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THEORY OF METALLIC CONDUCTIVITY
Ya. I. Frenkel, Leningrad.
§ 1. Drude’s Theory
The fundamental and most characteristic property of metallic bodies is their electrical conductivity. In this respect they not only surpass all other bodies quantitatively, but also differ from them qualitatively. The passage of current through metals is not accompanied by the transfer of matter in the ordinary sense of the word, as occurs in electrolytes. This peculiarity of metallic electrical conductivity was explained, soon after the rise of the electron theory, by the circumstance that the transport of electricity in metals is carried out not by ions, i.e., not by charged atoms or groups of atoms, but by “free” electrons, i.e., electrons detached from individual atoms and moving independently. The theory of “free” electrons first received its quantitative formulation from Drude. Drude based his theory on an analogy between the motion of free electrons in a metallic body and the motion of molecules in a gas. In both cases the particles under consideration (electrons or molecules) do not remain bound to definite positions of equilibrium, but move throughout the entire volume occupied by the corresponding body; the difference consists only in the fact that gas molecules are held within this volume by the solid walls of some vessel, whereas in the case of a metallic body the role of such walls is played by the surface of the latter.
The indicated circumstance by itself still gives no grounds for likening the free electrons in a metal to gas molecules (thus, for example, in the case of liquids the molecules likewise do not remain bound to any positions of equilibrium). Moreover: we have every reason to believe that “free” electrons, in contrast to gas molecules, are at all times subjected—from the atoms, positive ions, and one another—to enormous forces, continuously curving their trajectory, whereas gas molecules usually move—in the intervals between relatively very brief collisions—rectilinearly and uniformly. Thus the identification of the motion of “free” electrons with the motion of molecules in a gas has no reasonable foundation. It was introduced by Drude chiefly—if not exclusively—for the simplification of the calculation.
At the same time Drude left open the question of the number of free electrons (\(n\)—per unit volume) and of the length of the “free path” (\(l\)), fixing a priori only their root-mean-square velocity \(v\), namely determining the latter by the well-known equation
\[ \frac{1}{2} mv^2 = \frac{3}{2} kT, \tag{1} \]
where \(k\) is Boltzmann’s constant, and \(T\) the absolute temperature. From this, however, it immediately followed that the number of free electrons must be very small in comparison with the number of neutral atoms, which firmly retain their electrons. Otherwise the heat capacity of metals would be considerably greater than the value (6 calories per gram-atom) which it has according to the law of Dulong and Petit and which follows from the theorem of the equipartition of energy. In the case of solids at low temperatures this “theorem” of classical statistical mechanics loses its force; according to Nernst’s theorem, the heat capacity of solids tends to zero as the temperature decreases. On the contrary, the heat capacity of a monatomic gas retains a constant value (3 calories per 1 gram-atom). Thus the theory of an electron gas “locked up” in a solid-
dom metal, could be reconciled with the experimental facts only under the condition that the number of free electrons is small in comparison with the number of atoms.
Taking the number \(n\) as known, it is not difficult to calculate the specific electrical conductivity of the metal. The latter, according to Drude, is obtained as follows. In the presence of an external electric field of intensity \(E\), the free electrons move between two collisions with acceleration \(w=\dfrac{eE}{m}\) (\(e\) is the charge, \(m\) the mass of the electron), as if “falling” in the direction of this field. When an electron collides with some atom (neutral or positively charged), the former loses its kinetic energy, transferring it to the latter, which manifests it in the form of heat (Joule heat). The additional velocity of the electron at the end of the path is equal to \(wt\), where \(t\) is the time of flight. If this additional velocity is small in comparison with the mean velocity \(v\) of the random thermal motion, then one may put \(t=\dfrac{l}{v}\). Thus, for the mean value of the additional velocity communicated to the electron by the field \(E\), we obtain the expression:
\[ u=\frac{1}{2}wt=\frac{1}{2}\frac{eE}{m}\cdot\frac{l}{v}. \]
The product of this velocity by \(ne\) is nothing other than the density of the electric current, equal by definition to \(\sigma E\), where \(\sigma\) is the specific electrical conductivity of the metal. Thus for the latter the following formula is obtained:
\[ \sigma=\frac{e^{2}nl}{2mv}. \tag{2} \]
Comparison of this formula with experimental data is of no particular significance in view of the unknown character of both \(n\) and \(l\).
Drude, however, did not stop at the result given, but calculated that additional thermal conductivity which metallic bodies must possess thanks to the na-
the presence in them of an electron gas. In doing so he made the hypothesis that the additional electronic thermal conductivity coincides with the thermal conductivity of the electron gas itself, using for the latter the formula known from the kinetic theory:
\[ \chi=\frac{1}{3}vlc, \tag{3} \]
where \(c\) denotes the heat capacity of a unit volume of gas (at constant volume). In the case under consideration we have
\[ c=\frac{3}{2}kn \tag{4} \]
and consequently
\[ \chi=\frac{1}{2}kvnl. \tag{5} \]
Comparison of this formula with (2) gives
\[ \frac{\chi}{\sigma}=\frac{kmv^{2}}{e^{2}}, \]
i.e., according to (1)
\[ \frac{\chi}{\sigma}=3\left(\frac{k}{e}\right)^{2}T. \tag{6} \]
Metals are, as is well known, not only first-rate conductors of electricity, but also equally first-rate conductors of heat. Experiment shows that the ratio of the total thermal conductivity of various metals to their electrical conductivity is the same for all metals at one and the same temperature and is directly proportional to the latter. Thus this law, discovered by Wiedemann and Franz, is correctly expressed by formula (6), if in it one understands not the additional, but the total electrical conductivity of the metal. Moreover, the numerical value of the proportionality coefficient \(3\left(\frac{k}{e}\right)^2\) coincides exactly
with the experimental one (it should be noted that the ratio \(\chi/\sigma\) contains no unknown numbers \(n\) and \(l\)).
This result, which at first sight represents a triumph of Drude’s theory, is in fact connected with an internal contradiction. In order for the thermal conductivity of a metal to be reduced practically entirely to the thermal conductivity of the electron gas, the number of free electrons must be very large. In fact, formula (5) is based on the assumption that the temperature of a nonuniformly heated metal is determined by the kinetic energy of the free electrons located at the corresponding point, i.e., in other words, that the atoms, with respect to their thermal energy, “are equalized by the free electrons.” And this is evidently possible only if the number of the latter is comparable with the number of the former. But under such conditions the heat capacity of metals would be much greater than the value determined, in accordance with statistical mechanics, by the law of Dulong and Petit.
§ 2. Lorentz’s Theory.
Drude’s theory was improved by Lorentz, who took into account the distribution of velocities among the electrons, making use for this purpose of Maxwell’s well-known law. According to this law, the number of electrons whose velocity projections on the coordinate axes \(x, y, z\) lie in the intervals \((\xi,\xi+d\xi)\), \((\eta,\eta+d\eta)\), \((\zeta,\zeta+d\zeta)\) is equal to
\[ dn=Ae^{-\frac{W}{kT}}\,d\xi d\eta d\zeta, \tag{7} \]
where
\[ W=\frac{1}{2}m(\xi^2+\eta^2+\zeta^2)=\frac{1}{2}mv^2 \]
is the kinetic energy of the electron, and \(A\) is a constant equal to
\[ A=n\left(\frac{m}{2\pi kT}\right)^{3/2}. \tag{8} \]
The account of the Maxwellian distribution of velocities played an essential role in the theoretical explanation of the Richardson effect (emission of electrons by incandescent bodies); in the theory of the electrical and thermal conductivity of metals, however, it introduced only an inessential correction. Namely, for \(\sigma\) and \(\chi\) one obtains in this case the following values
\[ \sigma=\frac{4}{3}\frac{e^{2}ln}{\sqrt{2\pi mkT}} =\sqrt{\frac{8}{3\pi}\frac{e^{2}ln}{mv}} \tag{9} \]
\[ \chi=\frac{8}{3}\sqrt{\frac{kT}{2\pi m}}\,kln =\frac{4}{3}\sqrt{\frac{2}{3\pi}}\,klnv \tag{10} \]
where \(v\) is the mean square velocity, determined by formula (1). The ratio \(\dfrac{\chi}{\sigma}\) proves in this case to be equal to \(2\left(\dfrac{k}{e}\right)^{2}T\); thus the exact theory gives worse agreement with experiment than the approximate theory of Drude.
