ELECTRIC DISCHARGES IN GASES *

Carl T. Compton. Princeton, U.S.A.
Irving Langmuir. Schenectady, Research Laboratory

Part I. Survey of Elementary Processes **

A. PRODUCTION OF ELECTRONS AND IONS IN GASES

1. Ionization by electron impact . . . . . . . . . . . . . . . . . . . . . . . . 36

2. Photoionization. Probability of photoionization as a function of frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3. Cumulative ionization. Successive collisions, successive absorption, photo-impact, resonance radiation, metastable states, mean lifetime for excited and metastable states . . . . . . . . . . . . . . . . . . . . . . 44

4. Ionization by impact of positive ions . . . . . . . . . . . . . . . . . 50

5. Ionization by collisions of the second kind . . . . . . . . . . . . . . 51

6. Thermal ionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . —

B. LIBERATION OF ELECTRONS AND IONS FROM ELECTRODES

1. Thermionic emission of electrons. Dependence on temperature and the nature of the metal, work function, activated surfaces, emission rates, heat of evaporation of electrons . . . . . . . . . . . . . . . . . . . . . . 53

2. Thermionic emission of positive ions . . . . . . . . . . . . . . . . . . 58

3. Photoelectric emission of electrons. Critical frequency, emission rates, Einstein equation, cumulative emission, full emission, emission under the action of black-body radiation, thin films, selective photoelectric effect . . . . . . . . . . . . . . . . . . . . . . 60

* Reviews of Modern Physics, 2, 123, 1930.
** The survey of contents refers only to that part of the article which is printed in the present issue.

1. Relation to the contact potential difference.
   Thermionic and photoelectric work functions and contact potentials; the difference between homogeneous and “patchy” surfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

2. Electron emission in accelerating electric fields.
   The Schottky effect, surface forces, values of saturated emission, properties of nonuniform and activated surfaces, theories of the “mesh” action of patches and of “adions.” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3. Electron emission under the action of strong fields.
   Field currents, the effect of rough surfaces, dependence on temperature and field, local heating. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4. Electron emission under the action of electron bombardment and reflection of electrons.
   Secondary emission or emission of “delta rays,” dependence on the primary velocity, emission velocities, critical potentials, diffraction of electron waves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5. Electron emission under the action of impact of metastable atoms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

6. Electron emission under the influence of bombardment by positive ions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

7. Emission under the action of chemical reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

As a result of intensive investigations of phenomena in ionized gases, carried out over the past 30 years, most of the basic elementary processes occurring in discharges in gases are now recognized. The most important problem at the present moment is to analyze the complex phenomena of discharge in such a way that they can be quantitatively interpreted in terms of these basic processes for any region of each of the very diverse types of gas discharges. These basic processes are classified and briefly described in the present article. Some of them will be discussed in greater detail in Part II, which will be devoted chiefly to the study of discharge in gases in itself. In the present article special attention is given to those aspects of the subject which are still not entirely clear; certain very important phenomena are not discussed in detail, because they are widely known and completely understood. In all cases, however,

an attempt is made to give sufficiently extensive references to the literature of the subject, so that the interested reader may study it more deeply on his own.

A. Production of electrons and ions in gases.

1. Ionization by electron impact,

first discovered by Townsend,^1 plays a primary role in almost all gas discharges. In order to ionize any normal gas molecule, the impinging electron must possess a kinetic energy exceeding a known minimum, which is characteristic for each given type of gas molecule. This minimum energy is called the “minimum ionization potential” of the gas, since it is usually measured by the potential difference \(V_i\) through which the electron must pass in order to acquire this minimum energy of ionization. There also exist higher ionization potentials, corresponding to the simultaneous removal of two electrons, or to the removal of a second electron after the first has first been removed, or to the removal of one of the more deeply lying electrons from the molecule. Since the minimum energy required to remove an electron from a molecule is the same irrespective of whether it is supplied by the incident electron (impact ionization) or by the absorption of radiation (photo-ionization), we may equate these two energies and write:

\[ eV_i = h\nu_i, \]

where \(\nu_i\) is the frequency of the convergence limit of the absorption spectral series. This relation, the verification and application of which were so vigorously pursued by physicists 5–10 years ago, is at present so firmly established and well known that there is no need to discuss it here.

The probability of ionization of a gas molecule by electron impact increases from zero approximately linearly with increasing energy above the minimum energy

ionization up to approximately twice the value of this energy, and then more slowly.^4 Experimental determination of this probability requires measuring, first, the number of new ions formed per unit path of an electron of a given energy, and, second, measuring the total number of collisions undergone by the electron per unit length of path. As we shall see, this latter quantity is not only difficult to measure, but even to determine accurately. Thus, although it is often convenient to know approximately the probability of ionization upon impact, for the theory of gas discharges it is simpler to deal with a more precisely determined quantity—the number of new electrons produced by ionization per unit length of path at a given gas pressure by an electron of specified energy. This quantity may be called the probability of ionization per unit length of path at a pressure equal to unity.^3

Fig. 1. Number of Hg$^+$, Hg$^{2+}$, Hg$^{3+}$, Hg$^{4+}$, Hg$^{5+}$ ions per cm of path and mm pressure at 0°. On the right are given the effective cross-sectional areas for ionization of the Hg atom.

If the energy of the striking electron exceeds the value necessary for removing two or more electrons from the molecule, then the nature of the products of this ionization must be determined, i.e., it must be found whether they are singly, doubly, triply, etc., ionized. This determination was first carried out by Smyth^11 and quite recently by Bleakney,^3 who also determined the probability of obtaining each of the different types of mercury ions as a function of the energy of the incident electrons. His results are shown in Fig. 1.

The method by which the values of \(F\) in Table I were found consists in the fact that a doubly ionized molecule was counted as two singly charged ions, a triply ionized molecule as three ions, and so on. As for estimating the probability of multiple ionization in a single electron impact at higher voltages, here the figures of Table I are erroneous from the point of view of any theory of ionization by impact. For the investigation of this question, separate measurements of the single and multiple stages of ionization are desirable. Attempts at such measurements were made by Hippel and were finally carried out by Bleakney. But there is a high probability that a multiply charged ion, upon collision with a neutral molecule, will take an electron from the molecule, so that a state will very easily be reached in which multiply charged ions will be replaced by singly charged ions with an equivalent total charge. Further, since the probability of formation of initially multiply charged ions is always less than the probability of formation of singly charged ions, we may regard the values of \(P\) in Table I as correct as such, considering them in connection with theories of gas discharge, since for the latter only the total rate of ion formation is of interest.

The impacting electron and the new electron formed in the ionizing collision, in subsequent collisions with gas molecules, may form new ions. When the free paths of the electron are small in comparison with the dimensions of the gas space, or when the walls are negatively charged, so that the electron continues to ionize until its energy falls below the ionization potential, the total number of ions formed by each primary electron proves to be independent of the gas pressure \(^{7}\). Recently, improved methods have been developed for measuring the total ionizing ability of an electron at its various energies; in this connection figures have been obtained that are somewhat larger for helium and somewhat smaller for other gases, according to

K. Compton and I. Langmuir

Table I

Ionization of gases by electron impact

\(V_i\)—minimum ionizing potential.
\(P(V)\) is the probability that an electron with energy \(V\) volts will ionize a gas molecule over one cm of path at a pressure of \(0.01\) mm at \(29^\circ\).
\(C\)—the constant in the equation \(P(V)=C(V-V_i)\), which is observed accurately in the interval from \(V_i\) to \(2V_i\). In most cases ionization by direct electron impact in gas discharges occurs at velocities lying within these limits, except at very high potentials and very low gas pressures.

Gas	\(V_i\)	\(P(20)\)	\(P(30)\)	\(P(50)\)	\(P(100)\)	\(P(300)\)	\(C\)
Cs	3,88 volts
K	4,32 “
Na	5,12 “
Hg	10,39 “	0,067	0,146	0,195	0,213	0,193	0,00753
He	24,53 “	0,000	0,002	0,009	0,116	0,018	0,00039
Ne	21,47 “	0,000	0,003	0,010	0,026	0,035	0,00037
A\(_1\)	15,69 “	0,017	0,052	0,093	0,112	0,094	0,00365
*H\(_2\)	15,8 “	0,006	0,021	0,033	0,038	0,032	0,00156
*N\(_2\)\(^1\)	16,3 “	0,004	0,028	0,072	0,101	0,101	0,00223

* In the case of polyatomic gases the minimum ionizing potential cannot be determined as accurately as in the case of monatomic gases, since there is a probability that the molecule undergoes a change in the distance between nuclei simultaneously with ionization. This probability can be calculated by means of quantum mechanics, and the probability of large changes turns out to be small. The experimental values given in the table are therefore sufficiently accurate for ordinary purposes.

Table II

Total ionizing ability of electrons, ions per primary electron \(V\) volts

Gas	\(V=30\)	50	75	100	150	0
He		1,2		2,9
Ne			1,2	2,0
Ar	0,45	0,9		1,6	2,2
H\(_2\)				1,4		2,8
N\(_2\)			1,3	1,6
Hg	1,1	1,4		2,7		5,3

in comparison with those which were given in earlier works8. Some of these values are given in Table II.

If the impacting electron possesses a store of energy greater than the value of the minimum ionization energy, then the excess energy may be retained by this electron itself, or transferred to the ejected electron, or used for further ionization of the ion already obtained, or, finally, for any combination of these processes. From the laws of momentum transfer it is clear that the fraction of energy converted into the kinetic energy of the ion is an imperceptibly small quantity. There are data which indicate that the excess energy is not retained entirely by the incident electron, but may be distributed in any ratio between the two electrons9, although there are some indications that precisely the distribution between the electrons is considerably less probable than the retention of the greater part of this energy by one or the other10.

The immediate products of such ionization are the ejected electron and the positive ion, which is the original molecule minus an electron11. In many polyatomic molecules the internal energy of this “primary” ion is greater than the energy of the products of its dissociation, so that the ion may split into a positively charged and one or a greater number of neutral “secondary” products of ionization. Generally speaking, this secondary transformation requires some external stimulus, such as, for example, a collision, although energetically it is possible without any external aid. However, few cases are known in which this dissociation, following ionization, occurs spontaneously, and perhaps also instantaneously12. This capacity for subsequent dissociation is well explained from the point of view of the relation between the internal energy and the internuclear distance of the atomic components of the neutral molecule and of the primary molecular ion,*—a relation which may be found by studying the band spectrum13.

* This should be understood in the following way. In an unexcited neutral molecule the nuclei occupy a certain position of stable

Thus, for example, in hydrogen the primary product of ionization is \(H_2^+\). If the pressure of the gas is such that the \(H_2^+\) ions collide with molecules, then they tend either to dissociate into \(H + H^+\) or to associate\(^{14}\) into \(H_3^+ + H\), the relative amount of \(H_3^+\) becoming larger when the effective temperature (mean kinetic energy) of the ions is small.\(^{15}\) It was formerly thought that polar molecules after ionization always dissociate into positive and negative atomic ions, since the ionization potentials of hydrogen-halide compounds can be exactly calculated from a thermodynamic cycle based on this hypothesis.\(^{16}\) At the present time, however, it is known that this is an accidental coincidence; thus, for example, in HCl the primary product of ionization is \(HCl^+\), and no secondary products are observed.\(^{17}\)

2. Photoionization can occur if a normal molecule is subjected to radiation of frequency greater than the value \(\nu_i\) in the relation \(h\nu_i = eV_i\).\(^{1}\) The probability of such ionization is proportional to the density of the radiation and varies with frequency. A quantitative experimental study of this phenomenon is extremely difficult, since—with the exception of the case of alkali-metal vapors—the effective radiation lies in the far ultraviolet region, as a result of which an estimate of the radiation intensity is very difficult. On the other hand, when these spectral regions are illuminated by light—

equilibrium, corresponding to the minimum of potential energy. Upon excitation (or—in the limiting case—upon ionization) the bonds in the molecule, generally speaking, change; at the same time the position of stable equilibrium of the nuclei also changes. The process of electronic excitation occurs instantaneously, so that at the first moment the heavy nuclei find themselves displaced from the position of equilibrium. As a result, the molecule enters into strong vibration, and under certain conditions the bonds may be broken—the molecule dissociates. These qualitative considerations were first indicated by Franck. Condon gave them quantitative form and applied them to the interpretation of the intensity distribution in band spectra. On the Franck–Condon principle, see Kuhn, Diffuse Spectra and Chemical Data, Proceedings of the VII Physico-Chemical Conference. Journal of Physical Chemistry, II, no. 2 (1931). Editor’s note.

...of the walls, photo-electric emission of electrons from the walls of the apparatus masks the ionization of the gas, unless special precautions are taken.

The probability \(B_\nu\) of photoionization may be defined as the probability that an atom subjected to radiation of frequency \(\nu\) of unit density will be ionized in a time interval equal to unity. All theories agree that this probability is a maximum for the boundary of the arc-spectrum series (frequency \(\nu_i\)) and decreases rapidly with increasing frequency. In general, experiment confirms this. For very high frequencies (X-rays) there exists the well-known Owen law, according to which the absorption coefficient of a substance is inversely proportional to the cube of the frequency. Since the energy absorbed in each act of absorption is equal to \(h\nu\), Owen’s law may be formulated as follows: “the probability of a single act of absorption is inversely proportional to the fourth power of the frequency.” Since absorption produces photoionization, Owen’s law is equivalent to the following:

\[ B_\nu = C \nu^{-4} \]

With the aid of the statistical principle of detailed balance, the following relation is then derived\({}^{18}\)

\[ \frac{B_\nu}{q_\nu} = \text{const}\,\frac{\nu-\nu_i}{\nu^3} \]

between \(B_\nu\) and the effective cross section of the atom for recombination \(q_\nu\) (which is equal to the velocity of the electron \(v\), multiplied by the recombination coefficient \(\alpha\) of an ion and an electron of velocity \(v\), forming a neutral atom). But the effective recombination cross section \(q\) is not known experimentally with sufficient accuracy (see section C 2), and the theories that consider it have not been developed enough for rigorous application. Milne\({}^{16}\) assumed that the probability for an electron to be captured by an ion varies inversely as the square of their relative velocities,

i.e., that \(q_v\) “varies as” \(\dfrac{1}{v^2}\), which gives directly

\[ B_v = C v^{-3}, \tag{1} \]

since \(\dfrac{1}{2}mv^2 = h(\nu-\nu_i)\) (cf. equation 65, section C 2).

This assumption can be justified only as an approximation, for Milne showed that it represents the asymptotic form of \(q_v\) as the velocity approaches zero. Morse and Stueckelberg\(^{19}\) have recently shown that this follows from wave mechanics, and have also shown how \(q_v\) varies with velocity for various states of the hydrogen atom. These latter expressions, although exact, proved too complicated to be put into a form suitable for substitution into equation (1). Thus even in the simplest case—the case of hydrogen—an exact solution cannot be obtained. Kramers\(^{20}\) derived Oyen’s law; Becker\(^{18}\) proposed that \(q_v\) varies with velocity in such a way that the probability that the canal ray will capture an electron from a neutral molecule depends on the relative velocities,\(^{21}\) and obtained from equation (1)

\[ B_v = \frac{\text{const.}}{(\nu-\nu_i)\nu_i^3} \left[ 1-\frac{(\nu-\nu_i)^2}{4\nu_i^2} \right]. \tag{2} \]

This expression is not applicable for \(\nu \gg \nu_i\), but for \(\nu_i\) close to \(\nu\) it reduces to the following:

\[ B_v = \frac{\text{const.}}{(\nu-\nu_i)\nu^3}. \tag{3} \]

With the aid of the methods of the new quantum theory, Sugiura, Oppenheimer, and Reiche obtained expressions\(^{22}\) for the case of the hydrogen atom in which \(B_v\) varies as

\[ \frac{1}{\nu^5},\quad \frac{1}{\nu^{4.3}},\quad \frac{1}{\nu^5}. \]

It is evident that, from the theoretical point of view, the problem is still far from being regarded as solved with any degree of accuracy.

From the experimental point of view, the latter, highly careful works have convincingly shown that photoionization in calcium and rubidium vapors is well described by equations (2) and (3) and does not obey any other equations. But precisely for these cases only relative data for different frequencies are known. In the case of potassium vapor, the results available at present are in serious contradiction with all theories, although this may also be due to the complicating influence of photoionization of \(K_2\) molecules. Thus, as may be concluded on the basis of the cases considered, and also of certain others, at the present time we still cannot confidently assess the role of photoionization in discharge in gases. However, generally speaking, one may think that the absolute magnitude of photoionization is considerably smaller than the magnitude of ionization by electron impact.

It has been established that photoionization is observed in mercury vapors \(^{26}\) and in alkali metals \(^{23}\) for radiation with a frequency lower than the limit \(\nu_i\),—in particular for radiation belonging to the line absorption spectrum.* This is explained by the effect of accumulation of two or a greater number of portions of energy. However, this effect is apparently greater than is given by the theories proposed up to the present time.

A review of recombination and photoionization was recently published by Moller \(^{27}\).**

3. Cumulative ionization is ionization occurring by means of accumulation of energy. This may take place in various ways. By electron impact a molecule may be excited to a state close to complete ionization. Then it may re-

* Ionization itself, as indeed any process of decay in general, ordinarily corresponds to a continuous absorption spectrum. The frequency of the boundary of this continuous absorption, multiplied by \(h\), gives the minimum energy of photoionization; therefore the quanta corresponding to line absorption are smaller than this minimum energy. Editor’s note.

** In one of the forthcoming issues of Uspekhi a review by A. N. Terenin, “Photoionization of Gases,” will be printed. Ed.

return to a state of lower energy spontaneously, with the emission of radiation, or as a result of a subsequent collision, in which the energy may be emitted or transferred to the colliding molecule. Experiment shows that usually an atom or molecule, in the absence of perturbing collisions, remains in an excited state for a time interval of the order of \(10^{-8}\) sec. \(^{28}\) If this state proves to be “metastable,” i.e. a state from which a spontaneous transition accompanied by radiation is forbidden by the “selection rules” of quantum theory, then the lifetime in this state will be considerably longer. If the gas pressure is sufficiently high, so that molecular collisions occur frequently, then the lifetime may be limited by the average interval between two successive collisions. But under any conditions an excited molecule has some mean period of existence, denoted by \(\tau\). If, however, the excited molecule experiences the impact of another electron, or absorbs another portion of radiant energy, or collides with another excited molecule, it may acquire an additional amount of energy sufficient for ionization. Successive impacts, successive absorption processes, or a combination of absorption and impact—a combination that may be called a “photo-impact,” etc.—are the various ways in which this ionization by accumulation, cumulative ionization, can occur.

The rate of cumulative ionization is evidently proportional to the intensities of all the ionizing factors and to the mean lifetime \(\tau\) in the excited state. If the ionization is produced wholly by the current passing through the gas, then the rate of two-step cumulative ionization must be proportional to the square of the current strength, if the other factors remain unchanged.

Except in cases of very low pressures, there are two phenomena that increase the rate of cumulative ionization owing to an increase in the concentrations of excited molecules. The first of these phenomena is the strong absorption by the gas of “resonance” radiation, which

arises when the molecule returns from its first excited state to the normal one. A molecule that has absorbed light of a wavelength equal to its resonance radiation emits, after \(\tau\) seconds, its energy in the form of light of the same “resonance” wavelength. This radiation may be absorbed by another molecule, which in turn will emit it in the form of a resonance wavelength, etc. In this way the energy passes from molecule to molecule, thereby increasing the chances that each absorbed quantum of energy will entail ionization. This wandering of resonance radiation inside the gas can be treated as a diffusion problem and represented by[^29] the equation

\[ \iint \frac{1}{3}\cdot \frac{1}{\alpha^{2}\tau}\cdot \frac{dN'}{dn}\, ds = -\iiint R\, dx\, dy\, dz, \]

where \(\alpha\) is the absorption coefficient for the radiation, \(\tau\) is the mean lifetime of the excited state, \(N'\) is the concentration of excited molecules, \(n\) is the normal to the surface element \(ds\), and \(R\) is the total rate of production of new quanta of resonance radiation inside the closed surface.

