EXTRACTION OF ELECTRONS BY AN ELECTRIC FIELD

P. I. Lukirsky

1. THE CURRENT STRENGTH OF ELECTRON EMISSION

It was shown experimentally long ago that an electric current arises between two electrodes situated in a perfect vacuum if a strong electric field is created between them. This phenomenon was studied in detail and with great care in the works of Millikan and Cameron, who showed that the current between the electrodes cannot be due to ionization of the residual gas in the apparatus. The source of the current is electrons torn by the electric field from those points of the cathode at which the gradient of the electric field has the greatest value. The observed phenomenon does not depend on the temperature of the cathode and occurs at any low temperature; therefore it is often called “cold” electron emission.

Modern ideas about the nature of metals give an elegant quantum-mechanical explanation of this phenomenon, developed in the theory of Fowler and Nordheim1. At absolute zero temperature there exists in a metal an energy distribution bounded by a maximum energy \(W_f\). At the metal—vacuum boundary there is a potential barrier, the upper boundary of which is located by an amount \(\varphi_0\) (Fig. 1) higher than the greatest value of the energy at absolute zero temperature. This quantity \(\varphi_0\) is called the work function. In the absence of an electric field, emission is impossible. If, however, a strong electric field is created near the surface of the metal, the form of the potential barrier will have the appearance shown in our figure. The electrons of the metal, although their kinetic energy is less than the height of the barrier, can nevertheless pass “through” it and produce a current. The strength of this current will in general be equal to

Fig. 1.

\[ i=\int_{0}^{\infty} N(W)D(W,F)\,dW, \tag{1} \]

where \(N(W)\) is the distribution of electrons inside the metal over the normal components of the energy \(W\), and \(D(W,F)\) is the coefficient of penetration of electrons through the barrier—a function both of the electron energy and of the magnitude of the electric field \(F\). Fowler and Nordheim calculated the probability \(D(W,F)\) of penetration through the barrier and accordingly found the following expression for the current density:

\[ i=6.2\cdot 10^{-6}\, \frac{W_i^{1/2}}{(W_i+\varphi_0)\varphi_0^{1/2}}\, F^2 e^{-\frac{6.85\cdot 10^7}{F}\varphi_0^{3/2}} \ \text{amp cm}^{-2}, \tag{2} \]

where \(F\) is the field in volts per centimeter, and \(\varphi_0\) and \(W_i\) are in electron-volts. This fundamental Fowler–Nordheim formula was derived by them under the assumption that at the boundary of the metal there exists a rectilinear potential barrier. Assuming that the shape of the barrier is determined by the presence of electric image forces, Nordheim\(^2\), for such a barrier, obtained the corrected expression

\[ i=6.2\cdot 10^{-6}\, \frac{W_i^{1/2}}{(W_i+\varphi_0)\varphi_0^{1/2}}\, F^2 e^{-\frac{6.85\cdot 10^7\theta \varphi_0^{3/2}}{F}}, \tag{3} \]

where

\[ \theta=\theta\!\left(\frac{3.62\cdot 10^{-4}\sqrt{F}}{\varphi_0}\right) \]

is a function changing its value from zero to unity as the field changes from \(\infty\) to zero. For fields smaller than \(10^7\ \mathrm{V\,cm^{-1}}\), it is very close to unity. This function has been tabulated.

As formulas (2) and (3) show, the current density emitted by a metal under the influence of an electric field depends very strongly on the magnitude of the work function of the metal and on the field strength. Experimental investigation of the phenomenon of electron extraction by a field is very difficult, since the current density depends strongly on the magnitude of the electric field at the cathode surface. This circumstance accounts for the contradictory results obtained by a number of authors. Indeed, in the presence of roughnesses and points on the surface, the electric field near them will have a value many times greater than the field determined by the geometry of the apparatus and the electrodes. Emission will in practice proceed only from individual places on the cathode, and its magnitude will be very difficult to compare with the theoretical dependence.

