Theory of the Metallic State

II. Verification of Fermi Statistics in Equilibrium Effects*

L. Nordheim.

§ 1. Basic Concepts

The picture that we can give, on the basis of the preceding conclusions, of the state of the electrons in a metal is approximately the following. There is a very large number of free electrons. This number is \(z\) times greater than the number of atoms \(n\), and we shall be very close to the truth if we assume that \(z\) is equal to the valence of the corresponding element, i.e. equal to the number of those electrons which can easily be detached from the given ion. If one takes into account that, in the close packing of atoms in the metal lattice, the atoms are under the mutual influence of their force fields, it becomes obvious that the valence electrons can no longer belong only to their own ions. The expression “free electron,” as was noted above, should be understood only in the sense that the electron possesses sufficiently great mobility, as is required by the fact of the existence of high electrical conductivity. The kinetic energy of such electrons must, owing to the fulfillment of the Pauli principle, be very large; it must correspond to the absolute-zero energy of a Fermi–Dirac gas. In a very crude approximation we may disregard in our scheme the electrostatic forces of interaction between ions and electrons and apply the formulas obtained earlier that refer to noninteracting particles. The electron gas as a whole is held in by a potential discontinuity at the surface, which for this purpose must be greater than the absolute-zero energy.*

The circumstance that, in such a treatment, we have not taken account of the relatively very large electrostatic forces may nevertheless be considered rational, since on the average any element

* See Uspekhi fizicheskikh nauk, 15, 570, 1935; translated by S. G. Kalashnikov.

** In connection with this there is the circumstance that, for good conductors, low values of the ionization potential are characteristic.

*** When a system of charged particles is compressed, energy is released, so that the metal as a whole is a stable system. One may say that in this process the kinetic energy of the electrons in the bound state (in the classical picture—in their orbital motion) is transformed into the kinetic energy of translational motion.

the volume of the metal is neutral, and therefore the interaction forces must cancel. In any case, we must not take into account the full Coulomb force between the electrons, since the interaction of the electrons is screened by the ions of the metal. As for the qualitative behavior of electrons in a metal, the latter follows already from the Pauli principle and from the order of magnitude of the electron concentration and their mass. We may further assume that the adopted picture will correspond all the more closely to reality, the better the given metal conducts, i.e., the more mobile its electrons are in it.

We shall have to return later to the development and refinement of this picture. First, however, we shall see whether there are any relatively simple phenomena that would make it possible experimentally to detect the large difference found in the results of classical and quantum statistics, above all the large absolute-zero energy with the adjoining exponential distribution law.

This was shown quite directly experimentally by DuMond,^2 who compared the Compton effect on the electrons of a metal and on electrons bound in an atom.* In accordance with the broad distribution of the metal electrons over energies—from zero to the absolute-zero energy $\varepsilon$—the shifted Compton line should also be strongly broadened. DuMond indeed found good agreement in order of magnitude, although the broadening found in the experiment proved somewhat smaller than that predicted according to the elementary theory set forth in the preceding chapter. This would also follow if one takes into account that the electrons in a metal are not completely free (Ch. IV).

A further weighty confirmation of the correctness of the adopted picture is provided by the phenomena of electron emission: thermoelectronic emission, cold discharge, the photoelectric effect, as well as a number of other phenomena, which we shall consider in the present chapter.** From the analysis of all these phenomena it follows that the picture of completely free electrons represents, with good approximation, the actual conditions in a metal.

According to these simplest ideas, the state of the electrons in a metal is described by the following constants: first, by the critical value of the absolute-zero energy $\varepsilon_0=\mu$ (which can be calculated from the number of free electrons); second, by the height $C$ of the potential barrier at the boundary.

What has been said can best be explained with the aid of Fig. 1. The metal is represented as a potential well, the height of whose walls is equal to $C$. This well, even at $T=0$, is filled up to the level $\varepsilon=\varepsilon_0$ with electrons, so that there are always electrons with kinetic

* In calculating this effect it must be taken into account that only such processes can occur in which the final state of the electron is unoccupied (cf. Part III of the present article). Hence, among other things, it follows that this effect cannot be observed in visible light.^3

** A more detailed survey exposition of this question may be found in Nordheim’s article.^4

energy \(\varepsilon_0\). At higher temperatures there are still faster electrons, whose number even at the highest attainable temperatures is very small.

As a result of thermal motion the electrons encounter the surface of the metal. Emission of an electron can occur only in the case that the electron is able to pass through the surface. The theory of electron emission therefore falls into two parts: first, the calculation of the number of electrons \(N(\mathbf p)\) encountering the surface of the metal, as a function of their momentum \(\mathbf p\), and, second, the calculation of the probability \(D(\mathbf p)\) of transition through the surface. The total emission current will be

\[ i = e \int N(\mathbf p) D\, d\mathbf p, \tag{1} \]

where \(e\) is the elementary charge.

Fig. 1. Scheme of the potential distribution in a metal

Thermionic emission is obtained as a result of the presence of fast electrons produced by thermal motion. In cold discharge even slow electrons can, under the action of an external field, overcome the potential barrier at the surface and, finally, in the photoelectric effect sufficiently fast electrons are obtained as a result of excitation by light.

The circumstance that all these effects can be calculated comparatively simply is due to the following: first, in calculating the emission one must take into account mainly electrons with large kinetic energy, which may most likely be regarded as free; second, as we shall see below, the free path of the electrons, i.e. the segment over which they can move in the metal without disturbance, is very large (of the order of 100 atomic distances).

Let us first calculate the number of electrons which, moving from within, encounter the surface of the metal. Here only the normal component of the motion is of interest to us (just as in the classical

mechanics), i.e. the quantity \(W=\dfrac{p_x^2}{2m}\), if we assume that the surface of the metal is perpendicular to the \(x\)-axis. The number of electrons per unit volume whose momenta lie within the interval \(dp_x\,dp_y\,dp_z\) is given, according to Ch. I § 9 (8b), by the expression

\[ N(p)\,dp_x\,dp_y\,dp_z = \frac{G}{h^3} \frac{dp_x\,dp_y\,dp_z} {e^{\{(p_x^2+p_y^2+p_z^2)/2m-\varepsilon_0\}/kT}+1}, \]

so that the number of electrons \(n(p_x)dp_x\) with one definite component of the momentum will be

\[ n(p_x)dp_x = dp_x\,\frac{G}{h^3} \int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} \frac{dp_y\,dp_z} {e^{\{(p_x^2+p_y^2+p_z^2)/2m-\varepsilon_0\}/kT}+1}. \]

Introducing polar coordinates \(p_y=\rho\cos\varphi,\ p_z=\rho\sin\varphi\), we obtain

\[ n(p_x) = \frac{G}{h^3} \int_0^\infty\int_0^{2\pi} \frac{\rho\,d\rho\,d\varphi} {e^{\{(\rho^2+p_x^2)/2m-\varepsilon_0\}/kT}+1} = \]

\[ = \frac{2\pi G}{h^3} \int_0^\infty \frac{\rho\,d\rho} {e^{\{(\rho^2+p_x^2)/2m-\varepsilon_0\}/kT}+1}. \]

Introducing the new variable \(x=\dfrac{\rho^2}{2mkT}\) and the notations

\[ W=\frac{p_x^2}{2m};\qquad \beta=\frac{W-\varepsilon_0}{kT}, \]

we further obtain

\[ n(p_x)dp_x=n(W)dW = dW\,\frac{2\pi Gm^2kT}{h^3\sqrt{2mW}} \int_0^\infty \frac{dx}{e^{\beta+x}+1}. \]