For the calculation of \(\sigma\) and \(\chi\), Lorentz used the following method (which we consider it necessary to set forth, since it will be useful to us later). In the presence of a gradient of potential and temperature (or even only of the latter) in the direction of the \(x\)-axis, the Maxwellian distribution (7) must be slightly distorted. This distortion reduces, in first approximation, to the fact that at each point of space the electrons have not the velocity distribution that corresponds to the temperature or potential of this point, but the distribution corresponding to the temperature and potential of those points at which they last underwent a collision with atoms (we assume, consequently, that this or that distribution of electron velocities is established by such collisions). Electrons passing through some plane perpendicular to the \(x\)-axis, with velocities lying in the interval \(d\xi, d\eta, d\zeta\), underwent their last collision in a plane parallel to it, separated from it by the distance \(\Delta x=-\xi t\), where \(t=\dfrac{l}{v}\) is the time elapsed
from the moment of this collision \((v=\sqrt{\xi^2+\eta^2+\zeta^2})\). Thus the number of electrons of the kind under consideration, referred to unit volume, is not \(f_0(x,W)\,d\omega\), where, for brevity, \(Ae^{-\frac{W}{kT}}=f_0(x,W)\) and \(d\omega=d\xi d\eta d\zeta\), but
\[ f(x,W)d\omega=f_0(x+\Delta x,\, W+\Delta W)d\omega. \]
Here \(\Delta W\) denotes the change in the kinetic energy of the electron over the segment \(\Delta x\). In view of the smallness of the latter, we may put, with a sufficient degree of accuracy,
\[ f_0(x+\Delta x,\,W+\Delta W)=f_0(x,W)+ \left(\frac{\partial f_0}{\partial x}\Delta x+\frac{\partial f_0}{\partial W}\Delta W\right), \]
or, since
\[ \Delta W=eE\Delta x \]
(the change in energy is equal to the work of the acting force over the segment \(\Delta x\)),
\[ f(x,W)=f_0(x,W)-\left(\frac{\partial f_0}{\partial x}+\frac{\partial f_0}{\partial W}eE\right)\xi\frac{l}{v}. \tag{11} \]
This is the (approximate) expression for that distorted distribution of velocities which arises from the presence of a temperature gradient and a potential gradient in the direction of the \(x\)-axis. Knowing the function \(f(x,W)\), we can calculate the density of the electric and heat current (i.e. the amount of electricity or kinetic energy carried by electrons per unit time through an area of \(1\ \mathrm{cm}^2\), perpendicular to the \(x\)-axis) by the formulas:
\[ I=e\int \xi f\,d\omega,\qquad Q=\frac{m}{2}\int v^2\xi f\,d\omega. \tag{12} \]
The preceding expression for the electrical conductivity (9) is obtained directly from the first of these formulas, in connection
with the relation \(I=\delta L'\), under the condition \(\dfrac{dT}{dx}=0\). As for thermal conductivity, in computing it by the formula \(Q=-\varkappa\dfrac{dT}{dx}\) it is necessary to introduce the additional condition \(I=0\) (which is by no means satisfied automatically). This gives for \(\varkappa\) expression (10).
§ 3. Sommerfeld’s Theory.
As was already noted above, Lorentz’s theory did not introduce any substantial change into Drude’s theory and, in any case, did not eliminate the basic contradiction between the small number of free electrons required by the heat capacity and the large value of this number following from the thermal conductivity. This contradiction was resolved last year by A. Sommerfeld, who, while preserving all the ideas of Drude and Lorentz concerning the motion of electrons in metals, replaced only the Maxwellian distribution of velocities by the so-called Fermi distribution1:
\[ f_0=\frac{\left(\dfrac{m}{h}\right)^3}{\dfrac{1}{c}e^{\frac{W}{kT}}+1}. \tag{13} \]
Here \(h\) denotes Planck’s constant, and \(c\) is a certain, rather complicated function of the temperature \(T\) and of the concentration of electrons (i.e. their number per unit volume) \(n\). For very large values of \(T\) or very small values of \(n\), this function coincides with the coefficient \(A\) introduced above, divided by \(\left(\dfrac{m}{h}\right)^3\). In the opposite case—
case, i.e., for large values of \(n\) and small values of \(T\), the function \(c(n,T)\) is determined from equation \(^{1}\):
\[ \frac{4}{3\sqrt{\pi}}(\lg c)^{3/2} \left[1+\frac{\pi^2}{8(\lg c)^2}+\ldots\right] = \left(\frac{h}{m}\right)^3 A = \frac{nh^3}{(2\pi mkT)^{3/2}} . \tag{14} \]
In the first case \(\frac{1}{c}e^{\frac{W}{kT}}\) turns out to be very large in comparison with unity, so that the Fermi distribution is practically reduced to the Maxwellian one. In the second case \(\frac{1}{c}e^{\frac{W}{kT}}\), on the contrary, is small in comparison with unity, and we obtain an entirely different velocity distribution, and at the same time an entirely different character of the dependence of the mean kinetic energy on temperature than that which is determined by formula (1). Namely, it turns out that:
\[ \frac{1}{2}mv^2 = \frac{3}{2}kT\cdot \frac{2}{5}\lg c \left[ 1+\frac{1}{2}\frac{\pi^2}{(\lg c)^2}+\ldots \right]. \tag{15} \]
In particular, for \(T=0\) one obtains:
\[ \frac{1}{2}mv_0^2 = \frac{3}{10}\frac{h^2}{m} \left(\frac{3}{4\pi}n\right)^{2/3}. \tag{16} \]
Thus, according to the Fermi distribution, the mean kinetic energy of the electrons at absolute zero temperature does not vanish, but retains a finite value, which is the larger the greater the concentration of the electron gas. It is easy, further, to convince oneself that with increasing temperature this mean energy increases the more slowly, the greater its initial (“zero”) value; at high temperatures we return to the linear
\(^{1}\) The exact form of this equation is the following:
\[ \frac{2}{\sqrt{\pi}} \int_{0}^{\infty} \frac{c\sqrt{x}\,e^{-x}}{1+ce^{-x}}\,dx = \left(\frac{h}{m}\right)^3 A . \]
dependence of the classical theory \(\frac{1}{2}mv^2=kT\). These relations are represented graphically in Fig. 1.
Fig. 1.