As a result, electron emission under the action of a field occurs, for ordinary metals, at fields of \(10^6\ \mathrm{V/cm}\), whereas, according to the theory, practically noticeable emission should begin at fields greater than \(10^7\ \mathrm{V/cm}\).

Quantitative study of this phenomenon became possible only after a number of investigators succeeded in producing a smooth metal surface. If a very thin conical point is made from a metal and then heated in vacuum to a high

temperature, but still lower than its melting temperature, then the end of the point is rounded in the form of a regular smooth hemisphere. In various experiments the radius of such hemispheres had different values, from a micron to tenths and even hundredths of a micron. If such a point is placed inside a spherical vacuum vessel coated on the inside with a conducting layer, then, by applying a potential difference, we shall have near our hemisphere a very strong radial field with a possible intensity up to a value of \(10^8\ \mathrm{V/cm}\). Benjamin and Jenkins \(^{3}\) succeeded, for a molybdenum point, and Haefer \(^{4}\), for a tungsten point, in measuring the current at different fields and in showing complete agreement of the observed current magnitude with the value calculated from the Fowler and Nordheim theory. If on the ordinate axis one plots the value \(\lg \dfrac{i}{F^2}\), and on the abscissa axis \(1/F\), then the experimental data fit well on a straight line, as follows from the theory [see formulas (2) and (3)], and the slope coefficient agrees with the theoretical one, if for the work functions \(M_0\) and \(W\) one takes well-known values.

It should be noted that, because of the smallness of its dimensions, measurement of the radius of curvature of a point is very difficult to carry out. Haefer determined the shape and radius of the point, using for this purpose an electron microscope with magnification up to 20,000 times. In the same work Haefer, after measuring the current from tungsten, deposited barium, potassium, and cesium on its surface. Examination of the surface with an electron microscope showed that in a number of cases the deposited layers are formed in the form of crystals; this, apparently, is responsible for the erroneous data that were obtained by Miller \(^{5}\) for the dependence of the emission current on the work function. If observations are made of emission from tungsten coated with layers of barium, potassium, and cesium, both thick and optimal, under the condition that the surface is checked with an electron microscope (absence of crystallites), then within the limits of experimental error the Fowler–Nordheim law

\[ \lg \frac{i}{F^2}=\lg A-\frac{B\varphi_0^{3/2}}{F}, \tag{4} \]

for the dependence of emission on the work function \(\varphi_0\) is well justified. Thus we see that, in the main, the experimental data for metals for the total emission current, its dependence on the field and on the work function, are in good quantitative agreement with the Fowler and Nordheim theory.

In the electron laboratory of the Leningrad Physico-Technical Institute, B. G. Brezhnev investigated the phenomenon of electron ejection under the influence of a field from the surface of a cesium-antimony cathode. For the investigation a spherical vacuum device was constructed, the inner electrode of which consisted of a very thin, drawn-out glass point, rounded at the end in a flame.

burner. On this glass hemisphere, of diameter \(\sim 10\mu\), a sensitive antimony–cesium layer was formed. The second electrode was the outer glass sphere. It was coated on the inside by cathodic sputtering with a semitransparent layer of platinum, on top of which a layer of willemite was deposited. This made it possible, in addition to measuring the current intensity, also to make a visual observation of the current distribution. The measurements showed that, in the case of an antimony–cesium cathode, emission begins at values of the electric field \(\sim 10^{5}\ \mathrm{V/cm}\) and grows very rapidly with a further increase in the field strength. When the direction of the electric field was changed, even at the highest voltages, no current was present. Moreover, when the entire apparatus was cooled with liquid air for 8 hours, neither the visual pattern of emission nor the current intensity changed. This gives us confidence that in this case we are dealing with the extraction of electrons by the field. In Fig. 2 a graph is given in which the quantity \(\ln \dfrac{i}{F^{2}}\) is plotted as a function of \(1/F\).

Fig. 2.

