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Electron Extraction by a Strong Electric Field
A. A. Makalov, Leningrad.
Classical electron theory, in order to explain the phenomena of electrical conductivity, thermionic emission, thermoelectricity, etc., assumes the existence of free electrons in a metal. These free electrons, at a given temperature of the metal, possess the same energy of motion as the molecules of an ideal gas. Among them, at a given temperature, there exists the same distribution of velocities according to Maxwell’s law, and in exactly the same way they can fly out from the surface of the body, expending a certain energy to overcome the contact potential difference.
At ordinary room temperature, among the free electrons there are some which, on the basis of the Maxwellian velocity distribution, can leave the metal and form above it a volume density of electron gas (Debye¹). With increasing temperature the energy of motion of the electrons increases, the pressure of the electron gas above the metal also increases, and this electron gas can produce an electron current from the heated metal to another electrode, if some small potential difference is applied between it and the metal (the Richardson effect).
As early as 1914, Schottky² carried out experiments on increasing thermionic emission in a strong electric field. Considering the process of thermionic emission in a strong electric field to be analogous to emission under ordinary conditions, Schottky supposed that a strong electric field, being superposed on the contact field of the metal, can weaken it and facilitate the passage of electrons through the surface.
Let \(AB\) be the surface of the metal (Fig. 1), \(CD\) the contact field, and \(EF\) the external field. Then an electron, on leaving the metal and being accelerated by the external field, must perform work against the contact forces up to
¹ Debye. Ann. d. Phys., 33, 441, 1910.
² Schottky. Phys. ZS., 15, 872, 1914.
of a certain distance \(x_0\) from the surface of the metal, where the electric force acting on the electron and made up of the contact force and the intensity of the external field becomes equal to zero. Here Schottky assumes that the contact field is due chiefly to the appearance of induced charges on the surface of the metal when the electron moves away from it. The force acting on the electron in a field of this origin can be calculated by the method of “electrical images.” It will be equal to the force attracting the electron to an imaginary electron—its image—situated at an equal distance on the other side of the surface of the metal. Then the distance \(x_0\) at which the electron emerging from the metal does work in overcoming the resultant field is determined from the condition:
\[ \frac{e}{4x_0^2}=\left(\frac{dV}{dx}\right)_{x=x_0}, \]
and the magnitude of the decrease in the contact work will be
\[ eV'=e\int_0^{x_0}\frac{dV}{dx}\,dx, \]
where \(V\) is the potential difference between the electrodes.
Fig. 1.
Schottky carried out his experiments with a filament \(0.04\ \mathrm{cm}\) thick (all other authors carried out their investigations with filaments of the same order of thickness), stretched along the axis of a cylindrical anode of diameter \(0.8\ \mathrm{cm}\). Since \(x_0\) is very small in comparison with the radius of the wire \(x\), the function \(V(x)\) on this interval may be considered linear, and
\[ eV'=\sqrt{\frac{e^3}{4}\left(\frac{dV}{dx}\right)_{x=x_0}} =\frac{e^{3/2}}{2}\sqrt{\frac{V}{r\lg\frac{R}{r}}}, \]
since the field is cylindrical.
If the expression for the thermionic current in the absence of an external field is
\[ i=AT^n e^{-\frac{eV_k}{kT}}, \]
then in the presence of a strong external electric field it will be
\[ i'=AT^n e^{-\frac{V_k}{kT}} e^{\frac{e^{3/2}}{2kT}\sqrt{\frac{V}{r\lg\frac{R}{r}}}}. \]
EXTRACTION OF ELECTRONS BY A STRONG ELECTRIC FIELD
and the ratio of the thermionic current in the presence of the field to the current in the absence of the field will be
\[ \frac{i'}{i} = e^{\frac{e^{3/2}}{2kT} \sqrt{\frac{V}{r\lg\frac{R}{r}}}} \quad\text{or}\quad \lg\frac{i'}{i} = \frac{e^{3/2}}{2kT} \sqrt{\frac{V}{r\lg\frac{R}{r}}}. \]
Schottky asserts1 that his experiments confirmed this dependence. Such an increase of the thermionic current should theoretically also extend to ordinary temperatures, when there is practically no thermionic emission; i.e., in a very strong electric field electrons should fly out even from a cold metal, giving a measurable current. Concerning the magnitude of the electric field necessary for this, Schottky2 gives certain indications in another paper of his, in 1923. At the surface of the metal it must be of the order of \(10^7\) V/cm in order completely to compensate the contact field of the metal.