The integral is evaluated elementarily. It is equal to

\[ \int_0^\infty \frac{dx}{e^{x+\beta}+1} = \int_{e^\beta}^{\infty}\frac{dz}{z(z+1)} = \{\ln z-\ln(z+1)\}_{e^\beta}^{\infty} = \ln(1+e^{-\beta}) \]

and, consequently,

\[ n(W)dW = \frac{\pi Gm}{h^3}\sqrt{\frac{2m}{W}}\, kT\ln\left(1+e^{-\frac{W-\varepsilon_0}{kT}}\right)dW. \tag{2} \]

The number of electrons \(N(W)dW\) striking per unit time

per unit area of the plane \(x=\mathrm{const}\), we find by multiplying the preceding expression by the velocity \(v_x=\sqrt{2W/m}\):

\[ N(W)\,dW=\frac{G}{h^3}\,2\pi m kT\ln\left(1+e^{-(W-\varepsilon_0)/kT}\right)\,dW. \tag{3} \]

We shall use the following approximate expressions:

\[ N(W)\,dW= \begin{cases} \displaystyle \frac{G}{h^3}\,2\pi m \left\{(\varepsilon_0-W)+kT e^{\frac{W-\varepsilon_0}{kT}}+\cdots\right\}\,dW, & \displaystyle \text{for } \frac{W-\varepsilon_0}{kT}\ll 0, \tag{4a} \\[1.2em] \displaystyle \frac{G}{h^3}\,2\pi m kT \left\{\ln 2-\frac{1}{2}\frac{W-\varepsilon_0}{kT}+\cdots\right\}\,dW, & \displaystyle \text{for } |W-\varepsilon_0|\ll kT, \tag{4b} \\[1.2em] \displaystyle \frac{G}{h^3}\,2\pi m kT e^{-\frac{W-\varepsilon_0}{kT}} \left\{1-\frac{1}{2}e^{-\frac{W-\varepsilon_0}{kT}}+\cdots\right\}\,dW, & \displaystyle \text{for } \frac{W-\varepsilon_0}{kT}\gg 0. \tag{4c} \end{cases} \]

The graph of the function \(N(W)\) is given in Fig. 2. The straight line gives the distribution for \(T=0\), described by the first term in (4a); the distribution ends at the critical value \(W=\varepsilon_0\). The dashed line represents the distribution function for some higher temperature.* At energy values greater than the critical one, it passes into Maxwell’s exponential law. The number of electrons of interest to us in this latter region is very small in comparison with the case where \(W<\varepsilon_0\).

Fig. 2. Distribution law of electrons striking the surface of a metal

* As is clear from the expansion (4a), the distribution function for any temperature lies entirely above the straight line for \(T=0\). Thus the number of electrons striking the surface increases with temperature, which is connected with the increase in the mean velocity of the electrons.

§ 2. On the Calculation of the Transition Probability

The method of calculating the transition probability is provided by quantum mechanics. The metal–vacuum boundary may be characterized as a certain layer with a thickness of the order of atomic distances, in which the potential changes very rapidly and strongly (by several volts). If the distribution of the potential is known, then it is possible to calculate—at least in principle—the scattering, reflection, and refraction of the de Broglie waves corresponding to the incident electrons. The theory is entirely analogous to the analysis of optical reflection and refraction in an inhomogeneous medium, and the whole difference consists only in the fact that instead of Maxwell’s equations we have here Schrödinger’s equation. In doing so, we may, to a very good approximation, disregard the atomistic structure of the surface, since we shall be dealing mainly with electrons that leave the metal with very small velocities, so that the length of their de Broglie wave

\[ \lambda=\frac{h}{mv} \]

in the external space is sufficiently large in comparison with the lattice constant of the metal. Since, in the phenomenon of reflection, precisely the region of small velocities is essential, such a simplification is quite legitimate.

Fig. 3. Potential step at the surface of a metal

Suppose that the metal is situated to the left of the plane \(x=0\) (Fig. 3). Assuming that the periodic structure of the metal may be neglected, the potential \(U\) may be regarded as a function of only one variable \(x\). In the absence of an external field it is equal to a constant value \(C\) for \(x>0\), while for \(x<0\) it is also constant and equal to 0. Between these two regions there is a certain distribution of the potential, with respect to which, for various problems, definite assumptions must be made. The time-dependent Schrödinger equation

\[ \Delta\psi-\frac{8\pi^2 m}{h^2}U\psi-\frac{4\pi i m}{h}\frac{\partial \psi}{\partial t}=0, \]

in which now \(U\) is a function of only the single variable \(x\), can be satisfied by a solution of the type

\[ \psi=\varphi(x)e^{-\frac{2\pi i}{h}\left(\varepsilon t-y p_y-z p_z\right)}, \]

in which the variables \(t,y,z\) are separated. \(\varphi(x)\) satisfies the one-dimensional equation

\[ \frac{d^2\varphi}{dx^2}+\frac{8\pi^2 m}{h^2}\,[W-U(x)]\varphi=0, \tag{1} \]

where

\[ W=\varepsilon-\frac{p_y^2+p_z^2}{2m} \tag{1a} \]

gives the energy of the \(x\)-component of the motion.

In regions of constant potential the solution of (1) is given by an exponential function

\[ \varphi= \frac{a_1}{(W-U)^{\frac14}} e^{-\frac{2\pi i}{h}x\sqrt{2m(W-U)}}+ \frac{a_2}{(W-U)^{\frac14}} e^{\frac{2\pi i}{h}x\sqrt{2m(W-U)}} \tag{2} \]

and can therefore be represented as a superposition of waves propagating to the right (\(a_1\)) and to the left (\(a_2\)). The constant factor \((W-U)^{-\frac14}\) has been introduced for normalization. Taking into account that \(|\varphi|^2\) gives the electron density, we may write the following expression for the current strength:

\[ v_x|\varphi|^2= \sqrt{\frac{2}{m}(W-U)}\,|\varphi|^2 . \]

Since the modulus squared of the exponential factor is equal to 1, we find that the electron density is proportional to

\[ \frac{|a|^2}{|(W-U)|^{\frac12}}, \]

so that \(|a|^2\), up to a factor, gives the current strength for the corresponding wave.

Both the function \(\varphi\) itself and its derivative \(\frac{d\varphi}{dx}\) must be continuous throughout space. If there exist solutions (2) in the regions \(x\ll 0\) and \(x\gg 0\), then both these solutions are connected by equation (1); a specified pair of values of the coefficients \(a_1\) and \(a_2\) for \(x\ll 0\) passes into a perfectly definite pair of new coefficients \(b_1\), \(b_2\) for \(x\gg 0\). In this case the general solution will contain two arbitrary constants, since we are dealing with a second-order equation. The physical condition that the electrons pass only in one direction (from left to right) gives \(b_2=0\). Thus, to a given wave \(a_1\) coming from the left there corresponds uniquely a definite transmitted wave \(b_1\) and a reflected one \(a_2\). The transmission coefficient \(D\) (transition probability) and the reflection coefficient \(R\) are defined as

\[ D=\frac{|b_1|^2}{|a_1|^2};\qquad R=\frac{|a_2|^2}{|a_1|^2},\quad \text{with } R+D=1. \tag{3} \]

The solution (2) refers only to a region of constant potential; its applicability limits are therefore comparatively narrow. However, it can be shown that in a region with an arbitrary course of the potential \(U(x)\), provided only

the potential does not change too rapidly, the solution of (1) with very good approximation in the neighborhood of the point \(x_0\) is given by the expression

\[ \varphi = \frac{a_1}{[W-U(x)]^{\frac14}}\, e^{-\frac{2\pi i}{h}\int_{x_0}^{x}\sqrt{2m(W-U)}\,dx} + \frac{a_2}{[W-U(x)]^{\frac14}}\, e^{\frac{2\pi i}{h}\int_{x_0}^{x}\sqrt{2m(W-U)}\,dx}. \tag{4} \]

This is the well-known solution, given by Wentzel, Kramers, and Brillouin\(^5\), which is successfully used in a whole series of problems; it is a natural generalization of the solution (2). \(|a_1|^2\) and \(|a_2|^2\) give here as well the strength of the electron current in both directions, and we can operate with the expression (4) in exactly the same way as with (2).