The second phenomenon consists in the tendency of molecules in the excited state, under the influence of collisions with other molecules, to pass into a metastable state with a long lifetime. It is still entirely unclear which of these phenomena is more important in the cumulative ionization of mercury vapor and noble gases. In vapors of the alkali metals, which have no metastable states, only the first process can be effective.

The experimental results of Wien, given in Table III, may perhaps not give correct values of \(\tau\) for the excited states that interest us in the present case, since the conditions of the experimental measurements allow higher excitation levels, the transition from which to the lower excitation levels under consideration may also occur in several stages. Thus, for example, hydrogen atoms excited to the fifth quantum state may emit \(H_{\gamma}\) by directly returning to the second quantum state. But they may

also return successively to the fourth, third, and second states, emitting H\(_\alpha\) in the last transition. In this case the quantity \(\tau\), measured for H\(_\alpha\), must be equal to the sum of the values of \(\tau\) for the three transitions mentioned. Wien’s method gives a quantity \(\tau\) which is a weighted mean of all such processes. The probabilities of these transitions are such, however, that the experimental results are only slightly distorted by these complications. Methods free from these complications may be applied to determine the lifetimes of certain optically excited states\(^{31}\).

Table III

Molecule	Excited toward emission	\(\tau\)
H	H\(_\alpha\), H\(_\beta\), or H\(_\gamma\) . . . . . . . .	\(1.85 \cdot 10^{-8}\) sec
Hg	\(1s\,{}^1S_0 — 2p\,{}^3P_1\) (2536) . . . .	9.7
Hg	\(2p\,{}^3P_1 — 2s\,{}^3S_1\) (4358) . . . .	1.81
Na	\(1s\,{}^2S_1 — 2p\,{}^2P_{1/2}\) (NaD\(_{1/2}\)) . . . .	3.70
N	Arc lines . . . . . . . .	9.33
N	Spark lines . . . . . . . .	1.35

Maxwell\(^{32}\) measured \(\tau\) for lines of mercury atoms in various states of ionization and found values ranging between \(10^{-8}\) and \(9 \times 10^{-7}\) sec for the lines HgII, HgIII, HgIV* with a regular gradation of values, which indicates that the mean lifetime is greater in higher degrees of ionization. His calculations of the mean lifetime \(\tau\) of the excited state are based on finding the mean rate of “extinction” of molecules excited to this state. As a result one obtains the formula

\[ \tau = \left(\sum_{n',k'} A_{n,k}^{n',k'}\right)^{-1}, \tag{4} \]

* In spectroscopy it is customary to denote the non-ionized atom (giving an “arc” spectrum) by the Roman numeral I; the numeral II denotes a singly ionized atom, etc. — Ed. note.

where the summation extends over all states \(n, k\) between which transitions can occur; \(A_{n,k}^{n',k'}\) is the probability of such transitions. This expression, originally given by Tollman \(^{33}\) and others, indicates that the mean lifetime for all spectral lines originating from one and the same state must be the same. However, this expression does not take into account the population of the state under consideration from higher states.

The mean “natural” lifetime of metastable states is of a larger order of magnitude than the mean lifetime of other excited states. Saha and Kothari \(^{34}\) derived the theoretical expression

\[ \tau=\frac{3c^{5}m^{2}}{\pi^{2}e^{2}h\nu^{3}} \]

for the natural lifetime of a metastable atom for which the “forbidden” line has wave number \(\nu\). For \(\lambda=10\,000\ \text{Å}\), this formula gives \(\lambda=0.15\ \text{sec}\).

In a discharge tube in which an actual experiment is performed, the lifetime of metastable atoms is limited by impacts against the walls of the apparatus or by collisions with other molecules, so that the true lifetime is characteristic rather of the apparatus than of the atom itself. Under such conditions the lifetime is given by the relation

\[ \tau=\frac{A}{p}+Bp, \]

where \(p\) is the pressure, and \(A\) and \(B\) include effective collision radii, the dimensions of the apparatus, etc. The term \(A/p\) gives the rate of disappearance of metastable atoms at the walls of the apparatus, while the term \(Bp\) gives the rate of disappearance inside the apparatus. Culett \(^{36}\) showed that metastable Hg atoms diffuse through Hg according to the ordinary laws of diffusion of gases, provided that the effective radius of the metastable atom is 1.5 times greater than the radius of the normal atom. Much effort was spent in order to find the probability of return of a metastable atom to the normal-

state as a result of collisions with other atoms or molecules. Obviously, the most effective, in the sense of destroying metastable states, will be collisions with such molecules as can completely absorb the available store of energy by passing into an excited state or by dissociating—such, for example, is the dissociation of \(\mathrm{H}_2\) by the metastable molecule \(\mathrm{Hg}\).* Dorgello \(^{35}\) found that metastable neon atoms in neon exist for \(0.10\) sec. A critical review of recent work in this field, with an interesting theoretical analysis, has recently been published by Zemansky \(^{37}\).

It is very important to know the concentration of metastable atoms in a discharge tube, if such atoms are active agents in the ionization of the gas or of the electrodes. The only method suitable for measuring this concentration is an optical method, based on absorption or dispersion \(^{38}\). But this method requires extreme care in technique and in interpretation. For neon in the positive column of a glow discharge, Kopfermann and Ladenburg found that the concentration of the metastable states \(s_3, s_4, s_5\) is approximately expressed by the formula

\[ N_s=\frac{aI}{bI+c}; \tag{4} \]

where \(I\) is the current density, and \(b, c, a\) are excitation and damping factors which do not depend on the current but do depend on the pressure; for the state \(s_5\) they are \(a=3.5\times 10^{11}\), \(b=0.365\), \(c=1\), with \(I\) expressed in milliamperes per \(\mathrm{cm}^2\). Thus it follows from this that the concentration of metastable atoms

* Experiments with collisions of the second kind show that the conversion of quantum energy into kinetic energy of the translational motion of the colliding partners occurs with a very small degree of probability. The greatest probability is instead possessed by the transfer of excitation energy when the colliding unexcited atom or molecule has energy levels approximately equal to the energy store of the excited atom. Cf. J. Franck and P. Jordan. Anreg. d. Quantensprüngen durch Stösse. A theoretical interpretation of this peculiarity of energy transfer from the standpoint of wave mechanics was given by Kallmann and London. See Kallmann und London, Z. physikal. Ch. B., 1930. — Editor’s note.

may have an order of magnitude of \(10^{12}\) and consequently be comparable with the concentration of positive ions (as we shall see in Part II).

1. Ionization by impact of positive ions is well known in the case of \(\alpha\)-particles and canal rays. It has long been supposed \(^{39}\) that it also occurs in the case of ions of considerably smaller velocities. Ions, apparently, become very effective ionizing agents when they acquire the same velocities as the velocities of electrons under the action of the minimum ionizing potential \(^{40}\); owing to the considerable mass of the ions, this corresponds to positive ions with energies of several tens of thousands of volts. In those cases where the energies of the positive ions are considerably smaller, they are extremely inactive, so that the ionization caused by their impacts is, under such conditions, masked by the various complicating effects of secondary ionization—effects which, to a greater or lesser degree, are associated with every method of experimental investigation. In the final analysis, our direct knowledge of this phenomenon is highly insufficient, and many data are contradictory \(^{41}\).

In an apparatus that makes it possible to detect one act of ionization for every thousand primary positive ions, no indications were obtained of ionization of hydrogen by sodium or potassium ions up to velocities of about a thousand equivalent volts \(^{42}\). In the noble gases, ionization by potassium ions could be detected at small velocities of the order of one hundred volts, and the course of the figures in this region indicates that the probability of ionization falls, if not exactly to zero, then in any case to considerably less than \(0.1\%\) at velocities somewhat below one hundred volts. The ionization of certain gases by potassium ions at 750 volts, expressed as the number of ions formed per centimeter of path for each initial positive ion, at \(0.01\) mm pressure, is equal to \(^{43}\): argon—0.00288; neon—0.00112; nitrogen—0.00124; air—0.00098; hydrogen—0.0000. Perhaps the most direct proof of ionization at velo-

at growths of about 40 volts was found in the excitation of arc lines of the mercury spectrum in mercury vapor bombarded by sodium ions^44, but no quantitative conclusions can be drawn from these experiments. Experimental difficulties have hitherto made it impossible to draw any definite conclusions regarding the efficiency of ionization of gas molecules by ions of the same gas; it has only been shown that this efficiency is less by an order of magnitude than ionization by electron impacts. It is highly necessary to carry out direct experiments in order to discover the ionizing action of positive ions at energies of the same order of magnitude as those encountered in arcs, sparks, and glow discharges.

1. Ionization by collisions of the second kind is ionization caused by the transfer of energy from an excited or ionized molecule to some other molecule. The first condition is that the excess of energy be sufficient for ionization. The probability of this process apparently always reaches a maximum in those cases when the excess of energy is just sufficient for ionization of the colliding molecule and decreases with an increase in the excess energy. Thus, upon contact, a gas ion can capture the neutralizing electron from a neutral molecule of any gas possessing a lower ionizing potential^45, thereby ionizing the latter, and at the same time the captured electron can be raised to an excited orbit^46. In the case of homogeneous ions there occurs a simple exchange between the electron and the neutral molecule. But new ions may also be formed, as is observed under the action of metastable helium atoms in the \(2^3S\) or \(2^1S\) state (with energies respectively \(19.17\) and \(20.55\) V) on atoms of impurities upon contact (provided that these atoms possess a lower ionization energy^47).

2. Thermal ionization of a gas may be regarded as a case of thermal dissociation, in which the products are an electron and a positive ion, and the degree of ionization can be calculated as a function

of the temperature $T$ and pressure $P$ of the gas with the aid of Nernst’s heat theorem. The idea of such a calculation was applied with striking success to problems of ionization in stellar atmospheres, for which purpose the so-called “Saha equation” and its subsequent refinements were used[^48]. Attempts were made to apply this equation to the calculation of the degree of ionization of a gas in an enclosed space[^49], but serious errors appeared in the calculations, since in these attempts the very considerable role that may be played by the emission of electrons by the walls surrounding the enclosed space was ignored. A more general equation, equally well applicable both to stars and to laboratory conditions, has the form

\[ \log_{10}\frac{n_e n_p}{n_a} = -\frac{5040 V_i}{T} + \frac{3}{2}\log_{10}T + 15{,}385, \]

where $n_e$, $n_p$, and $n_a$ are, respectively, the numbers of electrons, positive ions, and atoms per cubic centimeter; $V_i$ is the ionization potential of the gas in volts, and $T$ is the absolute temperature[^50]. In particular, Langmuir and Kingdon showed that the proportional ionization of cesium atoms falling on a hot metallic surface is thermal ionization, which can be well calculated with the aid of this equation.

B. Liberation of Electrons and Ions from Electrodes

All processes of emission from electrodes depend strongly on the conditions at the surface, namely on the presence of surface films of other metals or of adsorbed gases. Therefore one cannot be certain to what extent the results obtained in special studies of these phenomena in pure form can be transferred to the actual conditions in gas discharges. This can probably be done, at least as a first approximation; however, from the standpoint of the theory of gas discharges it is desirable, insofar as possible, to investigate a larger number of such processes precisely under the conditions that exist during a discharge.

1. Thermionic emission of electrons can give currents limited only by the temperature to which the cathode can be heated. Thus, for example, at the melting temperature of tungsten the saturation emission is equal to 480 amp/cm². The saturation current density \(j_s\) is a function only of the temperature \(T\) and of the nature of the metal. The latter dependence is characterized by a certain constant \(\varphi_0\), called the “work function,” and by another constant \(A\). The expression for the saturation current density has the form

\[ j_s = AT e^{-\frac{e\varphi}{kT}} = AT e^{-\frac{b}{T}}, \tag{6} \]

If we put for \(\frac{e}{k}\) \(= 11\,606\) degrees per volt, we can express \(\varphi_0\) in equivalent volts. Equation (6) was originally derived on the basis of thermodynamic considerations \(^{51}\), based on the assumptions: 1) that the potential energies of the electrons inside and outside the surface of the metal differ by \(e\varphi\); 2) that \(e\varphi = e\varphi_0 + \frac{3}{2} kT\), where \(\varphi_0\) does not depend on temperature; 3) that the electrons outside the metal obey the gas equation \(pv = nkT\); 4) that all electrons striking the surface are absorbed by it. From these assumptions the 2nd follows as a first approximation from the consideration, based on assumptions 1 and 3, of Thomson’s coefficient; assumption 3 is based on the experimental fact that the emitted electrons possess a Maxwellian distribution of velocities characteristic of the temperature of the emitting metal \(^{52}\); assumption 4 is true only approximately, but the error introduced in this way probably almost does not depend on the temperature \(T\) (see reference \(^{57}\) below). Direct experimental confirmation of the relation

\[ e\varphi_0 = e\varphi + \frac{3}{2} kT \tag{7} \]

is very difficult. The available data indicate that the relation should be of this type, but with a somewhat larger temperature coefficient \(^{56}\).

Neglecting the contribution of the electrons to the specific heat of the me—

of a metal, neglecting the energy of the surface charge, taking the chemical constant of Zakur–Tetrode as the integration constant of the Clapeyron equation, and, finally, assuming zero reflecting power of electrons for all metals, one may predict that \(A\) must be a universal constant\({}^{54}\) and have the value \(A_0 = 60.2\ \text{amp}/\text{cm}^2\,\text{deg}^2\). This number apparently represents the correct value of \(A\) for metals similar to tungsten. However, there are some data which indicate that \(A\) itself is a function of \(\varphi_0\)\({}^{55}\) of the form

\[ A = A_0 e^{-\frac{\alpha}{k}}, \]

where \(\alpha\) is the negative derivative of \(\varphi_0\) with respect to temperature. For tungsten \(\alpha = 0\), for platinum \(\alpha\) is positive, and for thoriated tungsten it is negative.

On the basis of Sommerfeld’s new theory of metals\({}^{56}\), Sommerfeld and Fowler derived equation (6), but with a certain difference in interpretation. Fowler found that \(A = 2A_0(1-r)\), where the factor 2 enters because of the two possible orientations of the electron spin vector, and \(r\) is the fraction of electrons falling on the surface of the metal from the surrounding electron atmosphere and being reflected. According to this theory the variation in \(A\) for different metals is due to a characteristic difference in this reflecting power \(r\) (a fact also established by Richardson). Experiment would confirm this theory if it could be shown that \(r = \frac{1}{2}\) for tungsten (for which \(A = A_0\)), that \(r\) is less than one half for platinum and greater than one half for thoriated tungsten. There are some weak experimental indications that \(r \sim \frac{1}{2}\) for very slow electrons, but these experiments require corrections which become very uncertain at small velocities. Moreover, there are convincing theoretical arguments against assuming a noticeable value of \(r\) for very slow electrons (see in 7). Thus Fowler’s theory cannot be regarded as definitively confirmed experimentally on this point. Sommerfeld showed that the work function is the difference between

by the work necessary to remove an electron from the metal against the attractive forces toward the surface, and by the action of the pressure of the conduction electrons inside the metal, which tend to overcome these forces.

Despite this uncertainty in interpretation, equation (6) is one of the most accurately verified equations in all of physics, since it holds with the accuracy with which the temperature can be measured for a range of current intensities having a magnitude of about a million million times. Furthermore, there can be no doubt as to the essential correctness of the interpretation of \(\varphi\) as the work necessary to remove a unit charge of electrons from the metal.

More electropositive metals are characterized by a smaller work function and consequently emit larger thermionic currents at any given temperature than do electronegative metals. However, electropositive metals, generally speaking, are unsuitable for the manufacture of incandescent cathodes because of their volatility. Nevertheless, monatomic films of such substances on more electronegative metals are retained with remarkable stability and constitute a convenient source for emission at low temperatures. The best known of these are thorium films \(^{58}\) and cesium films \(^{59}\), which are characteristic of two methods of covering a cathode with a surface film.

Thorium films are obtained by diffusion of metallic thorium to the surface. This diffusion occurs especially rapidly along the boundaries of crystalline grains, but it can also occur from within the crystal toward these boundaries and probably also through the surface layer of the crystals. This diffusion proceeds at a convenient rate at \(2100^\circ\mathrm{K}\), and continues until the surface is covered with a monatomic layer of thorium. Such surfaces can give thermionic emission of about one ampere per \(\text{cm}^2\) at the maximum temperature at which thorium does not yet begin to evaporate faster than it is renewed by diffusion.

Cesium films are obtained by deposition of cesium atoms from its vapor on the surface of tungsten. Their stability

at temperatures up to \(900^\circ\mathrm{K}\) depends on the fact that the ionizing potential of cesium atoms is less than the work function of tungsten, so that tungsten takes the valence electrons from cesium atoms striking its surface and retains them as ions. If the temperature rises enough for the atoms to begin evaporating from the surface, they leave it as ions so long as the work function exceeds the ionizing potential of the atom. If one uses a metal with a higher work function than tungsten, e.g. oxidized tungsten, the cesium atoms are in this case held much more strongly, and the maximum temperature at which emission can be obtained without loss of the film is increased. Similar effects have been observed with vapors of rubidium and potassium \(^{60}\).

The degree of activation of such coated surfaces is conveniently expressed by the quantity \(\theta\), called by Langmuir “the fraction of the surface covered by adsorbed ions.” This quantity is an approximately linear function of the work function, so that \(\theta=(\varphi''-\varphi)/(\varphi''-\varphi')\), where \(\varphi,\ \varphi',\ \varphi''\) are the

Table IV

Some thermionic work functions \(^{63}\)

Metal	\(\varphi_0\) (volt)	Metal	\(\varphi_0\) (volt)
Hf	5.09	Th	3.35
* Pt	4.8—6.35	** Th	2.63
W	4.52	* U	2.84
Mo	4.41	** Ce	2.72
Ta	4.07	** La	2.72
Zr	4.50	*** BaO	1.68—3.44
** Zr	3.15	BaO + SrO	1.51—1.89
Cs	1.81	CaO	2.19
*** Cs	0.72	MgO	1.02

* Pt, perhaps owing to its electronegative character, is extremely difficult to free from hydrogen and electropositive contaminants. The characteristic value 6.35 can be attained only after the most thorough cleaning and degassing.

** Monatomic layers on tungsten.

*** The figures depend on the treatment of the surface.

of the work function of the surface in the state in which it was, almost completely covered, and of a perfectly clean surface \(^{58, 59}\).

In gases at low pressures these activated surfaces, such as, for example, a thorated surface, are easily destroyed by bombardment with positive ions when the anode voltage is above a certain critical value \(^{61}\).

The velocity distribution of thermally emitted electrons is Maxwellian and is characteristic of the temperature of the emitting surface \(^{62}\). The mean kinetic energy of electrons in equilibrium in a given volume is equal to \(\frac{3}{2}kT\), or \(\frac{1}{2}kT\) for each of the three dimensions. For electrons that cross the surface, the mean energy is only \(\frac{1}{2}kT\) for each of the two dimensions parallel to the surface; but the mean energy corresponding to the component of velocity normal to the surface is equal to \(kT\), and in all amounts to \(2kT\) for the mean energy of the emitted electrons. These energies can readily be expressed in equivalent volts by the equation

\[ Ve = kT. \tag{8} \]

It is convenient to express the quantity \(\frac{e}{k}\) \(^{64}\) as 11,606 degrees per volt \((10^8 \times 1.5911 \times 10^{-20} / 1.3709 \times 10^{-16})\). Thus the mean energy \(2kT\) of the electrons emitted at temperature \(T\) corresponds to \(V\), defined as follows:

\[ V = \frac{2kT}{e} = \frac{2T}{11606}\ \text{volts}. \tag{9} \]

This mean energy corresponds to only 0.052 volt at room temperature \((300^\circ\mathrm{K})\), 0.41 volt at \(2400^\circ\mathrm{K}\), and 0.63 volt at \(3655^\circ\mathrm{K}\)—the melting temperature of tungsten.

