RECENT RESEARCH IN THE FIELD OF THE EXTERNAL PHOTOELECTRIC EFFECT

L. B. Linford, Princeton, U.S.A.*

The present article is an abridged translation of Linford’s review.** This review is divided mainly into two parts, the first of which is devoted to general theories of the external photoelectric effect, while the second describes phenomena of a more special character, whose interpretation goes beyond the applicability of general theories. We have omitted several paragraphs from the second part, partly because in them the author gives viewpoints that are already obsolete at the present time, and partly because the questions are sufficiently well discussed in M. V. Savostyanova’s article “The Selective Photoelectric Effect.” In addition, we have not translated the introduction, since it sets forth generally known facts. In the text of the first part of the article, which is presented very concisely, we have found it appropriate to insert a number of remarks and additions. Indeed, the theories of the photoelectric effect, and especially those among them that make use of the apparatus of quantum mechanics, are presented rather poorly in our review literature. In contrast to Linford, who for the most part confines himself to the statement of final results, we have tried throughout to give derivations, and in cases where this proved difficult, to indicate the general course of the solution. All translator’s additions are marked at the beginning and at the end with the sign (×).

C. Ch.

I. BASIC THEORIES OF THE PHOTOELECTRIC EFFECT

We shall begin with a survey of the most important and most fruitful theories of the external photoelectric effect. The majority of them must of necessity base the description on simplified models, as a result of which the range of their application is limited to clean surfaces and absolute zero temperature.

A. Classical Theories

In the electron theory of metals preceding Sommerfeld’s, the model usually adopted was that according to which, inside a metal, free electrons, about 1 in number for each atom, formed a “gas” with a Maxwell–Boltzmann distribution of velocities. This theory explained well the high electrical conductivity and high thermal conductivity of a metal, but led to excessively large values for the heat capacity. A number of attempts were made to overcome this difficulty, but they all proved, to a large—

* Translation and adaptation by S. V. Cherdyn’tseva.
** Reviews of Modern Physics, 5, No. 1, p. 34, 1933.

or to a lesser degree unsuccessful. Theories of the photoelectric effect based on the indicated old model of the metal have already proved capable of explaining the fact that the spectral threshold of sensitivity and the magnitude of the photoelectric emission vary little with temperature. This result is connected with the fact that the change in the value of the most probable electron energy in the temperature range used in photoelectric experiments is small in comparison with the work function.

The theoretical distribution function over the spectrum was calculated by Richardson¹, J. J. Thomson², and Uspensky³. Richardson’s curve forms, at the long-wave boundary, a finite angle with the abscissa axis. The curves of the other two authors are tangent to the abscissa axis at this point. All three curves give, near the threshold, a sharper rise with frequency than the experimental curves. Richardson’s equation has a maximum at \(\nu = {}^{3}/_{2}\nu_{0}\)*; Thomson’s equation—at \(\nu = 2\nu_{0}\). Selective maxima were found experimentally precisely in this region. The maximum of Uspensky’s curve lies at \(\nu = 9\nu_{0}\). Hughes and DuBridge⁴ point out that if Thomson’s premises are modified and it is assumed that the coefficient of absorption of light depends on the frequency, then one can obtain a distribution function over the spectrum that satisfactorily fits the experimental data near the threshold. However, this equation will no longer have a maximum. Thus none of the theories listed gives agreement with experiment over a wide interval of frequencies.

Richardson⁵ derived an equation for photoelectric emission from a surface subjected to black-body radiation, without making any special assumptions concerning the theory of metals or the process of electron escape. Richardson’s equation for thermoelectronic emission is:

\[ I = A T^{r} e^{-\frac{\varepsilon_{0}}{kT}}; \tag{1} \]

here \(I\) is the emission current density, \(A\) is a constant, \(T\) is the absolute temperature, \(k\) is Boltzmann’s constant, \(r\) is a constant, equal, depending on the assumptions made in the derivation, either to \(1/2\) or to \(2\). The new theory of metals gives the value \(2\), which is now regarded as correct.

Equation (1) can be derived without any assumptions about the mechanism of emission, by determining the number of electrons emitted by a surface placed inside a black-body cavity**. The escape of these electrons may occur by virtue of either thermoelectronic or photoelectric emission under the action of black—

* \(\nu_{0}\) is the frequency of the long-wave boundary. — S. Ch.

** Such a derivation of Richardson’s equation was given by Deshman⁶. Richardson’s equation in the form written is, strictly speaking, not quite correct, since \(E_{0}\) depends on temperature. But for pure metals the deviations are small. For our purposes it will be quite sufficient to write the equation in this simplified form.

of radiation of the cavity. We have here an equilibrium process, and one may expect that, so long as the equilibrium conditions are fulfilled, one and the same equation remains valid independently of where the surface emitting electrons under the action of black radiation is located. Deviations from this equation may therefore serve as a measure of deviations from the equilibrium state. In this case \(A\) will evidently become smaller; \(T\) will represent the temperature of the black body.

This equation was tested experimentally by Roe\({}^{7}\), who used black-body radiation in his experiments, and by Suhrmann\({}^{8}\), who calculated the total emission from the experimentally found distribution function over the spectrum. Let us denote this function by \(F(\nu)\), and the energy-distribution function in the spectrum of a black body according to Planck’s law by \(E(\nu,T)\); then the total emission \(I_c\) will be equal to:

\[ I_c=\int_{\nu_0}^{\infty} F(\nu)E(\nu,T)\,d\nu . \tag{2} \]

Graphical integration makes it possible to determine the total emission for different temperatures.

Plotting along the abscissa axes \(1/T\), and along the ordinate axes \(I_c/T^2\), Roe and Suhrmann obtain straight lines, whose slope gives a good value for the work function. Later Suhrmann\({}^{9}\) reported that still better straight lines are obtained if the constant \(r\) in equation (1) is taken to be approximately equal to 4, and not to 2. This, however, cannot be justified theoretically.

B. Quantum-Mechanical Theories

1. New electron theory of metals. Applying Pauli’s exclusion principle to the free electrons in a metal and the Fermi statistics following from it, Sommerfeld and others succeeded in constructing a new theory that makes it possible to explain the electrical conductivity, thermal conductivity, and other properties of metals without ascribing excessively high values to the heat capacity. This provided, together with the methods of quantum mechanics, a new and very fruitful way of describing phenomena connected with the electron emission of metals\({}^{10}\).

It will be appropriate to precede the exposition of the theories of the photoelectric effect with a brief summary of the results of the new theory of metals.

Let us suppose that the surface of the metal emitting electrons coincides with the \((xy)\)-plane of our coordinate system, and that the outward normal to the surface of the metal coincides with the positive \(z\)-axis. Let \(\xi,\eta,\zeta\) be the components of the velocity of a certain electron along the \(x\)-, \(y\)-, and \(z\)-axes, respectively. In order to calculate the emission from a unit surface of the metal, it is necessary to find the number of electrons \(N(W)\,dW\) for which the normal component of the energy

\(W=\dfrac{1}{2}m\xi^2\) * lies within the limits between \(W\) and \(W+dW\), falling on a unit area of the metal surface per unit time, multiply this number by the probability \(D(W)\) of passage of an electron possessing the normal component of energy \(W\) through the metal surface, and integrate over all values of \(W\). Thus, for the current density we obtain:

\[ I=\int_0^\infty N(W)D(W)\,dW. \tag{3} \]

According to Fermi statistics, the distribution function of electrons in a metal with respect to velocities has the form **:

\[ f(\xi,\eta,\zeta)\,d\xi\,d\eta\,d\zeta = \frac{2m^3}{h^3}\, \frac{d\xi\,d\eta\,d\zeta} {e^{\frac{\frac12 m(\xi^2+\eta^2+\zeta^2)-\varepsilon}{kT}}+1}, \tag{4} \]

where \(f(\xi,\eta,\zeta)\,d\xi\,d\eta\,d\zeta\) is the number of electrons per unit volume whose velocity components along the \(x\)-, \(y\)-, and \(z\)-axes lie, respectively, within the limits from \(\xi\) to \(\xi+d\xi\), from \(\eta\) to \(\eta+d\eta\), and from \(\zeta\) to \(\zeta+d\zeta\), and \(\varepsilon\) is the maximum electron energy in the Fermi distribution at \(0^\circ\mathrm{K}\), equal to

\[ \varepsilon=h\nu=\frac{h^2}{8m}\left(\frac{3n}{\pi}\right)^{2/3}; \tag{5} \]

here \(m\) is the electron mass, \(n\) is the number of free electrons per unit volume ***. The values of \(\varepsilon\) vary within the limits of, approximately,

* Although energy cannot be decomposed into components in the literal sense of the word, nevertheless this mode of expression has proved convenient and has become widespread, since it is not the total kinetic energy \(\varepsilon=\dfrac{1}{2}m(\xi^2+\eta^2+\zeta^2)\), but the quantity \(W\) introduced by us that determines the electron’s ability to be torn out from the metal surface.

* An exposition of the foundations of Fermi statistics and a derivation of formula (4) are given, for example, in the article by K. Darrow, Uspekhi fizicheskikh nauk*, vol. X, issue 2, p. 225, 1930.

*** It is not difficult to verify equality (5). For this purpose let us imagine a rectangular piece of metal whose sides are equal to \(l_1,l_2,l_3\). For simplicity, let us assume that inside the metal the potential is constant and equal to \(V_0\), and that at its boundaries there are infinitely high potential barriers. The Schrödinger equation for an electron located inside the metal will have the form:

\[ \Delta\psi+\frac{8\pi^2m}{h^2}(E-V_0)\psi=0 \tag{5a} \]

(\(E\) is the total energy of the electron). The boundary conditions will be: \(\psi=0\) at the boundary of the metal. To this one must also add the normalization condition

\[ \int_0^{l_1}\int_0^{l_2}\int_0^{l_3}\bar{\psi}\psi\,dx\,dy\,dz=1 \]

2 electron-volts for alkali metals to values exceeding 10 V for some heavy metals.

The function \(f(\xi,\eta,\zeta)\), defined by equality (4), at \(0^\circ\mathrm{K}\) is constant and equal to \(\dfrac{2m^3}{h^3}\) for electron energies
\[ \varepsilon=\frac{1}{2}m(\xi^2+\eta^2+\zeta^2)<\varepsilon; \]
whereas for energies exceeding this value it becomes zero. At higher temperatures the function \(f(\xi,\eta,\zeta)\) in the region \(\varepsilon\simeq\varepsilon\) falls exponentially from \(\dfrac{2m^3}{h^3}\) to zero, reaching half its value at \(\varepsilon=\varepsilon\).

The distribution function of electrons with respect to energy has the form:
\[ F(\varepsilon)\,d\varepsilon= \frac{8\pi m}{h^3}\, \frac{(2m\varepsilon)^{\frac12}\,d\varepsilon}{e^{\frac{\varepsilon-\varepsilon}{kT}}+1}. \tag{6} \]

(x) To obtain distribution (6) from distribution (4), it is sufficient to integrate expression (4) over the entire volume of velocity space to which the values of the energy from \(\varepsilon\) to \(\varepsilon+d\varepsilon\) correspond. For this purpose it is convenient to pass to spherical coordinates \(v,\varphi,\theta\), where
\[ v=\sqrt{\xi^2+\eta^2+\zeta^2}=\sqrt{\frac{2\varepsilon}{m}}— \]

(the bar above denotes complex conjugation). Equation (5a) is solved in an elementary way (by separation of variables). Its solution is written below [see formulas (13a) and (13b)]. From these formulas it follows that we obtain a series of discrete, although very closely spaced, allowed energy levels of the electrons. The quantities \(k_j\), related to the energy of the electron [see formulas (13c)—(13d)], in each allowed state must take one of the values of the series:
\[ k_j=\frac{\pi\cdot n_j}{l_j},\quad j=1,2,3;\quad n_j=1,2,\ldots \tag{5b} \]

In each such state there may be, according to the Pauli principle, at most two electrons (with opposite spins). At absolute zero all states corresponding to low energies are filled with electrons “to capacity,” while all higher energy levels contain no electrons at all. If in momentum space (or, more precisely, in wave-number space \(k_1,k_2,k_3\)) one draws a sphere of radius \(k\), corresponding to the maximum energy \(\varepsilon\) [i.e. \(\varepsilon=\dfrac{h^2k^2}{8\pi^2m}\), cf. (13d)], then all electrons will lie inside the first octant of this sphere, in which all \(k_j>0\), filling it with uniform density of two per cell of volume \(\dfrac{\pi^3}{l_1l_2l_3}\) [see (5b)]. Dividing the volume of the octant \(\dfrac{1}{6}\pi k^3\) by the volume of a cell and multiplying the quotient by 2, we obtain the total number of electrons in the metal, which, on the other hand, is equal to \(l_1l_2l_3 n\). Equating one expression to the other, we obtain:
\[ k=\sqrt[3]{3\pi^2 n}, \tag{5c} \]
and substituting (5c) into (13d), we find formula (5). — S. Ch.

the total velocity of the electron. Indeed, then we obtain:

\[ F(\varepsilon)\,d\varepsilon = \frac{2m^3}{h^3} \int_{\varphi=0}^{2\pi} \int_{\theta=0}^{\pi} \int_{v=\sqrt{\frac{2\varepsilon}{m}}}^{\sqrt{\frac{2(\varepsilon+d\varepsilon)}{m}}} \frac{v^2\sin\theta\,dv\,d\theta\,d\varphi} {e^{\frac{\frac{mv^2}{2}-\bar{\varepsilon}}{kT}}+1} = \]

\[ = \frac{2m^3}{h^3}\,4\pi\,\frac{2\bar{\varepsilon}}{m}\, \frac{\sqrt{\frac{2}{m}}\left(\sqrt{\varepsilon+d\varepsilon}-\sqrt{\varepsilon}\right)} {e^{\frac{\varepsilon-\bar{\varepsilon}}{kT}}+1}, \]

which coincides with (6). (x)

The sharpest difference between the classical theory and the new views consists in the fact that, according to the former, all electrons at absolute zero of temperature are at rest, whereas according to the new theory they possess energy whose magnitude reaches the relatively large value \(\bar{\varepsilon}\). With increasing temperature, according to the old theory all electrons increased their energy; according to the new theory, however, this occurs only for a small number of the fastest electrons, whose energy becomes greater than \(\bar{\varepsilon}\).