If the number \(n\) is identified with the number of atoms per unit volume, i.e., if one assumes that each metallic atom contributes one electron, then for the velocity \(v_0\), according to formula (16), values of the order of \(10^8\ \mathrm{cm/sec}\) are obtained, i.e., values comparable with the velocities of revolution of the outer (valence) electrons in isolated atoms. The increase of the velocity in passing from absolute zero to ordinary temperatures proves to be very small. For the “atomic heat capacity” of such an extremely dense electron gas, one obtains a negligible figure (about one hundredth of a calorie). Thus the “electron gas” adds practically nothing to the total heat capacity of the metal. As for the electrical and thermal conductivities due to it, for them the following approximate formulas are obtained:
\[ \sigma=\frac{4\pi e^2l}{3h}\left(\frac{3n}{4\pi}\right)^{2/3} = \sqrt{\frac{3}{5}}\, \sqrt[3]{\frac{4\pi e^2nl}{3mv_0}}; \qquad (v \simeq v_0) \tag{17} \]
and
\[ \chi=\frac{4\pi^3lk^2T}{9h}\left(\frac{3n}{4\pi}\right)^{2/3} = \frac{\pi^2}{\sqrt{15}}\, \sqrt[3]{\frac{4\pi k^2Tnl}{3mv_0}}, \tag{18} \]
whence it follows that
\[ \frac{\chi}{\sigma}=\frac{\pi^2}{3}\left(\frac{k}{e}\right)^2\cdot T. \tag{19} \]
The last formula is in excellent agreement with experimental data. This cannot, however, be said of the first two, since they contain the unknown quantity of the mean free path \(l\). Let us note that formula (17) has approximately the same form as Drude’s formula (2) or Lorentz’s (9). The difference between them consists only in the fact that in the former formulas the velocity \(v\)
represented a relatively small quantity, proportional to the square root of the absolute temperature
\[ \left(\text{at normal temperatures } v=\sqrt{\frac{3kT}{m}}\simeq 6\cdot 10^6\ \text{cm/sec}\right), \]
whereas in formula (17) \(v_0\) is a constant of the order of \(10^8\ \text{cm/sec}\). If the number \(n\) of electrons is likewise regarded as constant, then the dependence of \(\sigma\) on temperature must reduce entirely to the temperature dependence of the mean free path \(l\). Experience shows that the electrical conductivity of various metals in the region of ordinary temperatures is inversely proportional to the absolute temperature. We must therefore have
\[ l=\frac{\mathrm{const}}{T}. \]
As for the absolute value of \(l\), substituting into formula (17) the experimental values of \(\sigma\), we obtain for it, at \(T=300\), a value of the order of \(10^{-5}\ \text{cm}\). Thus the mean free path of the electrons in a metallic body must have approximately the same magnitude as the mean free path of molecules in some gas at normal temperature and pressure, i.e. at a comparatively negligible concentration of molecules. Using the well-known formula:
\[ l=\frac{1}{n\pi r^2}, \tag{20} \]
where \(r\) is the effective radius of the atoms (we assume that the electrons do not collide with one another at all), it is easy to calculate that the “effective radius” of the atoms \(r\), which determines the magnitude of their deflecting action on an electron, is equal (approximately) to \(3\cdot 10^{-9}\ \text{cm}\), i.e. is 10 times smaller than that which is usually ascribed to them and which corresponds to the mean distance between neighboring atoms in some metallic body. Since, moreover, the number \(n\) does not depend on temperature, it is necessary to assume that this effective radius varies with temperature approximately in proportion to the square root of the temperature. The impression is obtained as though the matter were not one of the radius of the atoms, but of the amplitude of their thermal oscillations about the position of equilibrium. Let us note that the energy of these oscillations is equal to \(ar^2\), where \(a\) is the coefficient of proportionality-
ness, characterizing the strength of the bond of the atom with the position of equilibrium. Thus, from this point of view, the mean free path of the electrons \(l\) should have been inversely proportional to the thermal energy of the metal (per unit volume). The thermal energy of solids, as is known, is proportional to the absolute temperature only in the region of intermediate temperatures. As the temperature is lowered it decreases much more rapidly than the latter, so that the heat capacity, i.e. the derivative of the energy with respect to temperature, becomes zero at absolute zero. If, therefore, the preceding interpretation of formula (20) corresponds to reality, i.e. if the obstacle to the motion of electrons through a metallic body is not the atoms themselves, but only those spheres which are described by their centers owing to thermal oscillations, then on the basis of formula (17) we should have expected that the electrical conductivity, as the temperature is lowered, increases more rapidly than inversely proportional to the latter, namely inversely proportional to the thermal energy of metals. This conclusion is in qualitative agreement with experiment. In reality, however, it turns out that the energy must be replaced by the product of the (atomic) heat capacity \(c\) by the temperature. This law, found empirically by Grüneisen, is expressed by the formula
\[ \sigma = \frac{\mathrm{const}}{cT}. \tag{21} \]
It is seen from this that the preceding interpretation is confirmed by experiment only qualitatively.
§ 4. Distribution of velocities among electrons according to Pauli and Fermi.
We shall not enter into a more detailed analysis of formulae (17), (18), (19), and shall now briefly consider the physical essence of that law of velocity distribution from which they follow. This law was established in 1926 by the young Italian theorist E. Fermi by extending to gases the so-called Pauli principle, which deter-
dividing the structure of individual atoms. In more or less complex atoms, the electrons are arranged around the central nucleus in the form of a series of “shells” or groups, to which separate “terms” in the X-ray spectra of these atoms correspond.^1) The first group nearest the nucleus always consists of only two electrons (the “\(K\)-group”). The next, \(L\)-group contains 8 electrons. In the \(M\)-group—when it is fully developed—there are 18 electrons, etc. Moreover, the two electrons of the \(K\)-group move in one-quantum orbits (i.e. in orbits with principal quantum number 1), the electrons of the \(L\)-group in two-quantum orbits, the electrons of the \(M\)-group in three-quantum orbits, etc. Each group is, generally speaking, divided into subgroups, and it turns out that to each set of quantum numbers (principal, azimuthal, inner, and magnetic) there corresponds no more than one electron. This statement constitutes the Pauli principle. This principle can also be formulated as follows: in an atom there cannot exist two or more “equivalent” (in the sense of the character of their motion) electrons; every electron differs from the others in some respect (i.e. by at least one quantum number).
At absolute zero of temperature the atoms are in the normal state, corresponding to the minimum of their energy. However, the minimum is not an absolute one, but a relative one, satisfying the Pauli principle (or the “equivalence rule,” as Pauli himself calls it). Were it not for this restriction, then all the electrons, however great their number, would “settle” at the same—nearest—distance from the nucleus, i.e. would begin to move in identical one-quantum orbits. In reality, in these orbits there is room only for two electrons, provided that their magnetic axes have opposite directions.^2) The remaining electrons are compelled
^1) That is, the energies necessary to extract these electrons outward.
^2) On the magnetic properties of the electron and their significance for the mechanics of atoms (see my article “The Rotating Electron,” Advances in Physical Sciences, 1927, ...
therefore settle in more remote orbits. What they are compelled by and how, we do not know, but the fact remains a fact. Pauli’s principle has a purely regulative character, constituting the fundamental principle of electronic habitation, a kind of “housing law” of the electronic state. In this state, each pair of electrons is assigned a separate room, and the admission of a third electron into a room occupied by such an “electronic couple” is in no case permitted (just as the “unlawful cohabitation” of two electrons with identically directed moments is not permitted). Some electrons may, however, remain in a “bachelor” position, occupying a whole cell by themselves. The size of the elementary rooms allotted to individual electrons or pairs of electrons in the atom cannot be expressed in the form of any volume. It is therefore characterized in another way, by assigning to it a definite “weight” equal to unity (in the statistical sense of this word). Several “rooms” with one and the same energy (if such exist) are usually combined into one more extensive room, whose “weight” is equal to the sum of their weights.
The concept of such discrete “rooms” is wholly alien to classical statistical mechanics. It operates with a continuous manifold of spatial points and velocities, admitting a priori the possibility of any state, i.e. any position and any velocity of the particles under consideration. The state of some system consisting of a large number of identical particles, for example an electron gas, which was discussed in the preceding paragraph, is characterized by specifying the number of particles \(f(x, y, z; \xi, \eta, \zeta)\, dx\,dy\,dz\,d\xi\,d\eta\,d\zeta\), whose coordinates and velocities are contained in the intervals \(dx, dy, dz, d\xi, d\eta, d\zeta\), i.e. by the form of the distribution function \(f\). This method of characterizing the state—
p. 3, p. 202). If one abstracts from these magnetic properties, i.e. regards electron orbits that differ only in the direction of the magnetic moment as equivalent, then Pauli’s principle reduces to the statement that each orbit can be represented in the atom by no more than two specimens.
...of a complex system is also preserved in the new statistical mechanics connected with quantum theory—in those cases when (as, for example, occurs in the case of an electron gas or some other gas) each of the particles under consideration may have arbitrary coordinates and velocities within given sufficiently broad intervals. In this, however, the new statistics introduces the substantial addition that these intervals are subdivided into elementary cells, whose dimensions are determined by the formula
\[ \int dx\,dy\,dz\,d\xi\,d\eta\,d\zeta=\left(\frac{h}{m}\right)^3 . \tag{22} \]
These cells (whose shape remains indeterminate) also constitute, in the case under consideration, “elementary dwellings” for individual pairs of electrons in the sense indicated above. The circumstance that such cells correspond to separate quantum orbits, possessing the same statistical weight (1), was clarified by Planck as early as 1916. The transfer to them of the Pauli principle, i.e. the “residential law of the electron state,” is due to Fermi.
