Theory of the Metallic State*

L. Nordheim

III. Kinetic Interpretation of Statistics. The Lorentz-Sommerfeld Theory**

§ 1. Counting Collisions in Quantum Theory

In the preceding discussions we used the statistical method. The problem was, from among all possible states of the electron gas in a metal, to find the most probable one. The question of the mechanism by which such a most probable distribution is established was not touched upon at all, and therefore the whole method proves unsuitable for studying processes in which the state of the system changes. These include, in particular, all phenomena connected with the presence of currents. In classical physics this gap is filled by the kinetic theory, which makes it possible to study temporal changes of the distribution function. The aim of the present chapter is to set forth the corresponding theory for quantum statistics.

With the aid of Boltzmann’s H-theorem in the theory of gases it is proved that, in an isolated system, the Maxwell distribution is always established. This conclusion is based on two assumptions: the mechanical law of collisions and the hypothesis of molecular chaos, according to which neighboring molecular states occur, on the average, equally often. From these assumptions there follows the classical expression for the number of collisions, which necessarily leads to Maxwell-Boltzmann statistics.

It is known, however, that the latter must be replaced either by Einstein-Bose statistics or by Fermi-Dirac statistics. Consequently, the classical counting of collisions cannot be correct, and it is necessary to investigate how it must be modified in quantum theory. For this purpose we shall first formulate the old method of counting in the spirit of quantum mechanics.

Let the series of possible stationary states of an individual particle be characterized by the energies \(\varepsilon_1, \varepsilon_2, \ldots, \varepsilon_s\). Let, further, there be some distribution \(N_s\) over separate energy intervals \(\Delta \varepsilon_s\).

* Müller-Pouillet, IV, 11th ed., translated by S. G. Kalashnikov.
** See Uspekhi Fizicheskikh Nauk 15, 570, 1935; 15, 675, 1935.

the number of cells in the interval—\(A_s\). (As before, \(N_s=A_s n_k\), where \(n_k\) is the most probable number of particles in state \(k\). All the arguments can be carried out both with the quantities \(N_s\) and with \(n_k\).) The interaction between particles can be taken into account quite simply if one formally introduces a transition probability. For the case of interaction of two particles (collision) one may speak of the probability \(V_{r's'}^{rs}\) that, in the course of one second, a quantum jump will occur from the cells \(i,k\) to the cells \(i',k'\). The concept of an elementary probability is not yet specific to quantum theory and can be used with success to count collisions also in classical theory.

It is usually assumed that, for initial filling of the intervals (before the collision) \(N_r\) and \(N_s\), the number of transitions into the intervals \(r'\) and \(s'\) can be expressed as follows:

\[ W_{r's'}^{rs}=V_{r's'}^{rs} N_r N_s A_{r'} A_{s'} . \tag{1} \]

Here it is assumed that all \(V_{r's'}^{rs}\) almost do not differ from one another if \(i\) always lies inside the interval \(r\), and \(k,i',k'\)—inside \(s,r',s'\). The quantities \(A_r,A_s\) are regarded as given constants. Let us note that the number of transitions can also be written in the form:

\[ W_{i'k'}^{ik}=V_{i'k'}^{ik} n_i n_k, \]

where \(n\) has the meaning indicated above. In this case

\[ W_{r's'}^{rs}=A_r A_s A_{r'} A_{s'} W_{i'k'}^{ik}. \tag{1a} \]

As expression (1) shows, the number of collisions depends only on the degree of filling of the initial states.

The elementary probabilities \(V_{r's'}^{rs}\) may be calculated from the law of force interaction between the particles. Under very general assumptions one can show,* that \(V\ne 0\) only in those cases in which the law of conservation of energy is satisfied:

\[ \varepsilon_r+\varepsilon_s=\varepsilon_{r'}+\varepsilon_{s'}, \tag{2} \]

and that the probabilities of direct and reverse transitions are equal:

\[ V_{r's'}^{rs}=V_{rs}^{r's'} . \tag{3} \]

The latter result corresponds to the reversibility of mechanical processes in classical theory.

The fundamental expression (1) inevitably leads to classical statistics. Indeed, statistical equilibrium will be attained only when the frequency of direct and reverse processes is the same. This gives the relation:

\[ A_{r'} A_{s'} V_{r's'}^{rs} N_r N_s = A_r A_s V_{rs}^{r's'} N_{r'} N_{s'} . \]

* See, for example, Pauli’s paper\(^1\). There the question is also discussed of what, in quantum mechanics, should replace the hypothesis of molecular chaos. It should be noted that the corresponding assumption is already contained in relation (1).

which, on the basis of (3), can be written in the following form:

\[ \frac{N_r N_s}{A_r A_s}=\frac{N_{r'}N_{s'}}{A_{r'}A_{s'}}. \]

Taking relation (2) into account and then proceeding in the usual way, we arrive at the Maxwell–Boltzmann distribution:

\[ \frac{N_r}{A_r}=n_r=C e^{-\beta \varepsilon_r}. \]

For Fermi–Dirac statistics the characteristic feature is the restriction introduced by the Pauli principle. It follows from this that there can be no processes in which a particle could enter an occupied cell. If not all cells of the interval of final states are free, then only those processes which lead to unoccupied final states must be taken into account. This circumstance can be allowed for quantitatively if, in expression (1), one introduces an additional factor \(1-\frac{N_{s'}}{A_{s'}}\) (or \(1-n_{k'}\)), giving the ratio of the number of free cells to their total number in the given interval. Then for the number of collisions the following expression is obtained:

\[ W_{r's'}^{rs}=V_{r's'}^{rs} N_r N_s A_{r'}\left(1(-+)\frac{N_{r'}}{A_{r'}}\right)A_{s'}\left(1(-+)\frac{N_{s'}}{A_{s'}}\right). \tag{4} \]

The corresponding modification of the law in Einstein–Bose statistics cannot be carried out so visually. Qualitatively, however, it is clear that the correction must lead to an increase in the probability of transitions for which the final state is already occupied. Such a result is obtained because the complexes in which the final state captures more than one particle also have a larger statistical weight in comparison with Boltzmann statistics (cf. example 5, part 1, § 3); such distributions should therefore occur more often. The number of collisions here too will depend on the degree of filling of the final states. The simplest form of the possible correction is obtained by replacing, in the correction factor introduced above, the minus sign by a plus sign (see formula 4). A rationale for such a correction term can be given in Einstein’s theory of radiation² (photon statistics); its form is connected with the existence of induced emission, i.e. with the enhancement of the total emission in the case when photons of the emitted frequency are already present.

Let us now consider the general case of a process in which an arbitrary number of particles passes from the region of states \(s_1, s_2,\ldots\) into \(s_1', s_2',\ldots\). Then the number of collisions (the frequency of occurrence of the process) will be given by the expression:

\[ W_{s_1's_2'\ldots}^{s_1s_2\ldots} = V_{s_1's_2'\ldots}^{s_1s_2\ldots} N_{s_1}N_{s_2}\cdots (A_{s_1'}-\gamma_1 N_{s_1'}) (A_{s_2'}-\gamma_2 N_{s_2'})\cdots \tag{5a} \]

or

\[ W_{k_1'k_2'\ldots}^{k_1k_2\ldots} = V_{k_1'k_2'\ldots}^{k_1k_2\ldots} n_{k_1}n_{k_2}\cdots (1-\gamma_1 n_{k_1'}) (1-\gamma_2 n_{k_2'})\cdots . \tag{5b} \]

Here \(\gamma_i\) is the symbol introduced in part 1, § 4, having in each of the three statistics its own value. In the general case particles belonging to different statisti-

... (an example may be the interaction of photons and electrons). Therefore the symbol \(\gamma\) is supplied with an index. The elementary probability \(V\) depends only on the character of the interaction, but not at all on the special form of the distribution functions. For the elementary probability the reciprocity relation (3) must hold here as well.

The new expression for the number of collisions looks so unusual that distrust might perhaps arise even toward those vivid and simple arguments which we placed at the basis of its derivation. It is therefore extremely important that this law can also be derived directly from the general principles of quantum mechanics. Here we shall indicate only the general course of reasoning. The system as a whole may be described by a single Schrödinger function, into which all the coordinates of the individual particles \(q_1, q_2, \ldots q_i\) enter as variables (\(q_i\) is the coordinate of the \(i\)-th particle). The Hamiltonian function of the entire system must be symmetric with respect to the individual particles; it may be written in the following form:

\[ H(q_1 \ldots q_i \ldots) = \sum_i H_0(q_i) + \sum_{ik} H_1(q_i q_k) + \sum_{ikl} H_2(q_i q_k q_l) + \cdots \tag{6} \]

Here \(H_0\) is the energy of one single particle, on the assumption that there are no other particles at all. \(H_1\) expresses the interaction of each pair of particles (for example, Coulomb forces). The summation \(H_0\) is performed over all individual particles, \(H_1\)—over all pairs of particles, and so on. All the functions \(H_0\), all the functions \(H_1\), etc., are identical functions of their arguments; by this the symmetry of the complete function (6) with respect to all particles is realized. This same fact also ensures the splitting of the solution of Schrödinger’s equation into the different symmetry classes characteristic of each of the statistics. If the interaction is small, then in (6) only the first term will play the predominant role, while the higher terms may be regarded as a small perturbation. In this case the complete Schrödinger function may be represented as an expansion in the eigenfunctions of the unperturbed system. For the latter, one should take, of course, only functions of the required symmetry, i.e. either the functions (3) (Part 1, § 2), or (4), depending on which statistics we are dealing with. For the corresponding perturbation problem one may put:

\[ \psi(q_1 \ldots t) = \sum a_{k_1 k_2 \ldots}(t)\,\psi_{k_1 k_2 \ldots}(q_1 \ldots). \tag{7} \]

Each of the \(\psi_{k\ldots}\) corresponds to the case (Part 1, § 2) when one particle is precisely in the state \(k_1\), the second in the state \(k_2\), and so on; in other words, each function \(\psi\) describes one completely definite state of the complete system. The squares of the coefficients in (7) give the probability that the system at the time \(t\) will be in the corresponding state. Applying to these coefficients \(a_{k_1 k_2\ldots}\) the usual method of perturbation theory, it proves possible to express the result in terms of the transition probability, which in doing so is obtained in the form of relation (5). The factor \((1-\gamma n_k')\) enters here quite automatically. The result presented here for the case of Einstein–Bose statistics was obtained by Dirac\(^3\); the corresponding calculations for Fermi–Dirac statistics (and their generalization) were given by Jordan and Wigner.\(^{4,5}\)

§ 2. The \(H\)-theorem

It turns out to be possible to operate with the new expression for the number of collisions exactly as with the classical one.*

* See the works of Nordheim\(^6\) and Pauli\(^1\). In the latter work there is also considered a case important for photon statistics, when the number of particles during the process does not remain constant.

In what follows we shall consider only such processes in which a total of two particles participate, and which, moreover, obey one and the same statistics. In a state of equilibrium the condition of equal repetition of compensating processes

\[ W^{rs}_{r's'}=W^{r's'}_{rs} \]

may be written, in accordance with § 1 (5a), in the form:

\[ N_{r'}(A_{r'}-\gamma N_{r'})N_{s'}(A_{s'}-\gamma N_{s'}) = N_r(A_r-\gamma N_r)N_s(A_s-\gamma N_s), \]

or, what is the same,

\[ \left(\frac{A_{r'}}{N_{r'}}-\gamma\right) \left(\frac{A_{s'}}{N_{s'}}-\gamma\right) = \left(\frac{A_r}{N_r}-\gamma\right) \left(\frac{A_s}{N_s}-\gamma\right). \tag{1} \]

Taking § 1 (2) into account, we obtain, by the same route as before:

\[ \frac{A_r}{N_r}-\gamma=e^{\alpha+\beta\varepsilon_r}, \]

whence

\[ N_r=\frac{A_r}{e^{\alpha+\beta\varepsilon_r}+\gamma}. \tag{2} \]

Depending on the value of \(\gamma\) \((0,-1,1)\), the last formula gives the laws of distribution of Maxwell–Boltzmann, Einstein–Bose, and Fermi–Dirac.

The result obtained shows that all these laws (2) do indeed give stationary distributions for the corresponding method of counting collisions. At the same time it has been shown that the given form of the expression (5a) is a sufficient condition for the corresponding laws of distribution. The proof of the necessity of this condition may be given in the classical manner with the aid of the \(H\)-theorem. We shall set the function \(H\), as before, equal to \(-k\ln K\) (Part 1, § 4 (9)), which gives

\[ H=-k\sum_s\left\{(N_s-\gamma A_s)\ln(A_s-\gamma N_s)-N_s\ln N_s+\gamma A_s\ln A_s\right\}. \tag{3} \]

For the special case of an equilibrium distribution the latter expression is equal, as is known, to the entropy with the opposite sign.

