On the Photoelectric Effect

P. I. Lukirsky, Leningrad.

When radiant energy (optical rays, X-rays, and $\gamma$-rays of radioactive bodies) falls upon some body, a number of phenomena are observed. If we take a beam of X-rays, then, as it passes through a body, in the general case the following will occur:

1. Some of the quanta $h\nu_0$ incident on the body, in interacting with some atomic electron, will eject this electron beyond the limits of the atom. In this process the entire energy of the quantum $h\nu_0$ will be wholly expended on the work of removing the electron $P$ from the atom and on imparting to it the corresponding kinetic energy (photoelectric effect).

2. Bergen Davis has recently observed in the region of X-rays a phenomenon analogous to the Raman phenomenon. When graphite is illuminated by the $K\alpha$ line of molybdenum, Bergen Davis1 showed that the energy of the incident quantum, being expended in full, may go toward the work of removing an electron from the atom; however, the surplus energy is not imparted to the electron in the form of kinetic energy, as was the case in the photoeffect, but is emitted in the form of a new quantum with a frequency determined from the obvious equality

$$ h\nu' = h\nu_0 - P. $$

In both of the indicated cases the energy of the incident quantum is spent in full. As a result of both these phenomena, inside

of the atom there remains a free electron orbit. When it is filled by some electron, frequencies characteristic of the given body will be emitted (the phenomenon of fluorescence in X-rays).

1. When a quantum acts on some weakly bound electron (\(P\) is small in comparison with \(h\nu_0\)), part of the quantum’s energy is imparted to the electron in the form of kinetic energy, while another part is emitted in the form of a quantum \(h\nu'\). Both the energy of the new quantum and the energy \(E_p\) of the electron are determined by the law of conservation of energy and the law of conservation of momentum. In this phenomenon, known as the Compton effect, the frequency of the new scattered quantum depends on the angle of its scattering. The recoil-electron energy changes correspondingly as well.

2. In addition to the fluorescence rays and the rays scattered in the Compton effect, the so-called “classical scattering” of rays by the atoms of a body is also observed. In classical scattering the scattered rays have the same frequency as the incident rays. In the case of fluorescence we obtain rays characteristic of the atoms of the illuminated body, whereas in the Compton effect we obtain rays of lower frequency, and moreover one that is different at different angles. Another most essential distinction of classical scattering is that the phases of the rays scattered by different atoms are related to one another. This property is manifested, for example, in the phenomenon of X-ray diffraction. A relation between the phases is absent in the case of the other two radiations of the body. When \(\gamma\)-rays fall on a body, the same phenomena are observed. When optical wavelengths fall, however, in addition to the removal of an electron beyond the limits of the atom, the energy of the quanta may be wholly spent on exciting an atom or molecule, by transferring an electron to a higher energy level or by changing its vibrational energy in the molecule. In the latter case, sometimes only part of the energy of the quantum is expended on this process, while the remaining part is emitted in the form of a new quantum (the Raman effect).

An optically excited atom, upon returning to its former state, either emits a quantum characteristic of the given fre-

will emit light (fluorescence), or, in collision with neighboring particles, will transform the potential energy of excitation into the thermal energy of the body (collisions of the second kind).

Such, in the most general outline, are the basic processes observed in a body under the action of quanta of radiant energy.

Before proceeding to a detailed consideration of one of these processes, let us determine under what conditions one or another phenomenon is observed and how their probability changes with the frequency of the incident quantum.

The phenomenon of classical scattering is apparently observed at all wavelengths, since the phenomenon of diffraction by a crystal lattice has been observed not only for hard X-rays, but also for the \(\gamma\)-rays of radioactive substances; however, one may think that, as the frequency of the incident rays increases, the probability of this phenomenon, generally speaking, decreases.

The Compton phenomenon is observed at high frequencies on weakly bound electrons, and its probability increases as the ratio \(\frac{h\nu_0}{P}\) increases.

For the Raman effect this dependence is difficult to establish, even qualitatively for the present, since there are very few data. In the case of the photoelectric effect, on the contrary, this dependence has been elucidated in great detail. When the energy of the incident quantum is equal to, or only slightly greater than, the work \(P\) against the forces binding the electron, the probability of the effect is greatest. As the wavelength of the quantum decreases, the probability decreases, approximately in proportion to the cube of the wavelength. We have these data from the study of the curves of “true” absorption for X-rays. In what follows, considering various cases of the photoeffect, we shall at the same time clarify all the basic characteristic features of this phenomenon.

PHOTOEFFECT IN X-RAYS AND \(\gamma\)-RAYS.

When a substance is illuminated by X-rays and by the \(\gamma\)-rays of radioactive substances, the phenomenon of the photoeffect is observed,

whereas the velocities of the photoelectrons are determined by the Einstein relation

\[ \frac{1}{2}mv^2 = h\nu - P_1 - P_2, \tag{1} \]

where \(\frac{1}{2}mv^2\) is the kinetic energy of the electron torn out of the body, \(P_1\) is the work of removing the electron from the atom, and \(P_2\) is the work of passage of the electron through the surface layer of the body. The latter quantity, in the case of X-rays and \(\gamma\)-rays, does not play an essential role, since it is small (of the order of several equivalent volts) in comparison with the energy of the quantum (from \(10^4\) to \(10^6\) volts). On the contrary, in the photoelectric effect from visible light, as we shall see below, it plays an essential role.

It must be pointed out, however, that only in the case of the photoelectric effect in gases (for example, in observations with Wilson’s chamber) do the velocities of all photoelectrons satisfy equation (1); in those cases where we observe the photoelectric effect of solid or liquid bodies, only the maximum velocity of the observed electrons is equal to the indicated value, since, moving from within the body outward, the electrons spend their energy in various ways and emerge with all possible velocities—from the maximum down to zero.

The validity of the indicated relation for X-rays was confirmed by the work of de Broglie, Whiddington, and Robinson, who investigated the velocities of photoelectrons by their deflection in a magnetic field. If some body is illuminated with monochromatic X-rays of frequency \(\nu\), then the maximum velocities of the photoelectrons will be equal to

\[ h\nu - P_0, \]

where \(P_0\) is the energy of removal of an electron from the \(K\), \(L_1\), \(L_2\), \(L_3\), etc. levels of the atoms constituting our body, i.e. the magnitude of the so-called spectral term \(K_\sigma\), \(L_\sigma\), \(M_\sigma\), etc.