Fowler\(^{11}\) used the distribution (6) to construct a preliminary theory of the photoelectric effect. It had already been indicated that the electron’s ability to be torn out of the metal is determined solely by the normal component of the energy; Nordheim\(^{10}\) calculated the function \(N(W)\,dW\); it turned out to be equal to:

\[ N(W)\,dW = \frac{4\pi m}{h^3}\,kT\, \ln\left(1+e^{-\frac{W-\bar{\varepsilon}}{kT}}\right)dW. \tag{7} \]

(x) Indeed, the number of electrons per unit volume whose \(z\)-component of velocity lies between \(\zeta\) and \(\zeta+d\zeta\), according to formula (4), must be equal to:

\[ n(\zeta)\,d\zeta = d\zeta \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} f(\xi,\eta,\zeta)\,d\xi\,d\eta = \]

\[ = \frac{2m^3}{h^3}\,d\zeta \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} \frac{d\xi\,d\eta} {e^{\frac{\frac12 m(\xi^2+\eta^2+\zeta^2)-\bar{\varepsilon}}{kT}}+1}. \]

Introducing cylindrical coordinates \(\varphi,\rho,\zeta\) (where \(\xi=\rho\sin\varphi,\ \eta=\rho\cos\varphi\)), we have:

\[ n(\zeta)\,d\zeta = \frac{4\pi m^3}{h^3}\,d\zeta \int_{0}^{\infty} \frac{\rho\,d\rho} {e^{\frac{\frac{m}{2}(\rho^2+\zeta^2)-\bar{\varepsilon}}{kT}}+1}. \tag{7a} \]

Transforming the last integral to the variable \(y=\dfrac{m\rho^{2}}{2kT}\), we reduce it to the form:

\[ \frac{kT}{n}\int_{0}^{\infty} \frac{dy}{e^{\,y+\frac{W-\varepsilon}{kT}}+1}, \]

where \(W=\dfrac{1}{2}m\zeta^{2}\) is the normal component of the energy of the electrons under consideration. Introducing once more a new variable of integration according to the formula \(z=e^{\,y+\frac{W-\varepsilon}{kT}}\), we obtain:

\[ n(\zeta)\,d\zeta = \frac{4\pi m^{2}kT}{h^{3}}\,d\zeta \int_{e^{\frac{W-\varepsilon}{kT}}}^{\infty} \frac{dz}{z(z+1)} = \]

\[ = -\frac{4\pi m^{2}kT}{h^{3}} \ln\left(1+\frac{1}{z}\right) \bigg|_{e^{\frac{W-\varepsilon}{kT}}}^{\infty} d\zeta = \frac{4\pi m^{2}kT}{h^{3}} \ln\left(1+e^{-\frac{W-\varepsilon}{kT}}\right)d\zeta . \tag{7b} \]

The number of electrons per unit volume whose normal component of energy lies within the limits from \(W\) to \(W+dW\) must be equal to \(n(\zeta)\dfrac{d\zeta}{dW}\,dW\). Recalling the definition of the function \(N(W)\) given above, we easily verify the validity of the equality:

\[ N(W)=n(\zeta)\,\zeta\,\frac{d\zeta}{dW}=\frac{n(\zeta)}{m}. \]

Finally, we obtain Nordheim’s formula by substituting here, instead of \(n(\zeta)\), its expression (7b). (x)

For different intervals of values of \(\dfrac{W-\varepsilon}{kT}\), formula (7) may be replaced by one of the following three approximate formulas:

\[ N(W)\,dW=\frac{4\pi m}{h^{3}}(\varepsilon-W)\,dW \]

\[ \left(\text{for } \frac{W-\varepsilon}{kT}\ll 0\right), \tag{8a} \]

\[ N(W)\,dW=\frac{4\pi m}{h^{3}}\,kT\,dW \]

\[ \left(\text{for } \frac{W-\varepsilon}{kT}\simeq 0\right), \tag{8b} \]

\[ N(W)\,dW=\frac{4\pi m}{h^{3}}\,kT\,e^{-\frac{W-\varepsilon}{kT}}\,dW \]

\[ \left(\text{for } \frac{W-\varepsilon}{kT}\gg 0\right). \tag{8c} \]

At absolute zero we obtain:

$$ N(W)\,dW=\frac{4\pi m}{h^3}(\varepsilon-W)\,dW \quad \text{for } W<\varepsilon, \tag{9a} $$

$$ N(W)\,dW=0 \quad \text{for } W>\varepsilon. \tag{9b} $$

The difference in the form of the functions \(F(\varepsilon)\) and \(N(W)\) is illustrated in Fig. 1, where they are given for absolute zero (solid lines) and for a temperature of \(1000^\circ\text{K}\) (dotted line). \(W\) and \(\varepsilon\) are plotted on the same scale, but the scales for the ordinates \(F(\varepsilon)\) and \(N(W)\) are entirely different.

Fig. 1. Fermi distribution for electrons at absolute zero (solid line) and at \(1000^\circ\text{K}\) (dotted line): \(F(\varepsilon)d\varepsilon\) is the number of electrons with energy in the interval from \(\varepsilon\) to \(\varepsilon+d\varepsilon\); \(N(W)dW\) is the number of electrons with normal component of energy in the interval from \(W\) to \(W+dW\) and incident on a unit surface per unit time. For \(T>0^\circ\text{K}\), according to equation (9), \(N(W)\) for any value of \(W\) must be larger than for \(T=0\). The area under the curve \(N(W)\) is equal to the number of electrons falling on the surface in a time interval equal to 1 sec, and must increase with temperature. Nordheim11 draws the curve for \(T>0\) so that it lies below the curve for \(T=0\) at a value of \(W\) slightly smaller than \(\varepsilon\), and approaches the latter as \(W\) decreases. The same error was repeated by other authors (A. L. Hughes and L. A. Du Bridge, Photoelectric Phenomena, McGraw Hill, New York, 1932; S. Dushmann, Rev. Mod. Phys. 2, 381, 1930).

Fig. 2. Two kinds of potential barriers at the surface of a metal. \(OAB\)—a step barrier in the absence of an external field; \(OAC\)—the same in the presence of an accelerating external field; \(ODB\) and \(OES\) are the barriers created by electric image forces respectively in the absence and in the presence of an accelerating field.

To determine the transmission coefficient of the surface12 \(D(W)\), it is necessary to specify the form of the potential barrier retaining the electrons in the metal; then one must determine the wave function both inside and outside the metal. Usually the wave function is normalized so that the current density directed perpendicular to the surface, outward from the metal, is equal to unity, and one writes the functions with undetermined coefficients corresponding to the transmitted electron wave emerging from the metal outward, and the reflec-

LATEST STUDIES OF THE PHOTOELECTRIC EFFECT

wave returning into the metal. The coefficients must be chosen in such a way that the complete wave function and its first derivative are continuous everywhere. These conditions are sufficient for determining the coefficients in the expressions for the transmitted and reflected rays. The transmission coefficient is equal to the square of the absolute value of the coefficient in the term corresponding to the transmitted ray, provided that the analogous quantity for the incident ray is normalized to unity. The transmission coefficient thus obtained, being a function of the electron’s normal energy \(W\), depends on the form of the chosen potential barrier.

The simplest form of the potential barrier is a sudden increase of the potential from its value inside the metal to its value at infinity (\(OAB\) in Fig. 2). Closer to reality is a barrier satisfying the equation:

\[ V=h\nu_a-\frac{e^2}{4z}\quad (\text{for } z>z'), \tag{10} \]

which is derived under the assumption that at distances greater than \(z'\) (\(z'\) is of the order of atomic dimensions) the escaping electron experiences the action of the force of attraction to its electric image in the metal. The corresponding curve is denoted in Fig. 2 by the letters \(ODB\). In both cases the difference of potentials inside the metal and outside it is equal to \(h\nu_a=\varepsilon_a\). From the preceding it is clear that \(\varepsilon_a\) is equal to the sum of \(\bar{\varepsilon}\), i.e. the greatest electron energy in the Fermi distribution, and \(\varepsilon_0\)—the work function, i.e. the work which must be expended to tear from the metal an electron possessing the maximum energy \(\bar{\varepsilon}\):

\[ \varepsilon_a=\bar{\varepsilon}+\varepsilon_0;\qquad =h\nu_a=h\nu+h\nu_0. \tag{11} \]

Calculations by Nordheim and others have shown that for both barriers \(D(W)=0\) if \(W<h\nu_a\), and \(D(W)\) rapidly increases to unity when \(W>h\nu_a\). For the value \(W-h\nu_a\) corresponding to \(0.1\) electron-volt, we have \(D(W)>0.99\).

(x) We shall confine ourselves here to carrying out the calculation for the first case, as the simplest. Thus, suppose there is a potential barrier defined by the equalities:

\[ V=0\ \text{for } x<0,\qquad V=V_0\ \text{for } x>0,\qquad V_0>0. \]

In other words, at \(x=0\) we have a rectangular step, whose height is \(V_0\). The potential does not depend on \(y\) and \(z\), i.e. our problem is one-dimensional. Then the wave function of electrons having energy \(W\) will have the form:

\[ \text{I. } W>V_0: \]

\[ \text{a) } x<0\qquad \psi=A(W)e^{\,i\frac{2\pi p}{h}x}+B(W)e^{-i\frac{2\pi p}{h}x}, \]

\[ \text{b) } x>0\qquad \psi=C(W)e^{\,i\frac{2\pi p_0}{h}x}. \]

II. \(W<V_0\):

\[ \begin{aligned} \text{a) } x<0 \qquad \psi&=A'(W)e^{\,i\frac{2\pi p}{h}x}+B'(W)e^{-\,\frac{2\pi p}{h}x},\\ \text{b) } x>0 \qquad \psi&=C'(W)e^{\,\frac{2\pi p_0'}{h}x}. \end{aligned} \]

\[ p=\sqrt{2mW}\qquad (m\text{—mass of the electron}), \]

\[ p_0=\sqrt{2m(W-V_0)}, \qquad p_0'=\sqrt{2m(V_0-W)}. \]

(The factors depending on time have been omitted by us.) Recalling the statistical interpretation of the wave function and substituting the latter into the quantum-mechanical expression for the current density

\[ i=\frac{eh}{4\pi im}\left(\bar\psi\frac{d\psi}{dx}-\psi\frac{d\bar\psi}{dx}\right), \]

we find that the first term in the expression for \(\psi\) at \(x<0\) corresponds to the motion of electrons with momentum \(p\) from \(x=-\infty\) toward the barrier, and that, per unit time through a unit area perpendicular to the \(x\)-axis, there pass

\[ \frac{p}{m}A(W)\overline{A(W)} \]

electrons [the numerical value of the coefficient \(A(W)\) depends on how we agree to normalize the wave function]. In case I a part of these electrons, numbering

\[ \frac{pB(W)\overline{B(W)}}{m}, \]

will be reflected from the barrier and will move back from the barrier toward \(x=-\infty\) with momentum \(-p\) [see the second term in formula (Ia)], while the remaining

\[ \frac{p_0C(W)\overline{C(W)}}{m} \]

electrons will continue their motion in the same direction, i.e. from the barrier toward \(x=+\infty\), with momentum equal to \(p_0\) [see formula (Ib)]. In case II the probability of finding the electron anywhere to the right of the barrier will rapidly tend to zero with increasing distance [see formula (IIb)]. It is not difficult to verify formally that the current density will be the same on both sides of the barrier, i.e. that the number of incident particles is equal to the sum of the numbers of reflected and transmitted particles. Indeed, multiplying Schrödinger’s equation

\[ \Delta\psi+\frac{8\pi^2m}{h^2}(W-V)\psi=0 \]

by \(\bar\psi\), its complex-conjugate expression by \(\psi\), subtracting one from the other and integrating the result with respect to \(x\), we obtain:

\[ \bar\psi\frac{d\psi}{dx}-\psi\frac{d\bar\psi}{dx}=\text{const}; \]

i.e. \(i=\text{const}\). Computing, in case II, the current density at the points \(x=-\infty\) and \(x=+\infty\) and equating the results, we find that

\[ A(W)\overline{A(W)}=B'(W)\overline{B'(W)}; \]

consequently, in this case all electrons approaching the barrier will be reflected back.

The transmission coefficient \(D(W)\) of the barrier (i.e. the ratio of the number of electrons that have passed through the barrier to the number of electrons incident upon it) in case I will be equal to

\[ \frac{p_0}{p}\frac{C(W)\overline{C(W)}}{A(W)\overline{A(W)}}, \]

and in case II—to zero.

The reflection coefficient \(R(W)\) of the barrier (the definition is quite analogous) in case I is equal to \(\dfrac{B(W)B(\bar W)}{A(W)A(\bar W)}\), and in case II it is unity.

The ratio of the coefficients of the wave function can easily be found from the condition of continuity of \(\psi\) and \(\dfrac{d\psi}{dx}\) at the point \(x=0\). We obtain:

\[ A(W)+B(W)=C(W) \]

and

\[ p\{A(W)-B(W)\}=p_0 C(W), \]

and, consequently,

\[ A(W)=\frac{C(W)}{2}\left(1+\frac{p_0}{p}\right) \]

and

\[ B(W)=\frac{C(W)}{2}\left(1-\frac{p_0}{p}\right). \]

Hence, for case I we have:

\[ D(W)=\frac{4\frac{p_0}{p}}{\left(1+\frac{p_0}{p}\right)^2} \]

and

\[ R(W)=\left(\frac{1-\frac{p_0}{p}}{1+\frac{p_0}{p}}\right)^2 . \]

It is easy to see that as \(W-V_0\) increases upward from zero, \(D(W)\) increases very rapidly, tending to unity. (x)

If an accelerating field is applied to the surface, the potential barrier \(OAB\) takes the form \(OAC\), and the barrier \(ODB\) the form \(OEC\). In the first case the applied field should not affect the maximum height of the barrier; in the second case it lowers the height of the barrier and, correspondingly, the effective work function. As in the case of absence of an external field, \(D(W)\to 1\) when \(W\) exceeds the greatest height of the barrier; but in the case when \(W\) is less than the top of the barrier, \(D(W)\) remains finite, although small. This possibility for an electron to pass from one classically allowed region into another through a region where its energy is negative explains the emission of electrons by cold metallic surfaces in strong fields, as well as certain other phenomena which classical theory was unable to explain.

(x) The qualitative existence of this effect, which has no analogue in classical mechanics, is not difficult to verify by considering a barrier in the form of a rectangular wall of width \(a\), set perpendicular to the \(x\)-axis:

\[ V=0 \text{ for } x<0;\quad V=V_1 \text{ for } 0<x<a; \]

\[ V=V_0 \text{ for } x>a;\quad V_1>V_0>0, \]

If \(W < V_0\), then in the space to the right of \(x = a\) the wave function will again decrease according to an exponential law and \(D(W)\) will be equal to zero. If, however, \(V_0 < W < V_1\), we shall have:

\[ \psi(x)=A_1(W)e^{i\frac{2\pi p}{h}x}+A_2(W)e^{-i\frac{2\pi p}{h}x} \qquad (\text{for }x<0), \]

\[ \psi(x)=B_1(W)e^{\frac{2\pi p'}{h}x}+B_2(W)e^{-\frac{2\pi p'}{h}x} \qquad (\text{for }0<x<a), \]

\[ \psi(x)=C(W)e^{i\frac{2\pi p_0}{h}x} \qquad (\text{for }x>a), \]

where

\[ p'=\sqrt{2m(V_1-w)}, \]

and \(p\) and \(p_0\) have the same values as in the case of the step barrier. Determining the ratio of the coefficients of the wave function from the condition of continuity of \(\psi\) and \(\dfrac{d\psi}{dx}\) at the points \(x=0\) and \(x=a\), we obtain for \(D(W)\) a rather complicated expression, different from zero for any finite width and height of the barrier. In an analogous way it is not difficult to analyze the case when \(W>V_1\): as \(W\) increases from \(V_1\) to infinity, \(D(W)\) tends to unity, but not monotonically, as in the case of the step barrier, but with oscillations. (x)

Substituting the values \(N(W)dW\) and \(D(W)\) into equation (5) and integrating, one can obtain Richardson’s equation for thermionic emission and the empirically found equation for the autoelectronic effect.

The problem of photoelectric emission proves to be more complicated, since here it is necessary to determine the influence of the incident light on the distribution of electrons over velocities and to use the new distribution function obtained in order to determine the emission. The problem breaks up into the following stages: 1) to write the equation for light penetrating into the metal; 2) to determine the probability that an electron with velocity components \(\xi,\eta,\zeta\) will absorb a quantum of light of frequency \(\nu\) and finally acquire a normal component of energy equal to \(W'\) (this probability will depend, among other factors, on \(\xi,\eta,\zeta,\nu\), the intensity and polarization of the light, and the character of the electrostatic field in the neighborhood of the electron); 3) to calculate \(N'(W')dW'\), i.e. the number of excited electrons with normal component of energy lying within the limits between \(W'\) and \(W' + dW'\), which fall on unit surface area per unit time. The density of the emission current will be equal to:

\[ I=\int_0^\infty N'(W')D(W')\,d(W'). \tag{12} \]

Since it is impossible to obtain an exact solution according to this scheme, existing theories are forced to resort to various methods of approximation. We shall give a brief survey of the most important of them.

2. Wenzel’s Method

In the solution of the problem of photoelectric emission given by Wenzel[^13], the interaction of radiation and electrons was for the first time treated in a quantum-mechanical way. Wenzel considers free electrons enclosed in a rectangular piece of metal with sides equal to \(l_1, l_2, l_3\); the wave function inside this “box” is equal to*:

\[ \psi_0 = u_0 e^{\frac{2\pi i}{h}(mc^2+\varepsilon_0)t}, \tag{13a} \]

where

\[ u_0=\left(\frac{8}{l_1l_2l_3}\right)^{1/2}\sin k_1x\cdot\sin k_2y\cdot\sin k_3z, \tag{13b} \]

\[ k_j=\frac{\pi n_j}{l_j},\qquad j=1,2,3,\quad n_j=1,2,\ldots, \tag{13c} \]

\[ \varepsilon_0=\frac{h^2k^2}{8\pi^2m},\qquad k^2=k_1^2+k_2^2+k_3^2. \tag{13d} \]

Here \(\varepsilon_0\) is the kinetic energy of the electron before excitation by light. Wenzel further assumes that light has, in the metal, classical attenuation with absorption coefficient \(a\), so that the electric vector of the light wave in the metal, where the coordinate \(z\) is positive, is given by the expression**:

\[ \mathfrak{E}=\mathfrak{E}_0 e^{-az}\cos\{2\pi\nu t-(\mathfrak{K}\mathfrak{r})\}. \tag{14} \]

By means of this assumption he attempted to overcome the difficulties connected with the law of conservation of momentum. We shall return to this circumstance in more detail below, in connection with the theory of Tamm and Shubin.

