ELECTRONIC MULTIPLIERS

N. S. Khlebnikov, Moscow

I. INTRODUCTION

1. After the publication of the interesting results obtained by Kubetskii¹ and P. T. Farnsworth² with the first electronic multipliers, enthusiasm for the phenomenon of secondary emission led many to think that very broad prospects were opening up for its practical use. It was assumed that secondary-electron devices could replace electronic amplifying tubes in ordinary circuits and be used as generator tubes, not to mention the most varied applications in the amplification of photocurrents in electronic multipliers; in short, it was thought that secondary emission would bring about a complete revolution in the field of electrovacuum devices. Reality has by no means justified these hopes and expectations. This is natural, since they were not based on any sufficiently detailed study of the phenomena. Unfortunately, in many cases attempts were made to use secondary emission in practice without clarifying the fundamental possibility of such applications.

2. At the present time, those areas have become quite clearly outlined in which secondary emission is of undoubted value and may even be regarded as a very major step forward. These areas are incomparably narrower than was thought at first. It may be stated with confidence that, with the present state of our knowledge of secondary emission and with present-day possibilities, secondary-electron devices are incapable of replacing not only generator and amplifying electronic tubes, but, to their full extent, even ordinary photocells. Below we shall examine in detail the factors responsible for this state of affairs; here we shall note only that the sole area in which secondary-electron devices, in certain cases, offer fundamental advantages is that of photoelectric and similar devices. This is the area we shall chiefly consider below, and to which we shall confine ourselves. In addition, secondary emission permits certain improvements in the field of low-power tubes and other similar devices.

All this does not mean, of course, that secondary emission will not in the future find wider application. On the contrary, it is quite certain that every new achievement in the fundamental questions of this field will open up ever new possibilities also with respect to the practical use of secondary emission.

II. ADVANTAGES AND DISADVANTAGES OF SECONDARY EMISSION AS A METHOD FOR AMPLIFYING WEAK ELECTRON CURRENTS

A. Noises in electron multipliers

1. The most important advantage of the electron multiplier when used to amplify weak primary electron currents is the possibility of increasing the signal-to-noise ratio, which means lowering the sensitivity threshold of any photoelectric device. It is precisely this circumstance, and not at all the absolute value of the sensitivity of photoelectron multipliers, as is sometimes thought, that should serve as the principal measure of the sensitivity of these instruments and is the criterion for their applicability to weak currents in particular cases. Integral sensitivity in this respect is a characteristic of secondary importance. The best confirmation of the statement just made is provided by considering, on the one hand, the general theory of circuits containing ordinary photoelements with an external photoeffect and electron multipliers, which we shall carry out below, and, on the other hand, those cases of application of electron multipliers which may be called expedient in the sense that they have yielded fundamental improvements. We shall also discuss this below.

The question of the signal-to-noise ratio for an electron multiplier was first considered in the work of Zworykin, Morton, and Malter³; a general theory was given by Preisach⁴, and a detailed experimental investigation was carried out by P. A. Sinitsyn⁵.

2. To clarify this question, let us briefly consider the operating conditions of a circuit containing a simple photoelement and an electron multiplier having identical values of photocathode sensitivity.

Figure 1 shows an ordinary circuit in which a photoelement with an external photoeffect operates under a modulated light flux. Here \(R\) denotes the load resistance, and \(C\) the capacitance shunting the photoelement, consisting of the interelectrode capacitances of the photoelement and of the tube and the capacitance of the connecting leads.

Fig. 1.

The primary signal in such a circuit is a fluctuation of the current in the photoelement circuit, which, by means of the resistance \(R\), is converted into the corresponding voltage fluctuation, is applied through a coupling capacitor to the grid of the first tube, and can then be amplified in the usual way.

The sources of noise in this circuit are: a) the intrinsic noise of the photoelement, due to the shot effect of the photocurrent; b) the noise of the resistance \(R\), caused by the Johnson effect; c) the intrinsic noise of the first tube, which is a consequence of the shot effect in the thermionic emission of its cathode.

Since the last noise source can be represented in the form of a certain equivalent resistance connected in series with \(R\), the matter reduces to the first two noise sources; all the more so since the magnitude of this equivalent resistance for modern tubes proves small in comparison with the usual values of load resistances.

The mean-square value of the noise current in a vacuum photocell \(\overline{i^2}\) can, as is known, be expressed as follows:

\[ \overline{i^2}=2eI_a\Delta f, \tag{1} \]

where \(e\) is the charge of the electron, \(I_a\) is the current through the photocell, and \(\Delta f\) is the transmitted frequency band.