The passage of current between two very closely spaced points or spheres, where fields of this order may be expected when a large potential difference is applied to them, has been observed by many authors. But great experimental difficulties and the presence of side effects make it hard to obtain the phenomenon of electron emission by cold metals in pure form. It is very difficult to obtain an electric field of the order of \(10^6\) V/cm in air between closely spaced points, since with electrodes of such a shape the discharge potential is very low. If, however, everything is placed in vacuum, the latter must be such that the ionic current not only does not obscure the phenomenon but constitutes only an insignificant part of the observed electron current. The evolved gas will not only by itself, in the form of ions, increase the measured current, but by bombardment of the cathode will also cause additional electron emission, analogous to the emission of cathode rays in a discharge tube. This additional electron emission, being determined by the energy of ionic bombardment, will be determined not by the gradient of the electric field at the surface of the cathode, but by the potential difference between the electrodes.
Moreover, as will be seen below, in very strong electric fields particles of metal are torn out from the cathode and the anode; the chief cause of this is mechanical imperfections of the electrodes: poor polishing, destruction of the surface during preliminary degassing of the electrodes, and liberation of gas during the passage of current. Thus the necessary conditions are the best possible vacuum, careful polishing of the electrodes, and absence of gas evolution. Clarification of these circumstances, which determine the purity of the phenomenon, and study of their influence on the phenomenon constitute the main substance of the majority of works carried out in this field up to 1926.
Before proceeding to an account of the most recent works aimed at revealing the nature of this phenomenon, it is necessary to mention the fundamental arguments of Schottky1 concerning the influence of submicroscopic surface irregularities on the magnitude of the field gradient near the surface of a metal.
Let us imagine an absolutely smooth surface (Fig. 2) and on it a half-cylinder of radius \(\rho\). Near the surface of this cylinder, which represents a hump on the polished surface of the metal, the gradient of the electric field increases by more than a factor of two. If on the surface of this cylinder one places another half-cylinder of radius \(\frac{\rho}{2}\), then the gradient of the electric field again increases almost twofold (by analogy with the preceding case), and in all, consequently, by a factor of 4. Placing one upon another such
Fig. 2.
half-cylinders with radii decreasing each time by a factor of two, from \(2 \cdot 10^{-6}\) cm (the accuracy of optical polishing—\(1/20\) of a wavelength) down to \(2 \cdot 10^{-8}\) cm (the radius of an atom), it is necessary to double the gradient \(k\) times, where
\[ k=\frac{\lg \frac{\rho}{\rho_0}}{\lg 2} =\frac{\lg 100}{\lg 2} =6{,}7 \text{ times.} \]
Thus, a sharp hillock on the surface of a metal changes the magnitude of the gradient of the electric field so greatly that its true value is obtained from the calculated one by raising the latter to a certain positive power greater than unity.
Among the recent works pertaining to this subject, first of all one should set forth the investigation carried out by the Research Staff of the G. E. C. under the direction of Gossling2. The experiments were performed with well-pumped discharge tubes, in which the cathode, in the form of a loop of thin tungsten wire, was placed at a distance of 0.3 cm from a plane anode 2 cm in diameter. At such a distance between the electrodes the pla-
ELECTRON EMISSION BY A STRONG ELECTRIC FIELD
the anode can be regarded as infinite, and the gradient of the electric field at the surface of the wire could be calculated by the formula
\[ E=\frac{V}{r}\cdot\frac{1}{\lg \frac{1}{r}}. \]
Such a tungsten loop made of wire \(0.02\ \mathrm{cm}\) in diameter gave indications of the emission of electrons at a potential difference of \(6100\ \mathrm{V}\), i.e. at a field gradient \(E=7\cdot 10^{4}\ \mathrm{V/cm}\), while a wire of \(0.0016\ \mathrm{cm}\) gave a continuous electron current already at \(25000\ \mathrm{V}\), i.e. at \(E=4.4\cdot 10^{6}\ \mathrm{V/cm}\). Cathodes of other forms were also used: a brush of thin wires and the end of a wire of diameter \(0.002\ \mathrm{cm}\) and \(0.0002\ \mathrm{cm}\).