The fraction \(\frac{n}{n_0}\) of emitted electrons which are capable, in moving, of overcoming the retarding field of \(V\) volts can be calculated from the Boltzmann equation \(^{65}\),

\[ \frac{n}{n_0} = \varepsilon^{-\frac{Ve}{kT}}. \tag{10} \]

Thus, for example, one out of a thousand electrons of a hot cathode at a temperature of \(2400^\circ\mathrm{K}\) may pass into regions in which the potential is more than 1.43 volts negative with respect to the cathode.

The heat of evaporation of electrons from a metallic surface is the heat absorbed in the process of emission of an electron. It is analogous to the latent heat of evaporation of molecules from a liquid surface and is due to the fact that only the fastest electrons in the metal can leave the surface. This quantity and the corresponding heat of condensation were measured for electrodes in a high vacuum ^66, and the latter, i.e. the heat of condensation, also for electrons arriving at the anode during a discharge in gases ^67. This heat of evaporation proved to be identical with the work function \(e\varphi\). Determination of the heat of condensation may perhaps represent the best way of determining the work function of an electrode in a neutral or ionized gas, since the heat of condensation can be measured without subjecting the electrode to strong heating or to any other influence. In saturated emission the cooling effect caused by the loss of electrons increases exponentially with temperature, whereas the cooling due to thermal conductivity and radiation varies, roughly speaking, as the first and fourth powers of the temperature. Consequently, at high temperatures the cooling effect of thermionic emission may exceed the cooling caused by other causes.

1. Thermionic emission of positive ions usually consists in the emission of charged atoms of electropositive substances, such as alkali or alkaline-earth metals, which are present intentionally or accidentally as impurities on the surfaces of more electronegative metals ^68. Alkali atoms do not evaporate in the form of ions from the surfaces of their own metals. They evaporate in the form of ions from the surfaces of platinum, iron oxide, etc. It seems probable that here too, as in the case of films of alkali metals on tungsten ^59, ^60, the criterion for the possibility of evaporation in the form of an ion is

condition that the work function of the surface exceeds the ionization potential of the atom. If the surface conditions remain constant, this emission of positive ions varies with temperature according to an equation similar to equation (6) ⁶⁹.

Quite recently another type of emission of positive ions was found, namely, cases were discovered in which the ions are charged atoms of the heated metal itself, and not of its impurities. This was demonstrated by measuring the mass of these ions with the aid of a mass spectrograph. Walin ⁷⁰ was the first to describe this type of emission and found that such characteristic ions are emitted by chromium, molybdenum, tungsten, rhodium, ruthenium, tantalum, and columbium. Emission of this type was not observed from iron, nickel, cobalt, copper, silver, gold, iridium, platinum, zirconium, palladium, and antimony. With manganese the results obtained were doubtful. All these metals, upon first heating, gave off the usual alkaline contaminations. The first group gave an emission of characteristic ions, which persisted even after the contaminants had disappeared during prolonged heating.

L. P. Smith studied tungsten and molybdenum in greater detail ⁷¹. He found that these currents vary with temperature according to an equation of type (6), from which he calculated the work function for positive ions and obtained \(\varphi_{+0}=6.55\) volts for tungsten and 6.09 volts for molybdenum. These values are considerably smaller than those obtained by an estimate based on a simple thermodynamic energy cycle. Likewise the term corresponding to \(A\) in equation (6) does not agree with its theoretical value, unless one makes the assumption that these positive ions have a probability of reflection of about 0.99999 when they strike the metal surface in order to condense on it. Hence it is evident that the physical interpretation of this interesting type of emission is still far from clear.

It should be emphasized that these currents of “characteristic ions” are extremely small in comparison with the other currents of thermionic emission. For example, measurements made in the investi-

...the laboratory of the General Electric Company,* gave for tungsten about \(5.4 \times 10^{-10}\) amp/cm\(^2\) at \(2500^\circ\) K and about \(1.4 \times 10^{-8}\) amp/cm\(^2\) at \(2800^\circ\) K. These figures indicate that the rate of emission of these ions is considerably less than the rate of evaporation of neutral atoms. Thus, approximately one ion is emitted for every 2000 atoms evaporating at \(2000^\circ\) K, and approximately one ion for every 4200 atoms evaporating at \(2800^\circ\) K. It might be supposed that these ions are products of thermal ionization of the emitted vapor of the metal, but the application of equation (5) shows that such thermal ionization would give only about one ion per million evaporated atoms, and therefore it cannot explain the observed emission. Another supposition might be that this emission of ions is causally connected with the ionic nature of the metallic lattice, but at the present time such a supposition would be mere speculation.

For many purposes a heated filament of an electropositive metal in the presence of a vapor whose ionization potential is less than the work function of the filament is a very convenient source of positive ions \(^{59}\). For example, a tungsten filament at any temperature above \(1200^\circ\) K emits, as a positive ion, every cesium atom that strikes the filament. In this way an intense source of positive ions can be obtained, even at such a low vapor pressure that the vapor plays no noticeable role in the discharge, except that it supplies these positive ions upon contact with the filament. Such a source gives a current independent of the temperature of the filament, but proportional to the vapor pressure \(^{72}\).

3. Photoelectric emission of electrons occurs when a metal is illuminated by radiation with frequency \(\nu\) greater than the threshold value \(\nu_0\), determined by the condition

\[ h\nu_0 = e\varphi, \tag{11} \]

* In Schenectady, N.Y.

where \(h\) is Planck’s constant, and \(e\varphi\) is the work of removing an electron from the metal. The quantity \(\varphi\) is usually called the photoelectric work function of the metal. The number of electrons emitted per second is directly proportional to the intensity of the incident radiation, but the velocities of emission do not depend on the intensity.

The velocities of emission of photoelectrons are distributed between zero and a certain maximum, which depends on the frequency of the radiation and on the work function of the metal; this dependence is expressed by the well-known Einstein photoelectric equation 73

\[ \frac{1}{2}mv^2 = eV = h\nu - e\varphi, \tag{12} \]

which has been confirmed over the whole region of the spectrum, from the near infra-red to the region of X-rays 74. In this equation \(V\) denotes the smallest retarding potential difference which does not allow the electrons to pass from the emitting electrode to the receiving one. The usual interpretation of this equation amounts to saying that the electron receives the energy \(h\nu\) from the radiation and loses the energy \(e\varphi\) on leaving the surface.

Between zero and the maximum velocity there are various velocities, distributed in such a way that the most probable energy is approximately one half of the maximum energy 75 in the case of ordinary or ultraviolet light acting on a metallic surface. If the emission of photoelectrons occurs under the action of X-rays, then the greater part of the ejected electrons has an energy close to the theoretical maximum. The great generality of the relation \(h\nu =\) energy indicates that electrons emitted with energies smaller than the maximum lose the difference in energies in atomic collisions inside the metal during the interval of time between receiving the energy and being removed from the surface 76.

Table V gives some typical data concerning the initial energy of the electron, expressed in equivalent volts.

Table V

Maximum emission energies of photoelectrons for various metals and wavelengths

	Limit, Å	$V$ volts	$V$ volts	$V$ volts	$V$ volts	$V$ volts
		4000	3000	2500	2000	1000
Pt77 (degreased)	1945	—	—	—	—	6.00
Pt75 (ordinary)	2880	—	—	0.66	1.89	8.07
W78 (degreased)	2575	—	—	0.14	1.37	7.55
Hg79 (liquid)	2735	—	—	0.42	1.65	7.83
Zn75 (ordinary)	3760	—	0.83	1.65	2.88	9.06
K80 (pure)	7000	1.32	2.35	3.17	4.40	10.58
K80 (cesium hydride)	10000	1.85	2.88	3.70	4.93	11.11

Owing to the obvious discrepancy between the contact difference of potentials and the work functions of the two metals, Millikan for some time believed that thermionic electrons arise from free electrons or conduction electrons of the metal, whereas photoelectrons are initially bound electrons of the metal atoms81. This view was supported by the fact that temperature has very little effect on photoelectric emission82 (provided that the physical conditions at the surface do not undergo changes under the action of temperature)83. Later it was found that the discrepancies mentioned are due to complications in the interpretation of measurements of the contact potential difference, and the original conclusion of Richardson and Compton75 concerning the identity of the contact potential difference and the difference of the work functions of pure metals was fully confirmed (the complications, which will be discussed later, account for deviations from this identity in the case of metals with inhomogeneous surfaces). Furthermore, Sommerfeld’s theory of metals66 explains the independence of the photoelectric effect from temperature. Thus, at the present time nothing prevents us from considering that the same groups of electrons constitute both thermionic and photoelectric emission.

It is true that photoelectrons are also observed which arise from the deep layers of atoms, i.e., are “bound electrons.” For these electrons, liberated under the action of X-rays or gamma rays, \(\varphi\) in equation (12) includes the binding energies of the electrons at the various energy “levels” of atoms, so that equation (12) gives a convenient method for measuring these energies \(^{84}\). It is doubtful, however, that such energies play any appreciable role in the phenomena of discharge in gases.

At one time it was thought that evidence had been found for the existence of “cumulative” photoelectric emission \(^{85}\). This evidence was seen in the fact that, at frequencies lower than the usual limiting frequency \(\nu_0\), a weak emission is observed with a maximum velocity determined by Einstein’s equation (12), in which \(2h\nu\) stands in place of \(h\nu\). Recently, however, it has been found that this emission is due to a remarkable effect of even small accelerating fields, consisting in a lowering of the apparent work function for activated surfaces (see below, Section B 4).

The total photoelectron emission is not only proportional to the intensity of illumination, but is also a function of the nature of the metal and of the frequency. The more electropositive metals are also the more sensitive. The sensitivity, generally speaking, increases with increasing radiation frequency from zero at \(\nu_0\) to a maximum at

\[ \nu = \frac{3\nu_0}{2}, \]

after which it decreases, but then increases to one or several maxima at considerably shorter wavelengths \(^{86}\). Every maximum of light absorption entails a maximum in the photoelectron emission, but at least the first maximum apparently cannot be explained in this way. The only theory of the total emission as a function of frequency—the theory of Richardson \(^{87}\)—although correct as regards the order of magnitude and predicting a maximum of sensitivity at \(\frac{3\nu_0}{2}\), has nevertheless not proved correct in detail.

Complete photoelectronic emission under the influence of black-body radiation has proved to vary theoretically^88 and experimentally (89) with the temperature of the black body according to an equation similar to equation (6) for thermionic emission. This suggests the attractive hypothesis that thermionic emission is simply photoelectric emission caused by the density of radiation inside an incandescent metal, integrated over the entire spectrum and over the whole volume of the metal. A quantitative test of this hypothesis is very difficult, perhaps chiefly because of the uncertainty in estimating the depth from which photoelectrons can be ejected. The best estimates give, for integral photoelectronic emission, values approximately 5000 times smaller than for thermionic emission,^90 this difference being attributed to the influence of the factor \(A\) in equation (6).

Experiments with the photoelectric effect from thin films of metal on quartz have shown that photoelectrons can be ejected from depths of several atomic diameters within the metal.^91 The mean distance that an electron can travel in a metal without losing its ability to emerge beyond the surface is approximately \(2.7 \times 10^{-7}\ \text{cm}\) in platinum and \(5.0 \times 10^{-7}\ \text{cm}\) in gold, and the probability of traversing a given distance without losing the ability to escape decreases exponentially with distance.

Finally, mention should be made of the selective photoelectric effect,^92 which is observed in electropositive metals and usually occurs when the electric vector of the incident radiation has a component perpendicular to the surface of the metal. The selective photoeffect consists in an increased sensitivity, reaching a maximum at the very same frequency

\[ \nu = \frac{3\nu_0}{2}, \]

which above was regarded as the region of maximum photoelectric sensitivity. It was formerly thought that it was due to a mechanism different from that of the normal photoelectric effect. At present, however, it is believed that both effects are identical in nature,

but that the selective effect exists as a result of selective absorption in this spectral region and with light polarized in such a way that its electric vector is normal to the surface[^93]. This theory, however, cannot fully explain all the observations[^94].

1. The relation between the contact potential difference of thermionic and photoelectric emission is simple, but it is a very essential one. This relation is determined by the theoretical equation[^95]

\[ \varphi_1-\varphi_2=V_2-V_1+P_{12}, \tag{13} \]

where \(V_2 - V_1\) is the contact potential difference between metals 1 and 2, and \(P_{12}\) is the Peltier coefficient for their contact.

The magnitude \(P\) is negligibly small in comparison with the other terms. This expression has been confirmed, within the limits of experimental accuracy, for \(\varphi\) measured photoelectrically, thermionically[^97], and from the heat of condensation of electrons[^98]. Of all these methods, the thermionic one is the least accurate, since it is difficult to measure the contact potential difference between metals that are suitable for the simultaneous determination of thermionic constants.

In the case of metals with surfaces that are not atomically homogeneous, as, for example, metals with surface “patches” of adsorbed substance, the simple relation given above is not applicable for the following reasons. The property of the contact potential of a complex surface, measured by any standard method, is simply an average quantity, in which patches of different nature take part in proportion to their area. The photoelectric work function, when measured, as usual, from the long-wavelength threshold, is a characteristic of the most electropositive region of the surface. The thermionic work function, measured by studying emission as a function of temperature, is in turn an average quantity, but one in which the statistical weight of the electropositive portions of the area is considerably greater,

rather than electropositive ones. These differences are illustrated quite sharply if, as a simple example, we consider one square centimeter of the surface of a platinum plate on which there is a potassium spot one square millimeter in size. The latter area determines the thermionic and photoelectric work function of the surface, but its influence on the measured value of the contact potential amounts to only half a percent. Similar differences have often been observed in work with activated emitting surfaces, but their systematic investigation has only just begun^99.

Recent experiments and the considerations connected with them by Nottingham^99a appear very important in this connection. Nottingham determined the maximum wavelength at which photoelectric emission can still be detected for each of a number of applied fields (both accelerating and retarding) between the emitting and collecting electrodes. The emitting electrode was made of nickel, activated with a distilled film of an alkali metal. According to Einstein’s equation (12), the expected relation should be as indicated by the curve $AB$ in Fig. 2. Here $B$ is the point at which the field changes from retarding to accelerating, and $BC$ is the region of saturation current, in which the limiting frequency does not depend on the field (if one excludes the very small Schottky effect described in the following section B 5). $\nu'$ is the lowest frequency at which photoelectric emission can be detected in the presence of an applied potential difference $V$. Part of the curve has a slope

Fig. 2. Nottingham curve, showing the relation between the “effective” lower frequency limit $\nu'$ and the applied potential difference $V$. The curve $ABC$ shows the relation on the basis of Einstein’s equation and for homogeneous metallic surfaces. The curve $ABD$ was found for activated surfaces. $\varphi'$ is the “effective” work function.

at 45°, if \(\frac{hv'}{e}\) and \(V\) are expressed in one and the same units.

The curve \(ABC\) in fact represents the relations found with metals whose surface is homogeneous.

However, for activated surfaces the observed relations are represented by the curve \(ABD\). The limiting frequency is changed by the accelerating field; \(PP'\) represents

Fig. 3. \(F(x)\) is the surface force attracting an electron situated at a distance \(x\) from the surface of the metal; \(x_0\) is the distance at which this surface force is exactly equal and opposite to the force of the applied field.

the decrease of the effective work function caused by the accelerating field \(BP\). To illustrate the order of magnitude of this effect, the following example may be given: a field \(BP\) of 4 volts causes a decrease of the effective work function \(PP' = 2\) volts, but a further increase of the field to 700 volts causes a decrease of only a few tenths.

An analogous phenomenon is found at very small accelerating fields in thermionic emission from thoriated tungsten, as was reported by Reynolds\({}^{105}\) (see Fig. 3, Phys. Rev. 35, 164, 1930). There is every reason to suppose that a similar phenomenon is characteristic of any

emission from activated surfaces in weak accelerating fields.

There are several important consequences of this fact.

1. The values of the work function of activated surfaces (Table 4) all refer to an “effective” work function, determined for accelerating fields larger than \(P\) (Fig. 2) and extrapolated to zero field (as \(B'\)). Likewise, the photoelectric long-wave limit was, generally speaking, measured with considerable accelerating fields (see, for example, Millikan’s work). On the other hand, the long-wave limits calculated from Einstein’s equation or extrapolated from the region \(AB\) (Fig. 2) give values characteristic of the zero field, as in \(B\).

2. There is a theory of the change in the effective work function caused by an accelerating field. This theory satisfactorily explains experiments on emission from homogeneous surfaces, but it, as well as all its extensions that have been proposed, is entirely unsuitable for explaining the results obtained with activated surfaces. These theories are discussed in the next two sections (B 5, 6).

3. Electron emission in accelerating electric fields, strictly speaking, is never saturated, for there is always an increase of the current strength together with the field. For many purposes this effect is imperceptibly small, but in some cases it becomes very substantial. It makes it necessary to investigate just what we mean by saturation electron emission, as it is determined, for example, by equation (6). The influence of the accelerating field probably plays an essential role at the cathode in some forms of gas discharges. It was first discussed by Lenard \(^{100}\), but was presented in concrete form by Schottky \(^{101}\) and is usually known under the name of the Schottky effect. In its essential features the phenomenon consists in the following.

Each emerging electron \((-e)\) is attracted back by the metal with a force \(F(x)\), due to the induced positive charge on the surface of the metal, and ...

this force is approximately equal to the attraction of the “mirror image” of the charge \((+e)\). This force \(F(x)\) is exactly equal to \(\frac{e^2}{4x^2}\), if the electron is at a distance \(x\) that is large in comparison with the distance between neighboring atoms on the surface of the metal, for then the metallic surface acts as a conducting plane. But even at a distance of four or five interatomic distances this expression is valid as a rough approximation. On the other hand, it is obviously wholly inapplicable at distances of the order of atomic diameters and is replaced, for small distances, by the work function, which depends on the discrete atomic structure of the surface of metals and which approaches zero at the surface. The work function is expressed through this force by the formula

\[ e\varphi=\int_0^\infty F(x)\,dx. \tag{14} \]

Obviously \(F(x)\) cannot simply be the “image force \(\frac{e^2}{4x^2}\)” throughout the entire region from \(0\) to \(\infty\), since in that case the integral would be infinite. If we make the natural assumption that \(F(x)=0\) at \(x=0\), and that for small displacements \(F(x)\) increases proportionally to \(x\), then the simplest graphical representation of \(F(x)\) will be a parabola rising from \(F(x)=0\) at \(x=0\) to a maximum, and then, descending, merging with the image-force curve at \(x=x'\), as shown in Fig. 2b. Using such a law of force, Langmuir\(^{102}\) showed that equation (14) leads to

\[ \varphi=\frac{e}{2x'} \]

and that \(x'\) is of the order of an atomic radius. The values of \(x'\) in Ångström units are as follows: W 1.58; Pt 1.12; Mo 1.62; To 1.75; Th—2.13; Cs—3.95. These quantities, roughly speaking, vary as the cube roots of the atomic volumes and are approximately equal to the atomic radii.