Besides such velocities, the above-mentioned authors found velocities of electrons corresponding to the proper frequencies

OF THE PHOTOELECTRIC EFFECT

body; thus, when a body is illuminated with frequency \(\nu\), it itself emits fluorescence lines which, upon striking atoms, can likewise tear out electrons.

Analogous investigations for \(\gamma\)-rays were carried out by Ellis, Meitner, and later by Tibaux, with various bodies, both radioactive and non-radioactive, being irradiated by \(\gamma\)-rays. These works showed that the so-called line \(\beta\)-spectra, which are usually observed in the \(\beta\)-radiation of radioactive bodies, are of secondary origin. They are electrons torn from the \(K\), \(L\), \(M\), etc. levels of radioactive bodies by \(\gamma\)-rays issuing from atomic nuclei. In addition to these \(\gamma\)-rays, possessing quite definite velocities, there is a continuous \(\beta\)-spectrum. As the work of Gurney showed, this continuous spectrum is caused by primary \(\beta\)-rays, emitted spontaneously from the nuclei of radioactive bodies and thus giving rise to the phenomenon of isotopy.

Besides the method of magnetic deflection, the photoelectric effect in X-rays and \(\gamma\)-rays can be investigated with the aid of a Wilson chamber. In this case one can observe individual photoelectrons, thanks to the fog droplets deposited on those ions which these photoelectrons create along their path in the gas. From the total length of the photoelectron’s path one can judge its energy, since the path length is, at a given gas pressure, proportional to the square of the electron’s energy (Whiddington’s law). In this way the photoelectric effect and the Compton effect were studied in a whole series of cases.

Let us dwell somewhat more fully on the work of Auger. In order to investigate the photoelectric effect in a Wilson chamber, he filled it with hydrogen, to which a small quantity of some noble gas (argon, xenon, etc.) had been admixed. In this case the photoelectric effect will occur chiefly on the atoms of argon, since the energy of the incident X-rays is much closer in magnitude to the energy of the electron levels in argon than in hydrogen (see above—the probability of the photoelectric effect). But the electrons torn from argon will move in the chamber,

filled with almost pure hydrogen; consequently, at a given velocity they will traverse paths considerably longer than in the case of air (the range of electrons varies inversely with the atomic number of the gas). This increases the accuracy of measuring the length of the electron paths, from which, as indicated above, their energy can be found. Auger’s work showed that very often one can observe a series of electron paths from a single argon atom; sometimes two, sometimes four, of which the two shortest have the same length. To explain this Auger proposed the following scheme. A quantum of X-rays, being absorbed, tears an electron from the \(K\)-level of argon. The energy of this electron will be \(E_1 = h\nu - K_{\sigma}\), where \(K_{\sigma}\) is the work of removal from the level, in other words, the magnitude of the \(K\)-term. One of the \(L\)-electrons passes into the vacant place in the \(K\)-shell, and the atom emits the line \(K\alpha\) with frequency \(\nu_\alpha\), determined from the equality \(h\nu_\alpha = K_{\sigma} - L_{\sigma}\). This quantum, being absorbed in the same atom, tears an electron from the \(L\) group. The energy \(E_2\) of this electron will obviously be equal to

\[ E_2 = h\nu_\alpha - L_{\sigma} = K_{\sigma} - 2L_{\sigma}. \]

After this, in our atom there remain two unoccupied places in the \(L\)-shell. In their place two electrons from the \(M\)-shell will pass; in this process two quanta of frequency \(\nu_2\) (the \(L_\alpha\) line of the spectrum) will be emitted, which, obviously, is determined by the equality \(h\nu_2 = L_{\sigma} - M_{\sigma}\). These two quanta, being again absorbed in the same atom, will tear two electrons from the \(M\)-shell of the atom. Both of them will have identical energies \(E_3\), equal to

\[ E_3 = h\nu_{\sigma} - M_{\sigma}. \]

Measurements of the length of the resulting paths confirmed this picture.

This work gives us a vivid picture of the energy levels in the atom and confirms Bohr’s scheme, according to which we imagine the emission of a spectral line as the result of the transition of an atom from one energy state to another. At the same time it is curious to note, on the basis of Auger’s work, that the probability of absorption of the emitted-

the quantum by an atom in that very same atom is very large; it is approximately equal to \(1/5\). Of course, this fact can in no way be connected with the idea of a spherical wave propagating in all directions. An analogous phenomenon was observed by Ellis also in the absorption of \(\gamma\)-rays. In addition to the methods indicated for studying the photoeffect in the region of soft X-rays, the author applied the method of a retarding electric field in a spherical condenser (for a description of the method, see below). Determining the energies of photoelectrons ejected from zinc, the author, from their energies, found (see equation 1) a series of X-ray wavelengths lying in a region then (1924) not investigated, including the wavelength \(K\alpha\) of carbon. At the present time this region has been investigated with crystals having large lattice constants (organic fatty acids) by Thibaud (1927) and with a diffraction grating by Dauvillier (1927) and Thibaud (1927–1928). These authors found a coincident value for carbon \(K\alpha\).

Thus, in the author’s work mentioned, the phenomenon of the photoelectric effect in the case of soft X-rays was applied for purposes of spectroscopy. All these works show that the initial energy of the electron ejected in the photoeffect, in the case under consideration, is determined by equality (1) and therefore depends only on the frequency of the incident light.

The number of electrons ejected from an atom of the body in the photoeffect is directly proportional to the intensity of the incident light. It seems interesting to touch upon one more feature observed in the phenomenon of the photoeffect. If the gas in a Wilson chamber is illuminated by X-rays, then it can be shown that the photoelectrons fly out predominantly in directions perpendicular to the direction of the X-ray beam. If, however, the gas is illuminated by linearly polarized X-rays (they are not difficult to obtain by taking rays scattered by some body in a direction perpendicular to the direction of the primary beam), then in this case the initial velocities of the photoelectrons will lie in one plane, remaining in the same

time perpendicular to the beam. This preferred plane contains the electric vector of the wave. Thus these results lead to the conclusion that photoelectrons fly out in the direction of the electric vector of the incident rays. In the case of hard X-rays, however, this direction is somewhat displaced toward the direction of the beam, making with it an angle somewhat smaller than \(90^\circ\).