The data obtained fit a straight line well. However, if one calculates the work function of the cathode from the slope of this straight line, a very small value is obtained, \(\varphi_{0}=0.062\ \mathrm{V}\), whereas the value of the work function measured photoelectrically is \(1.3\ \mathrm{V}\). This is due to the fact that the true field at individual points of the cathode is many times greater than the field calculated from the dimensions of the electrodes. Indeed, visual observation shows that whereas the photoeffect from the cathode gives a uniform glow of the willemite over the entire sphere, emission under the influence of the field is concentrated in the form of separate spots and stripes with very different luminous intensities. If one turns to the work of S. A. Vekshinsky\(^6\) on the investigation of antimony layers, the resulting emission pattern is readily explained by the fact that spherulites of antimony partially separate from the glass and their edges bend in the form of sharp rims. When antimony is activated with cesium, this form of the layer is essentially preserved. Naturally, near these sharp edges of the spherulites the electric field reaches very large values, and it is from these places, mainly, that the extraction of electrons by the field occurs. It is curious to note that in some instruments it was possible to observe on the screen, at greatly increased magnification,

scale, whole individual spherulites and their mutual glow, studied in detail by S. A. Vekshinskii. If one estimates the ratio of the true field at individual points of the cathode to the value geometrically calculated, assuming that the value of the work function is equal to \(1.3\,V\), and applying the Fowler–Nordheim formula to this case, then we obtain a value \(\sim 100\). This quantity characterizes for us the magnitude of the roughness of our antimony–cesium method. We see that, while the question of the magnitude of the emission current for metals has been studied in detail, for complex cathodes the study of emission under the influence of a field is only beginning. Let us proceed to consider the question of the energy distribution of electrons extracted under the influence of a field.

2. DISTRIBUTION OF ELECTRON ENERGIES

As calculations show, the transparency coefficient of the potential barrier \(D(W,F)\) depends very strongly on the magnitude of the normal component of the electron energy \(W\). The energy distribution of electrons inside the metal is known; therefore an experimental study of the energy distribution would be valuable for the theory of the phenomenon of extraction of electrons by a field, since comparison of the energy distribution inside and outside the metal would give the direct value of the dependence of the transparency coefficient on the electron energy.

Fig. 3.

Such experiments to study the energy distribution were carried out by Henderson and Dahlstrom\(^7\). If a strong accelerating field, sufficient in magnitude for extracting electrons, is applied between the cathode and the grid surrounding it, and then a retarding field is created between the grid and the receiving electrode embracing it, then, by varying the magnitude of this field, one can study the energies of the electrons extracted by the field from the cathode. The scheme of such an experiment is shown in Fig. 3, where a graph of the potential energy is given. An electron from the cathode penetrates through a narrow potential barrier, the width of which is determined by the presence of a large field gradient at the cathode. It is then accelerated by the electric field in the direction of the grid. After passing through it, it will be slowed down and will approach the receiving electrode. However, it can reach the receiving electrode only if the potential of the latter is increased by the value \(\varphi_1\), since the field gradient at the receiving electrode is small and no penetration through the barrier can occur.

Therefore the current to the collecting electrode will begin only under the condition that the accelerating potential difference is greater than the retarding one by $\Delta V_i=\varphi_i$ volts. With a further increase of this value $\Delta V$, electrons from energy levels lying below the value $W_i$ will also penetrate into the collecting electrode; for them, however, $D(W,F)$ will be smaller. By measuring the current intensity as a function of the potential difference and differentiating this dependence, we shall evidently find the energy distribution of the electrons extracted by the field from the cathode.

Henderson and Dahlström carried out such measurements and obtained energy-distribution curves for different values of the field at the cathode surface. The maximum on these curves lies in the region of maximum energies; toward lower energies the curve gradually falls. Qualitatively these results agree with theoretical expectations, since electrons with energies smaller than $W_i$ have a lower probability of penetrating through the barrier.