The change with time of the function \(H\) will be:

\[ \frac{dH}{dt} = -k\sum_s \left\{\ln(A_s-\gamma N_s)+\gamma^2-\ln N_s-1\right\} \frac{dN_s}{dt}. \]

Since

\[ \sum_s \frac{dN_s}{dt}=\frac{dN}{dt}=0, \]

the term \(\gamma^2-1\) in the preceding expression does not appear at all, and we may write:

\[ \frac{dH}{dt} = -k\sum_s \left\{-\ln N_s+\ln(A_s-\gamma N_s)\right\} \frac{dN_s}{dt}. \tag{4} \]

We find the quantity \(\dfrac{dN_s}{dt}\), on the basis of § 1 (5), by summing over all processes that could lead to a change in \(N\):

\[ \frac{dN_s}{dt} = -\sum_{r,r's'} V^{r\ s}_{r'\ s'} N_r N_s (A_{r'}-\gamma N_{r'})(A_{s'}-\gamma N_{s'}) + \sum_{r,r's'} V^{r'\ s'}_{r\ s} N_{r'} N_{s'} (A_r-\gamma N_r)(A_s-\gamma N_s). \tag{5} \]

The first term in this expression gives the number of particles leaving, as a result of collisions, the interval \(s\); the second term, the number of particles entering this interval. The summation need be carried out over each pair of indices only once. The quantity \(V^{r\ s}_{r'\ s'}\), by definition, is symmetric both with respect to \(r'\), \(s'\), and with respect to \(r\), \(s\), and therefore

\[ \sum_s \frac{dN_s}{dt}=\frac{dN}{dt}=0 . \]

The last result was to be expected, since in the summation there must necessarily enter pairs of identical terms differing only in sign.

Substituting (5) into (4) and taking into account the symmetry property § 1 (3), we find

\[ \frac{dH}{dt} = -k\sum_s \sum_{r,r's'} V^{r\ s}_{r'\ s'} \left\{ -N_r N_s (A_{r'}-\gamma N_{r'})(A_{s'}-\gamma N_{s'}) \ln\left(\frac{A_s}{N_s}-\gamma\right) \right. \]

\[ \left. +N_{r'}N_{s'}(A_r-\gamma N_r)(A_s-\gamma N_s) \ln\left(\frac{A_s}{N_s}-\gamma\right) \right\}. \]

Collecting the terms in the last expression pairwise, we may replace the summation over the separate indices \(r\) and \(s\) by a summation over the pairs \((rs)\). Then

\[ \frac{dH}{dt} = \]

\[ = -k\sum_{rs,r's'} V^{r\ s}_{r'\ s'} \left\{ -N_rN_s(A_{r'}-\gamma N_{r'})(A_{s'}-\gamma N_{s'}) \ln\left(\frac{A_r}{N_r}-\gamma\right)\left(\frac{A_s}{N_s}-\gamma\right) \right. \]

\[ \left. +N_{r'}N_{s'}(A_r-\gamma N_r)(A_s-\gamma N_s) \ln\left(\frac{A_r}{N_r}-\gamma\right)\left(\frac{A_s}{N_s}-\gamma\right) \right\}. \]

Since, upon interchanging the primed indices with the unprimed ones, nothing changes in the last expression, we may also write it in the following form:

\[ \frac{dH}{dt} = -\frac{k}{2}\sum_{rs,r's'} V^{r\ s}_{r'\ s'} \left\{ -N_rN_s(A_{r'}-\gamma N_{r'})(A_{s'}-\gamma N_{s'}) \right. \]

\[ \left. +N_{r'}N_{s'}(A_r-\gamma N_r)(A_s-\gamma N_s) \right\} \left\{ \ln\left(\frac{A_r}{N_r}-\gamma\right)\left(\frac{A_s}{N_s}-\gamma\right) - \right. \]

\[ \left. -\ln\left(\frac{A_{r'}}{N_{r'}}-\gamma\right)\left(\frac{A_{s'}}{N_{s'}}-\gamma\right) \right\}, \]

or, finally:

\[ \frac{dH}{dt} = -\frac{k}{2}\sum_{rs,r's'} V_{rs,r's'}\,N_r N_s N_{r'} N_{s'} \left\{ \left(\frac{A_r}{N_r}-\gamma\right) \left(\frac{A_s}{N_s}-\gamma\right) - \left(\frac{A_{r'}}{N_{r'}}-\gamma\right) \left(\frac{A_{s'}}{N_{s'}}-\gamma\right) \right\} \left\{ \ln\left(\frac{A_r}{N_r}-\gamma\right) \left(\frac{A_s}{N_s}-\gamma\right) - \ln\left(\frac{A_{r'}}{N_{r'}}-\gamma\right) \left(\frac{A_{s'}}{N_{s'}}-\gamma\right) \right\}. \tag{6} \]

In the expression obtained, the terms standing in curly brackets have the same form \((a-b)(\ln a-\ln b)\) as in the ordinary \(H\)-theorem. This form is positive for any \(a\) and \(b\), except when \(a=b\), when it becomes zero. Since all the other quantities in (6) are also positive, we obtain from this:

\[ \frac{dH}{dt}=-\frac{dS}{dt}\leqq 0. \]

Here the equality sign will hold only when the third relation (1) is satisfied, i.e., when there is mutual balancing of compensating processes. By the preceding argument the \(H\)-theorem has been proved in full for all three statistics. This also shows the necessity of the corresponding method of counting collisions for the distribution functions given above.

The generalization of the result obtained to the case of interaction of particles of different types can be carried out according to exactly the same scheme. All the physical consequences of the \(H\)-theorem, for example the irreversibility of thermodynamic processes, etc., follow from this in exactly the same way as in the classical theory.

§ 3. The Fundamental Equation of the Electron Theory

We now pass to the most important part of the theory, namely to the analysis of the phenomena of electrical and thermal conductivity. For this, however, it is first necessary to make a few preliminary remarks.

Up to now we have considered only static problems: the calculation of equilibrium states and of the effects connected with them. Turning now to the phenomena of transport of heat and electricity, we come to processes in which an exchange of energy and charges takes place. Nevertheless, these processes too can be regarded as stationary, provided only that within a certain finite volume all quantities do not change with time.

The method of kinetic theory proves suitable for these cases as well. Any state of the system can always be described by some distribution function, which now, however, may no longer correspond to an equilibrium state. To solve the problem, it is necessary to take into account all processes that might cause a change in the state of an individual particle, and then to form the overall balance. In this way, for the distribution function one can obtain a functional equation, from which, under the corresponding initial and boundary conditions, it can also be determined. Here we have a complete analogy with Boltzmann’s well-known method in the classical theory of gases.

The general scheme which we take as the basis is as follows. In the zeroth approximation the electrons of a metal constitute an electron gas obeying Fermi statistics. This approximation proves sufficient for determining the equilibrium distribution and for describing all those phenomena that were considered in Part II of the present article. Applying the methods of kinetic theory, we can take into account that the electrons are not entirely free and that their motion is perturbed by the influence of the field of the positive ions. In addition, we must take into account that large forces act between the electrons themselves. All these interactions we can describe by means of the concept of collision processes introduced in the preceding paragraphs. Besides the interaction between particles, the form of the distribution function is also affected by a number of other factors: external forces, thermal contacts with sources of different temperatures, and so forth. In the case when it is possible to find a modified distribution function which, with all these factors taken into account, remains stationary, the problem is fully solved, and all macroscopic quantities can be readily calculated.

In solving the problem there arises a difficulty consisting in the fact that the state of the system, strictly speaking, is not the same in space. This occurs primarily because we usually have a spatially varying potential and temperature. It is not difficult to allow for this circumstance when all changes over atomic distances are small. In this case we may describe an individual particle (electron) not by a proper function of the entire region, but introduce only such functions which differ from zero only in a small volume (in comparison with macroscopic dimensions). Such a description can always be obtained with the aid of wave packets, which approximately (in the sense of the uncertainty relation) represent a particle with a definite position and momentum. Hence it is clear that in quantum theory too it makes sense to describe systems whose state depends on the spatial coordinates by a density distribution in the 6-dimensional phase space of a single particle. Such a description is effected by the distribution function

\[ \frac{G}{h^3} f(x,y,z,p_x,p_y,p_z,t)\, dx\, dy\, dz\, dp_x\, dp_y\, dp_z , \]

which may also depend on time. In doing so, of course, only such volume elements must always be considered as are larger than the cell \(h^3\). Such a choice of a volume element, which on the one hand is considerably smaller than the region within which the state changes appreciably, and on the other hand is considerably larger than atomic quantities, is the usual device of kinetic theory. The normalizing factor \(\frac{G}{h^3}\) is introduced so that for \(f=1\) in one cell \(\frac{h^3}{G}\) there is exactly one particle. The quantity \(f\) corresponds to the numbers \(n_k\) for each cell [see Part 1, § 6 (13)].

Instead of the momentum components \(p_x,\ p_y,\ p_z\), we could introduce the components of velocity:

\[ \xi=\frac{p_x}{m},\quad \eta=\frac{p_y}{m},\quad \zeta=\frac{p_z}{m},\quad dp_x=m\,d\xi,\ \text{etc.} \]

We shall now follow the usual method, which in the kinetic theory of gases leads to Boltzmann’s fundamental equation. Let us consider particles in some definite element of phase space. A change with time in the function \(f\) may occur, first, as a result of the presence of an external force \(K\) acting directly on all particles (in our case, electric and magnetic fields). This force will impart acceleration to the particles and thereby produce a change in the momentum \(p\). In the terms of wave mechanics this is expressed as a change in the mean values in the wave packet. According to a well-known theorem of wave mechanics,* the acceleration of a wave packet is obtained exactly the same as in classical mechanics:

\[ m\dot{\xi}=K_x,\ \text{etc.} \tag{1} \]

Further, the group velocity of propagation of the packet, according to de Broglie’s law, coincides with the classical value of the particle velocity. We therefore obtain in phase space the same mean changes of the density distribution as in the classical theory, and for the change of the distribution function (the excess of the number of departing particles) we may write:

\[ \frac{df^*}{dt} = \frac{\partial f}{\partial t} + \frac{\partial f}{\partial \xi}\dot{\xi} + \frac{\partial f}{\partial \eta}\dot{\eta} + \frac{\partial f}{\partial \zeta}\dot{\zeta} + \frac{\partial f}{\partial x}\dot{x} + \frac{\partial f}{\partial y}\dot{y} + \frac{\partial f}{\partial z}\dot{z} = \frac{\partial f}{\partial t} + \frac{K_x}{m}\frac{\partial f}{\partial \xi} + \frac{K_y}{m}\frac{\partial f}{\partial \eta} + \frac{K_z}{m}\frac{\partial f}{\partial \zeta} + \xi\frac{\partial f}{\partial x} + \eta\frac{\partial f}{\partial y} + \zeta\frac{\partial f}{\partial z}. \tag{2} \]

We mark this derivative with an asterisk in order to show that not all causes producing changes in the distribution function have yet been taken into account here.

The magnitude of the flux (2) will not depend on whether the particles obey Bose–Einstein or Fermi–Dirac statistics. Indeed, according to Liouville’s classical theorem, the distribution density in phase space will remain constant at all times; in order not to enter into contradiction with Pauli’s principle, we need only require that this density be initially less than unity. The quantum-mechanical justification is given by the circumstance that even in the presence of an external field (provided only that the interaction between particles is neglected) the energy function will contain only the terms \(H_0\) (see § 1 (6)). Therefore,

* Most simply proved by Ehrenfest (7).

** See the work of Kikuchi and Nordheim (5). It is shown there that the requirement that two particles always be in different states (Pauli’s principle) for wave packets (which do not correspond to states with precisely determined energy) is equivalent to the requirement of their orthogonality \(\left(\int \psi\psi' \, dV=0\right)\). This property is preserved also under any external perturbations.

different wave packets will remain completely unconnected and will not be able to influence one another.