Let us first see what the Pauli–Fermi theory gives when applied to an electron gas at absolute zero temperature. From the point of view of classical theory, the electrons should then be at rest in the place that corresponds to the minimum of potential energy, i.e. in one definite cell of “phase space” \((x, y, z; \xi, \eta, \zeta)\). But this is just as impossible as placing all the electrons in an atom on identical one-quantum orbits. In reality, only one pair of electrons is placed in the just indicated “most advantageous” cell; the other pairs are arranged as closely as possible in cells of higher energy, both kinetic and potential. If the latter is the same for the whole spatial volume under consideration, then the distribution of electrons in it remains uniform. Let us denote the maximum velocity possessed by one of the \(N\) electrons in the volume \(V\) at absolute zero by \(v_{\max}\).
In this case the points representing the electrons in velocity space (where the coordinates are the velocity components \(\xi, \eta, \zeta\)) must lie inside or on the surface of a sphere of radius \(v_{max}\), i.e. in a “velocity volume”
\[ \frac{4\pi}{3}v_{max}^{3}. \]
Combining this sphere with the spatial volume \(V\), we obtain for the total magnitude of the phase volume the product:
\[ \Phi=\frac{4\pi}{3}v_{max}^{3}\cdot V. \tag{23} \]
Since the volume of an elementary phase cell is equal to
\[ \left(\frac{h}{m}\right)^3, \]
the number \(Z\) of such cells in the volume \(\Phi\) is equal to:
\[ Z=\Phi\left(\frac{m}{h}\right)^3 =\frac{4\pi}{3}\left(\frac{m}{h}v_{max}\right)^3 V. \tag{24} \]
As for the different cells, we may represent them as a combination of the spatial volume \(V\) and concentric spherical layers with equal velocity volume:
\[ 4\pi v^2 \Delta v=\frac{1}{V}\left(\frac{h}{m}\right)^3. \tag{25} \]
At absolute zero temperature the number of cells must coincide with half the number of electrons \(N\) (not a single empty place!), which corresponds to the minimum value of the kinetic energy of the electron gas. Putting in (24)
\[ Z=\frac{N}{2} \]
and
\[ \frac{N}{V}=n \]
(the electron concentration), we obtain:
\[ \frac{m}{h}v=\left(\frac{3}{8\pi}n\right)^{1/3}, \tag{26} \]
or consequently:
\[ \frac{1}{2}mv_{max}^{2} =\frac{1}{2}\frac{h^2}{m}\left(\frac{3n}{8\pi}\right)^{2/3}. \tag{27} \]
This is the maximum energy of one of the electrons at \(T=0\). As for the mean energy \(\frac{1}{2}mv_0^2\), ...
it is obtained by summing the quantity \(\frac{1}{2}mv^2\) for the various spherical shells (25) and dividing by the number of electrons \(N\). For large values of \(N\), the summation may be replaced by integration, i.e., one may put:
\[ \frac{1}{N}\sum \frac{1}{2}mv^2 = \frac{1}{N}\int_0^{v_{\max}} mv^2 \cdot \frac{4\pi v^2\,dv}{\frac{1}{V}\left(\frac{h}{m}\right)^3} = \]
\[ = \frac{1}{n}\left(\frac{m}{h}\right)^3 4\pi m \int_0^{v_{\max}} v^4\,dv, \]
which gives, according to (26):
\[ \frac{1}{2}mv_0^2 = \frac{3}{10}\frac{h^2}{m} \left(\frac{3}{8\pi}n\right)^{2/3}. \tag{28} \]
The formula obtained differs from formula (16), taken by me from Fermi’s paper (ZS. f. Phys., 36, 910, 1926), by the replacement of \(4\pi\) by \(8\pi\). This circumstance is explained by the fact that Fermi, and after him Sommerfeld, applied the Pauli principle in a not quite exact form, namely by assigning elementary cells \(\left(\frac{h}{m}\right)^3\) to individual electrons, and not to electron pairs.
The considerations set forth are easily generalized to the case \(T>0\). In this case the number of phase cells \(Z\) may be greater than half the number of electrons \(N\); thus some cells remain unfilled. The probability of one or another distribution of the electrons (not contradicting the Pauli principle) with total energy \(W\) must then be proportional to \(e^{-\frac{W}{kT}}\) or \(e^{-\frac{W-W_0}{kT}}\) (where \(W_0 = N\frac{1}{2}mv_0^2\) is the energy at absolute zero). This principle, inherited by the new quantum statistics from classical statistical mechanics, leads to the formula
Fermi’s (13) for the velocity distribution (with a small correction, which reduces to replacing \(\left(\dfrac{h}{m}\right)^3\) in formulas (13) and (14) by \(\dfrac{1}{2}\left(\dfrac{h}{m}\right)^3\).
§ 5. Physical Picture of the Motion of Electrons in Metals.
In 1924, i.e. even before the emergence of the new statistics connected with Pauli’s principle, I developed an electronic theory of metals1, proceeding from the same fundamental conception as Sommerfeld’s theory, namely, that at absolute zero temperature free electrons possess a very large kinetic energy, almost not changing with an increase in temperature, and that the number of these electrons approximately coincides with the number of atoms.
The indicated conception followed from consideration of the process of formation of a solid (or liquid) metal by condensation of the vapor of this metal.
Metallic vapors are, as is known, nonconductors of electricity. This means that they contain no free electrons. Experience shows that the outer electrons of metallic atoms are more weakly bound to the latter than in the case of metalloids, which in the solid state form dielectric (nonconducting) bodies. Further, Bohr’s theory indicated that the outer electrons of metallic atoms revolve in very elongated orbits, resembling comets in this respect, whereas the outer electrons of nonmetallic atoms may be likened, with respect to the shape of their orbits, to ordinary planets of the solar system.
Metallic atoms, in their interaction with metalloidal ones, as is known, give up to the latter part of their
electrons. These easily surrendered electrons are usually called valence electrons, since their number determines the positive valence of the corresponding metal (or ion). The valence electrons are, evidently, nothing other than those “cometary electrons” of which we have just spoken. Their capture by a metalloidal atom is explained not so much by the weakness of their bond with the parent (metallic) atom as by the circumstance that, owing to the elongation of their orbits, they recede very far from its center. If the aphelion of their unperturbed elliptical orbit lies near the metalloidal atom, then they enter the sphere of attraction of the latter and no longer return. In an analogous way, comets of the solar system could be captured by some other system of the same type if it approached the sun sufficiently closely.
In a metallic vapor the average distances between atoms are very large in comparison with the dimensions of the orbits of the various electrons, including the cometary ones. The latter therefore remain firmly bound to the corresponding atoms. But when, upon condensation of the vapor into a solid body, these atoms find themselves in immediate proximity to one another, the cometary electrons must fly from one atom into a neighboring one. In doing so they can make in each atom only one, or—at most—a few revolutions (if the aphelion distance of their unperturbed orbit is not sufficiently large), and then pass to one of the nearest atoms. Thus, upon condensation of a metallic vapor, the cometary electrons are transformed into “wandering” ones, i.e. roaming throughout the entire volume occupied by the metallic body. These wandering electrons, having lost their connection with definite “owners” and continuously passing from hand to hand, are what are usually called free electrons. Their “freedom” is quite relative. From the individual property that they are in the case of metallic-vapor atoms isolated from one another, they become the collective property of the state formed by the соединե-
tion of these atoms into a liquid or solid body. But their bond with this state is by no means weaker than with their former hosts; on the contrary, it is strengthened even more, since each wandering electron, insofar as it is located inside the metal, is held in it not by one, but by several (neighboring) atoms¹).
In isolated atoms the outer—and among them the comet-like—electrons revolve, as is known, with a speed of the order of \(10^8\) cm/sec²).
Upon condensation of a metallic vapor they continue to move with a speed of approximately the same order of magnitude and even, as can easily be seen, somewhat greater. In fact, the greater the forces acting on the electrons, the greater the acceleration caused by them, and hence, generally speaking, the speed as well. This relation can easily be made precise with the aid of the so-called virial theorem. The latter states that in the case of a system of particles acting upon one another with forces inversely proportional to the square of the distance, and remaining at a finite distance from one another, the average kinetic energy of all the particles \(W\) must be numerically equal to their total energy with the opposite sign, i.e., in other words, to the work of dissociation of the system into its constituent elements³). Since upon condensation of a metallic vapor the total energy decreases by an amount
¹) These conditions are somewhat altered for electrons which have in some way emerged onto the surface of the metal. From there they can separate comparatively easily (with approximately half the expenditure of energy).
