \(\xi\), increases without bound, which is what causes superconductivity.

In its content Thomson’s chain theory is close to the old kinetic theory of conductivity, differing from it chiefly in the small magnitude of the electron’s thermal energy, especially at low temperatures, when the length of the chains increases considerably.

This formation of long chains at low temperatures does not correspond to the excessively low intensity of black radiation. For Li and Na at ordinary temperature, chains of length about \(0.0002\ \mathrm{cm}\) are obtained, containing about 10,000 electrons. At \(3^\circ\) abs. the length of the chains increases to \(0.02\ \mathrm{cm}\), which corresponds to 1 million electrons in one chain.

9. Borelius’s Work on the Theory of Metallic Conductivity.

The adoption, for free electrons, of the value of the thermal energy \(aT\) according to equation (1) makes it difficult to explain, quantitatively, the experimentally obtained value of the heat capacity of metals and the phenomena of thermoelectricity. On the other hand, without this assumption difficulties arise in explaining the Wiedemann–Franz law. Therefore Borelius [8] assumes that the large magnitude of the thermal conductivity of metals is due primarily to the regularity of their lattice, and only subsequently to the motion of the electrons. Among the shortcomings of the old classical theory one may also include the very large values for the mean free path \(l\) of the electrons. If the number of free electrons \(n\) is taken equal to the number of atoms \(N\), then for Ag at ordinary tempe-

...temperature one obtains a value equal to 30 times the distance between atoms. For \(n<N\) the quantity \(l\) is still larger. To get around the difficulties indicated above, Borelius assumes that the atoms are held in their positions by electrical and gravitational forces. Rejecting the law of equipartition of energy as applied to the motion of conducting electrons, he assumes that the energy of a conducting electron is considerably less than \(aT\) (see equation 1) and is equal to \(CT\), where \(T\) is the absolute temperature and \(C\) is a constant. Further, Borelius adopts the hypothesis of the first theory of J. Thomson, that an external electric field produces polarization inside the metal. Putting \(n=N\), he obtains for the moment of an elementary atomic dipole \(\varepsilon r\), and, applying Langevin’s magnetic scheme, obtains for the electric moment \(I\), under the action of an external electric force \(X\), the value

\[ I=I_m\frac{\varepsilon rX}{u}, \tag{28} \]

where \(u\) is the energy of one electron, and \(I_m\) is the moment of complete polarization. Owing to the polarization, a one-sided displacement of electricity arises. Through an area of \(1\ \mathrm{cm}^2\) there passes

\[ i_m=4\varepsilon n\nu, \tag{29} \]

where \(n\) is the number of electrons in \(1\ \mathrm{cm}^3\), \(\nu\) is the frequency of atomic vibrations, \(\varepsilon=1.59\cdot10^{-20}\) electromagnetic units is the elementary charge;

\[ r=\sqrt[3]{\frac{A}{\rho N_1}} \]

is the radius of the volume corresponding to one atom, \(A\) is the atomic weight; \(\rho\) is the density, \(N_1=6.06\cdot10^{23}\) is Avogadro’s number; \(n=N=\dfrac{1}{(2r)^3}\). Further, the author gives for the current strength \(i\) the expression

\[ i=\frac{4\varepsilon^2 r^2 n\nu X}{u}, \tag{30} \]

whence the magnitude of the electrical conductivity is equal to

\[ x=\frac{4\varepsilon^2 r^2 n\nu}{u}, \tag{31} \]

i.e. it is proportional to the square of the atomic dipole moment \(\varepsilon r\), to the frequency of atomic vibrations \(\nu\), to the number of electrons \(n\), and inversely proportional to \(u\), the energy of the electron.

