ELECTRICAL DISCHARGES IN GASES*

K. Compton and I. Langmuir

C. THE DISAPPEARANCE OF ELECTRONS AND IONS

1. Disappearance at electrodes and walls.
Heating, coefficient of accommodation, transfer of momentum, sputtering.

2. Disappearance by recombination.
Recombination at a surface, recombination of ions, capture of electrons, Langevin’s theory, Thomson’s theory, recombination of free electrons and ions, recombination spectra, dependence of the probability of recombination on velocity, connection with photoionization, the experiments of Davis and Barnes.

A given element of the volume of an ionized gas may lose electrons and ions: 1) by recombination; 2) by absorption or neutralization at surfaces; 3) as a result of motion.

1. The disappearance of electrons and ions at electrodes and walls is obviously connected with their motion toward the electrodes. In addition, certain phenomena of absorption and neutralization belong here. Of these phenomena, reflection and the secondary emission of new charged particles have already been considered by us (B 7, 8, 9).

Absorption of electrons by electrodes is connected with the heating of the latter by an amount \(e(\varphi+\overline{V})-kT\) per electron \(^{157}\), where \(\overline{V}\) is the mean energy of the incident electrons (in equivalent volts), and \(T\)—

* Continuation; see issue 1 of Uspekhi Fizicheskikh Nauk.

the excess of the electrode temperature in relation to its external connections. The quantity \(\overline V\) does not include the energy acquired near the surface of the electrode, since the latter is included in \(\varphi\). The term \(2kT\) (which is only \(e \cdot 1\) volt for \(T=5\,886^\circ\)) is, generally speaking, negligibly small. If the electrons reach the electrode in a retarding field, so that secondary emission and reflection of electrons occur, then these phenomena alter the heating appreciably. The heating effect per incident primary electron will then be

\[ H=e(\varphi+\overline V)-n_s e(\varphi+\overline V_s)-(1-n_s)\cdot 2kT, \tag{50} \]

where \(n_s\) is the number of secondary electrons per primary electron, and \(\overline V_s\) is the mean initial energy of the emitted secondary electrons. If the energy of the incident electrons \(\overline V\) is of the order of 10 volts, then the term \(n_s e(\varphi+\overline V_s)\) probably amounts to less than 10% of the term \(e(\varphi+\overline V)\). At higher incident energies \(n_s\) increases (to a maximum when \(\overline V\) is of the order of 300 volts), but \(\overline V_s\) remains so small relative to \(\overline V\) that it is very doubtful that the cooling effect due to the second term could ever become equal to the heating effect represented by the first term. The most significant terms of equation (50) have been tested both for electrodes in vacuum \(^{157}\) and in ionized gases \(^{158}\).

Fig. 10. Heat of neutralization of a positive ion.

Absorption or neutralization of positive ions at electrodes is characterized by more complex energy relations. Let us consider the cycle shown in Fig. 10. A positive ion with negligibly small kinetic energy reaches the surface, is neutralized, and releases the energy \(e\varphi_+\). Or, conversely, an electron leaves the surface,

absorbing the energy \(e\varphi_-\), combines with the ion, releasing the energy \(eV_i\), and the neutral atom may remain on the surface, releasing the heat of adsorption \(eL\) of the neutral atom. Comparing these equivalent processes, we obtain the relation

\[ \varphi_+ = V_i - \varphi_- + L, \tag{51} \]

which differs from that originally proposed by Schottky only by the term \(L\) (which has a vanishingly small value if there is no permanent deposition on the electrodes of ions-atoms).

Equation (51) correctly gives the total energy; however, it is not necessarily correct with respect to the heat obtained by the electrodes, since part of the energy \(V_i\) may be radiated during the recombination process. The neutralization process may in fact proceed approximately according to the scheme shown on the right in Fig. 10, since when the ion approaches the surface to a distance of approximately 10 atomic diameters, it attracts electrons with a force sufficient to extract them from the surface. The interval of time from this moment until the ion strikes the surface is of the order of \(10^{-12}\) sec, and therefore is considerably less than the mean time \(\tau\) of residence of a free atom in the excited state (A 3). Nevertheless, at the surface or near it the recombining ion must give up the energy \(V_i\), and it is quite possible that part \(\tau\) of it will be radiated. Perhaps this radiated energy is the cause of the faint continuous glow, often observed on the surfaces of cathodes \(^{159}\), a glow detected in some experiments by its photoelectric action and presumed by J. J. Thomson to be the cause of the emission of electrons that was attributed to the impact of positive ions \(^{160}\). Thus, for the heat of neutralization we must put

\[ \varphi_+ = (1-r)V_i - \varphi_- + L. \tag{52} \]

If the reflecting power and the magnitude of the solid angle under which the surface is seen are taken into account,

if it is a metal, then one should expect that the fraction \(r\) must be greater than 0.5. The conclusion of Compton and Van Voorhis\(^{158}\), concerning the fact that \(\varphi_+\) is very close to zero, must at present be recognized as incorrect, since subsequent investigations have shown that their original results are complicated by certain unexpected new factors, which will be indicated below. All that we can say at present is that for \(\varphi_+\) in equation (52) one must put \(1>r>0\).

The next factor which complicates and may considerably diminish the heating of electrodes by positive ions, and which apparently was not taken into account in earlier work, is the retention by the neutralized ion (atom) of part of its kinetic energy. Until now it has been assumed that the neutralized particle, if it is not adsorbed by the electrode, leaves it with negligibly small energy (as, for example, in thermal equilibrium with it).* However, such behavior is not typical of neutral molecules falling on a surface of another temperature, for it has been found that such molecules come only partially into thermal equilibrium with the surface (except in the case of adsorption accompanied by evaporation, which is probably not a property of neutralized ions, since the latter, generally speaking, possess kinetic energies far exceeding the heat of adsorption). The degree to which the reflected molecule equalizes its energy with the energy of the reflecting surface is called the accommodation coefficient and is defined by the fraction

\[ a=\frac{E_i-E_r}{E_i-E}, \tag{53} \]

where \(E_i\), \(E_r\), and \(E\) are respectively the kinetic energies of incidence, reflection, and the energy characterizing the temperature of the reflecting surface. Thus complete adjustment of the energy of the incident particle upon impact to the temperature of the surface, as occurs in the case of adsorption

* In other words, with velocities of the order of hundredths of a volt.

and subsequent evaporation, is described by the coefficient \(a=1\), whereas specular reflection corresponds to \(a=0\). Some values of the accommodation coefficient are given in Table X.

Table X

Accommodation coefficients \(a\)

Compton and Van Voorhis, in a still unpublished work, have shown that the heating of a cold tungsten cathode by 50-volt positive ions A, Ne, and He is of the order of only 0.8, 0.65, and 0.43 of the expected power, measured by the value of the cathode current multiplied by the accelerating potential drop. Such results are obtained under conditions in which there is no possibility for energy dissipation by collision of the incident ions with gas molecules. Likewise, in this case the heat of neutralization \(\varphi_+\) was not taken into account at all; inclusion of it would lead to a further decrease of the indicated fractions. These fractions are strikingly similar to the corresponding accommodation coefficients, which is probably an accidental coincidence, since the energy of the atoms in this case is approximately a thousand times greater than the energies to which the data of Table X pertain,

and it would be surprising if \(a\) did not change appreciably in this interval. The unexpectedly small magnitude of the heating is due to two factors: 1) the value of the accommodation coefficient being less than unity and 2) the fact that part of the current is not a current of incident ions, but a current of departing electrons arising under the action of excited atoms on the electrodes, as was shown by supplementary experiments \(^{142}\).