However, there is no quantitative agreement, since the width of the energy distribution, according to the experimental data of Henderson and Dahlström, is of the order of $10\ \mathrm{V}$, whereas the entire range of electron energies inside tungsten is only of the order of $5.7\ \mathrm{V}$. This is all the more contradictory because, when the electron energy changes from the value $W_i$ by $1\ \mathrm{V}$ downward, the transparency coefficient of the barrier, according to the theory, decreases by a factor of 600. In our view, such a discrepancy between the theoretically expected result and the experimental data of Henderson and Dahlström is explained by the fact that in their experiments they used a cylindrical case. Indeed, when passing through the holes of the grid, the electrons are deflected by some angle from their original direction. The resulting component of the energy along the generatrix of the cylinder in the retarding field will not change, and its magnitude will be equal to that energy interval which will be measured on the curve: current as a function of the retarding field. Since the electrons approach the grid with large energies, of the order of several thousand volts, a deflection through a small angle is sufficient to obtain a smeared-out energy distribution.

The situation will be different if, for the study of velocities, one uses the spherical case, namely, a double spherical condenser in which the electrons are first accelerated and then retarded. As was shown long ago, by the retarding-field method in a spherical condenser we measure the energy distribution of the electrons independently of the direction of the electron’s angle of emission relative to the normal. The resolving power of such a method, as was shown, is equal to

\[ \frac{\Delta V}{V}=\frac{r^2}{R^2}\sin^2\varphi, \tag{5} \]

where \(\Delta V\) is the inaccuracy in determining the energy of an electron having the energy value \(V\) electron-volts, \(r\) and \(R\) are the radii of the inner and outer spheres of the condenser, and \(\varphi\) is the angle between the normal to the inner sphere and the direction of flight of the electron. For a ratio \(r/R\) equal to \(1:10\), and for any value of the angle \(\varphi\), the resolving power of the method is equal to \(1/100\), which is quite sufficient for electrons of low energies (photoelectric effect, thermionic emission, etc.). In the case, however, of emission of electrons under the influence of a field, when the electrons approach the spherical grid with energies of the order of several thousand volts (at lower potentials the emission is very small), it is necessary that the angle of deflection \(\varphi\) should nevertheless be small. If, instead of a grid, one takes a sphere with separate small holes in it, then such openings will act as a diverging lens; moreover, the maximum angle of deflection of the electrons on passing through it will be approximately equal, as L. A. Artsimovich has shown, to

\[ \sin \varphi \simeq \frac{1}{4}\frac{\rho}{r}, \tag{6} \]

where \(\rho\) is the radius of the opening, and \(r\) is the radius of our sphere. Substituting for \(\rho = 0.02\ \text{cm}\), and for \(r = 1\ \text{cm}\), we obtain for \(\sin \varphi = 0.005\).

The total resolving power will then be equal (for \(r/R = \frac{1}{10}\)):

\[ \frac{\Delta V}{V} \simeq 10^{-2}\cdot 10^{-4} = 10^{-6}, \]

i.e. a value quite sufficient for an accurate investigation of the energy distribution. By this method of the double spherical condenser we have undertaken an investigation of the energy distribution in the extraction of electrons by a field.

Recently there came into our hands papers by Müller \(^{9,10}\), in which he studied the energy distribution of electrons when they are extracted by a field from tungsten by a somewhat similar method. After acceleration by the field, the electrons, instead of a grid, passed through a series of diaphragms, after which they entered a retarding field created between these diaphragms and a spherical receiving electrode. The presence of a series of diaphragms ensures the rectilinearity of the paths of the electrons passing through them, and the spherical shape of the receiving electrode ensures the orthogonality of the equipotential surfaces to the electron trajectories. Studying the energy distribution by this method, Müller found for the width of the energy spectrum a value equal to \(0.4\ \text{V}\), closely coinciding with that theoretically expected if the cathode temperature is taken to be \(500^\circ\text{K}\). However, in this work there is no proper control in the study of the distribution at different fields at the cathode surface and with different diaphragm apertures. The latter would be essential, since an exact accounting for the electron-optical properties of this series of diaphragms is difficult. In addition to the direct study of the electron energy distribution by the method indicated above, one can estimate the width