Otherwise this expression for the electrical conductivity may be represented also as

\[ x=\frac{1}{2}\varepsilon^2\frac{\nu}{ru}. \tag{32} \]

Since in the last equation only \(u\) depends appreciably on the temperature, \(r\) will be inversely proportional to \(T\). In passing from one metal to another, the electrical conductivity \(x\) changes more appreciably than \(r\). Such a change should be ascribed to the jump in the magnitude \(u\). The equation \(j_m = 4\varepsilon\nu\) leads to the result that for Ag Ohm’s law will be valid at current densities small in comparison with \(7\cdot 10^9\ \mathrm{A}/\mathrm{cm}^2\). Further, the author indicates that his theory gives for the energy of the electron \(u\) a small value in comparison with the energy of a gas molecule. For Ag and Bi at ordinary temperature he finds for the ratio:

\[ \frac{u}{\alpha T}=\frac{c}{z}\cdot 0.0106\ (\text{for Ag})\quad \text{and}\quad 0.015\ (\text{for Bi}). \]

Like Stark and Lindemann, Borelius explains superconductivity by the motion of electrons in the interatomic space without resistance. With regard to thermal conductivity, the author assumes the transfer of heat in the metal by elastic waves in a spatial lattice,

with part of the wave energy, equal to the ratio \(\dfrac{u}{2\alpha T}\), passing into the energy of the electrons. Applying Debye’s formula for thermal conductivity

\[ \lambda=\frac{1}{4}\rho c q L, \tag{33} \]

where \(\rho\) is the density, \(c\) the heat capacity, \(q\) the velocity of propagation of the wave, \(L\) the mean free path on which the wave energy decreases by the given amount, and putting

\[ \rho c=z\alpha N;\quad N=\left(\frac{1}{2r}\right)^3;\quad L=\frac{4raT}{u}, \]

Borelius obtains for the thermal conductivity

\[ \lambda=\frac{z^2T^2}{\mu}, \tag{34} \]

whence

\[ \frac{\lambda}{x}=2\left(\frac{\alpha}{\varepsilon}\right)^2 T, \]

i.e. the Wiedemann–Franz law in its usual form. As is evident from the preceding, Borelius’s work strongly reflects the constructions of other physicists and is partly similar to J. Thomson’s first theory.

10. R. H. Vereide on the Theory of Metallic Conductivity

Basing himself on Lenard’s experiments on the absorption and reflection of electrons from the surface of solid bodies, T. Vereide in 1918 \([23]\) attempted to sketch a scheme of the theory of metallic conductivity.

Vereide draws attention to the fact that the weak points of the old theory are: 1) the disagreement of the radiation formula with experiment, 2) the considerable thermal conductivity of insulators, 3) the impossibility of explaining the small heat capacity of free electrons.

The mean energy of atomic vibrations, as is known, is equal to

\[ U=\frac{3}{2}\frac{h\nu}{e^{\frac{h\nu}{kT}}-1}. \tag{35} \]

Vereide assumes the same quantity also for the energy of an electron emitted from an atom. Lenard’s experiments (Annalen der Physik, 12, 1903, p. 932) showed that the absorption of electrons incident on a solid body obeys the following law: \(\frac{dn}{n}=a\,dx\); here \(n\) is the number of moving electrons, \(dn\) the number absorbed over the distance \(dx\); \(a\) is the absorption coefficient. The probability of absorption of an electron over the distance \(dl\), if the number of atoms in \(1\ \mathrm{cm}^3\) is \(c\), is \(xc\,dl\). Hence Vereide finds for the mean free path of the electrons:

\[ l=\int_{0}^{\infty}xc\,e^{-xcl}\,l\,dl=\frac{1}{xc}. \tag{36} \]

Thus, the quantity \(l\) is determined through \(c\), the number of atoms in \(1\ \mathrm{cm}^3\), and \(x\), the magnitude of the absorptive capacity of one atom, independent of the velocity of the electrons. From statistical calculations one obtains for the concentration of free electrons \(n\)

\[ n=\frac{\varepsilon}{a v}, \]

where \(\varepsilon\) is the number of electrons which the atom ejects in 1 second, \(v\) is the mean velocity of motion of the free electrons. From Drude’s formula for electrical conductivity:

\[ d_1=\frac{1}{2}\frac{e^2}{m}\frac{nl}{v} \]

the author eliminates three unknown quantities: \(n\), \(l\), \(v\), and introduces instead of them: \(\nu\), the frequency of atomic vibrations, \(\varepsilon\), the radiating capacity of the atom in 1 second, \(a\), the absorptive capacity of the atom for electrons in 1 second. According to Lenard: \(a=cx\). As a result, for the electrical conductivity \(d_1\), Vereide obtains:

\[ d_1=\frac{e^2\varepsilon}{6ca^2}\, \frac{e^{\frac{h\nu}{kT}}-1}{h\nu} \tag{37} \]

and an analogous formula for the coefficient of thermal conductivity \(\lambda\):

\[ \lambda=\frac{k\varepsilon}{2ca^2}\, \frac{e^{\frac{h\nu}{kT}}\left(\frac{h\nu}{kT}\right)^2} {\left(e^{\frac{h\nu}{kT}}-1\right)^2}. \tag{38} \]