First of all it was observed that, under electron bombardment, only small spots on the surface of the anode become incandescent, i.e. that the electrons are emitted not by the entire surface of the wire cathode, but only by certain points on it, which also determine the measured current. That this circumstance is not due to the properties of the anode surface was shown in the following way. The cylindrical anode (Fig. 3) could rotate about the axis \(a\) while the inclination of the whole discharge tube was varied. Then, when the anode was rotated, the incandescent spot on it moved, remaining all the time opposite the cathode; the current strength, however, did not change noticeably. Such point emission of electrons is evidently explained by submicroscopic irregularities of the surface. Hence it follows quite obviously that the density of the electron current, as inferred from the dimensions of the cathode, does not at all correspond to the true current density. One can obtain some idea of this true density of the electron current if one takes into account that, from a cathode in the form of a wire \(0.0002\ \mathrm{cm}\) in diameter, placed perpendicular to the anode surface, a current of \(3\cdot 10^{-2}\ \mathrm{A}\) can be obtained. The area of electron emission here will be of the order of \(10^{-7}\ \mathrm{cm}^{2}\), and the current density is found to be greater than \(10^{5}\ \mathrm{A/cm}^{2}\). It is quite understandable that, with such a large current density, the individual places of the cathode surface emitting electrons become white-hot.
Fig. 3.
The characteristics (curves of the electron current as a function of the potential difference at the electrodes) obtained were as follows: if the quantity \(\sqrt{V}\) is plotted along the axis of abscissae, and \(\lg i\) along the axis of ordinates, straight or slightly curved lines are obtained (Fig. 4). Thus the emission of electrons by a cold cathode roughly follows the law
\[ i=Ae^{pV^{\frac12}} \]
over a range of current values from measurable ones up to \(10^{-1}\ \mathrm{A}\). The curves reproduced from the article represent the case of a loop cathode with a thickness of
wire \(0.0016\ \mathrm{cm}\). Curve 1 was taken before annealing the wire, curve 2 after heating it to \(2300^\circ\ \mathrm{K}\), curve 3 after further heating to \(2300^\circ\ \mathrm{K}\), and curve 4 after rupture of the wire. From the curves it is seen that degassing of the cathode leads to a parallel displacement of the characteristics and to higher values of the potential difference without
Fig. 4.
a significant change in the slope of the characteristics. Heating of the cathode by current from an external voltage source from \(300^\circ\ \mathrm{K}\) to \(1600^\circ\ \mathrm{K}\) in most cases caused a parallel displacement of the characteristics. Fig. 5 presents the characteristics at various temperatures for the same loop cathode before (a) and after (b) annealing. As
Fig. 5 a.
is seen from them, when the absolute temperature is doubled or tripled, the current changes approximately in the same ratio, i.e., as if the temperature changed by about 5 percent. The curves shift almost parallel toward smaller values of the potential difference, as required by the Schottky theory, but by an amount which, as already indicated, by no means corresponds to the one required.
tion. Moreover, this displacement is not proportional to the temperature, and it is as if for each state there exists a temperature at which the displacement is greatest.
This further complicates the question, all the more so because, after heating the cathode above this definite temperature, the emission differs from the initial one.
Millikan and Eyring\(^1\) investigated this in somewhat greater detail in their work. A thin tungsten filament of diameter \(0.00123\) cm was suspended and stretched by a load (iron) of \(4\) g along the axis of a copper cylinder with an internal diameter of \(1.625\) cm. When the wire was heated by current from an external voltage source to \(1100^\circ\) K, the lowering of the iron load due to the stretching of the wire was restrained by the magnetic field of a small electromagnet placed below. By this means the wire was protected from breaking. Everything was placed in a tube of “Pyrex” glass and was preheated. In addition, the cylinder was annealed by electron bombardment from the loosened spiral surrounding it, creating a potential difference of \(2000\) V between it and the spiral. The electron current to the cylinder (the filament was grounded) was measured by a quadrant electrometer or by a galvanometer with a variable shunt. A current from the cylinder of \(2 \cdot 10^{-12}\) A could be detected.
Fig. 5 b.