Further evidence for this conservation of energy by neutralized particles is given in the following section.

For the time being it is still impossible to obtain a quantitative estimate of the accommodation coefficient corresponding to the impact of ions on electrodes. Until this can be done, all estimates of the heating of the cathode by ion bombardment should be regarded as preliminary. Nevertheless, the equation for this heating per one positive ion with initial energy \(\overline{V}\) may be written in the following form

\[ H_+ = a\overline{V}_+ + \varphi_+ = a\overline{V}_+ + (1-\eta)V_i - \varphi_- + L. \tag{54} \]

Transfer of momentum to electrodes as a result of the impact of charged particles on these electrodes does not produce pressure on the electrodes if, as is usually the case, the particles acquire their momentum exclusively as a result of attraction by these electrodes, since the impulse acquired by the electrodes as a result of attraction is exactly neutralized upon impact. However, pressure on an electrode may be due to the impact of particles whose velocities were acquired in the field between other electrodes or as a result of the reaction of spontaneously emitted particles, such as thermoelectrons from a heated cathode \(^{166}\), or neutral atoms from an evaporating electrode \(^{167}\).

Further, if ions that are attracted to an electrode under the action of its field retain after neutralization part of their kinetic energy, as happens in cases where the accommodation coefficient is less than unity, then the impulse corresponding to the removal of these particles must be imparted to the electrode and create pressure on it. This

gives an explanation of the very large pressures on the cathode of a copper arc in vacuum which led Tanberg \(^{168}\) to the unpleasant conclusion that the copper atoms evaporating from the surface have a kinetic energy corresponding to approximately \(500\,000^\circ\) K. For example, the observed order of magnitude of this pressure can be accounted for if half the current at the cathode were carried by positive ions and if the latter retained, on the average, one tenth of their kinetic energy after neutralization. Langmuir \(^{169}\) discussed the related problems of the momentum imparted to the gas and of the resulting pressure in discharge tubes.

“Sputtering,” or the destruction of electrodes subjected to bombardment by positive ions, is a long-known and often very troublesome phenomenon. Although it was discovered \(^{170}\) as early as 1852, its peculiarities and explanation remain enigmatic.

Most investigations have been devoted to studying the rate of loss of weight of the cathode in a quiet discharge as a function of the nature of the metal, the nature of the gas, the cathode fall, current density, gas pressure, cathode temperature, or the geometry of the tube. The most striking result of the observations consists in the considerable difference obtained with different metals and gases. Some such observations are given in Table XI.

For most gases the following relation, giving the mass of sputtered cathode material per unit time, is obeyed with great accuracy,

\[ m=K(V_c-V_0), \tag{55} \]

where \(K\) and \(V_0\) are constants characteristic of the gas and the metal, and \(V_c\) is the cathode potential fall. Thus sputtering is approximately proportional to the excess of the cathode potential fall over \(V_0\). The quantity \(V_0\) lies between 350 and 550 V and is usually 450 V. The constant \(K\) is roughly proportional to the fourth root of the atomic weight of the gas \(^{176}\), provided that sputtering is not accelerated by che-

Table XI*

chemical reaction 177. The formula 172 was also proposed

\[ K=\frac{K_0\cdot A}{n}, \]

where \(A\) is the atomic weight of the metal, \(n\) is an integer between unity and 4, which in some cases (but not always) is equal to the valence, and \(K_0\) is a constant characteristic of the gas. The value \(K_0\) for \(N_2\) is approximately 0.000004.

The approximate validity of equation (55) does not necessarily mean that the sputtering rate is causally connected with the cathode fall \(V_c\), since in order to change the cathode fall, the gas pressure or the current density was changed. Günterschultze 177 found that the rate of destruction of the cathode depends on the distance of the anode, and also on the gas pressure and the geometry of the cathode in such a way that the supposition suggests itself that the loss of weight of the cathode occurs as a result of diffusion of metal atoms from the region of their maximum partial pressure above the surface of the cathode toward the anode, where their partial pressure is zero. According to Günterschultze, at constant current density the relation holds

\[ m=\frac{C\cdot V_c}{pD}, \tag{56} \]

* The numbers give the sputtering rate in mg/amp. hour, under conditions of cathode fall \(V_c=700\) volts and current density \(7\ \mathrm{MA}/\mathrm{cm}^2\).

where \(C\) is a constant characteristic of the gas, and \(p\) is the gas pressure and \(D\) the distance from the anode. All the relations and experiments cited above, although empirically useful for describing the rate of destruction of the cathode in ordinary glow discharges, do not make it possible to judge the basic causes of the phenomenon, owing to the complexity of the conditions under which it is observed. Much more clear-cut are experiments which give the rate of destruction under the influence of bombardment by ions of known energy under conditions in which collisions of the sputtered atoms with the gas molecules do not occur and do not retard the escape of the atoms. Such experiments were carried out by the laboratory of the General Electric Co. and by Kingdon and Langmuir\(^{17}\). In these experiments the electrons from an incandescent filament ionized a gas whose pressure was too low to have any appreciable influence on the sputtering of atoms from the cathode. The positive ions were attracted to the cathode with known and strictly uniform velocities. The results again led to the relation \(m = V - V_0\), similar to equation (55), but with a much smaller value of \(V_0\), which indicates that in the case of a glow discharge the positive ions striking the cathode lose, in collisions with gas molecules, a large part (at least three quarters) of the energy acquired in the cathode fall.

In the experiments of the General Electric Co. not a single case was found in which \(V_0\) exceeded 100 volts. The efficiency of the gaseous ions in destroying a tungsten cathode increases in the order \(\mathrm{H}_2\), \(\mathrm{He}\), \(\mathrm{N}_2\), \(\mathrm{Ne}\)—\(\mathrm{He}\), \(\mathrm{Hg}\), \(\mathrm{A}\). Kingdon and Langmuir investigated the effect of ion bombardment on the destruction of a monatomic layer of thorium on tungsten and found, for impacts at 150 volts, that one thorium atom is ejected for every 700,000 impacting helium ions, 45 neon ions, 23 mercury ions, 12 argon ions, 12 cesium ions, and that hydrogen ions are completely inactive. The extremely small sputtering ability of hydrogen and helium is attributed to the relatively great penetrating power of their atoms, so that their energy is dissipated too deep inside the metal to be able to cause

surface destruction. Fig. 11 illustrates some of the results presented.

Holst^180 observed the sputtering of tungsten in argon rectifiers at voltages as low as 25 volts.

Oliphant^141 indicates reasons which lead one to suppose that sputtering is caused not only by the kinetic energy of the striking ions, but that the fact that the ion is an electric charge is also of significance.

Fig. 11. a) Sputtering in tungsten. b) Sputtering of thorium atoms deposited on the surface of a tungsten filament (Kingdon and Langmuir). Number of sputtered atoms per one incident positive ion.

Spectroscopic observations and studies of magnetic deflection indicate that sputtering consists in the ejection of neutral atoms, whose velocities have the same order of magnitude as the velocities in evaporation processes^181, although these atoms, in their subsequent history, may acquire charges, group into aggregates, etc. Tungsten, molybdenum, and carbon, scattered from filaments of these substances in argon, may acquire both positive and negative charges. As a result of this peculiarity, the emergence of a very interesting type of discharge is possible—the “stream-

…of a discharge,” which can occur when these substances are sputtered in arcs, in argon (and also in neon)^182.