Wenzel regards the electric vector as the cause of the perturbation of the electron wave function (13a) and obtains a new wave function:

\[ \psi=\psi_0+\psi_1. \tag{15} \]

(x) The classical perturbation energy of the problem under consideration is equal to \(H'=-e\mathfrak{A}\mathfrak{v}\), where \(e\) is the electron charge, \(\mathfrak{A}\) the vector potential.

* Equations (13) represent a solution of the Schrödinger equation

\[ \Delta\psi_0+\frac{8\pi^2m}{h^2}(E-V)\psi_0=0, \]

where \(E>V=\text{const}\), with the conditions \(\psi_0=0\) on the boundaries of the box and with the normalization condition

\[ \int_{x=0}^{l_1}\int_{y=0}^{l_2}\int_{z=0}^{l_3}\bar{\psi}_0\psi_0\,dx\,dy\,dz=1 \]

(it is assumed that outside the “box” everywhere \(\psi_0=0\), which, strictly speaking, is correct only in the presence of infinitely high potential barriers on the boundaries of the “box”). The equation is solved in an elementary way (by separation of variables). — S. Ch.

** In formula (14), \(\mathfrak{K}\) is the wave-number vector, equal in magnitude to \(\dfrac{2\pi\nu}{C}\). The surface of the metal on which the light falls coincides with the \((xy)\)-plane. — S. Ch.

the potential of the light wave, \(v\) is the velocity of the electron. The corresponding quantum operator is obtained by the usual replacement of \(v\) by \(\dfrac{h}{2\pi i m}\operatorname{grad}\), i.e.

\[ H'=-\frac{eh}{2\pi i m}\,\mathfrak A\,\operatorname{grad}. \]

For the unperturbed system we have the wave equation:

\[ \frac{h}{2\pi i}\frac{\partial \psi_0}{\partial t}=H_0\psi_0, \tag{15a} \]

where

\[ H_0=-\frac{h^2}{8\pi^2 m}\Delta+mc^2 \]

(the constant potential inside the metal we consider as included in \(mc^2\)).

For the perturbed system we obtain:

\[ \frac{h}{2\pi i}\frac{\partial\psi}{dt} = H_0\psi+H'\psi = H_0\psi-\frac{eh}{2\pi i m}\mathfrak A\,\operatorname{grad}\psi, \tag{15b} \]

where \(\dfrac{\partial \mathfrak A}{\partial t}=\mathfrak E\), whence, taking (14) into account, we find:

\[ \mathfrak A= \frac{1}{4\pi i\nu}\mathfrak E_0 e^{-\alpha z} \left( e^{\,i[2\pi\nu t-(\mathfrak K\mathfrak r)]} - e^{-\,i[2\pi\nu t-(\mathfrak K\mathfrak r)]} \right). \tag{15c} \]

Substituting in equation (15b), instead of \(\psi\), its expression (15), and instead of \(\mathfrak A\), its expression (15c) (where the second term of this expression is discarded, since it does not produce the photoelectric effect), we shall obtain, if we also take into account equation (15a) and neglect the second-order small term \(H'\psi_1\), the equation from which the first-order correction \(\psi_1\) is determined: \((x)\)

\[ \frac{h^2}{8\pi^2 m}\Delta\psi_1 + \frac{h}{2\pi i}\frac{\partial\psi_1}{\partial t} - mc^2\psi_1 = \frac{eh}{8\pi^2 m\nu} e^{\alpha z-i(\mathfrak K\mathfrak r)+2\pi i\nu t} (\mathfrak E_0,\operatorname{grad}\psi_0). \tag{16} \]

\((x)\) The correction \(\psi_1\) is sought in the form of a series expansion:

\[ \psi_1 = \sum_{m_1m_2m_3} a_{m_1m_2m_3}(t) e^{\,2\pi i\left(\frac{m_1x}{l_1}+\frac{m_2y}{l_2}+\frac{m_3z}{l_3}\right)} e^{\frac{2\pi i}{h}(mc^2+\varepsilon)t} = \]

\[ = \sum_{\mathfrak f} a(\mathfrak f,t) e^{\,i(\mathfrak f\mathfrak r)+\frac{2\pi i}{h}(mc^2+\varepsilon)} ; \tag{16a} \]

where

\[ \varepsilon=\frac{h^2(\mathfrak f)^2}{8\pi^2 m}, \qquad \mathfrak f_x=2\pi\frac{m_1}{l_1}, \qquad \mathfrak f_y=2\pi\frac{m_2}{l_2}, \qquad \mathfrak f_z=2\pi\frac{m_3}{l_3}, \]

and the coefficients of the expansion are regarded as time-dependent. As long as \(\mathfrak E=0\), all \(a(\mathfrak f,t)=0\), but from the moment the illumination is switched on \((t=0)\) they begin to change slowly. In order to find, in the first—

In the zeroth approximation this change, it is sufficient to substitute the preceding expression into equation (16). We find:

\[ \frac{h}{2\pi i}\sum_R \frac{da(\mathfrak f,t)}{dt}\cdot e^{\,i(\mathfrak f\mathfrak r)+\frac{2\pi i}{h}(mc^2+\varepsilon)t} = \]

\[ =\frac{eh}{8\pi^2 m\nu}e^{-\alpha z-i(\mathfrak R\mathfrak r)} \cdot(\mathfrak E_0\operatorname{grad}u_0)\cdot e^{\frac{2\pi i}{h}(h\nu+mc^2+\varepsilon_0)t}, \]

and this is nothing other than the Fourier expansion of the expression standing on the right-hand side of the equation, the Fourier coefficients being equal to:

\[ \frac{h}{2\pi i}\frac{da(\mathfrak f,\mathfrak r)}{dt}= \]

\[ =\frac{1}{l_1l_2l_3}\frac{eh}{8\pi^2m\nu} \int_0^{l_1}\int_0^{l_2}\int_0^{l_3} dx\,dy\,dz\, e^{-\alpha z-i[(\mathfrak R+\mathfrak f)\mathfrak r]} (\mathfrak E_0\operatorname{grad}u_0)\times \]

\[ \times e^{\frac{2\pi i}{h}(h\nu+\varepsilon_0-\varepsilon)t}. \]

Integrating with respect to \(t\), we have:

\[ a(\mathfrak f,t)= \frac{1}{l_1l_2l_3}\frac{eh}{8\pi^2m\nu} \frac{e^{\frac{2\pi i}{h}(h\nu+\varepsilon_0-\varepsilon)t}-1} {h\nu+\varepsilon_0-\varepsilon} \times \]

\[ \times \int_0^{l_1}\int_0^{l_2}\int_0^{l_3} dx\,dy\,dz\, e^{-\alpha z-i[(\mathfrak R+\mathfrak f)\mathfrak r]} (\mathfrak E_0\operatorname{grad}u_0). \quad (x) \tag{16b} \]

According to the theory, \(|\psi|^2\) gives the probability of absorption of an energy quantum \(h\nu\) by an electron with initial energy \(\varepsilon_0\). This expression must be integrated over the whole volume and summed over all values of the initial energy \(\varepsilon_0\).

\((x)\) Indeed, quantum mechanics teaches that the squares of the moduli of the coefficients \(a(\mathfrak f,t)\) of the expansion (16a) are, up to a factor determined from considerations of normalization of those functions with respect to which the expansion is carried out, the probabilities that at the time \(t\) (only slightly different from \(t=0\)) an electron having, in the unperturbed state, the energy \(\varepsilon_0\), will be found in a state with energy

\[ \varepsilon=\frac{h^2(\mathfrak f)^2}{8\pi^2m}. \]

The total probability of excitation of the electron will be equal to

\[ P=l_1l_2l_3\sum_k |a(\mathfrak f)|^2 \]

(if the normalization factor is taken into account), which in turn is equal to

\[ \iiint dx\,dy\,dz\,|\psi_1|^2 \]

(according to the closure theorem). We may also write:

\[ P=\frac{1}{(2\pi)^3}\iiint d\mathfrak f_x\,d\mathfrak f_y\,d\mathfrak f_z\,|a(\mathfrak f)|^2, \tag{16c} \]

replacing the summation over the points \(m_1, m_2, m_3\) [see formula (16a)] by integration over momentum space. In order actually to de-

calculate this expression, it is necessary first to determine \(a(\xi)\) from formula (16b), where in the integration one may assume that \(e^{-\alpha l_3}=0\) (i.e., that the metal layer is sufficiently thick), and, having found \(a(\xi)\), take the triple integral over momentum space. This calculation is comparatively complicated, and for this reason we do not give it. (x)

Wenzel then assumes that all electrons possessing energy sufficient to escape from the metal actually emerge, independently of the direction of their motion, so that the transmission coefficient \(D\) is equal to unity, provided only that \(\varepsilon_0+h\nu>h\nu_a\).

(x) To compute the total photocurrent, it is necessary to sum the expression \(P\) over all electrons satisfying the last inequality; in other words, in momentum space one must take the integral

\[ \frac{2l_1l_2l_3}{\pi^3}\iiint dk_1\,dk_2\,dk_3\,P, \]

extending it over the spherical layer bounded by the radii \(k'\) and \(\bar{k}\), where \(\bar{k}\) has the same meaning as in formula (5c), while \(k'\) corresponds to an electron whose initial energy, when added to the photon energy, is just sufficient for ejection from the metal, i.e.

\[ \varepsilon_0+h\nu=h\nu_a;\quad \text{it is easy to see that } \quad k'=\sqrt{\frac{8\pi^2m(\nu_a-\nu)}{h}}. \]
(x)

In the final result Wenzel obtains the formula:

\[ I\sim \left[\nu^{5/2}-(\nu_a-\nu)^{5/2}\right]\nu^{-7/2}E_z^2 +\frac{1}{14}\left[\nu^{7/2}-(\nu_a-\nu)^{7/2}\right]\nu^{-9/2} \left[3E_z^2+2E_x^2+2E_y^2\right]+\cdots \tag{17} \]

Here \(E_x, E_y, E_z\) are the components of the electric vector, with the component \(E_z\) directed perpendicular to the surface.

Houston\(^{14}\) rigorously showed that only those electrons can leave the metal which, after their last collision inside the metal, have a component of energy normal to the surface exceeding \(\varepsilon_a\). He derives the formula:

\[ I\sim \left(\frac{\nu_0}{\nu}\right)^{1/2} \left\{ \frac{E_z^2}{\nu}\left(1+\frac{\nu_0}{2\nu}\right) + \frac{E_x^2+E_y^2}{3\nu} \left[\frac{\nu-\nu_0}{\nu}\right] \right\} \left(\frac{\nu-\nu_0}{\nu}\right)^2. \tag{18} \]

Taking Houston’s correction into account, Wenzel\(^{15}\) changes his formula, which assumes the form:

\[ I\sim \nu^{-7/2} \left\{ \frac{1}{3}\nu\left[\nu^{3/2}-(\nu_a-\nu)^{3/2}\right] -\frac{1}{5}\left(\nu^{5/2}-(\nu_a-\nu)^{5/2}\right) \right\}E_z^2+\cdots \tag{19} \]

Despite the great difference in equations (18) and (19), the corresponding distribution curves are similar in form.

The most important features of the spectral-sensitivity curves obtained from equations (18) and (19) are that: 1) the curves meet the abscissa axis at the point \(\nu=\nu_0\), forming a finite angle with it; 2) the curves have a maximum in the region \(\nu_a>\nu_{\max}>\nu_0\); 3) the equations show that the component \(E_z\) promotes the tearing out of the electron to a greater degree than \(E_x\) or \(E_y\). Although these curves do form an angle of finite magnitude with the abscissa axis at \(\nu=\nu_0\), their form in the region of high frequencies is in full agreement with the experimental curves.

3. Fröhlich’s theory for thin layers. Fröhlich\(^{16}\) pointed out the limited character of our knowledge concerning the behavior of light inside a metal. He therefore preferred to consider a layer whose thickness is sufficiently small that the absorption of light in the metal can be neglected. In this way he eliminates the factor containing the absorption coefficient \(\alpha\) from the equation for the electric vector of the light wave in the metal [see eq. (16)].

He divides space into three parts: I) inside the layer; II) outside the layer, on the side of the incident radiation (the direction of the positive \(z\)-axis); and III) behind the layer (the direction of the negative \(z\)-axis); for these three regions he obtains, respectively, the wave functions:

\[ 1)\quad u_0=2a\cos k_3z\sin k_1x\sin k_2y, \tag{20a} \]

\[ 2)\quad u_0=be^{ipz}\sin k_1x\sin k_2y, \tag{20b} \]

\[ 3)\quad u_0=be^{-ipz}\sin k_1x\sin k_2y, \tag{20c} \]

where the thin layer is assumed to be rectangular, with sides \(l_1\) and \(l_2\) and thickness \(l_3\). Here \(\dfrac{h k_j}{2\pi m}\) is the classical velocity in the directions of the \(x\)-, \(y\)-, and \(z\)-axes, respectively, for values of \(j\) equal to 1, 2, and 3; furthermore,

\[ p^2=k_3^2-\frac{8\pi^2m\nu_a}{h}, \tag{21a} \]

\[ b=2a\cos k_3l_3e^{-ipl_3}, \tag{21b} \]

\[ a=(2l_1l_2l_3)^{-\frac{1}{2}}. \tag{21c} \]

\((x)\) It is not difficult to derive formulas (21a)—(21c).

Let the potential in region I be equal to \(V_0=\mathrm{const}\), and in regions II and III equal to \(V_0+h\nu_a\), with the \((xy)\)-plane parallel to the surfaces bounding the metal layer and lying midway between them. Let, furthermore, \(2l_3\) be the thickness of the layer, and \(l_1\) and \(l_2\) its length and width. At the boundaries \(x=0,\ l_1,\ y=0,\ l_2\) the presence of infinitely high potential barriers is assumed. In view of the fact that the potential depends only on \(z\), the variables in the Schrödinger equation

\[ \Delta u_0+\frac{8\pi^2m}{h^2}(\varepsilon_0-V)u_0=0 \]

separate, and one may put:

\(u_0=X(x)Y(y)Z(z)\). Taking into account the boundary conditions, we obtain:

\[ \left. \begin{aligned} X(x)&=\sin k_1x,\quad k_1=\frac{\pi n_1}{l_1},\\ Y(y)&=\sin k_2y,\quad k_2=\frac{\pi n_2}{l_2}, \end{aligned} \right\} \quad n_1,n_2=0,1,2,\ldots \tag{21d} \]

\[ \left[k_1^2+k_2^2+k_3^2\pm k^2=\frac{8\pi^2m}{h^2}(\varepsilon_0-V)\right]. \]

For \(Z(z)\) one obtains:

\[ \frac{d^2Z(z)}{dz^2}+k_3^2Z(z)=0 \quad \text{in region I,} \]

\[ \frac{d^2Z(z)}{dz^2}+p^2Z(z)=0 \quad \text{in regions II and III.} \]

In the unexcited state \(p^2<0\) and

\[ p=i\sqrt{\frac{8\pi^2m}{h^2}\alpha-k_3^2}. \]

The boundary conditions \(Z(z)=0\) at \(\pm\infty\) give:

\[ \left. \begin{aligned} Z(z)&=a_1e^{ik_3z}+a_2e^{-ik_3z} && \text{(in region I),}\\ Z(z)&=b_1e^{ipz} && \text{(in region II),}\\ Z(z)&=b_2e^{-ipz} && \text{(in region III).} \end{aligned} \right\} \tag{21e} \]

The continuity conditions for \(Z(z)\) and \(\dfrac{dZ(z)}{dz}\) at \(z=\pm l_3\) give a homogeneous system of four equations with four unknowns \(a_1,a_2,b_1,b_2\), whence \(\dfrac{a_2}{a_1}\), \(\dfrac{b_1}{a_1}\), \(\dfrac{b_2}{a_1}\) can be determined. The system can have solutions different from zero only in the case when its determinant

\[ \left| \begin{array}{cccc} e^{ik_3l_3} & e^{-ik_3l_3} & e^{ipl_3} & 0\\ e^{ik_3l_3} & -e^{-ik_3l_3} & \dfrac{p}{k_3}e^{ipl_3} & 0\\ e^{-ik_3l_3} & e^{ik_3l_3} & 0 & e^{ipl_3}\\ e^{-ik_3l_3} & -e^{ik_3l_3} & 0 & -\dfrac{p}{k_3}e^{ipl_3} \end{array} \right| \]

is equal to zero. Expanding the determinant, we obtain the transcendental equation

\[ \tg k_3l_3=-\frac{ip}{k_3}, \]

whose solutions are the allowed values of \(k_3\). Now the homogeneous system gives:

\[ a_1=a_2=a,\quad b_1=b_2=b=2a\cos k_3l_3e^{-ipl_3}. \tag{21f} \]

Substituting (21f) into the system (21e), we are convinced of the validity of equations (20a) and (21b). Equality (21c) follows from the normalization condition:

\[ \int_{\infty}^{\infty} dz \int_0^{l_2} dy \int_0^{l_1} dx\, |u(xyz)|^2=1. \quad (x) \]

Further, Frenkel treats, similarly to Wentzel, the action of light as a perturbation and determines the number of electrons which after

by excitation by light acquire energy and direction that allow them to emerge from the metal.