First of all, Millikan observed that the electron current always produces a certain change in the structure of the surface of the wire, which becomes especially evident if the wire is fresh, i.e. the potential is applied for the first time after the thermal treatment (annealing) of the wire. Further extraction of electrons becomes more and more difficult, and the value of the critical potential difference (at which a noticeable emission—\(2 \cdot 10^{-12}\) A—begins) constantly increases. Subsequently, if the electron current does not exceed in magnitude the greatest value of the current in the preceding experiments, the curves become reversible. Such treatment of the surface by the electron current in the process of emission may be explained by the smoothing of irregularities on the surface of the wire by the positive ions of the residual gas, which decreases the local value of the electric-field gradient at the surface of the cathode. Sometimes, as the potential difference between the cylinder and the filament increases
\(^1\) Millikan a. Eyring, Phys. Rev. 27, 51, 1926.
the electron current suddenly increases by several thousand times, and luminous spots appear on the inner surface of the cylinder. These luminous spots constantly remained in one place, while their intensity fluctuated and in time became established. Correspondingly, the electron current also ceased to fluctuate. Apparently the liberation of gas, by increasing the ion current, also increased the bombardment of the cathode. The latter could not only form, at some point on the surface, an irregularity that entailed a strong increase of the local value of the gradient, but could also, by bombardment, produce strong electron emission.
In the experiments of Millikan and Eyring one can also determine the value of the potential gradient at the surface of the wire (at least as a lower limit of its possible true value). Since the field is cylindrical, the value of the field gradient at the surface of the wire is
\[ E = \frac{V}{r_1}\cdot \frac{1}{\lg \frac{R}{r}} = 228\ \text{V}. \]
Under these conditions the critical gradient was of the order of \(4\cdot 10^5\ \text{V/cm}\). In individual cases it depended on the treatment of the filament during annealing or in the process of electron emission. Thus a prolonged process of electron emission, as well as annealing to a very high temperature (\(1700^\circ\ \text{K}\)), raises the value of the critical potential difference; annealing to a temperature not so high (about \(1000^\circ\ \text{K}\)), on the contrary, lowers the critical potential difference. The reason is evidently that, when the wires are not annealed at a very high temperature, the liberated gas destroys the surface, creating new irregularities on it, which in the course of subsequent emission are smoothed out by ion bombardment, while during annealing at a very high temperature they are fused over. An excessive increase of the field gradient at the surface of the wire may also contribute to destruction of the surface and lead to a sudden increase of emission.
The characteristics (Fig. 6) obtained by Millikan and Eyring are always curves concave toward the axis of field voltage, not agreeing with the results of the preceding author’s work on the fulfillment of the law \(i = A e^{pV^{\frac{1}{2}}}\). The characteristics taken at a higher temperature—\(900^\circ\ \text{K}\)—show complete, to within 1%, independence of temperature; at a cathode temperature of \(1100^\circ\ \text{K}\), however, the characteristic shifted almost completely parallel toward lower voltage values, but by a very small amount (the current increased on the average by 12%), not corresponding to the change in temperature. Measurements at temperatures above \(1100^\circ\ \text{K}\) are not given in the work.
In comparing the characteristics of Millikan and Eyring with Gossling’s characteristics, it is evident that in order to obtain one and the same current (\(10^{-7}\ \text{A}\)), Millikan and Eyring had to apply
to the electrodes a potential difference 6 times smaller than Gossling’s. The suspicion arises whether this is not explained by a less perfect vacuum, where there would be enough gas to increase the current. Millikan believed that his pressure was less than \(10^{-6}\) mm Hg, since the MacLeod manometer (down to \(10^{-6}\) mm Hg) no longer gave a reading. As has already been mentioned, the presence of gas distorts the phenomenon not only by giving an ionic current, but also by the fact that ionic bombardment causes additional emission of electrons. Can the increase of current with temperature not be explained by the liberation of gas from the electrodes and the corresponding increase of ionic current? Unfortunately, Millikan and Eyring used wire of only one diameter, and therefore it was impossible to find separately how the increase of current depended on the potential difference and on the gradient of the electric field at the cathode surface.
