NEW THEORIES OF THE PHOTOELECTRIC EFFECT¹)

L. de Broglie

I. Introduction

The phenomenon of the emission of electrons from metallic surfaces under the action of light played a major role in the development of modern physics. Soon after the discovery of the photoelectric effect, Lenard’s work made it clear that some of its most important laws could not be explained from the standpoint of the electromagnetic theory of light. In 1905 Einstein boldly applied the quantum theory to the photoelectric effect and arrived at the famous equation which was later brilliantly confirmed in the well-known works of Hughes, Richardson and Compton, and Millikan¹. Thus the photoelectric effect became one of the cornerstones of the quantum theory of light. But now, in the course of the last several years, it has been found that photoelectrons liberated from a metal can provide data not only concerning the nature of the radiation that liberates them, but also concerning the behavior of electrons in the metal from which they have been obtained. Thus the photoelectric effect is at present closely connected with the new electron theory of metals, and this new theory has led to the necessity of reconsidering certain fundamental facts associated with the photoelectric effect. The purpose of the present monograph is to discuss these questions.

The foundations of the new electron theory of metals were laid by Sommerfeld in 1928². Until that time electrical phenomena in metals had been explained (insofar as possible) from the standpoint of the classical theory of the electron gas, formulated by Drude, Lorentz, and others. Although this theory had achieved considerable success in certain directions, it encountered serious difficulties in others. In the field of the photoelectric effect its role was somewhat negative, for, while not being in direct contradiction with any of the known facts, it at the same time gave no direct explanation of such questions as the dis—

¹) Actualités scientifiques et industrielles, No. 268, 1935. Translation and supplementary article by N. D. Morgulis (Kiev).

determination of the energies of photoelectrons and the variation of the photoelectric current with frequency—the two fundamental problems of this field. Indeed, the classical theory had practically no connection with the photoeffect, postulating that the mean thermal energy of the electrons is so small \(\left(\frac{3}{2}kT\right.\), equivalent to approximately \(0.04\ \mathrm{eV}\) at \(300^\circ\mathrm{K}\)\()\), that it may be neglected in comparison with the energy of the average quantum incident on the metal (\(4—5\ \mathrm{eV}\)). Neglecting this thermal energy, Einstein’s theory at once led to two important conclusions, which were usually regarded as quite exact:

1. There is a sharply defined maximum velocity of the photoelectrons, which is related to the frequency by the equation

\[ \frac{1}{2}mv_m^2=h\nu-p=h\nu-h\nu_0. \tag{1} \]

1. There is a sharply defined limiting frequency (\(\nu_0\) in the preceding equation), at which incident light of lower frequency than this, regardless of its intensity, will not produce photoelectric emission.

The new theory emphasizes that the conclusions just stated are exactly correct only for a metallic surface at a temperature of \(0^\circ\mathrm{K}\), whereas at higher temperatures there will be no sharply defined maximum velocity or limiting frequency. But it also goes further (and this is a major achievement), predicting the actual character of the energy-distribution and spectral-distribution curves at any temperature, at the same time providing a method for determining the quantities \(v_m\) and \(\nu_0\), which could be directly determined for the surface at \(0^\circ\mathrm{K}\) (provided, of course, that lowering the temperature did not cause other changes in the character of the surface). Moreover, the theoretical curves are in such excellent agreement with experiment that one may say that the theory, although not yet entirely complete, has met with considerable success here.

Since it seems evident that the further development of experimental investigations must be closely connected with the new theory, it appears desirable to set it forth in more detail in its present form. We must therefore begin with a review of some fundamental relations of Sommerfeld’s electron theory of metals \(^{2,3}\).

2. Sommerfeld’s Theory

Both the classical and Sommerfeld theories assume that in every conductor there is a swarm of “free” electrons capable of moving in the space between the atoms; the interior of the metal is thereby taken as a region with constant po-

potential (a refinement of this picture, taking into account the force fields around individual atoms, cannot be considered at the present stage of the theory). Classical theory held that the electrons have the same kinetic energy of thermal motion as the atoms, i.e. \(\frac{3}{2} kT\); the energies of the individual electrons are distributed about this average according to Maxwell’s distribution function. Sommerfeld’s theory, on the contrary, rejects the theorem of the uniform distribution of energy and regards the electrons as a degenerate gas obeying Fermi–Dirac statistics.

According to the latter, we arrive at unexpected conclusions concerning the behavior of the electron gas in a metal. Let us consider, for example, a piece of metal at \(0^\circ\)K. According to the old theory, all the electrons should then be at rest; however, according to Fermi statistics, their energies are distributed over an entire range of values—from zero up to a certain maximum value, which varies from one metal to another but is of the order of \(10\ \mathrm{eV}\). When the temperature is raised, according to classical theory, the energy increases proportionally to \(T\), but \(T\) would have to grow to \(75\,000^\circ\)K before the electron would have, on the average, an energy of \(10\ \mathrm{eV}\). According to the new theory, the mean energy does not change appreciably with temperature, but the upper boundary of the energies becomes less and less sharp. At the highest temperatures attainable with ordinary metals, the new theory predicts that the value of the mean energy will be 50 times greater than was obtained by the old theory; at low temperatures this ratio, of course, is still much greater. In other words, while according to the old theory it proved possible to neglect the energies of the electrons, according to the new theory, in the photoelectric effect they play a very important role.

Fig. 1. Components of velocities according to the Maxwell distribution function for \(T=1500^\circ\mathrm{K}\) and the Fermi–Dirac distribution for \(T=0^\circ\mathrm{K}\) and \(T=1500^\circ\mathrm{K}\).

It is shown in Fig. 1 what the difference is between these two distribution functions, where the Maxwell function is presented for a temperature of \(1500^\circ\mathrm{K}\), and the Fermi function both for \(0^\circ\mathrm{K}\) and for \(1500^\circ\mathrm{K}\) (attention should be paid to the fact that a gap must be left on the abscissa axis between the curves, since otherwise the Fermi curve may extend far beyond the limits of the page). For both curves the ordinates represent the number of electrons, and the abscissae the velocity components in any direction, expressed in electron-volts.

It is well known that at low temperatures electrons do not escape spontaneously from a metal in any considerable quantity. Therefore, in any theory it is necessary to postulate the existence of a “potential barrier” in order to retain the electrons in the metal. Let \(W_a\) denote the “height” of such a barrier, i.e. the total energy needed by an electron in order to emerge. Then, if an electron with initial kinetic energy \(E_k\) absorbs a quantum of energy \(h\nu\), the condition for its liberation from the metal will be

\[ E_k + h\nu \geq W_a . \tag{2} \]

If the initial kinetic energy is small in comparison with \(h\nu\), the smallest frequency capable of producing photoemission is equal to

\[ h\nu_0 = W_a . \]

Since for ordinary metals it is known that \(h\nu_0\) corresponds to approximately \(4\ \mathrm{eV}\), it is obvious that, according to the classical theory, the potential barrier must be of the same order of magnitude.

According to the new theory, however, even at \(0^\circ\mathrm{K}\) there are electrons with energies of the order of up to \(10\ \mathrm{eV}\). If this maximum energy is denoted by \(\psi_0\), then it is obvious that the limiting frequency at \(0^\circ\mathrm{K}\) will be equal to

\[ h\nu_0 = W_a - \psi_0 . \tag{3} \]

This means that the quantity \(W_a\) must be of the order of \(14\ \mathrm{eV}\), and this value is in agreement with that directly measured in electron-diffraction experiments \(^{4}\).

If the frequency of the incident radiation is greater than \(\nu_0\), then some electrons escape with an excess of energy, for which there will exist a sharp upper limit

\[ \frac{1}{2}mv_m^{\,2} = h\nu - (W_a - \psi_0) = h\nu - h\nu_0 . \tag{4} \]

This is precisely Einstein’s equation, and it is obvious that, from the point of view of any theory, it will hold only at \(0^\circ\mathrm{K}\), since at higher temperatures the energies of the electrons in the metal will not have a sharp upper limit.

Up to this point the results of the old and the new theories have coincided, since both led to Einstein’s equation. But further the clas—

classical theory proves untenable, for it leaves unanswered the following two very important questions:

1. How does the total number of electrons emitted per unit energy of radiation vary with the frequency of light and the temperature of the surface.

2. What is the character of the velocity distribution of the emitted electrons, and how does it vary with frequency and temperature (if the answer to this question is known, then the answer to the first can be obtained at once by integration over all velocities).

Our task now will be to give an answer to these questions, an answer obtained by applying Sommerfeld’s theory to this problem.

If \(f(\xi,\eta,\zeta)\,d\xi\,d\eta\,d\zeta\) denotes the number of electrons in a unit volume having velocity components \(\xi,\eta,\zeta\) in the intervals \(d\xi,d\eta,d\zeta\), then, according to Fermi–Dirac statistics,

\[ f(\xi,\eta,\zeta)= \frac{\dfrac{2m^{3}}{h^{3}}} {e^{\frac{\varepsilon-\mu}{kT}}+1}, \tag{5} \]

where

\[ \varepsilon=\frac12 mu^{2}=\frac12 m(\xi^{2}+\eta^{2}+\zeta^{2}), \]

and \(\mu\) is a constant depending on the number of electrons in a unit volume and given in the first approximation\(^1\) by the following expression

\[ \mu=\frac{h^{2}}{2m}\left(\frac{3n}{8\pi}\right)^{2/3}. \tag{6} \]

Hence it can be shown that the total number of electrons which, independently of direction, have total velocity \(u\) within the limits \(du\), is equal to

\[ f(u)\,du= \frac{\dfrac{8\pi m^{3}}{h^{3}}u^{2}du} {e^{\frac{\varepsilon-\mu}{kT}}+1}. \tag{7} \]

Fig. 2. Fermi–Dirac distribution for total velocities.

The function \(f(u)\) as a function of \(u\) is shown in Fig. 2.

In considering questions of photoelectric and thermionic emission it is often necessary to calculate the number of electrons which strike, in 1 sec., a unit surface of a metal

\(^1\) In the second approximation there appears here a term with \(T^{2}\), which is vanishingly small. However, \(\mu\) may nevertheless vary with \(T\), if \(n\) varies with \(T\) (see below). The value of \(\mu\) at \(T=0^\circ K\) is the value \(\mu_{0}\) from (3).

and which have a component of velocity normal to the surface lying within some interval. According to Nordheim\(^5\) this quantity can be calculated from equation (5) in the following way. Let \(\xi\) denote the component of velocity normal to the surface, and introduce a cylindrical coordinate system \(\xi,\rho\) and \(\theta\) such that \(\rho^2=\eta^2+\zeta^2\) and \(d\eta\,d\zeta=\rho\,d\rho\,d\theta\). Then the number of electrons per unit volume having the velocity component \(\xi\) in the interval \(d\xi\) is obtained by integrating over all values of \(\rho\) and \(\theta\), i.e.

\[ f(\xi)\,d\xi = \frac{2m^3}{h^3}\,d\xi \int_{0}^{\infty}\int_{0}^{2\pi} \frac{\rho\,d\rho\,d\theta} {e^{\frac{\frac12 m(\xi^2+\rho^2)-\mu}{kT}}+1}. \tag{8} \]

Integration with respect to \(\theta\) gives \(2\pi\). We now introduce, for abbreviation,

\[ e^{\frac{\frac12 m\xi^2-\mu}{kT}}=B; \]

\[ e^{\frac{\frac12 m\rho^2}{kT}}=z \]

\[ \rho\,d\rho=\frac{kT}{m}\frac{dz}{z}, \]

then

\[ f(\xi)\,d\xi = \frac{4\pi m^2 kT}{h^3}\,d\xi \int_{0}^{\infty}\frac{dz}{z(Bz+1)}. \tag{9} \]

This expression can be integrated at once, and, multiplying it by \(\xi\), in order to obtain the number of electrons incident on unit area in 1 sec., we obtain

\[ n(\xi)\,d\xi = \frac{4\pi m^2 kT}{h^3} \lg\left(1+e^{\frac{\mu-\frac12 m\xi^2}{kT}}\right)\xi\,d\xi . \tag{10} \]

The quantity \(\frac12 m\xi^2\) is the kinetic energy associated with the component of velocity normal to the surface. We may denote it by \(\varepsilon_n\) and (for lack of a better name) call it the normally directed energy, or, more simply, the normal energy. Since \(d\varepsilon_n=m\xi\,d\xi\), the preceding expression may be written as follows:

\[ n(\varepsilon_n)\,d\varepsilon_n = \frac{4\pi m kT}{h^3} \lg\left(1+e^{\frac{\mu-\varepsilon_n}{kT}}\right)d\varepsilon_n . \tag{11} \]

The graph of the function \(n(\varepsilon_n)\) as a function of \(\varepsilon_n\) is shown in Fig. 3 for temperatures 0 and \(1500^\circ\)K. For \(T=0^\circ\)K this expression takes a simple form, since in this case the exponential term is large compared with unity, and therefore

\[ n_0(\varepsilon_n)=\frac{4\pi m}{h^3}(\mu-\varepsilon_n), \tag{12} \]

which gives the straight line shown in the figure.