When thermionic currents of 50 to 100 volts pass through argon at pressures of several microns, the tungsten filament is rapidly sputtered, and the sputtered substance is deposited on the glass, chiefly behind the anode, whence it follows that it has a negative charge^183.

There are two principal theories of sputtering: the theory of local high temperatures and the theory of direct transfer of momentum. The theory of local high temperatures was developed chiefly by Hippel^184. Starting from the idea that the energy of the incident ion heats, for a short interval of time, the region immediately adjacent to the point of impact to a very high temperature, which rapidly falls, since the energy is immediately dissipated in the metal by thermal conduction, Hippel finds

\[ m \infty \frac{i}{e(1+C_e)T}\, e^{-\frac{q}{RT}} \Delta F \Delta t, \]

where \(i\) is the current density, \(e\) the charge of the electron, \(C_e\) the number of secondary electrons emitted per one impacting positive ion, \(T\) the mean temperature on the mean area element \(\Delta F\), from which evaporation occurs during the time interval \(\Delta t\), \(R\) the gas constant, \(q\) the heat of evaporation of the metal, and \(e\) the base of natural logarithms. \(\Delta F\), \(T\), and \(\Delta t\) can be estimated if the kinetic energy of the ion and the thermal conductivity of the metal are known^184. Although the equation written above correctly indicates the character of the relation between the amount of sputtered substance and the principal parameters, it remains, of necessity, a very crude approximation. A serious objection to any theory based on evaporation at high temperature is that, according to such theories, in the case of metals like tungsten there should be observed a colossal secondary (thermionic) emission of electrons, in parallel with the intensity of sputtering, whereas this has proved to be incorrect. It is possible, however, that this emission will be limited…

ELECTRICAL DISCHARGES IN GASES

reduced to a small value by the space charge, owing to the extremely small dimensions of \(\Delta F\)—the effective emitting area.

Kingdon and Langmuir\(^{179}\) showed that the rate of sputtering of thorium from the surface of tungsten can be explained with complete accuracy; moreover, one can roughly calculate the minimum sputtering voltage (approximately 50 volts), proceeding from the following assumptions: the first step of the process consists in an ion impinging on a surface thorium atom and carrying it into the underlying tungsten, thus producing a depression. When a second ion impinges on such a depressed thorium atom, this ion is elastically reflected and, on its return path, may strike one of the surrounding surface thorium atoms and tear it away, if the energy which this reflected ion can transfer to the atom according to the momentum-transfer law exceeds the atomic heat of evaporation. The total probability of removing a thorium atom as the result of the impact of \(N\) ions upon one definite atom is expressed by the formula

\[ (1-\theta)=\mu(3P_2+5P_3+6P_4+6P_5), \tag{57} \]

where

\[ P_n=\frac{1}{n!}(Np)^n e^{-Np} \]

is the probability that one and the same atom will receive \(n\) impacts while \(N\) ions strike the surface, where \(p\) is the probability that a given ion will impinge on the given atom. The quantity \(\theta\) is the fraction of the surface covered with thorium atoms, and \(\mu\) is the probability that an ion reflected from a depressed thorium atom will remove one of the surrounding thorium atoms. Similarly, the minimum voltage \(\overline V_s\) for sputtering is expressed in terms of the atomic heat of evaporation of thorium \(E_v\) from tungsten \((1.40\times10^{-11}\ \text{erg/atom})\) by the following formula

\[ V_s=\frac{300E_v}{4em_gm_c}\left[\frac{(m_g+m_c)^2}{m_g-m_c}\right], \tag{58} \]

where \(m_g\) and \(m_c\) are the masses of the incident ion and of the cathode atom.

The equation (57) exactly satisfies the experimental data if suitable values are chosen for the parameters \(p\) and \(\mu\). Equation (58) gives approximately correct values of \(V_s\) for Ne and A, but values that are too high for H, He, Cs, and Hg ions. The considerations given above indicate that the discrepancy in the case of H and He is probably due to their great penetrating power, whereas the discrepancy in the case of the heavy ions Hg and Cs may be caused by the fact that they carry the thorium atom too far into the tungsten at the first impact, so that the interaction at the second impact occurs rather with the whole tungsten plate than with a single thorium atom, as is assumed in equation (58).

Holst’s observations on the sputtering of tungsten in argon rectifiers at voltages as small as 25 volts led him to admit a mechanism different from that which had been proposed to explain the ejection of atoms from the surface of thorium. Holst assumed that sputtering is due to the transfer of the required amount of energy to the tungsten atom in a single direct impact. From this assumption it follows that

\[ V_s=\frac{300E_v}{4e}\cdot\frac{m_g+m_c}{m_g m_c}. \tag{59} \]

This formula gives very small values for \(V\) (for example, W in A, 16; W in Ne, 27; Pt in A, 10; Pt in Ne, 17; Cu in A, 8.3; Cu in Ne, 42). The accuracy of these values is very doubtful.

These remarks should emphasize the fact that, although cathode sputtering is one of the very important phenomena in discharge in gases, it has as yet been insufficiently explained. It may be significant that bombardment by ions destroys the surface crystalline layer (see Baume’s microphotographs \(^{181}\) and the conclusions on the basis of electron emission drawn by Reynolds \(^{105}\)). If submicroscopic dust is formed in this process, it may immediately evaporate, passing through the discharge region, under the influence of the heat of recombination of ions and electrons on the surface of the dust particles. In the case of tungsten, evaporation

very slowly, and such recombination centers can in fact be visible (and have been investigated optically by means of the Tyndall cone), when tungsten is sputtered in argon \(^{182}\). In the case of more volatile metals, such a process may possibly explain the appearance of atoms of the sputtered substance, which was considered proof of the theory of sputtering based on evaporation.

2. Disappearance of electrons and ions by recombination. The fact that electrons and ions can recombine has been known since the time when their presence in a conducting gas was discovered. Indeed, all emission of light in a silent or “glow” discharge was at one time attributed exclusively to this process. However, attempts to obtain light by mixing ions and electrons under controlled conditions proved entirely unsuccessful. In the last 10 years certain limitations on the possibility of recombination have become clear, and the recognition of these limitations has greatly simplified the interpretation of observations that had until then seemed contradictory. These limitations, whose establishment is due chiefly to Bonhoeffer and Franck, are based on the following idea.

Free electrons in an ionized gas possess kinetic energies of translational motion having all possible values, beginning with the minimum determined by the temperature of the gas; moreover, the magnitude of the kinetic energy depends on the electric field, the gas pressure, and the initial conditions of electron formation. When these electrons enter the region of attraction of positive ions, then according to classical theory one may expect that they will revolve around the ions in hyperbolic orbits. But a stable binding (recombination) occurs when the orbit becomes elliptical, which can happen only if the electron loses part of its energy. The ways of losing energy consist only either in collision with a third body at the moment of closest approach of the electron to the ion, or in radiation. The first of these paths is connected with a collision of three bodies, which

which is highly improbable, unless special conditions, indicated below, are fulfilled. The second method is connected with the emission of energy beyond the limits of the ordinary line spectrum, and the observed weakness of such radiation shows that recombination by this path is also highly improbable. Thus we arrive at the conclusion that recombination of electrons and ions is a much less probable event than had previously been supposed, when it was thought that every approach of an electron to an ion ends with the capture of the former. We shall now proceed to a more detailed consideration of recombination under these limiting conditions.