Expression (2) does not give the complete change of the distribution function, because we have not yet taken mutual collisions into account. As has already been mentioned, the interaction forces of the electrons in a metal can be divided into two types: the interaction of the electrons with one another (the Coulomb force) and the interaction with the ions of the lattice. The first type of force is highly inconvenient for calculation and is usually neglected; in the present article we shall likewise not take it into account. This neglect, despite the considerable magnitude of the Coulomb forces, can be physically justified to a certain extent by the fact that in collisions of this type momentum is conserved. Therefore neither the magnitude of the total kinetic energy nor the magnitude of the current, which is proportional to the momentum, changes in such collisions. In addition, it should be borne in mind that at considerable distances the interaction between electrons is screened by the positive ions lying between them.

Thus only the interaction with the ions of the lattice is to be taken into account. This latter must already necessarily be included, since otherwise the electrons could be accelerated arbitrarily, and the current would increase without limit. This interaction causes a continuous braking of the electrons and thereby ensures a finite value of the resistance of the metal. According to Lorentz these processes can be represented simply as reflection from ions, regarded as elastic spheres. The corresponding ideas of the modern theory will be examined by us in detail later. For the present, reasoning in the spirit of §§ 1, 2, we can use the following very general proposition. For each individual electron (wave packet) there exists a definite probability that the electron under consideration will undergo a collision per unit time. As a result of the collision the electron, while remaining in the same spatial element, will pass into a new interval of velocities \(d\xi' d\eta' d\zeta'\). According to the basic relation § 1(5), for the number of transitions (calculated per cell) we may write:

\[ W_{v'}^{v}=V_{v'}^{v} f(1-\gamma f'), \tag{3} \]

where, for brevity, \(f'\) denotes \(f(x,y,z,\xi',\eta',\zeta')\). The number of transitions \(W_{v'}^{v}\) is expressed as the product of the elementary probability \(V_{v'}^{v}\), independent of the number of electrons, the number of particles \(f\) in the initial state, and the factor \((1-\gamma f')\), which depends on the degree of filling of the final states.

We find the total change in the number of particles in the element \(dx\,dy\,dz\,d\xi\,d\eta\,d\zeta\) by summing all possible transitions. It can be represented as the difference of the number of outgoing particles:

\[ a\,dV\,d\tau=-dV\,d\tau\,\frac{Gm^3}{h^3}\iiint V_{v'}^{v} f(1-\gamma f')\,d\tau', \tag{4} \]

\[ (dx\,dy\,dz=dV,\ d\xi\,d\eta\,d\zeta=d\tau) \]

and the number entering:

\[ b\,dV\,d\tau=dV\,d\tau\cdot {Gm^3\over h^3}\int\!\!\int\!\!\int V_{v}^{v'} f'(1-\gamma f)\,d\tau'. \tag{5} \]

Thus the total change (decrease) of the distribution density will be:

\[ -\,{df\over dt}={df^*\over dt}+a-b. \tag{6} \]

This equation must be satisfied for any element of phase space.

We shall now require that the distribution be stationary. In this case expression (6) becomes zero, and the function \(f\) will not contain the time \(t\) explicitly. Then we obtain the desired fundamental equation in the following form:

\[ {K_x\over m}{\partial f\over \partial \xi} +{K_y\over m}{\partial f\over \partial \eta} +{K_z\over m}{\partial f\over \partial \zeta} +\xi{\partial f\over \partial x} +\eta{\partial f\over \partial y} +\zeta{\partial f\over \partial z} =b-a. \tag{7} \]

We thus arrive at an integro-differential equation for determining \(f\).

Once the distribution function has been found, the problem is completely solved. The current density is obtained by summing the quantities \(e\mathbf v\) over the individual electrons*:

\[ \mathbf i={eGm\over h^3}\int \mathbf v\left\{ {1\over \Delta V}\int_{\Delta V} f\,dV\right\}d\tau; \tag{8a} \]

the heat flux is calculated by an analogous summation (\(\varepsilon\) is the kinetic energy of an individual particle):

\[ \mathbf w={Gm^3\over h^3}\int \varepsilon\mathbf v\left\{ {1\over \Delta V}\int_{\Delta V} f\,dV\right\}d\tau. \tag{8b} \]

In doing this we may choose the volume element so small that, in the integration, \(f\) may be regarded as constant. Then the last two expressions are simplified, and we obtain:

\[ \mathbf i=e{Gm^3\over h^3}\int \mathbf v f\,d\tau, \tag{9} \]

\[ \mathbf w={Gm^3\over h^3}\int \varepsilon\mathbf v f\,d\tau. \tag{10} \]

Both of these quantities have a vector character.

From the fundamental equation (7) a number of conclusions can be drawn without resorting to its solution. When there are no external actions (neither an external field nor spatial inhomogeneities), the expression \((a-b)\) must vanish. This corresponds to compensation of the opposite transitions in the state of equilibrium (§ 1). Hence we immediately obtain from (4) and (5) the condition for the equilibrium distribution \((f_0)\):

\[ V_{v}^{v'} f_0(1-\gamma f_0')=V_{v'}^{v} f_0'(1-\gamma f_0). \tag{11} \]

\[ \text{* } e \text{ is the negative elementary charge.} \]

The quantities \(V\) here are not directly elementary probabilities in the sense of § 1, since they also depend on the state of the perturbing system (the crystal lattice); the latter itself changes its state upon collision with an electron.

We have already seen that, by using the appropriate method of counting collisions, we shall always arrive at one of the three statistics. We shall use this circumstance in order to obtain more detailed information about the quantities \(V_{v'}^{v}\) entering relation (11). Without yet specifying the question of the nature of the statistics (so as to include also the case of classical statistics), we may put:

\[ f_0=\frac{1}{e^{\alpha+\frac{\varepsilon'}{kT}}+\gamma} \tag{12} \]

(the term \(1-\gamma^2\) in the numerator for the case of Boltzmann statistics may be discarded as playing no role). Substituting this expression in (11), we find:

\[ V_{v'}^{v} e^{\frac{\varepsilon'}{kT}} = V_{v}^{v'} e^{\frac{\varepsilon}{kT}} . \tag{13} \]

This relation between \(V_{v'}^{v}\) and \(V_{v}^{v'}\) must always be fulfilled, provided only that the elementary probabilities are calculated from the fundamental principles of quantum mechanics. Hence:

\[ a-b=\frac{Gm^3}{h^3}\int V_{v'}^{v} \left\{ f(1-\gamma f')-e^{\frac{\varepsilon'-\varepsilon}{kT}} f'(1-\gamma f) \right\}d\tau' . \tag{14} \]

For the special case of processes in which, in collisions, the electron changes only the direction of its motion (but not its energy), the elementary probability is different from zero only for \(\varepsilon=\varepsilon'\). In this case, from (13) and (14) we obtain:

\[ a-b=\frac{Gm^3}{h^3}\int V_{v'}^{v}(f-f')\,d\tau' . \tag{15} \]

It is evident that precisely for the electrons of a metal the transfer of energy in collisions must be very small. This follows directly from the laws of conservation of energy and momentum, according to which in a collision only a part of the energy of order \(\frac{m}{M}\) (\(M\) is the mass of the atom) can be transferred. In what follows we shall therefore restrict ourselves* only to the special case (15).

* The theory can be developed in an analogous way without such a restriction as well [cf. Nordheim’s paper (8)]. It should be noted already now that, for the description of the properties of a metal at low temperatures, energy exchange is very significant. Whereas at high temperatures the special case considered above is a good approximation, at low temperatures the whole course of the argument becomes incorrect. The terms “high” and “low” temperature should be understood in relation to the Debye characteristic temperature \(\theta\).

Under the assumption made, any distribution function \(f_0\) depending only on the energy is compatible with (11). In particular, all three functions (12) satisfy this equation. Likewise, the basic equation will be the same for all three statistics. A difference arises only when the energy-transfer processes that we have neglected are taken into account. Let us note that here we are proceeding in the same way as with the interaction forces in calculating mean values; in the calculation we may neglect the interaction altogether, although its existence is unconditionally necessary for the establishment of definite values of the mean quantities.

§ 4. Lorentz–Sommerfeld Theory

The general solution (7) of § 3 in the presence of an external field may be obtained by the method indicated by Lorentz. Suppose that the true distribution \(f\) at any point of space differs only slightly from the equilibrium distribution \(f_0\). Then one may put:

\[ f=f_0+f_1,\quad f_1\ll f_0. \tag{1} \]

In accordance with what was said at the end of the preceding paragraph, from the logical point of view it is quite permissible to carry out the calculation with any initial function \(f_0\). However, the result will, of course, depend substantially on what we choose for \(f_0\). The reliably established empirical validity of Pauli’s principle compels us to adopt for \(f_0\) the Fermi–Dirac distribution as the only possible one; in this case we arrive at Sommerfeld’s theory. If for \(f_0\) we take the Maxwell distribution, we obtain Lorentz’s theory. In the chosen mode of exposition, the two theories are only two different special cases.

It should be noted that \(f_0\) is now already a function of the coordinates. It is defined as before by § 3 (12), but now we already allow the quantities \(\alpha\) and \(T\), in accordance with the presence of gradients, to vary in space.

Assuming \(f\) in the form (1), we simplify the problem as follows. The term \((a-b)\), which depends on collisions, vanishes identically for \(f_0\). Therefore only \(f_1\) remains on the right-hand side of the basic equation. Further, since \(f_0\) at every point possesses spherical symmetry with respect to the components of velocity,* according to § 3 (9), (10) the influence of \(f_0\) on the flux of heat and electricity disappears. In the left-hand side of the equation one may neglect the small

* We are dealing here with free electrons, for which

\[ \varepsilon=\frac{m}{2}(\xi^2+\eta^2+\zeta^2). \]

For non-free electrons the latter need no longer necessarily hold (see Part IV, §§ 3–5). However, \(\varepsilon(\xi,\eta,\zeta)\) will always have central symmetry, which is already sufficient for the assertion made in the text.

by the perturbation \(f_1\) relative to \(f_0\). In this approximation we obtain:

\[ \frac{d^{*}f_0}{dt}=-\frac{Gm^3}{h^3}\iiint V_{\nu'}^{\nu}(f_1-f_1')\,d\xi'\,d\eta'\,d\zeta', \]

where the unknown function \(f_1\) is contained only on the right-hand side.

Let us first consider the relatively simple case in which changes of state occur only along the \(X\)-axis. In this case we have:

\[ \frac{d^{*}f_0}{dt}=\frac{eF}{m}\frac{\partial f_0}{\partial \xi}+\xi\frac{\partial f_0}{\partial x} =-\frac{Gm^3}{h^3}\iiint V_{\nu'}^{\nu}(f_1-f_1')\,d\xi'\,d\eta'\,d\zeta'=b-a, \tag{2} \]

where \(F\) denotes the electric-field strength \((K_x=eF)\). The equation (2) written above is the starting point in the theories of Lorentz and Sommerfeld.

We now introduce for \(f_0\) its expression § 3 (12). In this case [Part I, § 9 (2a), (2b), (3), (4), (7)] for Boltzmann statistics (weak degeneracy, Lorentz theory)

\[ \alpha=\ln\frac{G}{nh^3}(2\pi mkT)^{\frac{3}{2}}, \tag{3a} \]

and for Fermi–Dirac statistics (strong degeneracy, Sommerfeld theory):

\[ \alpha=-\frac{\varepsilon_0}{kT}=-\frac{\mu}{kT}\left\{1-\frac{\pi^2}{12}\left(\frac{kT}{\mu}\right)^2+\cdots\right\}; \quad \mu=\frac{h^2}{2m}\left(\frac{3n}{4\pi G}\right)^{\frac{2}{3}}. \tag{3b} \]

Assuming, further,

\[ \varepsilon=-\frac{m}{2}(\xi^2+\eta^2+\zeta^2), \tag{4} \]

we have:

\[ \frac{\partial f_0}{\partial \xi} =\frac{\partial f_0}{\partial \varepsilon}\frac{\partial \varepsilon}{\partial \xi} =-m\xi\frac{\partial f_0}{\partial \varepsilon}, \]

\[ \frac{\partial f_0}{\partial x} = -\frac{df_0}{\partial\left(\alpha+\frac{\varepsilon}{kT}\right)} \frac{\partial}{\partial x}\left(\alpha+\frac{\varepsilon}{kT}\right) = kT\frac{\partial f_0}{\partial \varepsilon} \left(\frac{\partial \alpha}{\partial x} -\frac{\varepsilon}{kT^2}\frac{\partial T}{\partial x}\right). \]

Thus we obtain the following general expression:

\[ \frac{d^{*}f_0}{dt} = \frac{\partial f_0}{\partial \varepsilon}\xi \left(eF+kT\frac{\partial \alpha}{\partial x} -\frac{\varepsilon}{T}\frac{\partial T}{\partial x}\right). \tag{5} \]

Further simplification of the term depending on collisions is possible only under definite assumptions about the transition probability. We shall assume first that the metal is isotropic.* This isotropy may be connected either with the properties of the crystal itself (cubic system), or may have a statistical character if we have a disordered arrangement of microcrystallites. In the presence of isotropy the transition probability \(V_{\nu'}^{\nu}\)

* This assumption is not suitable for analyzing the Bridgman effect, which we shall consider in § 6.