Del Rosario\(^1\) drew attention to this circumstance. He carried out experiments with wires of different diameter and measured the pressure with a Dushman-type ionization manometer. At the best vacuum (in his case \(10^{-8}\) mm Hg), with a field gradient at the cathode surface of \(12.1 \cdot 10^{6}\) V/cm, he obtained a current of \(4 \cdot 10^{-11}\) A; however, when he admitted a small amount of air into the apparatus, at \(10^{-4}\) mm Hg, he obtained a reversible curve very similar to the curves of Millikan and Eyring.
Fig. 6.
Taking another wire diameter (twice as large), he observed the appearance of a current (\(2 \cdot 10^{-11}\) A) not at the same electric-field gradient, but at the same value of the potential difference between the electrodes. This compels one to suppose that, both in Gossling’s experiments and in those of Millikan and Eyring, the effect of electron extraction was completely masked by the ionic current and by ionic bombardment as a result of the not entirely careful degassing of the electrodes, and also in view of the insufficient vacuum and the presence of mercury vapor, the absence of which cannot be proven by the MacLeod manometer.
\(^1\) Del Rosario, Journ. of the Franklin Institute, 203, 243, 1927.
On this basis it is difficult to speak of a discrepancy between the temperature dependence of the electron current obtained by Millikan–Eyring and by Gossling and Schottky’s ideas concerning the nature of cold emission, and still less of such a temperature dependence establishing the untenability of the theory of free electrons.
But, moreover, even if every ionic current is actually absent, experiment can hardly give the true dependence of such electron emission in an electric field on temperature.
The point is that, as has already been mentioned, Schottky, from consideration of the influence of submicroscopic irregularities of the surface, came to the conclusion that the true value of the electric-field gradient at the surface is related to the gradient calculated from the assumption of its geometrically regular form by the relation
\[ \left(\frac{dV}{dx}\right)_{\text{true}} = \left(\frac{dV}{dx}\right)_{\text{calculated}}^{k}, \]
where \(1 < k < 10\) for an optically polished surface. Then in Schottky’s formula
\[ i=A e^{\frac{e^{3/2}}{2kT} \sqrt{\frac{-V}{\,r\,\lg \frac{R}{r}}} -\frac{b_k}{kT}} \]
the exponent takes the form
\[ \frac{ b_k - C \sqrt{ \left( \frac{-V}{\,r\,\lg \frac{R}{r}} \right)^{k} } }{kT}. \]
With increasing temperature \(T\), in this expression not only does the exponent \(k\) increase because of the growth of irregularities, but so also does \(r\)—the radius of the wire—owing to thermal expansion. To what degree an increase in these factors will affect the change of the whole expression with temperature is difficult to say; these subsidiary phenomena may explain an insufficient increase of electron emission with temperature.
But roughnesses on the surface may also be more significant. Thus Millikan and Eyring recorded a sudden increase of electron emission. It is possible that a bump appeared on the surface of the tungsten filament, having been torn out of the corresponding incandescent place on the anode. Likewise de la Rosa did not succeed in one case in producing on the surface of the cathode—a quartz filament platinized by cathodic sputtering—an electric-field gradient greater than \(1.1 \cdot 10^{6}\) V/cm, since all the sputtered platinum flew off the filament.
Such ejection of matter by electrodes in a strong electric field is studied in especially great detail in a very carefully executed work by Rother1 on electron emission.
This author’s apparatus was as follows. The glass cylinder of the discharge tube (Fig. 7a) was closed on one side by a corrugated platinum membrane 2.5 cm in diameter, at the center of which, on both sides, two small threaded cylinders were soldered. Onto the cylinder directed inward into the discharge tube an electrode was screwed; the other cylinder was screwed together with the movable part of a Fabry and Perot interferometer. The corrugated membrane made it possible to move this electrode by shifting the movable part of the interferometer with the aid of a micrometer screw. The displacement of the electrode could be determined to 0.00001 cm and checked by the displacement of equal-inclination fringes in the light of the mercury line
Fig. 7a.
between two mirrors coated by cathodic sputtering. The other electrode was screwed onto a small cylinder welded to the glass tube, clamped in the fixed part of the interferometer. The electrodes were rods 4 mm thick, made of various metals, spherical (radius of curvature 20 mm) or conical, the ends of which were optically accurately polished.