Recombination at surfaces is a triple collision, in which the surface plays the role of a third body, absorbing the excess of energy and momentum. Electrons and ions need not meet simultaneously at some point of the surface, but one may arrive first and remain on the surface as a surface charge until the other comes and combines with the first. This explains the effectiveness of surfaces in causing the disappearance of ions by recombination. Dust particles, submicroscopic particles, sputtered from electrodes, insulating surfaces—all act in this way. As will be explained in detail below, insulated surfaces are charged, with respect to the surrounding gas, to positive or negative potentials, depending on whether ions or electrons carry the maximum random current density. Except for quite special cases, which are observed directly in the vicinity of cathodes, the random current of electrons always exceeds the current of ions, owing to the small mass of the electrons and hence to their high velocity. Thus insulated surfaces, generally speaking, acquire negative potentials with respect to the surrounding gas until the electrons begin to be repelled and the positive ions to be attracted sufficiently strongly, so that electrons and ions strike the surface with equal velocity.

and recombine on it. Such surfaces in an intensely ionized gas can become very strongly heated, and small suspended metallic particles are often evaporated or else heated to incandescence. The energy that they absorb is made up of the heat of combination, the initial kinetic energy of the recombining charges, and the energy acquired by positive ions in falling through the potential difference between the surface and the surrounding space.

Recombination in the volume of the gas may occur by a method analogous to that described in the case of surfaces: by two successive collisions among three bodies (as distinct from a simultaneous collision of three bodies), or it may occur by direct union of an electron and an ion without the participation of a third body. The first of these processes takes place as follows: capture of electrons, i.e. the formation of negative ions, may occur when an electron collides with a neutral molecule. In such a case, if this negative ion collides with a positive ion, the electron is captured by the positive ion, freeing the neutral molecule. The neutral molecule may carry away the excess energy, if there is any; but we shall see that often in this case no excess energy is obtained. The speed of this recombination process is evidently limited by the rate of the slower of the following two processes participating in the phenomenon: 1) the attachment of an electron in the formation of a negative ion and 2) the combination of negative and positive ions.

Electron attachment leading to the formation of negative ions was studied in greatest detail by Loeb and his pupils[^185]. The probability of attachment upon collision differs so greatly in different gases and is comparatively so large in certain gases and vapors, which are always present as impurities and are very difficult to remove, that most conclusions are qualitative rather than quantitative. Let \(N\) be the total number of collisions experienced per second by an elec-

...tron in a gas at atmospheric pressure and ordinary room temperature, if it is assumed that the mean free path of the electrons has the classical value, \(4\sqrt{2}\) times greater than that of the gas molecules. The “constant” of capture is in fact not a constant, since it depends somewhat on the velocities of the electrons. Further, it is not always an additive property in gas mixtures, since water vapor apparently causes more intense capture when it is present as an impurity than would be expected from the value of \(h\) given in Table XII. Apparently \(p\) is not connected with any obvious property of the molecules, if one excludes the rough dependence on their electronegative character. The prevailing notion that the molecular dipole moment is the principal factor is evidently untenable; to cite the sharpest examples, one may point out that \(NH_3\) and \(H_2O\) possess large moments, whereas \(N_2\) and \(Cl_2\) possess no moment at all.

Table XII

From Table XII several interesting conclusions may be drawn. In a pure gas at atmospheric pressure the electron remains free on the average only about \(7\times 10^{-4}\) sec in \(CO\), about \(4.7\times 10^{-9}\) sec in \(Cl_2\), if strong fields do not accelerate the electron so much that attachment becomes unlikely. Obviously the majority of experimental...

work devoted to the mobilities of ions, and all the earlier work on recombination was carried out under conditions in which the carriers of negative charge are ions, not electrons. On the other hand, in discharges in “vacuum tubes” at pressures of the order of 1 mm or less, the mean free path of electrons in many gases is sufficiently long, so that most, if not all, phenomena of negative charge may be caused by free electrons, even if one neglects the increased effectiveness of the electric field observed at low pressures with respect to the preservation of electrons in the free state. (This effectiveness is a function of \(\frac{E}{p}\).) Likewise, the enormous influence of certain impurities on the capture of electrons is evident. Thus, for example, the rate of capture of CO increases by at least 100% in the presence of 0.025% oxygen or water vapor.

Combination of negative and positive ions is not restricted, since the neutral molecule which retains the electron can play the role of a third body and carry away the excess energy corresponding to a certain excited state of the combining positive ion and electron. We have no direct data concerning the probability of such a transfer of energy, and therefore we have no information on the probability of combination of two oppositely charged ions which collide with a relative velocity greater than that which they acquire in falling from infinity. The two principal theories of ion combination ignore this probability and assume that the combining ions lose part of the energy acquired by their mutual attraction in collisions with neutral molecules, so that, as soon as they enter each other’s sphere of influence, they remain in it until recombination occurs.

The rate of recombination is proportional to the concentrations \(n_1\) and \(n_2\) of positive and negative ions and is therefore equal to

\[ -\frac{dn_1}{dt}=-\frac{dn_2}{dt}=\alpha n_1 n_2, \tag{60} \]

where \(\alpha\) is the “recombination coefficient.” Table XIII gives some typical values of this quantity. From Table XIII we see that, if there are \(n\) positive and \(n\) negative ions in \(1\ \mathrm{cm}^3\) of air, then the recombination rate will be \(1.71\times 10^{-6} n^2\).

Table XIII

The number of collisions between gas molecules of two kinds—1 and 2—is determined, according to the kinetic theory of gases, by the expression \(^{187}\)

\[ \text{number of collisions} = 2\left(\frac{2\pi}{3}\right)^{\frac{1}{2}} n_1 n_2 \sigma_{12}^{2}(C_1+C_2)^{\frac{1}{2}} \ \mathrm{cm^3\ sec^{-1}}, \tag{61} \]

where \(C_1\) and \(C_2\) are the mean square velocities, and \(\sigma_{12}\) is the sum of the effective radii of collision. Hence we find that in air at room temperature the number of collisions per \(1\ \mathrm{cm}^3\) between \(2n\) molecules will be \(1.6\times 10^{-10} n^2\). Comparing this with the value \(1.71\times 10^{-6} n^2\), we find that the true recombination rate is approximately 10,000 times greater than the number of ordinary gas-kinetic collisions of positive and negative ions. It follows from this that mutual electrostatic attraction strongly accelerates the recombination process. This attraction effect is treated differently by Langevin and Thomson.