Since in the last equation only \(u\) depends significantly on the temperature, \(r\) will be inversely proportional to \(T\). In passing from metal to metal, the electrical conductivity \(x\) changes more significantly than \(r\). Such a change should be attributed to a jump in the magnitude \(u\). The equation \(r = 4 \varepsilon \eta \sigma\) leads to the result that, for Ag, Ohm’s law will be valid at current densities small in comparison with \(7 \cdot 10^9\ \mathrm{A}/\mathrm{cm}^2\). Further, the author indicates that his theory gives for the energy of the electron \(u\) a small value in comparison with the energy of a gas molecule. For Ag and Bi at ordinary temperature he finds, for the ratio:

\[ \frac{u}{xT}=\frac{c}{z}\cdot 0.0006\ \text{(for Ag) and }0.015\ \text{(for Bi).} \]

Like Stark and Lindemann, Borelius explains superconductivity by the motion of electrons in the interatomic space without resistance. With regard to thermal conductivity, the author assumes the transfer of heat in a metal by elastic waves in the spatial lattice, with a part of the wave energy, equal to the ratio \(\dfrac{u}{2xT}\), passing into the energy of the electrons. Applying Debye’s formula for thermal conductivity

\[ \lambda=\frac{1}{4}\rho cqL, \tag{33} \]

where \(\rho\) is density, \(c\) is heat capacity, \(q\) is the velocity of propagation of the wave, and \(L\) is the mean free path over which the wave energy decreases by the given amount, and assuming

\[ \rho c=\varepsilon aN;\quad N=\left(\frac{1}{2r}\right)^3;\quad L=\frac{4ra l''}{u}, \]

Borelius obtains for the thermal conductivity

\[ \lambda=\frac{a^2T}{\gamma u}, \tag{34} \]

whence

\[ \frac{\lambda}{x}=2\left(\frac{a}{\varepsilon}\right)^3T, \]

i.e. the Wiedemann–Franz law in its usual form. As is evident from the preceding, Borelius’s work strongly reflects the constructions of other physicists and is in part similar to J. J. Thomson’s first theory.

10. Vereide’s Works on the Theory of Metallic Conductivity

Basing himself on Lenard’s experiments on the absorption and reflection of electrons from the surface of solid bodies, T. Vereide in 1918 [23] attempted to outline a scheme for the theory of metallic conductivity.

Vereide draws attention to the fact that the weak points of the old theory are: 1) the disagreement of the radiation formula with experiment, 2) the considerable thermal conductivity of insulators, 3) the impossibility of explaining the small heat capacity of free electrons.

The mean energy of atomic vibrations, as is known, is equal to

\[ \bar{\varepsilon} = \frac{3}{2}\, \frac{h\nu}{e^{\frac{h\nu}{kT}}-1}. \tag{35} \]

Vereide assumes the same magnitude also for the energy of an electron emitted from an atom. Lenard’s experiments (Annal. d. Physik, 12. 1903, p. 932) showed that the absorption of electrons incident upon a solid obeys the following law: \(\frac{dn}{n}=a\,dx\); here \(n\) is the number of moving electrons, \(dn\) the number absorbed over the distance \(dx\), \(a\) the absorption coefficient. The probability of absorption of an electron over the distance \(dl\), if the number of atoms in \(1\ \mathrm{cm}^3\) is \(c\), is \(xc\,dl\). Hence Vereide finds for the mean free path of the electrons:

\[ l=\int_{0}^{\infty} xc\,e^{-xcl}\,l\,dl=\frac{1}{xc}. \tag{36} \]