Rother also used a discharge tube of another type (Fig. 7b). One electrode, in the form of a point or a rod with a rounded end, again with the aid of a corrugated membrane, could approach or recede from the other electrode—a platinum strip—when the carriage of the interferometer was moved. The platinum strip could be moved parallel to itself by means of two other membranes, so that a fresh metal surface, not damaged by the ejected particles, could be placed opposite the point. In addition, by this means it could always be kept taut after each annealing.
Experiments established that particles of metal are torn out both from the cathode and from the anode (spherical electrodes). On a copper anode microscopic pieces of iridium were obtained, and on an iridium cathode coarser
particles of copper. Both in air and in vacuum such expulsion of metal particles occurred in the same way. The crystalline structure of the torn-out particles and the entirely random distribution of the craters (the places of the torn-out particles) indicate that the surface structure at the crater site has a great influence on the expulsion of particles. But since the places of greatest accumulation and tearing out of particles often do not coincide with known surface irregularities, one may think that the particles are expelled by the gas being evolved; indeed, in general the very fact of matter being expelled should not necessarily be accompanied by a jump in current, as occurs in the experiment. By annealing the electrodes, Roter
Fig. 7b.
actually observed that degassing the electrodes can prevent the expulsion of metal particles from the electrodes.
After polishing the electrodes a second time after annealing, Roter obtained characteristics differing somewhat from those of Millikan—Eyring and Gossling. Plotting likewise along the abscissa axis \(\sqrt{V}\), and along the ordinate axis the logarithm of the electron current, Roter (in Roter’s paper the current is plotted directly as a function of the potential difference) obtained curves slightly convex with respect to the \(\sqrt{V}\) axis, and not concave as in Millikan. In Figs. 8(a) and (b) are shown curves, recalculated in this way, taken at various distances between the electrodes (indicated for each curve). 8a contains curves taken with a point opposite a spherical electrode; 8b—curves taken when both electrodes are spherical. Constructing, on the basis of these results, curves of equal current, Roter obtains not straight, but slightly curved lines. Apparently, at large distances the curvature of the electrodes has an effect. It is pos-
possibility of obtaining such constant-current curves indicates that the emission of electrons depends no longer on the potential difference between the electrodes, but on the potential gradient at the surface of the cathode, and that Rothe was thus able to obtain an effect free from the ionic current imposed on it in the presence of gas and from the additional emission of electrons due to ionic bombardment. The form which Rothe gave to his electrodes does not make it possible to calculate theoretically the gradient of the electric field at the cathode surface and to derive the characteristic equation for this case on the basis of Schottky’s ideas.
Fig. 8a.
Del Rosario’s doubt¹ that the electron current is determined by the gradient of the electric field at the cathode surface, expressed by him also in his second article (see below), gave rise to a number of works defending Millikan’s point of view. Thus Pierson², using an apparatus analogous to that of Millikan and Eyring, and a vacuum of \(10^{-8}\) mm Hg measured by an ionization manometer, confirmed the results of Millikan and Eyring.
Moreover, Millikan and Lauritsen³ found that their results, as well as the results of Millikan and Eyring, satisfy the dependence
\[ i = A (T + CE)^2 e^{-\frac{b}{T+CE}}, \]
¹ del Rosario, l. c.
² Piersol, Phys. Rev. 31, 441, 1928.
³ Millikan a. Lauritzen, Proc. of the Nat. Ac., Sc. 14, 45, 1928.
where \(E\) is the gradient of the electric field at the surface of the cathode. Into this dependence there fit well the points obtained by the authors in the investigation of currents from filament to cylinder and, others, from point to plane over a number of years.
Lauritsen’s expression for \(T=0\) gives the effect under consideration for \(E=0\)—the thermionic effect; while for \(T\) comparable with \(E\), it gives the complex effect also mentioned in the work of Millikan and Eyring; consequently, the application of an external field is equivalent to an increase in the “temperature” of the electrons, to an increase in their kinetic energy.
Moreover, Eyring, Mackeown, and Millikan\({}^{1}\) next give the results of investigating currents from a point to a plane at various distances between them. Calculating the electric field between the electrodes under the assumption that the point is a hyperboloid of revolution, they were able to compute an approximate value of the field gradient at the surface of the point. Then, plotting along the abscissa axis \(\frac{1}{E}\), and along the ordinate axis \(gi\), they obtained almost coincident straight lines for different distances between the point and the plane; whereas, plotting \(gi\) as a function of \(\frac{1}{V}\), they obtained parallel straight lines lying far apart from one another.