THEORY OF THE METALLIC STATE

will no longer depend on the absolute orientation of the vectors \(v\) and \(v'\), but will be a function only of their relative arrangement. Assuming, for simplicity, the absolute values of the velocities \(v\) and \(v'\) to be the same, we may regard \(V\) as a function only of \(v\) and of the angle \(\Theta\) between the vectors.

This permits (Lorentz) one to assume for \(f_1\) the following dependence on direction:

\[ f_1=\xi\chi(v), \tag{6} \]

where \(\chi\) now depends only on \(v\) (on the energy \(\varepsilon=\frac{m}{2}v^2\)). Relation (6) is justified by the fact that, with its aid, the integral equation for \(f_1\) can indeed be solved.*

For what follows it is expedient to introduce the polar coordinates shown in Fig. 1. With this choice of coordinates, integration over \(\xi'\), \(\eta'\), \(\zeta'\) may be replaced by integration over \(\Theta\) and \(\alpha\):

Fig. 1.

\[ \frac{1}{v^3}\,d\xi'\,d\eta'\,d\zeta'=\sin\Theta\,d\Theta\,d\alpha;\quad \chi(v')=\chi(v); \]

\[ \frac{Gm^3}{h^3}V_{vv'}=\frac{1}{v^3}V(v,\Theta); \]

and hence

\[ b-a=\int_0^\pi\int_0^{2\pi} V(v,\Theta)(f'-f)\sin\Theta\,d\Theta\,d\alpha = \]

\[ =\chi(v)\int_0^\pi\int_0^{2\pi} V(v,\Theta)(\xi'-\xi)\sin\Theta\,d\Theta\,d\alpha. \]

The quantity \(V(v,\Theta)\) appearing here gives the probability that some particle with velocity \(v\) will, in unit time, fall within the solid angle \(\sin\Theta\,d\Theta\,d\alpha\). From Fig. 1 we further find:

\[ \xi=v\cos\vartheta,\quad \xi'=v\cos\vartheta',\quad \cos\vartheta'=\cos\Theta\cos\vartheta+\sin\Theta\sin\vartheta\cos\alpha, \]

\[ \text{* According to Bohr }(^{9})\text{, one can avoid (6) by introducing as the sought function the total momentum of all electrons:} \]

\[ G(v)=\int_v^{v+dv}\iint_{\vartheta\ \varphi} vf_1 v^2\,dv\,\sin\vartheta\,d\vartheta\,d\varphi. \tag{6a} \]

However, the calculation proves possible only under the assumptions given in the text, and the results obtained are completely the same. Whether it is substantial, for the space integration, to put (6a) at the beginning of the calculation or at the end is immaterial.

and therefore

\[ \begin{aligned} b-a &=\chi\int_{0}^{\pi}\int_{0}^{2\pi} V(v,\Theta)\{v(\cos\vartheta\cos\Theta+ \\ &\quad +\sin\vartheta\sin\Theta\cos\alpha)-v\cos\vartheta\}\sin\Theta\,d\Theta d\alpha=\\ &=2\pi\chi\int_{0}^{\pi}V(v,\Theta)v\cos\vartheta(\cos\Theta-1)\sin\Theta d\Theta=\\ &=2\pi\xi\chi\int_{0}^{\pi}V(v,\Theta)(\cos\Theta-1)\sin\Theta d\Theta . \end{aligned} \tag{7} \]

In integrating with respect to \(\alpha\) it was taken into account that \(\alpha\) enters none of the quantities except \(\cos\alpha\).

The quantity entering (7),

\[ \frac{a-b}{\xi\chi} = 2\pi\int_{0}^{\pi}V(v,\Theta)(1-\cos\Theta)\sin\Theta d\Theta = \frac{v}{l}, \tag{8} \]

must be regarded as given, independent of the form of the distribution function. As we shall now see, \(l\) is nothing other than the electron’s mean free path [depending, according to (8), in a definite way on the velocity \(v\)]. The fundamental equation (2), together with (5), (7), and (8), then gives for \(\chi(v)\) an ordinary equation, from which one obtains:

\[ \chi = -\frac{l}{v}\frac{\partial f_0}{\partial \varepsilon} \left( eF+kT\frac{\partial x}{\partial x} -\frac{\varepsilon}{T}\frac{\partial T}{\partial x} \right). \tag{9} \]

Since \(f_1=\xi\chi\), relation (9) gives the solution of the problem posed.

That the quantity \(l\) must indeed be interpreted as a mean free path can be shown as follows. Suppose that, in the absence of external influences, at the time \(t=0\) there is a certain perturbation (6) of the distribution function \(f_0\). The fundamental equation [cf. § 3 (6)] gives:

\[ \frac{df}{dt} = \frac{d(f_0+f_1)}{dt} = -a+b = -\frac{v}{l}f_1, \]

and, since \(f_0\) is stationary \(\left(\dfrac{df_0}{dt}=0\right)\):

\[ \frac{df_1}{dt} = -\frac{v}{l}f_1,\qquad f_1(t)=f_1(0)e^{-\frac{v}{l}t}. \]

We see that the initial perturbation is indeed weakened by collision processes by a factor \(e\) over the distance \(l\).*

\[ \text{* Usually the mean free path is defined as the segment over which a ray of particles (of identical velocity) is attenuated by a factor } e. \]
Here, however, the question is the diminution of the additional term in the distribution function. The two definitions coincide completely when \(V\) does not depend on \(\Theta\). The quantity

\[ \tau=\frac{l}{v} \]

gives the relaxation time.

Having determined \(f_1=\chi \xi\), we can calculate, according to § 3 (9), (10), the heat and electric currents. For the electric current \(i_x\) (the remaining components are zero) we find:

\[ i_x=e\frac{Gm^3}{h^3}\int_{-\infty}^{+\infty}\!\!\int\!\!\int \xi f_1\,d\xi d\eta d\zeta = e\frac{Gm^3}{h^3}\int\!\!\int\!\!\int \xi^2\chi(v)\,d\xi d\eta d\zeta. \]

Introducing, instead of \(\xi,\eta,\zeta\), the new polar variables

\[ \xi=v\cos\vartheta;\quad d\xi d\eta d\zeta=v^2dv\sin\vartheta d\vartheta d\varphi \]

and carrying out the integration with respect to \(\vartheta\) and \(\varphi\), we obtain:

\[ i_x=e\frac{Gm^3}{h^3}\frac{4\pi}{3}\int_0^\infty v^4\chi dv = -e\frac{4\pi Gm^3}{3h^3}\int_0^\infty v^3 l\frac{\partial f_0}{\partial \varepsilon} \left(eF+kT\frac{\partial \alpha}{\partial x}-\frac{\varepsilon}{T}\frac{\partial T}{\partial x}\right)dv. \]

Introducing, finally,

\[ \varepsilon=\frac{m}{2}v^2;\quad v^3dv=\frac{2\varepsilon}{m^2}d\varepsilon, \]

we find for the electric current the following expression:

\[ i_x=-e\frac{8\pi}{3}\frac{Gm}{h^3}\int_0^\infty l\varepsilon\frac{\partial f_0}{\partial \varepsilon} \left(eF+kT\frac{\partial \alpha}{\partial x}-\frac{\varepsilon}{T}\frac{\partial T}{\partial x}\right)d\varepsilon. \tag{11a} \]

The magnitude of the heat flux [§ 3 (10)] is given by an analogous expression:

\[ w_x=\frac{Gm^3}{h^3}\int_{-\infty}^{+\infty}\!\!\int\!\!\int \xi\varepsilon f_1\,d\xi d\eta d\zeta= \]

\[ =-\frac{8\pi}{3}\frac{Gm}{h^3}\int_0^\infty l\varepsilon^2\frac{\partial f_0}{\partial \varepsilon} \left(eF+kT\frac{\partial \alpha}{\partial x}-\frac{\varepsilon}{T}\frac{\partial T}{\partial x}\right)d\varepsilon. \tag{11b} \]

The final results obtained may be written in the following, more convenient form:

\[ i_x=eK_1\left(eF+kT\frac{\partial \alpha}{\partial x}\right) -e\frac{K_2}{T}\frac{\partial T}{\partial x}, \tag{12a} \]

\[ w_x=K_2\left(eF+kT\frac{\partial \alpha}{\partial x}\right) -\frac{K_3}{T}\frac{\partial T}{\partial x}, \tag{12b} \]

where

\[ K_n=-\frac{8\pi Gm}{3h^3}\int_0^\infty l\varepsilon^n\frac{\partial f_0}{\partial \varepsilon}\,d\varepsilon,\quad (n=1,2,3). \tag{13} \]

Let us note that here \(\dfrac{\partial f_0}{\partial \varepsilon}\) is always negative, and therefore all \(K_n\) are positive quantities.

In formulas (12) and (13) both Sommerfeld’s theory and Lorentz’s theory are contained in full. These relations are valid for all three statistics, which affect only the form of the function \(f_0\). They are also valid for any law of interaction that affects only the magnitude \(l\). Formulas (12a, b) will remain valid even if we discard the assumption that there is no exchange of energy in collisions. In that case only the expressions for \(K_n\) will change, and these are calculated here in a considerably more complicated way (8).

Thus the problem has been reduced by us to the determination of the quantities \(K_n\). Let us note that the latter are functions of the temperature, which enters both into \(f_0\) and into \(\alpha\).

In the case of Fermi–Dirac statistics, and moreover of strong degeneracy (Sommerfeld’s theory), owing to the special form of the distribution function the integration can be carried out without exact knowledge of the mean free path (transition probability). For \(K_n\) in this case [§ 3 (12), § 4 (3b)] we obtain the expression:

\[ K_n=\frac{8\pi Gm}{3h^3}\int_0^\infty l(\varepsilon)\varepsilon^n \frac{d\varepsilon} {kT\left(e^{\frac{\varepsilon_0-\varepsilon}{kT}}+1\right)\left(e^{\frac{-\varepsilon_0+\varepsilon}{kT}}+1\right)}, \tag{14} \]

which, putting

\[ x=\frac{\varepsilon}{kT};\qquad \frac{\varepsilon_0}{kT}=a, \]

may be written in the following form:

\[ K_n=\frac{8\pi Gm}{3h^3}(kT)^n \int_0^\infty \frac{lx^n\,dx} {\left(e^{a-x}+1\right)\left(e^{-a+x}+1\right)}. \]

This expression is of exactly the same form as (5b) of § 8, part I, and therefore

\[ \begin{aligned} K_n &=\frac{8\pi Gm}{3h^3}(kT)^n \left\{ lx^n+2\left[c_2\frac{d^2(lx^n)}{dx^2}+\cdots\right] \right\}_{x=a} \\ &=\frac{8\pi Gm}{3h^3} \left\{ l\varepsilon^n+2c_2(kT)^2\frac{d^2(l\varepsilon^n)}{d\varepsilon^2}+\cdots \right\}_{\varepsilon=\varepsilon_0} \\ &=\frac{8\pi Gm}{3h^3} \left\{ l_0\varepsilon_0^n+\frac{\pi^2}{6}(kT)^2 \left.\frac{d^2(l\varepsilon^n)}{d\varepsilon^2}\right|_{\varepsilon=\varepsilon_0} +\cdots \right\}. \tag{15a} \end{aligned} \]

Here \(K_n\) depends only on the behavior of \(l(\varepsilon)\) near \(\varepsilon=\varepsilon_0\) and, in the first approximation, is determined by the value of \(l\) at \(\varepsilon=\varepsilon_0\). We have marked this circumstance by the index 0 on \(l\) and \(\varepsilon\). Thus, for the further calculation we do not yet need more detailed information about the law of scattering. Later, in part IV, we shall examine this question in greater detail.