²) Thus, for example, in the case of an electron revolving around a proton (a hydrogen atom) in a circle of radius \(r\), we have:
\[ \frac{e^2}{r^2}\ \text{(force of attraction)} = m \frac{v^2}{r} \]
(the centrifugal force), whence it follows:
\[ v = \sqrt{\frac{e^2}{mr}}, \]
which, for \(e \simeq 5\cdot 10^{-10}\), \(m = 10^{-27}\), and \(r = 10^{-8}\), gives \(v = 2\cdot 10^8\) cm/sec.
³) In the simplest case of the hydrogen atom, we have, for example,
\[ mv^2 = \frac{e^2}{r}. \]
The left-hand side of this equality is twice the kinetic energy, and the right-hand side is the potential energy with the opposite sign. The total energy is therefore equal to
\[ \frac{1}{2}mv^2 - \frac{e^2}{r} = -\frac{e^2}{er} = -\frac{1}{2}mv^2. \]
of the latent heat of condensation, then the kinetic energy of the electrons and protons must thereby increase by the same amount. This increase falls, if not entirely, then chiefly upon the cometary electrons, for the motion of the remaining electrons undergoes only an insignificant distortion, while the kinetic energy of the vibrational motion of the atoms as wholes may be neglected. Measuring the kinetic energy of the electrons in volts (i.e., by the potential drop in volts that is necessary for acquiring this energy), we obtain, for the kinetic energy of the electrons in isolated atoms, figures of the order of 5–7 volts, and for the additional energy acquired upon condensation of the vapor, 1–2 volts1.
Thus, applying the virial theorem to the process of “socialization” of the valence electrons which takes place during the condensation of metallic vapors, we obtain for them velocities of the same order of magnitude as those that follow from the Pauli–Fermi statistical theory (see above). It is necessary to note that the virial theorem, in the form given above, is applicable only to the absolute zero of temperature. For \(T>0\) there must occur a partial spontaneous disintegration of the metal both in the gaseous and in the solid state (owing to evaporation). In extending the virial theorem to this case it is also necessary to take into account the pressure of the vapor (or, more precisely, of the mixture of ions, electrons, and neutral atoms) on the walls of the enclosing vessel. — The agreement of the “zero energy” of the free electrons of the Pauli–Fermi–Sommerfeld theory with the value calculated by us from consideration of the condensation process is not completely exact, which, in all probability, is explained by the inaccuracy of this theory, which,
undoubtedly, is only a rather crude scheme. The sense and significance of this, albeit approximate, coincidence will be clarified below (see § 8).
§ 6. The theory of electrical and thermal conductivity arising from the preceding picture.
Starting from the above ideas about the number and motion of “free” electrons in a metal, in 1924 I developed a theory of the electrical and thermal conductivity of metals that differs substantially from the classical theories of Drude–Lorentz, and also from Sommerfeld’s theory constructed in an analogous manner.
There must exist a certain correlation between the motions of different electrons, amounting to the fact that the place vacated by one electron leaving a given atom must immediately be occupied by another. We shall not take this correlation into account, and shall try to trace the path of one particular electron through the metal. The latter can obviously be treated as a single molecule of gigantic dimensions; accordingly, the entire path of the electron might, it would seem, be regarded as one integral “quantized” orbit. It is easy, however, to convince oneself that this point of view is applicable only at absolute zero temperature. The thermal motion of the atoms, owing to its disordered character, must, so to speak, “break” the electron’s orbit into more or less short “quantized” segments, connected with one another only by the laws of chance. We shall call the rectilinear displacements corresponding to these segments the elementary displacements of the electron.
These “elementary displacements” evidently play the same role as the “free paths” in the theory of the electron gas. At absolute zero temperature they become infinite. In this case the metallic body offers no more resistance to the motion of the electrons than does an individual atom of it. The immediate cause of the electrical resistance of metals lies in those irregularities in
THEORY OF METALLIC CONDUCTIVITY
positions of atoms, which are caused by thermal motion.
At sufficiently high temperatures the elementary displacements of electrons must reduce to a minimum value equal to the distance \(d\) between neighboring atoms. This means that, in the absence of an external electric field, elementary displacements in different directions from some atom \(A_0\) to one of its \(s\) neighbors \(A_1, A_2, \ldots, A_s\) are equally probable, whatever the direction of the preceding elementary displacement \((A_k \to A_0)\). In the presence of an electric field of strength \(E\), elementary displacements in the direction of this field (or, more precisely, in the direction of the corresponding force \(F=eE\)) become more probable than displacements in the opposite direction. This change of probability in the case of the displacement \(\overrightarrow{A_0A_i}\) is proportional to \(e^{-\frac{U_i}{kT}}\), where
\[ U_i=-eEd\cos\theta_i \]
is the potential energy of the electron at \(A_i\) with respect to \(A_0\), and \(\theta\) is the angle between the segment \(\overrightarrow{A_0A_i}\) and the vector \(\vec E\) (or \(e\vec E\)).
This changed probability \(p_i\) can therefore be represented in the form:
\[ p_i=p_i^0\frac{e^{-\frac{U_i}{kT}}}{\displaystyle\sum_{k=1}^{s}e^{-\frac{U_k}{kT}}} \tag{29} \]
where \(p_i^0=\dfrac{1}{s}\) is its value at \(E=0\).
Denoting by \(t\) the time during which an elementary displacement is completed, we obtain for the mean velocity of displacement of the electron in the direction of the external force \(e\vec E\) acting on it the following formula:
\[ u=\sum_{i=1}^{s}p_i\frac{d\cos\theta_i}{t} \tag{30} \]
or, in the first approximation, putting \(e^{-\frac{U_i}{kT}} = 1 - \frac{U_i}{kT}\) and taking into account that \(\sum_1^s \cos \theta_i = 0\):
\[ u = \frac{e d^2}{t k T}\,\overline{\cos^2 \theta} \tag{31} \]
where \(\overline{\cos^2 \theta}\) denotes the mean value of \(\cos^2 \theta_i\) for all \(s\) displacements \(\overrightarrow{A_0A_i}\) (if they are regarded as equally probable). Since the number \(s\) is usually sufficiently large \((=12\) or \(8)\), one may put \(\overline{\cos^2 \theta} = \frac{1}{3}\). Taking into account that the current density is equal to \(neu=\sigma E\), we arrive at the following expression for the electrical conductivity of a metal:
\[ \sigma = \frac{e^2 n d^2}{3 k T t} \tag{32} \]
The same result can be derived from the well-known relation between the friction coefficient \(\vartheta\) and the diffusion coefficient \(D\):
\[ D\vartheta = kT, \tag{33} \]
first established by Einstein in the theory of Brownian motion. Here \(\vartheta\) is determined by the formula \(\vartheta=\frac{eE}{u}\), while the diffusion coefficient by the formula \(D=\frac{1}{3}d\cdot v'=\frac{d^2}{3t}\), where \(v'=\frac{d}{t}\) is the mean velocity of displacement of the electrons in the metal. This velocity is, generally speaking, somewhat less than the true velocity \(v\), approximately coinciding with the latter if the electron does not “linger” in individual atoms, but at each one makes only a single turn and immediately passes to the neighboring one.^1)
^1) If the concentration of electrons decreases in the direction of the \(x\)-axis, then in this direction we obtain a “diffusion” electric current with density \(-eD\frac{\partial n}{\partial x}\). In the presence of an electric field, to it is added...
THEORY OF METALLIC CONDUCTIVITY
In the case of alkali metals, whose atoms have only a single cometary electron with an extremely elongated orbit, this condition may be considered fulfilled. Taking \(d \lesssim 10^{-8}\) and \(v' \lesssim 10^{8}\), we obtain for the diffusion coefficient of the electrons a number of order 1 and, moreover, independent of temperature. Rewriting formula (32) in the form:
\[ \sigma=\frac{e^{2}nD}{kT}, \tag{34} \]
we see, consequently, that the electrical conductivity must be inversely proportional to the absolute temperature (in that temperature range for which the length of the elementary displacements reduces to the interatomic distance \(d\)) and is, in order of magnitude, equal to \(\dfrac{e^{2}n}{kT}\). Taking here \(e=4.7\,10^{-10}\), \(n \lesssim 10^{22}\), and \(k=1.3\cdot 10^{-16}\), we indeed obtain numbers sufficiently close to the experimental ones (in electrostatic units; to obtain the specific electrical conductivity in reciprocal ohms they must still be divided by \(9\cdot 10^{11}\)). I shall not give here more precise numerical data; the reader can find them in the work cited above. I shall only note that in the case of divalent (alkaline-earth) metals, the velocity \(v'\) turns out to be approximately twice smaller than \(v\); in this case, therefore, the cometary electrons describe around one and the same atom approximately two revolutions before passing to the next.