Fig. 3. Fermi-Dirac distribution for “normal energies”; above—the “tail” on an enlarged scale.

Now we may proceed to the application of the theory to the problem of photoelectric emission.

3. Spectral Distribution

4. Theory

A complete solution of the problem of photoelectric emission should lead to an expression giving the number of photoelectrons emitted from the surface, with any specified velocity in any specified direction, as a function of the following parameters of the phenomenon: 1) the intensity, frequency, character of polarization, and angle of incidence of the radiation exciting the phenomenon, and 2) the nature and temperature of the emitting surface. At present, however, our knowledge is still too limited to allow us to attempt to obtain a complete solution, and therefore we, for

To begin with, we shall considerably simplify our problem, excluding from consideration effects connected with the polarization and the angle of incidence of the light (the selective effect) and with the distribution over directions of the emitted electrons. In other words, we shall confine our attention only to the problem of determining the character of the velocity distribution function \(F(v)\), where \(F(v)\) represents the number of electrons emitted from the surface with velocities in the interval \(v, v+dv\). This function will contain the frequency \(\nu\) of the incident (unpolarized) light and will depend on the nature and temperature of the surface. Integration of this distribution function over all velocities will lead us to a new function \(f(\nu)\), called the spectral distribution function, which gives the total number of electrons emitted per unit energy of light as a function of its frequency. Both these functions are extremely important from the experimental point of view. It will be simpler to begin our analysis with a consideration of the spectral distribution function \(f(\nu)\) and then to extend the analysis to the more general problem of determining \(F(v)\).

From experiment it is known that the photoelectric current excited by unit intensity of incident monochromatic light varies with its frequency approximately in accordance with the curve of Fig. 4. Although in fact this curve approaches the frequency axis asymptotically, there is nevertheless a certain critical frequency \(\nu_a\), below which the current is immeasurably small, while above it the current grows rapidly with increasing frequency. The frequency \(\nu_a\) may for convenience be called the frequency of the apparent threshold. Let us now undertake the derivation of a theoretical expression for this curve of the spectral distribution. In the general case the derivation of such an expression would consist of the following successive operations:

Fig. 4. Typical curve of the spectral distribution, extrapolated (dashed line) for determining \(\nu_a\) (exaggerated).

1. One must specify the velocity distribution function of the electrons in the metal \(f(u)\)—this will be the Fermi–Dirac function given above.

2. One must calculate the probability \(P(\nu,u)\) that an electron with velocity \(u\) will absorb a quantum of frequency \(\nu\). The product of these two expressions will give the number of “excited” electrons, which will now have velocity \(u_1\), according to the equation

\[ \frac{1}{2}mu_1^2=\frac{1}{2}mu^2+h\nu . \tag{13} \]

1. It is necessary to determine the distribution function of the normal component of the velocity \(\xi\) for these excited electrons.

2. The expression obtained should be multiplied by the transparency coefficient \(D(\xi)\) for the surface potential barrier (this quantity is the fraction of electrons with velocity \(\xi\) that penetrate through the barrier).

3. Integrating now over all values of \(\xi\), we obtain the number of electrons emitted per second for the given frequency \(\nu\), i.e., the function of the spectral distribution.

The first attempt to give a theory on the basis of this scheme was made by Wentzel\({}^{6}\), whose work became the basis for the further development of the theory of the photoelectric effect. Wentzel came to the conclusion that the chief difficulty here consists in calculating the probability \(P(\nu,u)\), which can be done by applying quantum mechanics. His own calculations proved not yet wholly convincing, and therefore more complete theories were subsequently developed by Fröhlich\({}^{7}\) and Tamm\({}^{8}\). Unfortunately, although these theories turned out to be rather successful in some respects, the results to which they lead are mathematically so complicated that for the analysis of experimental data they could not be used with sufficient success\({}^{1}\). Furthermore, these theories did not take into account the influence of temperature on the energies of the electrons, and this turns out to be very important. Indeed, it may be said that one of the chief merits of the new theory is that it clarifies the question of the influence of temperature on photoelectric emission.

The most successful theory of the spectral distribution was developed by Fowler\({}^{10}\) in 1931. The central point of this theory is the recognition of the fact that, in experimental work with ordinary metals, the greatest frequency usually used (in the ultraviolet) does not often exceed the limiting frequency of these metals by more than 50%. The point is that the experimenter’s main interest is directed not so much toward an exact theory that gives the complete curve of the spectral distribution, as rather toward a theory that reproduces with sufficient accuracy the form of this curve near the boundary. If we restrict our attention to frequencies lying not far from the limiting one, then we immediately arrive at the following conclusions:

1. The electrons that can be emitted by the metal have initial energies differing only slightly from the maximum value of the energy according to Fermi statistics. However, these electrons will be very strongly affected by changes in temperature, and therefore this temperature effect will become so important that it will prove insuf—

\({}^{1}\) Quite recently Mitchell\({}^{9}\) published a considerably more general theory. It has not yet been quantitatively compared with experimental data, but promises in this respect to be fairly successful.

permissible to carry out the calculations under the assumption that the surface is at \(0^\circ\mathrm{K}\).

1. Since the initial velocities of the emitted electrons lie in a relatively narrow region, the probability factors depending on a small degree of this velocity may be regarded as constant.

2. Since the frequency range under consideration is also small, the factors depending on a small degree of \(\nu\) will be relatively insignificant in comparison with the factors depending on \(\nu-\nu_0\).

If all this is taken into account, the construction of the theory proves to be considerably simplified, and at the same time the restriction imposed by it on the frequency range is not very important from the experimental point of view (nevertheless, this restriction must be taken into account when extending the frequency range).

Taking the adopted simplifications into account, it is clear that there is no longer any need to calculate the probability function \(P(\nu,u)\), since it may now be regarded as constant. This eliminates the 2nd operation noted above and combines the 1st with the 3rd.

Passing to the 4th operation, we immediately encounter difficulties, and we still do not know the transparency coefficient \(D(\xi)\) sufficiently well. When this question has been clarified, it will not be difficult to introduce it into the theory. But it is now possible to make a very simple assumption, which for pure metals is very close to reality, namely that the transparency coefficient is equal to zero for electrons with energy (initial plus \(h\nu\)) less than the magnitude of the surface potential barrier \(W_a\), and then is equal to unity for electrons with energy exceeding \(W_a\).

Thus our calculation is now reduced to the following two operations1:

1. One must specify a function expressing the number of electrons incident on a unit surface of the metal in 1 sec. with normal energy \(\varepsilon_n\) in the interval \(d\varepsilon_n\).

2. Then it must be integrated over all values of the quantity \(\varepsilon_n\) greater than the critical one, determined from the relation

\[ \varepsilon_{n0} + h\nu = W_a . \]

The result of the integration gives the number of electrons \(N_B\), which

able to emerge; the number actually emitted must be proportional to \(N_B\).

Carrying out all that has been indicated, we obtain

\[ N_B=\int_{(W_a-h\nu)}^{\infty} n(\varepsilon_n)\,d\varepsilon_n, \tag{14} \]

where \(n(\varepsilon_n)\) is the function defined by equation (11). Therefore

\[ N_B=\frac{4\pi m kT}{h^3} \int_{(W_a-h\nu)}^{\infty} \lg \left[1+e^{\frac{\mu-\varepsilon_n}{kT}}\right]d\varepsilon_n . \tag{15} \]

Put

\[ \frac{\mu-\varepsilon_n}{kT}=x' \]

and

\[ \frac{\mu-(W_a-h\nu)}{kT}=x \]

and then make the substitution \(e^{x'}=u,\ e^x=u_0\) and \(d\varepsilon_n=-kTdx'=-kT\frac{du}{u}\); then

\[ N_B=\frac{4\pi m k^2T^2}{h^3} \int_0^{u_0}\lg(1+u)\frac{du}{u}. \tag{16} \]

The value of this integral is given in tables\(^1\) or can be obtained by expanding the logarithm in a series.

The result of the solution depends on whether the variable \(u\) in the region of integration is greater or less than unity. This, in turn, depends on the magnitude of the upper limit \(u_0\), and therefore we distinguish the following two cases:

Case I

\[ u_0\leqslant 1,\ \text{i.e. } x\leqslant 0 \text{ or } h\nu\leqslant (W_a-\mu). \]

\(^1\) Peirce’s tables; formula (439) gives, in our notation:

\[ \int \lg(1+u)\frac{du}{u} = u-\frac{u^2}{2^2}+\frac{u^3}{3^2}-\cdots \quad \text{for } u\leqslant 1 \]

\[ \int \lg(1+u)\frac{du}{u} = \frac{1}{2}(\lg u)^2-\frac{1}{u}+\frac{1}{2^2}\frac{1}{u^2} -\frac{1}{3^2}\frac{1}{u^3}+\cdots \quad \text{for } u\geqslant 1. \]

For \(u=1\) we obtain in both cases the series

\[ 1-\frac{1}{2^2}+\frac{1}{3^2}-\frac{1}{4^2}+\cdots=\frac{\pi^2}{12}. \]

In this case we immediately obtain

\[ N_B=\frac{4\pi m k^2T^2}{h^3}\left(e^x-\frac{e^{2x}}{2^2}+\frac{e^{3x}}{3^2}-\cdots\right);\quad x\leqq 0 . \tag{17a} \]

Case II

\[ u_0\geqq 1,\ \text{i.e. } x\geqq 0 \text{ or } h\nu\geqq (W_a-\mu). \]

In this case the integral must be divided into two parts: in the first, the integration is carried out within the limits from 0 to 1, and in the second—from 1 to \(u\). Substitution of the limits then gives

\[ N_B=\frac{4\pi m k^2T^2}{h^3} \left[\frac{x^2}{2}+\frac{\pi^2}{6} -\left(e^{-x}-\frac{e^{-2x}}{2^2}+\frac{e^{-3x}}{3^2}-\cdots\right)\right];\quad x\geqq 0 \tag{17b} \]

Since \(N_B\) represents the number of electrons arriving from within in 1 sec. at a unit surface, with normal energies sufficient for escape outward, we may assume that the current emitted by a unit surface will be proportional to \(eN_B\), i.e.

\[ I=\alpha eN_B, \tag{18} \]

where \(e\) is the charge of the electron, and \(\alpha\) is a coefficient of proportionality depending on the probability that an electron absorbs a quantum when unit radiation energy falls on the surface. Let us assume that \(\alpha\) is a constant quantity in our frequency region. Our equation for the spectral distribution can therefore be finally written in the following form

\[ I=\alpha AT^2\varphi(x), \tag{19} \]

where \(\varphi(x)\) is the function enclosed in brackets in (17a) and (17), and \(A\) is a universal constant equal to \(\dfrac{4\pi m e k^2}{h^3}\). It is interesting to note that this constant is identical with the constant \(A\) which appears in the equation for thermionic emission, whose numerical value is \(120\ \mathrm{A}/\mathrm{cm}^2\mathrm{grad}^2\). This, however, is not a simple coincidence, since the method of calculation in this case is analogous to that used in the derivation of the thermionic equation. Indeed, if we put \(\nu=0\) in (19), we immediately obtain Richardson’s equation

\[ I=AT^2 e^{-\frac{W_a-\mu}{kT}}, \]

neglecting the higher terms of the series, since the quantity \(W_a-\mu\) is always large in comparison with \(kT\).