In the region of this solution the current strength also remains constant. This shows clearly that the electron current passes through the region of variable potential without reflection. Thus, as our first result, we obtain that a potential which does not change very rapidly causes no reflection at all. We may further, without essentially limiting the generality, assume that over a sufficiently large distance on both sides of the transition layer no reflection also occurs, i.e. that (4) for large \(|x|\) is also a good approximation. Then the whole problem reduces to establishing the correspondence between the coefficients \(a_1\), \(a_2\) before the transition and \(b_1\), \(b_2\) after the transition (where \(b_2\) may at once be put equal to zero). As for reflection, it can be expected only in the case where there are regions inside which the solution (4) loses its force. This may be caused by two reasons:

1. \(U\) or \(\dfrac{dU}{dx}\) changes substantially already over an interval of the order of the length of the de Broglie wave. An exact solution of the problem in this case can be obtained only in rare cases. We shall consider here only one limiting case, when there is a discontinuity either of the potential itself or of its derivative. The case under discussion corresponds to the optical problem of reflection at the boundary of two media. Here in each of the regions with continuous \(U\) and \(\dfrac{dU}{dx}\) the solution will be (2) or, respectively, (4), and it is necessary to require that at the place of the jump of the potential both \(\varphi\) and \(\dfrac{d\varphi}{dx}\) remain continuous. In this way a large number of examples have been calculated\(^*\). In this case reflection is obtained every time the potential or its derivative undergoes a jump, even in the case where this jump is smaller than the kinetic energy of the electrons (according to classical mechanics, the electrons in this case should pass without any reflection). Nevertheless, the author believes that this result should not be ascribed much practical significance, since the potential curves encountered in nature at

* See the works of Nordheim, Fowler and Franck, and Junga\(^6\). These works also contain many examples relating to case II (a potential barrier), for which a result is obtained that in essential part coincides with that given in the present article.

in fact rounded, and therefore the reflection here cannot be large.*

It should be noted that by this method only those problems can be calculated exactly in which the curve of variation of the potential can be composed of rectilinear segments, since for a linear function \(U=ax+b\) the solutions of (1) are known; these are Bessel functions of order \(1/3\) (see the cited work of Fowler and Nordheim).

II. The situation will be quite different if the potential energy at some point is greater than the total energy of the electron (Fig. 4), i.e. when the curve of the potential energy intersects the level of the particle energy (the function \(W-U\) has zeros). At such places the wave function essentially changes its character: an exponential function with an imaginary exponent passes into an exponential function with a real exponent. If after the intersection the curve of the potential energy remains all the time above the energy level of the particle (curve \(A\), Fig. 4), then the physical requirement that there be no electrons moving to the right is equivalent to the mathematical requirement that only exponential functions with negative real exponents be chosen as the solution. The Schrödinger waves will then also penetrate into the forbidden region II, dying out there, however, according to an exponential law, exactly as occurs in optics in total internal reflection. It can be shown that in this case one always has \(|a_1|^2=|a_2|^2\), so that at such a potential threshold we indeed have total reflection.

Fig. 4. Potential curves (for calculating the transition probability)

If there is a distribution of potential of the type given by curve \(B\) (Fig. 4), i.e. a potential barrier beyond which motion with positive kinetic energy is again possible, then according to classical mechanics such a barrier nevertheless represents an absolute obstacle for electrons. According to quantum mechanics, however, in this case as well there exists a definite probability of passing through the barrier.** In region III there again exists a solution having the character of propagating waves, which

* For the theory of thermoelectronic emission (see § 4) the case of a potential jump rounded by the forces of the electrical image is especially important. Nordheim\(^7\) showed that even in this case the reflection is negligibly small, so that for a clean metal surface it can hardly play any role.

** This effect plays a fundamental role in Gamow and Condon’s theory of radioactive decay.

have an appreciable amplitude only in the case where the waves in region II do not have time to die away completely.

The transition probability is usually calculated by method I, comparing the solutions at jumps of the potential. However, with the Wentzel—Kramers—Brillouin method one can give a general solution, suitable for a potential barrier of arbitrary form.* For the transition probability (“transparency”) one then obtains the expression

\[ D=\frac{1}{\left(\cosh \frac{2\pi}{h}\int_{x_1}^{x_2}\sqrt{2m(U-W)}\,dx\right)^2}, \tag{5} \]

where \(U(x)\) is the prescribed course of the potential, and \(W\) is the energy of the normal component of the particle’s motion. The integration must be extended over the whole region between the two zeros of the function \(U-W\), which in Fig. 4 is shown by hatching. If the argument of the \(\cosh\) is sufficiently large, then the negative exponential function in it may be neglected; then expression (5) goes over into the following**:

\[ D=4e^{-\frac{4\pi}{h}\int_{x_1}^{x_2}\sqrt{2m(U-W)}\,dx} \tag{6} \]

Fig. 5. Potential barrier in the theory of the cold discharge

As an example, which we shall use below in the theory of the cold discharge, let us consider the triangular potential barrier shown in Fig. 5: \(U=0\) for \(x<0\), \(U=C-Fx\) for \(x>0\). The exponent of formula (6) in this case is equal to

\[ \frac{4\pi}{h}\int_{0}^{\frac{C-W}{F}}\sqrt{2m(C-W-Fx)}\,dx = \frac{4\pi[2m(C-W)]^{\frac{3}{2}}}{h\,m\,3F}; \tag{7} \]

* The proof of formula (5), which would lead us too far afield here, will be given elsewhere.

** In the literature many formulas are given, obtained for special forms of the potential barrier (rectangular, etc.), which differ from (6) only by a factor of order unity. This happens because, in deriving all these formulas, a discontinuous course of potential jumps is assumed. The extent to which the difference in these factors is real depends on how little the actual course of the potential differs from an abrupt jump. However, the essential part in all these formulas is the exponential part, which always coincides with our expression.

we obtain, therefore,

\[ D(W)=4e^{-\frac{4\pi}{hm}\frac{[2m(C-W)]^{\frac{3}{2}}}{3F}}, \tag{7a} \]

By the method of potential jumps, Fowler and Nordheim (the work cited above) obtained:

\[ D(W)=\frac{4W^{\frac{1}{2}}(C-W)^{\frac{1}{2}}}{C} e^{-\frac{4\pi}{hm}\frac{[2m(C-W)]^{\frac{3}{2}}}{3F}}, \tag{7b} \]

which, up to the inessential factor

\[ \frac{W^{\frac{1}{2}}(C-W)^{\frac{1}{2}}}{C}, \]

coincides with (7a).