According to the point of view set forth, the “free” electrons do not participate independently in the thermal motion of the atoms—
the ordinary conduction current
\[ neu=\frac{ne^{2}E}{\vartheta} \]
is combined with it. In the state of equilibrium these currents compensate one another. We have, therefore, denoting the potential energy of the electron by \(U\) \(\left(eE=-\dfrac{\partial U}{\partial x}\right)\):
\[ D\frac{\partial n}{\partial x}+\frac{n}{\vartheta}\frac{\partial U}{\partial x}=0, \quad \text{i.e.} \quad n=\text{const.}\, e^{-\frac{U}{D\vartheta}}. \]
On the other hand, according to Boltzmann’s theorem, it must be
\[ n=\text{const.}\, e^{-\frac{U}{kT}}. \]
Comparison of this formula with the preceding one gives (33).
... and therefore do not appreciably increase the heat capacity of a solid or liquid metal—just as they do not increase the heat capacity of metallic vapors (apart, of course, from very high temperatures, at which appreciable ionization of the atoms begins). Thus the specific heat capacity of a solid metal at ordinary temperatures remains equal to \(c = 3 k n_a\), where \(n_a\) is the number of atoms per unit volume, in accordance with the Dulong–Petit law.
Without taking an independent part in the thermal motion, the “free” electrons can, however, transfer thermal energy from one atom to a neighboring one, in approximately the same way as occurs in gases. In the presence of a temperature gradient in a metal, the distribution of temperature in it is determined by the well-known equation
\[ \frac{\partial T}{\partial t} = \frac{\chi}{c} \left( \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} + \frac{\partial^2 T}{\partial z^2} \right), \]
where \(\chi\) denotes, as usual, the coefficient of thermal conductivity.
Comparing this equation with the diffusion equation:
\[ \frac{\partial n'}{\partial t} = D \left( \frac{\partial^2 n'}{\partial x^2} + \frac{\partial^2 n'}{\partial y^2} + \frac{\partial^2 n'}{\partial z^2} \right), \]
which determines the change in space and time of the concentration \(n'\) of an arbitrarily chosen set of electrons (\(n'\) should not be confused with the total concentration \(n\)), we arrive at the equality:
\[ \frac{\chi}{c} = D, \tag{35} \]
which expresses the identity of the processes of transfer of electricity and heat in a metal (we note that this relation remains valid also in the case of gases). Putting in it \(c = 3 k n_a\) and \(D = \dfrac{\sigma kT}{n e^2}\), on the basis of (34), we obtain:
\[ \frac{\chi}{\sigma} = \frac{3 n_a}{n} \left( \frac{k}{e} \right)^2 \cdot T, \tag{36} \]
i.e. the Wiedemann–Franz law. It is necessary, however, to note that agreement with the experimental value of the coefficient of proportionality is obtained only for \(n=n_a\), and that, further, at low temperatures the heat capacity ceases to be constant, which leads to replacing \(T\) in the right-hand side of formula (36) by a more complicated function of temperature. Meanwhile experiment shows that the Wiedemann–Franz law is justified down to the very lowest temperatures.
At low temperatures the electrical conductivity also ceases to vary inversely proportionally to \(T\), but is expressed, as was already indicated above, by Grüneisen’s formula:
\[ \sigma=\frac{\mathrm{const}}{c\cdot T}, \]
where \(c\) is the specific heat capacity of the metal, or, more precisely, some function of temperature varying with the latter approximately in the same way as the heat capacity, but corresponding not to the given metal but, generally speaking, to some other one.
A qualitative interpretation of this formula from the standpoint of the theory set forth presents no difficulties. With lowering temperature the length of the elementary displacements constantly increases, becoming \(\infty\) at \(T=0\). Denoting this length by \(l\), we obtain for the diffusion coefficient the expression \(D=\frac{1}{3}lv'\), where \(v'\) remains approximately as before.
Hence it follows, according to formula (34),
\[ \sigma=\frac{\mathrm{const}}{\left(\dfrac{d}{l}\right)\cdot T} \]
Thus, in order to obtain Grüneisen’s formula, we must suppose that the ratio \(\dfrac{d}{l}\) varies directly
proportional to the heat capacity \(e\). However, it does not seem possible to derive this result in an exact quantitative form from the principles set forth above.
The theory presented differs from Sommerfeld’s theory in that the influence of an external field on the motion of electrons is reduced in it to a change in the probability of elementary displacements of an electron in different directions, whereas according to Sommerfeld—as also according to Drude and Lorentz—this influence is determined by the additional velocity acquired by the electron during an elementary displacement (i.e. between two collisions). According to the preceding ideas, this additional velocity plays no role whatever. Suppose, for example, that the electrons can move only parallel to the \(x\)-axis. In that case, in the presence of a field they will traverse the distance between two atoms \(A_1\) and \(A_2\) somewhat faster in one direction—say from \(A_1\) to \(A_2\)—than in the opposite direction. Since, however, both directions remain equally probable, and since, further, the length of the elementary displacements remains the same in one direction and the other, the electrons, on the average, cannot acquire any additional motion in the direction of the forces acting on them, whatever the above-mentioned additional velocity may be. The latter can manifest itself only in the case where the electrons “stumble” not at every step, as apparently occurs in metals at ordinary temperatures, but after several steps, and if the number of these steps (interatomic distances) in the direction of the acting force proves in the final count to be more considerable than in the opposite direction. It is precisely this conception (though not formulated explicitly) that underlies the calculation of Sommerfeld, Lorentz, and Drude. It is interesting to note that in the case of the last two theories, which assume that the mean velocity of the electrons is determined by the formula
\[ \frac{1}{2}mv^2=\frac{3}{2}kT, \]
the calculation of the mobility of the electrons by both methods (additional velocity or additional probability) leads practically to one and the same result.
§ 7. Principles of wave mechanics; cathode rays and waves.
I now turn to the exposition of new, highly peculiar conceptions of the motion of electrons in metals, arising from the new wave (or quantum) mechanics, which was created in recent years by L. de Broglie and E. Schrödinger, and also, in an equivalent but outwardly different form, by W. Heisenberg, M. Born, and P. Jordan.
The essence of wave mechanics reduces to a deepening of the analogy that exists between the motion of material particles and the propagation of light rays. This analogy is manifested, among other things, in the name “cathode rays,” which since Crookes’s time has been assigned to the stream of electrons emitted by the cathode of a discharge tube. In this case the role of rays is played by the trajectories of individual electrons (these trajectories may be either rectilinear or curvilinear).—On the other hand, the corpuscular theory of light, revived by Einstein, led to the likening of light waves to a stream of electrons, whose role is played here by “light quanta.” Such phenomena as the reflection and refraction of light, i.e. the so-called phenomena of “geometrical optics,” can be easily explained by the “quantum” theory of light, under certain very general assumptions about the interaction of light quanta with ordinary matter. However, the phenomena of interference and diffraction of light do not at all fit within the framework of the corpuscular theory and can be explained only by the wave theory of light. In this theory, light rays, just like light quanta whose trajectories they are, are geometrical functions; they should be regarded simply as lines perpendicular to the surfaces of light waves. The immediate reality, the “substance” of light, consists precisely in these waves, and not in rays.