Direct experimental data concerning the nature of \(F(x)\) have been obtained by investigating the influence of accelerating fields on electron emission. In the presence of an accelerating field \(E\) there exists a critical

the distance \(x_0\) from the surface at which the resultant force becomes zero. Within this distance the surface attractive force \(F(x)\) predominates, while outside it this force is smaller than the applied force \(Ee\). This critical distance is determined from the relation

\[ F(x)_{x=x_0}=Ee. \tag{15} \]

For all fields, with the exception of the most intense, it turns out that this critical distance \(x_0\) is sufficiently large that the approximation \(F(x)=\dfrac{e^2}{4x^2}\) is admissible, and we may use the relation \(\dfrac{e^2}{4x_0^2}\), whence

\[ x_0=\frac{1}{2}\left(\frac{e}{F}\right)^{\frac{1}{2}}. \tag{16} \]

The limits within which this approximation is justified may be estimated by taking the extreme case \(E=1\,000\,000\) volts/cm. In this case \(x_0=19\times 10^{-8}\) cm. The smallest distances between atoms of metals are as follows: Th \(3.54\times 10^{-8}\) cm \(10^3\); \(2.73\times 10^{-8}\) cm, K \(4.50\times 10^{-8}\) cm, etc. Thus, in this extreme case \(x_0\) is approximately six times greater than the interatomic distance. For \(E=10\,000\) volts/cm the approximation should be very good, for \(x_0\) will exceed the interatomic distance by approximately a factor of 60.

The applied accelerating field reduces in two ways the work which the electron must perform in order to leave the surface: 1) the integral of equation (14) must be taken only up to the distance \(x_0\), and not to infinity, since beyond \(x_0\) it cannot be evaluated, and 2) the force that must be overcome on the path from zero to \(x_0\) will be not \(F(x)\), but \(F(x)-Ee\). Calling \(\varphi'\) the “effective” work function, i.e. the true work that must be performed by the electron in order to be freed from the metal, we obtain

\[ e\varphi'=\int_0^{x_0}[F(x)-Ee]\,dx =\int_0^\infty F(x)\,dx-\int_{x_0}^\infty F(x)\,dx- \]

\[ -\int_0^{x_0}Ee\,dx; \]

the first integral is simply \(eq\), and in the second \(F(x)=\dfrac{e^2}{4x^2}\) within the limits from \(x_0\) to \(\infty\). Thus, introducing the value \(x_0\) from equation (16), we have

\[ \varphi'=\varphi-(Ee)^{-\frac12}. \tag{17} \]

The true emission—whether photoelectric or thermionic—is determined by the “effective” work function \(\varphi'\) to a greater degree than by the true work function \(\varphi\). Thermionic emission makes it easier to carry out a test of this Schottky effect, since it is relatively easy to obtain large fields on the surface of a filament of small diameter. Equation (6) in this case takes the form

\[ j=AT^2 e^{-\frac{e}{kT}\left[\left(\varphi_0-xe\right)^{\frac12}\right]} = AT^2 e^{-\frac{e\varphi}{kT}}\cdot e^{\frac{e(Ee)^{\frac12}}{kT}} = j_s e^{\frac{e^{\frac32}E^{\frac12}}{kT}} \tag{18} \]

or, if \(E\) is expressed in volts/cm, then

\[ j=j_s e^{\frac{4{,}389\,E^{\frac12}}{T}}. \tag{19} \]

This equation can easily be verified if the current is small and gas ionization is absent (in order to avoid distortion of the field by space charge,—see Part II), for then \(E\) is directly proportional to the applied potential difference, and the proportionality coefficient can be calculated on the basis of the laws of electrostatics and depends only on the geometry of the electrodes. Grouping this constant and the quantity \(e^{3/2}/kT\) into one “shape factor” \(S\), we have:

\[ j=j_s e^{SV^{\frac12}};\quad \text{therefore, }\lg j-\lg j_s=SV^{\frac12} \tag{20}. \]

The graph of \(\lg j\) against \(V^{\frac12}\) should therefore be a straight line, whose slope is computed from the geometry of the electrodes and whose intercept \(\lg j_s\) at \(V=0\) serves to determine the “saturation current.”

Experimental checks of equation (20) at the same time make it possible to clarify the surface conditions which strongly affect electron emission. With a clean metallic whisker, equation (20) is exactly confirmed over the whole range from two or three volts (below this potential, the limitation by space charge of the initial emission velocities and currents may serve as the cause of deviations) up to the most intense fields \((10^6\ \text{volts/cm})\) that were used in the study of thermionic emission \(^{104}\). Generally speaking, the shape factor \(S\) is from 5 to 100% greater than the value calculated from the geometry, but this deviation, as it turns out, is caused by microscopic surface roughnesses, which concentrate the field at protruding points. Abnormally large values of the slope \(S\) were found for wires that had been bombarded with positive ions, which indicates the appearance of “ripples” on surfaces subjected to impacts by ions of high velocity. For a tungsten wire with a clean surface and a fine-crystalline structure, annealed at high temperature in order to smooth surface inhomogeneities, equation (20) is satisfied at the theoretical value of the shape factor \(S\) with the accuracy with which the measurements can be made. This confirms that the surface force \(F(x)\) is identical with the force of the electrostatic image \(\dfrac{e^2}{4x^2}\) at distances greater than the minimum value \(x_0\) attained in these experiments.

Even if, contrary to all the data, the surface force is not the image force, the method indicated above can be applied to the investigation of the surface force, provided it is permissible to assume that the force is the same in a plane parallel to the surface. Writing equation (6) in the most general form indicated by equations (14) and (17):

\[ j = A T^2 e^{-\frac{1}{kT}\int_{0}^{x_0} [F(x) - (Xe)]\,dx} \tag{20} \]

and differentiating with respect to \(E\), we obtain

\[ \frac{d\lg i}{dE}=\frac{e x_0}{kT}. \tag{21} \]

Thus the rate of change of the logarithm of the thermionic current with change in the accelerating field gives the distance \(x_0\) from the surface at which the surface force \(F(x)\) is equal to the applied force. If fields over a wide range are used, then in this way one can find the surface force over a wide range of distances from the surface.

In general, emission from activated surfaces, such as the surfaces of cesium or thorium on tungsten or oxidized Wehnelt, does not obey the simple Schottky equation (19), but exhibits considerable discrepancies, especially at small fields. Moreover, these discrepancies are greater for surfaces only partly covered with the activating substance than for surfaces that are either completely clean or completely covered. Equation (21) was applied to the investigation of the peculiar surface force for such surfaces, as will be described below. It may further be noted that the degree of approximation to the Schottky equation (19) depends strongly on the previous history of the surface in the sense of the method of activation, ion bombardment, etc. \(^{105}\)

The only existing theory of these deviations from the Schottky equation is based on Langmuir’s assumption \(^{106}\), according to which electropositive atoms on activated surfaces give rise to electric forces near the surface; these forces vary from point to point and exert different effects on the removal of an electron from different regions, thereby causing imperfect saturation. The electropositive atoms thus form something like a positively charged “grid” that affects the emission. These electropositive atoms may act in groups or “patches,” which may be electropositive with respect to the surrounding pure metal,—in this form this theory was used by Richardson and Young \(^{107}\) and by Reynoldson \(^{105}\), or

they can act individually, as charges situated just beyond the surface, thus forming, together with their mirror image, a double-layer field in the form of the theory developed by Becker \(^{108}\). We shall consider these theories in somewhat greater detail, both because the phenomenon itself is of interest and because the analysis given below shows that none of the existing theories can explain the observed deviations from Schottky’s equation.

The theory of spots was proposed by Langmuir \(^{106}\) to explain the absence of a definite saturation, which is observed with filaments whose surface is only partly covered by an adsorbed film, for example, a film of thorium. At \(1500^\circ K\) the electron emission from the surface of tungsten completely covered with a layer of thorium atoms is approximately \(126\,000\) times greater than from the surface of pure tungsten. If half the surface of tungsten is covered with such a thorium film, while the other half remains clean, then the total emission will be approximately \(63\,000\) times greater than from pure tungsten. If, however, the thorium is uniformly distributed over the surface of the filament, then the work function will be a linear function of the amount of thorium on the surface and, consequently, according to equation (7.5), the electron emission will be

\(365\) times greater than the emission of pure tungsten \((126\,000^{\frac{1}{2}})\).

In the first case we have an emission corresponding to the arithmetic mean, whereas in the second we have the geometric mean. In both cases the current must increase with the applied accelerating field according to Schottky’s equation (20). Thus, with an amount of thorium sufficient to cover half the surface, the line \(AB\) in Fig. 4 represents the emission as a function of the field for a uniform distribution, while the parallel line \(CD\) represents currents \(356\) times larger when the thorium gathers into large spots. If the linear dimensions \(b\) of the spots of active substance on the cathode are sufficiently small in comparison with \(x_0\), the distance to the critical surface indicated by equation (16),

then the distribution must be effectively homogeneous and the current will be represented by \(AB\), but if \(b \gg x_0\), then the current will be represented by \(CD\). When the field increases, \(x_0\) decreases, and thus one may expect a transition from \(AB\) to \(CD\), illustrated by the line \(AGHD\), so that the region \(GH\) is that in which the Schottky effect will be abnormally large.

Fig. 4. Effect of the sizes of a “spot” on electron emission.

A quantitative estimate of the field strength which is necessary in order to make the electronic emission depart noticeably from the Schottky line \(AB\), for example at \(G\), can be made if one considers the following simplified case. Let the surface consist of spots of two kinds, between which there is a contact potential difference \(V_0\), and let these spots be distributed in the form of squares, alternating like a chessboard, each square having side length \(b\). If the potential of the more negative squares is \(0\) and there is no applied field, then the distribution of the potential in space near the chessboard is given by the following expression:

\[ V=\frac{1}{2}V_0+\frac{4V_0}{\pi^2}\sum_{n=1,2,3}\left[\frac{1}{n^2}e^{-\frac{n\,2^{1/2}\pi z}{b}}\cos\frac{n\pi x}{b}\cos\frac{n\pi y}{b}\right], \]

where \(z\) is the distance from the surface, and \(x\) and \(y\) are distances measured from the center of one of the more electropositive squares in directions parallel to the sides of the squares.

In order to determine the motion of an electron, we

must add this potential \(V\) to the potential \(E\) and to the fictitious potential \(\frac{e}{z}\), caused by the electric image force. For a scalar quantity whose gradient, in any direction and at any point, represents the component, calculated per unit charge, of the force acting on an electron or ion, the name “motive”\({}^{109}\) has been proposed. Thus \(M\) will be expressed by the formula

\[ M=M_{\infty}+\frac{e}{4z}+Ez+V, \]

where \(M_{\infty}\) is a negative quantity, numerically equal to the work function for zero field and expressed in volts. Thus for tungsten \(M_{\infty}=-4.52\) volts.

Differentiating \(M\) with respect to \(z\) and equating to zero, we obtain an equation which, when solved with respect to \(z\), gives the distance \(z_M\) where the “motive” becomes a minimum, i.e. gives the position of the critical surface. This distance \(z_M\) is a function of \(x\) and \(y\), but if \(V_0\) is set equal to zero, then \(z_M\) gives the same value \(x_0\) as is given by equation (16). The value \(z_M\) corresponding to \(V_0=0\) we shall call \(z_0\), and we may put

\[ z_M=(1+\lambda)z_0, \tag{22} \]

where \(\lambda\) is a small quantity if \(z_0\gg b\). The equation \(\frac{dM}{dz}=0\) can in this case be represented in the form

\[ 1-(1+\lambda)^{-2}=\alpha \varepsilon^{-\beta}, \]

\[ \alpha=\alpha_0\cos\left(\frac{\pi x}{b}\right)\cos\left(\frac{\pi y}{b}\right) \tag{23} \]

\[ \alpha_0=\frac{4\cdot 2^{1/2}V_0}{\pi Eb}\varepsilon^{-\beta} \tag{24} \]

\[ \beta=\frac{\pi}{b}\left(\frac{e}{2E}\right)^{1/2}=\frac{4.44\,z_0}{b}. \tag{25} \]

This expression can be expanded in a power series in \(\lambda\), and then, by inversion, we obtain the following series for \(\lambda\), expressed in terms of \(\alpha\),

\[ \lambda=\frac{1}{2}\alpha+\frac{\alpha^2}{8}(3-2\beta)+\frac{\alpha^3}{16}(5-9\beta+3\beta^2). \]

This equation thus allows us, with the aid of equation (22), to determine the distance to the critical surface of the minimum “image” as a function of \(x\), \(y\), and \(V_0\). The “image” \(M_\mu\) on the critical surface may also be expressed in terms of \(\lambda\), and therefore also in terms of \(\alpha\). The final expression for the change of the “image” on the critical surface, due to \(E\) becoming different from zero, will be the following:

\[ \Delta M_\mu=(eE)^{\frac12} \left[ 1+\frac{\alpha}{2\beta}-\frac{\alpha^2}{8} -\frac{\alpha^3}{16}(1-\beta) \right]. \]

The total electron emission can now be calculated* from equation (6) by reducing the quantity \(b\) by \(\dfrac{e}{KT}\Delta M\) and averaging the resulting current density over the whole surface of the cathode. In this case the terms containing even powers of \(\dfrac{e}{KT}\Delta M\) drop out, and the total current density from the surface may be found from the expression:

\[ j=j_n\left[ 1+\frac{\alpha_0^2}{32} \left( \frac{1}{\beta^2}-1 \right) \right], \tag{26} \]

where \(j_n\) is the current density corresponding to the normal Schottky effect, expressed, for example, by equation (19).

For convenience the quantity \(\alpha_0\) may be calculated by eliminating \(E\) from equations (24), (25). Then

\[ \alpha_0=2.55\times 10^6\, V_0\beta^2 b e^{-\beta}, \tag{27} \]

where \(V_0\) is expressed in volts.

As an example illustrating the use of these equations, we may consider the case of a tungsten filament having radius \(0.01\ \mathrm{cm}\) inside a cylinder of radius \(1\ \mathrm{cm}\). For the cylinder, at three different positive voltages \(V_a\) relative to the filament (the first line

* It is assumed here that the electrons which reach the critical surface from the cathode have a Maxwellian distribution of velocities. This would be true independently of the distribution of the image in whose vicinity they are located, if the critical surface were approximately plane, i.e. \(\lambda \ll 1\) (see Mott, Smith and Langmuir, Phys. Rev. 28, 754–758, 1826, especially the theorem on p. 756).

table VI), the accelerating field voltage on the filament is given in line 2. Line 3 indicates the normal increase of electron emission according to Schottky’s theory, expressed by equation (19). The normal distance \(z_0\) of the critical surface from the cathode is given in line 4.

Table VI

Influence of local fields, caused by a checkerboard arrangement of thorium spots on tungsten, on the absence of saturation. Size of the square
\(b = 10^{-6}\ \mathrm{cm}\); contact potential difference \(V = 1.9\) volts

1 \(V_a\) (volts) . . .	0	100	250	500
2 \(E\) volts/cm\(^{-1}\) . .	0	2170	5430	10850
3 \(j_n/j_s\) . . . . .	1.000	1.146	1.242	1.357
4 \(Z_0\) (\(10^{-6}\ \mathrm{cm}\)) . .		4.0	2.6	1.8
5 \(\beta\) . . . . . .		18.0	11.4	8.0
6 \(\alpha_0\) . . . . .	0	2.10	0.007	0.10
7 \((j-j_n)/j_n\) . . .	0	−10	−10	−3.10
8 \(\lambda_m\) . . . . .	0	−10	0.003	0.05

Let us now suppose that the surface of the filament is half covered, in a checkerboard pattern, with square spots of closely packed thorium atoms, the side of each square being \(10^{-6}\ \mathrm{cm}\). Since a tungsten surface completely covered with thorium contains approximately \(7 \times 10^{14}\) thorium atoms per square centimeter\(^{110}\), each spot contains 700 atoms. The quantities \(\beta\) and \(\alpha_0\), given by equations (25) and (27), are presented in lines 5 and 6, while line 7 gives the change in current relative to \(j_n\), caused by the local fields of the spots, calculated from equation (26).

Thus, even with such a highly nonuniform distribution of thorium atoms as we have assumed, with clusters of 700 atoms, local fields cannot cause changes in emission that would be at all comparable with the emission due to the normal Schottky effect.

With spots of somewhat larger size, containing about 7000 atoms and occupying squares with side
\(3 \times 10^{-6}\ \mathrm{cm}^2\), the quantities \(\dfrac{j-j_n}{j_n}\), calculated from the equation

(26), become much more significant and opposite in sign, so that the Schottky effect increases, as shown by the curve \(GH\) in Fig. 4, but in such a case \(\frac{1}{\beta}\) and \(\beta\lambda\) become so large that the approximations made in deriving these equations become no longer applicable. It is, however, highly improbable that such large accumulations of adsorbed atoms could exist on plane surfaces. Becker\(^{108}\) showed, for example, that a thorium layer at emission temperatures behaves like a two-dimensional gas on the surface.

A random distribution of adsorbed atoms also gives appreciable irregularities in the emission even from large surfaces. For example, if the average number of atoms in identical square spots (each of area \(b^2\)) is \(q\), then the mean difference between the numbers of atoms in any two squares chosen at random will be \((2q)^{\frac{1}{2}}\), so that the mean contact potential difference between neighboring squares will be \(5.2 \times 10^{-8} V_0 b\), where \(V_0\) corresponds to the difference between completely covered and uncovered surfaces. Thus, if we put \(b = 4 \times 10^{-6}\) (approximately equal to \(z_0\)) and \(V_0 = 1.9\), we obtain a potential difference of only 0.025 volt. For a filament heated to \(1500^\circ\), this should produce a difference of 20% in the normal emission of the two surfaces. In this case the arithmetic mean and the geometric mean differ by only 0.5%, so that an abnormally large Schottky effect cannot be observed.

Finally, if there were attractions between the thorium atoms, so that they formed clusters of \(n\) atoms each, then these clusters, being distributed at random, would produce an average contact potential difference between neighboring squares of area \(b^2\), equal to

\[ \Delta V = 5.2 \times 10^{-8}\,\frac{V_0 \cdot n^{\frac{1}{2}}}{b}. \]

In this case the difference between the arithmetic and geo-

of the arithmetic mean, which is proportional to \((\Delta V)^2\), will be 50% instead of 0.5% at \(n=100\).

Langmuir, Richardson, and Becker experimentally observed, with filaments whose surface was partially covered with an adsorbed film, a case in which the change of emission with voltage was far more significant than the change depending on which surface was involved—clean or completely covered.

Kingdon and Langmuir, in certain unpublished works in 1923, and later Reynolds \(^{105}\), measured the emission as a function of voltage and temperature for thoriated filaments of different activities. They usually had good vacuum conditions, so that the metallic parts were completely degassed and the bulbs were pumped down to the limit and immersed in liquid air. The filaments were arranged along the axis of a cylinder provided with guard ends, so that the field strength was accurately known.

Fig. 5. Electron emission from a heated thoriated filament as a function of the field. The figure shows the approach to the Schottky curve \(A_1B\) above 10,000 volts/cm for a uniformly activated filament and a different relation \(C\) for a filament that had been subjected to bombardment by positive ions (Reynolds).

Reynolds, working with thoriated filaments that had been annealed at very high temperatures and then activated, found that the change in the dependence on voltage agrees excellently with Schottky’s theory for fields with strengths exceeding approximately 10,000 volts/cm, as is also evident from curves \(A\) and \(B\) in Fig. 5 for two different degrees of activation. If, however, the filament

was well activated and then slightly deactivated by bombardment with positive ions; moreover, deactivation proceeds slowly when the liquid air is removed and a high voltage is applied, so that the emission does not coincide with the Schottky curve above 10,000 volts/cm, as shown by curve C in Fig. 5. This bombardment can produce roughness of the surface and, consequently, destroy the surface layer of tungsten metals. Subsequent annealing at high temperature, accompanied by activation, usually brings the filament to the conditions represented by curves A and B.

Fig. 6. Electron emission from a thoriated filament as a function of the activity θ and of the field at various temperatures (Kingdon and Langmuir).