§ 3. Thermionic Emission

We now have everything necessary for constructing the theory of electron emission. Thermionic electron emission, as already mentioned, occurs for the reason that a certain fraction of the electrons, owing to thermal motion, acquires kinetic energy sufficient to overcome the potential jump at the surface. Since for \(W<C\), owing to total reflection, the transition probability becomes zero, we obtain, according to § 1 (1) and (3), for the density of the emission current the expression (what is meant, of course, is the saturation current that occurs when the action of the space charge is eliminated):

\[ i=\frac{G}{h^3}2\pi mekT\int_C^\infty D(W)\ln\left(1+e^{\frac{\varepsilon_0-W}{kT}}\right)dW \tag{1} \]

Since the potential jump must be considerably greater than the energy of the absolute zero \(\varepsilon_0\), we can use here the approximation of § 1 (4c). Introducing the kinetic energy of the \(x\)-component of the motion in external space, \(x=W-C\), we obtain

\[ i=\frac{G}{h^3}2\pi mekT e^{\frac{\varepsilon_0-C}{kT}}\int_0^\infty D(X+C)e^{-\frac{X}{kT}}\,dX. \]

For further calculations let us introduce a new variable \(x=\dfrac{X}{kT}\), i.e., let us measure the energy of the emitted electron in units of \(kT\). Then

\[ i=\frac{G}{h^{3}}\,2\pi m e (kT)^{2} e^{-\frac{C-\varepsilon_{0}}{kT}} \int_{0}^{\infty} D(C+xkT)e^{-x}\,dx . \]

Introducing, finally, the mean transition probability for the given temperature \(T\)

\[ \bar D(T)=\frac{1}{kT}\int_{0}^{\infty}D(C+X)e^{-\frac{X}{kT}}\,dX =\int_{0}^{\infty}D(C+xkT)e^{-x}\,dx, \tag{2} \]

we obtain finally:

\[ i=\frac{2\pi G m e k^{2}}{h^{2}}\,\bar D\,T^{2} e^{-\frac{\chi}{kT}}; \qquad \chi=C-\varepsilon_{0}. \tag{3} \]

The coefficient \(\bar D\) may also depend on the temperature \(T\); however, always \(\bar D\leqslant 1\):

\[ \bar D\leqslant \int_{0}^{\infty} e^{-x}\,dx=1. \]

The limiting value 1 is attained only for complete transparency. On the basis of what has been set forth above, one may assume that the influence of reflection at the surface is very small, so that \(\bar D\) differs only very little from unity.

Formula (3) coincides with the well-known Richardson formula. In this form it was obtained by a purely thermodynamic route by Laue and Dushman[^8]. In the thermodynamic derivation it is sufficient only to assume that the entropy of the electrons is given by the Stern–Tetrode expression (see Part 1, § 9 (6)); any additional model conceptions thereby prove entirely superfluous. Richardson himself, using classical statistics, first obtained an almost equivalent formula:

\[ i=A T^{\frac12}e^{-\frac{\chi}{kT}}; \qquad A=\frac{N}{V}e\sqrt{\frac{k}{2\pi m}}. \tag{3a} \]

In practice formulas (3) and (3a) differ from one another, as is known, extremely little, since a slowly varying function of the temperature in front of the exponential factor can hardly be noticed. However, the correctness of the new formula, and not the old one, has been proved by the fact that according to the new theory the constant

factor must have the same value for all metals, equal to

\[ \frac{2\pi G m e k^2}{h^3}=G\cdot 60.2\ \mathrm{A/cm^2} \]

(whereas the quantity \(A\) in (3a) also depends on the as yet unknown value of the electron concentration in the metal, \(\frac{N}{V}\)). For pure metals whose surface can be well cleaned by annealing in vacuum (tungsten and molybdenum), this value appears to have been well confirmed. It is true that it is claimed that agreement with experiment is obtained better without the factor \(G=2\). However, to prove this conclusively is very difficult; moreover, here it is also necessary to take into account the circumstance that partial reflection from the surface (the factor \(\overline D\)), which up to now has not yet been directly measured experimentally, should improve the agreement of the theory with experiment. The required value of the reflection coefficient, of the order of \(50\%\), could be obtained if there were a sharp potential jump at the surface. However, an exact calculation, carried out with allowance for the forces of the electric image (which, undoubtedly, must be taken into account here), gives, for such a smoothed course of the potential, values of \(\overline D\) differing only insignificantly from unity.

It may seem that the derivation of formula (3), by itself quite satisfactory, is not yet especially conclusive for our purposes, since the same formula can also be obtained without any model assumptions. However, the considerations presented here give an entirely new interpretation of the work function, an interpretation whose correctness can be confirmed by other methods. The work function is the name given to the constant \(\chi=C-\varepsilon_0\) in the exponent of Richardson’s formula. Whereas in the old theory this constant was taken to be simply equal to the magnitude of the potential jump at the surface, in the new theory this constant turns out to be equal to the difference between the magnitude of the jump and the energy of the absolute zero, \(\varepsilon_0\). True, this quantity still turns out to be equal to the energy required to tear an electron out of the metal, since in the latter there are always electrons with the energy of the absolute zero. However, the magnitude of the total potential jump \(C\) itself in the new theory turns out to be equal to the sum of the work function, which has a value of the order of \(2\text{–}6\ \mathrm{V}\), and the energy of the absolute zero (\(3\text{–}10\ \mathrm{V}\)); it is therefore much greater than in the old theory.

The order of magnitude of the potential difference \(C\) can be determined by an independent method from measurements of the refractive index of electron waves. Denoting by \(X\) the energy of an electron in external space, we find for the latter inside the metal

quantity \(X+C\); the corresponding de Broglie wavelengths will be

\[ \lambda_a=\frac{h}{\sqrt{2mX}};\qquad \lambda_i=\frac{h}{\sqrt{2m(X+C)}}, \]

and we obtain the refractive index equal to

\[ n=\frac{\lambda_a}{\lambda_i}=\sqrt{\frac{X+C}{X}}. \tag{4} \]

Thus, when an electron enters a metal, the length of the electronic wave is shortened in the ratio \(1/n\).*

In calculating the positions of the diffraction maxima in whose formation the inner layers of the crystal participate (diffraction by a space lattice), we must find the corresponding displacement of them. This displacement was in fact found by Bethe and Hartree\(^9\) in treating the experimental data of Davisson and Germer. According to Rupp’s measurements\(^ {10}\) we obtain the following values for the potential jump:

	K	Cu	Ag	Au	Zr	Mo	W	Ni
\(C\) observed	7.3	13.5	14.0	14.0	10.2	13.5	12.4	16
\(\chi\) observed	\(\sim 1\)	\(\sim 4.4\)	\(\sim 4.1\)	\(\sim 4.4\)	3.8	4.4	4.5	\(\sim 4.4\)
\(C-\chi=\varepsilon_0\)	6.3	9.1	9.9	9.6	6.4	9.1	8	11.6

The work function has been determined experimentally (to be sure, for most metals very unreliably). Therefore experimental values can also be obtained for the energy of the absolute zero \(\varepsilon_0\); these values are given in the appended table. The value \(\varepsilon_0\) obtained in this way agrees in order of magnitude with the theoretical value [Part 1, § 9 (4)], so that an attempt was even made to determine the number \(Z\) of free electrons per atom*. The best value turns out to be that obtained for nickel (Davisson and Germer’s measurements). Here the theoretical value \(\varepsilon_0=11.5\) \((z=2)\) coincides completely with the experimental one, whereas for \(z=1\) and for \(z=3\) one would obtain manifestly impossible—

* In order to prevent a possible misunderstanding, it should be noted that, in bombarding a metal with electrons from outside, we do not risk coming into conflict with the Pauli principle. The energy level \(\varepsilon_0\) lies much lower than \(C\), so that for the electrons penetrating from outside there will always be a sufficient number of free places.