The fundamental idea of the de Broglie–Schrödinger theory is that also in the case of a material stream, such as cathode rays, we are in fact dealing with ...
not with individual particles similar to light quanta, but with waves of a special kind, analogous to light waves. This by no means implies that electrons as such (i.e., as discrete material particles) do not exist at all. It means only that their motion in space cannot be derived from the principles of classical “corpuscular” mechanics. We must replace the concept of cathode rays by the concept of cathode, or (as de Broglie calls them) phase, waves; establish the connection between the wave and corpuscular conceptions, analogous to the connection between light waves and quanta; and, finally, find the laws of propagation of cathode waves in space—again guided by their analogy with light waves. Such is the program. As regards its execution, it reduces to the following. The actions of light on material bodies can be described in terms of corpuscular theory if one regards light quanta as particles with energy \(h\nu\) and momentum
\[ \frac{h}{\lambda}=\frac{h\nu}{c} \]
(\(c\) is the speed of light)\(^1\). It follows from this that, in the case of cathode rays, we must determine the frequency of oscillations and the wavelength of the corresponding cathode waves by the formulas:
\[ h\nu = V,\qquad \frac{h}{\lambda}=mv, \tag{37} \]
where \(V\) is the total energy of the electron, and \(mv\) is its momentum. The energy \(V\) should be defined in such a way that an electron at rest, in the absence of external forces, has the energy \(mc^2\). Under this condition the velocity of the cathode waves \(\lambda\nu\) turns out to be approximately equal to \(\dfrac{c^2}{v}\), i.e., greater than the speed of light by as many times as the latter is greater than the speed of the electrons. Restricting ourselves to small (in comparison with \(c\)) values of \(v\), we may use for the kinetic energy the ordinary expression \(\dfrac{1}{2}mv^2\). Denoting by
\(^1\) See my article in Uspekhi Fiz. Nauk, vol. 2, 1927.
...the sum of this kinetic energy and the potential \(U(x,y,z)\), we may put \(m^{2}v^{2}=2m(W-U)\), and consequently (37):
\[ \frac{h}{\lambda}=\sqrt{2m[W-U(x,y,z)]}. \tag{38} \]
With the aid of this formula, the length of the cathode waves is determined unambiguously as a function of the coordinates (the frequency of the oscillations \(\nu\), however, remains independent of the latter, i.e. the same at all points of space).
The amplitude \(\psi\) of light waves of a definite frequency, propagating in some isotropic but inhomogeneous medium, in which their wavelength \(\lambda\) changes continuously in passing from one point to a neighboring one, is determined as a function of the coordinates by the well-known equation:
\[ \frac{d^{2}\psi}{dx^{2}}+\frac{d^{2}\psi}{dy^{2}}+\frac{d^{2}\psi}{dz^{2}}+\frac{4\pi^{2}}{\lambda^{2}}\psi=0. \tag{39} \]
The light oscillations at each point are expressed by the product of this amplitude and \(\cos 2\pi\nu t\). As for the square of the amplitude (or, more precisely, of its absolute value), it determines the mean energy of the waves or the intensity of the rays at the corresponding point.
Schrödinger assumed that the propagation of cathode waves is determined by the same differential equation (39) as the propagation of light waves, if the wavelength \(\lambda\) is expressed as a function of the coordinates according to formula (38). Thus Schrödinger’s equation has the form:
\[ \frac{d^{2}\psi}{dx^{2}}+\frac{d^{2}\psi}{dy^{2}}+\frac{d^{2}\psi}{dz^{2}}+\frac{4\pi^{2}m}{h^{2}}(W-U)\psi=0. \tag{40} \]
Born, however, expressed the idea that the “energy” of cathode waves, i.e. the square of the function \(\psi\), represents a measure of the intensity of the cathode rays, or, more precisely, of the “density” of the electron flux forming them. The intensity of the latter is obtained from this energy \((\psi)^{2}\) (which may also be interpreted as a measure of the probability of finding the electron in the corresponding place) by multiplying it by the velocity of the “rays” \(v\), i.e. by \(\frac{h}{m\lambda}\).
§ 8. Application of the Theory of Cathode Waves to Metals1
Passing to the application of the ideas set forth above to the motion of “free” electrons in metallic bodies, we first of all obtain an extremely simple and intuitive interpretation of the distribution of velocities at absolute zero temperature which follows from the Pauli–Fermi principle. Namely, putting in formula (26) \(\frac{m v_{\max}}{h}=\frac{1}{\lambda_{\min}}\), according to (37), we obtain:
\[ \lambda_{\min}=\sqrt[3]{\frac{\delta\tau}{3m}}. \tag{41} \]
This means that the minimum length of the cathode waves wandering through the metal in all directions is approximately equal to the mean distance between neighboring electrons, if they are imagined as arranged in the form of a regular cubic lattice. Further, it is not difficult to convince oneself that condition (25) is equivalent to the following: at absolute zero temperature the motion of the electrons in a metal can be described by the superposition of a system of \(\frac{1}{2}N\) standing cathode waves, which are resonant oscillations for a body of the volume and shape under consideration2. In the case, for example, of a metal having the shape of a rectangular parallelepiped with edges \(a_1, a_2, a_3\), the length of the various waves is expressed by the formula
\[ \frac{1}{\lambda} = \sqrt{ \left(\frac{k_1}{a_1}\right)^2 + \left(\frac{k_2}{a_2}\right)^2 + \left(\frac{k_3}{a_3}\right)^2 } \tag{42} \]
where \(k_1, k_2, k_3\) are integers. Let us note that entirely analogous relations are obtained when representing the thermal vibrations of the atoms of a metal (or of any other monatomic solid) by superposing a system of elastic standing waves—as is done, for example, in Debye’s well-known theory of heat capacity.
It is also curious to note the following circumstance. The dimensions of the circular orbit described by an electron in an isolated atom are determined by Bohr’s well-known condition:
\[ \text{angular momentum } mvr = k \frac{h}{2\pi}, \]
where \(k\) is an integer and \(r\) is the radius of the orbit. Putting in this formula \(mv = \dfrac{h}{\lambda}\), we obtain:
\[ \frac{2\pi r}{\lambda} = k \;(= 1, 2, 3 \ldots) \tag{43} \]
This equality, first obtained by de Broglie, shows that in isolated metallic atoms the length of the orbit of the valence electrons, if not exactly equal to, is at least comparable with the wavelength of the corresponding phase waves. If, upon condensation of a metallic vapor, all valence electrons are transformed into “free” ones, then the distance between them \((n^{-1/3})\) proves to be comparable with interatomic distances. Since, on the other hand, interatomic distances in solids are comparable with the dimensions of the orbits of the outer electrons, we arrive at the conclusion that the minimum or mean length of the cathode waves in a solid metal has the same order of magnitude as in isolated atoms. And from this it follows that the mean velocity of the electrons remains approximately the same in both cases.
The indicated relations change somewhat with increasing temperature. First of all, shorter cathode waves appear than those whose length is determined by formula (41). Further—and this is especially important for us—owing to the irregular arrangement of the atoms associated with their thermal motion,
... the cathode waves, as they propagate, begin to undergo diffuse scattering, which diminishes their intensity, i.e. causes an apparent absorption, and is the direct cause of the electrical, as well as the thermal, resistance of the metal. Here by “resistance” we mean the reciprocal of the conductivity. At absolute zero of temperature the electrical and thermal conductivities of metals are equal to infinity, i.e. the corresponding resistances are equal to zero. This means, from the standpoint of wave mechanics, that the cathode waves coursing through a metal propagate in it without any “absorption” or diffuse scattering whatever, i.e. in the same way as light waves propagate in ideally transparent bodies.
It is known, however, that even ideally transparent bodies—for example, perfectly pure air—at temperatures different from absolute zero become more or less “turbid,” i.e. begin to scatter the light rays passing through them. The visibility of atmospheric air, or of the “sky,” is due to just such scattering of the sun’s rays. The cause of the “turbidity” of air is the slight local condensations and rarefactions that arise from the irregular, disordered motion of its molecules. The coefficient of scattering of light in the case of gases whose molecules move quite independently of one another, as theory and experiment show, does not depend on temperature. In solids, on the contrary, it decreases rapidly as the temperature is lowered, while in the region of moderate temperatures it increases approximately in direct proportion to the temperature. It is necessary, however, to stipulate that this holds only for visible light or, more precisely, for such light waves whose wavelength is large in comparison with interatomic distances. In the case of short X-ray waves, whose wavelength is of the same order of magnitude as, or even smaller than, these distances, the coefficient of scattering at moderate temperatures remains constant, as in gases, and likewise in liquids and in solids.