The results of Kingdon and Langmuir were as follows. A thoriated filament was annealed for one minute at 2740° and activated for 25 minutes at 2120°. It was free of carbon and therefore could be completely deactivated by heating for 10 sec at a temperature of 2600°K; partial deactivation was achieved in four stages by heating for 30 or 60 sec at 2300° and 2400°. At each stage the emission was measured at 5 voltages and approximately 6 different temperatures. Fig. 5 gives several typical curves for \(T \log_{10} j\) as a function of

...of \(\sqrt{E}\), where \(E\) is expressed in volts per cm. According to equation (19), the lines obtained should be straight and should all have a slope of 1.906. At the highest field strengths (about 10,000 volts/cm), as can be seen, the curves approach the theoretical slope. In Fig. 6 they are displaced in the vertical direction by various amounts \(\Delta\) in order to show their asymptotic approach to the Schottky line.

These results show that, with an almost completely thorated surface \((\theta = 0.91)\) and with a clean surface \((\theta = 0.00)\), the approach to the Schottky curve* is very considerable, but relatively larger deviations occur with incomplete thorating. In any case, the deviations from the Schottky line become larger when the temperature is lowered, whence it follows that these deviations are not caused by space charge.

Table VII

Deviations \((\eta)\) from the Schottky law at fields of 1840 volts/cm

| \(\theta\) | \multicolumn{2}{c}{\(T = 1500^\circ\)} | \(\eta\) at the temperature giving \(j = 17 \times 10^{-6}\) amp. cm\(^{-2}\) |
|---:|---:|---:|---:|
| | \(\eta\) | \(j\) (amp. cm\(^{-2}\)) | |
| 0.91 | 0.81 | \(9200 \times 10^{-6}\) | 0.69 |
| 0.76 | 0.64 | \(1800 \times 10^{-6}\) | 0.46 |
| 0.60 | 0.59 | \(765 \times 10^{-6}\) | 0.44 |
| 0.24 | 0.72 | 3.75 | 0.77 |
| 0.00 | 0.80 | 0.23 | 0.89 |

Table VII gives the values of \(\eta\), the ratio of the observed emission at a field of 1840 volts/cm (50 volts on the anode) to that calculated from the Schottky curve by extrapolation to \(E = 9200\), as shown in Fig. 6. It is seen from the table that

* A close approach to the Schottky curve, and moreover over a considerably wider range of fields, was obtained by Reynolds with clean wires. The small deviations in the present case are apparently due to the influence of slight surface roughness on the shape factor of equation (20).

the greatest deviation is obtained when the tungsten is coated with thorium by approximately 70%.

If tungsten filaments are heated at a very high temperature, evaporation occurs, as a result of which the smooth surface is destroyed and dodecahedral faces appear. The latter form angles of \(120^\circ\) with one another. Sometimes the edges thus formed are sufficiently large that they can be seen under a microscope. The effect of such edges and depressions on the change of emission with voltage may be approximately calculated from the theory that we developed for the case of patches.

If we consider a flat surface level with edges protruding in the form of peaks, then the potential at points of the surface above the centers of the depressions may be roughly estimated as \(Fh\), where \(h\) is the effective height of the edge above the depression. We may therefore use equations (26) and (27), simply substituting \(V_0 = Fh\). The increase of \(V_0\) with increasing \(F\) in such a case causes deviations from the normal Schottky effect, which increase considerably more rapidly with increasing voltage than in the case of patches on a flat surface. The influence of such geometrical irregularities on the surface apparently also gives a curve of the type \(AGHD\) (Fig. 4).

The final result of this discussion is that the theory of “patches” predicts variations of emission with the field which, although they occur in the required direction, nevertheless, upon more detailed consideration, prove to be completely inconsistent with the observed changes, as is shown in Figs. 5 and 6. There are two difficulties here. First of all, in order to obtain deviations from the Schottky curve comparable with those observed, the patches must contain several thousand atoms. Secondly, the theory of “patches” predicts deviations from the Schottky curve which should be small at small fields and increase at large fields, whereas in reality the opposite is found.

The reason for this inconsistency is not difficult to see. From Eq. (21) it can be seen that the rate of increase of the current

with the field depends only on the conditions at the critical \(x_0\) (or \(z_0\)), where the internal surface forces are exactly balanced by the applied field. For weak fields \(x_0\) is large, approaching infinity for a zero field. Evidently, local inhomogeneities of the field caused by spots or adsorbed ions, which may be substantial near the surface, are smoothed out and become imperceptible at large distances. Thus any theory based on such surface inhomogeneities must prove untenable in explaining the observed relatively large deviations from the Schottky equation at small fields, and must predict large deviations in strong fields, which again contradicts the facts. In order to obtain noticeable deviations from the Schottky equation, these theories require that the scale of the surface inhomogeneities be comparable with the critical distance \(x_0\), i.e. they require that the spots contain thousands of atoms. Even if this were true (which is very unlikely), we would still be faced with the principal difficulty of explaining the fact that deviations from the Schottky equation at weak fields are especially sharply expressed in the case of activated surfaces.

We now turn to that form of the “lattice theory” of activated surfaces which was investigated and extended in an interesting way by Becker \(^{108}\), and which we may call the “adion-field theory,” borrowing from Becker the term “adion” as an abbreviated designation of an “adsorbed ion.” This theory aims to explain three features of activated surfaces: 1) a small work function, 2) the absence of saturation in emission, and 3) the electropositive character of the surface. The explanation reduces to the following.

Consider two identical electrodes \(C\) and \(A\) (Fig. 7a), separated from one another by a distance \(d\), and suppose that the cathode \(C\) is more or less covered by a surface layer of positive ions at a distance \(d_\alpha\). We may assume that the electrodes are made of tungsten, while the ions belong to some activating substance,

e.g., barium (\(\mathrm{Ba}^{+}\)). The fact that barium evaporates in the form of ions serves as evidence of its existence in the form of ions on the surface, since the surface forces that tend to ionize it probably have their greatest magnitude at the very surface or in its immediate vicinity. The influence of this layer of “adions” on the field between the plates can, in a first approximation, be taken into account as the influence of a layer of positive charge with surface density \(\sigma = Ne\), where \(N\) is the mean number of adions per \(\mathrm{cm}^{2}\). The field produced by this layer is uniform and very

Fig. 7. Adion theory of activated surfaces.

close to \(E_e = 4\pi Ne\) on the cathode side and \(4\pi Ne\,\dfrac{d_c}{d_a}\) on the anode side. The rise of potential from the cathode to the layer will be \(4\pi Ne\,d_c\), and is equal to the potential drop over the distance from the layer to the anode.

If the electrodes were connected by a wire (\(V=0\)), then the layer of adions would have no effect on the emission from the cathode, since the total work performed by an electron in passing from \(C\) to \(A\) would be exactly the same as if the adion layer were absent. However, the field \(E_a\) is interpreted by the observer as the contact potential difference between \(C\) and \(A\), and he compensates it with an equal and opposite field \(V\), applied externally, in order to obtain what he considers to be the true saturation emission from \(C\).

at zero field (this agrees exactly with observations and with what is in fact done experimentally in the case of emission between heterogeneous electrodes). Under such circumstances, the work which an electron must perform in order to be liberated will be diminished by this apparent contact potential difference \(V = E_a d_a = E_c d_c\), in comparison with that observed in the absence of an adion layer. Therefore the emission will be greater in accordance with this decrease in the effective work function.

These considerations are represented graphically in Fig. 7 b, where \(F(x)\) represents the surface force at various distances from the cathode, and the area bounded by it is equal to the work function of the normal surface of the cathode. The layer of adions creates an additional force \(E_c\) within the distance \(d_c\) and a force \(E_a\)—in the opposite direction—outside this distance. These forces are added to \(\varphi_1\), the energy of removal of the electron within the distance \(d_c\), and are subtracted, giving the quantity \(\varphi_2\), outside this distance. The total work performed in removal thus remains unchanged, since it is equal to \(\varphi - \varphi_1 + \varphi_2 = \varphi\). If, however, the external field \(E_a\) is compensated by an applied external field, in order to correct the contact potential difference, then the effective work function is decreased by the difference between the areas \(\varphi' = \varphi - \varphi_1\). Thus this theory gives a plausible interpretation of the influence of adions on the work function and on the contact potential difference.

Up to now the theory still has not been able to explain the deviations of emission in an accelerating field from the Schottky equation (19). In order to attempt to do this, we pass to the second approximation and consider the layer of adions no longer as a homogeneous positive layer, but as discrete positive charges, each of which, together with its negative mirror image in the metal, forms an electric doublet. Electrons leaving from different points pass through fields of different intensities, extending into the region outside the adion layer. Thus the effective work function changes above the surfac—

...and the nature of the force \(F\) changes with \(x\) in a different way than in the case of a clean or uniformly covered surface. Qualitatively, this is precisely what is needed to explain the deviations from Schottky’s equation. However, a quantitative investigation shows that this theory, like the patch theory, is unable to explain the fact that deviations from Schottky’s equation increase as the fields are decreased. In Fig. 7b, for example, such doublet fields would have importance at distances \(d_c\) or \(2d_c\) from the surface, but not at the relatively large distances from the surface \(X_0\), which according to equation (21) determine the change of the current with field for all fields ordinarily obtained.

Dr. Becker has kindly informed us that he intends shortly to publish in Physical Review an analysis of the “addition theory,” extended to the case of activated rough surfaces, as distinct from the infinite plane surface considered above. Becker indicates that this analysis leads to results in significantly better agreement with the facts. A “rough surface” is treated as a surface composed of crystal faces, the linear dimensions of which may be of the order of the critical distance \(x_0\) for weak fields.

In connection with this theory there arises a further interesting question: to what extent is the specific emission capacity of an adsorbed substance preserved in a monatomic layer? A layer of cesium 10 or 5 atoms thick on the surface of tungsten undoubtedly possesses the emission capacity characteristic of cesium. The question arises to what extent this specific property disappears for a monatomic layer, being replaced by the electrostatic effect of the ion charges.

In conclusion, one should note one assumption underlying both forms of the “lattice” theory: the explanation of the deviations from Schottky’s theory in the case of activated surfaces is ascribed to the influence of the adsorbed layers on the work function, and no account is taken of...

possible peculiarities of the constant \(A\) in the thermionic equation (6). For example, equation (21), which was used by Becker and Müller \({}^{108}\) to compute the character of the surface force \(F(x)\), is based on the assumption that \(A\) is a constant. However, Fowler \({}^{111}\) has recently shown that \(A\), generally speaking, is not a constant and that, in particular, in the case of activated surfaces a noticeable dependence of \(A\) on temperature is possible. We shall see in the following section that strong electric fields can produce electron emission (alongside thermionic emission) in exactly the same way as high temperature produces thermionic emission. It is therefore possible that activated surfaces are characterized by such a dependence of \(A\) on the field, which may explain the abnormally weak emission in weak fields.

1. Currents under the influence of strong fields. According to experimental data, Schottky’s theory (equation 18) becomes more accurate at large field intensities. From equation (18) it is evident that when the field increases to a value determined by the condition

\[ Ee=\varphi_0^2 \qquad \left(\text{whence } E_m=\frac{\varphi_0^2}{e}\right), \tag{28} \]

the work function and the cooling effect must disappear, and the current density must increase to the limiting value

\[ I(\max)=AT^2. \tag{29} \]

We can measure the rates of increase of the current \(I\) for a given relative increase of \(T\) or \(E\) by the exponents \(n_T\) or \(n_E\), defined as follows:

\[ n_T=\frac{d(\log j)}{d(\log T)} \]
\[ n_E=\frac{d(\log j)}{d(\log E)}. \tag{30} \]

Carrying out logarithmic differentiation of equation (18) with respect to \(T\) or \(E\), we obtain

\[ n_T=2+\frac{e}{kT}\left[\varphi_0-(eE)^{\frac12}\right] =2+\frac{b-4.389E^{\frac12}}{T}. \tag{31} \]

and

\[ n_E=\frac{1}{2}\frac{e}{kT}(Ee)^{\frac{1}{2}}; \tag{32} \]

for tungsten \(b=52\,600\). At \(T=1500\), according to equation (31), \(n_T\) decreases from 37 when \(k=0\) to \(n_T=2\), when \(E\) approaches the limiting value \(E_M\), given by equation (28), which for convenience may be represented in the form

\[ E_M=0.0519\,b^2 . \tag{33} \]

For tungsten the limiting field will therefore be

\[ E_M=1.44\times 10^8\ \text{volts cm}^{-1} \tag{34} \]

At this field the current density according to equation (29), with \(A=60.2\), will be \(1.36\times 10^8\ \text{amp cm}^{-2}\).

According to equation (18), even at room temperature (\(T=300^\circ\)) it is possible to obtain currents of \(5.4\times 10^7\ \text{amp/cm}^{-2}\), increasing the field up to the indicated maximum voltage.

The temperature coefficient of the electron current which would thus be caused from the metal by an intense field proves to be negligibly small in comparison with the temperature coefficients of the usual characteristics of electron emission[^112].

Equation (32) shows that the relative increase of the current strength with the field voltage increases as the field is increased until finally (when \(E\) rises to \(E_M\)) \(n_E=b/2\). Thus, for tungsten at room temperature the maximum value of \(n_E\) will be 88. When the field approaches the value \(E_M\), an increase of the field by 1% causes \(j\) to increase by a factor of 2.4. When \(E\) has a value equal to 90% of its final value, i.e., when \(E=0.9E_M\), the emission amounts to only about \(10^{-4}\) of the value which it reaches at \(E=E_M\).

If there are small geometrical irregularities, such as, for example, edges or corners of crystals on the surface of the cathode, then at the tops of these elevations the field will be considerably stronger. In regions where the field is only 10% greater, the current density may be so much greater that

practically all the emission will occur from these points.

In the strong fields necessary for tearing electrons out of cold metals, the ponderomotive force, due to the electrostatic field, can be very considerable at the surface. This force is equal to \(4.4 \times 10^{-7} E^2\) dyn. \(\mathrm{cm}^{-2}\), when \(E\) is expressed in volts per cm. Thus, when \(E = E_M\), for tungsten the force at the surface is equivalent to a negative pressure of approximately 9000 atmospheres. This pressure, however, lies considerably below the limiting stress for such a substance as tungsten, and therefore it cannot destroy the surface, except in those cases when the temperature is very high or when there are small pieces weakly bound to the main mass of the electrode.

Experimentally, phenomena connected with the ejection of electrons from cold metals have been observed and studied in great detail in recent years. R. Wood\(^{113}\) passed the discharge of an induction coil between platinum spheres 1.5 mm in diameter, placed at a distance of 1–5 mm from one another in a relatively good vacuum. In this case X-rays arose at the anode sphere, whence it follows that the voltage across the gap was very high. Here we evidently have an example of a cold discharge from the cathode. In these experiments a considerable transfer of platinum from the anode to the cathode was observed. This was probably due to mechanical destruction of the anode as a result of temperature fluctuations caused by the intermittent discharge, i.e., fluctuations of the same kind as arise at the anode of a Coolidge X-ray tube operating on alternating current. The mechanical force caused by high local fields throws the destroyed metal toward the cathode. When the poles are reversed, a whole stream of small sparks is ejected from the new anode, consisting of incandescent particles torn away from the anode.

F. Rother\(^{114}\) investigated the emission of electrons from the cathode in an intense field obtained when the surfaces of the cathode and anode were placed extremely close to one another, so that their separation was from \(10^{-6}\) to \(10^{-8}\) cm.

Thus it became possible to obtain large field strengths at such low potential differences that measurements could be carried out at atmospheric pressure. Lilienfeld, using high potential and high vacuum, applied electron currents extracted from point cathodes and, on this principle, constructed an X-ray tube. Millikan and Eyring[^115] quantitatively measured electron currents extracted from tungsten filaments at fields in the range from \(0.4\) to \(1.1 \times 10^6\) volts/cm and studied the effect of the temperature of the filament and of the conditions at its surface.

The past history of the surface has a very great influence on the field strength necessary to produce electron currents. A fresh surface, generally speaking, exhibits a large emission, which may change with time, so that the results at first prove irreproducible. If the surface has already been “conditioned” by passing currents at intense fields, then the emission at lower fields usually becomes reproducible. Heating the filament to \(2700^\circ\mathrm{K}\) changes the surface in such a way that the field required to obtain a given electron current from the filament at room temperature is considerably increased. These observers found that the temperature coefficient of electron emission at high fields is not only much smaller than at low fields, but obtained the unexpected result that the “field currents” are completely independent of the temperature of the filaments between \(300^\circ\mathrm{K}\) and \(900^\circ\mathrm{K}\), within the experimental error of about \(5\%\). However, at \(1100^\circ\) the field necessary to produce the given current (\(10^{-11}\) ampere) was \(30\%\) less than the field required at \(300^\circ\), if the condition of the filament was such that strong fields were required. If, however, the conditions were such that a large electron emission was obtained (fresh filament), then the critical field even at \(1100^\circ\) was the same as at \(300^\circ\).

In the following work Eyring, Mackeown, and Millikan[^116] studied field currents in high vacuum from points of tungsten, platinum, nickel, and steel. The points were made-

were made, as far as possible, in the form of hyperboloids of revolution and were placed close to a flat tungsten disk serving as the anode. The field strength on the surface of the point was calculated according to the indicated theory. Within the experimental error (2%) the field at the surface needed to produce the given current was independent of the distance between the point and the plane and thus was independent of the total voltage applied between these electrodes.

In all these experiments, as well as in the experiments published by the research laboratory of the General Electric Comp. in London[^117], the results satisfy the empirical equation[^118]

\[ j = A(T+cE^{2})e^{-\frac{b}{T+cE}}, \tag{35}* \]

according to which, for strong fields and low temperatures, the logarithm of the electron current is a linear function of the reciprocal field strength. This relation is in much better agreement with experiment than the relation based on Schottky’s theory, according to which \(\log I\) is a linear function of \(E^{\frac12}\). The fact that the temperature coefficient of field currents is practically equal to zero, in contradiction with the classical theory, which gives a finite value for it [equation (29)], indicates that the energies of the electrons do not increase in proportion to the absolute temperature, as they should according to the classical gas laws. This fact, however, apparently agrees with the modern quantum theory of the degenerate electron gas developed by Fermi, Dirac, Nordheim, and Sommerfeld[^119]. On the basis of the premises of this theory, a theory was developed for the present case, according to which field currents from a cold metal are expressed by the formula[^120]

\[ j = aE^{2}e^{-\frac{D}{E}}, \tag{36} \]

where \(a, D\) are constants of the metal. The data of Eyring, Mackeown, and Millikan[^116], obtained with a steel

* \(e\), as in the preceding formulas, is the base of natural logarithms.

with a point, give straight lines if one plots \(\log j\) as a function of \(\dfrac{1}{E}\); the slope of these straight lines gives, for \(D\), the value \(D=6.7\times 10^{6}\) volts cm\(^{-1}\). If we assume that the value of \(b\) for steel is 67,000, then this gives for \(c\) in equation (35) the value \(0.010\) degree cm volt\(^{-1}\).

By logarithmic differentiation of equation (36) we find, for the quantity \(n_E\) defined by equation (30), the value

\[ n_E = 2+\frac{D}{E}. \tag{37} \]

In the experiments described above with steel points, for which the mean field strength was \(E=0.33\times 10^{6}\), \(n_E=22\).

In some unpublished works of Coolidge and Langmuir, dating from 1922, a V-shaped tungsten filament of diameter \(0.0216\) cm was mounted inside a small glass bulb opposite the center of a molybdenum disk 3 cm in diameter, the distance being set with a micrometer. The molybdenum disk was heated to incandescence and the filament was treated at a temperature of about \(2700^\circ\) K. One side of the angle formed by the filament was then broken off, for which purpose the plate was brought into contact with the filament; after this there remained a straight piece of filament opposite the molybdenum disk.