On the other hand, the mean square of the voltage \(\overline{v^2}\) produced by the Johnson effect across the resistance \(R\) is equal to:

\[ \overline{v^2}=4kTR\Delta f, \tag{2} \]

where \(k\) is Boltzmann’s constant and \(T\) is the absolute temperature. Neglecting the capacitance \(C\), which is in general small (\(\sim\) several centimeters), i.e., assuming the load of the photocell to be purely ohmic, we can determine the ratio of the two noise signals:

\[ \frac{\overline{i^2}R^2}{\overline{v^2}}= \frac{I_aR}{\frac{2kT}{e}}= \frac{I_aR}{2V_T}, \tag{3} \]

where \(\frac{kT}{e}\), replaced by \(V_T\), is the “temperature voltage,” characterizing the thermal energy of electrons in a purely ohmic resistance at absolute temperature \(T\). This last equality shows that the ratio of the noise signals does not depend on the width of the frequency band and is determined exclusively by the ratio of the voltage drop across the resistance to twice the “temperature voltage” of the same resistance. At room temperature (\(T=300^\circ\text{K}\)) \(V_T\) proves equal to \(0.025\ \text{V}\), so that the square of the ratio of the shot-effect noise to the Johnson-effect noise will be:

\[ \frac{\overline{i^2}R^2}{\overline{v^2}}=\frac{I_aR}{50}, \tag{4} \]

where \(I_aR\) must be expressed in millivolts. Since we are dealing with very weak signals, it is clear that \(I_aR\) will be much less than \(50\ \text{mV}\), and thus we see that, generally speaking, the decisive significance in the circuit under consideration is the Johnson noise produced by the load resistance.

The magnitude of the Johnson noise can be determined by means of formula (2). If the value of the resistance \(R\) is specified, then the noise increases with an increase in the frequency band transmitted by the amplifier. The upper limit of this band cannot be raised without limit, since \(R\) is shunted by the capacitance \(C\). In those cases when the frequency is comparable with \(\frac{1}{2\pi RC}\) and greater than this value, the simplification,

introduced above and based on the assumption that the photoelement is loaded purely ohmically is no longer permissible, and accordingly formula (2) must be replaced by another one that takes into account the shunting effect of the capacitance \(C\). In this case

\[ \overline{v^2}=4kT\int_{f_1}^{f_2}\frac{R\,df}{1+(2\pi fRC)^2}, \tag{5} \]

where the difference of the limiting frequencies \(f_2-f_1\) is equal to the bandwidth \(\Delta f\). We shall obtain the largest value of the noise signal by taking the bandwidth to be equal to infinity. Then \(f_1=0\) and \(f_2=\infty\), and, carrying out the integration, we obtain

\[ \overline{v^2}_{\max}=\frac{kT}{C}. \tag{6} \]

The maximum value of the voltage across the resistance \(R\), arising due to the Johnson effect, \(V_{R\max}\), is

\[ V_{R\max}=\sqrt{\overline{v^2}_{\max}}=\sqrt{\frac{kT}{C}}. \]

At room temperature \((T=300^\circ K)\) this gives

\[ V_{R\max}=10\sqrt{\frac{41}{C}}, \tag{7} \]

if the voltage is expressed in microvolts and \(C\) in micromicrofarads.

In the case where a broad frequency interval is amplified, in order to avoid distortion it is necessary that the load in the photoelement circuit should not vary with frequency. For a specified band width \(\Delta f\), the band \(R\) must be chosen in accordance with the value of the capacitance \(C\). Since the load resistance of the photoelement, taking into account the shunting action of the capacitance, is equal to

\[ \frac{R}{\sqrt{1+(2\pi fRC)^2}}, \]

it is convenient to introduce the concept of the “upper limit” of frequency, defining it as the frequency

\[ f_2=\frac{1}{2\pi RC}, \]

at which the load resistance becomes equal to

\[ \frac{R}{\sqrt{2}}, \]

i.e. falls by approximately \(30\%\).

When using the exact formula (5) for a frequency band specified in a definite manner, the noise values due to the shot effect turn out to be \(20\)–\(30\%\) smaller than when the simplified formula (6) is applied. Convenient by virtue of its simplicity, formula (6) is quite suitable for practical purposes, since it gives the order of magnitude of the Johnson noise; it, as is evident, is determined exclusively by the magni-

...capacitance, i.e., the sum of the intrinsic capacitances of the photocell, the tube, and the wires. Taking \(C=20\,\mu\mu F\), we obtain for Johnson noise \(15\,\mu V\)—a value not exceeded by circuits with normal tubes and photocells, and to which the actual values are close in the case when one aims to obtain high sensitivity.