THEORY OF METALLIC CONDUCTIVITY
In applying these principles to the scattering of cathode waves in metals, caused by the thermal motion of their atoms, it is necessary, first of all, to decide whether these waves should be treated as “long” or as “short” in comparison with interatomic distances. In the first case they must be scattered approximately in the same way as waves of ordinary light, i.e. approximately proportionally to \(T\) in the range of moderate temperatures, and in the second—as X-rays, i.e. approximately independently of \(T\), in that same temperature range, of course. At very low temperatures their scattering coefficient must vanish in both cases1.
The scattering coefficient of waves in some turbid medium is determined, generally speaking, by the formula
\[ \mu=\frac{1}{J}\frac{dJ}{dx} \tag{44} \]
where \(J\) is the intensity of the waves, and \(dJ\) is its change over the interval \(dx\) in the direction of their propagation. Integrating this equation, we obtain
\[ J=J_{0}e^{-\mu x}. \tag{45} \]
In the case of cathode waves the intensity \(J_{1}\), measured by the square of the amplitude \(\psi\), represents a measure of the number of electrons taking part in the motion that is described by these waves. Thus formula (45) expresses the law of decrease of the number of electrons in the cathode beam owing to the scattering that they undergo by “colliding” with atoms. Here the obstacle to their rectilinear motion is not the atoms themselves (for at \(T=0\) scattering ceases), but the thermal vibrations of the latter [cf. the interpretation of formula (20)]. Be that as it may, the relative number of electrons “scattered,” i.e. thrown out of the beam under consideration over an interval of length \(dx\), is equal to \(\mu\,dx\). Hence
it follows that the scattering coefficient $\mu$ is nothing other than the reciprocal of the average free path of the electrons or their “elementary displacement” $l$:
\[ \mu = \frac{1}{l}. \tag{46} \]
Thus, knowing $\mu$, we can calculate, by the formulas given in the preceding sections, the electrical and thermal resistance of metals (or the corresponding conductivities).
Let us note that an entirely analogous theory of the thermal conductivity of solid dielectrics was proposed in 1914 by Debye. Experimentally, the fact had been established that at low temperatures this thermal conductivity becomes extraordinarily large, exceeding the thermal conductivity of metals. Describing the thermal motion of atoms in solids by superposing a system of elastic waves, Debye arrived at the idea that the cause of thermal resistance lies in the scattering experienced by these waves due to fluctuations in the density of the corresponding body. Since, however, these fluctuations were caused in turn by elastic waves, Debye’s theory led to a contradiction with the principle of superposition of elastic waves on which it was based. The theory of the electrical and thermal resistance of metals sketched above is evidently free from such a contradiction, since in it the scattered (cathode) waves are distinct from the scattering (elastic) waves.
Without pausing for the moment over the determination of the numerical value of the coefficient $\mu$ (see below), we may state the following propositions of a qualitative character.
If the cathode waves in metals are treated as long waves, then in the range of medium temperatures one may put:
\[ \mu = \operatorname{const} T. \tag{47} \]
In order that the electrical conductivity $\sigma$ should then be inversely proportional to $T$, it must be expressed by Sommerfeld’s formula (17), which does not explicitly contain the temperature; then
as application of my formula (34) would give, in connection with the equality \(D=\frac{1}{3}lv'=\frac{1}{3}\cdot\frac{v'}{\mu'}\), \(\sigma=\frac{\mathrm{const}}{T^{2}}\).
If, conversely, we treat cathode waves as short (likening them to X-rays), then under the same conditions we obtain:
\[ \mu=\mathrm{const}, \tag{48} \]
and consequently, according to my formula \(\sigma=\frac{\mathrm{const}}{T}\), while according to Sommerfeld’s formula \(\sigma=\mathrm{const}\).
The question of which of these two formulas corresponds to reality does not admit of a simple solution. As we have seen above, the mean length of cathode waves in a metal is precisely comparable with interatomic distances. Thus these waves, strictly speaking, cannot be treated either as long or as short. A theory of the scattering of light waves whose length has an intermediate value does not yet exist; the construction of such a theory presents very great difficulties.
There are, however, two circumstances which testify rather in favor of Sommerfeld’s formula than of mine.
According to the theory set forth above, the scattering coefficient of cathode waves \(\mu'\), in the case of a not entirely pure metal, i.e. a metal containing a small quantity of some impurities (even of another, better-conducting metal), must always be greater than the scattering coefficient of the same metal in a perfectly pure state \(\mu\).
Indeed, such irregularly distributed impurities must act on cathode waves in approximately the same way as dust suspended in air acts on the scattering of light waves. We may therefore put:
\[ \mu'=\mu+\Delta\mu \tag{49} \]
where \(\Delta\mu\) is an essentially positive quantity, practically independent of temperature. Substituting here the value of \(\mu\)
from (47) and, using Sommerfeld’s formula for electrical conductivity, we obtain
\[ \rho'=\frac{1}{\sigma'}=\rho+\Delta\rho=\mathrm{const}\,T+\Delta\rho, \]
where \(\rho \sim T\) is the resistance of the pure metal, and \(\Delta\rho\) is an additional resistance, proportional to the amount of impurities and independent of temperature. This result is in complete agreement with the experimental facts expressed by the so-called Matthiessen rule. According to my formula, however, the additional resistance should have been directly proportional to the absolute temperature.
Further, it is known that the resistance of most metals increases sharply (approximately twofold) upon melting. This fact is directly explained from the point of view of “long cathode waves” by the circumstance that the inhomogeneities due to density fluctuations in the case of a liquid are greater (approximately twice) than in the case of the corresponding solid. From the point of view of “short waves,” however, the scattering coefficient should already attain in a solid metal near the melting temperature its maximum value,
\[ \mu=\frac{1}{d}\simeq 10^8\ \mathrm{cm}^{-1}, \]
so that its considerable increase upon melting becomes incomprehensible.
These considerations undermine the correctness of the calculation of the mobility of electrons which was made in § 6, proceeding from the conception of the change in the probability of various elementary displacements under the influence of an electric field (and which is directly connected with the Boltzmann law of distribution of electrons in a given force field). As for the numerical value of \(\mu\) (at ordinary temperatures), both points of view—of both “short” and “long” cathode waves—lead to figures agreeing with the experimental data on the electrical conductivity of various metals, if the latter is computed in the first case by my formula, and in the second by Sommerfeld’s formula.
I have no possibility of dwelling here on the details of the calculation. The calculation of \(\mu\) proves possible with the aid of Schrödinger’s equation (40), if by \(u\) in it one understands the potential energy of the electron with respect to all the atoms and regards the latter as neutral. The energy of cathode waves scattered by one such atom, as the new theory of “collisions,” developed by Born and Wentzel, shows, is approximately equal to
\[ \mu = 4\pi \left(\frac{e^2}{mv_0^2}\right)^2 \frac{4k^2a^2}{1+4k^2a^2} \left(k=\frac{2\pi}{\lambda}=\frac{2\pi mv}{h}\right), \tag{50} \]
if the energy \((\varphi)^2\) of the incident waves is taken to be equal to 1. Here \(a\) denotes the effective radius of the atom in the usual “Bohr” sense, i.e. a quantity of order \(10^{-8}\,\mathrm{cm}\). In order to obtain the complete scattering coefficient of a (pure) metal containing \(n_a\) atoms per unit volume, it is necessary to multiply the preceding expression by \(n_a\) in the case of “short” waves and by the mean-square fluctuation \(\overline{(\Delta n_a)^2}\) in the case of long ones1. This gives, in the first case, \(\mu \sim 10^8\), i.e. \(l \lesssim 10^{-8} \lesssim d\), and in the second \(\mu \sim 10^5\), i.e. \(l \sim 10^{-5}\,\mathrm{cm}\) (for \(T \sim 300\)). Both figures agree with the experimental data concerning the magnitude of the electrical conductivity, if the latter is calculated by my formula in the first case and by Sommerfeld’s in the second. I think, however, that the temperature dependence of \(\sigma\) at low temperatures agrees better with Sommerfeld’s formula, i.e. with the theory of an electron gas formed by long cathode waves, than with my theory of wandering comet-like electrons, to which very short cathode waves correspond. It remains, however, an undoubted fact that these waves actually have a length comparable with interatomic distances, and therefore are rather short than long.