(x) The initial equation in this calculation is again equation (15b); in expression (14) for the electric vector in the metal the real exponential factor, which accounts for the absorption of light, is now omitted. The solution is sought in the form:

\[ \psi=\psi_0+\lambda_1\psi_1+\lambda_2\psi_2+\lambda_3\psi_3, \tag{21g} \]

where \(\lambda_i=\dfrac{e}{h\nu}\mathfrak{E}_i\) (by \(\psi\) we now mean the complete wave function). If (21g) is substituted into equation (15b), then, neglecting squares of the small quantities \(\lambda_i\), it is not difficult to obtain the following three equations:

\[ \Delta\psi_i+\frac{4\pi i m}{h}\frac{\partial\psi_i}{\partial t} -\frac{8\pi^2m}{h^2}V(z)\psi_i = \frac{\partial u_0}{\partial x_i} \left\{ e^{-i(\mathfrak{K}r)-\frac{2\pi i}{h}(\varepsilon_0+h\nu)t} + e^{i(\mathfrak{K}r)-\frac{2\pi i}{h}(\varepsilon_0-h\nu)t} \right\}; \]

here \(u_0\) is the time-independent factor of the unperturbed wave function; \(x_1=x,\ x_2=y,\ x_3=z\). In the stationary state we must have:

\[ \psi_i=\varphi_i^{+}e^{-\frac{2\pi i}{h}(\varepsilon_0+h\nu)t} +\varphi_i^{-}e^{-\frac{2\pi i}{h}(\varepsilon_0-h\nu)t}, \]

where \(\varphi_i^{+}\) and \(\varphi^{-}\) contain no time. The second term is of no importance for the photoeffect and may be discarded. Then for \(\varphi_i^{+}=\varphi_i\) we obtain the equation:

\[ \Delta\varphi_i+\frac{8\pi^2m}{h^2} \left[\varepsilon_0+h\nu-V(z)\right]\varphi_i = \frac{du_0}{dx_i}e^{-i(\mathfrak{K}r)}, \]

which can still be simplified by assuming that \(\mathfrak{K}\) is small in comparison with \(k\), and therefore by dropping the exponential factor on the right. Substituting here \(U_0\) from the solutions (20), we obtain, for example, for \(\varphi_3\) the following equations:

\[ \begin{aligned} &\text{In region I:}\\ &\qquad \Delta\varphi_3+\left(k^2+\frac{8\pi^2m}{h^2}\nu\right)\varphi_3 = -2ak_3\sin k_3z\,\sin k_2y\,\sin k_1x.\\[6pt] &\text{In region II:}\\ &\qquad \Delta\varphi_3+\left(k^2+\frac{8\pi^2m}{h^2}(\nu-\nu_a)\right)\varphi_3 = ipb e^{ipz}\sin k_2y\,\sin k_1x.\\[6pt] &\text{In region III:}\\ &\qquad \Delta\varphi_3+\left(k^2+\frac{8\pi^2m}{h^2}(\nu-\nu_a)\right)\varphi_3 = -ipb e^{-ipz}\sin k_2y\,\sin k_2x. \end{aligned} \tag{21h} \]

The equations written down can be solved exactly, by separation of variables, which is possible because the inhomogeneous term satisfies the boundary conditions for \(x\) and \(y\). In an analogous way the equations for \(\varphi_1\) and \(\varphi_2\) can be solved.

We shall not, however, enter into the details of these rather simple calculations.

The final solution is obtained in the form:

\[ \psi=u_0 e^{-\frac{2\pi i}{h}\varepsilon_0 t} +(\lambda_1\varphi_1+\lambda_2\varphi_2+\lambda_3\varphi_3) e^{-\frac{2\pi i}{h}(\varepsilon_0+h\nu)t}. \tag{21i} \]

Frehlich finds the strength of the photocurrent by substituting (21i) into the formula:

\[ I=\frac{eh}{4\pi mi}\int_0^{l_1}\int_0^{l_2} \left(\psi\frac{\partial\psi^{*}}{\partial x} -\psi^{*}\frac{\partial\psi}{\partial x}\right)\,dx\,dy \]

and carrying out the summation over all electrons in the same way as Wendell does. (x)

As a result the following equations are obtained.

For \(\nu_0<\nu<\nu_a\):

\[ \left. \begin{aligned} I&=\frac{e^2\nu_a^2 E_x^2}{16\pi h^2}\nu^{-4} \left\{A_1\frac{\nu-\nu_0}{\nu_a} +A_2\left(\frac{\nu-\nu_0}{\nu_a}\right)^2+\ldots\right\},\\ \text{where}\qquad A_1&=-\frac{7\nu_0}{\nu_a} +\left(\frac{21}{2}-\frac{7}{2}\frac{\nu}{\nu_a}\right)\frac{\nu}{\nu_a} -\left(\frac{21}{8}+\frac{5}{4}\frac{\nu}{\nu_a}\right) \left(\frac{\nu}{\nu_a}\right)^2+\ldots,\\ A_2&=\left(-\frac{11}{2}+\frac{2\nu}{\nu_a}\right) +\left(3+\frac{3}{4}\frac{\nu}{\nu_a}\right)\frac{\nu}{\nu_a} +\left(\frac{9}{16}+\frac{11}{8}\frac{\nu}{\nu_a}\right) \left(\frac{\nu}{\nu_a}\right)^2+\ldots . \end{aligned} \right\} \tag{22a} \]

For \(\nu>\nu_a\):

\[ \left. \begin{aligned} I&=\frac{e^2\nu_a^2 E_x^2}{16\pi h^2}\nu^{-4} \{B_1+B_2+\ldots\},\\ \text{where}\qquad B_1&=\left(\frac{1}{2}\frac{\nu_a}{\nu}-1\right)\frac{\nu}{\nu_a} +\left(\frac{5}{2}+\frac{1}{8}\frac{\nu_a}{\nu}\right)\\ &\quad+\left(\frac{7}{4}-\frac{3}{8}\frac{\nu_a}{\nu}\right)\frac{\nu_a}{\nu} -\frac{3}{2}\left(\frac{\nu_a}{\nu}\right)^2+\ldots,\\ B_2&=\left(-2+\frac{\nu}{\nu}\right) +\left(\frac{15}{4}\frac{\nu}{\nu_a}-\frac{9}{16}\right)\frac{\nu_a}{\nu}\\ &\quad+\left(\frac{21}{16}-\frac{5}{8}\frac{\nu}{\nu_a}\right) \left(\frac{\nu_a}{\nu}\right)^2+\ldots . \end{aligned} \right\} \tag{22b} \]

The general form of the curves of spectral sensitivity remains the same as in Wendell and Houston. But the velocity distribution is obtained differently. According to this theory, the most probable value of the energy of the electron being ejected differs little from the maximum energy, a phenomenon confirmed experimentally by Lukirskii and Prilezhaev,\(^{17}\) who showed in general that the thinner the layer, the closer the observed energy distribution approaches that required by the theory under consideration.

The positions of the spectral maxima, determined by Frehlich with the aid of his theory for certain alkali metals, agree well with the experimental data. The numerical values of \(\nu\) were

found from formula (5) under the assumption that there is one free electron per atom. For the frequency of the spectral threshold, values were taken that were not entirely exact, but in any case of the right order. The calculated frequency of maximum emission \(\nu_{\max}\) differed from the observed value by no more than \(10\%\). The best agreement between theory and experiment was obtained for potassium; moreover, the experimental values of the frequency of the spectral threshold and of the spectral maximum were taken from the data of Suhrmann and Teichsing\(^{18}\) for a film of distilled potassium on platinum. The calculated \(\nu_{\max}\) was approximately \(5\%\) smaller than the corresponding experimental value.

(x) Since the theory does not take into account the surface structure of the metal, we have the right to expect good agreement with experiment only in the case when the de Broglie wavelengths of the emitted electrons are sufficiently large that the surface of the metal may be regarded as quite smooth, or, in other words, when the work function of the metal surface is sufficiently small. From this it becomes clear why the theory gives good agreement with experiment for the alkali metals, which, as is known, have an especially small work function. (x)

4. The theory of Tamm and Shubin. The calculations of Fröhlich were, in their main features, repeated independently of him by Tamm and Shubin\(^{19}\), who, however, went further and gave a more satisfactory interpretation of a number of experimental results. Tamm and Shubin point out that a free electron cannot completely absorb the energy of a light quantum while simultaneously conserving both energy and momentum*. If the electron is in an electric field, then momentum can be transferred through the mediation of the field to its source, in the present case—the crystal lattice of the metal. The ions composing the lattice possess considerable mass, and therefore the excess momentum can be transferred to them with little loss of energy.

There are two types of fields in which conduction electrons may be found: first, the fields of potential barriers at

* This can easily be shown by the following example. Let a photon of frequency \(\nu\), carrying momentum \(\dfrac{h\nu}{c}\) and energy \(h\nu\), be wholly absorbed by an electron at rest in a field-free space. Then the conditions of conservation of energy and momentum lead to the simple equalities:

\[ h\nu = m_0 c^2\left(\frac{1}{\sqrt{1-\dfrac{v^2}{c^2}}}-1\right), \qquad \frac{h\nu}{c}=\frac{m_0 v}{\sqrt{1-\dfrac{v^2}{c^2}}} \]

(\(m_0\) is the rest mass of the electron, \(v\) is the velocity of the electron after absorption of the photon), which must be satisfied simultaneously. It is not difficult to see that this is impossible, whence follows the assertion made in the text.—S. Ch.

of the metal boundary; second, the fields of the crystal lattices. But since the latter are small in comparison with the former, we are entitled to neglect the volume effect due to the action of the lattice when calculating the surface effect associated with the potential barrier at the surface of the metal.

The way in which Wentzel attempts to satisfy the law of conservation of momentum is not free from objections. It is true that a uniformly damped wave can be expanded in a Fourier series in such a way that, formally, the requirement of simultaneous conservation of energy and momentum in photoelectric processes can be satisfied. But Tamm and Shubin \(^{19}\), and later Bloch \(^{20}\), draw attention to the fact that the absorption of light in a metal is, in essence, photoelectric, i.e. discontinuous, in character. It is difficult to determine the influence of this discontinuity in the absorption of light on the results of the theory. Therefore the construction of a theory free from this defect is highly desirable.

Frohlich’s calculations, which assumed that the electric vector of the light wave does not decay in the metal and that the electrons inside the metal are free, give emission due to the surface effect. Tamm and Shubin point out the erroneousness of Frohlich’s generalization to the case of a solid metal, which he carries out by summing the effects for separate layers and using a single absorption coefficient for light and electrons. This method assumes that photoelectric excitation inside the metal has exactly the same character as at the surface. The emission from the surface of a thick metal layer calculated by Tamm and Shubin is equal to:

\[ I=\frac{e^2\lambda\,(E_z)^2}{2ch^2\nu^4\cos\theta}\,(\nu\Delta)^2(3\Delta-\nu)-(\Delta-\nu)(3\Delta+\nu)\ln\frac{\nu^{1/2}+\Delta^{1/2}}{(\Delta-\nu)^{1/2}} \tag{23} \]

where \(\Delta=\nu-\nu_0\), and \(\theta\) is the angle of incidence of the light. The spectral distribution (23) has the same form as the distribution found experimentally by Zourman and Teissing \(^{21}\) for a thick layer of potassium.

From the relation given it is evident that, although only a very small fraction of the light is absorbed in the surface layer of the metal, it nevertheless supplies a large part of the photoelectrons in the frequency region adjacent to the long-wave boundary of the photoelectric effect. A large part of the light is absorbed inside the metal, but only a small number of electrons escape from there to the outside. This may occur either because of inelastic collisions of the electrons in the metal before leaving it, or because inside the metal electrons are excited with less energy than near the surface. The latter circumstance must lead to the result that the number of excited electrons will be large, but their energy will be insufficient for escape.

To determine the conditions for absorption of light inside the metal, Tamm and Shubin use the eigenfunctions determined by Bloch.

functions of an electron moving in the periodic field of the metal lattice:

\[ \psi_{k_1'',k_2'',k_3''} = e^{\,i(2\pi \nu_{k_1''k_2''k_3''}t-k_1''x-k_2''y-k_3''z)} U_{k_1''k_2''k_3''}(x,y,z) \tag{24} \]

Here

\[ k''^{2}=k_1''^{2}+k_2''^{2}+k_3''^{2}=\gamma \nu;\quad \gamma=\frac{8\pi^{2}m}{h}, \]

\(U\) is a certain periodic function of the coordinates with a period equal to the lattice constant. \(k''^{2}\) has the dimension of inverse length and is proportional to the energy of the state under consideration*.

* Let us give the derivation of formula (24), given by Bloch (F. Bloch, Zs. f. Phys. 52, 555, 1929), who solves the one-electron problem for a space with a potential periodically varying in the three principal directions. For simplicity we shall assume that these three directions are mutually perpendicular and coincide respectively with the directions \(x\), \(y\), and \(z\) of the axes of our coordinate system (although the same result remains valid for any triclinic crystal as well). Then the periodicity condition for the potential in the Schrödinger equation

\[ \Delta\psi+\frac{8\pi^{2}}{h^{2}}(\varepsilon-V)\psi=0 \tag{I} \]

can be written in the form:

\[ V(x,y,z)=V(x+g_1a,y+g_2b,z+g_3c), \tag{II} \]

where \(a,b,c\) are the periods of variation of \(V\) in the directions of the coordinate axes, and \(g_1,g_2,g_3\) are arbitrary integers.