** In the work of Rosenfeld and Witmer\(^ {11}\). However, here the results of Rupp’s first measurements were used, which are unreliable.

*** Cf. the table for \(\mu Z^{3/2}\) in Part 1, § 9.

values. It should be noted, however, that here we cannot under any circumstances expect very good agreement (for Ni it must be accidental), and therefore we cannot regard this method of determining \(Z\) as reliable. The expression for \(\varepsilon_0\) was derived under the assumption of free electrons, and taking account of the atomic fields inside the crystal may (and must) change it.*

What is most essential here for our purposes is that for the potential jump one obtains a quantity considerably larger than the work function; this fact may be regarded as reliably established. The theoretical estimate of this quantity, carried out by Bethe, likewise led to agreement in order of magnitude.

The work function, as is known, depends very strongly on the state of the surface. Fowler \(^{13}\) showed that these effects too are amenable to calculation.**

§ 4. Influence of the electric field on emission
Cold discharge

Further confirmation is furnished by the study of the influence of an external electric field. From the point of view of our ideas, this influence reduces to a deformation of the potential distribution in the surface layer. This distribution is shown qualitatively in Fig. 6. The dotted curve gives the potential distribution in the absence of an external field, the solid curve—in the presence of a field. The behavior of the potential curve of interest to us can be given in a very good approximation. The upper part of this curve is determined by the force of the electric image (i.e., by the force with which the conducting plane acts on a point charge), whose potential is

\[ U_B=-\frac{e^2}{4x}. \tag{1a} \]

Fig. 6. Change in the shape of the potential barrier by an external electric field

Since only the upper part of the curve is essential, the calculations may be carried out under the assumption (which

* See the article by I. Tamm \(^{12}\), in which an exact definition is given of the quantities under discussion. Using the results of Part IV of the present article, we could say that the energy of the absolute zero gives the width of the occupied part of the critical region of electron states, whereas \((\varepsilon_0+\chi)\) coincides with the lower edge of this region; \(C\) is the mean value of the potential for an electron that has flown in from outside. These two quantities by no means need coincide.

** See Nordheim’s communication \(^{14}\). At the present time the author is inclined to ascribe the principal role to the change of the work function with temperature. See also the work of Zwikker \(^{15}\) on this.

gives, apparently, a quite good approximation) of the following form of the curve: on the right—consisting of the potential of the electric image forces and a constant value \(C\) (a jump at the surface), while on the left—passing with a kink into a constant value of the potential inside the metal*.

Thus, in the absence of an external field we may put

\[ U=0 \text{ for } x<x_0;\quad U=C-\frac{e^2}{4x}\text{ for }x>x_0. \tag{1b} \]

The quantity \(x_0=\dfrac{e^2}{4C}\) enters here as a parameter.

When an external field \(F\) is applied in the outer space (this field may always be regarded as homogeneous over small regions), the complete course of the potential will be

\[ U=0 \text{ for } x<x_0;\quad U=C-\frac{e^2}{4x}-xF \text{ for }x>x_0;\quad x_0=\frac{e^2}{4C}, \tag{2} \]

as is also shown in Fig. 6. The lower kink, of course, has no physical reality; however, it has no effect on the results of the calculation.

The influence of the external field reduces, first, to a lowering of the work function by the amount \(e\sqrt{F}\). This is the well-known Schottky correction**, which is very well confirmed in the region of not too strong fields. Secondly, there arises the possibility of emission of slow electrons as a result of the effect analyzed in § 2. This effect can reach an appreciable magnitude only when the width of the potential barrier is not very large, which will occur only at sufficiently large fields. The fields required for this, as we shall see below, are so much larger than those at which the Schottky correction was observed that these two effects do not overlap. At very strong fields the effect of the “transparency” of the barrier becomes predominant, so that electron emission can occur at any low temperature; the possibility of a cold discharge is obtained.

In the limiting case \(T=0\), the distribution is given by (4a) of § 1. The transition coefficient is determined by (5) of § 2. In this case, to a first approximation, the electric image forces may be disregarded altogether*** and the calculation may be carried out under the assumption of a triangular poten-

* Zwickker\(^ {16}\) showed that the potential curve is not changed substantially by the space charge. See also the works of Frank,\(^ {17}\) which showed that this result is almost unchanged also when using quantum mechanics and the new statistics. Therefore, in what follows we shall not enter into the details of the influence of the space charge.

** See, for example, the article cited in the preceding paragraph.

*** In more detail in the work of Fowler and Nordheim\(^1\), p. [[unclear: page/reference]]. The correction introduced by taking the image forces into account was calculated by Nordheim\(^ {18}\). Its influence proved to be indeed small.

...tial barrier shown in Fig. 5. Using formula (7a) of § 2, we obtain, according to § 1 (1), for the emission current the expression:

\[ i=e\frac{2\pi Gm}{h^3}\,4\int_0^{\varepsilon_0}(\varepsilon_0-W)\, e^{-\frac{4\pi}{h}\frac{[2m(C-W)]^{3/2}}{3Fm}}\,dW . \]

An approximate evaluation of this expression gives*

\[ i=\frac{Ge}{4\pi h\chi}\,F^2 e^{-\frac{4\chi\lambda^{3/2}}{3F}}; \qquad \chi^2=\frac{8\pi^2 m}{h^2}, \tag{3} \]

which coincides with the law found by Millikan and Lauritsen[^19]

\[ i=\mathrm{const}\,F^2e^{-\frac{\alpha}{F}} . \]

Substituting numerical values for the constants and putting \(G=2\) (\(i\)—in A/cm\(^2\), \(\chi\)—in volts, \(F\)—in V/cm), we find

\[ i=6.2\cdot10^{-6}\frac{F^2}{\chi} e^{-\frac{6.8\cdot10^7\chi^{3/2}}{F}} . \tag{3a} \]

Noticeable emission occurs for values \(F\sim 10^7\) V/cm. In experiment, however, discharge begins at lower fields (of the order of \(10^6\) V/cm), and the magnitude of the required field turns out to depend substantially on the preliminary treatment of the surface. Accordingly, it is necessary to suppose that under ordinary conditions cold emission is determined not by the constants of the pure metal, but is connected with the presence of especially active spots on the surface. At such spots either the work function is lowered (a decrease from 4 to 1 V could already give a factor of 10), or, owing to surface irregularity, fields arise there that are considerably higher than follows from the geometry of the experimental apparatus. The latter effect can also be analyzed quantitatively,[^20] and here too one may assert that the theory correctly describes the observed phenomena.

At higher temperatures, fast electrons begin to play a role; these pass more easily through the barrier and thereby increase the total emission. Taking this circumstance into account,

* In Fowler and Nordheim there enters the factor \(\varepsilon_0^{1/2}/C\lambda^{1/2}\) instead of \(1/\lambda\) in (3). This wholly inessential difference is explained by the use of formula (7b) of § 2 instead of (7a). The expression given here is more convenient, since it does not contain the insufficiently well-known quantities \(\varepsilon_0\) and \(C\).