THE PHOTOEFFECT WHEN GASES ARE ILLUMINATED BY OPTICAL FREQUENCIES.

Passing to the case of optical wavelengths, we shall first of all consider the photoeffect in gases.

The question of the existence of the photoeffect in gases long remained disputed, since some of the light scattered by the walls of the apparatus could always fall on the metallic electrodes. Metal, however, emits considerably more electrons than a gas; therefore the effect from the gas could always be masked, and there was never certainty of its existence.

To avoid this, Mohler (F. Mohler, 1926) proceeded as follows. He filled an apparatus having two electrodes with cesium vapor, one of the electrodes being a filament of thoriated tungsten heated to incandescence. Without illumination, between these electrodes, at a small applied potential difference (about \(0.5\)—\(1\ \mathrm{V}\)), there flowed a pure electron current of very small magnitude, since the field was distorted by the negative space charge and, near the filament, had the opposite direction. If the gas is illuminated and the photoeffect is produced in it, then the electrons torn from the electrodes will change the current only slightly. But, with a photoeffect from the gas, positive ions will be formed as a result of the removal of electrons. An insignificant number of these positive ions will suffice to change radically the distribution of potential caused by the space charges and thus to amplify the main electron current. The amplification of this current serves as a measure of the number of positive ions, and consequently also as a measure of the photoeffect. These experiments of Mohler undoubtedly establish the presence of the photoeffect in gases.

In the case of the removal of electrons from the inner (X-ray) levels of the atom, we must always remove them beyond the limits of the atom, since all inner shells are filled with electrons. But when a valence (optical) electron is removed, we can not only remove it beyond the limits of the atom (continuous absorption beyond the series limit), but also transfer it to one of the possible optical orbits. In this case we shall have resonance absorption, not associated with the photoeffect. Whereas in the preceding cases absorption was always accompanied by the photoeffect, Mohler, varying the wavelength of the light illuminating cesium, noticed that as it decreased the photoeffect changed in the following way. When the energy of the incident light is equal to the difference of the levels \((1s — 2p)\), corresponding to the first line of the principal series of cesium, the photoeffect begins; then it becomes equal to zero; later, when \(h\nu\) is equal to \((1s — 3p)\), it appears again, then disappears again, and so on. Finally, when the magnitude of the quantum becomes greater than the limit of the principal series of cesium, the photoeffect always occurs, although its intensity gradually decreases as the incident quantum increases.

Such a course of the phenomenon is due to the fact that, when the energy of the quantum is equal to \((1s — 2p)\), resonance absorption of light occurs and the cesium electron is transferred from the normal state \(1s\) to the level \(2p\). Such an excited cesium atom, colliding with other atoms, can, by collisions of the second kind (at the expense of the kinetic energy of the thermal motion of the atoms), eject its electron from the \(2p\) orbit—this gives us the observed photoeffect. When \(h\nu > (1s — 2p)\), no absorption occurs and there can be no photoeffect, since the electrons remain in their normal orbits, from which it is difficult to eject them by a thermal collision, since this requires a considerable portion of energy. At \(h\nu = 1s — 3p\) we again have absorption, transition to the \(3p\) orbit, and the photoeffect by a collision of the second kind. But when \(h\nu \geqq 1s\), the electron is ejected beyond the limits of the atom at once, since \(h\nu\) is greater than the work of ionization of the atom; in this case we have simultaneously continuous absorption and the photoeffect.

The gradual decrease of the photoeffect and of continuous absorption with further increase of the energy of the quantum, just as in the case of X-rays, is the fundamental law of the probability of action of a quantum on electrons in the photoeffect.

THE PHOTOEFFECT WHEN A METAL IS ILLUMINATED BY OPTICAL FREQUENCIES.

As in the case of X-rays, so also in Mohler’s experiments, the phenomenon of the photoeffect was observed on atomic electrons. When metals are illuminated by optical frequencies, however, the emitted electrons are apparently not ordinary atomic electrons. Therefore the general character of the phenomenon indicated is somewhat different.

When metals are illuminated by visible or ultraviolet light, electrons are torn out. This phenomenon begins at a certain wavelength, characteristic for the given metal specimen and called the excitation limit of the photoeffect. With further decrease in the wavelength of the incident light, the photoeffect becomes more and more intense. However, in order to characterize the increase of the photoeffect with decreasing wavelength, it is necessary to consider the influence of the polarization of the incident light. When light is incident normally on the surface of a metal, the electric vector of the light wave always lies in the plane of the metal. In this case the number of photoelectrons, beginning from the excitation limit, increases smoothly as the wavelength of the light decreases (the normal photoeffect). But when light falls on the metal in such a way that the electric vector of the wave is perpendicular to the surface of the metal, the so-called selective photoeffect occurs. The selective photoeffect begins at the same wavelength, then increases as the wavelength decreases, reaches a maximum, falls, and then slowly, smoothly increases. It is possible that such a course of the phenomenon is connected with the fact that the direction of the electric vector in one way or another determines the direction of the initial velocity of the photoelectro-

of electrons inside the metal, which is also what we have in the case of X-rays.

The number of photoelectrons that manage to escape from the metal to the outside (this number is what we observe experimentally), of course, will be determined by how their initial velocities are directed with respect to the surface of the metal.

In what follows we shall speak only of the normal photoelectric effect.

A number of works have been devoted to the study of the photoelectric effect in metals. These include the works of Millikan1, Ramsauer2, Becker3, Richardson and Compton4, and many others.

Millikan illuminated sodium and lithium with monochromatic spectral lines and determined the maximum velocities of the emitted electrons by the method of a retarding electric field. He succeeded in demonstrating the validity of Einstein’s relation (1) and, with great accuracy, in finding the numerical value of Planck’s constant
$h = 6.57 \cdot 10^{-27}$ erg·sec. Measuring, in separate experiments, the contact potential difference that exists between sodium or lithium, on the one hand, and an auxiliary electrode, on the other, he found that for sodium and lithium the contact potential difference is equal to the difference of the excitation limits of the photoeffect. From this he came to the conclusion that, in the photoelectric effect, an equalization of the free electrons of the metal takes place. However, analysis of this question will show us that such a conclusion may prove to be erroneous. In his method Millikan had no possibility of investigating the velocity distribution of the photoelectrons, since he placed a Faraday cylinder opposite the surface of the metal and created a retarding electric field between them. In this case the field is so complicated that to judge from the curve of current decrease at different retarding

potentials, it is rather difficult to judge the velocity distribution.