The substitutions

\[ \begin{aligned} Rg_1:\;&x'=x+g_1a,\;y'=y,\;z'=z,\\ Sg_2:\;&x'=x,\;y'=y+g_2b,\;z'=z,\\ Tg_3:\;&x'=x,\;y'=y,\;z'=z+g_3c \end{aligned} \tag{II} \]

belong to the substitution group of the Schrödinger equation of our problem; this means that if \(\psi(xyz\varepsilon)\) is a solution of this equation belonging to the eigenvalue \(\varepsilon\), then \(\psi(x'y'z'\varepsilon)\) is also its solution, belonging to the same eigenvalue \(\varepsilon\). But every such solution can be represented in the form of a linear combination of a complete set of linearly independent solutions belonging to the eigenvalue \(\varepsilon\); therefore one may write:

\[ \begin{aligned} \psi_j(x+a,y,z,\varepsilon)&=\sum_k a_{kj}\psi_k(xyz\varepsilon),\\ \psi_j(x,y+b,z,\varepsilon)&=\sum_k b_{kj}\psi_k(xyz\varepsilon),\\ \psi_j(x,y,z+c,\varepsilon)&=\sum_k c_{kj}\psi_k(xyz\varepsilon). \end{aligned} \tag{III} \]

Between the matrices appearing on the right-hand side of the system (III) and substitutions of type (II) one can establish a one-to-one correspondence. In this case, to the element of the group \(R^{g_1}S^{g_2}T^{g_3}\) there will correspond the product of matrices:

\[ (a_{kj})^{g_1}(b_{kj})^{g_2}(c_{kj})^{g_3}. \]

If, as a boundary condition, one introduces the requirement that the eigen-

If an electron, absorbing an energy quantum \(h\nu\), passes from the state \(k''\) into the state \(k'\), then the energy relation must be satisfied:

\[ \nu_{k'_1 k'_2 k'_3}=\nu_{k''_1 k''_2 k''_3}+\nu . \tag{25} \]

Moreover, the discreteness of the energy levels in the metal results in the following diffraction condition:

\[ k'_j=k''_j \pm \frac{2\pi n_j}{a} \tag{26} \]

\[ (n_j=1,2,\ldots;\; j=1,2,3), \]

where \(a\) is the lattice constant of the metal.

(x) The selection rule (26) is not difficult to obtain from the general law according to which a transition from the state \(k''\) to the state \(k'\) under

new functions are periodic functions of the coordinates with periods \(G_1a\), \(G_2b\), \(G_3c\) (\(G\) are large integers), then to the substitutions

\[ R^{G_1},\; R^{G_2},\; R^{G_3} \]

there will correspond the matrices:

\[ (a_{kj})^{G_1}=(b'_{kj})^{G_2}=(c_{kj})^{G_3}=\delta_{kj}\ldots, \tag{IV} \]

where \(\delta_{kj}\) is the unit matrix.

Instead of the randomly chosen linearly independent solutions we may substitute into the right-hand side another system of solutions \(\psi'_j=\sum_k t_{kj}\psi_k\), choosing the matrix \(T=(t_{kj})\) so that the new matrices

\[ (a'_{kj})=T^{-1}(a_{kj})T,\quad (b'_{kj})=T^{-1}(b_{kj})T,\quad (c'_{kj})=T^{-1}(c_{kj})T \]

are diagonal. Simultaneous reduction of all three matrices to diagonal form is possible, since they commute with one another. It now follows from equation (IV) that

\[ a'_{jj}=e^{\frac{2\pi i}{G_1}k_j},\quad b'_{jj}=e^{\frac{2\pi i}{G_2}l_j},\quad c'_{jj}=e^{\frac{2\pi i}{G_3}m_j},\ldots, \tag{V} \]

where \(k_j,l_j,m_j\) are integers. Equation (III) gives:

\[ \psi_\lambda(x+a,y,z,E)=e^{\frac{2\pi i}{G_1}k_j}\psi_\lambda(xyz\varepsilon) \tag{VI} \]

and two other analogous expressions. The relations (VI) are equivalent to the equality:

\[ \psi_{klm}(xyzt)= e^{2\pi i\left(\frac{kx}{aG_1}+\frac{ly}{bG_2}+\frac{mz}{cG_3}+\nu_{klm}t\right)} U_{klm}(xyz), \tag{VII} \]

where \(U_{klm}(x,y,z)\) is a periodic function of the coordinates with periods \(a,b,c\), whose form depends on the course of the potential curves and cannot be determined from the general theory. In equation (VII) the usual periodic time factor characteristic of stationary states has been introduced. Thus, we are dealing with plane de Broglie waves on which a modulation with the frequency of the lattice structure is superposed. If the electron field is small (i.e. the electron is weakly bound), then one may assume that

\[ \frac{2\pi k}{aG_1}=k''_1, \]

where the wave number \(k''_1\) is connected with the frequency of the electronic oscillation in the same way as in the case of a free particle.

The wave function (VII) must be normalized to unity in the volume of the “fundamental parallelepiped” with sides \(aG_1\), \(bG_2\), \(cG_3\).—S. Ch.

is possible under the action of light only in the case when the matrix element of the perturbing energy of the light wave corresponding to this transition,

\[ (k'|H'|k'')=\iiint \overline{\psi}_{k'}H'\psi_{k''}\,dx\,dy\,dz \tag{26a} \]

is different from zero (the integration is over the volume of the fundamental parallelepiped). The operator of the perturbing energy, as we have seen, has the form:

\[ H'=-\frac{eh}{2\pi i}\left(A_x\frac{\partial}{\partial x}+A_y\frac{\partial}{\partial y}+A_z\frac{\partial}{\partial z}\right), \]

where the spatial periodicity of the components of the vector potential \(A_x, A_y, A_z\) may be neglected, since for most conduction electrons in a metal the de Broglie wavelength is very small in comparison with the wavelength of the light wave; we shall also neglect the absorption of light in the metal (one may assume that the coefficient of absorption of light is small in comparison with the wave numbers of the electrons). The Bloch wave functions may be represented in the form:

\[ \psi_k=e^{-i(k_1x+k_2y+k_3z)} \sum_{r_1=-\infty}^{+\infty} \sum_{r_2=-\infty}^{+\infty} \sum_{r_3=-\infty}^{+\infty} b_{r_1r_2r_3k_1k_2k_3} e^{2\pi i\left(\frac{r_1x}{a}+\frac{r_2y}{b}+\frac{r_3z}{c}\right)}, \tag{26b} \]

where \(a, b, c\) are the lattice periods for the three principal directions.

Acting on \(\psi_{k''}\) with the operator \(H'\) and substituting the result into (26a), we obtain a sum of expressions containing, as factors, integrals of the type

\[ \iiint \overline{\psi}_k\psi_{k''}\,dx\,dy\,dz, \]

\[ \int_{0}^{aG_1} e^{i(k_1'-k_1'')x}\,dx \]

and

\[ \int_{0}^{aG_1} e^{i\left(k_1'-k_1''\pm\frac{2\pi n_1}{a}\right)x}\,dx; \]

\[ aG_1=\frac{2\pi s}{k'} \]

is the length of one of the sides of the fundamental parallelepiped; \(s\) is an integer, and \(g_1\) and \(n_1\) are positive integers. The first of the integrals written down vanishes by virtue of the orthogonality of the eigenstates. The remaining integrals can be different from zero only in the case when the imaginary exponents of the integrands vanish, i.e., when \(k_1''=k_1'\) (a case not corresponding to any transition) or when

\[ k_1'=k_1''\pm\frac{2\pi n_1}{a}, \]

which is what had to be proved.

The frequency of the absorbed light \(\nu_0\) must satisfy the condition:

\[ \nu_0=\sum_{j=1}^{3}\left\{\left(k_j'\pm\frac{2\pi}{a}n_j\right)^2-k_j''^{\,2}\right\}, \]

or \(\nu_0\) must be equal to the change in the frequency of the electronic oscillation [cf. (24) and (26)]. Introducing the notation \(k_0=\dfrac{2\pi}{a}\) and \(\nu_1=\dfrac{k_0^2}{\gamma}\), we obtain:

\[ \nu_0=\nu_1 \sum_{j=1}^{3} n_j^2\left(1\pm \frac{2k_j''}{k_0 n_j}\right). \tag{26c} \]

It is not difficult to show that \(\dfrac{2k_j''}{k_0}<1\). Indeed, for \(T=0^\circ\) we have: \(k_j''<\bar{k}\), where \(\bar{k}\) is taken from formula (5c); on the other hand, taking into account that in a cubic lattice \(n=\dfrac{1}{a^3}\), we may write:

\[ \frac{2\bar{k}}{k_0}=\left(\frac{3}{\pi}\right)^{1/3}<1, \]

whence the validity of the inequality being proved follows. (x)

We shall determine the smallest absorbed frequency by putting, in equality (26c), one of the numbers \(n_j\) equal to 1 and the other two equal to zero *.

(x) Putting, for example, \(n_1=1,\ n_2=n_3=0\), we find: \(\nu_{\min}=\nu_1\left(1-\dfrac{2k_1''}{k_0}\right)\); hence it is clear that for electrons in different initial states, \(\nu_{\min}\) will be different. In the region of low frequencies (the visible spectrum), a quantum \(h\nu\) can be absorbed by an electron only in the case when the relation

\[ k_1''=\frac{k_0}{2}\left(1-\frac{\nu}{\nu_1}\right) \tag{26d} \]

is satisfied, or the same relation for \(k_2''\) or \(k_3''\). (x)

The smallest of the frequencies capable of imparting to the electron an energy sufficient for escape from the metal, and satisfying the preceding conditions, is equal to:

\[ \nu_0'=2\sqrt{\nu_1\nu_a}-\nu_1, \]

where

\[ \nu_1=4\left(\frac{\pi}{3}\right)^{2/3}\nu. \]

(x) This is not difficult to verify by substituting into the equality

\[ \frac{h k_1''{}^2}{\gamma}+h\nu=h\nu_a, \]

which expresses the fact that the initial energy of the electron, added to the photon energy, must be equal to the height of the potential barrier \(h\nu_a\) at the surface of the metal, the allowed value \(k_1''\) taken from (26d), and remembering that \(k_0^2=\gamma\nu_1\). (x) Thus we obtain a second spectral threshold at \(\nu_0'\); this new threshold is sen—

* The absorption of lower frequencies in a metal is connected with collisions of electrons with lattice ions and with one another.—S. Ch.

tivity may be called the threshold for the volume effect. For a number of metals the frequency \(\nu_0^1\) is almost twice as high as the long-wavelength boundary for the surface effect; for alkali metals it is greater than the frequency corresponding to the first maximum of the spectral-distribution curve. The volume threshold is not sharp, and its position cannot be established exactly; a small volume effect is observed in the range of frequencies between the surface and volume thresholds. In the range of frequencies greater than \(\nu_0^1\), emission from within the metal can cause a secondary rise of the spectral-distribution curve. This phenomenon was in fact discovered on the surfaces of alkali metals. Fig. 3, due to Tamm and Shubin, shows how the observed spectral distribution is composed of emission due to the surface effect (broken line) and emission due to the volume effect (dashes). The solid curve is taken from the data of Suhrmann and Theissing\(^{18}\) for a thick layer of potassium*.

5. Penny’s theory. Penny\(^{22}\) attempts to give a more exact theory of emission in a thin layer of metal, using a model possessing a lattice structure and introducing an absorption coefficient for light inside the metal. What is most important in his theory is that it requires the presence in the metal of a number of discrete levels and bands of allowed electron energies, separated from one another by forbidden energy regions**. Unoccupied by electrons, allowed

Fig. 3. The observed spectral-distribution curve is composed of curves corresponding to the surface effect (broken line) and to the volume effect (dashes). The solid line represents the emission observed by Suhrmann and Theissing on a thick layer of potassium (Tamm and Shubin).

* The spectral maxima described here should not be confused with those observed in the selective photoeffect from sensitized surfaces (for example, from a potassium surface sensitized by a glow discharge in a hydrogen atmosphere). The two effects are entirely different both in their nature and in their order of magnitude (in the case of the selective photoeffect the maxima prove to be much more sharply expressed and the photoelectric currents are ten times greater than in the photoeffect of the corresponding clean surfaces; cf., for example, R. Suhrmann and H. Theissing, loc. cit., Fig. 2 and Table 1). — S. Ch.

** This general result of the quantum theory of crystal lattices was obtained by Peierls (R. Peierls, Ann. d. Phys., 4, 121, 1930) and by Kronig and Penny (R. de L. Kronig and W. G. Penney, Proc. Roy. Soc., L. A., 130, 499, 1931). For an electron situated in a potential field,

the energy bands extend from \(\varepsilon\) to \(\varepsilon_a\) and beyond. The existence of these bands cannot be confirmed by existing experimental methods. The other results of Pennie’s theory are analogous to the results of the theories considered above.

6. General remarks. The best among the works developing a general theory of the photoelectric effect should be recognized as the work of Tamm and Shubin. Despite a certain incompleteness, this theory nevertheless gives a good explanation of a whole series of phenomena observed experimentally. Summing up the results of the quantum-mechanical theories, one may make the following general summary.

All the theories give a curve of spectral distribution forming a finite angle with the abscissa axis at \(\nu=\nu_0\). All of them proceed from the Fermi distribution at \(0^\circ\mathrm{K}\). They give a maximum at a frequency \(\nu_{\max}\), such that \(\nu>\nu_{\max}>\nu_0\).

According to the theory of Tamm and Shubin there should be a maximum, a minimum, and a secondary rise. For heavy metals the second spectral threshold, calculated theoretically, is situated farther in the ultraviolet part of the spectrum than that found experimentally. All the theories lead to the result that the component of the electric vector normal to the surface plays a greater role in the ejection of electrons than the components parallel to the surface*. In the approximation given here, the formulae of Fröhlich and of Tamm and Shubin depend only on \(E_z\).

In conclusion we note that Frenkel’s criticism of the theory of Tamm and Shubin,\(^{23}\) as well as the other explanation of the spectral maximum given by him in the same article, proved to be erroneous.\(^{24}\)

II. Individual photoelectric phenomena

The general theories considered by us above give a correct description of the most general and characteristic properties of the photoelectric

* possessing spatial periodicity, forbidden energy bands exist for any nonzero value of the potential barriers, but their width tends to zero when the barriers are decreased without bound. — S. Ch.

** Following Tamm and Shubin, one can give the following simplified explanation of this phenomenon. As we have seen, the transition \(k'' \to k'\) is possible only in the case when

\[ \iiint \psi_{k'}^{*}\left(\mathfrak{E}\,\mathrm{grad}\,\psi_{k''}\right)\,dx\,dy\,dz \ne 0. \]

Let us have a metal bounded by the \((xy)\)-plane. Then the wave functions of the electrons in the metal will have the form
\[ w(zt)e^{-i(k_1''x+k_2''y)}, \]
and the dependence of the first factor on \(z\) will be determined by the height of the surface barrier. Neglecting the dependence of \(\mathfrak{E}\) on the coordinates [see the derivation of formula (26)] and setting \(E_z=0\), we obtain, taking into account the orthogonality property of the wave functions,

\[ \iiint \psi_{k'}^{*}\left(\mathfrak{E}\,\mathrm{grad}\,\psi_{k''}\right)\,dx\,dy\,dz = -i\left(E_x k_1''+E_y k_2''\right) \iiint \psi_{k'}^{*}\psi_{k''}\,dx\,dy\,dz =0, \]

i.e. the tangential components give no emission at all. The small effect observed experimentally must be ascribed to the action of the lattice field, which we have not taken into account, and also to the roughness of the metal surface. — S. Ch.

effect; nevertheless, there exists a whole series of phenomena of a more special character, mainly connected with surfaces of complex structure, which remain unexplained by them. Often, however, in this case as well we can be helped by one or another application of quantum-mechanical theories, and sometimes even by classical considerations. Some of these considerations must subsequently enter into an improved general theory of the photoelectric effect.

A. The influence of temperature on photoelectric emission

In the quantum-mechanical theories considered, we proceeded from the velocity distribution at \(0^\circ\mathrm{K}\), with the work function \(\varepsilon_0=h\nu_0\) being determined from the equality \(\varepsilon_0=\varepsilon_a-\varepsilon\). If the temperature of the metal is above absolute zero, then there exist electrons whose energy is greater than \(\varepsilon\). Owing to the absence of a definite upper limit for the energy of the electrons which could serve as a characteristic energy in the case under consideration, we shall have to retain the former definition of the work function for all temperatures. Thus, at temperatures above absolute zero, electrons can be torn out of the metal by light quanta whose frequency is less than the frequency of the spectral threshold. With the terminology we have adopted, the work function will depend on temperature only if the energy required to tear out an electron possessing a normal component of energy equal to \(\varepsilon\) changes with temperature, i.e., in other words, if \(\varepsilon_a\) depends on \(T\).

Lawrence and Linford \(^{25}\), working with potassium films, found a discrepancy between the theoretical curves of Wentzel and Houston, corresponding to equations (19) and (20), and the spectral-distribution curves they observed, which approached the abscissa axis by touching it. They report that this discrepancy can be explained by a temperature effect.