Langevin’s theory of ion recombination \(^{188}\) is based on the assumption that neighboring oppositely charged ions attract one another with velocities determined by the attractive forces and by the mobilities* of these ions \(\mu_1\) and \(\mu_2\). If the distance between the ions is \(r\), then the field strength—

* “Mobility” is the constant \(\mu\) in the relation \(\bar v=\mu E\) between the field strength and the mean velocity of increase \(\bar v\) in the direction of the field. As will be indicated below, \(\mu\), generally speaking, is not a constant, but is itself a function of \(E\). At high gas pressures and weak fields \(\mu\) does not depend on \(E\).

the field strength will be \(\dfrac{e}{r^2}\), and the relative velocity of approach \(\dfrac{e(\mu_1+\mu_2)}{r^2}\), assuming that there are no other ions at distances comparable with \(r\). If \(1\ \mathrm{cm}^3\) contains \(n_2\) negative ions, then the velocity with which the negative ions cross the surface of a sphere of radius \(r\), drawn about the positive ion, will be

\[ 4\pi r^2 n_2 \cdot \frac{e(\mu_1+\mu_2)}{r^2} = 4\pi e n_2(\mu_1+\mu_2), \]

i.e. it will be independent of \(r\). Thus the mean time required for a negative ion to reach the given positive ion will be \(\dfrac{1}{4\pi e n_2(\mu_1+\mu_2)}\) sec. But since there are \(n_1\) positive ions, the mean interval of time between combinations with any positive ion will be \(\dfrac{1}{4\pi e n_1 n_2(\mu_1+\mu_2)}\); the quantity reciprocal to this expression gives the rate of recombination and, consequently, the quantity \(\alpha\)

\[ \alpha n_1 n_2 = 4\pi e(\mu_1+\mu_2)\cdot n_1 n_2. \tag{62} \]

This theory implicitly assumes that ions recombine every time they are attracted to one another. This is quite natural, since mobilities can be used only in the case when the moving ions lose energy in collisions so rapidly that they always possess the limiting velocity, and under such conditions the kinetic energy retained by an ion is insufficient to cause separation of the recombined ions as soon as they come sufficiently close to one another. This consideration indicates that Langevin’s theory should be most applicable at high gas pressures and should fail at low pressures. Consequently, since \(\mu_1\) and \(\mu_2\) vary inversely with the pressure, \(\alpha\) must vary in the same way according to this theory. This is confirmed at high pressures (several atmospheres), but not at low pressures. The absolute values of \(\alpha\), calculated from equation (62), are approximately correct at high pressures, but, in the extreme

at least 100 times greater than the required value at those low pressures at which the measurements of $\alpha$ and $\mu$ are carried out (approximately 0.1 atm.).

Townsend$^{188}$ extended Langevin’s theory so that it approximately took into account the influence of thermal motion, for, until the ions approach so closely that the work of separating them will considerably exceed the energy of thermal motions, it is always possible for an ion to escape combination. This fact and the influence of neighboring ions introduce into Langevin’s theory corrections with the correct sign. However, the most elegant account of thermal energies was given by Thomson.

Thomson’s theory of ion recombination$^{189}$ is based on the idea that combination occurs in the case when one of two ions of opposite sign that have met collides with a neutral molecule and loses so much kinetic energy that it cannot escape the attraction of the other. If $v$ is the relative velocity of two ions with masses $M_1$ and $M_2$ at a distance $r$ from one another, then they will describe closed orbits about one another and will ultimately unite if $r$ is less than $r_0$, determined by the relation

\[ \frac{1}{2}\cdot \frac{M_1M_2}{M_1+M_2}\cdot v_2=\frac{e^2}{r_0}. \]

In this case Thomson assumes that the mean velocities $u_1$ and $u_2$, immediately after one of the ions undergoes a collision with a molecule, are the velocities of thermal motion at the given temperature. (This is, of course, not true when the ions approach to a short distance, but it may be considered sufficiently justified for the critical distance $r_0$, which alone is important in the present case.) Then, taking into account that $v^2=u_1^2+u_2^2$, we obtain

\[ \frac{1}{2}M_1u_1^2=\frac{1}{2}M_2u_2^2=\frac{3}{2}kT \]

\[ v^2=3kT\left(\frac{M_1+M_2}{M_1M_2}\right); \]

therefore the critical distance $r_0$ is expressed by

\[ \frac{e^2}{r_0}=\frac{3}{2}kT \tag{68} \]

and, as is evident, will be inversely proportional to the absolute temperature \(T\).

Further, the probability that the ion suffers a collision with a molecule while it passes by ion 2 at the critical distance \(r_0\) is expressed through the mean free path \(\lambda_1\) by the formula*

\[ w=1+2\left[\frac{e^{-\frac{2r_0}{\lambda_1}}-1}{\frac{2r_0}{\lambda_1}}+\frac{e^{-\frac{2r_0}{\lambda_1}}}{\frac{2r_0}{\lambda_1}}\right] \]

and the corresponding expression holds for the probability \(w_2\) that ion 2 suffers a collision. The total probability that two ions which approach to the distance \(r_0\) suffer a collision within this distance is expressed as

\[ w_1+w_2-w_1w_2, \]

where the last term characterizes the probability that both ions suffering a collision are already included in \(w_1\) and \(w_2\).

Finally, the number of times that an ion gets within the distance \(r_0\) of each other ion will be approximately

\[ \pi r_0^2 n_1 n_2 (u_1^2+u_2^2)^{\frac12}\ \mathrm{cm^3\,sec^{-1}}. \]

Since it is assumed that each ion suffering a collision enters into recombination with the neighboring ion, the recombination rate will be

\[ \alpha n_1 n_2=\pi r_0^2 (u_1^2+u_2^2)^{\frac12}(w_1+w_2-w_1w_2); \]

substituting for \(r_0\) the value from equation (63), we obtain

\[ \alpha=\frac{4\pi e^4}{9k^2T^2}(u_1^2+u_2^2)^{\frac12}(w_1+w_2-w_1w_2). \tag{64} \]

At high pressures \(w_1\) and \(w_2\) approach 1, so that \(\alpha\) should increase with pressure, but approach a constant value at high pressures. This proved to be true\({}^{190}\), with the correction that at very

* Everywhere in the formulas \(\varepsilon\) denotes the base of natural logarithms.

at high pressures $\alpha$ again begins to decrease with increasing pressure. This marks the point from which Langevin’s theory begins to describe the phenomena more accurately than Thomson’s theory.

Equation (64) quantitatively satisfies the facts over a considerable range of pressures with the accuracy that can be expected if one takes into account the very considerable uncertainty in the values of the mean free path of the ion and the insufficiently precise information on the nature (mass) of the ion,¹⁹¹ which often may be an ion of some impurity, or even an entire cluster.

In Langevin’s theory the rate of recombination is limited by the rate at which the ions are attracted to one another; in Thomson’s theory it is limited by the rate of diffusion of ions into the region where the attractive forces can hold them close to one another. A new confirmation of Thomson’s theory, at the same time serving as an objection to Langevin’s theory for pressures of the order of atmospheric and below, was recently indicated by Loeb and Marshall¹⁹² in the form of the conclusion, from observations¹⁹³, that recombination occurs abnormally rapidly in a short interval after the formation of ions, especially when an ionizing agent such as an $\alpha$-particle creates a nonuniform distribution of ions. This was correctly interpreted¹⁹⁴ as a consequence of the fact that the true effective initial concentration of ions is considerably greater than the value calculated from the assumption of a uniform distribution throughout the whole volume, and it was shown that the correct values of $\alpha$ are obtained after a time sufficient for a uniform distribution to be established. Loeb and Marshall point out that the very fact of the establishment of a uniform distribution testifies that diffusion is more essential in the life of an ion than mutual attraction. Further, Loeb and Marshall analyzed the problem by considering the ions as undergoing Brownian motion and subjected to the action of their electric fields, and showed that at atmospheric pressure and room temperature the ions’ own fields are

is an insignificant factor in comparison with diffusion as regards the displacement of an ion through a distance of the order of the mean free path from another. However, at several atmospheres the field becomes an important factor.