We can now proceed to analyze equation (19) and compare it with experiment. First of all, it is of interest to note the form taken by the equation in the case of a metal surface maintained at \(T=0^\circ\mathrm{K}\). In this case \(x=\pm\infty\) respectively

therefore, on whether \(h\nu\) will be greater or less than \(W_a-\mu\). Then we obtain:

\[ \left\{ \begin{aligned} I_0&=0 &&\text{for } h\nu \leq W_a-\mu,\\ I_0&=\frac{1}{2}\,\frac{\alpha A}{k^2}\,[h\nu-(W_a-\mu)]^2 &&\text{for } h\nu \geq W_a-\mu . \end{aligned} \right. \tag{20} \]

The curve of the spectral distribution is therefore a parabola with its vertex on the frequency axis at the point
\[ \nu=\nu_0=\frac{W_a-\mu}{h}. \]
Thus at \(0^\circ\mathrm{K}\) there is a sharply defined limiting frequency, below which there will be no photoelectric emission. At higher temperatures, however, this will no longer be the case, since then the curve approaches the axis asymptotically, as shown in Fig. 5, and consequently we shall no longer obtain a sharply defined boundary in the ordinary sense of the word. For convenience, however, let us introduce

Fig. 5. Spectral distribution curves according to Fowler’s theory.

Fig. 6. Fowler function \(\Phi(x)\).

the notion of a “characteristic” frequency (which in what follows we shall also call the limiting frequency), defined by

\[ h\nu_0=W_a-\mu. \tag{21} \]

Obviously, \(\nu_0\) is the frequency for which \(x=0\) [from equation (15)], and is the frequency below which there would be no emission if the surface temperature could be lowered to \(0^\circ\mathrm{K}\), without changing \(W_a\) or \(\mu\). As will be seen below, the quantity \(\nu_0\) can be determined from experimental data obtained at any temperature; moreover, it then turns out to be approximately independent of \(T\), although the values of the “apparent” boundary obtained by extrapolation may show considerable changes with temperature. \(\nu_0\) is therefore a quite definite characteristic of the surface.

The benefit of Fowler’s theory is to a considerable extent associated with an interesting graphical method which he proposed

is suitable for the analysis of experimental data. If equation (19) is written in the form

\[ \lg \frac{I}{T^2}=B+\Phi(x), \tag{22} \]

where \(B=\lg A=\mathrm{const}\), and is independent of \(\nu\) and \(T\), and \(\Phi(x)=\lg \varphi(x)\), then it is obvious that if one plots the experimental values of the quantity \(\lg \frac{I}{T^2}\) as a function of \(x=\frac{h(\nu-\nu_0)}{kT}\), then the resulting curves, apart from an additive constant, will be the same for all metals and all temperatures and will be superposed on the theoretical curve of the dependence of \(\Phi(x)\) on \(x\). \(\Phi(x)\) will be a universal function of the argument; its calculated values are presented in Table 1, and its exact graph is in Fig. 6. However, in

TABLE 1

\(x\)	\(\lg x\)	\(\Phi(x)\)	\(x\)	\(\lg x\)	\(\Phi(x)\)
\(-8,0\)	\(+0,903\)	\(-3,475\)	\(+2,0\)	\(+0,301\)	\(+0,546\)
\(-6,0\)	\(0,778\)	\(-2,606\)	\(3,0\)	\(0,477\)	\(0,785\)
\(-5,0\)	\(0,699\)	\(-2,171\)	\(4,0\)	\(0,602\)	\(0,983\)
\(-4,0\)	\(0,602\)	\(-1,739\)	\(5,0\)	\(0,699\)	\(1,150\)
\(-3,0\)	\(0,477\)	\(-1,308\)	\(6,0\)	\(0,778\)	\(1,293\)
\(-2,5\)	\(0,398\)	\(-1,090\)	\(8,0\)	\(0,903\)	\(1,527\)
\(-2,0\)	\(0,301\)	\(-0,884\)	\(10,0\)	\(1,000\)	\(1,713\)
\(-1,5\)	\(0,176\)	\(-0,674\)	\(12,0\)	\(1,079\)	\(1,866\)
\(-1,0\)	\(0,000\)	\(-0,469\)	\(14,0\)	\(1,146\)	\(1,998\)
\(-0,5\)	\(-0,301\)	\(-0,268\)	\(16,0\)	\(1,204\)	\(2,113\)
\(-0,2\)	\(-0,699\)	\(-0,160\)	\(20,0\)	\(1,301\)	\(2,305\)
\(0,0\)	\(-\infty\)	\(-0,085\)	\(25,0\)	\(1,398\)	\(2,497\)
\(+0,2\)	\(-0,699\)	\(-0,015\)	\(30,0\)	\(1,477\)	\(2,655\)
\(0,4\)	\(-0,398\)	\(-0,055\)	\(35,0\)	\(1,544\)	\(2,788\)
\(0,6\)	\(-0,222\)	\(+0,125\)	\(40,0\)	\(1,602\)	\(2,904\)
\(1,0\)	\(0,000\)	\(0,249\)	\(50,0\)	\(1,699\)	\(3,097\)
\(1,5\)	\(+0,176\)	\(0,400\)	—	—	—

practice the values of \(x\) can actually be calculated for given values of \(\nu\) and \(T\) only in the case where the limiting frequency \(\nu_0\) is known, which itself must be determined from these data. Therefore Fowler proposed measuring \(I\), as usual, as a function of \(\nu\) for a given metal at constant temperature and then representing these data in the form of a graph of the dependence of \(\lg \frac{I}{T^2}\) on \(\frac{h\nu}{kT}\). The curve thereby obtained should have the same form as the theoretical one, except that its origin will be shifted along the \(x\)-axis by the amount \(\frac{h\nu_0}{kT}\). Hence the value of \(\nu_0\) can be determined if one measures how far the experimental curve must be shifted along this axis in order for it to coincide

with the theoretical one. The experimental curve will be displaced relative to the theoretical one also along the vertical axis by an amount equal to \(B\), the value of which depends on the units used in measuring \(I\) (the current and the radiation intensity) and on the (at present unknown) value of the probability coefficient \(\alpha\). \(B\) may vary from metal to metal, but for a given metal it will be independent of \(T\), and consequently the curves obtained for a given surface at different temperatures will all show the same vertical displacement.

Fig. 7. Fowler function \(\Phi(x)\), plotted on a semilogarithmic scale.

The second method of analysis was proposed by the author\(^{11}\) and is useful in cases where, instead of measurements of \(I\) as a function of \(\nu\) for a whole series of constant temperatures (isothermal curves), one uses the more convenient method of measuring \(I\) as a function of \(T\), at constant values of \(\nu\) (isochromatic curves). In this case one plots the dependence of

\[ \lg \frac{I}{T^2} \]

on \(\lg T\). The curves thus obtained, after a parallel displacement, must coincide with the theoretical curve of the dependence of \(\Phi(x)\) on \(\lg x\). The vertical component of this displacement will again depend on \(B\), while the horizontal one will be equal to

\[ \frac{h(\nu-\nu_0)}{k}, \]

since

\[ \lg x=\lg \frac{h(\nu-\nu_0)}{k}-\lg T. \]

Since \(\nu\) is known, for the measurements here are made at this single frequency, the quantity \(\nu_0\) can again be determined. This method obviously eliminates the need to measure the relative intensity of incident radiation of different frequencies, as was necessary in the first method. The second method is, of course, inapplicable if for some reason (for example, a change in the state of the surface) \(\nu_0\) is no longer independent of \(T\). A plot of the dependence of \(\Phi(x)\) on \(\lg x\) is given in Fig. 7.

B. Experimental verification

Fowler himself was the first to show how successfully his theory predicted the form of the spectral-distribution curves obtained experimentally, and how useful it proved to be in analyzing experimental data for the purpose of determining the threshold frequency \(\nu_0\). Fowler

developed this theory of his during his stay at the University of Wisconsin (USA), where Mendenhall, together with his collaborators, had collected a large quantity of carefully obtained data on the photoeffect of clean metallic surfaces in a high vacuum. Analysis of these data showed that the experimental curves of the spectral distribution correspond quite accurately to the theoretical curve over the entire range of frequencies and temperatures in which the experiments were carried out. The horizontal displacement required in order to bring the experimental and theoretical curves into coincidence gave values of \(\nu_0\) which, within the limits of experimental error, proved independent of temperature. The values of the frequency of the apparent threshold \(\nu_a\), however, proved, as was to be expected, to be somewhat smaller than \(\nu_0\), and showed an undoubted decrease with increasing temperature. At the present time it is already clear that this apparent decrease of the threshold frequency (and therefore also of the work of extracting an electron from the surface) is not an actual change in the state of the surface, but is caused simply by an increase in the thermal energy of the electrons, and that the value \(W_a-\mu\), which is a measure of the true work of extraction, probably depends hardly at all on temperature (this question will be discussed in greater detail in the next chapter).

Fig. 8. Test of Fowler’s theory (Fowler).

In Fowler’s work there are published curves representing data obtained for gold by Morris, for silver by Uinchem, and for tantalum by Cardwell. He also reports that the data of Lawrence and Linford for tungsten and of Goetz for tin show the same good agreement. In the case of tin, the analysis clearly showed a change in the limiting frequency with change of the crystal structure, as had been found by Goetz. Since the publication of Fowler’s theory, his method has been used for the analysis of data obtained for tungsten (Warner\(^{12}\)), copper and iron (Glasoe\(^{13}\)), molybdenum, palladium, and zinc (DuBridge and Roe\(^{14,15,16}\)), thoriated tungsten (DuBridge and Smith\(^{17}\)), and a number of other metals (Welch\(^{18}\)). A number of typical curves are given in Figs. 8 and 9. In each case the curves observed at different temperatures were displaced by the amount necessary to superpose them on the theo-

theoretical curve—the vertical displacement proved to be the same for all temperatures. Very complete data of DuBridge and Roe for palladium are represented by the curve in Fig. 10, which shows

Fig. 9. Verification of Fowler’s theory for pure W (Warner). The points are given both in the initial and in the displaced positions.

Fig. 10. Verification of Fowler’s theory for Pd (DuBridge and Roe).

especially well how exactly the experimental points lie on the predicted curve. The observations were carried out at eight different temperatures, covering an interval of approximately \(700^\circ\), and extending to temperatures at which the thermionic current already becomes appreciable. The region of wavelengths exten-

...ranged in some cases up to 600 Å (approximately 1.2 eV) from the boundary. In some unpublished experiments of DuBridge and Smith with thoriated tungsten it was found that the theory is applicable for wavelengths extending up to 1.3 eV from the boundary, and beyond this point it, as might be expected, already begins to give deviations.

Fig. 11. Isochromatic analysis of data for Pd
\(\bigl(\Phi(x)\) as a function of \(\lg x\bigr)\).

Fig. 12. Isochromatic analysis of data for Au.

The second (isochromatic) method of analysis was used by the author\(^{11}\) for data obtained for palladium and gold, with the results shown in Figs. 11 and 12. The agreement is again very good, and the resulting values of \(\nu_0\), within the accuracy of the measurements, coincide with the values obtained from these same data by means of the first method.