The potential difference between the point and the plane was set until the field current reached \(10^{-5}\) ampere from the point at room temperature. The distance was varied, and in each case the voltage was changed until the current reached the same value. Fig. 7 shows the results obtained in this way. Here the abscissa axis gives the voltages, and the ordinate axis gives the logarithms of the distance between the end of the wire and the disk.

Maxwell\(^{121}\) calculated the distribution of potential between two coaxial, confocal paraboloids of revolution maintained at a given potential difference. The potential at any point \(P\) turned out to be a linear function of

\[ \log\left(r^{\frac{1}{2}}\cos\frac{\theta}{2}\right), \]

where \(r\) is the distance of the point \(P\)

from the focus \(O\) and \(\theta\) is the angle \(POT\), where the line \(OT\) lies on the axis of the paraboloid. The electric field normal to the surface of the inner paraboloid at its vertex is expressed as

\[ E=\frac{V}{p\ln \frac{c}{p}}, \]

where \(V\) is the potential difference between the two paraboloids, \(p\) is the distance from the focus to the vertex for the inner paraboloid, determined by the equation \(y^{2}=4px\), and \(c\) is the corresponding distance for the larger paraboloid. If \(c\) is considerably greater than \(p\), so that the curvature of the outer paraboloid is negligibly small in comparison with the inner one, then the inner paraboloid may be replaced, without appreciable error, by a plane surface. Thus we obtain a method for calculating the potential gradient around a point placed opposite a plane surface. In fact, this theory should be more applicable in the case of a point and a plane than the theory of confocal hyperboloids. If \(a\) is the radius of curvature at the point (at the vertex of the inner paraboloid), and \(c\) the distance between the point and the plane, then the equation takes the form

\[ E=\frac{2V}{a\ln \frac{2c}{a}} . \tag{38} \]

If the currents from the tungsten point, which are plotted in Fig. 8, depend primarily on the electric field at the surface of the point, then—since the current is kept constant—the field must likewise remain constant as the distance increases. According to equation (38), the logarithm of the distance \(c\) should be a linear function of the voltage \(V\). In fact, Fig. 8 shows that the experimental points lie, within the errors of the experiment, on a straight line.

Because the wire breaks along a plane surface and does not have a surface resembling a paraboloid, the true field strength at the end of the wire must vary greatly over different parts of the surface and should not be expressed by equation (38).

However, this equation gives a good approximation to the mean field strength at the end of the wire. Thus, since the ratio between the fields at the given points near the surface of the end of the wire is approximately independent of the distance between the point and the plane, our conclusion that the voltage for a given field strength must vary linearly with the logarithm of the distance remains valid.

From the slope of the straight line in Fig. 8 we find that \(\dfrac{dV}{d(\ln c)} = 1910\) volts, and hence, from equation (38), putting \(a\) equal to the radius of the wire (\(0.0108\ \mathrm{cm}\)), we find that the mean field strength is \(350\,000\ \mathrm{volts}/\mathrm{cm}^{-1}\).

Fig. 8. Emission with a cold cathode. Relation between the potential and the distance between the point and the plane for constant current.

The preceding results, together with those of Eyring, Mackeown, and Millikan \(^{116}\), show that the field currents are indeed a function only of \(E\) and do not depend on the applied potential difference \(V\). Thus these results apparently refute del Rosario’s conclusion, according to which the current depends on \(V\), and not on \(E\).

The following qualitative observations by Coolidge and Langmuir in connection with these experiments throw light on the nature of field currents. When the current strength exceeds several microamperes, spots of fluorescence appear on the glass, and sometimes also on the anode. These spots show that a large part of the field currents comes from small regions of the cathode, for the fluorescence spots usually subtend angles of only ...

only a few degrees, if counted from the place of their occurrence at the cathode, and sometimes even less than one degree. In some experiments the cathode was a thread mounted on long wires. When the cathode was shaken, the fluorescence spots moved together with it.

As the current and the voltage were increased, the fluorescence spots became brighter, and the walls of the tube began to heat up intensely at the corresponding points. Sometimes even a puncture of the glass was observed. These phenomena do not depend on the presence of gases, and the fluorescing spots prove capable of continuously receiving electrons only because they become positively charged as a result of secondary emission of electrons[^123]. As the field is increased, not only do the already existing spots become brighter, but new ones continuously arise as well. Thus, experimental observations of the change in field currents with voltage do not agree exactly with the theory, which leads to equation (36), since the latter was derived on the assumption that the current is uniformly distributed over the entire surface.

Field currents are obtained considerably more easily from surfaces covered with adsorbed films of an electropositive substance. This effect was studied by Dushman, Langmuir, and Hull in June–October 1914[^124].

In developing the design of high-voltage kenotrons it was found necessary to degas the anodes by heating them to incandescence at voltages of 30,000 or 40,000 volts, while the cathode temperature was lowered so much that the currents were limited by saturation emission, whereas in normal operation the current is limited by space charge and the voltage drop from cathode to anode is relatively small. In 1915 kenotrons for 100,000 volts were built. During evacuation the anode was charged positively up to 100,000 volts with respect to the cathode, and currents of 10 or 2 milliamperes were drawn from the cathode. Under this regime the “cold effect” of the cathode became very noticeable at approximately about 50,000 volts, but with progressive degassing and heating of the anode and cathode,

and, by heating the glass walls, it was finally possible to apply 100,000 volts without a noticeable cold-cathode effect. When this effect is present, its existence is revealed by spots of fluorescence on the glass and by intense local heating at points on the anode, which become white-hot, while the remaining part of the anode is below red heat. In order to avoid this effect completely at 100,000 volts, it was found necessary to eliminate all sharp points in the electrode parts when the most intense electric fields were used.

In 1914 some experiments were carried out with the aim of testing the practicality of using thoriated tungsten filaments in kenotrons. At first it was impossible to obtain greater electron emission than with a pure tungsten filament, but with continuous degassing of the electrodes and the walls of the tube, and with activation of the filament by heating it to 2100°, it became possible to obtain the typical increase of current from the adsorbed film of thorium. Then it was initially found that at 30,000 volts, without any heating current through the filament, electron currents sufficient to maintain the anode at red heat could be obtained from the thoriated filament. Initially the tube could not start by itself; i.e., if the anode was cooled by removing the anode voltage for several minutes, the tube did not begin to operate again when the voltage was reapplied. However, in the end, after further heat treatment, the tube was brought to such a state that it could begin to operate as soon as 30,000 volts were applied to the anode; the currents were initially small, but increased rapidly as the cathode was heated by radiation from the anode. These and several other analogous experiments showed that the cold-cathode effect is observed with an activated thoriated cathode at voltages approximately from one-third to one-half of that required to obtain the effect with pure tungsten. For this reason, for several years very careful efforts were made to avoid all traces of thorium in the ca—

cathodes and anodes in Kenotrons and Coolidge X-ray tubes.

In other experiments, a V-shaped tungsten filament containing 1.5% ThO$_2$ was mounted in a tube, the point of the V being placed at a distance of 5 mm from a flat tungsten anode, which was degassed by heating to white heat by electron bombardment. With the activated filament, giving currents of the order of $10^{-4}$ ampere, the saturation current at 5000 volts was more than ten times greater than at 100 volts. Then a capsule containing metallic potassium was broken inside the tube, so that it would be easier to maintain the thoriated filament in an active state. Under these conditions, at 7000 volts on the tungsten anode, a magnificent fluorescence was observed in the form of spots and bands, while on the glass, behind the anode, a green fluorescence was observed. After the discharge had heated the anode to red heat, no further fluorescence was observed on the anode. With the filament heated so that it gave small currents at low voltages, the voltage was then raised to 10,000 and 20,000 volts. Then spots and bands of green fluorescence appeared on the glass, which moved together with the cathode, if it could be shaken. With a further increase of the voltage, there occurred inside the tube what seemed to be sudden flashes or explosions, as if the charges on the glass were regulating themselves. After each flash the position of all the fluorescent spots proved to be shifted, and the intensity usually increased strongly. Soon, however, the spots returned to their former place, and the intensity decreased. When the voltage was raised to 35,000 volts, a sudden flash was obtained, accompanied by a crackling sound, and then on the anode there appeared a blindingly bright speck, which rapidly heated the entire anode to incandescence. This spot moved over the surface of the anode when the cathode was moved, which shows that it was caused by a thin bundle of cathode rays issuing from a small region of the cathode. If the heating current of the filament was then broken, a current of 5 mA was obtained at 30,000 volts and with a cold cathode. In this case it was not possible

be a noticeable bombardment of the filament by positive ions, since no heating was ever observed at the cathode. Instantaneous heating of the filament to \(1800^\circ\mathrm{K}\) made the filament inactive, so that at \(30\,000\) volts no cold cathode emission was obtained; but at \(35\,000\) volts a similar flash again appeared, which made the cathode active.

Coolidge in 1914 and Coolidge and Langmuir in 1922 observed that when the field currents from the point in a high vacuum increased to several milliamperes, incandescent sparks were thrown off from the cathode, in general in the direction of the anode. These particles described sharply curved trajectories, the curvature being convex toward the anode, so that the majority of the particles did not reach the anode. Many of these particles, striking the glass, were reflected; sometimes it was also observed that these particles, striking the disk-shaped anode, rebounded from its surface. This effect is probably due to local heating caused by the enormous current densities in the spot from which the electrons were emitted. Such particles, on leaving the cathode, acquired a strong negative charge and were attracted to the anode. However, owing to their high temperature they emit electrons, become positively charged, and are repelled from the anode. It is difficult, however, to understand how these particles can acquire positive charges of such magnitude as to account for the often observed very considerable curvature.

The increase in temperature \(\Delta T\), caused by a current \(i\) amperes flowing from a small spot in which the current density is \(I\ \mathrm{amp}/\mathrm{cm}^{-2}\), may be calculated approximately from the formula:

\[ \Delta T=\frac{\rho I i}{8\pi\lambda}, \tag{39} \]

where \(\rho\) is the specific resistance of the substance (ohm-centimeters), and \(\lambda\) is the thermal conductivity \((\mathrm{watt}\cdot \mathrm{cm}^{-1}\cdot \mathrm{deg}^{-1})\). For example, for tungsten at room temperature \(\rho=5\times 10^{-6}\), and \(\lambda=1.5\), so that the rise in temperature will be \(\Delta T=1.3\times 10^{-7}Ii\). Thus, if the current density is \(I=10^8\ \mathrm{amp}\cdot \mathrm{cm}^{-2}\), then the temperature rise in the emitting spot will be

only \(1.3^\circ\) with an electron emission of 10 mA. The heated particles emitted from the tungsten cathode therefore probably arise only at those points of the surface where the surface metal is in insufficiently good thermal and electrical contact with the basic substance of the cathode. It is also possible that the increase in heating is caused by the Bridgman effect.*

Schottky calculated\({}^{126}\) that the field at any surface irregularity is proportional to the average field referred to the whole surface, but is approximately ten times greater in magnitude if these surface irregularities are small in comparison with the linear dimensions of the surface. This agrees with the fact that the observed values of \(E\), capable of giving field currents, are 10 or even 100 times smaller than those calculated according to Schottky’s theory, or according to the newer theories based on wave mechanics.

The distribution of the potential near the apex of a protrusion above the surface may be roughly computed in the following manner. Let us consider the distribution of the potential in the neighborhood of an infinitely conducting solid body which is bounded by two intersecting planes forming a dihedral angle \(\beta\), this angle being less than \(\pi\). Let \(\alpha\) be the exterior angle \(\alpha = 2\pi - \beta\). Then it can be shown that the potential at any point of space outside the solid body is expressed by

\[ V = K r^n \sin n\theta, \tag{40} \]

where

\[ n = \frac{\pi}{\alpha}. \tag{41} \]

Here \(K\) is a constant, \(r\) is the distance from the point to the line of intersection \(O\) of the two planes, and \(\theta\) is the angle between the radius vector and its projection onto one of the planes. The electric field \(E\) on the surface of any plane of the solid body is expressed by

\[ E = K n r^{\,n-1}. \tag{42} \]

* Bridgman\({}^{125}\) found that the resistance of gold and silver increases by approximately \(1\%\) when the current density is \(4 \times 10^{-6}\) amp. cm\(^{-2}\), and this increase in resistance, roughly speaking, is proportional to the square of the current density.

Let us now consider a symmetric ridge rising to the same height \(h\) above a horizontal plane and formed by two plane surfaces making an angle \(\beta\), with the bisecting plane being vertical. Let the line \(OC\) be the vertical line passing through any given point \(O\) on the crest of the ridge. At any given height \(c\) above the \(O\) field, the field produced by the ridge (if the horizontal plane is disregarded) is directed along the vertical line and has the magnitude

\[ E = K n c^{\,n-1}, \tag{43} \]

whereas the field produced by the plane (not counting the field due to the ridge) will be

\[ E = E_0, \tag{44} \]

where \(E_0\) is the field strength that exists above the plane in the absence of the ridge. Similarly, we can find the potential at a point at distance \(c\) above \(O\), first disregarding the plane, and then abstracting from the existence of the ridge. These potentials will respectively be

\[ V = K c^n \tag{45} \]

\[ V = E_0(h+c). \tag{46} \]

We can now eliminate \(c\) and \(K\), assuming that the fields given by equations (43) and (44) are equal and that the potentials expressed by equations (45) and (46) are equal to each other.* Thus we find that the field strength on the surface of the ridge at a distance \(r\) from the crest has approximately the magnitude

\[ E = E_0\left[\frac{nh}{(1-n)r}\right]^{1-n} \tag{47} \]

or

\[ \log\left(\frac{r}{h}\right)=\log\left(\frac{n}{1-n}\right)-\frac{1}{1-n}\log\left(\frac{E}{E_0}\right). \]

* That this approximation gives satisfactorily accurate results follows from the fact that the field \(E\) at the top of a hemispherical protuberance on a plane surface, computed in this way, has the value \(4E_0\), whereas the exact calculation, which can be carried out in this case, gives \(3E_0\).

Table VIII gives the values of \(\frac{r}{h}\), calculated from equation (48) for ridges having different angles at the apex \(\beta\). The values of \(\frac{r}{h}\) have been calculated for two values of \(E\), namely \(10E_0\) and \(100E_0\). Thus, for a ridge with an angle of \(90^\circ\) (inclination of each side \(45^\circ\)), the field varies inversely proportional to the cube root of \(r\)—the distance from the apex—and the field is more than 100 times greater than \(E_0\) for a distance \(2 \times 10^{-6} \times h\) from the apex, and more than \(10E_0\) for a distance \(2 \times 10^{-3}h\).

Table VIII

Field \(E\) at a distance \(r\) from the apex of a ridge of height \(h\) above a plane;
\(\beta\)—angle at the apex of the ridge; \(E_0\)—field at a large distance from the ridge.

\(\beta\)	\(h\)	\(\frac{r}{h}\), \(E = 10E_0\)	\(\frac{r}{h}\), \(E = 100E_0\)
\(0^\circ\)	0.500	0.01	10
\(30^\circ\)	0.545	0.0075	\(4.8 \times 10\)
\(60^\circ\)	0.600	0.0048	\(1.5 \times 10\)
\(90^\circ\)	0.667	0.0020	\(2 \times 10\)
\(120^\circ\)	0.750	0.0003	\(3 \times 10\)
\(150^\circ\)	0.857	\(6 \times 10\)	\(6 \times 10\)
\(180^\circ\)	1.000	0	0

These results may be applied to calculating the field distribution above the square end of the wire in the experiment of Fig. 8. The real wire may be approximately replaced by a cylinder with a rounded end, on which a ridge is superposed with \(\beta = 90^\circ\) and \(h = \frac{1}{10}\) of the wire diameter, i.e. \(h = 0.0022\ \mathrm{cm}\). The mean field above the surface was \(350\,000\ \mathrm{volt}/\mathrm{cm}^{-1}\). The field was at least 10 times greater than the field at a distance \(4.4 \times 10^{-6}\ \mathrm{cm}\) from the apex of the ridge, which gives a total area of \(6 \times 10^{-7}\ \mathrm{cm}^2\). Since the current was \(10^{-5}\) amp, the mean current density on this area is equal to \(17\ \mathrm{amp}/\mathrm{cm}^{-2}\). The field \(100E_0\), i.e. \(3.5 \times 10^{-7}\ \mathrm{volt}/\mathrm{cm}^{-1}\), extended to a distance \(r = 4.4 \times 10^{-9}\ \mathrm{cm}\) according to the data of Table VIII, and in order to give

required current, the current density must be \(1.4 \times 10^{-4}\) amp.

Obviously, we cannot expect this theory to be applicable in those cases where the value of \(r\) is less than the diameter of the atoms forming the surface. In fact, the emission of a thin beam of cathode rays shows that this emission cannot take place from an edge, but occurs from individual protruding points along this edge. The theory, however, helps to form an idea of the magnitude of the effect to be expected from surface irregularities.

Some experiments under conditions of exceptionally good vacuum were carried out by Coolidge in 1923 (unpublished work). Two short V-shaped filaments were mounted inside a small bulb, so that the points of the V were separated by approximately \(0.1\) mm. One of these filaments had a diameter of \(0.0025\) cm, and the other—\(0.020\) cm. The tube was well evacuated, and the filaments were degassed at a very high temperature. During the greater part of the work with the tube it was completely immersed in liquid air, in order to improve the vacuum still further.

Below are given some of Coolidge’s observations. When the tube had been immersed in liquid air, after the filament had been heated to a high temperature and then cooled, the filament proved to be quite inactive. As the voltage was gradually increased, when the latter reached approximately 10,000 volts, a noticeable field current could be detected by means of a galvanometer; then, at constant voltage, the current gradually increased faster and faster, but suddenly became constant and remained so. In this case the filament was in an “active” state. The filament remained active even when the voltage was disconnected for a prolonged time and then reapplied. When the filament was heated to \(1600^\circ\) for 10 or 15 seconds, or to \(2400^\circ\) for one second, the filament became inactive, but could again be made active by applying a sufficiently high voltage, until the current again rose to its limiting value. With prolonged experimentation the voltage required to activate the filament gradually increased. Heating

ELECTRICAL DISCHARGES IN GASES

anode-filament (a filament of large diameter), even up to 2500°, when the tube was immersed in liquid air, had no effect on the field currents obtained with a thicker filament. After the liquid air was removed and the temperature of the bulb rose, the field currents at a given voltage first decreased, and then, when the temperature approached room temperature, the current increased greatly. Heating the filament above 1600°K caused a strong decrease in activity. In some experiments oxygen at a pressure of several microns was introduced into the tube. Usually this did not cause any noticeable increase in activity when the tube was immersed in liquid air at room temperature, but when the tube was then moderately heated, the filament became very active. This indicates that traces of alkali metals are released from the glass, which become coated with a monatomic film of oxygen on the tungsten. Heating the filament above 1600° removes a certain amount of oxygen from the tungsten surface and makes the filament inactive.

In general, a filament can be made active by raising the voltage sufficiently to extract large currents. This activation is often accompanied by crackling, indicating that discharges along the surface of the glass cause the release of gas, which alters the cathode in such a way as to make it active. The cathode could be made insensitive by heating to 1600 or 1800°K, and sometimes by extracting exceptionally large currents (several milliamperes).

The voltage difference required to obtain a current from an inactive and from an active filament differs in a ratio of 10 to 1, or even 20 to 1.

These results show how important it will be, in future work on field currents, to exercise the utmost care in controlling the conditions on the cathode surface. After each measurement of the field current it is desirable to check the condition of the filament by measuring its electron emission at the corresponding control temperature. Finally, attention should be drawn to the two most recent papers^127, which treat field currents on the basis of wave mechanics and, in particular, consider

considering the modifications introduced into Eq. (36) by the presence of a thin layer of adsorbed electropositive substance. Comparison with experiment shows the possibility of a satisfactory interpretation of the complications introduced by the surface layer, and also indicates the existence of stable films of thickness of the order of \(2.5 \times 10^{-8}\,\mathrm{cm}\).