For the case of Maxwell–Boltzmann statistics (Lorentz’s theory), more exact data on the mean free path are necessary. Here we shall restrict ourselves to the simplest assumption and shall consider that \(l\) does not depend on the velocity at all. In this case it is necessary—

the required integration can be carried out, and for \(K\) one obtains [cf. (13) and Part I § 9 (2a)] the following expression:

\[ K_n=\frac{8\pi N}{3m^2}\left(\frac{m}{2\pi kT}\right)^{\frac{3}{2}}\frac{l}{kT} \int_0^\infty \varepsilon^n e^{-\frac{\varepsilon}{kT}}\,d\varepsilon = \]

\[ =\frac{8\pi N}{3m^2}\left(\frac{m}{2\pi kT}\right)^{\frac{3}{2}} l(kT)^n\int_0^\infty x^n e^{-x}\,dx \]

(\(N\)—the number of electrons in \(1\ \mathrm{cm}^3\)). The last integral is known; it is simply equal to \(n!\), and we finally find:

\[ K_n=\frac{4}{3}\frac{Nl}{\sqrt{2\pi mkT}}(kT)^{n-1}n! \tag{15b} \]

The assumption we have used that \(l\) is independent of the velocity corresponds to Lorentz’s original model, in which the ions are regarded as infinitely heavy elastic spheres. In this case

\[ l=\frac{1}{n\pi R^2}, \tag{16} \]

where \(n\)—is the number of ions in \(1\ \mathrm{cm}^3\), and \(R\)—the radius of the ion. In what follows we shall all the time compare the results of the two theories, marking the corresponding formulas with the letters S (Sommerfeld) and L (Lorentz).

With regard to the physical interpretation of the entire calculation, the following must be noted. We have everywhere neglected energy exchange between the electrons and the lattice. Correspondingly, we should first have obtained a transition of the mechanical energy of the electrons into the thermal energy of the same electron gas. However, this thermal energy must be equalized and must pass, in part, also to the ionic lattice. Thus the energy accumulated by the electrons in the field will ultimately pass into the thermal energy of the entire metal. We again leave open for the moment the question of the mechanism of this transition, since it is immaterial for the distribution function. We shall discuss this question in greater detail in Part IV.

§ 5. Electrical conductivity and thermal conductivity. The Wiedemann–Franz law

The quantities \(F\), \(T\), \(\dfrac{d\alpha}{dx}\), \(\dfrac{\partial T}{\partial x}\) in the equations of § 4 (12a, b) are determined by experimental conditions. Therefore we can now calculate the flows of heat and electricity.*

* Certain discrepancies encountered in the literature in the statistical interpretation of the coefficients of the Thomson and Peltier effects are explained by differences in the conditions taken as the basis of the calculation. On this question we refer to the work of Sommerfeld and Frank. Formulas (7) and (8) of these authors coincide completely with our § 4 (12a), (12b).

Suppose that in the direction of the current no other changes of state occur. In this case \(\dfrac{\partial T}{\partial x}=0\), \(\dfrac{\partial \alpha}{\partial x}=0\), and

\[ i_x=e^2K_1F=\varkappa F. \]

The electrical conductivity \(\varkappa\) is obtained from § 4 (15a, b) as equal to

\[ \varkappa=e^2K_1= \begin{cases} \dfrac{8\pi e^2Gml_0\varepsilon_0}{3h^3}, & (1)\ \mathrm{Z}\\[1.2em] \dfrac{4}{3}\,\dfrac{e^2Nl}{\sqrt{2\pi mkT}}, & (1)\ \mathrm{L}. \end{cases} \]

As noted above, only the first approximation § 4 (15a) has been used here. Since

\[ \varepsilon_0 \cong \mu=\frac{h^2}{2m}\left(\frac{3N}{4\pi G}\right)^{2/3}, \qquad (2)\ \mathrm{Z} \]

\[ \bar v=\sqrt{\frac{3kT}{m}}=\text{the mean velocity of the Maxwellian distribution}, \qquad (2)\ \mathrm{L} \]

the preceding expressions for \(\varkappa\) can be rewritten in the following form:

\[ \varkappa= \begin{cases} \dfrac{e^2l_0N}{mv_0}, & (1a)\ \mathrm{Z}\\[1.2em] \dfrac{4}{\sqrt{6\pi}}\,\dfrac{e^2lN}{m\bar v}. & (1a)\ \mathrm{L} \end{cases} \]

In both cases we obtain expressions which agree, up to a numerical factor, with the result of the primitive Drude theory [part 1, § 1 (3)]. However, in the formulas of the Lorentz theory \(\bar v\) is small, and therefore \(N\) also turns out to be too small. In Sommerfeld’s theory, on the other hand, \(v_0 \gg \bar v\), and \(N\), in order of magnitude, may be taken equal to the number of atoms in \(1\ \text{cm}^3\). The temperature dependence of the electrical conductivity proves to be entirely connected with \(l_0\). In part IV we shall see that the model assumptions here too lead to good results.

Taking \(N\) (and together with it also \(v_0\)) as known, we obtain from (1) Z and (2) Z the expression

\[ l_0=\frac{3h^3\varkappa}{e^2 8\pi Gm\varepsilon_0} =\frac{\varkappa h}{e^2}\left(\frac{3}{4\pi G}\right)^{1/3}N^{-2/3} =\frac{\varkappa h}{e^2}\left(\frac{3}{4\pi G}\right)^{1/3}n^{-2/3}z^{-2/3}, \qquad (3) \]

which makes it possible to find the free path \(l_0\) from empirical data. From this, for various metals at room temperature, the following quantities are obtained:

Metal	Na	K	Cu	Ag	Au	Hg	Pb	Fe	Ni	Pd	Pt
\(lz^{-2/3}\cdot 10^7\ (\text{cm})\)	35	36	48	59	38	12	6.5	7	6	8	6
\(\tau z\cdot 10^{15}\ (\text{sec.})\)	33	42	30	42	28	10	5.8	4.7	3.7	5.8	5.4

\(l_0\) turns out to be of the order of 100 atomic distances. And this latter value is well explained by the model theory. We see how far from reality is the idea of elastic spheres, which gives for \(l\) a value of only the order of atomic distances. The table also gives the relaxation time

\[ \tau=\frac{l_0}{v_0}=\frac{\chi m}{e^2N}=\frac{\chi m}{e^2nz}, \]

which provides a direct measure of the rapidity with which the stationary state is restored. The correctness of the order of magnitude for \(\tau\) can be checked in the following way. In electrical (optical) oscillations with frequencies of the order \(\nu=\frac{1}{\tau}\) and higher, the electrons will no longer be able to give the equilibrium current corresponding to the existing field. We should expect that, beginning with these frequencies, deviations from the formulas of the phenomenological Maxwell theory will be observed. Such deviations are in fact observed*; they begin precisely at frequencies of the order \(\nu=10^{13}\).

We shall now examine the question of the limits of applicability of Ohm’s law. The latter should hold only so long as the assumption that \(f_1 \ll f_0\) [§ 4 (1)] is valid. Since in all calculations only the region near \(\varepsilon=\varepsilon_0\) is essential, it is sufficient for us to compare the corresponding quantities only for the critical value \(\varepsilon_0\). According to § 4 (6) and (9) we have:

\[ f_1=-\frac{\xi}{v}\,l\,\frac{\partial f_0}{\partial \varepsilon}eF, \]

and from § 3 (12):

\[ f_0(\varepsilon_0)=\frac{1}{2};\quad -\left.\frac{\partial f_0}{\partial \varepsilon}\right|_{\varepsilon=\varepsilon_0} =\frac{1}{4kT}. \]

Putting \(\frac{\xi}{v}\simeq 1\), we obtain for the inequality \(f_1\ll f_0\) the following expression:

\[ \frac{l_0eF}{4kT}\ll\frac{1}{2}. \]

On the other hand, according to § 5 (1):

\[ l_0eF=\frac{3h^3i}{16\pi em\varepsilon_0}. \]

Therefore we obtain the required condition for the validity of Ohm’s law in the following form:

\[ i\ll\frac{32\pi m\varepsilon_0ekT}{3h^3}. \]

* A more exact theory of metal optics was developed by Kronig on the basis of Bloch’s ideas and led to good results.

Substituting \(\varepsilon_0=10^{-11}\) erg \((=6\ \mathrm{V})\) and \(T=300^\circ\), we find for the right-hand side of the inequality, in round figures, \(2\cdot 10^{19}\ CGSE \(=6\cdot 10^9\ \mathrm{A}/\mathrm{cm}^2\). We can expect deviations from Ohm’s law even by \(1\%\) only at currents of about \(5\cdot 10^7\ \mathrm{A}/\mathrm{cm}^2\), which are nevertheless almost 10 times larger than in Bridgman’s well-known experiments. Therefore the results obtained by Bridgman can hardly be considered correct.

To calculate the heat flow we shall assume that the electric current is zero. Putting \(i=0\) in § 4 (12a), we see that in this case the electric field may nevertheless be nonzero. This latter circumstance, as we shall see below, is connected with the presence of thermoelectric effects. We obtain from this the relation:

\[ \left(eF+kT\frac{\partial z}{\partial x}\right) = \frac{K_2}{K_1T}\frac{\partial T}{\partial x}, \]

substitution of which in § 4 (12b) gives:

\[ w_x=\frac{1}{T}\frac{K_2^2-K_1K_3}{K_1}\frac{\partial T}{\partial x} = -\gamma\frac{\partial T}{\partial x}. \]

By definition, the quantity

\[ \gamma=-\frac{1}{T}\frac{K_2^2-K_1K_3}{K_1} \tag{4} \]

is equal to the coefficient of thermal conductivity (we note that here we obtain only that part of the thermal conductivity which is due to the free electrons; for good conductors the heat transfer by the crystal lattice may be neglected).

In calculating the numerator of (4) for the case of Sommerfeld’s theory, we find that the terms of the first approximation drop out altogether. Such a result is not unexpected, since the first approximation corresponds to the distribution at \(T=0\). For purely electrical phenomena this circumstance plays no essential role, but for thermal processes it is precisely the influence of temperature on the state of the system (the distribution function) that proves important. Therefore here we must use the second approximation [neglecting already only terms \((kT)^4\)], for which from § 4 (15a) we find:

\[ K_3K_1-K_2^2 = \left(\frac{8\pi Gm}{3h^3}\right)^2 \left\{ \left(\varepsilon^3l+\frac{\pi^2}{6}(kT)^2\frac{\partial^2(l\varepsilon^3)}{\partial \varepsilon^2}\right) \left(\varepsilon l+\frac{\pi^2}{6}(kT)^2\frac{\partial^2(l\varepsilon)}{\partial \varepsilon^2}\right) - \left(\varepsilon^2l+\frac{\pi^2}{6}(kT)^2\frac{\partial^2(l\varepsilon^2)}{\partial \varepsilon^2}\right)^2 \right\}_{\varepsilon=\varepsilon_0} \simeq \left(\frac{8\pi Gm}{3h^3}\right)^2 \frac{\pi^2}{3}(kT)^2l_0^2\varepsilon_0^2. \]

The derivatives of \(l_0\) drop out in this approximation as well, and only the influence of temperature on the distribution function proves essential. We therefore find:

\[ \gamma= \begin{cases} \displaystyle \frac{8\pi Gm}{3h^3}\,\frac{\pi^2}{3}\,k^2T\,l_0\varepsilon_0, & \tag{5)З} \\[1.2em] \displaystyle \frac{8}{3}\,\frac{Nl}{[[unclear: denominator partly cut off]]}\,k^2T. & \tag{5)Л} \end{cases} \]

For experimental verification it is more convenient to use the Wiedemann–Franz law:

\[ \frac{\gamma}{\chi T} = \frac{1}{e^2 T^2}\, \frac{K_1K_3-K_2^2}{K_1^2} = \begin{cases} \dfrac{\pi^2}{3}\left(\dfrac{k}{e}\right)^2, & \text{(6) З}\\[6pt] 2\left(\dfrac{k}{e}\right)^2. & \text{(6) Л} \end{cases} \]

This ratio in both theories turns out to be a universal constant for all metals. However, the new, larger value of the numerical coefficient \(\left(\dfrac{\pi^2}{3}=3.3\right)\) agrees considerably better with the experimental data. The theoretical numerical value (6) З is equal to \(7.56\cdot 10^{-11}\) CGSE \(= 6.83\cdot 10^{-6}\ \text{watt}\cdot\text{ohm}\cdot\text{degree}^{-1}\) at \(T=273^\circ\mathrm{C}\).

In conclusion it should be especially emphasized that the results derived here apply only to sufficiently high temperatures and to genuinely good conductors. If these conditions are not fulfilled, significant deviations from the theory may occur.