A summary of the published data is presented in Table 2, in which the values of the limiting wavelength \(\lambda_0 = \frac{c}{\nu_0}\), the corresponding work function \(\varphi = h\nu_0\), and the extrapolated values

TABLE 2

Metal	\(T\) (°K)	\(\varphi\) (eV)	\(\lambda_0\) (Å)	\(\lambda_a\) (Å)	Thermionic \(\varphi\) (eV)
Ag	296	4.71	2620	\(2610^{10}\)
Ag	673	4.76	2592
Ag	873	4.75	2590	2700
Ag		avg. 4.74	2604
Au	296	4.86	2530	\(2650^{10}\)
Au	733	4.92	2500
Au	1013	4.92	2500	2610
Au		avg. 4.90	2510
Mo	303	4.14	2992		\(4.15^{15}\)
Mo	940	4.16	2983
Pd	305	4.96	2490	2490
Pd	400	4.97	2486
Pd	550	4.97
Pd	730	4.97		2550
Pd	830	4.98	2482
Pd	925	4.98
Pd	1008	4.96
Pd	1078	4.97		2660	\(4.99^{14}\)
Pd		avg. 4.97	2486
Ta	293	4.13	3010	3050	\(4.13^{30}\)
Ta	973	4.18	2970	3150	4.20
Ta		avg. 4.15	2990
W (filament)	790	4.70	2626
W (filament)	900	4.65	2650
W (filament)	1100	4.71	2619	\(2575^{12}\)
W (filament)		avg. 4.69	2632
W (deposited layer)	295	4.54	2720	2725	\(4.54^{12}\)

of the apparent limiting wavelength \(\lambda_a\). In this connection, the constancy of the values of \(\lambda_0\) for each individual metal deserves special interest.

As regards the experiments themselves, it should be noted that in most cases they were carried out with clean metallic surfaces that had been carefully degassed in a very high vacuum. Only under such conditions can one be certain that the nature of the surface itself does not change with temperature. It was found, however, that even a partially degassed palladium surface remained so constant that it showed good agreement with theory. Uda then found that, if the measurements are limited to room temperature, then surfaces that have not been completely degassed lead to curves coinciding with theory.

Therefore the theory is not limited in its applications only to perfectly clean surfaces. Moreover, the data obtained for thorated tungsten indicate that the theory is equally well applicable also to complex surfaces. The only limitation connected with the theory is merely the assumption made at the very beginning, namely, that the frequency region used must not extend too far beyond the limiting frequency; and, as we have already seen, this limitation is not serious in most actual cases.

C. Temperature Effects

Inspection of Table 2 shows that the values of \(\lambda_0\) and \(\varphi_0\) obtained by means of Fowler’s method are approximately independent of temperature. The small changes obtained in this way lie within the experimental errors associated with the process of graphical superposition of the curves, and, in any case, do not exceed the expected changes associated with a small change in the amount of gas contained in the surface film. In the case of Pd, for which the most complete data were obtained, these changes are quite small and have no systematic character.

It is interesting to raise the question to what extent the greatest influence of temperature could be detected if, in the future, it proved possible to reduce the experimental inaccuracy still further. Such an influence may show itself in a change either of \(W_a\), or of \(\mu\), or of both of these quantities. As regards the quantity \(\mu\), Sommerfeld showed that, to a high degree of accuracy, it can be calculated from equation (6) (further refinement leads to an additional term containing \(T^2\), whose numerical value at \(1000^\circ\ \mathrm{K}\) is only \(10^{-4}\%\) of the whole expression and which may therefore be neglected). It is therefore evident that \(\mu\) can change only with a change in the number of free electrons per unit volume \(n\). Assuming that the number of free electrons per atom remains constant, \(n\) may nevertheless change in connection with the volume expansion of the metal. Denoting

\[ n=\frac{N}{v} \]

and taking

\[ v=v_0[1+\beta(T-T_0)], \]

we obtain approximately

\[ \mu=\mu_0[1+\beta(T-T_0)]^{-\frac{2}{3}}, \tag{23} \]

where \(\mu_0\) is the value of \(\mu\) at \(T_0\). Hence, in the first approximation, one obtains

\[ \frac{1}{\mu_0}\frac{d\mu}{dt}=-\frac{2}{3}\beta . \tag{24} \]

In the typical case this leads to a relative temperature

coefficient by an amount of the order of \(3\cdot 10^{-5}\) V per degree. If \(\mu=5\) V, and \(W_a=10\) V, then this would give an increase of the quantity \(\varphi=W_a-\mu\), approximately, by \(3\%\) for a temperature interval of \(1000^\circ\). Such a change could probably have been detected experimentally. It follows from this that either the total number of free electrons \(N\) must also increase with \(T\), or \(W_a\) must decrease with \(T\). In fact, one should expect that the surface electric field, and hence also \(W_a\), will decrease owing to thermal expansion, as a result of the increase in the distance between the metallic ions at the surface of the metal. Gerifield estimated the magnitude of this effect and found that it may cause a change in \(W_a\) of the order of \(0.1\) V, which would reduce \(\varphi\) in the example given above by \(2\%\). The combined change of \(W_a\) and \(\mu\) may therefore cause an increase of \(\varphi\) by \(1\%\) when the temperature changes by \(1000^\circ\), and this is in fact the limit of the experimental uncertainty. Although this calculation is very rough, it is nevertheless of interest to increase the accuracy of the measurements in order to see whether such a change can be observed.

A number of authors have pointed out that the presence of a small temperature variation of \(\varphi\) would have a significant influence on the observed value of the coefficient \(A\) in Richardson’s thermionic equation. This question was recently again analyzed in detail and clearly set forth by Becker and Brattain\(^{19}\). Briefly speaking, if one assumes that the work function \(\varphi\) is independent of \(T\), then Sommerfeld’s electron theory leads directly to the following theoretical equation for thermionic emission

\[ I=AT^2 e^{-\frac{\omega}{T}}, \tag{25} \]

where \(\omega=\frac{1}{k}(W_a-\mu)=\frac{\varphi}{k}\), and \(A\) is the universal constant

\[ \frac{4\pi me k^2}{h^3}=120\ \text{A}/\text{cm}^2\ \text{deg}^2. \]

On the other hand, if \(\omega\) varies with temperature and if we denote

\[ \frac{d\omega}{dT}=\delta, \]

then the theory leads to the equation

\[ I=A_1T^2 e^{-\frac{b}{T}}, \tag{26} \]

where \(b=(\omega-\delta T)\) and \(A_1=Ae^{-\delta}\). The empirical values of \(A_1\) and \(b\) are determined by measuring the intercept on the ordinate axis and from the slope of the Richardson straight line \(\left(\lg \frac{I}{T^2}\ \text{as a function of}\ \frac{1}{T}\right)\). If the intercept thus obtained is equal to \(A=120\), then it may be concluded that \(\delta=0\) and therefore \(\omega\) is a constant quantity. However, experiments with many pure metals have led—

led to the empirical value \(A_1 = 60 = \dfrac{A}{2}\). Hence \(e^{-\delta} = \dfrac{1}{2}\) and approximately \(\delta = 0.7\). For example, in the case of tungsten \(\omega = 52000^\circ\), and therefore the relative temperature coefficient \(\dfrac{\delta}{\omega} = 1.35 \cdot 10^{-5}\) per \(1^\circ\), which corresponds to a change of \(1.35\%\) over a range of \(1000^\circ\). This is also of the same order of magnitude as that obtained from the calculation given in the preceding paragraph, and could have been detected experimentally if it had been possible to increase the accuracy of the measurements. In fact, analyzing the data given in Table 2, Becker and Brattain come to the conclusion that it contains indications of a similar temperature change, which both in its direction and in its magnitude corresponds to that required to explain the observed values of \(A_1\). Taking into account, however, the inaccuracy of the experiment, it is hardly possible to attach such great significance to these small changes of \(\varphi\), which is why further experimental investigations are necessary. Until this has been done, one may approximately neglect the effect associated with the presence of a small temperature change of \(\varphi\), although the corresponding corrections may, if desired, be easily introduced.

Although the quantities \(\varphi\) do not depend on \(T\), the question of the temperature dependence of the photocurrent \(I\), which in past years was the subject of a number of discussions, has continued up to the present to remain very interesting. Earlier experiments led to indications that the photoelectric current does not depend on temperature, although in most cases the temperature range was small and the measured photocurrents were excited by spectrally unresolved light. However, when measurements were made over wider ranges with approximately monochromatic radiation, Ives\(^{20}\) in the region of very low temperatures and the author\(^{21}\) in the region of high temperatures found an undoubted temperature effect. At first it seemed natural to ascribe this effect to the temperature change of the work function, although the author also supposed that they might be caused by the thermal energy of the electrons. With the development of Fowler’s theory this question was subjected to detailed experimental investigation, which led to results in complete agreement with the theory.

If we return again to Fowler’s equation, it is obvious that the dependence of \(I\) on \(T\) will be strongly affected by the frequency of the light producing the effect. We may consider three cases:

1. \(\nu = \nu_0\), i.e. \(x = 0\): equation (19) then leads to \(I = \dfrac{aA\pi^2}{12}T^2\), i.e. \(I\) is proportional to \(T^2\).

2. \(\nu \gg \nu_0\). Then \(x \gg 0\), and in this case

\[ I = AT^2 \left(\frac{x^2}{2} + \frac{\pi^2}{6}\right) \tag{27} \]

or, as a function of \(\nu\),

\[ I=\frac{A}{2}\left[\frac{h^2(\nu-\nu_0)^2}{k^2}+\frac{\pi^2}{3}T^2\right]. \tag{28} \]

Even for wavelengths differing by only \(100\ \text{\AA}\) from the limiting one, the first term in brackets is considerably larger than the second, and for high frequencies \(I\) changes only slightly with \(T\).

1. \(\nu \ll \nu_0\). In this case \(x \ll 0\), and we obtain approximately

\[ I=\alpha A T^2 e^x=\alpha A T^2 e^{\frac{h(\nu-\nu_0)}{kT}}, \tag{29} \]

which will lead to a very strong change of \(I\) with \(T\), just as in the case of thermionic emission.

Fig. 13. Change of photocurrent with temperature at different frequencies for Pd.

Figure 13 gives the dependences of \(I\) on \(T\), obtained experimentally for Pd, for light of different frequencies. These are the same data that had already been analyzed by means of the isochromatic method in Fig. 11, where it was evident that they are in complete agreement with Fowler’s theory. Indeed, all the data presented in Figs. 8, 9, and 10 for different temperatures show that the dependence of \(I\) both on temperature and on frequency is described quite successfully by the theory. This is direct and convincing proof that the distribution of electron energies in metals is correctly given by Fermi statistics.

4. Distribution of the Energies of Photoelectrons

The methods used in the development of Fowler’s theory of the spectral distribution can also be extended to the problem of the distribution of the energies of photoelectrons[^22]. The success achieved as a result of applying them to this new problem may be regarded as an undoubted achievement of the new electron theory of metals, for the classical theory proved altogether unsatisfactory when applied to this problem, except for the introduction of certain ad hoc assumptions concerning the nature of electronic collisions inside the metal.

The energy distribution of photoelectrons is usually determined experimentally by measuring the current \(I\), produced by photoelectrons reaching the collector electrode as a function of the magnitude of the retarding potential \(V\), applied between this electrode and the emitting surface. The form of the volt-ampere curve obtained in this way will depend strongly on the geometrical arrangement of the two electrodes, and only in certain special cases can these curves be used to determine the energy distribution function. We shall consider the following two cases:

Case I. Parallel plates (Fig. 14). If the plates are large compared with the distance between them, then the electric

Fig. 14. Plane-parallel capacitor.
Fig. 15. Spherical capacitor.

field between them is uniform and normal to the surface. If an electron leaves plate \(A\) with a velocity whose component normal to the plate is equal to \(v_n\), then it can reach plate \(B\), in the presence of a retarding potential \(V\), only if

\[ \frac{1}{2} m v_n^{2} \geq eV. \]

If \(f(E_n)\,dE_n\) is the number of electrons leaving the plate with normal energy \(E_n=\frac{1}{2}mv_n^{2}\) in the interval \(dE_n\), then the number of electrons \(F(V)\) reaching the plate in the presence of a retarding potential \(V\) will be

\[ F(V)=\int_{eV}^{\infty} f(E_n)\,dE_n. \]

Therefore

\[ f(E_n)=-\frac{1}{e}\left[\frac{dF(V)}{dV}\right], \tag{30} \]

where the derivative is taken at the point at which \(eV=E_n\). From the experimental curves we can thus determine the distribution function of the normal energies.