Thus, the quantity \(l\) is determined through \(c\), the number of atoms in \(1\ \mathrm{cm}^3\), and \(x\), the absorptive capacity of a single atom, which does not depend on the velocity of the electrons. From statistical calculations one obtains for the concentration of free electrons \(n\)

\[ n=\frac{\varepsilon}{av}, \]

where \(\varepsilon\) is the number of electrons which the atom ejects in 1 second, and \(v\) is the mean velocity of motion of the free electrons. From Drude’s formula for the electrical conductivity,

\[ d_1=\frac{1}{2}\frac{e^2}{m}\frac{nl}{v}, \]

the author eliminates the three unknown quantities: \(n\), \(l\), \(v\), and introduces in their place: \(\nu\), the frequency of atomic vibrations; \(\varepsilon\), the radiating capacity of the atom in 1 second; \(a\), the absorptive capacity of the atom for electrons in 1 second. According to Lenard: \(a=cx\). As a result, for the electrical conductivity \(d_1\), Vereide obtains:

\[ d_1 = \frac{c^2\varepsilon}{6a^2}\, \frac{e^{\frac{h\nu}{kT}}-1}{h\nu} \tag{37} \]

and an analogous formula for the coefficient of thermal conductivity \(\lambda\):

\[ \lambda = \frac{k\varepsilon}{2ca^2}\, \frac{ e^{\frac{h\nu}{kT}} \left(\frac{h\nu}{kT}\right)^2 }{ \left(e^{\frac{h\nu}{kT}}-1\right)^2 }. \tag{38} \]

On the basis of observing the magnitude of the temperature coefficient of resistance, Vereide assumes that \(\varepsilon\)—the emissive capacity of the atom—changes little with temperature. Vereide finds that his formula (37) can explain the following regularities:

1. At high absolute temperatures \(T\), the specific resistance of metals \(\rho\) increases approximately in proportion to \(T\).

2. The change in resistance is above all due to the change in the energy of the body, which depends on the magnitude

\[ \frac{h\nu}{e^{\frac{h\nu}{kT}}-1}. \]

This explains the change in resistance upon melting.

1. As \(T\) decreases, the resistance tends to the limiting value 0.

2. At low temperatures, for different substances the curves of the change in resistance

\[ \frac{\rho}{\rho_{273}} \]

go in the order of the atomic frequencies,—the rule given by Schimank \([55]\).

1. In the periodic system of the elements, the curve of electrical conductivity is analogous in its course to the curve of atomic volumes.

2. Other conditions being equal, the electrical conductivity is the greater, the greater the capacity of atoms to emit electrons, characterized by the quantity \(\varepsilon\).

3. For each group of the periodic system the quantity

\[ \frac{\varepsilon}{a^2} \]

changes in the same way as the frequency of atomic vibrations \(\nu\).

1. Within one group of the periodic system \(a\) changes little, and therefore \(\varepsilon\) changes like the frequency \(\nu\) of atomic vibrations.

Vereide takes for \(a\) a value of the order \(10^{-16}\) and obtains for Fe the following values:

\[ \begin{array}{rcl} c & \text{about} & 10^{23},\\ l & \text{”} & 10^{-7},\\ \nu & \text{”} & 10^{23},\\ \varepsilon & \text{”} & 10^{14}. \end{array} \]

The most essential result in Vereide’s work is the introduction into the expression for metallic conductivity of the energy and frequency of atomic vibrations.

In this respect Vereide’s theory is close to Wien’s theory. Very interesting is Vereide’s attempt to outline a series (1–8) of new empirical regularities connected with the magnitude of metallic conductivity.

11. P. Bridgman’s theory.

A great step forward in the development of questions of metallic conductivity is represented by the theoretical works of P. Bridgman, which very fully reflect the experimental results obtained by him.