§ 7. Thermoelectricity

Let us now consider the process of heat liberation by an electric current, which is produced at the expense of the work performed by the field on the electrons. For the calculation it is wholly unimportant what the mechanism is by means of which the disordered thermal energy of the electrons is transferred to the metal lattice. Likewise there is no contradiction in the fact that, in deriving the fundamental equation, we neglected the interaction of the electrons with the lattice, while now we are considering the heating of the entire metal as a whole. The circumstance that all the work of the field ultimately passes into heat is simply one of the consequences of the law of conservation of energy.

If there is a heat flux, then part of the energy is continuously carried away. The excess energy which is transformed into heat, for an element of volume, will be:

\[ QdV=(A-\operatorname{div}\mathbf{W})\,dV. \tag{1} \]

Choosing the volume element in the form of a cylinder of height \(dx\) and base area \(1\ \mathrm{cm}^2\) in the plane \(yz\) (all quantities depend only on \(x\)), we obtain for the amount of heat:

\[ Qdx=\left(A-\frac{dw_x}{dx}\right)dx. \tag{2} \]

The magnitude of the work is computed as follows. On a single electron from the velocity interval \(d\xi\,d\eta\,d\zeta\), during the time \(dt\), the work \(eF\xi\,dt\) is performed. In the volume element, however, there are

\[ \frac{Gm^3}{h^3}\,f\,dx\,d\xi\,d\eta\,d\zeta \]

electrons with the velocity under consideration; therefore the total work is obtained by integrating over all values of the velocity:

\[ A\,dx\,dt = eF\,\frac{Gm^3}{h^3} \left\{\iiint \xi f\,d\xi\,d\eta\,d\zeta\right\} dx\,dt. \]

Comparing the expression obtained with § 3 (9), we find the well-known relation of the phenomenological theory:

\[ A=iF. \tag{3} \]

After this, for the amount of heat liberated in each \(\mathrm{cm}^3\), we obtain:

\[ Q=iF-\frac{\partial w_x}{\partial x}. \tag{4} \]

Here for \(F\) one must take the true value of the field, taking into account also the field which is created by the electrons themselves. From § 4 (12a) we have:

\[ F=\frac{i}{e^2K_1}-\frac{kT}{e}\frac{\partial \alpha}{\partial x} +\frac{K_2}{eK_1T}\frac{\partial T}{\partial x}. \tag{5} \]

In exactly the same way the heat flux can, according to § 4 (12b), be expressed in terms of the electric one:

\[ w_x=\frac{i}{e}\frac{K_2}{K_1} +\frac{K_2^2-K_3K_1}{K_1T}\frac{\partial T}{\partial x} = \frac{eiK_2}{\varkappa} -\gamma\frac{\partial T}{\partial x}. \]

Substituting this expression in (4), we finally find:

\[ Q=\frac{i^2}{\varkappa} +i\left\{ -\frac{kT}{e}\frac{\partial \alpha}{\partial x} -\frac{1}{e}\frac{\partial}{\partial x}\left(\frac{K_2}{K_1}\right) +\frac{K_2}{eK_1T}\frac{\partial T}{\partial x} \right\} +\frac{\partial}{\partial x}\left(\gamma\frac{\partial T}{\partial x}\right) = \frac{i^2}{\varkappa} \]

\[ -\frac{i}{e}\left\{ kT\frac{\partial \alpha}{\partial x} +kT\frac{\partial}{\partial x}\left(\frac{K_2}{kTK_1}\right) \right\} +\frac{\partial}{\partial x}\left(\gamma\frac{\partial T}{\partial x}\right). \tag{6} \]

The expression obtained contains three terms proportional to \(i^2\), \(i\), and \(i^0\). These terms may be interpreted as follows. The term \(\frac{i^2}{\varkappa}\) represents the ordinary Joule heat. The term not containing \(i\) at all gives, evidently, the amount of heat supplied as a result of thermal conduction \((Q=\operatorname{div}\gamma\operatorname{grad}T)\). And, finally, the term with \(i\) describes thermoelectric phenomena. Following the usual terminology, we classify the latter as follows:

I. Thomson effect. The phenomenon consists in the reversible liberation of heat which takes place in a homogeneous but nonuniformly heated conductor. The magnitude of the effect is proportional to the current (which is taken as positive in the direction opposite to the motion of the electrons). The Thomson coefficient \(\rho\) is, by definition, given by the multiplier of \(i\frac{\partial T}{\partial x}\). Since

\[ \frac{\partial}{\partial x} = \frac{\partial T}{\partial x}\frac{\partial}{\partial T}, \]

then for the Thomson coefficient we obtain:

\[ \rho=\frac{kT}{e}\,\frac{\partial}{\partial T}(\alpha+\delta). \tag{7} \]

Here, for brevity, the following notation has been introduced:

\[ \delta=\frac{K_2}{K_1 kT}. \tag{8} \]

II. Peltier effect. The Peltier heat is proportional to the current (it is considered positive in the same case as above) and is released when current flows between two different metals that are at the same temperature. If 1 and 2 are two points located on different sides of the interface, then:

\[ Q_{12}=i\,\frac{kT}{e}\int_{1}^{2}\frac{\partial}{\partial x}(\alpha+\delta)\,dx =i\,\frac{kT}{e}(\alpha+\delta)\bigg|_{1}^{2}. \]

Fig. 2.

The Peltier coefficient is the factor multiplying \(i\), and therefore

\[ \Pi_{12}=\frac{kT}{e}\{(\alpha_2+\delta_2)-(\alpha_1+\delta_1)\}. \tag{9} \]

From (7) and (9) there follows the well-known Thomson relation:

\[ \frac{\partial}{\partial T}\left(\frac{\Pi_{12}}{T}\right)+\frac{\rho_1-\rho_2}{T}=0, \tag{10} \]

which can also be obtained by a purely thermodynamic method.

III. Thermoelectromotive forces. As a third effect we shall consider the occurrence of thermoelectromotive forces developing in a circuit containing different metals at unequal temperature.

By definition, the electromotive force is equal to the potential difference in an open circuit. Let us consider the circuit shown in Fig. 2, containing two junctions \(b\) and \(c\), which are at different temperature; closure of the circuit could be effected, for example, by connecting the points \(a\) and \(d\).

The thermoelectromotive force will be:

\[ F_{12}=\varphi_a-\varphi_d=-\int_a^d Fdx. \]

Since in the effect under consideration the current \(i=0\), on the basis of (5):

\[ F=\frac{k}{e}\left(-T\frac{\partial \alpha}{\partial x}+\delta\frac{\partial T}{\partial x}\right), \]

and, consequently:

\[ F_{12}=-\frac{k}{e}\int_a^d\left(-T\frac{\partial \alpha}{\partial x}+\delta\frac{\partial T}{\partial x}\right)dx. \]

In the last expression we may integrate by parts, and the values at the limits will vanish, since \(T_a=T_d=T_1\). Then one obtains:

\[ F_{12}=-\frac{k}{e}\int_a^d(\alpha+\delta)\frac{\partial T}{\partial x}dx =-\frac{k}{e}\int_a^d(\alpha+\delta)dT. \]

Finally, dividing the interval of integration into the segments \(ab\), \(bc\), \(cd\) (since \(\int_a^b=0\) owing to \(T_a=T_b\)), we find the following relation:

\[ \begin{aligned} F_{12} &=-\frac{k}{e}\left\{\int_{T_1}^{T_2}(\alpha_2+\delta_2)dT +\int_{T_2}^{T_1}(\alpha_1+\delta_1)dT\right\} \\ &=\frac{k}{e}\int_{T_1}^{T_2}\{(\alpha_1+\delta_1)-(\alpha_2+\delta_2)\}\,dT =-\int_{T_1}^{T_2}\frac{\Pi_{12}}{T}\,dT. \end{aligned} \tag{11} \]

The thermoelectromotive force proves to be related to the Peltier coefficient. The relation obtained here, too, coincides exactly with the conclusions of phenomenological thermodynamics.

Formulas (10) and (11) follow as a consequence of the fundamental equations of § 3 (12a, b), and for their derivation the exact value of the quantities \(K_n\) is unnecessary. As was already mentioned, these formulas retain their validity also under much more general assumptions (for example, when energy exchange in elementary acts is admitted). Thus statistical theory, as indeed it should, reproduces with great generality the results of thermodynamics. Statistical theory, however, also makes it possible to give an interpretation of the coefficients, something thermodynamics cannot do. We shall calculate here the quantity \(\alpha+\delta\), which enters into relations (7) and (9).

1. (Sommerfeld theory). According to the definition of these quantities (8) and § 4 (15a), in the second approximation [up to terms containing \((kT)^4\)] we have:

\[ \delta=\frac{K_2}{K_1 kT} =\frac{1}{kT} \left\{ \frac{ \varepsilon^2 l+2c_2(kT)^2\dfrac{\partial^2(\varepsilon^2 l)}{\partial\varepsilon^2} }{ \varepsilon l+2c_2(kT)^2\dfrac{\partial^2(\varepsilon l)}{\partial\varepsilon^2} } \right\}_{\varepsilon=\varepsilon_0} = \]

\[ =\frac{1}{kT} \left\{ \varepsilon+2c_2\frac{(kT)^2}{\varepsilon l} \left( \frac{\partial^2(l\varepsilon^2)}{\partial\varepsilon^2} -\varepsilon\frac{\partial^2(l\varepsilon)}{\partial\varepsilon^2} \right) \right\}_{\varepsilon=\varepsilon_0} = \]

\[ =\frac{1}{kT} \left\{ \varepsilon+\frac{\pi^2}{3}(kT)^2 \left( \frac{1}{\varepsilon}+\frac{1}{l}\frac{\partial l}{\partial\varepsilon} \right) \right\}_{\varepsilon=\varepsilon_0}. \]

Since, by definition, \(\alpha=-\dfrac{\varepsilon_0}{kT}\), we find:

\[ \alpha+\delta=\frac{\pi^2}{3}kT \left( \frac{1}{\varepsilon}+\frac{1}{l}\frac{\partial l}{\partial\varepsilon} \right)_{\varepsilon=\varepsilon_0}. \tag{12)Z} \]

The terms of the first approximation vanish here, as they do for all the other thermal effects. It should also be noted that \(\varepsilon_0\) itself, by means of a small correction term, depends slightly on the temperature [the temperature-independent part of \(\varepsilon_0\) gives the quantity \(\mu\); see Part 1, § 9, formulas (7), (4)].

L (Lorentz theory). From (8) and § 4 (15b), (3a), one obtains:

\[ \delta=\frac{K_2}{K_1 kT}=2;\qquad \alpha=\ln\frac{G}{nh^3}(2\pi mkT)^{\frac{3}{2}} \]

\[ \alpha+\delta=\ln\frac{G}{nh^3}(2\pi mkT)^{\frac{3}{2}}+2. \tag{12)L} \]

The possible dependence of \(l\) on the velocity is not taken into account here at all, whence in formula (12)L the corresponding term accounting for this dependence is absent; conversely, in formula (12)Z this circumstance is taken into account by the second term.

The magnitude of the Thomson-effect constant is obtained from (7); we have:

\[ \rho=\frac{\pi^2}{3}\frac{kT}{e}\frac{d}{dT}kT \left( \frac{1}{\varepsilon}+\frac{1}{l}\frac{dl}{d\varepsilon} \right)_{\varepsilon=\varepsilon_0}. \tag{13)Z} \]

Assuming that the electron density does not depend on \(T\), we must consider that \(\varepsilon_0\) also changes little with changing temperature. Further, from the temperature dependence of the electrical conductivity it follows that \(l\) may be regarded as proportional to \(T\). Therefore

\[ \rho=\frac{\pi^2}{3}\frac{k}{e}kT \left( \frac{1}{\varepsilon}+\frac{1}{l}\frac{dl}{d\varepsilon} \right)_{\varepsilon=\varepsilon_0}. \tag{13a)Z} \]

In the case L we have:

\[ \rho=\frac{kT}{e} \left( \frac{3}{2T}-\frac{\partial\ln n}{\partial T} \right) = \frac{3k}{2e} \left( 1-\frac{2}{3}T\frac{\partial\ln n}{\partial T} \right). \tag{13)L} \]

The expressions obtained for \(\rho\) show that the order of magnitude of the coefficient is quite different in the two cases. The value З proves to be much smaller, approximately by a factor of \(\dfrac{2\pi^2 kT}{9\varepsilon_0}\). The correct order of magnitude is given by the value З, whereas the theory Л gives too large a value. The proportionality to temperature also apparently agrees with experimental data*.