Case II. Concentric spheres (Fig. 15). If the emitting sphere \(A\) is very small in comparison with the collector sphere \(B\), then an electron leaving \(A\) in any direction will move radially and will reach the outer sphere, in the presence of a retarding potential \(V\), only if its total velocity \(v\) is such that \(\frac{1}{2}mv^2 \geq eV\). If we put \(E=\frac{1}{2}mv^2\), and if \(f(E)\,dE\) denotes the number of electrons emitted with energies in the interval \(E,\ E+dE\), then, as before,

\[ f(E)=-\frac{1}{e}\left[\frac{dF(V)}{dV}\right]_{eV=E}. \tag{31} \]

and we can now determine the distribution function of their total energies. We now turn to the derivation of theoretical expressions for both distribution functions and for the current-voltage curves.

A. Normal energies

The equation for the current-voltage curve can in this case be obtained directly from Fowler’s theory. Indeed, the creation of a retarding potential \(V\) between parallel plates is exactly equivalent to increasing the potential barrier at the surface by the amount \(eV\). Therefore in Fowler’s equation (19) we replace the quantity \(W_a\) by \(W_a+eV\), and obtain immediately for the current in the collector circuit

\[ I=\alpha AT^2\varphi(x_1), \tag{32} \]

where \(\varphi(x_1)\) is the Fowler function already defined above, and

\[ x_1=\frac{h\nu-(W_a+eV-\mu)}{kT}. \tag{33} \]

For a given frequency of the incident radiation \(\nu\) and temperature \(T\), \(x_1\) and \(\varphi(x_1)\) will be functions only of \(V\), and, consequently, the preceding equation (32) is an expression for the current-voltage curve. We can write \(x_1\) in the following form:

\[ x_1=\frac{h\nu-(W_a-\mu)-eV}{kT}=\frac{e(V_m-V)}{kT}, \tag{34} \]

where we put

\[ eV_m=h\nu-(W_a-\mu)=h(\nu-\nu_0). \]

For simplicity let us first consider the case when \(T=0^\circ\mathrm{K}\). Then, as in equation (20),

\[ \left\{ \begin{array}{ll} I=0 & \text{for } x_1 \leqslant 0 \text{ or } V \geqslant V_m,\\[4pt] I=\dfrac{1}{2}aAe^2(V-V_m)^2 & \text{for } x_1 \geqslant 0 \text{ or } V \leqslant V_m. \end{array} \right. \tag{35} \]

Thus the current–voltage curve is in this case a parabola with its vertex on the voltage axis at \(V=V_m\), as shown in Fig. 16.

Fig. 16. Theoretical current–voltage curve for plane-parallel electrodes.

At this temperature there are no electrons emitted with an energy exceeding \(eV_m\), and therefore there is a sharply defined maximum of their energies, determined from

\[ eV_m-h\nu-(W_a-\mu)=h(\nu-\nu_0), \tag{36} \]

where \(\nu_0\) is determined, as before, from equation (21). This is precisely Einstein’s equation, which previously held at all temperatures, but according to the present theory is exactly valid only at \(T=0^\circ\mathrm{K}\). At higher temperatures the current–voltage curves, given by equation (32), approach the axis asymptotically, and a sharply defined maximum energy will no longer exist. We can nevertheless define \(V_m\) as a characteristic energy, and this characteristic energy (which is not a maximum) will satisfy Einstein’s equation. As we shall see below, this energy can be determined empirically from data obtained at any temperature. Thus \(V_m\) is the maximum energy that could be observed if it were possible to lower the temperature of the surface to absolute zero, provided that the quantity \(W_a-\mu\) remained unchanged.

The equation for the distribution of the normal energies of the electrons can, according to (30), be obtained by direct differentiation of equation (32). However, it proves simpler to return directly to equation (15), by integration of which equation (32) is obtained. Replacing in (15) \(W_a\) by

\(W_a+eV\), we immediately obtain the equation for the expression \(F(v)\), i.e., the number of electrons capable of passing against the retarding field \(V\). We then obtain

\[ F(V)=\alpha \frac{4\pi mkT}{h^3}\int_{(W_a+eV-h\nu)}^{\infty} \lg \left[1+e^{\frac{\mu-E_n}{kT}}\right]\,dE_n, \tag{37} \]

where \(\alpha\) is the probability factor introduced in equation (18). But since

\[ f(E_n)=-\frac{1}{e}\left[\frac{dF(V)}{dV}\right], \]

we immediately obtain

\[ f(E_n)=\alpha \frac{4\pi mkT}{h^3} \lg \left(1+e^{\frac{\mu-W_a+eV-h\nu}{kT}}\right) = \]

\[ =\alpha \frac{4\pi mkT}{h^3} \lg \left(1+e^{\frac{E_m-E_n}{kT}}\right), \tag{38} \]

where we have put

\[ E_m=eV_m=h\nu-(W_a-\mu)\quad \text{and}\quad E_n=eV. \]

Equation (38) is the normal-energy distribution function we sought. At \(T=0^\circ\mathrm{K}\) it reduces to

\[ f_0(E_n)=\alpha \frac{4\pi m}{h^3}(E_m-E_n), \tag{39} \]

which is represented by a straight line intersecting the energy axis at \(E_m=eV_m\), thereby giving a sharply defined maximum energy. At higher temperatures the curve falls off asymptotically, as shown in Fig. 17. These curves could in fact be obtained directly from Fig. 3,

Fig. 17. Theoretical distribution of normal energies.

which represents the distribution of normal energies inside the metal, for the present curves are simply parts of the preceding ones, lying to the right of the point on the abscissa \(E_n=W_a-h\nu\). According to our assumptions, electrons with energy greater than this value can escape, and their energy distribution will therefore not be distorted, except for an identical diminution by the amount \(W_a-h\nu\).

In reality the curves of Figs. 16 and 17 do not exactly resemble the experimental curves. This is probably caused by our assumption that the transparency coefficient is equal to unity for all electrons, whereas, probably, for slower electrons the transparency coefficient may be smaller. Near the value \(V_m\), however, the theoretical curves must have the correct form. All this was subjected to direct verification in the experiments of DuBridge and Hergenrother\(^{23}\), who used the method of parallel plates in measuring current–voltage curves for a carefully degassed molybdenum surface, for a whole series of frequencies and temperatures.

Fig. 18. Analysis of current–voltage curves with plane-parallel electrodes for Mo at \(T=300^\circ\mathrm{K}\).

Since the theoretical equation (32) is identical with Fowler’s equation for the spectral distribution, the same graphical method can be used for analyzing the experimental data, except that the variable is now no longer \(\nu\), but \(V\). Consequently, if one plots the dependence of \(\lg \dfrac{I}{T^2}\) on \(\dfrac{eV}{kT}\), the curve obtained should have the same form as Fowler’s curve, but be shifted along the axes. The vertical shift must depend on the coefficient \(\alpha\) and on the intensity of the incident light. For a certain specified intensity this shift, however, must be independent of frequency and temperature. The horizontal shift must be equal to \(\dfrac{eV_m}{kT}\), as is immediately evident from the definition of the parameter \(x_1\) given in equation (34).

A series of experimental current–voltage curves for Mo, treated graphically in this way, is presented in Figs. 18 and 19. Fig. 18 presents observations obtained at the constant temperature \(300^\circ\mathrm{K}\) for a whole series of wavelengths of the incident light, whereas Fig. 19 presents the results of obser-

measurements at one frequency, but for a whole series of values of \(T\). It is quite obvious that the experimental points fall exactly on the theoretical curves, giving remarkable confirmation of the theory over a wide range of values of \(\nu\) and \(T\). The vertical shift necessary for agreement is the same for all the curves in both figures. From the horizontal shift, however, the values \(V_m\) were determined, and it was found that: 1) for a definite constant frequency \(V_m\) does not depend on \(T\), and 2) for a constant temperature \(V_m\) varies with \(\nu\) in accordance with Einstein’s equation. In each case the resulting measurement error was of the order of \(1\%\). Thus the theory gives a good explanation of the distribution of the normal energies of photoelectrons, at least for energies not much smaller than \(V_m\). In fact, the points shown in the figures refer to the entire lower half of the experimental curves, so that the region considered proves not to be too restricted, despite the approximations that were made. Still more important, the theory gives a definite answer to the question of the existence of a sharply defined maximum energy and reveals the clear meaning of Einstein’s equation. This becomes still clearer when we try to extend the theory to the more general problem of the distribution of total energies of photoelectrons.

Fig. 19. Analysis of current–voltage curves for plane-parallel electrodes, for Mo at different temperatures.

B. Total energies

The calculation of the distribution of the total energy of photoelectrons is considerably more complicated, since it is then necessary also to take into account the components of velocity parallel to the emitting

surface. Nevertheless, this is the most general problem in the field of photoelectric emission, since if it is solved, the answers to the problems already considered above can be obtained at once as special cases.

We must begin our calculation with the general Fermi–Dirac expression for the number of electrons per unit volume of the metal with velocities (independently of direction) lying in the interval \(u\) and \(u+du\), namely

\[ n(u)\,du=\frac{\dfrac{8\pi m}{h^3}\,u^2du}{e^{\frac{\frac12 mu^2-\mu}{kT}}+1}. \tag{40} \]

Suppose that, under the influence of unit intensity of the incident light, some fraction \(p\) of these electrons absorbs an energy quantum, thereby acquiring a velocity \(u_1\), determined from

\[ \frac12 mu_1^2=\frac12 mu^2+h\nu. \tag{41} \]

The velocity distribution of the excited electrons, apart from the additional term \(h\nu\), will be of the same character as that of the unexcited ones, and the distribution over directions will remain homogeneous, if we assume that there is no preferred direction for the additional velocity imparted by the light.\(^1\) Of the excited electrons, however, only those can escape whose velocity \(u_1\) is directed in such a way that its component \(\xi\) normal to the surface is greater than the minimum value \(\xi_c\) required to overcome the potential barrier, with \(\frac12 m\xi_c^2=W_a\). This means that the vector \(u_1\) must lie inside a cone with its axis normal to the surface and with vertex semi-angle equal to \(\arccos \frac{\xi_c}{u_1}\), and which therefore subtends the solid angle \(2\pi\left(1-\frac{\xi_c}{u_1}\right)\). The probability that this vector will lie inside this cone is the ratio of this solid angle to \(4\pi\) and is therefore equal to \(\frac12\left(1-\frac{\xi_c}{u_1}\right)\), provided that \(u>\xi_c\), and, of course, is zero for \(u<\xi_c\).

\(^1\) Fowler assumed that the additional velocity was acquired in the direction normal to the surface. Although these two assumptions seem substantially different, they lead to the same result, since only the fastest electrons are included in the integration (i.e. those having the greatest normal velocities), to which belong electrons that have received the additional velocity in a direction almost normal to the surface. One would expect a small difference in the results if the second approximation were calculated, and in this case the assumption of a homogeneous distribution of the additional velocities would appear a priori more plausible.