Recombination of free electrons and positive ions[^195] belongs to the class of phenomena which, together with recombination at surfaces, are of exceptional interest for gas discharges. The existence of such recombination cannot be inferred simply on the basis of luminosity in a gas discharge, for it is easy to show that in the majority of cases the light is due chiefly to excitation of atoms, and not to recombination. For example, the positive column of a glow discharge emits more intense light than the negative glow. Nevertheless, measurements show that the concentrations of electrons and ions are approximately 100 times greater in the negative glow than in the positive column, while their velocities in the negative glow are smaller. If the light in both regions were due to recombination, then in the negative glow it would have to be at least \(10^{2}\) times greater than in the positive column, and even greater still, since the low velocities of the electrons in the negative glow favor recombination. Hence it follows with certainty that recombination plays a negligible role in the formation of the luminosity of the positive column. As a further confirmation of this conclusion we shall see in the second part of the present review that the radial concentrations and potential gradients, and also the relation between the axial potential gradients and the tube diameters in cylindrical positive columns, show that all kinds of recombination in the positive column at low gas pressures play a vanishingly small role in comparison with recombination at the tube walls.

This, however, still does not prove that the light of the negative glow is not produced to a noticeable extent, and perhaps even chiefly, by recombination. That recombination

plays an essential role here; this is proved by the fact of Doppler broadening of the spectral lines of negative luminosity[^197], the magnitude of this broadening being just what would be expected if the radiating atoms moved with velocities of the same order of magnitude as the ion velocities, i.e., with velocities considerably greater than molecular velocities. Along with these, the lines of the positive column have a broadening corresponding to emitters with ordinary thermal velocities. From the experiments of Miss Dewey one may even conclude that, at least in her apparatus, practically all the negative luminosity is due to recombination.

There are two additional criteria by means of which a recombination spectrum can be distinguished. One of them, used chiefly by Miss Gayner[^198] and Kent[^199], consists in the relatively great intensity of the higher members of spectral series, as compared with their intensities in spectra due to excitation. This is explained by the fact that in recombination each electron has a chance to make transitions both between higher and between lower energy states, whereas in excitation the majority of atoms are excited to the lower states. This fact is proved by direct observation of the intensities of the excitation spectrum under conditions of so low an ion concentration that recombination becomes negligibly small.

Kent’s observations with argon are especially illustrative. He ionized the gas intensely by means of a low-voltage arc with a hot cathode, then removed the voltage (or reduced it to a value too small to maintain the arc), and during this second period simultaneously photographed the spectrum and measured the distribution of electron velocities by the Langmuir probe method. The use of a rotating sector and a commutator made it possible to repeat these intervals rapidly and thus to carry out continuous observations. The intensity of the higher members of the series measured the rate of recombination, and

it was found that this velocity decreases considerably when the mean velocity of the electron is increased by means of the low potential mentioned above. In the absence of voltage the mean energy of the electrons was \(0.4\) V, the concentrations of electrons and ions were of the order of \(10^{12}\) per \(\mathrm{cm}^3\), and the recombination coefficient \(\alpha\) was found to be about \(2 \times 10^{-10}\). The number of kinetic collisions between positive ions and electrons can be calculated from equation (61), assuming that the effective radius of collision coincides with the radius of the argon atom. In this way it was found that the true rate of combination is \(1/375\) of the rate at which, according to this calculation, electrons and ions should collide; this number may therefore be taken roughly as the upper limit of the probability of combination of argon ions with electrons at \(0.4\) V. We have seen that positive and negative ions can combine 10,000 times faster than the frequency of gas-kinetic collisions, whereas positive ions and electrons combine several hundred times more slowly than follows from the frequency of kinetic collisions, even under the exceptionally favorable conditions of very small electron velocities.

Table XIV

Table XIV gives an example of the relatively great intensity of the higher series lines in recombination spectra.

Another criterion of recombination consists in the existence of a continuous band extending from the series limit toward shorter wavelengths. Fig. 12 gives the results of a photometric measurement of the intensity distribution in these

bands in the case of cesium vapors adjacent to the absorption limits \(3D_{2,3}\), \(2P_{12}\), and \(1S_0\).

The interpretation of these bands is simple. The frequency associated with the state of an atom with energy \(h\nu_i\) is the limit of a spectral series caused by the falling of electrons into the state \(i\) from states of higher energy. If an electron whose kinetic energy is equal to 0 at a great distance combines with an ion and falls directly into the energy state \(i\),

Fig. 12. Intensity of continuous recombination spectra beyond the absorption limits in cesium vapors.

state \(i\), the emitted light will have the frequency \(\nu_i\) of this series limit. But in reality the electrons which recombine have energies \(\frac{1}{2}mv^2\) greater than 0. According to classical theory they should not recombine, since they would have to describe a hyperbolic orbit about the ion. But according to quantum theory there is a definite probability, which is a function of \(v\), that recombination will occur. An electron possessing velocity \(v\), upon recom-

bination gives emission of light with frequency \(\nu\), determined by the relation

\[ h\nu=h\nu_i+\frac{1}{2}mv^2. \tag{65} \]

Since the velocities are distributed continuously (often obeying the Maxwell distribution), it is evident that \(\nu\) has all values beginning with \(\nu_i\) and higher, which therefore give a continuous band.

The distribution of intensity in the band evidently depends on two factors: 1) the distribution of electron velocities and 2) the probability of combination, or the recombination coefficient, characteristic of a given velocity. The first factor \(N(v)\) can be measured by the Langmuir probe method, so that measurements of the band intensity can be used to estimate the recombination coefficient \(\alpha(v\nu_i)\) for electrons of a given velocity \(v\), forming atoms in the energy state \(i\). The equation for this coefficient may be written as follows:

\[ I(\nu)\,d\nu=h\nu\cdot \alpha(v\nu_i)\,N^+N^-(v)\,dv, \tag{66} \]

where \(h\,d\nu=mv\,dv\), and \(I(\nu)d\nu\) represents the energy for frequencies lying between \(\nu\) and \(\nu+d\nu\), emitted in one second by a unit volume, while \(N^-(v)dv\) is the concentration of electrons with velocities between \(v\) and \(v+dv\). The recombination coefficient \(\alpha(v\nu_i)\) is often written \(vq(v\nu_i)\), where \(q(v\nu_i)\) is the effective cross section for recombination of electrons of velocity \(v\) passing into the \(i\)-electron energy state of the atom. The complete recombination coefficient \(\alpha(v)\) for electrons of velocity \(v\) will evidently be \(\alpha(v)=\Sigma(v\nu_i)\), and the ordinary total recombination coefficient

\[ \alpha=\int_0^\infty \alpha(v)\,dv. \]

Moler \({}^{202}\) carried out a careful study of the spectrum of cesium and helium by this method and showed that the recombination coefficient \(\alpha(v\nu_i)\) varies with the electron velocity according to an equation of the form

\[ \alpha(v\nu_i)=\frac{K}{v(\nu-\nu_i)^{\frac{1}{2}}}\ \text{or}\ \frac{K}{v^2(\nu-\nu_i)^{\frac{1}{2}}}. \tag{67} \]

and depends so much more significantly on the term \((\nu-\nu_i)^{1/2}\) than on the term \(\nu^n\) that it is impossible to say definitely whether \(n=1\) or \(2\), although the agreement is somewhat better for \(n=2\). Thus the recombination coefficient approaches infinity at the limit of the series, but rapidly decreases at higher frequencies. In equations (66) and (67) \(\nu\) and \(v\) are evidently related by equation (65).