The quantitative meaning of formula (3) (for wide-band amplification, provided the practical rule \(f=\dfrac{1}{2\pi RC}\) is observed, the shunting effect of the capacitance may be neglected, and this formula remains valid) can now be understood with greater clarity. In accordance with formula (3), the noise of the photocell itself exceeds Johnson noise only at such large light fluxes when the constant component of the current in the photocell circuit proves to be greater than \(50\) mV, which is more than three thousand times greater than the Johnson-noise value found by us. The ratio of the constant component to the alternating component is given by the modulation factor \(M\) of the light. Even in the case when this factor is only \(0.1\), the ratio of the signal to Johnson noise exceeds 300, under conditions when the intrinsic noise of the photocell exceeds Johnson noise. Since in practice, generally speaking, considerably smaller signal-to-noise ratios are permissible, the noise becomes significant only at much smaller values of the light fluxes. Thus, for small values of the light fluxes, when the signal-to-noise ratio approaches the permissible limit, the intrinsic noise of the photocell may be neglected.

All the considerations presented show that, in wide-band amplification of photocurrents (for example, television), the sensitivity threshold is determined exclusively by Johnson noise arising in the load resistance. From this it is clear that, for the purpose of reducing noise, it is desirable to amplify the photocurrent directly, without converting it, by means of the load resistance, into a potential difference. Such direct current amplifiers, making it possible to increase the signal to a value greater than Johnson noise, are precisely electron multipliers.

In considering the operation of electron multipliers, we shall take into account as their intrinsic noise only the noise produced by the shot effect of photoemission, i.e., we shall regard the multiplication process as an ideal process of current amplification that introduces no intrinsic noise. Zworykin, Morton, and Molter³ showed that this corresponds the more closely to reality, the larger the coefficient of secondary emission \(\sigma\) in the first stages of the multiplier.

To calculate the gain in threshold sensitivity that can be achieved with the aid of an electron multiplier, we shall specify a definite value of the signal-to-noise ratio \((\rho)\) and calculate, for a given \(\rho\), the minimum photocurrent corresponding to the case a) of a simple photocell and b) of an electron multiplier. The ratio of the minimum signals will give us the ratio of the threshold sensitivities in the two cases.

According to what was set forth above, in the first case we may neglect the noise of the photoelement, and the minimum photocurrent, in accordance with formula (2), will be

\[ I_c=\rho \sqrt{\frac{4kT\Delta f}{R}} . \tag{8} \]

For the second case we shall assume that the electron multiplier gives a total amplification by a factor of \(\gamma\), and that it introduces no noise of its own. We shall take \(\gamma\) to be sufficiently large that the Johnson noise in the load resistance of the multiplier may be neglected. To determine the minimum value of the photocurrent in this case, we shall use formula (1), which expresses the current of the photocathode of the multiplier \(I_a\). The signal-to-noise ratio will be

\[ \rho=\frac{I_m}{\sqrt{2eI_a\Delta f}} . \]

Since the relation between the alternating \((I_m)\) and direct \((I_a)\) components of the current has the form

\[ I_m=\frac{1}{\sqrt{2}}MI_a, \]

where \(M\) is the modulation factor, we obtain

\[ I_m=\rho^2\frac{\sqrt{2}}{M}\,2e\Delta f . \tag{9} \]

Comparing formulas (8) and (9), we find the ratio of the minimum useful signals in the two cases:

\[ \frac{I_c}{I_m}=\frac{M}{\rho}\sqrt{\frac{kT}{2eR\Delta f}} . \tag{10} \]

Since in practice (transmission of audio and television frequencies) the lower frequency limit is small in comparison with the bandwidth, the quantity \(R\Delta f\) may be replaced by \(Rf_2\) (\(f_2\) is the upper limit of the band). The value of this product depends only on the capacitance \(C\) shunting the load resistance \(R\), which must be sufficiently small so that no appreciable reduction of gain occurs at high frequencies, and at the same time sufficiently large for the output of the photoelement to be fully utilized. As indicated above, it is convenient to set \(Rf_2=\dfrac{1}{2\pi C}\); in this case we have

\[ \frac{I_c}{I_m}=\frac{M}{\rho}\sqrt{\frac{\pi kTC}{e^2}} =\frac{M}{\rho}\sqrt{\pi}\sqrt{\frac{V_T}{\dfrac{e}{C}}}. \tag{11} \]