The retarding-field method in a plane capacitor also gives no answer to the question, since in this case, as is easy to see, one can find only the distribution of the normal components of the velocities, whereas the photoelectrons, as is known, fly out in all possible directions.

For studying the velocities of photoelectrons, Richardson and Compton applied a somewhat different method, subsequently used by Becker for studying the photoelectric effect under the influence of black-body radiation. The inner illuminated electrode was made in the form of a small plate, and the outer one in the form of a sphere. In such a case the field will be approximately radial—especially if the plate is taken to be of very small dimensions. Therefore the “current strength—retarding field” curve will give an approximate distribution of the velocities of the photoelectrons. But in this case it is difficult to say how much the result will be affected by those distortions of the field which will occur near the plate. These distortions at the center of the sphere, where the greatest field gradient exists, may very strongly distort the trajectories of the motion of the electrons. In addition, in the above-mentioned work of Richardson and Compton the measurement of the velocities was carried out imprecisely, as is evident at least from the fact that for Planck’s constant they obtained numerical values from \(3.5\) to \(5.8 \cdot 10^{-27}\) erg·sec.

In the work carried out by the author jointly with S. Prilezhaev, the method of the spherical capacitor was applied. Below I shall dwell in more detail on this work.

Method of the Spherical Capacitor

Let the inner sphere of a spherical capacitor be illuminated by monochromatic radiation of frequency \(\nu\), under the influence of which it emits electrons in all possible directions. In the radial field of the capacitor the electrons will move along curves of the second order. Let us consider those of them which leave the surface of the inner—

of the third sphere, possessing an initial energy \(\frac{1}{2}mv^2\). Varying the retarding field \(eV\), we shall observe that for all values

\[ eV \geq \frac{1}{2}mv^2 \]

the electrons will return to the inner sphere. In this case the trajectories of their motion will be ellipses whose aphelia will lie closer than the outer sphere. For values of the retarding field

\[ eV < \frac{1}{2}mv^2 \]

their trajectories will be hyperbolas, or parabolas, or ellipses with distances to aphelion greater than the distance to the outer sphere. Electrons moving along hyperbolas and parabolas will always reach the outer sphere, just as will those electrons whose elliptical paths have a distance to aphelion greater than the radius of the outer sphere. The remaining electrons will return back.

It can be shown1 that, when the ratio of the radii of the spheres is \(\frac{b}{a}=7.5\) (and this case occurred in our apparatus), with the retarding field

\[ eV = 0.98 \cdot \frac{1}{2}mv^2 \]

all electrons will reach the outer sphere; whereas with the field

\[ eV = \frac{1}{2}mv^2 \]

they will all, as we have seen, return back. Thus the current–retarding-field curve will give us, with the indicated accuracy, the true velocity distribution of the photoelectrons. However, as is easy to see, for determining the boundary of the maximum velocities the method will be absolutely exact.

For measuring photoelectrons by the indicated method, an apparatus was constructed, shown in Fig. 1. A glass sphere \(S\), silvered on the inside, served as the outer plate of a spherical condenser. The inner electrode was a little sphere \(E\) of the metal under investigation. This little sphere could be easily changed, since it was screwed onto a rod that was removed from the apparatus together with the ground joint. The rod was enclosed inside a tube of fused quartz, sealed into the ground joint. The latter was done to improve insulation and to eliminate the harmful influence of creeping discharges over the glass. The diameter of the outer sphere was \(11\ \mathrm{cm}\), and that of the inner little sphere was \(1.5\ \mathrm{cm}\).

Fig. 1.

In the outer sphere, opposite the inner little sphere, there was an opening \(1\ \mathrm{cm}\) in diameter, closed by a quartz plate; through this opening the little sphere was illuminated by a monochromatic line, selected by the quartz monochromator \(M\) from the spectrum of the mercury arc \(L\). Between the arc and the first slit of the monochromator, filters \(F\) (mica, gelatin, celluloid, etc.) were placed. The filters were used in those cases when there was not complete certainty as to the monochromaticity of the spectral lines employed (chiefly in the region of longer wavelengths).

The air was pumped out of the apparatus in the most thorough manner. A potential \(V\) from the potentiometer \(R\) was applied to the outer sphere \(S\). The average current strength during the measurements was about \(10^{-13}\ \mathrm{A}\). The accuracy of determining the current strength was several tenths of a percent.

It should be noted that no so-called “photoelectric fatigue” was observed, as indeed should be the case under stationary conditions, which could be established by prolonged evacuation of the apparatus before the measurements.

The balls made of the metals under study were subjected to the most thorough mechanical treatment, whereby their surface was cleaned of oxides and other contaminants.

Measurements and Typical Results.

A ball made of the metal under study was illuminated with one of the monochromatic lines of the mercury spectrum, and the curve of the dependence of the photoelectric current on the applied electric field (potential \(V\)) was recorded.

A typical curve, obtained for nickel when illuminated by its spectral line \(\lambda = 2537\ \text{Å}\), is shown in Fig. 2. At potentials \(V \geqq 1.0\) volt the curve runs parallel to the axis of abscissas. In this case the field inside the condenser is accelerating and the current strength remains constant, since all photoelectrons, whatever velocities they may have upon leaving the metal, reach the outer sphere.

Fig. 2.

At \(V = V_1 = 1.0\) volt the curve begins to decline. At this magnitude of the field, those electrons which have velocities equal to zero upon leaving the metal begin to be held back. The circumstance that in this case the applied potential \(V_1\) is not equal to zero is due to the presence, in the space between the electrodes, of a contact field \(K\), exactly equal to the applied potential difference \(V_1\), taken with the opposite sign. The sum of the contact field and the applied field gives the zero field at which electrons of zero velocity begin to be held back. When the same metal is illuminated with different wavelengths of incident light, the decline of the curve always begins at one and the same value of the field \(V = V_1\) (see the following curves).

The value of the potential \(V = V_1\) gives us the magnitude of the contact potential difference between our electrodes, thanks to which we can correct all observations for the contact potential difference \(K\) and obtain the true value of the velocities of the emitted electrons. It is important to note that we find this value from the same curve, whereas Millikan had to make special measurements of the contact potential difference, which, generally speaking, changes with time if the surface properties of the metal change (for example, absorption of vapors and gases by the surface, or its oxidation).