Now electrons with initial velocity in the interval \(u, u+du\), after emission, will have velocity in the interval \(v, v+dv\), where

\[ \left\{ \begin{aligned} \frac{1}{2}mv^2&=\frac{1}{2}mu_1^2-W'_a=\frac{1}{2}mu^2+h\nu-W_a,\\ v\,dv&=u_1\,du_1=u\,du . \end{aligned} \right. \tag{42} \]

If we multiply the number of electrons in unit volume that have velocity \(u_1\) by the value of the velocity component \(\xi_1\) normal to the surface, we obtain the number of electrons striking a unit surface per unit time. That part of them which emerges outward will be determined by the transparency coefficient \(D\) of the surface for electrons of this velocity. Then, finally, the number of electrons emitted through a unit surface per unit time with velocity in the interval \(v, v+dv\) will be determined from

\[ N(v)\,dv=\xi_1 pD\,\frac{1}{2}\left(1-\frac{\xi_c}{u_1}\right)n(u)\,du . \tag{43} \]

Taking the value of \(n(u)\) from equation (40) and expressing everything as a function of \(v\), using equation (42) for this, we obtain

\[ N(v)\,dv= \frac{8\pi m^3}{h^3}\xi_1 pD\,\frac{1}{2} \left[1-\frac{\xi_c}{(v^2+\xi_c^2)^{1/2}}\right]\times \]

\[ \times \left[v^2+\xi_c^2-\frac{2h\nu}{m}\right]^{1/2} \frac{v\,dv}{ e^{\frac{\frac{1}{2}mv^2-h\nu+W_a-\mu}{kT}}+1 }. \tag{44} \]

This is the general equation for the velocity distribution; in deriving it no assumptions have yet been made. It will be considerably simplified if we make assumptions of the same character as in the theory of the spectral distribution. This means, first of all, the assumption that for small ranges of values of \(\nu\) and \(v\) the factors \(p\) and \(D\) are essentially constant. Then, since \(v^2\ll \xi_c^2\) \(\left(\frac{1}{2}m\xi_c^2\right.\) is of the order of \(15\) eV, whereas \(\frac{1}{2}mv^2\) usually does not exceed \(1\) eV), we may expand the first term in brackets in a series, obtaining approximately \(\frac{v^2}{2\xi_c^2}\). Similarly, for frequencies close to \(\nu_0\), the quantity \(\xi_1\) will be only slightly greater than \(\xi_c\), while the second term in brackets will be slightly less than \(\xi_c\), so that their product will be equal approxi-

…limited by \(\xi_c\). Therefore we obtain

\[ N(v)\,dv=\frac{\alpha A_{11}}{4}\, \frac{v^{3}dv}{e^{\frac{\frac12 m(v^{2}-v_m^{2})}{kT}}+1}, \tag{45} \]

where \(\alpha\) is a proportionality coefficient containing \(p\) and \(D\) and taken to be constant; \(A_{11}\) is a universal constant, equal to \(\dfrac{8\pi m^{3}}{h^{3}}\), and \(v_m\) is determined from the relation

\[ \frac12 mv_m^{2}=h\nu-(W_a-\psi), \tag{46} \]

which again represents Einstein’s equation. Since it is usually convenient to express energy in units of potential differences, then

\[ eV_1=\frac12 mv^{2}; \]

\[ eV_m=\frac12 mv_m; \]

\[ e\,dV_1=mv\,dv, \]

whence it follows that

\[ f(V_1)dV_1= \frac{\alpha A_1 V_1\,dV_1}{e^{\frac{e(V_1-V_m)}{kT}}+1}, \tag{47} \]

where

\[ A_1=\frac{2e^{2}}{m^{2}}\cdot\frac{A_{11}}{4}\cdot\frac{4\pi me^{2}}{h^{3}}. \]

Fig. 20. Theoretical curve of the distribution of total energies and current–voltage curves.

This is the final equation for the energy distribution of photoelectrons, from which all other equations of interest from the experimental point of view can be obtained. This equation is represented graphically in Fig. 20 for a typical value of \(V_m\) and three different temperatures. It is quite obvious that, as before, at \(0^\circ\text{K}\) there is a sharply defined maximum energy equal to \(V_m\), but at higher temperatures we obtain a gradual falling off of the curves.

Despite the fact that the energy-distribution curve at room temperature is, in general, similar to the curves obtained…

Usually, in experiment, there are nevertheless two important differences: 1) the experimental curves, contrary to the prediction, are linear at lower energies, and 2) the most probable energy (the maximum of the curve) lies closer to the apparent intersection with the axis (the apparent maximum value \(V\), which can be obtained by extrapolation) than has usually been reported. The first discrepancy is possibly explained by the fact that the transparency coefficient \(D\) is not equal to unity for very slow electrons. This part of the curve, being experimentally the least accurate, is at the same time not the most interesting, and therefore this discrepancy is not essential. The second discrepancy is probably not real, since the most recent experiments, which will be described below, have shown good agreement with the theory. To carry out a quantitative experimental check it is desirable to derive a theoretical current–voltage curve, which could then be compared directly with observations. For this purpose we use the relation

\[ I = e \int_V^\infty f(V_1)\,dV_1, \tag{48} \]

where \(I\) is the current reaching the collector in the presence of a retarding potential \(V\). Consequently,

\[ I = eaA_1 \int_V^\infty \frac{V_1\,dV_1}{e^{\frac{e(V_1 - V_m)}{kT}} + 1}. \tag{49} \]

We now make the following substitutions: \(x = \frac{eV}{T}\), \(x_0 = \frac{e}{kT}\) and \(x_1 = \frac{eV_1}{kT}\) [from the fact that \(V_m = \frac{h}{e}(\nu - \nu_0)\), it follows that \(x_0\) is the quantity denoted by \(x\) in equations (17a) and (17b)]. Making the further substitution \(e^{x_1}=\omega\), \(e^{-x_0}=a\), and \(dx_1=\frac{d\omega}{\omega}\), we arrive at the following simpler form of the integral

\[ \int_{e^x}^{\infty} \frac{\lg \omega\, d\omega}{\omega(a\omega + 1)}, \]

which can be integrated by parts. The character of the result depends on whether, in the region of integration, the quantity \(x_1\) is smaller or greater than \(x_0\), and this in turn depends on whether the lower limit \(x\) is greater or smaller than \(x_0\). We obtain, accordingly, two cases, which may be analyzed in the same way as was already done for

in deriving equation (16), as a result of which we obtain

\[ I=\alpha AT^2\left\{\frac{\pi^2}{6}-\frac{1}{2}(x^2-x_0^2)+x\lg[1+e^{(x-x_0)}]- \right. \]

\[ \left. -\left[e^{(x-x_0)}-\frac{e^{2(x-x_0)}}{2^2}+\frac{e^{3(x-x_0)}}{3^2}-\ldots\right]\right\} \quad \text{for } x \ll x_0; \tag{50} \]

\[ I=\alpha AT^2\left\{x(x-x_0)+x\lg[1+e^{(x-x_0)}]+ \right. \]

\[ \left. +\left[e^{-(x-x_0)}-\frac{e^{-2(x-x_0)}}{2^2}+\frac{e^{-3(x-x_0)}}{3^2}-\ldots\right]\right\} \quad \text{for } x \gg x_0, \tag{51} \]

where we have put

\[ A=\frac{A_1 k^2}{e^2}=\frac{4\pi m e k^2}{h^3}, \]

which is identical with \(A\) in Fowler’s equation (19) and with \(A\) in the thermionic equation, and is numerically equal to \(120\ \mathrm{A/cm^2\ grad^2}\).

Since

\[ x_0=\frac{h(\nu-\nu_0)}{kT}, \]

it is evident that the preceding equations represent a general relation for the photoelectric emission of a surface as a function of all three variables \(\nu\), \(V\), and \(T\). We can write it briefly in the following form

\[ I=\alpha AT^2\Psi(x,x_0), \tag{52} \]

where \(\Psi(x,x_0)\) denotes the function enclosed in braces in equations (50) and (51). We shall first consider two special cases:

1. \(V=0\). If there is no retarding potential, then the current will be equal to the full saturation current, and the preceding equations must pass into others giving the dependence of \(I\) on \(\nu\) and \(T\). Making the substitution, we at once obtain

\[ I=\alpha AT^2\left[\frac{\pi^2}{6}+\frac{x_0^2}{2}-\left(e^{-x_0}-\frac{e^{-2x_0}}{2^2}+\ldots\right)\right] \quad \text{for } x\gg 0 \text{ or } \nu\gg\nu_0; \]

\[ I=\alpha AT^2\left[e^{x_0}-\frac{e^{2x_0}}{2^2}+\frac{e^{3x_0}}{3^2}-\ldots\right] \quad \text{for } x_0\ll 0 \text{ or } \nu\ll\nu_0. \]

These same equations prove to be identical with Fowler’s equations (17a) and (17), and, consequently, the general equation (52) includes Fowler’s equation as a special case, which of course is a necessary condition.

1. \(T=0^\circ\mathrm{K}\). In this case the volt-ampere curves have a very simple form, since, carrying out the corresponding substitutions, and in particular for the values of \(x\) and \(x_0\), we obtain

\[ I=\frac{\alpha A e^2}{2k^2}(V_m^2-V^2)\quad \text{for } V\ll V_m; \]

\[ I=0\quad \text{for } V\gg V_m, \]

which gives a parabola intersecting the \(V\)-axis at the point \(V_m\), as is shown in Fig. 20.

The theoretical volt-ampere curves for any temperature can be represented graphically by using equation (52). However, \(\Psi(x, x_0)\) is, unfortunately, not a universal function of \(x - x_0\), and consequently its character will depend on the numerical value of \(x_0\) in each particular case. In Fig. 21 curves are shown for five selected values of \(x_0\). Since \(x_0\) depends both on \(\nu - \nu_0\) and on \(T\), these curves represent the dependence on \(\nu\) at constant \(T\), or, conversely, the dependence on \(T\) at constant \(\nu\). If, for example, \(T = 300^\circ\mathrm{K}\), then the different

Fig. 21. Theoretical current—voltage curves
for different values of \(x_0\). 1) \(x_0 = 5\); 2) \(x_0 = 10\);
3) \(x_0 = 20\); 4) \(x_0 = 40\); 5) \(x_0 = \infty\).

curves correspond to different values of

\[ V_m = \frac{h}{e}(\nu - \nu_0), \]

namely:

\[ \left\{ \begin{array}{rrrrrr} x_0: & \infty & 40 & 20 & 10 & 5 \\ V_m\,(V): & \infty & 1.04 & 0.52 & 0.26 & 0.13. \end{array} \right. \]

On the other hand, for a constant frequency corresponding, for example, to \(V_m = 0.5\,V\), these curves correspond to the following different temperatures:

\[ \left\{ \begin{array}{rrrrrr} x_0: & \infty & 40 & 20 & 10 & 5 \\ T\,(^\circ\mathrm{K}): & 0 & 150 & 300 & 600 & 1200. \end{array} \right. \]

In connection with this dependence of the shape of the curve on the numerical value of \(x_0\) (and consequently also on \(V_m\)), the question of comparing theory with experiment is more complicated than in the case of Fowler’s theory, where a single theoretical curve is sufficient. Of course,

if \(V_m\) were known in each individual case of experimental data, then the corresponding theoretical curve could be calculated at once. But, in general, the value of \(V_m\) can be determined, when applying the theory, only from these data themselves. It is possible, however, to develop an approximate method of analysis that is sufficiently accurate in a small range of values of \(V\). This method is based on the fact that, for \(x > x_0\), equation (51) becomes (when the logarithm is expanded in a series)

\[ I=\alpha A T^{2}\left[(x+1)e^{-(x-x_0)}-\frac{1}{2}\left(x+\frac{1}{2}\right)e^{-2(x-x_0)} +\frac{1}{3}\left(x+\frac{1}{3}\right)e^{-3(x-x_0)}-\cdots\right] \]

and if then \(x-x_0>1\) and \(x_0>10\), this corresponds rather closely to the equation

\[ I=\alpha A T^{2}x\left[e^{-(x-x_0)}-\frac{1}{4}e^{-2(x-x_0)}\right]. \]

This may be written in the following form

\[ \lg \frac{I}{xT^{2}}=B+\chi(x-x_0), \]

where

\[ B=\lg(\alpha A)\quad \text{and} \]

\[ \chi(x-x_0)=\lg\left[e^{-(x-x_0)}-\frac{1}{4}e^{-2(x-x_0)}\right]. \]

Therefore the quantity \(\lg \frac{I}{xT^{2}}\) is a universal function of \(x-x_0\), and this relation, as it turns out, is observed with sufficient accuracy over a fairly wide range of values of \(x\) near \(x_0\). By determining the dependence of \(\chi(x-x_0)\) on \(x-x_0\), we obtain a theoretical curve which must have the same form as the experimental curves, if the dependence of \(\lg \frac{I}{xT^{2}}\) on \(x\) is plotted. The horizontal displacement necessary for coincidence with the theoretical curve must then be equal to \(x_0=\frac{eV_m}{kT}\), whence \(V_m\) can also be determined.

C. Experimental verification

An experimental verification of this theory of energy distribution, using the method of analysis given above, was carried out by Reim\(^{24}\) for Mo at various temperatures and by DuBridge and Hill\(^{25}\) for Na at room temperature. In both cases

the results proved to be in good agreement with the theory over the entire range of frequencies and temperatures used, and for all values of \(V\) from approximately \(\frac{1}{3}V_m\) and higher. In Fig. 22 are presented the current–voltage curves obtained by Rapp for Mo at room

Fig. 22. Analysis of current–voltage curves for Mo at 300° K (Rapp).