of the spectrum of this distribution by another method. If we take the simple spherical case in which the emission of electrons under the influence of a field is observed (an apparatus of the type of B. G. Brezhnev), then from the resolving power of this apparatus one can form an idea of the magnitude of the tangential components of the energy of the electrons. Indeed, electrons torn out by the field from some point on the surface of the sphere will produce a blurred spot on the outer sphere if the magnitude of the tangential energy is different from zero. Suppose that electrons with an energy corresponding to the value of the tangential component fly out in the direction tangent to the inner sphere. They will move along the hyperbola shown in Figure 4. Determining the point of intersection of the hyperbola with the outer sphere, we shall find the value of the diameter of the spot \(D\) as a function of the magnitude of the tangential energy \(V_t\) and of the magnitude of the total potential difference \(V_0\) applied to the capacitor. The approximate value of this quantity, given by L. A. Artsimovich, is:

Fig. 4.

\[ D = 4R \sqrt{\frac{V_t}{V_0}}, \tag{7} \]

where \(R\) is the radius of the outer sphere. It is obvious that such an apparatus, operating as an electron microscope, resolves two points on the surface of the inner sphere if the distance \(d\) (see Fig. 4) between them is equal to

\[ d = 4r \sqrt{\frac{V_t}{V_0}}, \tag{8} \]

where \(r\) is the radius of the inner sphere. In addition to the presence of a tangential component of the velocity, the resolving power of such a spherical electron microscope will also be determined by the diffraction of electron waves emerging from two neighboring points. The resolution due to diffraction will be the smaller, the smaller the potential \(V_0\) (the greater the de Broglie wavelength) and the larger the radius of the inner sphere \((d^2 \sim r\lambda)\). It must be noted, however, that the resolving power is determined mainly by the magnitude of the tangential energy, while diffraction plays a considerably smaller role under ordinary observational conditions. In the above-mentioned work of B. G. Brezhnev on the study of electron emission under the influence of a field from an antimony–cesium cathode, separate luminous spots were observed on the fluorescent screen.

If one measures the size of the smallest of these spots, then from the data given above one can estimate the maximum value of the tangential ...

...of the tangential component of the energy. This quantity was found to be equal to \(0.02\text{ V}\). Hence it is easy to conclude that the total energy of the electrons is of the same order, and that there can be no question of any \(10\text{ V}\) found by Henderson and Daltstrom. Müller gives an expression for the resolving power of his electron microscope and compares his theoretical calculations with observations of individual bright spots which are sometimes obtained in emission from pure metals. The data he gives agree well with his calculations. This is all the more surprising since Müller uses an approximate formula which differs by a factor of two from the certainly correct formula (7) given by us.

In conclusion I shall mention that such a spherical device, in which we have electron emission under the action of a field, is an electron microscope with a magnification of several hundred thousand times and with great resolving power.

3. EMISSION OF ELECTRONS AT ARBITRARY FIELDS AND TEMPERATURES

Above we considered questions concerning the total emission of electrons under the action of a field and the velocity distribution of these electrons. In this consideration we dealt with truly “cold” emission. Indeed, the Fowler–Nordheim theory was given by them for a metal temperature equal to absolute zero, while the cited experiments were carried out at low temperatures at which it may be assumed that the distribution of electron energies inside the metal practically does not differ from the Fermi–Dirac distribution at \(T=0^\circ\).

However, as is easy to see, the emission of electrons under the influence of a field and the distribution of energies in this emission must, generally speaking, depend on the temperature of the cathode.

Fig. 5.