At potentials \(V < V_1\) the resultant field will be retarding for the electrons emitted from the small sphere, and, as it increases, electrons of ever greater velocities will return to the small sphere. Finally, when the true retarding field reaches the value of the maximum velocity of the electrons:

\[ e(V_2 + K) = \frac{1}{2} mv_{\max}^2 \tag{2} \]

(where \(V_2\) is the applied field), all photoelectrons emitted from the small sphere will be stopped. At this potential \(V_2\) we should have a current equal to zero, just as for all fields \(V < V_2\). Meanwhile, as is seen from Fig. 2, in these fields there is a small current in the reverse direction, which slowly increases as \(V\) decreases. This current is caused by the fact that part of the light is reflected from the small sphere and falls on the outer sphere. Under the influence of the illumination the outer sphere emits photoelectrons, which move in the field, accelerating for them, from the sphere toward the small sphere.

To clarify the character of the dependence of the reverse current on the applied electric field, special control measurements were carried out. These measurements show that at the potential \(V = V_1\) (absence of field) the reverse current vanishes.

Since in all cases the reverse current goes to zero at \(V = V_1\), it is very easy to exclude it from the ordinary photoeffect curves.

The following metals were investigated by the indicated method:

\[ \mathrm{Al,\ Zn,\ Sn,\ Ni,\ Cd,\ Pb,\ Cu,\ Pt,\ Ag.} \]

In Fig. 3 are shown the curves of the photoelectric effect for nickel. The curves for the other listed metals (zinc, copper, aluminum, tin, cadmium, lead, platinum, and silver) have an analogous character.

From all these curves it is evident that, for a given metal, the decrease of the current begins when it is illuminated with different wavelengths always at one and the same value of the potential \(V = V_1\). In this case, as was indicated, the true field is equal to zero, and consequently the value \(V = V_1\) determines for us the magnitude of the contact potential difference between the metal under investigation and the external sphere. That point of the curve at which the current is equal to zero gives us the value of the maximum velocity of the photoelectrons corresponding to the given wavelength. In order to determine the true value of the maximum velocity, it is necessary to take, instead of the applied field, values reckoned not from zero but from the point \(V - V_1\), since these values—the applied potential plus the contact field—will characterize for us the true difference of potentials traversed by the electron.

Fig. 3.

In this way we are able to determine the contact potential differences between the external sphere and the metals under investigation, and to find the values of the maximum velocities when illuminating them with various wavelengths. The latter enables us to determine the numerical value of Planck’s constant \(h\) and the threshold of excitation of the photoelectric effect for these metals.

The intermediate course of the curve allows us to find the distribution of velocities of the photoelectrons. As is easily seen from the curves presented, in the photoelectric effect all possible velocities are always present, from zero up to the maximum.

Determination of Planck’s constant \(h\).

According to Einstein’s equation, for the maximum velocity of the photoelectrons we have:

\[ \frac{1}{2}mv_{\max}^{2}=e(V_{2}+K)-h\nu-\eta_{1}-\eta_{2}, \tag{3} \]

where \(V_{2}\) is the applied potential corresponding to a current equal to zero, and \(K=V_{1}\) is the contact potential. Thus, \(V_{2}+K\) is the true potential difference corresponding to the absence of current.

Plotting on the axis of ordinates the values of the potentials \(V_{2}+K\), obtained for a given metal when it is illuminated by various wavelengths, and on the axis of abscissas the corresponding values of the frequencies \(\nu\), we obtain a straight line, whose angular coefficient will determine for us the value of \(\frac{h}{e}\). Knowing the numerical value of the electron charge \(e\), we determine the magnitude of Planck’s constant \(h\).

It must be noted, however, that the current curves gradually (see Fig. 3) approach the zero value, as a result of which it is not possible to find their points of intersection with the axis of abscissas. In view of this, special measurements of the ends of the curves were made, the sensitivity of the electrometer being brought to the highest possible limit. Particular attention was paid to the strict monochromaticity of the wavelengths used. As an example, Fig. 4 gives curves obtained for aluminum when it was illuminated by wavelengths: \(2534.77\ \text{\AA}\), \(2653.66\ \text{\AA}\), \(2967.28\ \text{\AA}\), \(3021\ \text{\AA}\), and \(3125.66\ \text{\AA}\). It is easy to see that here the curves proceed considerably more steeply than in the preceding cases. Nevertheless, even in this case the exact determination of the points of intersection with the axis of abscissas is somewhat difficult. In view of

for this we made use of a regularity in the course of the curves which it was possible to observe in all measurements. Namely, it was found that the strength of the photoelectron current at a distance sufficiently close to the end of the curve satisfies the empirical formula \(i=A(V_2-V)^2\), where \(A\) is a constant for the given curve. Thus, plotting along the ordinate axis, instead of the current strengths, the values \(\sqrt{i}\), we obtain the straight lines marked in Fig. 4 by dotted lines. Their point of intersection with the abscissa axis is then taken as the value of the maximum velocity of the electrons. Such an extrapolation inevitably entails some error. However, it must be thought that, with the precautions adopted, it has been reduced to a minimum.

Fig. 4.

The values of the maximum velocities obtained in this way are plotted along the ordinate axis in the graph of Fig. 5. Along the abscissa axis are plotted the values of the frequencies \(\nu\). The curves are entered in the following order from left to right: aluminum, zinc, tin, nickel, cadmium, copper, platinum. The values obtained from their slope for the most carefully studied metals are collected in the following table:

The mean value of Planck’s constant \(h\) from our measurements is obtained as

\[ h=6.546\cdot 10^{-27}\ \text{erg sec.} \]

From optical data, as is known, we have:

\[ h = 6.545 \cdot 10^{-27}\ \text{erg sec}. \]

The agreement is very close; however, it must be noted that the accuracy for which we can vouch in our determinations is about \(0.1\)—\(0.2\%\).