Fig. 23. Analysis of current–voltage curves for Mo at 1000° K (Rapp).

temperature and treated in the manner indicated, while Fig. 23 presents the data for this same surface at \(1000^\circ\mathrm{K}\)—the highest temperature at which thermionic current is still imperceptible. The same data, presented, as usual, in the form of the dependence \(I=f(V)\), are given in Fig. 24; here the influence of temperature on the tail of the curves is clearly visible and therefore also on the apparent (extrapolated) value of the retarding potential.

In this case the apparent retarding potential, even at room temperature, is approximately 10% greater than \(V_m\).

The curves for a distilled sodium surface at room temperature (\(300^\circ\mathrm{K}\)) are shown in Fig. 25 (in order to put them all on one and the same drawing, the individual curves were shifted both horizontally and vertically by arbitrary amounts). Rather unexpected in this case is the fact that, contrary to the assumptions adopted in the theory, the experimental points agree well with the theory over a rather wide range of wavelengths, from 2400 to 5790 Å.

Fig. 24. Effect of temperature on the current–voltage curves.

Fig. 25. Analysis of the current–voltage curves for Na.

An independent verification of this theory was recently published by Brady\(^{26}\) for thin potassium films. The measurements are in very good agreement with the theory for all wavelengths used, and for film thicknesses from as small as 3–4 atomic layers up to “thick layers.” At the same time, no change in the energy distribution with film thickness was observed, as had previously been reported by Lukirskii and Prilezhaev.\(^{27}\)

Some unpublished data obtained by Rar for a distilled zinc surface show fairly success-

agreement with the theory, provided that the zinc film had been obtained under conditions of an extremely good vacuum. The slightest traces of gas completely altered the form of the volt-ampere curves to such an extent that they could in no way be brought into agreement with the theoretical curve. The influence of the gas led chiefly to an increase in the tail of the curves, much as might have been expected if the temperature of the surface had increased. Indeed, the observations would have agreed rather well with the theory if it could have been assumed that the surface temperature was several hundred degrees higher than it actually was. This strange result has not been explained, but Nottingham has reported a similar phenomenon for thermionic electrons from thoriated tungsten. It is as though the surface films filtered out the slower electrons.

Nevertheless it is evident that, for clean surfaces, the present theory gives a good representation of the general character of the energy distribution of photoelectrons. As we see, the theory of the energy distribution also includes the theory of the spectral distribution, so that we obtain a general theory of the photoelectric effect which not only qualitatively but also quantitatively correctly describes the general character of this phenomenon for all values of the temperatures of the emitting surface and throughout the most important interval of frequencies and energies. In extending the theory to wider regions of these variables it will obviously be necessary to make still greater use of the methods of quantum mechanics and of the electron theory of metals in its most exact form.

The success of the theory in its present form has a threefold significance: 1. From the point of view of experiment, the theory gives the long-awaited quantitative description of the observed phenomena and leads to a useful method for analyzing experimental curves of the energy distribution and of the spectral distribution. 2. The ambiguities connected with the concepts of the “maximum energy of photoelectrons” and the “threshold frequency” have been clarified; these quantities are now replaced by more precisely defined quantities \(V_m\) and \(\nu_0\), and at the same time a more exact characterization of the meaning of Einstein’s equation has been given. 3. The success thereby achieved provides further direct evidence for the correctness of Fermi–Dirac statistics in its application to free electrons in a metal.

It remains for us to consider only two important propositions of the theory, which have not yet been subjected to a complete experimental investigation, but which promise to lead to results of considerable importance.

5. Photoelectric determination of \(\dfrac{h}{e}\)

As has already been emphasized earlier, at ordinary temperatures there is no sharply defined maximum energy of photo-

electrons. In such a case the question arises of the interpretation of the experiments of Millikan, Olpin, and Lukirskii and Prilezhaev on the precise determination, by the photoelectric method, of the universal constant $\frac{h}{e}$. These experiments were based on the assumptions of the existence of a similar maximum energy, connected with the frequency by Einstein’s equation. In particular it is of interest to pose the following questions: 1) can the influence of temperature, not taken into account in these experiments, give rise to a systematic error in them, 2) what is the probable magnitude of such an error, and 3) what method is the most exact for eliminating this error in future experiments.

First of all it should be noted that, in principle, these experiments need not necessarily lead to the correct value of the quantity $\frac{h}{e}$, since there is no exact theoretical connection between the frequency of light and any extrapolated or apparent maximum energy $V_a$. As we see, Einstein’s equation contains only the characteristic energy $V_m$, which would be equal to the extrapolated value $V_a$ only in the case where the emitting surface were at $0^\circ\mathrm{K}$. Nevertheless, the observers in fact obtained values of $\frac{h}{e}$ which agreed closely with the values obtained by other methods. This means that empirically $V_a$ is also a linear function of $\nu$ with a slope approximately equal to $\frac{h}{e}$. In other words, it turns out by chance that $V_a - V_m$ must be almost independent of $\nu$.

From the theory, however, it is difficult to determine the exact relation between $V_a - V_m$ and $\nu$, since $V_a$ evidently depends on each separate method of extrapolation used and on the accuracy of the measuring instruments. The best of the extrapolation methods used was given by Lukirskii and Prilezhaev,^27 who found that, if their current-voltage curves are plotted graphically in the form of a dependence of $I^{1/2}$ on $V$, then in the lower region they are quite accurately straight lines, and therefore can be extrapolated to the axis in order to obtain exact and reproducible values of $V_a$. Prilezhaev has recently analyzed this method from the point of view of the new theory and has come to the conclusion that the values of $V_a$ obtained in this way differ from $V_m$ by an amount which is almost independent of $\nu$ in the range of frequencies used in the experiments. Therefore the error in their final value of $\frac{h}{e}$ which can be ascribed to this cause is small in comparison with the other experimental errors (according to Birge’s analysis, they are of the order of 0.25%, as may be estimated from their internal agreement). This result is quite unexpected and fortunate. However, quite by chance their curves (obtained for surfaces which had not been degas—

were) proved to be of such a fortunate form that it led to a linear dependence between \(I^{1/2}\) and \(V\). Such a relation is not predicted by theory, and measurements made in our laboratories (on carefully degassed surfaces) did not show agreement with such a linear relation. In fact, in some cases we found a difference of \(1\%\) between the values of \(\dfrac{h}{e}\) obtained by extrapolation and by theoretical analysis (Fig. 26). Although this difference may perhaps be of no significance (in view of other errors in these experiments), it seems quite possible that, at least in some cases, the extrapolation method may introduce a noticeable systematic error into the photoelectric determination of \(\dfrac{h}{e}\). Therefore, at the very least it would be highly desirable to undertake a new experimental determination in which this source of error would be eliminated. For this purpose, however, it is necessary to develop a more accurate method of analyzing the data than the one described above.

Fig. 26. Photoelectric determination of \(h\) for Na, \(I\)—\(V_m\) as a function of \(\nu\), \(II\)—\(V_a\) as a function of \(\nu\).

6. Absolute value of the photoelectric yield

The photoelectric yield of a surface for a given frequency of light is defined as the absolute value of the photoelectric current excited by unit intensity of absorbed light of that frequency. It may be measured, for example, in amperes per watt, in coulombs per calorie, or in electrons per quantum. This yield can in no case exceed one electron per quantum, which for \(2536\ \text{\AA}\) corresponds to \(0.206\ \mathrm{A/W}\). In reality, the greatest yield ever reported (so far as I know) was equal to \(1/14\) electron per quantum, although values of the order of \(10^{-3}\) to \(10^{-5}\) are more usual. Very large values are obtained only for surfaces that show strong “spectral selectivity.” For nonselective surfaces one may take, as a characteristic value of the yield, \(10^{-3}\) electron per quantum with a frequency differing ...

from the threshold by 1 V (although it has in fact been accurately measured only in very few cases).

Since the new theory predicts the correct dependence of the photoelectric yield on frequency and temperature, the interesting question arises as to the extent to which it gives the correct order of magnitude and absolute value. To answer this question it is necessary to calculate the absolute value of the probability factor \(\alpha\) in Fowler’s equation

\[ I=\alpha A T^{2}\varphi(x). \]

However, as has already been indicated, this factor cannot be calculated exactly until a more exact wave-mechanical theory of the interaction of electrons and light quanta in a metal has been developed. It is possible, however, from very simple considerations to estimate the order of magnitude of the absolute yield, or rather its possible upper limit, for a simple (non-selective) surface.

As is immediately clear from a simple dimensional analysis, the quantity \(\alpha\) represents the fraction of the number of electrons falling on unit surface in 1 sec. which absorb an energy quantum, if the intensity of the incident light is equal to unity. If we take as our unit of intensity \(1\ \text{quant}/\text{cm}^{2}\cdot\text{sec}\), then, evidently, no more than one electron can be excited in 1 sec. The total number of electrons falling on the surface in 1 sec. can be calculated at once (for \(T=0^\circ\mathrm{K}\)) by integrating equation (11) from 0 to \(\mu\), whence one obtains

\[ n_{0}=\frac{2\pi m\mu^{2}}{h^{3}} \]

or, from equation (6),

\[ n_{0}=\frac{\pi h}{2m}\left(\frac{3n}{8\pi}\right)^{4/3} =0.092\,\frac{hn^{4/3}}{m}, \]

where \(n\) is the number of electrons per unit volume.

For the typical case (\(\mu=15\ \mathrm{eV}\)) we find \(n_{0}\simeq 1.5\cdot 10^{31}\), and therefore \(\alpha=\frac{1}{n^{0}}=6.7\cdot 10^{-32}\). Since Fowler’s equation contains the thermionic constant \(A=120\ \mathrm{A}/\text{cm}^{2}\cdot\text{degree}^{2}=7.55\cdot 10^{20}\ \text{electrons}/\text{cm}^{2}\cdot\text{sec}\), the factor \(\alpha A\simeq 5\cdot 10^{-11}\ \text{electrons}/\text{quant}\cdot\text{degree}^{2}\). For \(T=300^\circ\mathrm{K}\) and for a frequency 1 V away from the threshold \((x\simeq 40)\), \(\varphi(x)\simeq 1000\), and therefore \(I\simeq 5\cdot 10^{-3}\ \text{electrons}/\text{quant}\). This value is somewhat greater than the observed one, being only its upper limit. Nevertheless, it is evident that the fact that the photoelectric yield is of the order of \(10^{-3}\) (rather than 1) electrons/quant can be explained from fairly simple considerations.

*

More interesting in this problem is the fact that the quantity \(\alpha\) varies only proportionally to \(n^{-4/3}\). This means that for a given metal \(\alpha\) must be constant, independently of the nature or state of activation of the surface, whereas for different metals the quantity \(\alpha\) should vary only within small limits (for example, by a factor of 4), since the values of \(n\) for different metals do not differ greatly from one another. In other words, the enormous differences in the photoelectric yield of different surfaces for a definite incident light frequency must be attributed directly to differences in the values of \(\nu_0\) (and hence also of \(x\)), and only to a small extent—to differences in the values of \(\alpha\). This is quite analogous to the situation observed in the case of thermionic emission.

It would be very interesting to measure experimentally the values of \(\alpha\) for different metals with different surface conditions in order to test the conclusions indicated above. The constancy of \(\alpha\) for a definite metal was confirmed by the experiments of DuBridge and Smith. They found that for thoriated tungsten the constant \(B=\lg \alpha A\) proved to be independent of the degree of surface activation, although the work function varied in this case from 2.5 to 4.5 V. Measurements of the absolute value of \(B\) for other metals are now being carried out.

7. Conclusion

As was already indicated at the beginning, our discussion has been restricted only to the theory of the normal (nonselective) photoelectric effect, excited by light with a frequency not differing greatly from the threshold frequency \(\nu_0\). This region is very interesting from the experimental point of view, and it is precisely in it that temperature effects are important. Therefore the theory, although incomplete, is nevertheless very useful.