1. Emission of electrons under the influence of electron bombardment is one of several phenomena described under the general name of secondary electron emission, although it was originally called the emission of delta rays \(^{128}\). We cannot establish whether the secondary electron is simply a reflected primary electron or whether it is an electron produced in the metal and ejected under the influence of the primary one. For practical purposes there is no need to distinguish these two possibilities. However, the fact that the number of secondary electrons sometimes exceeds the number of primary ones shows that emission actually takes place.

The most important facts relating to this secondary emission may be summarized as follows. a) Secondary emission depends substantially on the treatment of the metal surface; generally speaking, it is decreased by heat treatment and degassing and is increased in the presence of contamination, especially by electropositive substances. b) The emission increases from a small value (possibly from zero) for zero velocities of the primary electrons, rises to a maximum for primary electrons with energies of several hundred volts, and then slowly decreases with further increase of the energy of the primary electrons. c) The maximum ratio of the number of secondary electrons to the number of primary ones lies between 1.0 and 1.5 for well-degassed ordinary metals; it may reach values of 3 or 4 for metals that have not undergone special treatment. For electropositive metals this ratio is larger. For films of alkali metals on an oxidized metallic surface, the number of secondary electrons per primary electron may reach 8 or 10.

d) Although the general course of the dependence of secondary emission on voltage is such as described in item (c), nevertheless this course is superposed with small variations in the character of the secondary maxima, indicating the existence of critical potentials at which secondary emission arises as a result of certain new processes. e) The velocities of the secondary electrons are, for the most part, very small and amount to only a few volts, even for primary electrons with velocities of 1000 volts. f) The directions of emission of the secondary electrons are distributed more or less at random. g) With the exception of a possible influence on the surface conditions, moderate changes in the temperature of the surface have little effect (there is, however, an indication ¹²⁹ that the critical potentials mentioned in item d are sharply expressed if the surface is exceptionally free from adsorbed gases, which can be achieved by keeping it continuously hot in a good vacuum). The facts enumerated are firmly established by many investigators ¹³⁰. Some of them, in agreement with one another, indicate that about 0.50 of the incident electrons of very small velocities are reflected from ordinary metallic surfaces, although the interpretation of the results is especially uncertain for very small velocities.

In view of the importance of reflectivity in the theory of thermionic emission (see section B 1), it is of interest to note the following serious objection to a reflectivity of 50% for small velocities. Langmuir and Jones ¹³¹, in studying ionization phenomena between an axial tungsten cathode in a cylindrical molybdenum collector, measured the fraction of the primary electrons emitted by the cathode that were collected by the cylinder without loss of energy by inelastic collisions (see also section 2 B j). If the logarithm of this fraction is plotted along the ordinates, and the pressure along the abscissae, straight lines are obtained, the slope of which is equal to \(R/\lambda_1\), where \(R\) is the radius of the cylinder, and \(\lambda_1\) is the free path of an electron for inelastic collisions at unit pressure.

The intersection of this line with the abscissa axis gives the fraction of electrons that would have been collected if there had been no collisions. The latter would be equal to unity if there were no reflection of electrons incident on the cylinder. In fact, the intercepts found experimentally had values between 0.77 and 0.88 and were independent of the nature of the six gases used. This result shows that, for molybdenum surfaces cleaned by bombardment with positive ions, the coefficient of reflection for electrons incident normally on the surface, with energies corresponding to 10 or 20 volts, lies between 0.17 and 0.23.

The character of the curves obtained by Langmuir and Jones is in good agreement with the empirical equation

\[ \frac{1}{1-r} = 1 + 0.0645 V^{\frac{1}{2}}, \]

where \(r\) is the coefficient of reflection of electrons having an energy corresponding to \(V\) volts. The data of Langmuir and Jones are completely incompatible with any increase of \(r\) as \(V\) decreases. The results thus indicate that the coefficient of reflection becomes very small for electrons of small velocities.

The theory of the forces acting on an electron near a surface, based on electrical images, also gives weighty arguments in favor of the view that the coefficient of reflection should approach zero as the velocity of the incident electrons becomes very small. Consider, for example, electrons in equilibrium with a surface at \(2000^\circ\), which strike a tungsten surface. These electrons have on average an energy of about 0.2 volt, but as they approach the metal an attractive force, corresponding to the electrical image, begins to act on them. If the work function, 4.5 volts, is entirely due to the electrical force of the image, the electrons must possess an energy of approximately 4.7 volts by the time they reach the actual surface of the metal. If, in doing so, they acq—

ELECTRIC DISCHARGES IN GASES

are excited only with a velocity of 0.2 volt, or about 4% of their energy, then they cannot leave the surface. According to some of the newest theories, based on wave mechanics, the electrical image force may be equivalent to energies of 10 or 15 volts, which are partly balanced by the pressure of the electrons inside the metal. On this basis, a loss of energy amounting to only 4% prevents reflection of the electrons.

Langmuir^132 showed that from the thermal conductivity of a metal one can calculate the relaxation time of an electron. For tungsten this time is of the order of magnitude \(10^{-15}\) sec. This is approximately the time required for an electron having unusually large energy to lose \(\frac{1}{e}\) part of its excess energy. An electron with an energy of 0.2 volt, which acquires an energy of 4.7 volts when it approaches the surface of the metal, moves with a velocity of \(0.2 \times 10^8\) sec. When such electrons penetrate into the metal to a distance of only one atomic diameter, they remain inside the metal for approximately \(0.4 \times 10^{-15}\) sec and therefore lose approximately half of their energy, owing to the normal thermal conductivity of the metal. Thus only an exceptionally small fraction of the electrons lose less than 4%, and only these electrons can be reflected from the surface.

However, there is also another effect which lowers just as strongly the probability of reflection of electrons of small velocities. When the incident electron penetrates into the surface layer of atoms, all the other free electrons are repelled by it, leaving a slightly larger positive charge than that which existed before the arrival of the incident electron. This is a kind of internal image force, analogous to the force postulated by Debye and Hückel in their theory of electrolytes. The interval of time necessary to produce this redistribution of charge can be roughly estimated in the following way. In a metal with specific resistance \(\rho\), consider a charge \(e\) placed at the center of a sphere. Let

*

\(t\) will be the time required for the charge inside the sphere to decrease to \(\frac{1}{e}\) of its initial value. We thus find

\[ t = 10^{9}\,\frac{\rho}{4\pi c^{2}}, \]

where \(c\) is the speed of light. For tungsten \(\rho = 5 \times 10^{-6}\), and therefore we find \(t = 4 \times 10^{-19}\) sec. Thus redistribution inside the metal occurs in \(0.001\) of the time during which the electron remains inside the metal. At some small distance from the surface, where the concentration of electrons is smaller, the relaxation time will be comparable with the rate of passage of the electron, and therefore relatively significant hysteresis phenomena will arise, leading to a noticeable dissipation of energy and thus hindering reflection.

The reflection of electrons, apart from the importance which these processes have in certain special types of discharge [the Lilienfeld X-ray tube \(^{133}\), Hull’s dynatron \(^{134}\)] and in equilibrium theories of thermionic and photoelectric emission, has recently acquired great interest in connection with the interpretation of the nature of the electron itself. The beginning of this cycle of work was laid 6 or 8 years ago by Davisson and Kunsman \(^{135}\), in a study of the angular distribution of that very small fraction of fast primary electrons falling on the surface of a metal which are reflected with practically unchanged initial energy. Thereafter the phenomenon was carefully investigated by Davisson and Germer \(^{136}\), who used reflection from the faces of a nickel crystal. These experiments revealed intense reflection at certain definite angles of incidence and reflection, obeying the ordinary laws of diffraction of waves by a crystal lattice. Thus Davisson and Germer showed that electrons behave like trains of waves with wavelength

\[ \lambda = \frac{h}{p}, \]

where \(h\) is Planck’s constant and \(p\) is the momentum of the electrons. It was further found that the refractive index of the crystal for these waves is not equal to unity, but that the crystal possesses a kind of “disper-”

...“dispersion curve” for electrons of different equivalent wavelengths. This “wave nature” of electrons was remarkably confirmed by the experiments of G. P. Thomson, who showed that electrons, when passing through thin metal films or falling at small angles on a film deposited on quartz, are scattered as though diffraction were taking place by an atomic spatial lattice[^137] of waves of length \(\lambda=\frac{h}{p}\).

Under the influence of electron bombardment, secondary electrons can be emitted not only from metallic surfaces, but also from the surfaces of insulators. High-velocity electrons, striking a glass surface, can cause the emission of more than one secondary electron for each incident electron. Thus the glass surface becomes positively charged, not negatively. In most high-vacuum tubes intended for discharges at high potentials, especially when they operate at voltages above normal, spots of green or blue fluorescence are observed on the glass; these owe their origin to beams of cathode rays, deflected from the cathode by strong fields. In such fluorescent spots a considerable development of heat is detected, indicating that these spots are charged to a relatively high positive potential with respect to the cathode. Since these phenomena occur in tubes in which appreciable currents cannot be carried by positive ions, it follows from this that the positive charge is maintained by the emission of secondary electrons.

Langmuir[^137a] studied these phenomena of the emission of secondary electrons from glass surfaces. Generally speaking, two types of discharges are possible. First, a discharge in which the walls acquire a negative charge and therefore repel all further electrons, except for a few which balance the weak currents of positive ions. Under these conditions, fluorescence in the presence of traces of gas can be excited by bombardment ...

positive ions. There are some indications that the red fluorescence sometimes observed at low oxygen pressures is caused by a similar bombardment by positive ions. In the second type of discharge, the emission of secondary electrons maintains the walls at a positive charge, so that they continuously receive electrons; in such a case a considerable part of the discharge energy may be released as heat at the walls.

A repetition[^137a] of Lilienfeld’s[^138] experiments shows that the discharge he observed in high vacuum in long glass tubes is due exclusively to the emission of secondary electrons from the walls of the tube.

1. Emission of electrons as a result of impacts by metastable atoms. In recent years it has been pointed out that these processes account for that part of the electron emission from the cathode in the presence of an excited gas which was formerly attributed entirely to the photoelectric action of radiation. The liberation of electrons by impacts of metastable atoms was first definitely proved by Webb[^139] and Messenger.[^140] These investigators placed screens of quartz and fluorite of measured optical transparency between the excited gas and the electron-emitting electrode, and, from the decrease in the number of emitted electrons, they were able to estimate the relative share of the emission caused by impacts of metastable atoms and by the action of light from different parts of the spectrum. They showed that the influence of metastable atoms is comparable with, and in some cases considerably exceeds, the photoelectric action with respect to the number of emitted electrons.

Recently Oliphant[^141] showed that a beam of positive helium ions passing through an aperture in the negatively charged plane collector of Langmuir can be converted, for the most part, into a beam of metastable helium atoms of approximately the same velocity, if these ions are made to strike the surface of a metal at a glancing angle. These metastable atoms, in turn, prove capable of liberat-

to expect electrons upon impact with the second metallic surface. The probability of liberation was, at the extreme limit, several percent, and the simplest explanation of the results consists in the fact that each metastable atom which returns to the normal state at the metallic surface liberates an electron. A certain percentage of these metastable atoms is reflected from the surface, and this percentage decreases with the depth of penetration, i.e. with the velocity and with the glancing angle, and increases when the surface is heated. The fraction reflected generally lies between 10 and 50%. Electrons liberated by these metastable atoms possess kinetic energies between the maximum value \(V_0-\varphi\) (where \(V_0\) is the energy of the metastable state and \(\varphi\) is the work function of the metal) and a minimum value of about two volts, which may be significant in connection with Sommerfeld’s theory of metals. Evidently, the energy of emission of the electrons has no relation whatever to the kinetic energy of the impacting metastable atoms.

Fig. 9. Thomson and Harrington’s apparatus for studying electron emission under the action of positive ions and metastable atoms. Nickel electrodes.

Very important data concerning the secondary emission of electrons from a negative cold electrode during a discharge were recently obtained by Thomson and Harrington\(^{142}\). They introduced into a region of uniformly ionized gaseous neon, the ionization of which was maintained by auxiliary electrodes, the system of electrodes shown in Fig. 9. The round disk-shaped electrode \(D\) was surrounded by a broad guard ring \(G\) and was drilled through at the center. Immediately behind the aperture was placed the Faraday cylinder \(F\). Between \(D\) and \(G\) there was a narrow space, visible in the drawing. Between the region of the main discharge and the region situated below

of these electrodes, there was no connection other than this gap and the aperture in electrode \(D\). Directly opposite \(D\) there was placed another Faraday cylinder \(B\), surrounded by glass, except for the open end facing \(F\). The experiments of immediate significance were as follows:

1) Whatever the potential of the electrodes \(DG\) relative to the ionized gas above them, between \(F\) and \(G\) there always flowed a considerable current, which had one direction or the other depending on the sign of the field between \(F\) and \(G\), and which was completely independent of the potential of \(DG\) relative to the gas. Furthermore, the application of a field between \(D\) and \(G\) had a very small effect on this current. These observations show that the conductivity between \(F\) and \(G\) was produced by uncharged particles passing from the region of the main discharge into the region below \(DG\). Equally, the currents were of such a magnitude that the possibility of explaining them by the photoelectric effect was excluded, especially if one takes into account the complications of reflection that would have to be associated with obtaining photoelectric emission from the lower side of \(G\). The only possible explanation of the currents between \(F\) and \(G\), apparently, is that metastable atoms with a long lifetime diffuse through the gap.

2) The collector \(B\) was used in order to prove the existence of a very considerable electronic emission from the upper surface of disk \(D\), which was situated in the region of the main discharge. To do this, the collector was charged negatively relative to the surrounding space up to a certain potential, say up to 100 volts. Then around it there arose a layer \(S_2\) with a positive space charge, and only ions, but not electrons, from the surrounding ionized gas could be collected in the collector. Keeping these conditions constant, the experimenters then began to vary the potential \(V\) of the electrodes \(DG\) from a small to a large positive value. As long as \(V\) was less negative than 100 volts, the electrodes \(DG\) had practically no effect on

ELECTRICAL DISCHARGES IN GASES

current through \(B\). As soon as \(V\) was raised to 100 volts, a strong negative current was obtained in \(B\). If \(V\) is made still more negative, this increases the effect relatively little. This negative current through \(B\) must be an electron current coming from \(D\), and the very fact that it appears precisely when the potential \(D\) becomes more negative than \(B\) proves that these electrons must arise at the very surface of \(D\). If they arose inside the space-charge layer \(S_1\), and not at the electrode \(D\), their velocity distribution would not be so uniform.

These electrons may be emitted under the action of light, of metastable atoms, or of positive ions. There is no doubt that all three processes play a role. But the following considerations indicate that metastable atoms are of the greatest importance. The magnitude of the density of this electron current amounted to approximately half the total current density in \(D\), which indicated that approximately half the current \(D\) was caused by these secondary electrons, and the other half by the positive ions passing through. The data reported in the following section (B 9) lead to the conclusion that the emission of electrons under bombardment by positive ions is considerably less intense than under the action of metastable atoms. The same conclusion can be drawn from the fact that the results proved to be essentially identical regardless of whether the potential was 50, 100, or 200 volts. Likewise, a photoelectric effect of this order of magnitude was not registered. Consequently, the only correct explanation that remains is the influence of metastable atoms.

This work is still in progress; additional data are being obtained for various gases, and, in addition, the free paths of electrons and the fraction of the cathode current carried by positive ions are being measured.

An earlier work by Yutergeven \(^{143}\) with an apparatus analogous to parts \(D\) and \(G\) (Fig. 9) led to results for He, Ne, and A which, in their main features, can be

are explained in an analogous manner, and whose significance in relation to the electron emission caused by the impact of positive ions is discussed in the following section (B 9). These conclusions were supported by Found,^143 who used the characteristics of a cylindrical electrode to estimate the fraction of electron emission under the action of metastable neon atoms in the positive column of a discharge in neon. The current of positive ions to the negative collector is proportional to the area of the surrounding layer, whereas the electron emission from it, due to metastable atoms, depends only on the area of the collector itself and is therefore constant. In a positive column 5 cm in diameter in neon at a pressure of 0.225 mm and at a current of 0.4 amp, the emission of secondary electrons from a negative tungsten collector is equal to 7.5 mA/cm\(^{-2}\), whereas the current density of positive ions is 0.82 mA cm\(^{-2}\). Thus the current in the collector at 100 volts, negative with respect to the discharge, is caused approximately two thirds by secondary emission and one third by positive ions.

1. Electron emission under the action of bombardment by positive ions is another phenomenon whose existence had long been assumed, but which has only recently been detected. As yet we know very little about it. Practically all earlier work has been discredited by various complicating effects.

The simplest conditions are realized when a metallic surface is bombarded by positive ions in a high vacuum and the resulting electron emission is measured with allowance for the reflection of positive ions. Klein^144 and Jackson^145 proceeded in this way, using, as the source of ions, hot strips of alkali metals emitting positive ions. The results of these two authors are in serious contradiction with one another, caused, as we believe, by an unrecognized complication connected with the arrangement

of the experiment in the first work. Jackson found no appreciable electron emission (i.e., the emission could have been less than 0.5%) for K ions falling perpendicularly on Al, Ni, and Mo, for ions with velocities below 200 volts in the case of Al, 300 volts in the case of Ni, and 600 volts for Mo, provided that these metals were degassed by heat treatment. Above these velocities the electron emission rises regularly to 7.0% for Al, 4.2% for Ni, and 3.8% for Mo at velocities of 1000 volts. Without heat treatment the emission could be detected at velocities equal to approximately one half of the minimum velocities indicated above, and it was approximately twice the stated values at 1000 volts. Positive Cs ions are somewhat more effective, while Na and Rb ions are less effective than potassium ions. The electrons emitted in this way were very slow; practically all of them were retarded by a retarding field of 1 volt. ^146

In the experiments considered above, no more than four percent of the incident ions were reflected. However, at more grazing angles, and especially with degassed surfaces, the percentage of reflection may be considerably greater, and the reflected ions possess a complex distribution of velocities and energies, which they retain. ^147

A quite definite indication of the effectiveness of alkali ions in liberating secondary electron emission from a bombarded metallic surface may be based on the remarkable constancy of ion currents from a sufficiently heated filament in contact with vapors of alkali metals. The reason for this constancy is that such currents depend only on the frequency with which the metal atoms strike it, and are completely independent of the applied voltage, provided only that the latter is sufficient to overcome the space charge. ^148 It was established that if such a current of cesium ions from a filament to the surrounding coaxial cylinder is a saturation current at 50 volts, then an immediate increase of the voltage to 250 volts reveals no noticeable change in the current. This shows that in

secondary electron emission show no changes exceeding a fraction of a percent in the range from 50 to 250 volts, whence it follows with very great probability that secondary electron emission itself is negligibly small. It is especially important to know the magnitude of this phenomenon in mercury vapor. Langmuir and Mott-Smith \(^{149}\) found that the current in a negative nickel collector bombarded by Hg ions up to 1160 volts shows no signs of secondary electron emission under the influence of the bombardment, although their experimental conditions were such that an emission not exceeding \(10\%\) could not be detected because of the uncertainty of the “edge correction” to the collector current. Dellenbach, Gerecke, and Stoll \(^{150}\) showed that \(\mathrm{Hg}^{+}\) ions, falling on an iron electrode with energies up to 3000 volts, are capable of releasing electrons from the electrode in an amount not exceeding \(1\%\).