For the Peltier coefficient we obtain from (9) and (12):

\[ \Pi_{12}=\frac{(kT)^2}{e}\frac{\pi^2}{3}\left\{\left(\frac{1}{\varepsilon^{(2)}}+\frac{1}{l^{(2)}}\frac{\partial l^{(2)}}{\partial \varepsilon^{(2)}}\right)_{\varepsilon^{(2)}=\varepsilon_0^{(2)}}- \left(\frac{1}{\varepsilon^{(1)}}+\frac{1}{l^{(1)}}\frac{\partial l^{(1)}}{\partial \varepsilon^{(1)}}\right)_{\varepsilon^{(1)}=\varepsilon_0^{(1)}}\right\}, \tag{14)З} \]

\[ \Pi_{12}=-\frac{kT}{e}\ln\frac{n_1}{n_2}. \tag{14)Л} \]

Although both formulas look quite different and give a different temperature dependence, the numerical magnitude of the effect is of the same order in both cases (tenths of a volt). However, the experimental data on the temperature dependence agree with the new theory.

The thermoelectromotive force can be expressed, according to (11), through the Peltier coefficients. Taking \(\varepsilon\) and \(\dfrac{1}{l}\dfrac{dl}{d\varepsilon}\) to be independent of \(T\) (which is a good approximation), we obtain:

\[ F_{12}=\frac{\pi^2 k^2}{6e}\left\{\left(\frac{1}{\varepsilon^{(1)}}+\frac{1}{l^{(1)}}\frac{\partial l^{(1)}}{\partial \varepsilon^{(1)}}\right)_{\varepsilon^{(1)}=\varepsilon_0^{(1)}}- \left(\frac{1}{\varepsilon^{(2)}}+\frac{1}{l^{(2)}}\frac{\partial l^{(2)}}{\partial \varepsilon^{(2)}}\right)_{\varepsilon^{(2)}=\varepsilon_0^{(2)}}\right\}(T_1^2-T_2^2), \tag{15)З} \]

\[ F_{12}=-\frac{k}{e}(\ln n_1-\ln n_2)(T_1-T_2). \tag{15)Л} \]

For high temperatures the dependence of the thermoelectromotive force on temperature in (15)З is well confirmed by experiment.

The expressions for \(\rho\), \(\Pi_{12}\), and \(F_{12}\) given by the new theory also satisfy Nernst’s theorem, vanishing for \(T=0\). The latter does not hold in the Lorentz theory. As for the form of the temperature dependence at low temperatures, here we cannot expect agreement between theory and experiment, since the assumption of the theory concerning the existence of a definite mean free path (§ 3) ceases to be valid.

* There is, however, a considerable difficulty here, consisting in the fact that the sign of expression (13)З, precisely for good conductors, proves to be negative. Taking into account the dependence of the mean free path on the velocity [the second term in (13)З] does not clarify the matter, since according to the model theory (Part IV) the mean free path should increase with energy. Nevertheless, it was recently shown by Blokhintsev and Nordheim [14] that this effect too can be explained with the aid of corresponding model representations.

The Bridgman effect. In conclusion we shall add a few remarks concerning the Bridgman effect. This effect consists in the fact that Peltier heat is observed between identical single crystals at the same temperature, provided only that the crystals are oriented differently. As Ehrenfest and Rutgers [15] have shown, this effect is directly entailed by the preceding discussion. Formula (14) of § 3 for the Peltier effect contains two terms. The first term plays no role for the effect under consideration, since the absolute-zero energy in both crystals is the same independently of their orientation \((\varepsilon_0^{(1)}=\varepsilon_0^{(2)})\). On the contrary, the second term, which depends on the change of the mean free path with velocity, can give an explanation of the effect. For this it is only necessary to assume that the character of the dependence of the mean free path on the velocity also depends on the direction of motion.

This may be explained clearly in the following way. Let in two conductors I and II, with the same electron density, the mobility (i.e., the mean free path) for different velocities \(v_1\) and \(v_2\) (let \(v_1 > v_2\)) be especially large in comparison with other nearby values of the velocity. In that case, in conductor I the current will be produced mainly by electrons with velocity \(v_1\), and in conductor II by electrons with velocity \(v_2\). But then, in the presence of a current at the boundary of separation, there will necessarily be released an energy

\[ \frac{m}{2}(v_1^2-v_2^2), \]

which will be interpreted by us as Peltier heat. The considerations just set forth are applicable also to two identical single crystals in the Bridgman effect, if the current in different directions is due predominantly to electrons of different velocity. Nevertheless, formula (14) cannot be transferred directly to the Bridgman effect. Under the assumption made, one can no longer write, following Lorentz, the relation § 4(6), and therefore the whole subsequent method of integration is no longer possible. Here also, however, at least formally, one succeeds in solving the corresponding generalized integral equation, and indeed the desired effect is obtained. A more detailed numerical comparison of orders of magnitude is impossible here, since unknown functions enter in the integration.*

One might think that the effect under discussion should be very small for a Fermi-Dirac gas, since here only electrons with energy very close to \(\varepsilon_0\) should take part in the calculation; therefore operating with two different velocities \(v_1\) and \(v_2\) could give rise to doubt. In this connection we shall point out still another possible explanation of the Bridgman effect, one which has not yet been mentioned in the literature. Namely, according to the band theory (Chap. IV), electrons with the same energy create in the crystal a current that depends on the direction of their motion. Therefore, in two crystals in Bridgman’s experiment, charges could be transported also by electrons of the same energy (namely, the energy of absolute zero), while, however, the number of electrons participating in the transport must now be different. This too could lead to the occurrence of thermoelectric effects.

We shall also mention one last purely thermoelectric phenomenon—the Benedicks effect.** The effect consists in the appearance of a thermocurrent in a homogeneous but nonuniformly heated substance. The basic formula § 4(12) gives no indication of the existence of such an effect. Thus, together with the kinetic point of view, the Benedicks effect could be connected only with some conditions that do not correspond to the basic assumptions of our calculations. No explanation can be found in the approximate method of integration either (the assumption of smallness of the perturbation,

\[ \text{,} \]

* A complete solution of the problem for an anisotropic medium is, to the highest degree, difficult and has not yet been given. Houston’s attempt [13], as Rutgers has shown, is not free of objections.

** See a review of Benedicks’s presentation and a critique of the available experimental material [17].

§ 4 (1) and in neglecting \(f_1\) in the left-hand side of the fundamental equation), since if the latter were inadmissible, we would not obtain Ohm’s law. The only remaining possibility is to regard the selected elements of volume not as homogeneous, that is, to assume the existence of temperature gradients already appreciable over segments of the order of the mean free path. Under these conditions, however, the thermoelectric coefficients would have to depend on these gradients.

§ 7. Magnetic Effects

Until now we have everywhere assumed only the presence of an electric field. We shall now also consider magnetic effects, among which the most important are the Hall effect and the influence of a magnetic field on the resistance. For the elementary theory, great difficulties arise here. It is precisely for these effects that the wave-mechanical refinement of the theory (Part IV) is of very great importance, whereas the formal theory set forth above proves insufficient. We shall now analyze the principle of calculation, which, on the one hand, will serve as the basis for the development of a more complete theory, and, on the other, will already allow us to reveal those points at which the theory needs modification. In what follows we shall consider only the so-called transverse effects*.

Let the magnetic field \(H\) be directed along the \(z\)-axis. In this case the force acting on an electron is equal to:

\[ \mathbf{K}=\frac{e}{c}[\mathbf{vH}];\quad K_x=\frac{e}{c}\eta H,\quad K_y=-\frac{e}{c}\xi H,\quad K_z=0. \]

The existing magnetic field gives rise to an additional electric field in the direction \(y\) (besides the field in the direction of the current, \(x\)), and the complete form of the fundamental equation [cf. § 3 (2) and § 4 (2)] will be:

\[ \frac{df^*}{dt} = \frac{\partial f_0}{\partial \varepsilon}(\xi G_x+\eta G_y) + \frac{eH}{mc}\left(\eta\frac{\partial f}{\partial \xi}-\xi\frac{\partial f}{\partial \eta}\right) \]
\[ = \int\!\!\int\!\!\int V(f'-f)\,d\xi' d\eta' d\zeta', \tag{1} \]

where

\[ G_x=eF_x+kT\frac{\partial \alpha}{\partial x}-\frac{\varepsilon}{T}\frac{\partial T}{\partial x};\quad G_y=eF_y+kT\frac{\partial \alpha}{\partial y}-\frac{\varepsilon}{T}\frac{\partial T}{\partial y}. \tag{1a} \]

In the terms expressing the influence of the field, we shall for the time being keep the exact distribution function

\[ f=f_0+f_1. \]

One might think that, for the theory of magnetic effects, the existence of electron spin would play a special role**. It turns out, however, that this is not the case. Since no forces at all act on a dipole in a homogeneous field (apart from the time of setting up the dipole), the influence of the field reduces only to a certain

* A complete analysis of all transverse effects from the point of view of the Sommerfeld theory is given by Sommerfeld and Frank [10]. The present state of the wave-mechanical theory may be found in Peierls’ papers [18]. In comparison with the indicated surveys, our presentation is very brief.

** This was pointed out by Bloch [19].

change of the distribution function \(f^0\)*. Therefore the influence of spin can be easily taken into account; it turns out to be much smaller than the effects being analyzed (see the paper by Sommerfeld and Frank, loc. cit., p. 23).

Equation (1) can be solved if we put

\[ f=f_0+\xi \chi_x+\eta \chi_y, \tag{3} \]

which represents a natural generalization of the Lorentz relation § 4 (6). Here both \(\chi_x\) and \(\chi_y\) must depend only on the absolute value of the velocity \(v\) (on the energy), but not on the direction. Taking into account that, for any function \(g(\varepsilon)\),

\[ \frac{\partial g(\varepsilon)}{\partial \xi}=m\xi\,\frac{dg(\varepsilon)}{d\varepsilon}, \]

we find:

\[ \frac{\partial f}{\partial \xi} =\chi_x+m\xi\left(\frac{df_0}{d\varepsilon} +\xi\frac{d\chi_x}{d\varepsilon} +\eta\frac{d\chi_y}{d\varepsilon}\right), \]

\[ \frac{\partial f}{\partial \eta} =\chi_y+m\eta\left(\frac{df_0}{d\varepsilon} +\xi\frac{d\chi_x}{d\varepsilon} +\eta\frac{d\chi_y}{d\varepsilon}\right). \]

Substituting these expressions in (1), we obtain:

\[ \frac{\partial f_0}{\partial \varepsilon}(\xi G_x+\eta G_y) +\frac{eH}{mc}(\eta\chi_x-\xi\chi_y) = \]

\[ =\int\!\!\int\!\!\int V\{\chi_x(\xi'-\xi)+\chi_y(\eta'-\eta)\}\,d\xi'\,d\eta'\,d\zeta' =-\frac{v}{l}(\xi\chi_x+\eta\chi_y). \tag{4} \]

The calculation of the integral in the term depending on collisions can be carried out in the same way as in § 4, and therefore \(l\) will be expressed as before, § 4 (8). From this it is immediately clear that the magnetic field by itself still does not disturb statistical equilibrium, since when \(G_x=G_y=0\), equation (4) is satisfied for \(\chi_x=\chi_y=0\).