Proceeding from general considerations based on the principle of detailed statistical equilibrium between the probability of recombination \(a(\nu\nu_i)\) and the probability of absorption \(B(\nu\nu_i)\) for the frequency \(\nu\), corresponding to \(v\) of equation (65), Milne \(^{203}\) derived the relation

\[ a(\nu\nu_i)=\frac{K'B(\nu\nu_i)\nu^3}{(\nu-\nu_i)^{1/2}}. \]

It is further well known \(^{204}\) that \(B(\nu\nu_i)\) is, to a considerable approximation, proportional to \(\frac{1}{\nu^4}\), at least in the region of X-rays beyond the edge of the absorption band. Consequently, we should approximately expect

\[ a(\nu\nu_i)=\frac{K}{\nu(\nu-\nu_i)^{1/2}}, \]

which agrees quite satisfactorily with equation (67). The quantity \(B(\nu\nu_i)\) determines the probability of photoionization (see Section A, 2). Experiments on recombination are in better agreement with the theory than are experiments on photoionization \(^{205}\). One may consider that the theory outlined above is built on a quite satisfactory foundation, since it permits calculating the recombination coefficient from spectral data, and that the existing experimental discrepancies may be due to treacherous errors which are extremely difficult to avoid in work with such refined technique. Experimental work in this area has only just begun, and we still do not have data that could be regarded as definitive. Meanwhile other investigators \(^{206}\) have limited themselves

interesting qualitative observations, Moller alone obtained the necessary data for calculating the recombination probabilities by this method.

In Fig. 13 are given the experimental values of the probabilities of recombination into the \(2P\) state of the Cs atom as a function of the mean electron velocity. The absolute figures are given

Fig. 13. Probabilities of electron recombination into the state of cesium as a function of the kinetic energy of the electron in volts (calculated from the intensity of the recombination spectra—curve I; and of photo-ionization—curve II).

roughly with the aid of the estimate \(\alpha = 5.4 \times 10^{-14}\) at \(V = 0.2\) volt. This quantity means that approximately one collision out of 5,000 ends in recombination into the \(2P\)-state for electrons at 0.2. Curve 1, found from the recombination spectrum, agrees very well with equation (67). Curve 2 was calculated from measurements of photoionization. The reason for the discrepancy between these curves is not clear.

Further theoretical works relating to the problem of recombination can only be mentioned here. Milne\(^{207}\), from a consideration of statistical equilibrium, came to the conclusion that for small electron velocities the effective recombination cross-section \(q(v)\) must have

form \(\frac{C}{v^2}\). Since the effective cross section is related to the absorption coefficient by the equation \(a(v)=v\cdot q(v)\), we have

\[ a(v)=\frac{C}{v}. \]

If \(v\) is expressed in terms of \(\nu\) and \(\nu_i\) from equation (65), we obtain directly equation (67), only without the factor \(\nu\) (or \(\nu^2\)), which is practically constant for small electron velocities, when it exceeds only slightly.

This also agrees satisfactorily with the conclusions of Morse and Stueckelberg \(^{208}\), who calculated \(a(vv_i)\) by means of wave mechanics and found, for any state,

\[ a(vv_i)=\frac{C}{v}=\frac{D}{(\nu-\nu_i)^{1/2}} \]

as the asymptotic value for small velocities. They further obtained numerical values of \(C\) (or \(D\)) for lower states and showed that the sum, taken over all states, is finite and varies as a function of \(\frac{1}{v}\), but cannot be expressed in simple form.

Kramers \(^{209}\), Eddington \(^{210}\), and Ruchard \(^{211}\) proposed theories of the combination of free electrons with atomic nuclei deprived of electrons, the theory of Kramers containing only the fundamental physical constants. The first two theories found the most interesting applications in the field of X-ray physics and in the theory of the internal structure of stars; the latter was used to interpret the neutralization of canal rays.

Incidentally, it may be noted that, in order to increase the intensity of recombination spectra, three arrangements have been used with the greatest success. Namely: 1) the Schüler \(^{212}\) hollow cathode, in the presence of which the glow discharge emits light characteristic of Paschen recombination \(^{206}\); 2) an electrodeless ring discharge, at the center of which recombination light was emitted (Gerzberg, Mir-

in Del206, 3) a low-voltage arc between a hot cathode with intensive electron emission at very small dimensions and a surrounding anode with a large surface area. All these arrangements are especially convenient for obtaining large concentrations of electrons and ions in regions with so small an electric field that the velocities of the ions in them are negligible.

Finally, the capture of electrons by rapidly moving α-particles has recently been investigated by Davis and Barnes213 with very interesting and unexpected results. It was known214 that α-particles do not capture electrons from molecules through which they pass until their velocity has decreased below a certain limiting value215. Hence it was natural to conclude that capture is impossible if the relative velocity exceeds this critical value. For example, it was found that α-particles whose initial velocity was \(2.06 \times 10^9\) cm/sec do not capture any electrons at all until their velocity falls approximately to \(0.28 \times 10^9\) cm/sec, and do not capture a second electron for complete neutralization until their velocity falls to a value less than \(0.31 \times 10^9\) cm/sec. In order to explain this circumstance in more detail, Davis and Barnes passed a beam of α-particles into an evacuated vessel in which there also passed a stream of electrons from a hot cathode, the electrons moving with an adjustable velocity in the very same direction. In this way each α-particle, during its flight, was surrounded by a cloud of electrons whose velocities relative to the α-particle could be adjusted at will. The particles that had captured electrons were then separated from the rest by means of a magnetic field, and the number of particles that had not captured electrons was counted on a fluorescent screen.

In this way it was found that there exists a sufficiently high probability of capture when the relative velocity of the electrons and α-particles has certain characteristic values, and that these characteristic velocities are related

with the orbital velocities of the electrons in He+, in the simple image illustrated by Fig. 14. The conditions for the capture of a free electron by an α-particle consist in the fact that \(v_0 = v_1 - u\) or \(v_0 = u - v_2\). In other words, the relative velocity of the electron and the α-particle must be equal to the velocity which the electron would have in an orbit if it were captured. But this relative velocity exists under the conditions before that, since the electron is accelerated by the Coulomb field in the process of capture. Thus, another way of formulating it consists

Fig. 14. Conditions for the capture of electrons by an α-particle. \(U\) is the velocity of the α-particle; \(V_0\) is the orbital velocity which the electron would have in some orbit. \(V_1\) and \(V_2\) are the velocities of free electrons.

in saying that capture is probable if the total energy relative to the α-particle is equal to twice the energy in one of the Bohr orbits. In addition, capture is possible if the relative velocity is zero. Analogous energy relations hold for the simultaneous capture of two electrons; moreover, the following interesting fact is observed: the condition for recombination consists simply in the sum of the energies of the two electrons being equal to the total negative energy of the final state of the helium atom.