Determination of the excitation limits of the photoelectric effect

As is easily seen from Fig. 5, the point of intersection of our straight lines (for the true values of \(V_2 + K\)) gives us the frequency \(\nu_0\) at which the maximum velocity of the photoelectrons is equal to zero. At lower frequencies \(\nu < \nu_0\), the photoelectric effect will not occur. According to Einstein’s equation (3), this frequency is obtained by putting \(V_2 + K = 0\), which gives:

Fig. 5.

\[ h\nu_0 = r_1 + r_2. \tag{4} \]

Thus Einstein’s equation may be rewritten as follows:

\[ e(V_2 + K) = \frac{1}{2}mv_{\max}^{2} = h\nu - h\nu_0. \tag{4'} \]

The data obtained in this way for various metals are given in Table I. Here, in the first column, are given the values of the contact potentials \(K\) between the metal under investigation and the outer sphere. In the following columns are given the values of the frequencies \(\nu_0\) and the corresponding values for \(\lambda_0\), and the values of the energy \(h\nu_0\), expressed in equivalent

volts. It should be noted that the values we obtained for the limits are shifted, in comparison with the data of other authors, toward the red part of the spectrum. Although, generally speaking, different values always should be obtained on different specimens, in our case this shift is of a systematic character.

This is explained by the fact that the determinations of the maximum velocities of the photoelectrons were carried out with the greatest possible accuracy. With cruder measurements, of course, as is easy to see, smaller values of the quantities \(\nu_0\) should be obtained.

TABLE I.

From the values of the contact potential differences \(K\) between the metals under study and the outer sphere, we can find the values of the contact potential differences \(K_{1,2}\) between any two metals taken. For this it is sufficient to take the difference of two values of \(K\); in such a case we, according to Volta’s rule, exclude the value of the outer sphere and find the magnitude of the contact potential difference \(V_1 = K\) between metals 1 and 2. For this it is necessary that the outer sphere remain under identical conditions, which could always be verified.

Comparing the value of the contact potential difference \(K_{1,2}\) thus obtained with the difference of the values of the corresponding thresholds and of the excitation of the photoeffect, we can see that, always with an accuracy to hundredths of a volt, the relation

\[ h\nu_1 - h\nu_2 = K_{1,2}. \tag{5} \]

holds.

Its existence can be checked in the following way. The observed potentials \(V_2\) and \(V'_2\) for the values of the maximum velocity, obtained at one and the same wavelength, must be identical for all metals. Indeed, writing the equalities for two metals:

\[ e(V_2 + K_1)=h\nu-h\nu_1, \]

\[ e(V'_2 + K_2)=h\nu-h\nu_2 \]

and subtracting them, we obtain:

\[ e(V_2 - V'_2)-eK_{1,2}=h\nu_2-h\nu_1, \]

since \(K_{1,2}\), according to Volta’s rule, is equal to \(K_{1,2}=K_1-K_2\). Thus, by virtue of equality (5), we have

\[ V_2 - V'_2=0. \]

Table II gives the contact potential differences \(K_{1,2}\) between aluminum and the other metals investigated, as compared with the corresponding differences of the excitation thresholds.

TABLE II.

On the Binding of Electrons

If any two metals, which may be characterized by saying that the mean work of removal of an electron from the first metal is \(\Phi_1\), and for the second metal \(\Phi_2\), are brought into contact, then a contact potential difference will be established between the free ends of these metals, numerically equal to

\[ K_{1,2}=\frac{\Phi_1}{e}-\frac{\Phi_2}{e}, \]

since the electrons will pass from that metal where the work is smaller into the other metal, where it is larger. Such a transition will always exist, regardless of the nature of those factors which determine the existence of the work of removal of an electron \(\Phi\). It should be noted, however, that in doing this we neglect the diffusion potential difference concentrated at the place of contact of the metals, which accounts for the absorption and liberation of Peltier heat. But since the magnitude of this jump is of the order of hundredths or thousandths of a volt, with the accuracy available experimentally in determining contact potentials it may, of course, be neglected. In the general case we shall have \(K_{1,2}=\frac{\Phi_1}{e}-\frac{\Phi_2}{e}+\pi_{1,2}\), where \(\pi_{1,2}\) is the diffusion potential difference. By the threshold of excitation of the photoelectric effect, \(h\nu_0\), we determine precisely the work of removal of an electron from the metal, since the photoelectric effect ceases to exist at those values of the quantum energy which are insufficient to tear the electron out. Moreover, this is true only in the case where it is precisely those electrons that are torn out whose free transition is responsible for the appearance of the contact potential difference, i.e. the so-called conduction electrons. If, in the photoelectric effect, electrons more strongly bound to the atoms were torn out rather than conduction electrons, then \(h\nu_0\) would be greater than \(\Phi\), and equality (5) would not be satisfied, for example, in the case of X-rays,

Comparing the value of the contact potential difference \(K_{1,2}\) thus obtained with the difference of the values of the corresponding thresholds of excitation of the photoeffect, we can see that, always to within hundredths of a volt, the relation holds:

\[ h\nu_1 - h\nu_2 = K_{1,2}. \tag{5} \]

Its existence can be checked in the following way. The observed potentials \(V_2\) and \(V'_2\) for the values of the maximum velocity, obtained at one and the same wavelength, must be identical for all metals. Indeed, writing the equalities for two metals,

\[ \begin{aligned} e(V_2 + K_1) &= h\nu - h\nu_1,\\ e(V'_2 + K_2) &= h\nu - h\nu_2, \end{aligned} \]

and subtracting them, we obtain:

\[ e(V_2 - V'_2) - eK_{1,2} = h\nu_2 - h\nu_1, \]

since \(K_{1,2}\), according to Volta’s rule, is equal to \(K_{1,2} = K_1 - K_2\). Thus, by virtue of equality (5), we have

\[ V_2 - V'_2 = 0. \]

Table II gives the contact potential differences \(K_{1,2}\) between aluminum and the other metals investigated, as against the corresponding differences of the excitation thresholds.

TABLE II.

On the binding of electrons.