In the first paper \([{}^3]\), Bridgman takes into account the following results of the change in resistance of the 22 metals he investigated under pressures up to \(12000\ \mathrm{kg}/\mathrm{cm}^2\): 1) the temperature coefficient of conductivity changes hardly at all with increasing pressure; 2) the change in resistance with pressure is the smaller, the greater the pressure; 3) the curvature of the resistance–pressure curves in most cases increases as the temperature is lowered; 4) two metals—Bi and Sb—behave anomalously: their resistance increases with pressure. On the basis of these experimental results, Bridgman assumes that, when the atoms of a solid body are in close contact, electrons pass from atom to atom without resistance. Such a transition becomes impossible when the atoms are sufficiently removed from one another. A “gap” is formed between the atoms, across which the electrons cannot get. Such gaps are formed in a metal upon heating, when the distances between atomic centers become greater than some definite magnitude \(y\). The resistance \(w\) of the metal is proportional to the number of such gaps. The number of gaps depends on the amplitude of the atoms. Near \(\tau = 0^\circ\) abs. the gaps disappear and the electrons pass adiabatically from atom to atom. This explains superconductivity. For intermediate temperatures Bridgman assumes for the magnitude of the vibrational energy of an atom the law of equipartition of energy. This gives:

\[ \frac{\nu^2 a^2}{\tau}=\mathrm{const}. \tag{39} \]

Here \(\nu\) is the frequency of atomic vibrations, \(a\) is their amplitude, and \(\tau\) is the absolute temperature. In addition, Bridgman assumes that

\[ \left(\frac{\partial \nu}{\partial \tau}\right)_v=0 \tag{40} \]

and also introduces the relation given by Grüneisen \([{}^{22}_3]\):

\[ \frac{1}{\nu}\left(\frac{\partial \nu}{\partial p}\right)_s = \frac{1}{C_p}\left(\frac{\partial v}{\partial \tau}\right)_p . \tag{41} \]

Here: \(s\) is entropy, \(p\) is pressure, \(C\) is heat capacity, \(v\) is volume. These three relations, according to Bridgman, are approximate. At low \(\tau\), the quantity \(\nu^2 a^2\) grows faster than \(\tau\). From (40, 41) Bridgman obtains the following law for the change of the frequency \(\nu\) with temperature and pressure:

\[ \frac{1}{\nu}\left(\frac{\partial \nu}{\partial \tau}\right)_p = \left(\frac{\partial \nu}{\partial \tau}\right)_v^{2} \bigg/ C_v\left(\frac{\partial v}{\partial p}\right)_\tau \tag{42} \]

and

\[ \frac{1}{\nu}\left(\frac{\partial \nu}{\partial p}\right)_\tau = \left(\frac{\partial \nu}{\partial \tau}\right)_v \bigg/ C_v . \tag{43} \]

and for the change of the amplitude:

\[ \frac{1}{a}\left(\frac{\partial a}{\partial p}\right)_{z} = -\frac{\left(\frac{\partial r}{\partial z}\right)_{p}}{C}, \tag{44} \]

and

\[ \frac{1}{a}\left(\frac{\partial a}{\partial z}\right)_{p} = \frac{1}{2z} - \frac{\left(\frac{\partial r}{\partial z}\right)_{p}^{2}}{C\left(\frac{\partial r}{\partial p}\right)_{z}} . \tag{45} \]

Bridgman assumes that the principal cause of the change in the specific resistance \(r\) of a metal is the change in the amplitude \(a\) of the atomic vibrations. This is justified by the fact that, on the one hand, the preceding equations give:

\[ \frac{2}{a}\left(\frac{\partial a}{\partial z}\right)_{p} = \frac{1}{z}, \tag{46} \]

while on the other hand, experiment shows that \(\dfrac{\partial r}{\partial z}\) for many metals is inversely proportional to \(z\). Further, Bridgman gives the very important relation:

\[ \frac{1}{r}\left(\frac{\partial r}{\partial p}\right)_{z} = -\frac{2}{a}\left(\frac{\partial a}{\partial p}\right)_{z} + \frac{1}{3}\frac{1}{r}\left(\frac{\partial r}{\partial p}\right)_{z}. \tag{47} \]