Since equation (4) must be fulfilled for all values of \(\xi\) and \(\eta\), the corresponding coefficients in both parts must be equal. Hence we obtain for \(\chi_x\) and \(\chi_y\) two ordinary equations:

\[ \frac{\partial f_0}{\partial \varepsilon}G_x -\frac{eH}{mc}\chi_y+\frac{v}{l}\chi_x=0, \]

\[ \frac{\partial f_0}{\partial \varepsilon}G_y +\frac{eH}{mc}\chi_x+\frac{v}{l}\chi_y=0. \]

Their solution gives:

\[ \chi_x=-\frac{l}{v}\frac{\partial f_0}{\partial \varepsilon} \frac{G_x+\dfrac{hl}{v}G_y}{1+\left(\dfrac{hl}{v}\right)^2}, \]

\[ \chi_y=-\frac{l}{v}\frac{\partial f_0}{\partial \varepsilon} \frac{G_y-\dfrac{hl}{v}G_x}{1+\left(\dfrac{hl}{v}\right)^2}, \tag{5} \]

\[ \text{* } f_0 \text{ will explicitly depend on } H. \text{ The expression for } f_0 \text{ will be:} \]

\[ f_0=\frac{1}{2}\left\{ \frac{1}{e^{(\varepsilon-\varepsilon_0+\mu H)/kT}+1} + \frac{1}{e^{(\varepsilon-\varepsilon_0-\mu H)/kT}+1} \right\}. \tag{2} \]

where, for brevity, we have denoted

\[ h=\frac{eH}{mc}. \]

For the electric and heat currents we now obtain two components, not equal to zero, namely:

\[ \begin{aligned} i_x&=e\left\{L_1\left(eF_x+kT\frac{\partial\alpha}{\partial x}\right) +HM_1\left(eF_y+kT\frac{\partial\alpha}{\partial y}\right) -L_2\frac{1}{T}\frac{\partial T}{\partial x} -HM_2\frac{1}{T}\frac{\partial T}{\partial y}\right\},\\ i_y&=e\left\{L_1\left(eF_y+kT\frac{\partial\alpha}{\partial y}\right) -HM_1\left(eF_x+kT\frac{\partial\alpha}{\partial x}\right) -L_2\frac{1}{T}\frac{\partial T}{\partial y} +HM_2\frac{1}{T}\frac{\partial T}{\partial x}\right\}, \end{aligned} \tag{6a} \]

\[ \begin{aligned} w_x&=L_2\left(eF_x+kT\frac{\partial\alpha}{\partial x}\right) +HM_2\left(eF_y+kT\frac{\partial\alpha}{\partial y}\right) -L_3\frac{1}{T}\frac{\partial T}{\partial x} -HM_3\frac{1}{T}\frac{\partial T}{\partial y},\\ w_y&=L_2\left(eF_y+kT\frac{\partial\alpha}{\partial y}\right) -HM_2\left(eF_x+kT\frac{\partial\alpha}{\partial x}\right) -L_3\frac{1}{T}\frac{\partial T}{\partial x} +HM_3\frac{1}{T}\frac{\partial T}{\partial y}. \end{aligned} \tag{6b} \]

Here

\[ L_n=-\frac{8\pi Gm}{3h^3}\int_0^\infty \frac{l\varepsilon^n\,\dfrac{\partial f_0}{\partial \varepsilon}} {1+\left(\dfrac{hl}{v}\right)^2}\,d\varepsilon;\qquad M_n=-\frac{8\pi Ge}{3h^3c}\int_0^\infty \frac{\dfrac{l^2}{v}\,\varepsilon^n\,\dfrac{\partial f_0}{\partial \varepsilon}} {1+\left(\dfrac{hl}{v^2}\right)^2}\,d\varepsilon. \tag{7} \]

From these very general formulas (valid both in Sommerfeld’s theory and in Lorentz’s theory) all the effects of interest to us follow as special cases. The integrals \(L_n\) and \(M_n\) could be evaluated according to the scheme given in Part I, § 8. The presence of spin can be taken into account by choosing for \(f_0\) the expression (2), and the general expressions for \(i\) and \(w\) retain their form. When the magnetic field is eliminated (\(h\to 0\)), the quantities \(L_n\) go over into the integrals \(K_n\) of § 4 (13).

As an example of the application of the formulas obtained, let us consider the isothermal Hall effect.*

This effect consists in the appearance of a transverse electric field (\(F_y\)) in the presence of a current \(i_x\) and a magnetic field \(H_z\). We shall put \(i_y=0\) and calculate the potential distribution for the case in which the circuit in the \(y\)-direction (perpendicular to the current) is not closed. From (6) it follows:

\[ \left(i_y=0;\ \frac{\partial}{\partial x}=0;\ \frac{\partial}{\partial y}=0\right), \]

\[ F_y=H\frac{M_1}{L_1}F_x, \]

\[ i_x=e^2F_xL_1+e^2F_yHM_1 =e^2F_x\left(L_1+H^2\frac{M_1^2}{L_1}\right)=\chi F_x. \tag{8} \]

The Hall-effect constant (\(R\)) is defined as the potential drop over a segment of \(1\ \mathrm{cm}\) when \(i_x\) and \(H\) are equal to unity. Therefore:

\[ R=\frac{F_y}{i_xH}=\frac{F_y}{\chi F_xH} =\frac{M_1}{\chi L_1} =\frac{M_1}{e^2L_1\left(L_1+H^2\frac{M_1^2}{L_1}\right)}. \tag{9} \]

* The isothermal effect should be distinguished from the adiabatic one, for which \(w_y=0\) (thermal impermeability of the lateral walls). Because of the high thermal conductivity of metals, comparatively usually the conditions of the adiabatic effect are fulfilled. For more detail on this, see Sommerfeld and Frank (l. c.).

THEORY OF THE METALLIC STATE

It should already be noted here that the wave-mechanical refinement of the whole picture (Part IV, § 7) also makes it possible to explain the so-called anomalous Hall effect.

The scheme of the experiment discussed above is the same as in the investigation of the change of resistance in a transverse magnetic field. This effect has recently been studied in detail by Kapitza, up to the highest attainable fields. From (8) and § 5 (1) we obtain, for the ratio of the electrical conductivities in the field \(\chi\) and outside the field \((\chi_0)\):

\[ \frac{\chi}{\chi_0} = \frac{L_1}{K_1} \left\{1+H^2\frac{M_1^2}{L_1^2}\right\} = \frac{L_1}{K_1} \left\{1+H^2\chi^2 R^2\right\}. \tag{10} \]

For the numerical value of the Hall-effect constant we shall already obtain a sufficient approximation if we use only the first term of the expansion, Part I, § 8 (7). This gives:

\[ R=\frac{e}{mc}\frac{l_0}{v_0\chi}.^{*} \tag{11} \]

Replacing \(\chi\) by \(\chi_0\) (which is quite permissible for not very large fields) and using § 5 (1a) 3, we arrive at the following expression for the constant \(R\) in Sommerfeld’s theory:

\[ R=\frac{1}{ecN}. \tag{12}З \]

Somewhat more transparent calculations according to Lorentz lead to the expression

\[ R=\frac{3\pi}{8}\frac{1}{ecN}, \tag{12}Л \]

almost coinciding with (12) З.

The two expressions differ only by an inessential numerical factor. They give for \(R\) an order of magnitude in good agreement with the experimental data, if \(N\) is taken equal to the number of atoms.**

The elementary theory gives no indication of a change in the resistance. In the first approximation, according to (7) and § 4 (13):

\[ L_1=\frac{K_1}{1+\left(\frac{h l_0}{v_0}\right)^2}; \qquad \frac{M_1}{L_1}=\frac{e}{mc}\frac{l_0}{v_0}, \]

and therefore from (10) we obtain \(\frac{\chi}{\chi_0}=1\); when operating with the distribution function \(f_0\) (for \(T=0\)) no dependence of the resistance on the magnetic field is obtained at all. To obtain this dependence it proves necessary

* We use the Gaussian system of units instead of the electromagnetic one, as in Sommerfeld; hence the factor \(\frac{1}{c}\).

** We draw particular attention to the results of the work of Kikoin and Fakidov\(^{20}\) with molten alkalis. For the latter, the basic assumptions of the theory correspond most closely to reality.

use the second approximation. Elementary considerations give in this case:

\[ \frac{\chi}{\chi_0}=1-\frac{BH^2}{1+CH^2}, \]

and for the relative change of the resistance \(\rho=\frac{1}{\chi}\) one obtains the expression

\[ \frac{\Delta\rho}{\rho}=-\frac{\Delta\chi}{\chi_0}=\frac{BH^2}{1+CH^2}. \tag{13} \]

\(B\) and \(C\) have the following values:

\[ B=\frac{\pi^2}{24}\frac{e^2}{mc^2}\left(\frac{kT}{\varepsilon_0}\right)^2, \tag{14a} \]

\[ C=\frac{l_0^2e^2}{2\varepsilon_0 mc^2}=\chi_0^2R^2. \tag{14b} \]

The dependence (13) apparently agrees well with the experimental data. For small \(H\) a quadratic law is obtained; for large \(H\) we must expect the appearance of “saturation.” Thus, in order to explain the deviations from the quadratic law there is no need to introduce new hypotheses. According to Frank, Kapitza’s curves are well explained: for weak fields they give a quadratic law, passing over for strong fields into a linear one. The bend of the curves associated with saturation should be observed only in very strong fields \((CH^2>1)\). At present there are already indications of the existence of such an effect for Te and Ge. The value of the constant \(C\), which according to (14b) can be obtained from the Hall effect, also proves satisfactory. Let us note that the whole theory has been developed for the case of sufficiently high temperature. The available experimental data, however, usually refer to low temperatures, where the dependence of the resistance on the field is especially strong. Therefore we cannot expect very good agreement of the theory with experiment.

The value of the constant \(B\), on the contrary, comes out considerably smaller than is needed (approximately by a factor of \(10^4\)); moreover, the temperature dependence also proves incorrect. The formal theory is here insufficiently accurate. However, as Peierls has shown (loc. cit.), one may expect that a consistent carrying through of the wave-mechanical theory will lead to better results. The reason for the smallness of \(B\) lies in the disappearance of the first approximation. According to wave theory, however, the dependence of the energy on the velocity will no longer possess spherical symmetry, and the group velocity in the field of the crystal lattice will also depend on the direction of motion. Taking this circumstance into account already in the first approximation leads to a substantially definite effect. In calculating there enters a factor of order

\[ \left(\frac{\varepsilon_0}{kT}\right)^2, \]

which compensates the quantity

\[ \left(\frac{kT}{\varepsilon}\right)^2 \]

in (14a) and leads to an acceptable value for \(B\). According to Peierls’s estimate, an insignificant anisotropy \((\sim 10\%)\) could already produce the corresponding effect. Here, however, relation (3) can no longer be used, and the problem becomes very complicated*.

As for the influence of a longitudinal magnetic field, it is clear from the preceding considerations that the magnetic field itself cannot disturb the stationary state; it therefore also cannot affect the resistance. This is true, in any case, so long as

* For one special case the theory was given by Bloch and Nordheim \(^{14}\).

for the time being one may neglect the presence of spin. Indeed, a magnetic field in the \(x\) direction affects only the \(\eta\)- and \(\zeta\)-components, so that the distribution with respect to \(\xi\) will still be determined only by the field \(F_x\). The longitudinal effects, like the transverse ones, are extremely small.

Starting from formula (6), one can give a theory of the remaining magnetic and thermomagnetic effects as well. In this case the following general result is obtained. For those effects that appear in the first approximation, the theoretical relations between the coefficients agree well with experiment. The order of magnitude of the coefficients also comes out correctly. For effects requiring a second approximation in the distribution function, such agreement no longer occurs. This result becomes quite understandable if one takes into account the considerations presented in the present paragraph.

LITERATURE

1. W. Pauli, Sommerfeld-Festschrift, Leipzig, 1928.
2. A. Einstein, Phys. Z. 18, 121, 1917.
3. P. A. M. Dirac, Proc. Roy. Soc. London, A 114, 293, 1927.
4. P. Jordan u. E. Wigner, Z. Physik. 47, 631, 1928; H. Weyl, Gruppentheorie und Quantenmechanik, Lpz., 1931; Kikuchi u. L. Nordheim, Z. Physik 60, 652, 1932.
5. L. Nordheim, Proc. Roy. Soc., London, A 119, 689, 1928.
6. F. Ehrenfest. Z. Physik 45, 455, 1927.
7. L. Nordheim, Ann. d Phys. 9, 607, 1931.
8. N. Bohr. Diss., Kopenhagen, 1911.
9. A. Sommerfeld u. N. H. Franck, Rev. mod. Phys. 3, 1, 1931.
10. R. de L. Kronig, Proc. Roy. Soc., London, A 124, 409, 1929; 133, 255, 1931.
11. S. Schubin. Z. Physik 68, 97, 1931.
12. Y. Fujioka, Z. Physik 76, 537, 1932.
13. D. Blochinzew u. L. Nordheim, Z. Physik 84, 168, 1933.
14. P. Ehrenfest u. A. J. Rutgers, Roc. Amstr. 32, 698, 883, 1929; A. J. Rutgers, Dissertation Leinden, 1930.
15. Houston. Z. Physik 48, 449, 1928.
16. C. Benedicks, Ergebn. d. exakt. Naturwiss. 48, 28, 1929; Hdb. d. Phys. 13, 200, Berlin, 1928.
17. R. Peierls, Leipziger. Vortr., 1930, 75; Ann. d. Phys. 10, 97, 1931.
18. F. Bloch, Z. Physik 53, 216, 1929.
19. J. Kikoin u. J. Fakidow, Z. Physik 71, 393, 1931.