It now seems possible to outline the directions in which the theory may be developed; there are three of them:

1. A more detailed study of the theory of the interaction of electrons with light quanta inside a metal, based on Sommerfeld’s simple theory. This development has already been begun by Mitchell, who carried out calculations for frequencies far from the threshold (where temperature effects are not important). His theory has not yet been applied to the problem of the energy distribution.

2. A study of the question to what extent deviations from the simple Fermi distribution, caused by atomic fields inside the metal, may affect the preceding theory. Although these deviations may play an important role, they have not yet been studied, except in a few cases\(^29\), and no attempt has yet been made to investigate their influence on photoelectric emission.

3. Further study of the selective photoelectric effect. This problem is the most difficult. It has long been suspected that pronounced

...spectral selectivity (as, for example, in alkali metals under certain surface conditions) is caused by a resonance effect of a definite character at the surface. Fowler’s theory of selective transparency of the surface is incomplete, since it cannot explain the enhanced emission (over and above that given by the simple theory) for a certain frequency region. Recently Zener \(^{30}\) suggested that selective absorption of light by surface films of atoms (in the case of complex surfaces) may explain many of the observed effects. A more detailed understanding of the nature of metallic surfaces (both simple and complex) is also one of the most important desiderata in the field of electron emission.

ADDENDUM

to L. DuBridge’s article—New Theories of the Photoelectric Effect

N. D. Morgulis, Kiev

In DuBridge’s article printed above, the material covered is that dealing with the photoelectric effect of pure metals and published approximately up to 1935. During the last three years, however, a whole series of papers on this question has appeared in print, the discussion of which is the purpose of the present addendum. Although it is not possible to agree with DuBridge’s opinion that “the experimenter’s chief interest is directed not so much toward an exact theory, which gives the complete curve of the spectral distribution, as rather toward a theory which conveys with sufficient accuracy the form of this curve near the threshold,” we shall nevertheless confine the material set forth below to the circle of questions directly touched upon in DuBridge’s article, referring the reader for other matters to the literature already available \(^{31}\). We shall begin with the theoretical works, and then pass to the experimental ones.

In his fundamental work Fowler \(^{10}\), in calculating the strength of the photoelectron current \(I\), proceeded from the number of electrons per unit volume \(N\) with velocity normal to the surface in the interval \(\xi\), \(\xi+d\xi\). In this connection DuBridge notes in this article that it seems to him more logical to proceed not from the quantity \(N\), but from the number of electrons \(N'\) striking unit surface area per second. From this entirely natural inference DuBridge arrives at expression (19), which differs from that given by Fowler himself and very much resembles Richardson’s equation for thermionic emission. It should be noted that a similar conclusion was also reached by the author of the present article in a completed but unpublished work begun in 1934. The calculation carried out in this work...

calculation for \(I\) proceeded exactly from equation (15), as a result of which, of course, the same Richardson-type expression for \(I\), similar to Dobretsov’s (19), was obtained, with the constant \(A = 120\ \mathrm{A/cm^2\,deg^2}\). On the basis of this expression obtained by him, the author was also able to determine the electron work function \(\Phi\) by the method of graphical matching, for silver and palladium, on the basis of the experimental data of Winch \(^{32}\) and Dobretsov and Rep \(^{33}\).

It should also be noted that in Blokhintsev’s work \(^{34}\) an improvement of Fowler’s theory was made at the time, in which the starting point was the so-called “surface photoeffect,” calculated by Tamm for \(0^\circ\mathrm{K}\); these calculations were extended to arbitrary temperatures with allowance for Fermi–Dirac statistics, for \(\nu \sim \nu_0\). As a result of the calculations, the relation was obtained

\[ \lg \frac{I\nu^4}{T^{5/2}} = \lg F\left(\frac{h\nu - h\nu_0}{kT}\right) + \lg E_x^2 + \lg C, \tag{53} \]

where \(E_x\) is the component of the electric vector normal to the surface, and the function \(F(\psi)\) is given by the author in tables. A comparison of the obtained formula (53) with experimental data, using the usual method of graphical matching, for Au, Ag, Ta, W, and Zn, presented in Fig. 27, led (except for the last case, Zn) to good agreement and to a value \(\Phi = e\varphi_0\) close to that obtained from Fowler’s theory.

Fig. 27. Comparison of the obtained formula (53) with experimental data.

In their calculations of the photoelectric current \(I\), Fowler and Dobretsov proceeded from a rectangular potential barrier at the surface—

...and took the transparency coefficient to be \(D=0\) for \(E<W_a\) and \(D=1\) for \(E>W_a\). Meanwhile, such an assumption is, of course, very approximate, and primarily with respect to slow photoelectrons with \(E\sim W_a\). Further refinement of the theory must, of course, proceed along the line of transition to the barrier given by the forces of the electrostatic image, namely

\[ V=-\frac{e^2}{4x_0}=\mathrm{const}\ \text{ for } x\leq x_0;\quad V=-\frac{e^2}{4x}\ \text{ for } x\geq x_0, \tag{54} \]

if \(x=x_0\) corresponds to the surface of the metal. Such a refinement of the theory of the photoeffect near the boundary was carried out by Mitchell\({}^{35}\), who also obtained, as the result of the calculation, an expression of the already known type

\[ \lg \frac{I}{T^2}=\lg B+\lg F(\mu), \tag{55} \]

where \(\mu=\dfrac{h\nu-h\nu_0}{kT}\). The constant \(B\) does not depend on temperature, but may depend on the frequency \(\nu\). The use of formula (55) leads to the preceding, graphical type of method for analyzing experimental data. Taking into account, however, the possible dependence of \(B\) on \(\nu\), one should here prefer DuBridge’s isochromatic method to Fowler’s isothermal one. One may also start from an even more complicated form of the potential barrier, such as, for example, that proposed by Nottingham\({}^{36}\) on the basis of his investigations of the electron emission of pure and thoriated tungsten, i.e.,

\[ R=1-D=e^{-\frac{p_x^2}{2m\omega}}, \tag{56} \]

where \(\dfrac{p_x}{m}=u_n\) is the normal component of the velocity of the photoelectron. In this case a function of the old type is obtained, but the value of the work function \(\Phi\) turns out to be overestimated by \(\Delta\varphi=8+0.138\,T-5\cdot10^{-5}\,T^2\) mV. Therefore, on the basis of the data of DuBridge and Roe, the quantity \(\Phi\) for Pd has a temperature coefficient equal to \(4.5\cdot10^{-5}\ \mathrm{eV/grad}\), which leads to the value \(A=60\ \mathrm{A/cm^2\cdot grad^2}\).

In the calculations of Fowler and DuBridge it was assumed that the probability of absorption of a quantum by an electron does not depend on the energy of the latter. However, as Houston\({}^{37}\) indicates, according to quantum-mechanical data it must be proportional to \(u_n^2\). Houston developed Fowler’s calculation with this probability taken into account and arrives at results coinciding with the experimental data. In this case, for determining \(\Phi\), the function \(F(\mu)\) is still obtained, while the indicated probability affects only the vertical displacement, from which one can obtain data concerning it.

The processes associated with photoemission in the region of interest to us have been analyzed in considerable detail in the theoretical work of Rudberg[^38]. If the kinetic energy of an electron inside and outside the metal is denoted by \(W\) and \(E\), where \(E = W + h\nu - W_a\) (\(W_a\) is the magnitude of the potential jump at the surface), then the distribution function \(f(E)\) will be:

\[ f(E)\,dE = F(W)\sum P_k, \]

where

\[ F(W)=\frac{1}{e^{\frac{W-\psi}{kT}}+1}. \tag{57} \]

\(P_k\) is the probability of emission of an electron from level \(k\) in the metal with energy in the interval \(W,\ W+dW\); \(F(W)\) is the Fermi distribution factor. Since the momentum \(p=\sqrt{2mW}\), the process associated with photoemission may be represented as a series of the following transitions of the electron in momentum space, shown in Fig. 28, where the geometric \(x\)-axis is directed along the normal to the surface of the metal outward.

Fig. 28. The process associated with photoemission, represented as a series of transitions of the electron in momentum space.

Fig. 29. Theoretical curves \(f(E)\) for different values of the constant \(n\).

An electron that was initially in the elementary volume \(p_1,\ p_1+dp\), where \(p_1=\sqrt{2mW}\), upon absorbing a quantum \(h\nu\), passes into the volume \(p_2,\ p_2+dp\), where \(p_2=\sqrt{2m(W+h\nu)}\), and finally, having flown out, finds itself in the volume \(p_3,\ p_3+dp\), where \(p_3=\sqrt{2m(W+h\nu-W_a)}\). The last transition, associated with the change of momentum from \(p_2\) to \(p_3\), must be represented by a vertical straight line, since in the investigation of the distribution of initial velocities the retarding field in the spherical condenser is directed normal to the surface. Therefore photoemission is restricted here to the initial region in the first zone, lying between the two vertical straight lines shown by dashed lines in Fig. 27. Depending on the nature of the assumption concerning the magnitude \(P_k\),

made by different authors in deriving the formula for the velocity distribution, the distribution function \(f(E)\, \(dE\) will have a different form, namely, for

\[ f(E)\,dE = AE^n F(E-h\nu), \]

where

\[ F(E-h\nu)=F(W)=\frac{1}{e^{\frac{E-h\nu-\mu}{kT}}+1}. \tag{58} \]

In Fig. 29 the theoretical curves \(f(E)\) are presented for different values of the constant \(n\); in Fig. 30 the solid curve is the result of Roper’s analysis\({}^{39}\) of experimental data on the investigation of the velocity distribution of photoelectrons from Mo at different temperatures in the interval \(300\)—\(1000^\circ\mathrm{K}\), by the method of the spherical condenser, while the dotted curve gives the theoretical curves for \(n=\frac{3}{2}\) (Mitchell). We see that the experimental data of Fig. 29 correspond best of all to the case \(n=\frac{3}{2}\) (Mitchell), although it should be noted that it would have been still better for \(n=2\)—\(2.5\). In general, in the region of large energies the data of all theories approximately coincide, since they are all masked according to (57) by the Fermi function \(F(W)\).

Fig. 30. Solid curve—the result of an analysis of Roper’s experimental data; dotted curve—the theoretical curves for

\[ n=\frac{3}{2}. \]

In DuBridge’s equation (19) there enters the coefficient \(\alpha\), characterizing the probability of absorption by an electron at the surface of a light quantum and equal to \(\alpha=CN^{-4/3}\), where \(N\) is the concentration of electrons in the metal. From the vertical displacement of the experimental curve

\[ \lg \frac{I}{T^2}=f\left(\frac{h\nu}{kT}\right) \]

one can determine the experimental value of this quantity \(\alpha\), and then \(\frac{\alpha}{\alpha_1}\) will determine the fraction of the incident quanta absorbed by the electrons at the surface. This quantity was determined by Mann and DuBridge\({}^{40}\) for Mg, Be, and Na and is presented in Table 3, together with the values for W and Ba obtained by other authors. From this table we see that the ratio

\[ \frac{\alpha_1}{\chi}\approx 10^{-2}\text{—}10^{-3}, \]

i.e. only a fraction of a percent of all quanta incident upon it is absorbed by the electrons at the surface. It turns out that if one takes \(\alpha\sim 10^{-33}\), then for \(x=40\), \(\varphi(x)=800\), and at \(T=300^\circ\mathrm{K}\), \(I=5\cdot 10^{-5}\ \mathrm{eV/quantum}\). Further, it is interesting to note that

films K—Ag, K—Pt, and Na—Al, investigated [[unclear: cut-off text]]

1. This calculation differs slightly from that given by Fowler, who proceeded from an expression for the number of electrons in a unit volume with normal velocity in the interval \(d\xi\). It seems somewhat more logical to proceed here from an expression for the number of electrons striking in 1 sec. a unit surface, which, fortunately, leads to a simpler result that does not contain the unpleasant factor \((W_a-h\nu)^{1/2}\), which Fowler regarded as constant, although in fact it could vary by 15% or even more in the region where the theory was tested. ↩