Fig. 8b.
Incidentally, the results of Millikan and Eyring, as well as those of Gossling, represented in this way, also give straight lines.
It should be noted that del Rosario\({}^{2}\) obtains precisely the opposite results. The curves \(g\left(\frac{i}{r}\right)=f(E)\) give completely noncoinciding straight lines, while the points of the curve \(g\left(\frac{i}{r}\right)=f(V)\) for different \(r\) (\(r\) is the radius—
\({}^{1}\) Eyring, Mackeown and Millikan, Phys. Rev. 31, 900, 1928.
\({}^{2}\) del Rosario, Journ. of Franklin Inst. January, 1928.
wires of the cathode and, consequently, \(\frac{i}{r}\)—a quantity proportional to the current density—lie on one straight line.
On the other hand, Fort1 investigated the increase of the saturation current caused by a strong electric field. He confirmed the validity of Schottky’s theory within the limits \(1400—2100^\circ K\) and electric fields up to \(7\cdot 10^5\ \mathrm{V/cm}\).
The experimental investigations presented here show that the increase of the saturation current caused by an external electric field fully follows Schottky’s theory. Further, the experiments of Millikan and his collaborators lead to the dependence
\[ i = A(T + CE)^2 e^{-\frac{b}{T+CE}}, \]
which is unjustified when \(T\) is comparable with \(E\) (Fort).
Rother’s characteristics more nearly satisfy the dependence
\[ i = Ae^{BE}, \]
than Schottky’s expression, and, finally, del Rosario concludes from his experiments that the electron current is determined to a greater extent by the potential difference between the electrodes than by the electric field at the surface of the cathode.
Such divergence in the results of individual authors served as an impetus for the theoretical development of this phenomenon and for the appearance of more modern theories of it.
Thus, Richardson2 attempted to give a theory of cold discharge, using Schrödinger’s wave mechanics. For the density of the electron current as a function of the electric-field strength he obtained the equation
\[ I = A\left(\frac{1}{\sqrt{E}} + C\right)^2 e^{-\frac{B}{\sqrt{E}}}, \]
where \(A\), \(B\), and \(C\) are constants into which temperature does not enter explicitly. Using the experimental data of Millikan and Eyring to compare with the results of his theory, he notes a slight systematic divergence from his theory. But the results of Millikan and Eyring, as already indicated above, cannot serve to test the theory of the phenomenon. Another theory was given by Houston3. Basing himself on the new elec-
of Sommerfeld’s electron theory of metals1, which retains the classical conception of free electrons but applies to them the new quantum statistics of Fermi—Dirac, he obtains the following formulas for the emission of electrons.
If the external field does not fully compensate the contact field, then the density of the electron current is given by the formula
\[ I=\frac{4\pi e m}{h^{3}}\,(kT)^{2}\,e^{-\frac{W_{a}-W_{i}}{kT}}, \]
where \(W_{a}\) is the contact work. This dependence, for \(W_{i}\)—the change of the contact work—equal to zero, passes into the Dushman formula.
But when the external electric field compensates the contact field or exceeds it, the dependence has the form:
\[ I=\frac{\pi e m}{h^{3}}\left\{(W_{i}-W_{a})^{2}+\frac{\pi^{2}}{3}(kT)^{2}-(2kT)^{2}e^{-\frac{W_{i}-W_{a}}{kT}}+\cdots\right\} \]
Houston gives in his work a table of values of the current density as a function of \((W_{a}-W_{i})\), for various temperatures.
The dependence
\[ i=A\,(T+CE)\,e^{\frac{bk}{T+CE}} \]
obtained experimentally by Millikan and Lauritsen was derived theoretically by Nordheim and Fowler2, using wave mechanics and Fermi–Dirac statistics. The numerical value of the constants entering into their formula is such that appreciable emission is obtained at electric fields not much greater than \(10^{7}\ \mathrm{V/cm}\). In Millikan, Eyring, and others it begins earlier—already at \(10^{6}\ \mathrm{V/cm}\); in Petter it begins later. However, as the authors themselves note, it is still impossible to compare the results of Houston’s or Nordheim and Fowler’s theory with the experimental data, because of the above-mentioned considerations concerning the influence of roughnesses on the metal surface.