Later Fowler \(^{26}\) developed a theory that gives various methods for determining the spectral boundary at \(0^\circ\mathrm{K}\) from experimental data. He constructs his theory for three different cases corresponding to different initial assumptions and determines, among other things, the dependence of the emission \(I_\nu\), corresponding to the long-wavelength boundary, on temperature. Fowler’s assumptions are as follows:

1) All electrons whose energy is greater than \(\varepsilon_a\) escape outward. The result at \(T=0\) is analogous to the first term of Wentzel’s first equation [equation (17)]. It is found that \(I_\nu \sim T\) and that for \(T>0\) the curves reach zero, forming a finite angle with the abscissa axis. The calculated temperature effect proved smaller than the observed one, so that the case considered was rejected as not corresponding to reality.

2) All electrons with a normal component of energy exceeding \(\varepsilon_a\) escape outward. For \(T=0\) one obtains a result similar to the first term of Houston’s equation (18) and to Wentzel’s corrected formula (19). In this case \(l_{\nu_0}\sim T^2\).

3) A term is introduced to take account of the probability of excitation of an electron by light. For \(T=0\) one obtains a result analogous to the first term of Fröhlich’s formula (22a). In this case \(l_{\nu_0}=T^2\).

In the last two cases the curve of the spectral distribution approaches the abscissa axis, touching it. The temperature corrections in both cases come out correctly, and the degree of accuracy of the existing experimental data does not yet permit an unambiguous decision in favor of one or another assumption.

Jung and Franck \(^{27}\) determine the probability of an electron’s escape from the metal by a method different from that applied by Fowler in the last of the variants of the theory considered; they obtain \(l_{\nu_0}\sim T^{5/2}\). This expression has not yet been checked against experimental data, but it is doubtful that the difference in the temperature corrections is so significant that it could be detected.

In calculating the distribution function over the spectrum for different temperatures on the basis of the second assumption, Fowler was interested mainly in the region of frequencies near the long-wave boundary. This made it possible for him to introduce the simplifying assumption that the probability of excitation of an electron does not depend on the frequency of the light or on the energy of the electron. He then determines the number of electrons which, on receiving the energy \(h\nu\), acquire a normal component of energy sufficient for escape.

Integrating equation (4), which gives the distribution of electrons over velocities, over all values of \(\xi\) and \(\eta\), we obtain the number \(n(\zeta)\,d\zeta\) of electrons in unit volume whose \(z\)-components of velocity lie between \(\zeta\) and \(\zeta+d\zeta\). The number \(\overline{N}\) of electrons per unit volume which, under the action of light of frequency \(\nu\), can acquire a \(z\)-component of energy sufficient for ejection will be equal to the sum of the expressions \(N(\zeta)\,d\zeta\), taken over all velocities satisfying the inequality \(\frac{1}{2}m\zeta^2 \geq h(\nu_a-\nu)\), i.e.:

\[ N=\frac{2m^3}{h^3} \int_{\zeta=-\infty}^{+\infty} \int_{\eta=-\infty}^{+\infty} \int_{\frac12 m\zeta^2=h(\nu_a-\nu)}^{+\infty} \frac{d\zeta\,d\eta\,d\xi} {e^{\frac{\frac12 m(\xi^2+\eta^2+\zeta^2)-\varepsilon}{kT}}+1}. \tag{29} \]

Depending on the sign of the quantity \(\mu=\dfrac{h\nu-h\nu_0}{kT}\), we obtain one of the two following series expansions:

\[ \overline{N}= \frac{\pi(2m)^{3/2}}{h^3} \frac{k^2T^2}{(h\nu_a-h\nu)^2} \left[ e^\mu-\frac{e^{2\mu}}{2^2}+\frac{e^{3\mu}}{3^2}-\cdots \right] \quad (\mu<0) \tag{30} \]

and

\[ \bar N = \frac{\pi (2m)^{1/2}}{h^3}\, \frac{k^2 T^2}{(h\nu_a-h\nu)^{1/2}} \left[ \frac{\pi^2}{6}+\frac{1}{2}\mu^2 - \left\{ e^{-\mu}-\frac{e^{-2\mu}}{2^2}+\frac{e^{-3\mu}}{3^2}-\cdots \right\} \right] \quad (\mu \geq 0). \tag{31} \]

As \(T \to 0\) we have:

\[ \bar N \sim \frac{(h\nu-h\nu_0)^2}{(h\nu_a-h\nu)^{1/2}} \quad \text{for } \nu>\nu_0, \tag{32a} \]

\[ N=0 \quad \text{for } \nu<\nu_0, \tag{32b} \]

\[ N \sim T^2 \quad \text{for } \nu=\nu_0 . \tag{33} \]

The assumption that the emission is proportional to the number of electrons with such a normal component of energy which, when increased by \(h\nu\), allows the electron to leave the metal, may be written in the form: \(I \sim N\). Then equations (30) and (31) give:

\[ \frac{I\,(h\nu_a-h\nu)^{1/2}}{T^2} = A f(\mu) = A f\!\left(\frac{h\nu-h\nu_0}{kT}\right), \tag{34} \]

where \(A\) is a constant independent of \(\nu\) and \(T\), and \(f(\mu)\) denotes the terms enclosed in square brackets in equations (30) and (31), the choice of one or the other equation depending on the sign of \(\mu\).

Near the threshold \((\nu_a-\nu)\) is large, and a small change of \(\nu\) hardly changes the value \((\nu_a-\nu)^{1/2}\), which may therefore be included in the constant \(A\). Denoting \(\varphi(\mu)=\log f(\mu)\) and taking the logarithm of equality (34), we obtain:

\[ \log \frac{I}{T^2} = B+\varphi(\mu) = B+\varphi\!\left(\frac{h\nu-h\nu_0}{kT}\right). \tag{35} \]

The graph of the dependence \(\varphi(\mu)\) may be regarded as a modified curve of the spectral distribution; its form will be the same for all surfaces. If, on the same scale, one constructs from the experimental data the graph of the dependence of the quantity \(\log \frac{I}{T^2}\) on \(\frac{h\nu}{kT}\), then a curve of the same form is obtained, but shifted relative to the theoretical curve. The vertical displacement of the curves necessary to bring the two curves into coincidence is a measure of the constant \(B\), which depends on the choice of units for the current and for the light intensity, and also on the probability that an electron is torn out by a light quantum. But since the latter, for a given surface, is practically independent of temperature, the curves for one surface at different temperatures must have one and the same vertical displacement relative to the theoretical curve.

In constructing the theoretical curve, we plotted along the abscissa axis

\[ \mu=\frac{h\nu-h\nu_0}{kT}; \]

on the abscissas of the experimental curve the quantity \(\frac{h\nu}{kT}\) is plotted; thus the horizontal displacement of the curves relative to one another must be equal to \(\frac{h\nu_0}{kT}\); from this \(\nu_0\) and the work function are calculated.

Fig. 4 shows the results of an analysis of the data for degassed palladium, carried out by the method described by DuBridge and Roe1. The theoretical curve is drawn as a solid line; the points correspond to the results of observations at different temperatures, and here a parallel shift has already been made in order to bring both curves into coincidence.

Fig. 4. Analysis of the results of photoelectric measurements for pure palladium by Fowler’s method. The points show observational data at different temperatures; they have already been shifted to bring the experimental curve into agreement with the theoretical one (DuBridge and Roe).

The method described requires determination of the relative spectral intensities of the light source for different frequencies. DuBridge2 showed that it is possible to dispense with this as well. He proposes plotting the dependence \(\varphi(\mu)\) on \(\log |\mu|\), and then determining experimentally the emission \(I\) at different temperatures for light of some single frequency and plotting the dependence \(\frac{I}{T^2}\) on

\[ \log \frac{I}{T}(=-\log T). \]

To bring both curves into coincidence, one applies—

again make a parallel shift in both directions, the vertical shift not playing a large role, while the horizontal shift, now equal to

\[ \log \frac{h\nu-h\nu_0}{k} \]

(since

\[ \log \mu=\log \frac{h\nu-h\nu_0}{k}-\log T \]

), will allow \(\nu_0\) to be determined if the frequency \(\nu\) of the incident light is known.

To show the compatibility of both methods and the degree of agreement of the results of the analysis of the experimental data carried out by both methods, Table 1 gives the results obtained for pure palladium.

Both methods give, in essential respects, one and the same result. In principle, Fowler’s method is perhaps more irreproachable. In both cases the error in determining the spectral threshold from observational data considerably exceeds the error arising from slight contamination of the electron-emitting surface. For a clean surface, the choice of method should be dictated by the convenience of obtaining the required experimental data.

TABLE 1

Fowler’s method \(^{29}\)	Fowler’s method \(^{29}\)	DuBridge method \(^{29}\)	DuBridge method \(^{29}\)
Surface temperature (°K)	Work function (V)	Wavelength of incident light (Å)	Work function (V)
305	4.96	2482	4.96
400	4.97	2399	4.95
550	4.97	2378	4.94
730	4.97	2345	4.94
830	4.98	2302	4.98
925	4.98	2225	4.98
1005	4.96
1078	4.97
Average . . . 4.97 V	Average . . . 4.97 V	Average . . . 4.96 V	Average . . . 4.96 V

Table 1, in agreement with other data \(^{18}\) for clean surfaces, shows no shift of the threshold with temperature in the range of temperature variation from room temperature to \(1100^\circ\) K. Meanwhile, for palladium it would have been easy to detect a change of the spectral threshold by \(1\%\), which corresponds to 0.05 V. Thus it is established that, for the metals investigated, the height of the potential barrier \(h\nu_a\) remains constant to within \(1\%\).

The magnitude of the vertical shift necessary for matching the experimental curve with Fowler’s theoretical curve is equal to the constant \(B\) from equation (35), and the antilogarithm of \(B\) is equal to the constant \(A\) from equation (34). For a given surface, \(A\) is proportional to the probability that the incident light quantum has torn an electron out of the metal. The reflection coefficients for most metals are practically constant, so that an appreciable change in \(A\) signifies a change in emission. Thus Fowler’s method makes it possible to consider separately changes in the long-wavelength limit and in the emissive ability. If \(B\), and consequently also \(A\), changes with temperature, then in Fowler’s method this will show itself in a change of the relative vertical shift

curves with temperature. The DuBridge method has the disadvantage that, in the case of a dependence of \(B\) on \(T\), it will no longer be possible to bring both curves into coincidence. The change in \(B\) upon contamination of the surface and upon a change in temperature can be followed from the data of Welch and Warner.

Welch \(^{30}\) determines the spectral distribution function for various metals at various instants of time after mechanical cleaning of the surface.

The changes occurring in this process must evidently be ascribed to the action of gases and vapors adsorbed on the surface. The work function increases with time; the magnitude of this increase reaches \(0.16\ \mathrm{V}\); the exception is germanium, for which the work function decreases slightly. Changes in the vertical displacement corresponded in all cases to a decrease in the emissivity, which fell to approximately one half of its initial value.

Warner \(^{31}\) used Fowler’s method to analyze experimental data obtained for a tungsten filament. His plot (see Fig. 2 of his paper) shows a change in the vertical displacement, which corresponds to an increase in the emissivity by about a factor of 10 when the temperature is changed from \(790^\circ\mathrm{K}\) to \(1100^\circ\mathrm{K}\). This result was explained by contamination of the tungsten.

In the cases considered we are dealing with large changes in the emissivity, which show up clearly in the spectral distribution curve. When the changes are small and are accompanied by a displacement of the threshold, analysis by this method acquires special significance.

For the convenience of analyzing the experimental material by both of the methods set forth, DuBridge gave a table of values of \(\log_{10}\psi\) and \(\varphi(\mu)\) for values of the argument \(\mu\) from \(-8.0\) to \(+50.0\).

B. Saturation phenomena

In contrast to clean surfaces, where the saturation phenomenon is observed already in weak fields, some complex surfaces require, in order to obtain the saturation current, the application of an anomalously large accelerating potential. Ives \(^{32}\) reports that saturation in alkali metals sets in at larger accelerating fields the thinner the metal layer. Zhurman \(^{33}\) confirms this result and finds, in addition, that when the surface is illuminated by light whose frequency differs little from the long-wave limit of the photoelectric effect, saturation requires a stronger field than when illuminated by light of high frequencies. Haxford \(^{34}\) likewise observed both these effects, investigating oxide cathodes of the type used in ordinary thermionic emitters.

The phenomena under consideration are in close connection with the discover-

deviations observed for activated surfaces from Schottky’s equation[^35] for thermionic emission in an accelerating field. Schottky’s equation is derived under the assumption that the electrons emitted from the metal are in the field of their electric image in the metal surface [see equation (10)]; it proved valid for clean surfaces[^36],[^37].

Debye[^38] showed, assuming the law of electric images to be valid, that in an accelerating field equal to \(E \dfrac{CGSE}{cm}\), the effective work function \(\varepsilon_e\) obeys the equation:

\[ \varepsilon_0 - \varepsilon_e = c(\varrho E)^{\frac{1}{2}} . \tag{36} \]

Lawrence and Linford[^25] found that the threshold shift for thick layers of potassium on tungsten approximately follows this relation. In weak fields, some surfaces give a greater threshold shift than that required by the theory.

Deviations from Schottky’s equation for thermionic emission were investigated by Becker and Mueller[^39] and by Reynolds[^40]. Strong deviations from equation (36) were found by Linford[^41] for thoriated tungsten and by Huxford[^34] for oxide surfaces.

Becker and Mueller showed that, if the effective work function is known as a function of the applied field voltage, then the field strength at the surface, against which the emitted electrons must move, can be calculated from the formula:

\[ \frac{d \varepsilon_e}{dE} = - z_1 e, \tag{37} \]

where \(E\) is the applied field voltage in absolute electrostatic units, and \(z_1\) is the distance from the surface to the point where the applied field is compensated by the surface field, giving a resultant field equal to zero. The field voltages calculated by the authors were of the order of the electric image fields at distances less than \(2 \cdot 10^{-6}\,\mathrm{cm}\), but were considerably greater than the latter at large distances. Thus, at a distance of \(10^{-4}\,\mathrm{cm}\) the calculated field was of the order of \(1000\,\dfrac{\mathrm{V}}{\mathrm{cm}}\), whereas the electric-image field should have been only \(3.6\,\dfrac{\mathrm{V}}{\mathrm{cm}}\).

Formula (37) is more conveniently written in the form:

\[ \frac{d \nu_e}{dE_v} = - z_1 e \, 300 h, \tag{38} \]

where \(\nu_e\) is the effective spectral threshold and \(E_v\) is the applied field in volts per centimeter.

To explain the retarded saturation, Langmuir[^42] proposed that the substance covering the surface is situated on it

not as a homogeneous layer, but as spots that cover the other places more densely. The work function for a complex surface depends on the thickness of the film covering the surface, and therefore it will be different for different regions of an inhomogeneous surface. This explanation has been used by a number of investigators for the interpretation of experimental results.

In order to carry out a quantitative comparison of theory with experiment, it is necessary to calculate the magnitude of the local fields of the spots mentioned and to determine the expected emission when this field is superposed on the field of the electric image.

If two metals \(A\) and \(B\), having respectively work functions equal to \(\varepsilon_A\) and \(\varepsilon_B\), are brought into contact, then between them there arises a contact potential difference equal to \(V_B - V_A\). From energy considerations it is evident that

\[ \varepsilon_A-\varepsilon_B = V_B - V_A + P_{AB}, \tag{39} \]

where \(P_{AB}\) is the Peltier coefficient for the pair of metals under consideration; because of its smallness in comparison with the contact potential difference it may be neglected. Thus the contact potential difference between two metals may be taken as equal to the difference of their work functions. The metal possessing the smaller work function is more electropositive. The presence of other metals in the circuit between \(A\) and \(B\) should not affect relation (39).

Fig. 5. Potential barrier of the electric-image field and the effect on it of the local field of inhomogeneities in the film and of the external accelerating field: \(ADB\) is the potential of the electric image, \(N\) and \(M\) are the potentials of spots possessing respectively the greatest and the smallest work function, \(NB\) and \(MB\) are the potentials of the spot field. The resulting potential barriers are \(AFB\) and \(AEB\). The broken lines show these same barriers in an accelerating field of \(1000\ \mathrm{V\cdot cm^{-1}}\) (the curves are calculated for \(MN=V_0=0.36\ \mathrm{V}\) and a spot diameter equal to \(b=1.8\cdot 10^{-4}\ \mathrm{cm}\). See p. 171 and Fig. 6).