In Fig. 5 a graph is given of the potential barrier near the surface of a metal placed in an electric field. Under the influence of the electric field there occurs, first of all, a lowering of the height of the barrier and, as a consequence, a decrease in the work function from the metal (the Schottky effect). The magnitude of this decrease in the work function \(\Delta\varphi_0\) in the field is, according to Schottky, equal to

\[ \Delta\varphi_0 = e^{3/2}\sqrt{F}, \tag{9} \]

where \(e\) is the electron charge, and \(F\) is the field at the metal surface. The distance \(x_0\) at which the potential energy has a maximum is equal to:

\[ x_0=\frac{e^{1/2}}{2\sqrt{F}} . \tag{10} \]

These formulas have been derived under the assumption that the forces acting on an electron outside the metal are forces of electric image attraction. In addition to the lowering of the barrier height under the influence of the external field, its shape changes, which in strong fields leads to emission of electrons “through” the barrier.

When the temperature of the metal changes from \(T=0\), the distribution of electron energies inside the metal changes owing to the appearance of a “Maxwellian tail” in the Fermi–Dirac distribution; therefore the energy distribution will extend appreciably above the boundary \(W\).

At high temperatures and not very strong fields, i.e. fields in which there is practically no penetration “through” the potential barrier, we shall observe ordinary thermionic emission from region \(A\), obeying the well-known Richardson–Dushman law. Added to it will be the escape of electrons from region \(B\), since these electrons, owing to the lowering of the work function, will now also possess an energy greater than the height of the barrier. In this case, as the external field increases, the lowering of the barrier will become greater and greater, which will lead to an expansion of region \(B\) and a corresponding increase in emission. The increase in thermionic emission due to the inclusion of electrons from region \(B\) is known as the Schottky effect.

If, conversely, strong fields and not too high a temperature are taken, then above the level \(W_i\) there will be a certain number of electrons owing to the “Maxwellian tail.” Since in strong fields the barrier is permeable to electrons, “cold” emission, obeying the Fowler–Nordheim law, will be observed from the main region \(D\); but at a temperature \(T\) different from absolute zero, emission from region \(C\), due to penetration through the barrier, will be added to it. This gives us a definite temperature dependence of the “cold” emission. Moreover, the energy spectrum of the electrons will also depend on the magnitude of the temperature, which, of course, can be established and studied experimentally.

If we take the general case of a large field and high temperature, then emission will occur from all four indicated regions \(A\), \(B\), \(C\), and \(D\); in this connection it must be said that between the emission from regions \(B\) and \(C\) there will be a continuous transition, since the probability of passage “through” the barrier for the upper part of the distribution of region \(C\) will be close to the probability of passage of electrons from region \(B\). It should additionally be noted that, when the barrier narrows in large electric fields, for a number of electrons from the complete

EXTRACTION OF ELECTRONS BY AN ELECTRIC FIELD

the distributions of their energies will satisfy the interference condition, as a result of which a periodically varying term will appear in the total emission. Unfortunately, this intermediate region of mean temperatures, as well as of high temperatures in very strong fields, has not yet been investigated, although at present all the necessary experimental prerequisites for this exist.

A precise study of the emission of electrons from metals, whether thermionic emission or the emission of electrons under the influence of a strong electric field, is at present complicated by the fact that, as experience shows, the work function is a function of the crystallographic direction. Indeed, a whole series of studies on the thermionic emission from various crystallographic faces of one and the same single crystal, as well as studies determining the threshold of the photoelectric effect, have established differences in the magnitude of the electron work function. Thus, for example, in investigating thermionic emission from a single-crystal tungsten wire, Nichols¹¹ found different values for different crystallographic directions. (Here, as below, by a crystallographic direction with indices \(h, k, l\) we shall mean the direction perpendicular to the plane \(|h, k, l|\).)

As can be seen from the appended table, the values of the work functions fluctuate within the limits of \(0.3\,V\). Differences of this order were also found in the experiments of other investigators and for other single crystals. It is easy to see that, when studying emission from polycrystalline bodies, this presence of different work functions for different faces must affect the measured emission current.

In this case the results of observation will depend on the electric fields in which the phenomenon is observed. This problem is quite analogous to that which we have in considering “spots” on the surface of a metal, possessing a different work function (thorium spots on tungsten).