From a calculation of experimental data Bridgman obtains that the change of \(r\) with \(p\) is caused chiefly by the change in the atomic amplitudes. The last term of formula (47) is a correction term and gives only a small change in \(r\) (from \(5\%\) to \(10\%\)). The old electron theory of metallic conductivity could not explain the change of \(r\) with \(p\), and therefore Grüneisen gave \([^{22}]\) the relation:

\[ \frac{1}{u}\left(\frac{\partial u}{\partial p}\right)_{z} = \frac{1}{n}\left(\frac{\partial n}{\partial p}\right)_{z} - \frac{1}{N}\left(\frac{\partial N}{\partial p}\right)_{z} + \frac{1}{r}\left(\frac{\partial r}{\partial p}\right)_{z} - \frac{1}{C_{p}}\cdot\frac{1}{r}\left(\frac{\partial r}{\partial z}\right)_{p} \left[ 1+\frac{1}{u}\left(\frac{\partial u}{\partial z}\right)_{p}z \right]. \tag{48} \]

Here \(u\) is the velocity of the free electrons, \(N\) their number per unit volume.

Bridgman generalizes equation (47) for the change of \(r\) not only with pressure, but also with other physical factors, and gives it the following form:

\[ \frac{dr}{r}=2\frac{dz}{a}+\frac{1}{3}\,dv. \tag{49} \]

When the amplitudes of two neighboring atoms increase, and for a corresponding value of the phases of their vibrations, the distance between two neighboring atoms becomes greater than a certain limiting value and a “gap” arises. This causes an increase in \(r\).

Another important factor, according to Bridgman, influencing the resistance is the frequency of the atomic vibrations. The change in frequency is affected by the time of contact and separation of the atoms. From equation (42) it is evident that the change of \(\nu\)—the frequency—with pressure is of the same order as the change in amplitude. According to the old theory of electrical conductivity, the atoms of the body were passive, representing walls between which the free electrons move; but by the modern theory, the motion of the atoms strongly affects the motion of the electrons. The influence of frequency is manifested in the fact that, with an increase in frequency, the time of contact and separation of atoms decreases; hence the number of gaps increases, and consequently the resistance increases. The change in frequency with pressure must therefore also lead to a change in resistance. According to Bridgman, this influence is much smaller than that of the amplitude, but in some cases it may be appreciable.

The second term in formula (49), which contains \(dv\), expresses just this influence of the change in frequency. Thus Bridgman’s formula connects the change in resistance with the changes in the amplitude and frequency of the atomic vibrations. It gives a satisfactory explanation of the experimental facts, especially of the negative pressure coefficient of the resistance of many metals.

...free electrons were involved. Bridgman’s theory closely links the questions of metallic conductivity with the theory of the solid state. Unfortunately, Bridgman leaves in the shade a whole series of essential questions concerning the motion of electrons.

In a second paper, devoted to metallic conductivity \(\left[{}^{3}_{2}\right]\), Bridgman somewhat expands and partly modifies his theory. In doing so he takes into account the results of his new experimental investigations: 1) At large current densities \((5\cdot10^{6}\ \mathrm{A}/\mathrm{cm}^{2})\) in thin layers of Au and Ag, deviations from Ohm’s law of about \(1\text{–}2\%\) were found, in accordance with the prediction of J. J. Thomson (Corpuscular theory of Matter). If it is assumed that the free path of the electrons is of the same order as the distances between atoms, then such deviations from Ohm’s law are possible only at current densities of about \(10^{11}\ \mathrm{A}/\mathrm{cm}^{2}\). Therefore the observed deviations make a longer free path of the electrons probable. 2) Bridgman’s earlier experiments on the change of resistance with pressure showed that 5 metals (Bi, Sb, Li, Ca, and Sr), unlike other metals, do not decrease but increase their resistance with pressure

\[ \left(\frac{dw}{dp}>0\right). \]

Investigating the change of resistance of these metals under the action of tension, he found that these metals can be divided into two types: the Bi type (Bi, Sr), for which \(w\) decreases with tension, and the Li type (Li, Ca, Sb), for which \(w\) increases with tension. 3) Finally, Bridgman investigated the influence of pressure up to \(12\,000\ \mathrm{kg}/\mathrm{cm}^{2}\) on thermal conductivity.