Houston* found, as a further approximation instead of (3), the following expression:

\[ i=\frac{Ge}{4\pi h\gamma}\,e^{-\frac{4\chi^{3/2}}{3F}} \left(F^2+\frac{32mk^2}{3h^2}\chi T^2\right). \tag{4} \]

On substituting numerical values, however, it turns out that the correction term manifests itself appreciably only at high temperatures, of the order of \(1000^\circ\) and above, so that cold emission may practically be regarded as independent of temperature. At still higher temperatures the phenomenon under consideration is joined by the ordinary thermionic emission, which is superposed on the effect produced by the field, and indeed additively \(^{22}\).

§ 5. Photoelectric Effect

The special properties of the Fermi–Dirac distribution make it possible to understand, as Fowler \(^{23}\) first showed, the fundamental peculiarities of the photoelectric effect in metals. The latter consists in the fact that an electron receives from an incident light quantum the energy \(h\nu\), which is transformed into kinetic energy and enables the electron to overcome the potential barrier at the surface.

The number of electrons with energy greater than the critical value \(\varepsilon_0\) will, however, be vanishingly small in comparison with the number of electrons having energy less than \(\varepsilon_0\). If we also assume that the probability of transfer of the photon energy does not depend very strongly on the kinetic energy of the electrons (and this, as we shall see below, can indeed be justified), then we arrive at the conclusion that an appreciable photocurrent can arise only in the case where the energy of the incident photons exceeds \(C-\varepsilon_0=\chi\); and this means that the work function of the photoelectrons is identical with the work function of the thermoelectrons. Thus we obtain a natural interpretation of the well-known experimental fact. The sudden appearance of a photocurrent at the threshold frequency of the photoelectric effect occurs because, at the critical photon energy, the number of electrons capable of leaving the metal increases extremely rapidly.

From this quite crude picture one can also derive another consequence. At higher temperatures we can no longer expect the existence of a very sharp threshold of the photoelectric effect, since a certain number of electrons (namely, with energy \(\varepsilon>\varepsilon_0\)) could be

* See Houston’s paper. \(^{21}\) Here we have again replaced the factor \(\dfrac{\varepsilon^{1/2}}{C\chi^{1/2}}\) by \(\dfrac{1}{\gamma}\).

For a more detailed discussion of the phenomenon, see Nordheim (Phys. Z., l. c.); there the question of the transition of a cold discharge into thermionic emission is also treated.

torn out of the metal by smaller quanta. This effect, as we shall see further on, is amenable to a more exact analysis. With a gradual increase in the frequency \(\nu\) of the incident light, we must expect that the number of electrons capable of leaving the metal will at first also increase strongly. It will be equal to the number of those electrons whose energy lies in the interval between \(C-h\nu\) and the maximum value \(\varepsilon_0\) (Fig. 7). However, beginning with \(h\nu>C\), the number of electrons of interest to us ceases to increase (at any rate so long as ionization processes on tightly bound electrons do not occur). Since the probability of an elementary act in all photoprocesses decreases with increasing \(\nu\), on passing through the boundary \(h\nu=C\) the photocurrent should decrease. We thus obtain the possibility of a selective photoeffect. The maximum should lie near \(h\nu=C\), which in fact is approximately observed for alkali metals; for other metals (which all have larger \(C\)) this maximum is shifted into the far ultraviolet and therefore cannot be observed.

Fig. 7. Energy relations in the photoeffect

Scattering processes must play a comparatively very insignificant role here, since the mean free path of electrons in a metal is sufficiently large. The circumstance that photoelectrons are distributed over a very wide interval of velocities is connected not with scattering processes inside the metal, as was previously supposed, but with the fact that from the very beginning the electrons are already distributed over a broad region of initial states with different energy values.

For a more exact investigation of these phenomena, more detailed statements about the elementary acts are necessary. However, the general conclusions made above remain entirely valid.

At the present time there is already a large number of works\(^{24}\) on this question, which lead to the following picture of the phenomenon. The probability of the elementary process of absorption of a light quantum by an electron, according to the general laws of quantum mechanics, is proportional to the square of the matrix element

\[ M_{\mathbf{k}\mathbf{k}'}=\int \psi_{\mathbf{k}'}(\mathbf{A}\operatorname{grad}\psi_{\mathbf{k}})\,dV. \tag{1} \]

Here \(\mathbf{A}\) is the vector potential of the incident light wave, i.e.

\[ \mathbf{A}=\mathbf{a}e^{2\pi i\nu t};\qquad \mathbf{E}=-\frac{\partial \mathbf{A}}{\partial t};\qquad \mathbf{H}=c\,\operatorname{rot}\mathbf{A}, \]

where \(\mathbf{a}\) is a constant amplitude vector, \(\psi_{\mathbf{k}}\) is the eigenfunction of the initial state, and \(\psi_{\mathbf{k}'}\) of the final state of the electron, pri-

rather, the energies of these two states differ precisely by the quantity \(h\nu\). The integration extends over all space. If for \(\psi_k\) one substitutes the expression for a simple plane wave [Part 1, § 6 (2)], then from (1) we obtain, to within a constant normalizing factor, the integral over the spatial coordinates

\[ M_{kk'} \sim \int (aK)e^{2\pi i r\left(\frac{K-K'}{K}\right)}\,dV = 0, \tag{1a} \]

which is equal to zero as a consequence of the pure periodicity of the integrand. This corresponds to the fact that in reality a free electron does not absorb, but only scatters. True absorption can occur only when there is a definite binding of the electron in the metal. This binding is already brought about by the presence of the potential jump at the surface. Therefore here, as the solution, one takes not a simple plane wave, but the combined eigenfunction considered in § 2, i.e., for the initial state of an electron inside the metal—a wave with complete internal reflection from the surface (with allowance for the exponentially decreasing part adjoining it in the external space), and for the final state—a correspondingly chosen combination of incoming and outgoing waves. If such eigenfunctions are used, then the integral (1) already gives a value different from zero, and a quite definite finite yield is obtained. As Tamm and Shubin have shown, the correct order of magnitude is thereby obtained. Knowing the probability of an elementary act of excitation of an electron, the total emission is found by integrating over all the electrons of the Fermi distribution.

From these, for the time being still only qualitative, considerations there immediately follows a series of characteristic features of the photoeffect. In planes parallel to the surface of the metal, the eigenfunction behaves like an ordinary plane wave (as a consequence of the separation of variables indicated in § 2), and the whole problem may be regarded as one-dimensional. As a result, only the normal component of the motion will enter everywhere, and only this normal component will receive all the energy \(h\nu\). On the same grounds, only the normal component of the light vector will be effective (because in the scalar product \(A\operatorname{grad}\psi_k\) there remains only the term \(A_x\dfrac{\partial\psi_k}{\partial x}\)). We obtain the well-known vectorial dependence for the so-called selective photoeffect, which, as we see, is determined by those bindings imposed on the electrons by the potential jump at the surface.

The best test of these considerations is achieved by studying the temperature dependence of the photocurrent near the threshold of the effect. If the frequency of the incident light differs only slightly from the frequency

...boundary, then only electrons in a certain small interval of velocities can be leveled. As a consequence, here one may neglect the dependence of the probability of leveling on the frequency. Complete photoemission in this case must be simply proportional to the number of electrons in the corresponding initial state, i.e., to the number of electrons* for which the energy \(W\) of the normal component of motion is greater than \(C-h\nu\).