If the magnitude of the applied field is small, so that the distance at which the external field is equal to the force retaining the electron is greater than the magnitude of the inhomogeneities of the metal (the size of the spots or the size of the crystal of a polycrystalline body), then we shall observe an effect determined by the mean value of the work function. In this case the barrier near the surface will be determined not only by the work function, but also by those contact fields which compensate the difference in work functions. Conversely, in large fields, when the distance \(x_0\) is considerably smaller than the size of the regions with a definite work function, the current will be emitted independently by different portions of the surface and will be equal to

\[ i=\sum f_A i_n, \]

where \(f_n\) is the part of the area with work function \(\varphi_n\), and \(i_n\) is the corresponding current density. This circumstance leads to the fact that, in large fields, emission will occur chiefly from places with a low work function; only the effective emitting area will be smaller. The Schottky dependence will in this case be satisfied. At small fields there will be another law of emission, which, as is known, leads in weak fields to a deviation from the Schottky straight line. It should be noted that a deviation from the Schottky law in weak fields is also observed for pure polycrystalline metals. Naturally, in the case of studying the emission of electrons under the action of strong electric fields, we always have the case in which \(x_0\)—the distance at which the external field equals the retaining force—is very small; therefore we have separate emission from different places on the surface, provided only that the regions with different work functions do not have dimensions of the order of atomic distances. In the latter case, even for strong fields we would have an effect corresponding to the averaged value of the work function.

4. EMISSION OF ELECTRONS UNDER THE ACTION OF AN ELECTRIC FIELD BY SINGLE CRYSTALS

Let us turn to the experiments on the study of electron emission under the action of a strong electric field, which were carried out by a number of authors on single crystals. As was set forth above, the most convenient methods for studying electron emission in strong fields were those in which the cathode was a small hemisphere, obtained by heating thin conical tips. If such a tip is placed in a vacuum and heated, its shape changes and it turns into a hemisphere, as Heffer showed when studying these tips in an electron microscope. At the same time, however, recrystallization also occurs in this tip, and since the dimensions of the tip are of the order of the dimensions of a single crystal of the polycrystalline wire taken (\(\sim 10^{-4}\) cm), a single-crystal hemisphere is in practice formed at the tip. This can readily be verified in the following way. If such a cathode is placed inside a glass sphere, the inner walls of which are coated with a conducting semitransparent layer with willemite deposited on it, then at large potential differences, of the order of several thousand volts, one can visually observe the luminescence caused by electrons torn out from the cathode. In all experiments with such observations a symmetrical pattern is obtained, whose symmetry fully coincides with the symmetry of the crystal of the cathode material. This serves as unconditional proof that the cathode is a single crystal. Such observations have been made in many laboratories and give coincident results. This pattern is easily obtained immediately after heating the cathode; for a long time, however, it can be observed only under the condition of very careful degassing of the apparatus. The most

detailed studies of the resulting patterns were carried out in the excellent work of Benjamin and Jenkins. Let us first turn to the patterns that are obtained in the study of pure metals: tungsten and molybdenum. Since the observed pattern has the symmetry of the crystal, it can be easily indexed. In the main, the resulting patterns amount to the following: 1) in regions lying near different principal crystallographic directions, the intensities of the glow are different, differing from one another in order of intensity by a factor of 2, and 2) in some directions the glow is very small (dark spots), so that it is a hundred times smaller than the glow of the nearest region. Unfortunately, these observations were made visually, and the current intensity in the various directions was not measured. The simplest explanation that can be put forward for interpreting these results would be that different electron emission occurs along different crystallographic directions, owing to differences in the work function. Indeed, as we have seen, the emission of electrons under the influence of a field depends on the work function \((\varphi_0^{3/2}\) in the exponential function of Fowler–Nordheim). However, the change in the work function that occurs for different crystallographic directions can in no way explain a hundredfold change in intensity.