These new data compel the introduction of certain additions and corrections to the previous theory of metallic conductivity. For the path of electrons in a metal Bridgman distinguishes 2 types: 1) the normal type, when with increasing pressure \(w\) decreases. In this case the electrons pass directly from one atom to the neighboring atom; 2) the anomalous type, when \(w\) increases with increasing pressure. For such metals two mechanisms of passage are possible: either the same as above, but the law of interaction of the atoms is anomalous; or else the electrons move in channels between atoms (Li type). In the first case (Bi type) the amplitudes change anomalously with pressure. For metals of the Li type the electrons move in channels between atoms. When the distances between atoms and their amplitudes change, the dimensions of these channels change. When the temperature changes, in addition, the velocity of the electrons also changes. A detailed representation of the form of the channels requires knowledge of the details of the crystal structure. Bridgman gives the forms of the channels for Li (body-centered cubic lattice), Ca (face-centered cubic), Sb, and others. He assumes the length of the free path of the electron to be proportional to the cross-sectional area of the channel:

\[ w=\frac{\mathrm{const}}{c^{2}}, \tag{50} \]

where \(c\) is the cross-section of the channel. The thermal motion of the atoms produces an apparent increase in the volume occupied by the atoms and a narrowing of the channels. At constant temperature \(\theta\):

\[ \frac{1}{w}\left(\frac{\partial w}{\partial p}\right)_{\theta} = -\frac{2}{c}\left(\frac{\partial c}{\partial p}\right)_{\theta}, \qquad (51), \quad \text{whence} \quad c=-2\left(\frac{\partial c}{\partial p}\right)_{\theta} \bigg/ \frac{1}{w}\left(\frac{\partial w}{\partial p}\right). \tag{52} \]

If \(L\) is the distance between the centers of neighboring atoms, then

\[ \left(\frac{\partial L}{\partial p}\right)_{\theta} = 2\left(\frac{\partial a}{\partial p}\right)_{\theta} + \left(\frac{\partial c}{\partial p}\right)_{\theta} \quad \text{and} \quad \frac{1}{L}\left(\frac{\partial L}{\partial p}\right) = \frac{1}{3}\left[ \frac{1}{v}\left(\frac{\partial v}{\partial p}\right)_{\theta} \right]. \tag{53} \]

Taking the maximum kinetic energy of an atom to be equal to twice the kinetic energy of a gas molecule at the same temperature, Bridgman, for \(300=T\) abs. and frequency \(\nu\), obtains

\[ 4\pi^{2}\nu^{2}a^{2}\frac{m}{2} = 2\cdot 300\cdot 2\cdot 10^{-16}. \]

Thus, the problem of finding \(a\) was reduced to finding \(\nu\), for which Bridgman proposes Lindemann’s formula

\[ \nu=3.08\cdot 10^{-12}\sqrt{T_{s}/m}^{\,3}. \]

Further, Bridgman gives a theory of the change of resistance with tension. Tension causes a narrowing of the interatomic channels. For the change of atomic amplitudes with tension \(T\), Bridgman finds:

\[ \frac{1}{a}\left(\frac{\partial a}{\partial T}\right)_{\theta} = -\left(\frac{\partial l}{\partial \theta}\right)_{T}\bigg/ C_{T}, \tag{54} \]

where \(l\) is the mean free path of the electrons, and further

\[ \left(\frac{\partial c}{\partial T}\right)_{\theta} = \left(\frac{\partial L}{\partial T}\right)_{\theta} = -\frac{L s}{E} \quad \text{or} \quad \frac{1}{L}\left(\frac{\partial L}{\partial T}\right)_{\theta} = -\frac{s}{E} \tag{55} \]

(\(E\) is Young’s modulus, \(s\) is Poisson’s ratio).

Whence

\[ \frac{1}{w}\left(\frac{\partial w}{\partial T}\right)_{\theta} = \frac{2}{c}\frac{Ls}{E}. \tag{56} \]

The temperature coefficient \(r\), according to Bridgman, is composed of two quantities: 1) the change in the velocity of the electrons, 2) the change in their mean free path. Both of these factors are inversely proportional to the temperature. Therefore

\[ \frac{1}{w}\left(\frac{\partial w}{\partial \theta}\right)_{c} = -\frac{1}{\theta}\left(\frac{1}{2}+2\frac{a}{c}\right). \tag{57} \]

(The conclusion follows.)