According to § 1 (2), up to a certain constant we have

\[ i=\operatorname{const} kT \int_{C-h\nu}^{\infty} W^{-\frac12} \ln\left(1+e^{-\frac{W-\varepsilon_0}{kT}}\right)dW = \]

\[ =\operatorname{const} kT \int_{0}^{\infty} [x+(C-h\nu)]^{-\frac12} \ln\left(1+e^{-\frac{x+(\chi-h\nu)}{kT}}\right)dx, \tag{2} \]

where \(\chi\) is the work function. In the case when \(h\nu \sim \chi\), the terms of the integral will have a noticeable value only for \(x \sim kT\). Since under these same assumptions \(C-h\nu \gg x\), the quantity \(x\) in the square brackets may be neglected. Then, introducing the notation

\[ -\frac{\chi-h\nu}{kT}=\delta;\qquad z=\frac{x}{kT}, \tag{3} \]

we finally obtain

\[ i=\operatorname{const}(kT)^2(C-h\nu)^{-\frac12} \int_{0}^{\infty}\ln(1+e^{-z+\delta})\,dz. \tag{4} \]

To calculate the integral one may use expansion in a series; in doing so, two cases should be distinguished:

1. For \(\delta \leqslant 0\), i.e. for \(h\nu<\chi\) (the photoelectric-effect boundary has not yet been fully reached), we expand \(\ln\) in a series and integrate term by term. Then

\[ i=\operatorname{const}(kT)^2(C-h\nu)^{-\frac12} \left\{e^\delta-\frac{e^{2\delta}}{2^2}+\frac{e^{3\delta}}{3^2}-\ldots\right\}, \tag{5a} \]

and this expression in the limit \(T=0\) gives, as was to be expected,

\[ i=0. \tag{5b} \]

1. For \(\delta \geqslant 0\), i.e. for \(h\nu \geqslant \chi\) (the photoelectric-effect boundary has been passed), it is convenient to divide the region of integration into parts.

We have:

\[ \int_{0}^{\infty}\ln(1+e^{-z+\delta})\,dz = \int_{0}^{\delta}\ln(1+e^{-z+\delta})\,dz + \int_{\delta}^{\infty}\ln(1+e^{-z+\delta})\,dz = \]

\[ = \int_{0}^{\delta}(\delta-z)\,dz + \int_{0}^{\delta}\ln(1+e^{-z})\,dz + \int_{0}^{\infty}\ln(1+e^{-z})\,dz = \]

\[ = \frac12\delta^2 -\left(e^{-\delta}-\frac{e^{-2\delta}}{2^2}+\frac{e^{-3\delta}}{3^2}-\ldots\right) +\frac{\pi^2}{6}. \]

* Here, evidently, what is essential is precisely the number of electrons in a definite state, but not the number of electrons § 1 (3) passing through unit surface area.

The last result was obtained by expanding again \(\ln\) in a series, taking into account § 8 (8b) of Part I, and integrating the expansion term by term. Hence we find

\[ i=\operatorname{const}(kT)^2(C-h\nu)^{-\frac12} \left\{ \frac{\pi^2}{6}+\frac{\delta^2}{2} -\left(e^{-\delta}-\frac{e^{-2\delta}}{2^2}+\cdots\right) \right\} \tag{6a} \]

For \(T=0\) (i.e., for \(\delta\to\infty\)) the expression obtained gives

\[ i=\operatorname{const}\,\frac{(h\nu-\chi)^2}{(C-h\nu)^{\frac12}} . \tag{6b} \]

As Fowler showed \(^{25}\), formulas (5a) and (6b) very well describe the experimentally found dependence of the photoeffect on \(T\) and \(\nu\) for K, Ag, Au, Ta, and Sn. If, however, one makes the assumption that not only the normal component of the electron velocity changes, but others as well, the agreement of the theory with experiment is lost.

The ordinary photoeffect discussed above is a surface effect, which will always occur in pure metals as a consequence of the presence of a potential jump at the surface. For such a photoeffect, what is essential is the behavior of the eigenfunctions only in the immediate vicinity of the surface. It therefore becomes quite understandable that the photoeffect must depend very strongly on the structure of the surface. In particular, one may suppose that a layer of foreign atoms on the surface of a metal very strongly changes the course of the potential and thereby also the eigenfunctions. Thus a very strong influence of the surface structure is not surprising. At the same time, however, an exact calculation of this influence is made very difficult, since here we can no longer in any case expect that the assumption of a sharp potential jump will lead us to a correct description of the experimental data. We shall therefore not attempt to carry out any calculations here.

Nevertheless, it may be said that the shape of the photoelectric-yield curve as a function of frequency is, in its essential features, determined by the Fermi distribution. This follows first of all from the good results obtained above for phenomena near the threshold of the photoeffect. As for the subsequent course of the dependence, we refer to Fig. 1 of the work of Tamm and Shubin. The absolute magnitude of the maximum of the curve (and its position, if the work function is also changed by the surface layer) is determined by the unknown matrix element ((1)) and is very sensitive to all changes in the course of the potential inside the surface layer.

The experimentally obtained curve of photoelectron yield proves to be somewhat more compressed in comparison with that which is calculated from the Fermi distribution for free electrons. In this one might see an indication of the existence of a coupling of the electrons with the lattice (deviations from the properties of free electrons). The assumption of such not entirely free electrons, as this

will be shown in Part IV of the present article, always leads to a compression of the distribution curve, i.e., to a lowering of the zero-point energy.

Apart from the surface effect discussed above, one can also obtain a certain volume effect, if one takes into account that the eigenfunction inside the metal differs in its properties from a plane wave. Thus, for example, if one uses the Bloch eigenfunction (Part IV, § 2), then possibilities arise for additional transitions. These latter are responsible for the ordinary absorption and dispersion[^26]. According to Tamm and Shubin, these same processes may also take part in photoemission, and they play a noticeable role only at high frequencies. It further turns out that the influence of these processes will be stronger for heavy metals (Ag, Au, etc.) than for the alkali metals. Moreover, since deeper layers of the metal are involved here (the penetration depth of the optical wave is of the order of \(10^{-5}\) cm, which is greater than the mean free path of the electrons), we must also expect a greater role for scattering processes in the volume effect. From this it is also clear that the influence of the orientation of the electric vector and of the incident wave must here diminish. However, we still do not yet have a completed solution of the problem.

§ 6. Contact Potentials. Concluding Remarks

Let us now pose the question of the conditions under which two metals in contact with one another are in electrical equilibrium (it is assumed here that both metals are maintained at the same temperature; the theory of thermoelectric phenomena will be given in Part III). So long as the two metals are isolated from one another, each of them is characterized by its potential well, filled with electrons up to the critical level \(\varepsilon_0\). The relative position of the potential wells is not yet determined, since it depends on those charges which are present on the metals. The schematic Fig. 8a, in which the course of the potential is shown graphically, illustrates this case. Inside the metals (regions I and II) there are depressions of depths \(C_1\) and \(C_2\), which are

Fig. 8. Distribution of the potential between two metals

measure of the potential jump at the surface of the metal. Between them we find a linear course of the potential (it is assumed that the surfaces of the pieces are flat and parallel to one another), so that a potential “mountain” arises between the metals.