If the work function for certain regions of the surface differs from the work function of the remaining part of the surface, then between both parts of the surface there must arise a certain contact potential difference, which in turn entails the appearance of local electrostatic fields. The contact potential of the entire surface as a whole can therefore be considered only relative to a surface sufficiently far away that the action of the local fields may be neglected. The contact potential determined in this way defines the work function in the absence of a field. It

represents the mean value of the contact potentials for the various parts of the surface.

In the absence of an accelerating field, the emission of electrons from parts of the surface possessing the smallest work function, i.e., from the most electropositive parts, is hindered, in addition to the field of the electric image, also by the field of the aforementioned patches. Electrons, on the other hand, which leave electronegative parts of the surface, do work against the forces of the electric image, diminished by the field of the patches. In Fig. 5 the potential barriers at surfaces of both types are represented schematically for the case in which the field of the patches extends to a much greater distance from the surface than the field of the electric image; it is assumed that the areas of patches of both types are equal. The horizontal straight line \(OB\) represents the potential in the space outside the surface in the absence of a field; we may take it equal to zero. The curves situated above and below the straight line \(OB\) refer respectively to parts of the surface with greater and lesser work function. \(MN\) is equal to the contact potential difference between parts of both types. The curves \(MB\) and \(NB\) represent the course of the potential for the centers of patches of both kinds. The sum of this potential and the potential of the field of the electric image (the curve \(ADB\)) gives the resultant potential barrier, which is represented for the two cases by the corresponding curves \(AEB\) and \(AFB\). The potential curves for other points of the surface have a form intermediate between these two. The height of the straight line \(OB\) determines the long-wavelength limit in the absence of a field; it is then clear that electrons whose energy is exactly equal to the energy required to tear them from the surface will not be able to pass through the most electronegative parts of the surface. The effective work function at the centers of the most electropositive parts in the absence of a field is greater than the work function that would exist in the absence of local fields by the amount \(OM\).

If an accelerating electric field is applied, the potential is represented by the straight line \(OC\) (Fig. 5), whose slope will determine the field strength. The resultant potential barriers (dotted lines) are obtained by adding the potential of the field to the potential barrier in the absence of the field. The change in the work function is equal to the difference between the maximum heights of the barrier in the presence of the field and without it. The drawing shows that the effective work function for electropositive patches (curves \(AEB\) and \(AEC\)) decreases by a considerably larger amount than for electronegative patches (curves \(AFB\) and \(AFC\)). The work function for a clean surface, whose potential barrier (curves \(ADB\) and \(ADC\)) is the barrier of the field of the electric image, also decreases somewhat more strongly than the work function for electronegative patches. Thus, in accelerating fields the effective work function of electropositive regions determines the behavior of the entire

surface as a whole, and the greater part of the thermoelectrons is emitted precisely by these regions.

For small changes in the voltage of the applied field, the change \(\Delta \varepsilon\) of the greatest height of the potential barrier is, in the first approximation, equal to the change of the potential of the applied field at the point corresponding to the top of the barrier. This, in turn, is equal to the product of the change of the applied field \(\Delta E\) by the electron charge \(e\) and by the distance \(Z_1\) from the surface to the top of the barrier, i.e. to the point where the field intensities of the surface field and of the field applied from outside become equal in magnitude and opposite in direction, i.e. \(\Delta \varepsilon = Z_1 e \Delta E\). Hence, on passing to the limit, equation (37) is obtained.

Compton and Langmuir\(^ {37}\) made the assumption that the local inhomogeneities are arranged on the surface in a checkerboard pattern, each spot having the form of a square with side \(b\ \mathrm{cm}\), and that between two neighboring spots there exists a contact potential difference equal to \(V_0\). Under these assumptions the authors compute the local fields and express the result in the form of a Fourier series. Assuming that the spots are clusters of closely packed thorium atoms, they arrive, on the basis of certain considerations, at the numerical values \(b = 10^{-6}\ \mathrm{cm}\) and \(V_0 = 1.9\ V\). The dependence calculated for such a surface of the thermoelectron emission on the applied accelerating field follows the Schottky law in weak fields, but departs from this law more and more as the field voltage is increased; experiment, however, gives precisely the opposite result. They therefore conclude that the spot theory is not capable of explaining the observed emission phenomena.

However, an analysis of the thermoelectron observations carried out by Becker and Rojansky\(^ {13}\), and of the photoelectric observations made by Linford\(^ {11}\), shows that, with a proper choice of \(V_0\) and \(b\), the computations lead to results in agreement with experiment. For \(b\) one obtains values of the order of \(10^{-4}\), while \(V_0\) proves to depend on the amount of adsorbed substance. As to the relative arrangement of the spots, the assumption made by Compton and Langmuir is retained. In space near the surface, the potential \(V_p\) of the field of the spots differs somewhat from the potential of the surface itself. Introducing a correction for this change, we obtain the expression:

\[ V_p = \frac{1}{2}V_0 + \frac{8V_0}{\pi^2} \sum_{jk} (-1)^{j+k} e^{-\{(2j+1)^2+(2k+1)^2\}^{1/2}\frac{\pi z}{b}} \times \]

\[ \times \frac{\cos(2j+1)\frac{\pi x}{b}}{2j+1} \cdot \frac{\cos(2k+1)\frac{\pi y}{b}}{2k+1}, \qquad j,k=0,1,2\ldots \tag{40} \]

where the origin of coordinates is placed at the center of an electropositive spot.

The photoelectric method for determining the surface field is associated with measurements of the long-wavelength threshold. Electrons with the lowest energy can be emitted from points on the surface located at the centers of spots having a smaller work function; these points will correspond to field strengths determined by the indicated method. To calculate the field strength \(E_p\) near the center of an electropositive spot, one must put, in equation (40), \(x=y=0\) and \(E_p=-\dfrac{dV}{dz}\). In accordance with the data of the photoelectric measurements, in formula (40) only the first term of the expansion was retained, i.e. it was assumed that \(j=k=0\). The surface field \(E_s\) is composed of the field of the electric image \(E_i\) and the field of the local inhomogeneities \(E_p\):

\[ E_s = E_i + E_p = \frac{e}{4z^2} - \frac{8V_0 z^{\,[[unclear: exponent]]}}{[[unclear: denominator]]\,b} \cdot e^{-\frac{2^{1/2}z}{b}} \tag{40a} \]

Taking \(V_0=0.36\ \mathrm{V}\) and \(b=1.8\cdot 10^{-4}\ \mathrm{cm}\), the calculated and observed values for the field strengths can be brought almost into complete agreement. Becker and Rojansky found that numerical values of the same order are also required to explain the data of thermionic observations.

For thoriated tungsten the work function is approximately \(1.5\ \mathrm{V}\) lower than for pure tungsten. The small contact potential difference between the spots therefore indicates that thorium atoms are distributed over the whole surface, but with nonuniform surface density. The sizes of the spots are, in order of magnitude, the same as the sizes of the tungsten crystals in the filaments used. Dr. Becker assumes a definite connection between the sizes of the former and the latter.

The general character of the field of local inhomogeneities is shown in Fig. 6, where the dependence of the surface field strength on the distance from the surface is plotted on a logarithmic scale. The straight line \(AB\) gives the field of the electric image. The dotted curve represents the field of the spots, calculated under the assumptions made by Compton and Langmuir, with numerical values of the constants \(V_0=0.36\ \mathrm{V}\) and \(b=10^{-6}\ \mathrm{cm}\). The solid curve above it, merging with the curve of the electric-image field at distances from the surface of the order of \(10^{-5}\ \mathrm{cm}\), gives the resultant surface field. The dashed curve represents the field of the local inhomogeneities for the values of the constants \(V_0=0.36\ \mathrm{V}\) and \(b=1.8\cdot 10^{-4}\ \mathrm{cm}\); the resultant surface field corresponding to these values is depicted by the solid curve, which merges with the curve of the electric-image field in the region of strong fields and with the curve of the spot field in the region of weaker fields. In both cases the arrows indicate the distance from the surface equal to the diameter of the spot.

The circles mark the values of the surface-field strength,

certain ones from those photoelectric measurements⁴¹ for which the constants were determined. The triangles indicate the field strengths for 70% thoriated tungsten, calculated by Becker and Mueller³³ from thermionic-emission data.

Fig. 6. Curves of surface fields, plotted on a logarithmic scale. \(AB\) is the electrostatic image force. The broken line represents the field of spots at \(V_0 = 0.36\ \mathrm{V}\) and \(b = 1.8 \cdot 10^{-4}\ \mathrm{cm}\). The solid line above it, merging with \(AB\) in the region of small and strong fields, represents the resultant field above a spot with a small work function. The circles denote fields determined from photoelectric measurements (Linford). From these data the spot constants were calculated. The triangles give the field strengths determined from thermoelectronic measurements with thoriated tungsten (Becker and Mueller). The crosses correspond to fields near an oxidized cathode, found from Huxford’s photoelectric measurements. The dotted line and the solid line above it represent the corresponding field of spots and the resultant field near the surface, for the values of the spot constants given by Compton and Langmuir: \(V_0 = 1.9\ \mathrm{V}\) and \(b = 10^{-6}\ \mathrm{cm}\). The arrows mark the points of the curve corresponding to a distance from the surface equal to the diameter of the spot.

From equation (40a) and Fig. 6 the following facts follow: 1) The field of local inhomogeneities at the surface is proportional to \(V_0\) and inversely proportional to \(b\). 2) On moving away from the surface, the field decreases approximately 10-fold at a distance \(\frac{1}{2}b\), 100-fold at a distance \(b\), and 1000-fold at a distance \(2b\). The figure shows why the numerical values of the constants chosen by Compton and Langmuir lead to departures from Schottky’s equation at large fields.

It should be remembered that the calculations given above were based on an idealized model, and that the calculated constants therefore represent only certain average values. If, instead of assuming that all spots are approximately the same in magnitude, one introduces the assumption that in the film covering the metal the fluctuations of the layer density are often superposed on one another at spots of large dimensions, then we must obtain a curve with two maxima, like the uppermost solid curve in Fig. 6. Using Huxford’s photoelectric measurements³⁶ with oxidized cathodes and carrying out calculations by formula (38), we in fact obtain a similar result; in Fig. 6 the points of the curve constructed in this way are shown by crosses. The course of this curve can

explain by the superposition of spots with a diameter of about \(10^{-5}\) cm on spots with a diameter of \(2\cdot10^{-4}\) or \(3\cdot10^{-4}\) cm, although for a definitive clarification of the question there is a lack of experimental data in the region of sufficiently strong fields. To explain the absence of saturation of the photocurrent observed for some surfaces in weak fields, it is enough to assume the presence of spots for which \(b\) is of the order of \(10^{-4}\)—\(10^{-3}\) cm and \(V_0\) is of the order of several tenths of a volt. Light whose frequency only slightly exceeds the frequency of the long-wave limit can tear an electron only from surfaces possessing a smaller work function. Weak accelerating fields will cause comparatively large changes in the work functions of these surfaces, and as a result, in the spectral region adjacent to the long-wave limit, we shall have large relative changes in emission.

If we illuminate the surface with light of higher frequency, electron emission will occur from all parts of the surface. The long-wave limit of those parts of the surface with the larger work function will depend to a somewhat lesser degree on the applied field than for a clean surface. The relative change in emission from parts of the surface possessing the smaller work function will be smaller than in the region of low frequencies, owing to the fact that a given magnitude of the absolute change will constitute a smaller fraction of the total emission. If the frequency of the light approaches the frequency corresponding to the spectral maximum, then the absolute change in emission with the effective work function will decrease. As a result, in the region of high frequencies we shall observe a much smaller relative change of the total emission with the applied field voltage than near the long-wave limit.

An especially strong change of the work function with the applied field voltage was observed by Nottingham \(^{45}\). One of the films of an alkali metal on a heavy metal showed a decrease of the effective work function by \(1.9\ \mathrm{V}\) at an accelerating potential equal to \(4\ \mathrm{V}\). The author gives no details of the experiments; it is indicated, however, that when using plane-parallel electrodes at a distance \(d\) cm from one another, the field required for such a shift must be equal to \(\frac{4}{d}\ \mathrm{V/cm}^{-1}\). Turning to equation (37), we see that if a change in the work function by \(1.9\ \mathrm{V}\) is caused by a field of \(\frac{4}{d}\ \mathrm{V/cm}^{-1}\), then the mean distance \(z_1\) from the surface to the point where the surface field and the applied external field become equal must be of the order of \(\frac{1}{2}d\), i.e., half the distance between the electrodes. No uniform distribution of charges over the surface can produce surface fields at such distances, and it remains to admit that in the film there are inhomogeneities whose linear dimensions have the same

order as the distance between the electrodes. These inhomogeneities may arise from the nonuniform deposition of the alkali metal on the film. In work carried out by the author of the present article jointly with Laurensen, devoted to the investigation of a thin film of potassium on a tungsten filament, where there certainly were not and could not be spots comparable in size with the distance between the electrodes, no such effect could be observed. The abrupt jump in the rate of change of the long-wavelength limit with the field strength—from a value characteristic of the retarding field to a value approximately twice as large as that required by the theory of electrical images—was used to determine the zero field between the anode and cathode.

Zurman and Teising \(^{4)}\) report that they have established in a potassium film on platinum the presence of regions with widely differing work functions. The authors use Zurman’s method \(^{8}\) for calculating the total emission \(I_c\) from equation (2) and the observed curves of the spectral distribution. With the aid of equation (1) they construct a graph of the dependence of \(\log I/T^2\) on \(1/T\). The theory requires that the dependence be represented by a straight line whose slope is equal to \(-\dfrac{\varepsilon_0}{k}\). The authors found that the slope changes with temperature. In the temperature range \(1200^\circ\mathrm{K} < T < 2000^\circ\mathrm{K}\), where \(T\) is the temperature characteristic of black-body radiation, the slope of the curve corresponds to a work function of \(2.02\ \mathrm{V}\). In the range \(2400^\circ\mathrm{K} < T < 4000^\circ\mathrm{K}\) the slope corresponded to \(2.98\ \mathrm{V}\). They explained this phenomenon by the fact that at low temperatures of the radiation source electrons could be torn out only from surface regions with a small work function, whereas with a high-temperature source electrons could be torn out from everywhere, and the work function determined under these conditions referred to the entire surface as a whole.

If one assumes \(r = 2\), then the slopes will correspond best to the values \(2.46\) and \(2.93\ \mathrm{V}\), while for \(r = 5\) a good straight line is obtained. Zurman therefore indicates that the best results are given by values \(r > 2\).

Analysis of the experimental data by Fowler’s method shows that for two thicker films (see Figs. 7 and 8 in the cited article) the slope of the straight line practically does not change on going from room temperature to the temperature of liquid air. A very thin film, containing considerably less than one atomic layer (see Fig. 6 in the cited article), gave, at the temperature of liquid air, a decrease in the work function by \(0.2\ \mathrm{V}\). But here it should be borne in mind that small changes in the amount of alkali metal cause, in such thin films, significant changes in the work function, and that therefore the observed change in the work function may be attributed to condensation of a small amount of potassium on cooling the surface. The form of all the curves very closely approaches Fowler’s theoretical curve—

— a result which would have been difficult to expect if the effective work function changed appreciably with frequency.

The curvature of the graph of the dependence of $\log \dfrac{I}{T^{2}}$ on $\dfrac{1}{T}$ can be explained by the fact that the photoelectric emission does not exactly obey Richardson’s equation. The spectral distribution function which Richardson$^{1}$ obtains by equating the right-hand sides of equalities (1) and (2) gives a steeper slope of the curve near $\nu = \nu_0$ than is required by the experimental data. If the total emission, calculated from Richardson’s spectral distribution function, gives a rectilinear dependence of $\log \dfrac{I}{T^{2}}$ on $\dfrac{1}{T}$, then it should be expected that the observed curves of spectral distribution will have the form described by Suhrmann and Teichsing, even if the surface had one definite work function.