In some recent unpublished work Langmuir and Switzer measured the changes of current in a negative collector and their dependence on voltage with much greater accuracy than in the work of Langmuir and Mott-Smith. The collector was of molybdenum, of the type with a guard ring. The outside diameter of the guard ring was \(1.92\ \mathrm{cm}\), while the diameter of the inner part of the collector was \(0.64\ \mathrm{cm}\), and the distance between the two collectors was less than \(0.005\ \mathrm{cm}\). Both parts of the collector were always kept at the same potential, but the current was measured only for the central part. The small change in current upon changing the collector potential by one volt—let us call this change \(\beta\)—was measured under the most varied conditions. The largest observed change \((\beta = 0.001)\) was found only under such conditions when the thickness of the layer \((x)\) was very considerable \((\beta = 0.20\ \mathrm{cm})\). When \(x\) was changed by varying the ionization intensity, for example by changing the cathode current, it was found that \(\beta\) changes as a function of \(x\) according to the equation

\[ \beta=\beta_{0}+1.65\frac{x^{2}}{Vr_{0}^{2}}, \tag{49} \]

where \(V\) is the negative voltage on the collector, and \(r_{0}\)—ra

plus the outer perimeter of the guard ring. In this equation the term containing \(x^2\) has the form that should be expected for the residual edge correction due to the insufficient size of the guard ring. As it turned out, this equation represents the results for the discharge both in argon and in mercury vapor.

At a greater intensity of ionization the term containing \(x^2\) becomes small in comparison with \(\beta_0\), and therefore the magnitude \(\beta_0\) must be found with sufficient accuracy. For discharges in mercury vapor \(\beta_0 = 0.0002\ \mathrm{volt}^{-1}\). For argon \(\beta_0 = 0.0006\ \mathrm{volt}^{-1}\). In these experiments the collector voltage varied from 50 to 150 volts. Thus, over this entire interval of 100 volts one may conclude that, if there is electron emission under the action of bombardment by positive ions, it varies approximately from 2% for mercury to 6% for argon, these fractions being expressed as percentages of the total current of positive ions (experiments with collectors intended to capture electrons emitted from the surface of the negatively charged collector are at present being carried out by Tonks and Switzer). Similar experiments by Jürgensen^143 at considerably lower pressures do not give exact agreement with equation (49), but if this equation is used as a rough approximation, the quantity can be calculated. Typical values of this quantity are as follows: He—Ni—0.0020; A—Ni—0.00095; A—C—0.00012.

It should be noted that these values are “apparent,” since they are based on the assumption that the total collector current is the current of positive ions. These values would have to be increased if part of the current were produced by liberated electrons; for example, they would be doubled if half the current were produced by electrons, as in the case He—Ni investigated by Jürgensen and Harrington^142.

The most direct investigation of electron emission under the action of positive ions of a gas discharge was carried out by Penning^151, who directed neon ions from the discharge region through a tube and made them fall

to strike a copper, silver, or iron electrode after they had acquired the desired velocity under the action of an auxiliary field. The probabilities, found in this way, of emission of electrons as a function of the energy of the ions are given in Fig. 10. It is interesting to note that emission occurs even at zero impact velocities. Extrapolation of the experimental curve gives a probability at zero velocity somewhere between 2 and 2.5%. Penning assumes that this occurs whenever the ionization potential of the gas atom exceeds twice the work function of the electrode, \(V_i > 2\varphi\), since such an ion possesses sufficient energy to tear out an electron and thus neutralize itself and, in addition, to tear out one more electron. Oliphant \(^{141}\) indicates that in Penning’s results there is probably also a share belonging to metastable atoms, obtained as in his own work. Be that as it may, the increase of emission with the velocity of the ion indicates that Penning observed true electrode emission under the action of ions of low velocities, although its values may be extremely large if the complications indicated by Oliphant are taken into account.

Fig. 10. Probability of electron emission under bombardment by positive ions at \(V\) volts (neon ions, copper electrodes).

In support of Penning’s views one may point out, apparently, that in a cold discharge in Hg unexpectedly high potentials are required in comparison with other gases in which ionization is more difficult. This may occur because the inequality \(V_i > 2\varphi\), in the case of mercury, is expressed considerably less sharply than in the case of other gases, so that the positive ions are less effective in ejecting

of electrons from the cathode. If this is true, then the discharge in potassium vapor should be considerably more difficult when the cathode is made of a substance such as oxygen-free zinc, which does not adsorb potassium atoms or ions and therefore does not lower its work function.

Finally, the emission of electrons released from the cathode by the impact of positive ions was estimated indirectly, and moreover on an insufficiently reliable basis, by Klemperer^152 from data on the minimum sparking potential between parallel metallic plates, combined with the values of the “effective” (not minimum) ionizing potential of gases. This estimate was based on the assumption that each electron ionizes every time it passes through this “effective” ionizing potential, which is the characteristic and explanation of the minimum sparking voltage (an assumption and assertion that are not obvious). This estimate agrees in order of magnitude with calculations based on the theory of the spark discharge of Townsend and Osterreis (see Part II). Klemperer’s data on the percentage yield of electrons per impact of a positive ion are given in Table IX.

Table IX

Yield of secondary electrons per positive ion (indirect estimate)

Gas →	Air	CO₂	H₂	A
Yield in % . . .	0.66	0.09	0.74	0.7
\(U^*\) . . .	5.8	8.4	2.4	3

\(U = \mathrm{kV/cm}\); pressure in cm Hg is a measure of the average energy acquired by an ion when passing through the gas between flat parallel electrodes.

10. Emission under the influence of chemical reactions probably plays an insignificant role in gas discharges. This phenomenon itself may have different origins. For example, air, especially humid air, when in contact with phosphorus, becomes charged with both positive and

with negative ions of such low mobilities that they obviously contain very complex products of chemical reactions. Incandescent platinum in contact with phosphorus vapors emits positive ions, but gives no appreciable quantity of negative ions or electrons. The gaseous products of electrolysis are ionized, although this is probably due to their passage in the form of bubbles through the electrolyte, and not to chemical action.

Electrons are emitted under the chemical action of certain gases on such electropositive metals as sodium, potassium, and amalgamated aluminum. In this respect the following gases have been investigated: H₂S, HCl, CO₂, H₂O, COCl₂, CSCl₂, O₂, Cl₂, Br₂. Cases were observed in which one electron is emitted for every 1600 reacting molecules ^154. The emitted electrons had velocities of the same order of magnitude as those observed in the photoelectric effect, but did not show the saturation which, as we have shown, is a consequence of the presence of a complex surface.

Undoubtedly, some cases of emission attributed to chemical action are in fact thermionic emission caused by the high temperatures developing during the reaction, or else have a photoelectric origin, being associated with chemiluminescence. However, there is no reason to exclude electron emission under the influence of chemical action as a real phenomenon, since it is known that activated molecules exist as intermediate products of chemical reactions ^155, and these activated molecules can cause the emission of electrons if their energy is sufficient for this, in exactly the same way as such emission occurs when metastable atoms come into contact with electrodes.

An excellent review of ionization and chemical action is given by Richardson ^156 in his book Emission of Electricity from Hot Bodies.

(To be continued.)

LITERATURE

1. Townsend. Phil. Mag. 1, 198 (1901); Electricity in Gases, Chap. VIII (Oxford 1915).

2. On methods for determining ionizing potentials see Compton and Mohler, “Critical Potentials,” National Research Council Bulletin, N. 192; Franck and Jordan, Anregung von Quantensprüngen durch Stösse (Springer, 1926); Geiger and Scheel, “Handbuch der Physik,” XXIII (1927); L. Bloch, Ionisation et Resonance des Gaz et Vapeurs (Société Française de Physique, 1925); Mohler, Critical Potentials of Atoms and Molecules (International Critical Tables VI, pp. 69—74, 1929).

3. Compton and Van Voorhis (Kompton and Van Voorhis). Phys. Rev. 27, 724, 1926) give values for a number of gases up to 400 volts; Jones (Phys. Rev. 29, 822, 1927) and Bleakney (Ibid. 35, 139, 1930) give values for mercury vapor. Buckmann (Ann. d. Physik 87, 509, 1918) gives values for air up to electron energies of 25,000 volts.

4. This contradicts the conclusions of Hippell (Ann. d. Physik 87, 1935, 1928), whose results apparently cannot be reconciled with the results of other observers. Evidently Hippell’s method of measurement does not take into account certain unexpected complicating factors that play a substantial role. Cf. also Funk, Ann. d. Physik 4, 149, 1930.

5. Compton and Van Voorhis, Phys. Rev. 27, 724, 1926; Jones, ibid. 29, 822, 1927. Jones, who used a more refined method for determining the bombardment energy by ionizing electrons, obtains values of \(P\) for Hg close to the results of Compton and Van Voorhis, but approximately 25% higher at small values of \(V\). This indicates that it may be that the constant \(C\) should be 25% larger than that given in Table I for all gases. The values for \(V\) greater than 100 volts are probably quite accurate.

6. Condon. Phys. Rev. 32, 858 (1928).

7. Langmuir and Jones. Phys. Rev. 31, 402—3 (1928).

8. Jonson. Phys. Rev. 10, 609 (1917); Lehmann and Osgood. Proc. Roy. A 111, 609 (1927).

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16. Born. Verh. d. Phys. Ges. 21, 13, 679 (1919); Grimm. Z. f. Physikal. Ch. 102, 113, 141, 504 (1922); Foote and Mohler. Origin of Spectra, Chap. VIII (Chem. Cat. 1922).

17. Barton. Phys. Rev. 30, 614 (1927).

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19. Morse and Stueckelberg. In press in Phys. Rev.

1. Kramers. Phil. Mag. 46, 836 (1923).

2. Rückhardt. Ann. d. Physik 71, 377 (1923); Z. Physik 15, 164 (1923).

3. Suguira. Journ. de Phys. 8, 113 (1927); Oppenheimer. Z. Physik 41, 268 (1927); Reiche, ibid. 53, 168 (1929).

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6. Mohler. Phys. Rev. 28, 46 (1929).

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11. Wien. Ann. d. Physik 60, 597 (1919); 483 (1924); Kerschbaum, ibid. 79, 465 (1926); Dempster. Astrophys. Journ. 67, 193 (1928).

12. Cario. Z. Physik 10, 185 (1922); Turner. Phys. Rev. 23, 464 (1924).

13. Maxwell. Phys. Rev. 34, 199 (1929).

14. Tolman. Phys. Rev. 23, 693 (1924).

15. Saha and Kothari. Naturwiss. 17, 271 (1929).

16. Dorgelo. Physica. 5, 429 (1925).

17. Metastable neon in neon: Meissner und Graffunder. Ann. d. Physik. 84, 1009 (1927); in N₂, N₂, A, Eckstein, ibid. 87, 1003 (1928). Metastable Hg in Hg, Zemansky. Phys. Rev. 29, 513 (1927); Coulliétte, ibid. 32, 636 (1928); in N₂, Pool. Phys. Rev. 33, 22 (1929); Asada. Phys. Zeits. 29, 708 (1929).

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19. De Groot. Z. Physik 55, 52 (1929); Kopfermann und Ladenburg, ibid. 48, 15, 26, 51 (1928).

20. Townsend. Electrician, April 3 (1903); “Electricity in Gases”, Chap. IX (Oxford 1915).

21. J. J. Thomson. “Rays of Positive Electricity”, 1, 36.

22. Loeb. Science 66, 627 (1927).

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25. Tate. Phys. Rev. 23, 293 (1924).

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27. Duffendack and Smith. Nature, May 21, 1927; Duffendack, Henshaw and Goyer. Phys. Rev. 34, 1132 (1929).

28. Frank and Jordan. Anregung von Quantensprüngen durch Stösse, p. 119 (Springer 1926); see also Penning, Z. Physik 57, 723 (1929). Ionization of argon by metastable neon.

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30. Saha. Phil. Mag. 46, 534 (1923).

31. Langmuir and Kingdon. Proc. Roy. Soc. 107, 61 (1925); Fowler, Statistical Mechanics, p. 281 (Cambridge, 1929).

ELECTRICAL DISCHARGES IN GASES

1. Richardson. Emission of Electricity from Hot Bodies, Chap. II (1916); Phil. Mag. 23, 601, 619 (1912); ibid. 24, 740 (1912); ibid. 28, 633 (1914).

2. Richardson and Brown. Phil. Mag. 16, 353 (1908); (cf. also reference 57 below).

3. Davisson and Germer. Phys. Rev. 20, 300 (1922).

4. Dushman. Phys. Rev. 21, 633 (1923); cf. also Richardson. Emission of Electricity from Hot Bodies, p. 39.

5. Du Bridge. Proc. Nat. Acad. Sc. 14, 788 (1928); Bridgman. Phys. Rev. 31, pp. 90, 862 (1928).

6. Sommerfeld. Z. Physik 47, 1 (1928).

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8. Langmuir. Phys. Rev. 22, 357 (1923).

9. Langmuir and Kingdon. Science 57, 58 (1923); Proc. Roy. Soc. A 107, 61 (1925); Becker. Phys. Rev. 28, 341 (1926); 29, 364 (1927); Bell Telephone Laboratories Reprint No. B—412, August 1929.

10. Killian. Phys. Rev. 27, 578 (1926).

11. Kingdon and Langmuir. Phys. Rev. 22, 148 (1923); A. W. Hull. Transaction Amer. Inst. of Electrical Engs. 49, 763 (1928).

12. Richardson and Brown. Phil. Mag. 16, 353 (1908); Schottky. Ann. d. Physik 44, 1101 (1914); Jones. Proc. Roy. Soc. A 102, 734 (1923); Germer. Phys. Rev. 25, 795 (1925).

13. Dushman ref. 54; Dushman, Dennison and Reynold. Phys. Rev. 29, 903 (1927); Du Bridge. Phys. Rev. 31, 236 (1928); Dushman. Int. Critical Tables VI, p. 53.

14. Birge. Phys. Rev. Supp. 1, 1 (1929).

15. Langmuir and Mott-Smith. Ir. G. E. Rev. 27, 449 (1929); Mott-Smith and Langmuir. Phys. Rev. 28, 756 (1926).

16. Richardson and Cooke. Phil. Mag. 20, 173 (1910); ibid. 21, 404 (1911); Cooke and Richardson. ibid. 25, 624 (1913); Wehnelt and Jentzsch. Verh. d. D. Phys. Ges. 10, 610 (1908).

17. Schottky and Issendorff. Z. Physik 26, 85 (1924); Van-Voorhis. Phys. Rev. 30, 318 (1927).

18. Richardson. Phil. Mag. 16, 740 (1908); Proc. Roy. Soc. 89, 507 (1914); Davisson. Phil. Mag. 23, 121 (1912); Barton, Harnwell and Kunsman. Phys. Rev. 27, 739 (1926).

19. Richardson. Phil. Mag. 4, 98 (1902); Kunsman. Proc. Nat. Acad. Sc. 12, 656 (1926).

20. Wahlin. Phys. Rev. 34, 164 (1929) and unpublished data.

21. L. P. Smith. Phys. Rev. 34, 1496 (1929); 35, 381 (1930).

22. Bainbridge. Phys. Rev. 34, 752 (1929).

23. Einstein. Ann. d. Physik. 17, 145 (1905).

24. Richardson and Compton. Phil. Mag. 24, 575 (1912); Millikan. Phys. Rev. 7, 355 (1916); Duane and Hunt, ibid. 6, 166 (1915); Webster, ibid. 7, 599 (1916).

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

26. Richardson. Proc. Roy. Soc. A 94, 269 (1918).

27. Du Bridge. Phys. Rev. 31, 236 (1928).

28. Warner. Proc. Nat. Acad. Sc. 13, 56 (1927).

29. Kazda. Phys. Rev. 26, 643 (1925).

30. Richardson and Joung. Proc. Roy. Soc. A. 107, 377 (1925).

31. Millikan. Phys. Rev. 7, 18 (1916); ibid. 18, 236 (1921).

32. Millikan and Winchester. Phys. Rev. 24, 16 (1907); Burt, ibid. 24, 207 (1924); Koppius, ibid. 18, 443 (1921); Nielson, ibid. 25, 30 (1925).

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2. De Broglie. Jour. d. Phys. et de Rad. 2, 265 (1921); Robinson. Phil. Mag. 50, 241 (1925); Ellis. Proc. Cambr. Phil. Soc. 22, 369 (1924).

3. Nottingham. Phys. Rev. 33, 633 (1929).

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5. Richardson. Phil. Mag. 23, 619 (1912).

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7. Wilson. Proc. Roy. Soc. A. 93, 359 (1917); Roy. Soc. Proc. A. 112, 599 (1926).

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13. Richardson. Phil. Mag. 23, 261 and 615 (1912).

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99a. Nottingham. Journ. Frank. Inst.; also Proc. Am. Phys. Soc. February (1930) meeting.

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16. Millikan and Eyring. Phys. Rev. 27, 51 (1926).

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2. Research Staff of the Gen. Elec. Co, London, Phil. Mag. (7) 1, 609 (1926).

3. Millikan and Lauritsen. Proc. Nat. Acad. of Sci. 13, 45 (1928).

4. E. Fermi. Z. Physik 36, 902 (1926); P. A. Dirac. Proc. Roy. Soc. London, 112, 661 (1926); L. Nordheim. Z. Physik 46, 833 (1928); A. Sommerfeld. Z. Physik. 47, 33, 38, 43 (1927).

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6. Maxwell. Collected papers. Vol. 1, p. 242.

7. Del Rosario. Ir. Frank. Inst. 203, 243 (1927).

8. Langmuir. Gen. Elec. Rev. 23, 513 (1920).

9. Gossling. Phil. Mag. 1, 609 (1926).

10. Bridgman. Proc. Nat. Acad. Sc. 10, 297 (1921); Proc. Amer. Acad. of Arts and Sc. 57, 131 (1922).

11. Schottky. Z. Physik. 14, 63 (1928).

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13. Campbell. Phil. Mag. 22, 276 (1911); 25, 803 (1913); 28, 286 (1914); 29, 369 (1915).

14. Krefft. Phys. Rev. 31, 199 (1928).

15. Gehrts. Ann. d. Physik 36, 995 (1911); Hull. Phys. Rev. 7, 1, 141 (1926); Mc Allister, ibid. 21, 122 (1923); Farnsworth, ibid. 20, 358 (1922); 25, 41 (1925); Petry, ibid. 26, 346 (1925); 28, 362 (1926); Gill. Phil. Mag. 45, 864 (1923); Horton and Davies. Roy. Soc. Proc. 97, 23 (1920); Phil. Mag. 46, 129 (1923); Stuhlman. Phys. Rev. 25, 234 (1925); Tate, ibid. 17, 394 (1921).

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1. Ann. d. Physik 32, 673 (1910); ibid. 53, 24 (1914); Leipziger Ber. 63, 34 (1911).

2. Webb. Phys. Rev. 24, 113 (1924).

3. Messenger. Phys. Rev. 28, 962 (1926).

4. Oliphant. Proc. Roy. Soc. A 124, 228 (1929).

5. Uyterhoeven and Harrington. Phys. Rev. 35, 124 (1930).

6. Uyterhoeven. Proc. Nat. Acad. Sc. 15, 32 (1929); also thesis for Cal. Inst. Tech. (1929); Found. Phys. Rev. 34, 1625 (1929).

7. Klein. Phys. Rev. 26, 800 (1925).

8. Jackson. Phys. Rev. 28, 524 (1926); 30, 473 (1927).

9. See also Baerwald. Ann. d. Physik. 60, 26 (1919).

10. Read. Phys. Rev. 31, 155 (1928); Gurney, ibid. 32, 467 (1928).

11. Langmuir and Kingdon. Science. 57, 58 (1923); Killian. Phys. Rev. 27, 578 (1926).

12. Langmuir and Mott-Smith. G. E. Rev. 27, 544 (1924)

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3. Klemper. Physik. 52, 650 (1928).

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6. Smyth. Proc. Nat. Acad. Sc. 11, 679 (1925).

7. Richardson. Emission of Electricity from Hot Bodies. Chap. IX.