There is as yet no satisfactory explanation of these experiments. The very fact that the experiments have yielded any result at all is unexpected, if one takes into account that, for the observed capture to occur, it is necessary that the electron being captured, with the appropriate velocity, pass at a distance of the order of \(10^{-6}\) cm from the α-particle, i.e. at a distance much greater than the wave-mechanical radius. In other words, these experiments indicate that the effective

the cross-section of the atom for recombination is approximately a million times greater than the ordinary kinetic cross-section, under the condition that the energy has certain relative values, including zero. For recombination under the condition that the relative velocity is close to zero, this was indeed to be expected from the theories set forth, but recombination at other energies is apparently connected with processes essentially different from those investigated up to now ^216.

(To be continued.)

REFERENCES

1. Richardson and Cooke, Phil. Mag. 20, 73 (1910); 21, 440 (1911).

2. Schottky und von Issendorff, Z. Physik 26, 85 (1924); Compton and Van Voorhis. Proc. Nat. Acad. Sc. 13, 336 (1927); Van Voorhis. Phys. Rev. 30, 318 (1927).

3. Geiger und Scheel, Handbuch der Physik, XIV, pp. 173, 183--188, 234.

4. Thompson, Phil. Mag. 48, 1, 1924; 2, 675, 1926.

5. Knudsen, Ann. d. Physik, 34, 593, 1911.

6. Soddy and Berry, Proc. Roy. Soc. 84, 576, 1911.

7. Langmuir, J. Am. Chem. Soc. 37, 425, 1915.

8. Knudsen, Ann. d. Physik 46, 641, 1915.

9. Compton and Van Voorgis. Reported at the April session of 1930 of the American Physical Society.

10. Duffield, Burnham and Davis, Proc. Roy. Soc. Lon. 97, 326, 1920.

11. Beer and Tyndall, Phil. Mag. 42, 956, 1921; Tyndall, ibid. 42, 972, 1921; Sellerio, ibid. 44, 765, 1922.

12. Tanberg, Phys. Rev. 35, 294, 1930.

13. Langmuir, Journ. Frankl. Inst. 196, 751, 1923.

14. Grove, Phil. Trans. 1852.

15. Crookes, Proc. Roy. Soc. Lon. 50, 88, 1891.

16. Kohlschütter, Z. Elektrochem. 15, 316, 1909; Jahrbuch d. Radioakt. 9, 355, 1912.

17. Blechschmidt, Ann. d. Physik 81, 999, 1926.

18. Güntherschulze, Z. Physik 36, 563, 1926; 38, 575, 1926.

19. Grandquist, Kgl. Akad. Stockholm, 54, 595, 1897; Holborn and Austin, Wiss. Abhand. d. Phys. Tech. Reichsanstalt, 4, 101, 1903.

20. Kohlschütter and Müller, Z. Elektrochem. 12, 365, 1906; Kohlschütter and Goldschmidt, ibid., 14, 221, 1908; Blechschmidt, Ann. d. Physik 81, 999, 1926.

21. Güntherschulze, Z. Physik 38, 98, 1923.

1. General Electric Co., Ltd. Phil. Mag. 45, 98, 1923.

2. Kingdon and Langmuir, Phys. Rev. 22, 148, 357, 1923.

3. Holst, Physika 4, 68, 1924; see also A. W. Hull, Trans. Amer. Inst. E. E. 49, 753, 1923.

4. Von Hippel, Ann. d. Physik 80, 672, 1926; Baum—Z. Physik 40, 686, 1927.

5. I. Langmuir, Science 60, 392, 1924.

6. I. Langmuir, Phys. Rev. 2, 475, 1913.

7. Von Hippel, Ann. d. Physik 81, 1043, 1926; Blechschmidt und von Hippel, ibid. 86, 1006, 1928.

8. Loeb, Kinetic Theory of Gases, Ch. 507—515; J. Frankl. Inst. 197, 45, 1924; Wahlin. Phys. Rev. 19, 173, 1922; Cravath, ibid. 33, 605, 1929.

9. Computed from data collected by Thomson—Conduction of Electricity through Gases, 3rd ed., p. 40.

10. See Jeans, Dynamical Theory of Gases, 2 ed., p. 27; Thomson (Cond. Elec. through Gases, 3 ed., p. 43) uses an equation whose coefficients are 2.45 times larger.

11. Langevin, Ann. de Chim. et de Phys. 28, 287, 433, 1903; Townsend, “Electricity in Gases”, 206—220.

12. Thomson, Phil. Mag. 47, 337, 1924; Conduction of Electricity through Gases, 3 ed., p. 44—57.

13. Langevin, Comptes Rendus 137, 177, 1902; Mk Clung Phil. Mag. 6, 283, 1902.

14. Loeb, Phys. Rev. 32, 81, 1928.

15. Loeb and Marshall, J. Frankl. Inst., 208, 371, 1929.

16. Plimpton, Phil. Mag. 25, 65, 1913; Rümelin, Ann. d. Physik 45, 821, 1914.

17. Plimpton, Phil. Mag. 25, 65, 1913; Townsend, Electricity in Gases, p. 220.

18. For an excellent review of the question see Seeliger, Phys. Z. 30, 329, 1929; cf. also Atkinson, Z. Physik 51, 188, 1928.

19. Compton, Turner and McCurdy, Phys. Rev. 24, 597, 1924.

20. Dewey, Phys. Rev. 32, 918, 1928.

21. Haynor, Z. Physik 35, 365, 1926.

22. Kenty, Phys. Rev. 32, 624, 1928.

23. Mohler, Phys. Rev. 31, 187, 1928; cf. especially Phys. Rev. Suppl. 1, 216, 1929.

24. Robertson and Dewey, Phys. Rev. 31, 973, 1928.

25. Mohler, Phys. Rev. 31, 187, 1928; Research Paper No. 46, U. S. Bureau of Standards, March 1929.

26. Milne, Phil. Mag. 47, 209, 1924.

27. Compton, X-rays and Electrons, pp. 189—198. Oppenheimer, Z. Physik 41, 268, 1927. It should be noted that the absorption coefficient is proportional to \(\nu^{-3}\). The energy absorbed in one act of absorption is \(h\nu\). Consequently, the probability of absorption varies as \(\nu^{-4}\),

1. Mohler, Foote and Chenault, Phys. Rev. 27, 37, 1926.

2. Paschen, Berlin, Berichte, 135, 1925; Herzberg, Ann. d. Phys. 84, 553, 1927; Mierdel, ibid. 85, 612, 1928; Balasse, Comptes Rendus, 184, 1002, 1927.

3. Milne, Phil. Mag. 47, 209, 1925; cf. also Becker, Z. Physik 18, 325, 1923.

4. Morse and Stueckelberg, Phys. Rev. 35, 116, 1930; Oppenheimer, Z. Physik 55, 725, 1929.

5. Kramers, Phil. Mag. 46, 836, 1923; Wentzel, Z. Physik 27, 257, 1924.

6. Eddington, Inner. Constitution of the Stars.

7. Richardt, Z. Physik, 16, 164, 1923.

8. Schüler, Z. Physik, 35, 323, 1926; 37, 728, 1926.

9. Davis and Barnes, Phys. Rev. 34, 152, 1929; Barnes, ibid. 34, 1224, 1929; 35, 217, 1930.

10. Henderson, Proc. Roy. Soc. A102, 496, 1923.

11. Davis, Nature, May 26, 1923.

12. Stueckelberg and Morse, Phys. Rev. 35, 115, 1930; Adams, ibid. 34, 537, 1929.