Such a change could be accounted for by a difference of two or several times (the total intensity of the regions). A more trivial explanation would be that small residual impurities, which strongly alter the work function, are adsorbed differently in different crystalline regions, which accounts for the presence of a sharp pattern. However, upon repeated annealing the pattern is reproduced and is very stable, which leads one to think that the phenomenon cannot be explained by the presence of impurities. The observed effect might be caused by the fact that electron waves traveling from inside the crystal for certain crystallographic directions, according to Bragg’s conditions, are strongly reflected backward, as a result of which dark spots are obtained for these directions.

In addition to the possible causes indicated, attention should be paid to one further additional circumstance. We have seen that when tips are heated, a regrouping of atoms occurs in them so that, as a result, a single crystal is formed in the shape of a sphere, i.e. a figure with minimal surface. But since the free energy per unit surface (surface tension) for crystals is a function of direction, the equilibrium form of the surface corresponding to the minimum of free energy must be a polyhedron with smoothed and rounded edges. If this is so, then near the faces of such a polyhedron the field must be smaller than at the edges, and the emission from them must be smaller. Unfortunately, Haefer, in studying the shape of tips, only roughly determined their sphericity and did not carry out a special study of the details of their external—

of their shape. Other authors have not done this either. However, experiments with large single crystals of rock salt, taken in the form of spheres, show the formation of polyhedra with rounded edges. It is difficult to say to what extent this phenomenon is expressed in molybdenum and tungsten, but it should be pointed out that, in order to explain a hundredfold difference in electron emission, a considerable difference in the field is needed and, consequently, a strongly pronounced change in shape, which, as indicated, is not directly observed. Such is the situation with the pattern of electron emission under the influence of the field from single crystals of pure metals.

In addition, Benjamin and Jenkins2 studied the effects of prolonged heating on single crystals in an electric field, as well as the change in the pattern that results if foreign atoms—alkali and alkaline-earth metals—are deposited on tungsten or molybdenum and then the temperature of the cathode is gradually raised. In these observations, the apparatus with electron emission under the action of the field is a splendid microscope, with the aid of which one can study the details of the phenomena of migration and evaporation of atoms.

Having considered the main data on the phenomenon of electron extraction under the influence of an electric field, we can see that the basic theoretical conceptions of the nature of this phenomenon are undoubtedly confirmed. At the same time, the experimental data are limited almost exclusively to a very small number of experiments with tungsten and molybdenum tips. Very little has been studied concerning the distribution of velocities, and the study of emission from single crystals has hardly been touched upon, unless one counts the qualitative observations of Benjamin and Jenkins. It should be noted that the main observations were made with pure metals. The emission of semiconductors and dielectrics and of more complex surfaces (for example, composite cathodes) has not been touched upon at all. Meanwhile, there is no doubt that investigation of the extraction of electrons under the action of the field—one of the most elementary phenomena—provides information on the electronic levels in bodies and on the nature of the potential barrier at the boundary between these bodies and vacuum.

LITERATURE

1. R. H. Fowler and L. Nordheim, Proc. Roy. Soc., A 119, 173, 1928.
2. L. Nordheim, Proc. Roy. Soc., A 121, 626, 1928.
3. M. Benjamin and R. O. Jenkins, Proc. Roy. Soc., 176, 262, 1940.
4. R. Haefer, Z. Physik, 116, 604, 1940.
5. E. Müller, Z. Physik, 102, 734, 1936.
6. S. A. Vekshinskii, ZhTF, X, 1359.
7. L. E. Henderson and R. K. Dahlstrom, Phys. Rev., 55, 473, 1939.
8. P. I. Lukirsky, ZhRFKhO, VII, 463, 1924.
9. E. W. Müller, Z. Physik, 120, 261, 1943.
10. E. W. Müller, Z. Physik, 120, 270, 1943.
11. M. H. Nichols, Phys. Rev., 59, 944, 1941.
12. M. Benjamin and R. O. Jenkins, Proc. Roy. Soc., 180, 225, 1942.

2. M. Benjamin and R. O. Jenkins, Proc. Roy. Soc., 180, 225, 1942. ↩