If any two metals, which may be characterized by the fact that the average work of removing an electron from the first metal is \(\Phi_1\), and for the second metal \(\Phi_2\), are brought into contact, then between the free ends of these metals there will be established a contact potential difference numerically equal to

\[ K_{1,2}=\frac{\Phi_1}{e}-\frac{\Phi_2}{e}, \]

since the electrons will pass from that metal where the work is smaller into the other metal, where it is greater. Such a transition will always exist, irrespective of the nature of those factors which determine the existence of the work of removal of the electron \(\Phi\). However, it should be noted that in doing so we neglect that diffusion potential difference, concentrated at the place of contact of the metals, by which the absorption and liberation of Peltier heat are determined. But since the magnitude of this jump is of the order of hundredths or thousandths of a volt, then, with the experimentally available accuracy of determining contact potentials, it can, of course, be neglected. In the general case we shall have \(K_{1,2}=\frac{\Phi_1}{e}-\frac{\Phi_2}{e}+\pi_{1,2}\), where \(\pi_{1,2}\) is the diffusion potential difference. By the threshold of excitation of the photoelectric effect \(h\nu_0\), we precisely determine the work of removing an electron from the metal, since the photoeffect ceases to exist at those values of the quantum energy which are insufficient for tearing the electron outward. Moreover, this is true only in the case where precisely those electrons are torn out whose free transition determines the occurrence of the contact potential difference, i.e. the so-called electrons of electrical conductivity. If, in the photoelectric effect, electrons more strongly bound to the atoms than the electrons of electrical conductivity were torn out, then \(h\nu_0\) would be greater than \(\Phi\), and equality (5) would not be satisfied, for example, in the case of X-rays.

On this basis it must be asserted that the validity of equality (5), shown by the experimental data, inevitably leads to the conclusion that in the photoelectric effect at optical wavelengths, the electrons torn out of metals are those whose motion accounts for its electrical conductivity. However, on the basis of equality (5) it is not possible to decide what the nature of these electrons is, whether they are bound or free.

Let us consider both cases. If the electrons are free and behave inside the metal like gas particles (the Drude–Riecke–Lorentz theory), then in this case, at the point of contact of the metals, a diffusion potential difference is formed

\[ \pi_{1,2}=\frac{kT}{e}\lg\frac{n_1}{n_2}, \]

where \(n_1\) and \(n_2\) are the concentrations of free electrons in the first and in the second metals. This difference must be of the order of hundredths of a volt, and its existence is responsible for the Peltier heat. To explain contact potential differences (of the order of several volts) there exist a number of theories. For example, the contact potential difference may be due to the presence, on the free surfaces of metals, of a double electric layer (Richardson), caused either by the fact that the surfaces of metals are covered with a layer of protruding orbital electrons of atoms (Frenkel), or by an adsorbed layer of electron gas. Finally, its existence may be caused by the fact that, when moving away from the surface layer of the metal, an electron does work against those forces by which it polarizes the nearby atoms (“image effect,” according to Langmuir). However, in all these cases, as follows from general consideration, independently of the origin of the works \(\Phi_1\) and \(\Phi_2\), the contact potential difference will be equal to

\[ K_{1,2}=\frac{\Phi_1}{e}+\frac{\Phi_2}{e}+\pi_{1,2}, \]

and, consequently, down to small values of \(\pi_{1,2}\), it will coincide with the difference \(h\nu_1-h\nu_2\).

Thus the equality \(h\nu_1 - h\nu_2 = K_{1,2}\) will always hold if we define the work \(h\nu\) for the emission of conduction electrons. But from this relation we cannot learn the nature of their connection. Likewise, we cannot answer the question of the origin and localization of the contact potential step. Depending on one or another point of view regarding the origin of \(\Phi\)—the work of removal of the electron—we obtain one picture or another. If we assume that the existence of the work \(\Phi\) is due to the presence of double electric layers on the surfaces of metals, then the contact steps will be concentrated on the outer free surfaces of the metals in contact, and their difference will give that external contact field which in all experiments we observe between the free ends.

If, however, the work \(\Phi\) arises from the electrostatic interaction between the departing electron and the remaining metal (image effect), then between the metal and the void there will be no potential step. When two metals are brought into contact, in this case a contact difference of potentials will be established, equal to the difference of the works \(\Phi_1 - \Phi_2\), the potential step being concentrated at the place of contact of the metals; but we shall still detect it between the free surfaces of the metals. Neither the relation (5) given above, nor all the known dependences and observations of contact potentials, answer the question of the localization of the steps, since under both possible assumptions they are explained equally well. As for the Peltier step, corresponding to a hundredth of a volt, it is due to the diffusion of electrons and is always concentrated where heat is evolved, i.e. at the place where the metals are in contact.

In the case of double layers at the place of contact there is only this diffusion step; the other Volta steps are outside, on the free surfaces. In the case of the Langmuir picture, both the Volta and Peltier steps are concentrated at the place of contact, but the evolution of heat corresponds only to the diffusion Peltier step, since

an electron, according to Langmuir’s picture, at the point of contact performs the difference of works $\Phi_1-\Phi_2$ and undergoes a Volta potential jump

\[ \frac{\Phi_2-\Phi_1}{e}; \]

the sum of the works in this case is equal to zero, and heat is neither evolved nor absorbed.

However, it must be pointed out that the absolute value of the quantity $h\nu_0$ sets an unconditional limit to the binding energy of the electron inside the metal, if such a binding exists. The binding energy cannot be greater than $h\nu_0$, and, if it exists at all, it must be less than this quantity.

The question of whether the electrons of electrical conduction, and consequently also the electrons of the photoelectric effect, are free, or whether they may be likened to orbital electrons, is difficult to decide. The difficulties which the classical theory of electrical conduction (Drude–Riecke–Lorentz) encountered in explaining the heat capacity of metals disappear in Sommerfeld’s theory1.

Distribution of the velocities of photoelectrons.

As was already indicated, the velocities of photoelectrons when a metal is illuminated by a monochromatic spectral line have all possible values, beginning with the maximum, determined by equality (1), down to zero velocity. By differentiating any of the curves of Fig. 3, which give us the current strength as a function of the retarding field, we obtain the velocity-distribution function; in other words, we obtain the dependence of the number of electrons on their velocities. In Fig. 6 we have velocity-distribution curves obtained by differentiating the curves in Fig. 3. All these curves have been referred to one and the same number of electrons, i.e. to identical saturation currents, and to equal values of the maximum velocities corresponding to different wavelengths.

In Fig. 6 the curves are shown for aluminum and for the following wavelengths: curve 1—2302 Å, 2—2537 Å, 3—3130 Å.

From these curves it is seen that the most probable velocity of the electrons approximately corresponds to the value of one half of the maximum velocity. As for the distribution of velocities, for the given metal it depends little on the wavelength of the incident radiation. However, it is somewhat different for different metals, and we cannot, as Ramsauer does, assert that the distribution curve, expressed in relative values \(\dfrac{V}{V_m}\), does not depend on the kind of metal.