When the metals are brought closer together, the height of the potential mountain will become smaller and smaller and, when the metals come into direct contact, will disappear altogether; this obvious result follows at least from the experimental fact that the transition resistance can be lowered arbitrarily. In this case the electrons can pass freely from one metal into the other, and consequently, in the end a definite state of equilibrium must set in, characterized by a quite definite mutual arrangement of the potential wells.

This state of equilibrium is determined by the fact that in both directions across the separating surface the same number of electrons must pass. It is quite clear that this condition will certainly not be fulfilled if the bottoms of the wells are at the same level (Fig. 8b); in this case an enormous excess of electrons could pass from the metal with the larger zero energy. Therefore, when the metals are in contact, there must remain between them a certain internal potential difference. One may immediately expect that equilibrium will be reached when the electrons with the maximum energy of the absolute zero are at the same level (Fig. 8c). Indeed, this proposition can be proved at once.

The number of passing electrons belonging to a definite energy interval is given by § 1 (3). In this relation \(W\) is the energy of the normal component of the motion, measured from the level of the bottom as from zero. If we now denote the potential of the bottom by \(\varphi\), then in the indicated expression \(W\) must everywhere be replaced by \(W-\varphi\). Hence it is immediately seen that equality of both electron fluxes, and moreover for each of the energy intervals separately, will occur when

\[ \varphi_1+\varepsilon_0^{(1)}=\varphi_2+\varepsilon_0^{(2)}; \tag{1} \]

whence, for the internal potential difference, we obtain

\[ \varphi_1-\varphi_2=\varepsilon_0^{(2)}-\varepsilon_0^{(1)}, \]

which proves the assertion made (electrons of the metal with the larger \(\varepsilon_0\), having an energy smaller than \(\varepsilon_0^{(1)}-\varepsilon_0^{(2)}\), will at all times undergo total reflection). The internal potential difference cannot be measured directly experimentally; it figures here as a purely auxiliary quantity. However, the magnitude of the internal potential difference plays a definite role in thermoelectric phenomena.

Relation (1) can also be justified by a purely thermodynamic route. We saw [Part I, § 5 (20) and § 9 (7)] that \(\varepsilon_0\) is nothing other than

chemical potential of the electron in the metal. We could regard both metals as two electron phases in equilibrium with one another. If, in the transition from one phase to the other, a certain work is performed at the expense of potential energy, then it is known that the conditions for thermodynamic equilibrium of two such phases require that the sum of the potentials (chemical and mechanical, and in our case—chemical and electrical) be the same in both phases. This directly leads to relation (1).

Since the height of the potential wells of the two metals is different, there must also arise between the metals an external potential difference, whose magnitude, as is especially clear from Fig. 8c, is equal to

\[ C_1-\varepsilon_0^{1)}-\left(C_2-\varepsilon_0^{2)}\right)=\chi_1-\chi_2. \tag{2} \]

Fig. 9. Circuit of different metals and the distribution of potential in it

Thus one obtains a natural explanation of the appearance of a contact potential difference (the Volta effect), whose magnitude turns out to be equal to the difference of the work functions of thermoelectrons. This is in good agreement with experiment (it should be borne in mind that it is not easy to determine contact differences experimentally, since surface contaminations, which strongly affect the magnitude of the work function, are very difficult to remove at room temperatures).

The external potential difference that arises must be understood in the following way. If one imagines, for example, a ring made of two metals (Fig. 9a), which is not closed at one place, then between the points \(B\) and \(C\) above the surface of the metal the indicated potential difference arises. The course of the potential (for example along the dashed line in Fig. 9a) is shown in Fig. 9b. For a complete circuit of the electron along the entire dashed circle the resultant work is, of course, equal to zero. It should be especially emphasized that in all arguments of this kind it is necessary to distinguish internal potentials from external ones*.

* A deeper analysis of the question of the work function leads to a somewhat different treatment of contact potentials (see the work of I. Tamm and D. Blokhintsev \(^{27}\)). However, relation (2) remains valid as before.

It should perhaps also be noted that metallic contact at point A is not at all necessary. Above the surface of any metal there is an electronic atmosphere, whose density, under normal conditions, decreases very rapidly. Thus, with sufficiently close approach of the metals, even before their direct contact, electrons could pass from one metal into the other. This effect, if only it is complicated appreciably by exchange interaction, is already sufficient to ensure the establishment of equilibrium values of the potential.

Up to now we have considered the theory of such phenomena for whose explanation it was sufficient to know only the equilibrium distribution of the electrons. We shall consider one more example: the explanation of the temperature-independent paramagnetism of the alkali metals.^28 The picture which we used in the preceding exposition is known to be incomplete, since until now we have left out of consideration one of the fundamental properties of the electron—its magnetic moment. As a result of taking this property into account, only the statistical weight \(G\) was introduced by us into the calculations. However, if the truly free electrons are analogous to small magnets, we should expect that the enormous number of elementary magnets which we had in metals would have to produce strong paramagnetism; the latter would in fact be observed if classical statistics were correct. However, the degeneracy of the electron gas prevents the ordering of the electronic magnets and thereby eliminates the full manifestation of this effect. With ordered orientation of the magnets, the number of cells of phase space could no longer contain a multiple number of electrons, and, consequently, the value of the absolute-zero energy would have to increase very greatly. As a result it turns out that only an insignificant part of this paramagnetism remains, which proves, within wide limits, to be independent of the temperature. It is precisely this phenomenon, which had earlier seemed quite mysterious, that provided one of the most convincing proofs of the correctness of the new theory.

In order to give a general survey of the results obtained, we shall conclude by listing the phenomena discussed.

1. Broadening of Compton scattering on the electrons of a metal (DuMond). Requires a broad distribution of electron velocities in the metal.

2. Thermionic emission, Richardson’s formula. Gives a new interpretation of the work function and, together with electron diffraction, makes it possible to estimate the magnitude of the potential jump at the surface.

3. Cold emission. The dependence of the emission on the field is correctly given. The temperature dependence of the effect and the additivity of thermionic emission indicate the presence of a large number of slow electrons and a small number of fast electrons excited thermally.

4. Photoelectric effect. A justification is given for the fact that the photoelectric and thermoelectric work functions coincide. The dependence of the photoelectric effect on frequency and temperature is explained, and the possibility of explaining the selective effect is indicated.

5. Contact potentials. The magnitude of the contact potential difference proves to be equal to the difference of the thermionic work functions.

6. Paramagnetism of the alkali metals. Requires for its explanation a strong degeneracy of the electron gas.

THEORY OF THE METALLIC STATE

7. Specific heat of metals. It should be recalled once again that this difficulty of the classical theory is completely removed by the degeneracy of the electron gas.

All these effects can be quite naturally interpreted with the aid of the assumption that, even at \(T = 0\), the electrons are distributed over a broad interval of kinetic energies. One may confidently assert that the width of this interval is of the same order as that calculated from Fermi statistics for free electrons. It may perhaps also be said that there are many indications that the width of this interval is somewhat smaller than what is obtained for free electrons (Compton effect, photoelectric effect, paramagnetism). A final opinion, however, cannot yet be formed here.

It is possible that some parts of the theory may not seem entirely convincing (in the details many questions still remain to be solved; this is especially true of the numerical values of the quantities); nevertheless, the totality of all the material unambiguously decides the question in favor of the basic assumption we have made. In any case, we obtain a clear explanation of all the phenomena considered. Finally, it may also be asserted that the conception of completely free electrons already gives, to a certain extent, a good approximation to the actual conditions.

LITERATURE

1. K. E. Herzfeld, Sommerfeld-Festschrift 1928.
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