After this article had already been written, Nottingham$^{17}$ published another theory explaining the experimentally observed change of the work function with the applied field for certain complex surfaces. As already indicated, this change cannot be explained by the theory of electric images.

Nottingham analyzes both his own data and observations made by other authors on the thermoelectronic and photoelectric emission of various complex surfaces in weak accelerating and retarding fields. Studying the thermoelectronic emission of a thoriated tungsten filament as a function of the field voltage, he finds that the observed velocity distribution obeys Maxwell’s law, but corresponds to a temperature higher than that of the filament. In addition, he notes that the coefficient $A$ in Richardson’s equation (1), which is a measure of emissive ability, decreases as the field voltage is increased. Thus, at an accelerating potential greater than 6 V its value amounted to only about one tenth of its value for a clean surface.

To explain these phenomena the author introduces into consideration a potential barrier consisting of the potential of the electric image $\tau$, at $BDO$ in Fig. 2, at distances exceeding the distance from the surface to the thorium atom layer, the minimum of the potential at this layer, and a maximum, having approximately the form of a parabola, in the space between the thorium layer and the metal serving as the substrate. It is assumed that this last maximum is higher than $AB$. If the anode has the potential $AB$, then electrons possessing just enough energy to reach the anode must overcome the barrier.

If a retarding field is applied, then the form of the potential barrier, on moving away from the thorium layer, will approach a straight line with a positive angular coefficient, in contrast to the straight line $AC$ in Fig. 2, whose angular coefficient is negative. If the anode potential is above the top of the barrier between

with a film and the substrate (film barrier), then any electron with energy sufficient to reach the anode will freely pass over the barrier. In this case the observed emission turned out, as was to be expected, to be the same as if the film were absent altogether. If the anode potential falls to the potential of the electron source, then slow electrons will overcome the barrier in the film, whose transmission coefficient is less than unity. Fast electrons, however, will still pass unhindered over the barrier. As a result we obtain a preferential emission of electrons of high velocities, and, consequently, the velocity distribution of the electrons will correspond to a higher temperature than the temperature of the filament. Nottingham succeeded in constructing a model of a film barrier of atomic dimensions that filters out slow electrons.

He then used this model to explain the experimental data he had obtained, which we have already mentioned above \(^{45}\). He found that in retarding fields the displacement of the threshold follows Einstein’s equation:

\[ h\nu-\varepsilon_0=\frac{1}{2}mv^2=eV \tag{41} \]

and that in moderately strong fields the magnitude of the displacement approaches the value predicted by the theory of electric images. When the anode potential differs little from the potential of the electron source, the observed displacement proves too large to be explained by the action of the field of electric images, and not large enough to satisfy Einstein’s equation.

The explanation considered assumes that Einstein’s law remains valid so long as the anode potential is above the barrier in the film. Otherwise the slower electrons will be forced to overcome the barrier. The author notes that, since we are able to measure only currents of a definite finite magnitude, we can obtain an appreciable current only in the case when there are electrons whose excitation energy is greater than the minimum energy necessary to reach the anode after passing the barrier in the film. This effect will be the more noticeable the lower the anode potential falls; therefore the observed change in the spectral threshold will be smaller than the change in the anode potential.

Despite the complete satisfactoriness of this explanation for the given case, it ceases to be correct when an external accelerating field is applied. Extending Nottingham’s explanation to the case of weak accelerating fields, we must expect that, analogously to the preceding case, the observed displacement of the spectral threshold will be smaller than the change in the height of the barrier of the electric image outside the film, which decreases in accordance with the laws of electric images. Experiment shows, however,

angle: both Nottingham’s observational data and the data cited by other authors give a shift equal to, or even greater than, the shift predicted by the theory of electrical images.

In his latest paper Nottingham gives a diagram of his apparatus. The source of electrons was a nickel cylinder placed inside a cylindrical anode, from which it could be removed for cleaning off dirt and applying a layer of sodium. When the film on the cylinder is thin, it is difficult, or even impossible, to cover the cylinder with a uniform layer. A small change in thickness causes, in such a film, a large change in the work function. In the film there may be inhomogeneities of the same order of magnitude as the diameter of the cathode, and such inhomogeneities may, as we have seen, be the cause of the indicated effects.

The application of the theory under consideration to thermionic emission lies outside the scope of the present article. It should be noted, however, that the theory of the film barrier, whose validity in the region of weak fields is beyond doubt, will hardly remain applicable to emission in fields such as those used by Becker and Mueller[^39], for we are entitled to expect from it a result analogous to that to which it leads in the case of the photoelectric effect. We shall dwell on this point somewhat more fully.

The theory of local inhomogeneities in the film explains the “filtering out” of part of the slow electrons in thermionic emission, as well as the small values of the constant \(A\) in Richardson’s equation in strong fields. To be convinced of this, we must recall that each of the spots must be divided into areas corresponding to different potential barriers, whose form will be intermediate between the limiting barriers \(AFB\) and \(AEB\) in Fig. 5. To determine the emission it is necessary to calculate it separately for barriers of each type and then sum over the whole surface, in the same way as Becker and Rozhanskii[^44] did. If the anode potential is greater than \(F\), then all parts of the surface will have normal emission. If it lies within the limits between \(F\) and \(B\), then the emission of the electropositive parts of the surface will remain as before, while the electronegative parts will now emit only those electrons whose energy is sufficient to pass the barrier outside the film.

In other words, here we shall have a predominance of electrons of higher velocity. With a further fall of the anode potential the field becomes accelerating, and the increase in emission occurs at the expense of the centers of the electropositive spots. This entails a decrease in that part of the filament surface which emits electrons and, consequently, a decrease in \(A\).

Comparing the two theories, we see that the theory of the film barrier can give a qualitative explanation of the photoelectric

emission in very weak fields and ceases to be valid in the region of stronger fields. The theory of local inhomogeneities explains the phenomena occurring in strong fields, and, with additional assumptions of a quite acceptable character, also the phenomena observed in weak fields. In the region of thermoelectronic phenomena, the film-barrier theory likewise gives an explanation of the observational results in the region of weak fields, while the theory of local inhomogeneities explains the results obtained in strong accelerating fields.

The possibility of applying the film-barrier theory to emission in a strong field is doubtful; on the other hand, the theory of local inhomogeneities is capable of giving a qualitative explanation of a number of properties of emission also in weak fields. For a final solution of the question, more thorough experimental and theoretical investigations of this problem are necessary.

C. Influence of Space Charges

An investigation of the possible influence of space charges on the effective field of the surface and, consequently, on the photoelectric emission of this surface is of undoubted interest. Indeed, one should recall the great role played by space charges in strong thermoelectronic emission.

Bartlett and Waterman^48 calculated the surface fields under the assumption that the space charge plays the principal role even in the case of small emission. In doing so they neglect the force of interaction of the electron with its electric image, which undoubtedly exists, and use Poisson’s equation, which assumes a continuous distribution of charges, applying it also to those parts of space where the electron density is very small.

At low temperatures the space charge cannot be the cause of the appearance of fields at distances of the order of \(10^{-5}\) cm from the surface, and therefore their calculations cannot be applied to such temperatures.

Zwickker^49 carries out analogous calculations, taking into account simultaneously both the space charge and the field of the electric image. He is unable to obtain an exact solution, but approximate calculations show that the space charge plays only a small role in the formation of the field at the surface.

The influence of the space charge on the photoelectric effect at absolute zero can be described very simply. Independently of the cause of the formation of the field at the surface, we have a certain potential barrier, the top of which exceeds the energy \(\varepsilon\) of the fastest electron in the metal by an amount \(\varepsilon_0\), equal to the work function. In the absence of added energy, none of the electrons can move away from the metal surface farther,

than at that critical distance to which the height of the potential barrier \(\varepsilon\)* corresponds. Consequently, at distances from the surface exceeding the critical distance, there can be no space charges, and the latter cannot influence the potential barrier in this region. Closer to the surface the action of the space charges may become noticeable.

It is known that the law of electrical images for clean surfaces is certainly valid at large distances, where space charges cease to play a role; therefore one may consider that at distances greater than the critical distance the surface field is the field of an electrical image. To explain the observed values of the work function, the critical distance must be assigned a value of the order of \(10^{-7}\,\mathrm{cm}\); at this distance the field proves to be approximately \(4 \cdot 10^{6}\ \mathrm{V}/\mathrm{cm}^{-1}\). The effective change in the long-wavelength limit of the photoelectric effect in an accelerating field depends on the character of the field at the surface. Since the field strengths used in the experiments are much smaller than the strengths of the surface fields in those parts where space charges can play a role, changes in the spectral threshold should not depend on space charges.

In the limiting case of weak photoelectric emission, the space charge of the electrons torn out by light will be small, and therefore the observed emission characteristic at absolute zero should not depend on space charges.

When the temperature is raised, the number of electrons having sufficient energy to pass beyond the critical distance increases. If the field outside the region of space charges at absolute zero is the field of an electrical image, then it should not depend on temperature. The influence of space charges on the field in this part of space should be manifested in an increase of the work function with increasing temperature. Analysis of the data of photoelectric measurements by Fowler’s method shows that the work function in the temperature range from room temperature to \(1100^\circ\mathrm{K}\) changes by less than \(1\%\). Thus, if the space charge does exert some influence on photoelectric emission in retarding fields, this influence is in any case extremely small.

In a retarding field, the phenomenon attributed to the action of space charge was found and investigated by Marx and Meyer,\(^{10}\) who worked with a photocell in which a thick film of potassium served as the cathode. They measured, with a string electrometer, the maximum potential to which the anode is charged, and found that Einstein’s equation (41) proves to be satisfied if one uses

* From the point of view of quantum mechanics, there is a small probability that the electron will be at a distance from the metal greater than the critical distance. This effect, however, may be neglected.

monochromatic light. When using light of high frequency, the anode reaches some definite potential; if to this light one adds light of another, lower frequency, but in any case exceeding the long-wavelength frequency limit, then the anode potential decreases by an amount proportional to \((\nu_1-\nu_2)\dfrac{n_2\nu_1}{n_1\nu_2}\), where \(\nu_1\) and \(\nu_2\) are the frequencies satisfying the condition \(\nu_1>\nu_2>\nu_0\), and \(n_1\) and \(n_2\) are the numbers of electrons ejected, respectively, by light of frequencies \(\nu_1\) and \(\nu_2\).

The theoretical explanation of the phenomenon described, given by the authors, is based on the fact that, when light of lower frequency is added, there occurs a redistribution of the volume charges in the space between the electrodes, which entails an increase in the energy required by an electron in order that, after being torn from the cathode, it may reach the anode. The decrease in the maximum potential was found to depend on the ratio \(\dfrac{n_2}{n_1}\) and not to depend on the absolute values of these quantities. However, at low light intensities the electron density in the space between the electrodes will be so small that the Poisson equation used by the authors ceases to be valid, because the electric charge can no longer be regarded as distributed continuously. Further theoretical investigation of the question is of no particular significance, since the data of the experiments described can be explained in a simpler way.

For the results set forth in the first note to appear in print, where information on the details of the experiments was still lacking, Olpin\(^{51}\) gives the following explanation. The potential reaches equilibrium in monochromatic light at the moment when the number of electrons with energy sufficient to reach the anode becomes equal to the number of electrons emitted by the anode under the action of scattered light. In the case of a potassium photoelectric cell, a thin film of potassium is formed on the anode, which makes it photoelectrically sensitive and, probably, even more so than the cathode.

When light of lower frequency is added, the number of electrons reaching the anode remains the same, but the scattered light of the lower frequency will tear electrons from the anode, as a result of which its potential will fall until the necessary additional number of electrons, torn from the cathode by light of the higher frequency, compensates for this fall. Using a photoelectrically insensitive anode, Olpin was unable to detect any effect.

From the subsequent, more detailed description of the experiments of Marx and Meyer it is evident that their anode was completely shielded from scattered light, and they believed that by this they had eliminated every effect connected with scattered radiation. But since there was potassium in their photoelectric cell, a photoelectrically sensitive layer must have formed on all

glass surfaces, and these latter must have played the role of the anode in Olpin’s explanation. Photoelectrons torn from the cathode could reach the anode only in the case when the potential of the surrounding glass became equal to the potential of the anode; thus the entire interior of the photocell acted exactly as the anode would have acted had it not been screened from the light. With these additional remarks, Olpin’s explanation remains valid as before.

LITERATURE

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2. Thomson J. J., Phil. Mag., 2, 674, 1926.
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4. Hughes A. L. and Du Bridge L. A., Photoelectric Phenomena, Mc. Graw. Hill, New-York 1932; p. 200.
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6. Dushman S., Phys. Rev. (2), 21, 623, 1923.
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10. Nordheim (Nordheim L., Phys. Ztschr., 30, 177, 1929) gave an excellent review of the theory of metals, thermoelectronic and autoelectronic emission, as well as a review of the older theories of the photoelectric effect.
11. Fowler R. H., Proc. Roy. Soc., A, 118, 229, 1928.
12. In addition to the review of the older works on this question indicated in footnote 10, there exists a more recent and detailed review made by Condon (Condon E. U., Rev. Mod. Phys., 3, 43, 1931). Of later works we shall indicate the following: Frank N. H. and Young L. A., Phys. Rev. (2), 38, 80, 1931; Wentzel W., Phys. Rev. (2), 38, 1205, 1931; Rojansky and Wetzel W. Phys. Rev. (2), 38, 1979, 1931.
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15. See Lawrence E. O. and Linford L. B., Phys. Rev. (2), 36, 496, 1930.
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17. Lukirsky P. and Prilezaev S., Ztschr. f. Phys., 49, 236, 1928.
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19. Tamm I. and Schubin S., Ztschr. f. Phys., 68, 97, 1931.
20. Bloch F., Phys. ZS., 32, 881, 1931.
21. Suhrmann R. and Theissing H., Ztschr. f. Phys., 52, 453, 1928.
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31. Warner A. H., Phys. Rev. (2), 38, 1871, 1931.
32. Ives H. E., Astroph. J., 60, 209, 1924.

1. Suhrmann R., Naturwiss., 16, 336, 1928.

2. Huxford W. S., Phys. Rev. (2), 38, 379, 1931.

3. Schottky W., Phys. Ztschr., 15, 872, 1914.

4. Dushman S., Rev. Mod. Phys., 2, 381, 1930; the author gives a good review of the thermoionic studies.

5. Compton K. T. and Langmuir I., Rev. Mod. Phys., 2, 123, 1930.

In the section devoted to contact potentials and electron emission in accelerating electric fields (pp. 144–160), a description is given of a number of phenomena related to this topic.

1. Debye P., Ann. d. Phys., 33, 441, 1910.

2. Becker J. A. and Mueller D. W., Phys. Rev. (2), 31, 341, 1928.

3. Reynolds N. B., Phys. Rev. (2), 35, 158, 1930.

4. Linford L. B., Phys. Rev. (2), 36, 1100, 1930.

5. Langmuir I., Gen. Elec. Rev., 23, 504, 1920.

6. Becker and Rojansky (Becker J. A. and Rojansky V.) kindly informed the author of this result from their unpublished work:

7. Linford L. B., Phys. Rev. (2), 37, 1018, 1931.

8. Nottingham W. B., Phys. Rev. (2), 35, 669, 1930.

9. Suhrmann R. and Theissing H., Ztschr. f. Phys., 73, 709, 1932.

10. Nottingham W. B., Phys. Rev. (2), 41, 793, 1932.

11. Bartlett R. S. and Waterman A. T., Phys. Rev. (2), 37, 279, 1931; Bartlett R. S., Phys. Rev. (2), 37, 959, 1931; (2), 38, 1566, 1931; Waterman A. T., Phys. Rev. (2), 38, 1497, 1932.

12. Zwickker C., Physica, 11, 161, 1931.

13. Marx E., Naturwiss., 17, 806, 1929; Phys. Rev. (2), 35, 1059, 1930; Meyer A. E. H., Ann. d. Phys., 9, 787, 1931. A theoretical study of the question is given in the article by Marx E. and Meyer A. E. H., Phys. Ztschr., 32, 153, 1931.

14. Olpin, A. R., Phys. Rev. (2), 35, 112, 1930.

1. DuBridge and Roe. ↩

2. DuBridge. ↩