We shall now try to clarify what, generally speaking, accounts in the photoelectric effect for the existence of electrons of all possible velocities. Obviously this may occur either because the electrons, having received within the metal equal amounts of energy from the incident quanta, expend it in different ways while moving inside the metal (a phenomenon analogous to the release of Joule heat in the process of electrical conduction), or else the difference in velocities may be due to the fact that, when the electrons are torn away, different amounts of work are performed, which is possible in the case when the binding of individual electrons may be different.

Fig. 6.

To shed light on this question, special experiments were undertaken to investigate the photoelectric effect from thin, semitransparent metallic films. Glass beads, 4.5 cm in diameter, were coated by cathode sputtering, under completely identical conditions, with thin layers of different metals, beginning with a thickness \(\simeq 10^{-6}\) cm and up to a completely opaque layer of great thickness. The sputtering was carried out at our request

Yu. P. Maslakovets. The beads were placed in the apparatus (Fig. 1), and the photoelectric effect was investigated. In this case the curves have a different form than in the case of thick metal layers, since in the region of small retarding fractions they run more horizontally, which is caused by a decrease in the number of low-velocity electrons. The contact potential and the excitation threshold of the photoeffect remain as before. This can be verified in the following ways. Since the curves, current strength—retarding potential, run almost horizontally, for short wavelengths it is impossible to find the exact value \(V=V_1=(K)\) at which the current begins to fall. For long waves this can be done more accurately, since in this case the curves run more steeply. The best method for determining the contact potential is the study of the reverse current from the external sphere, which, as we have seen, always goes to zero when the value of the true field is equal to zero. For comparison of the velocity distributions obtained upon illumination with one and the same wavelength (\(2537\ \mathring{\mathrm A}\)), Fig. 7 gives three curves for silver films: 3—for a film of thickness \(\simeq 10^{-6}\) cm, 2—for a film 3 times thicker, and 1—for a thick opaque film deposited in the same way.

Fig. 7.

Whereas the contact potentials for them remained the same, the distribution curves have different forms. The thicker the films, the more asymmetric the curves become. The region corresponding to slow electrons gradually disappears, while the most probable velocity of the electrons approaches more and more their maximum velocity, which in all three cases is, of course, the same. From this we see that

with decreasing film thickness we have increasingly monochromatic velocities of the photoelectrons, approaching the value of the maximum velocity.

Completely analogous results were obtained in studies of platinum films, beginning with a thickness of the order of \(10^{-6}\) cm up to thick opaque films.

Thus we arrive at the conclusion that photoelectrons, receiving one and the same amount of energy upon absorption of quanta, expend it gradually as they move out from different depths of the metal.

On the basis of the experiments indicated, we may note that layers with thickness \(10^{-6}\) cm and greater fully determine the contact properties of the metal, since all of them have a contact potential equal to the contact potential of a continuous thick layer. However, the conditions for the escape of electrons from these layers are different. Namely: light penetrates to depths greater than \(10^{-6}\) cm. The electrons torn out from these greater depths possess, for the most part, those low velocities which are absent in the case of very thin layers. Thus we may assert that the depth from which the electrons emerge is equal to \(10^{-6}\) cm, since in this case their velocity distribution is already different from that in the case of a continuous metal.

Fig. 8.

In addition to those indicated, experiments were performed with thin layers of metal deposited on another metal. Specifically, experiments were performed with a thin layer (\(<10^{-6}\) cm) of platinum deposited on an aluminum sphere. In Fig. 8 are shown the velocity-distribution curves obtained under illumination by the line \(2537\) Å for the following cases: 1—for

of a platinum layer on aluminum, 2—for solid platinum, and 3—for platinum deposited on glass. In all three cases we have entirely different velocity distributions, while the contact potential and the boundary remain the same. In the case of platinum deposited on glass, as we have seen, electrons of low velocities are absent, i.e., they are produced upon emerging from great depths. In the case of platinum on aluminum (1), the number of low-velocity electrons is not only no smaller, but, on the contrary, even greater than in the case of thick platinum (2), so that the maximum of the curve is shifted toward zero velocities. Let us compare the results obtained for a platinum film on aluminum, first with the results for a film on glass, and then with the curve for thick platinum. In both cases of thin platinum films, the number and the distribution of velocities from the platinum film are the same. But when the film is deposited on aluminum, aluminum electrons are added to these electrons as well. The total current thereby becomes considerably larger, while the electron velocities are small, since these electrons from the aluminum, before emerging outward, must pass through the layer of platinum.

In comparing curves 1 and 2 we see that, in the case of platinum, more low-velocity electrons are torn out of aluminum than in the case of solid platinum. This is due to the fact that aluminum is more photoelectric than platinum. Therefore we obtain the principal number of electrons from aluminum; nevertheless they must pass through the platinum layer, in which they lose velocity. The photoeffect from the platinum layer proceeds as usual, but it is small; therefore the platinum layer on aluminum adds very few electrons with high velocities. By making such layered metals, we can obtain the most varied curves for the velocity distribution of photoelectrons.

These experiments also confirm that a thickness of approximately \(10^{-6}\) cm fully determines the boundary of the photoeffect and the contact potential difference of the metal and strongly changes the velocity distribution, since the electrons come from great depths of the metal.

Such, in general outline, is the character of the photoelectric effect. We have dwelt in more detail on the case of the photoef-

the photoelectric effect in metals, since this case has been studied in the greatest detail. The photoelectric effect in dielectrics, the influence of the polarization of light on the distribution of electron velocities, and a whole series of other questions that naturally arise have as yet been little studied. Likewise, a whole series of fundamental features in both of the indicated phenomena—features which, apart from the photoelectric effect, are observed in the interaction of quanta of radiant energy with electrons—has so far not been studied.

1. A. Sommerfeld, Naturwissenschaften. 825, Heft 11, 1927; see also A. Sommerfeld, УФН. 8, 765, 1928. ↩↩↩↩

2. Ramsauer, Ann. d. Phys. 45, 1120 (1914). ↩

3. Becker, Ann. d. Phys. 78, 83 (1925). ↩

4. Richardson and Compton, Phil. Mag. 24, 575 (1912). ↩