This expression gives us, in general form, the magnitude of the gain in threshold sensitivity attained by the use of an electron multiplier. It should be noted that this formula, apart from constants, contains only the specified value of the signal-to-noise ratio \(\rho\), the light modulation factor \(M\), and the capacitance \(C\). In the radical expression we have the ratio of two potential differences: the temperature potential \(V_T\)

and the peak value of the potential difference produced on the capacitance by one electron. At room temperature \(V_T=0.025\ \mathrm{V}\), and, taking \(C=20\ \mu\mu\mathrm{F}\), we find:

\[ \frac{e}{C}=0.8\cdot 10^{-8}\ \mathrm{V}, \]

so that the ratio of the two voltages proves to be equal to \(3.2\cdot 10^6\). Substituting these quantities in formula (11), we obtain for the gain in threshold sensitivity:

\[ \frac{\text{minimum signal of the photocell}}{\text{minimum signal of the multiplier}} \simeq 3\,200\,\frac{M}{\rho}. \tag{12} \]

Taking \(M=1\) (100% modulation) and \(\rho=5\), we find that the gain is equal to 640. With less stringent requirements on \(\rho\), taking it equal to 2, we have 1,600.

The preceding consideration is not entirely rigorous for the reason that we have not taken into account two circumstances: 1) additional sources of noise, which are, on the one hand, the amplifier tube and, on the other, the noise produced in the electron multiplier by fluctuations in secondary emission, and 2) the existence of thermoelectronic current from the photocathode, which, for example, for oxygen-silver-cesium cathodes already has a rather appreciable magnitude at room temperature (\(\sim 10^{-12}\ \mathrm{A/cm^2}\)). To show the validity of our calculations, it is sufficient to estimate these factors approximately.

In order to take account of the intrinsic noise of the amplifier tube, we must in formula (8) replace \(R\) by

\[ \frac{R}{1+\dfrac{R_{\text{экв}}}{R}}, \]

where \(R_{\text{экв}}\) is the additional equivalent resistance mentioned above. For very wide-band amplification (high-quality television), taking \(\Delta f=2\cdot 10^6\), taking \(R=4\,000\ \Omega\), and bearing in mind modern tubes with large transconductance, it is easy to find that the ratio given by formula (12) increases by 10%. In the case of smaller load resistances or tubes with greater noises, the increase will be still larger.

On the other hand, if one takes into account the additional noise produced in the multiplier by fluctuations in the coefficient of secondary emission, this ratio must be reduced. Zworykin, Morton, and Malter showed that, if \(\sigma\) is sufficiently large, one may confine oneself to taking fluctuations into account only at the first stage of multiplication, using formula (1). This leads to an increase of the noise at the output of the multiplier by

\[ \sqrt{\frac{\sigma}{\sigma+1}} \tag{13} \]

times, where \(\sigma\) denotes the amplification at the first stage. For \(\sigma=5\) the additional noise amounts to 10%. Thus, the two additional sources of noise approximately compensate one another. From

From formula (13) it is clear that, in order to reduce the noise at the output of the multiplier, it is desirable to have, in the first stages, values of \(\sigma\) as large as possible. This indicates, in particular, that it is advisable to have higher potential differences in the first stages.

The presence of thermoelectronic current can be taken into account as an increase in the constant component, i.e., as a decrease in the modulation factor \(M\). Since these currents are nevertheless small, their role becomes substantial only at very small primary fluxes. These currents are factors determining the sensitivity threshold of the multiplier when operating with unmodulated light (dark currents; see below).

An essential question is what magnitude the total gain of the multiplier must have in order to ensure a substantial increase in the signal-to-noise ratio, i.e., that magnitude of gain at which the assumption made above regarding the negligibility of Johnson noise becomes approximately valid. Denoting by \(a\) the ratio of the shot-effect noise to the Johnson noise at the output of the multiplier, we have

\[ a^2=\gamma^2\frac{I_a R}{2V_T}, \tag{4a} \]

where, as in formula (3), \(I_a\) is the constant component of the photocurrent and \(\gamma\) is the total gain of the multiplier. From (4a) we find the expression for \(\gamma\) in terms of \(a\):

\[ \gamma=a\sqrt{\frac{2V_T}{I_aR}}. \tag{14} \]

The greatest value of the total gain \(\gamma_{\max}\) is necessary for the smallest primary signals. Applying (9) and proceeding in the same way as in the derivation of (12), we obtain

\[ \gamma_{\max}=a^2\frac{M}{\rho}\,3\,200. \tag{15} \]

Thus, the required magnitude of the total gain is determined, on the one hand, by the operating conditions (the modulation factor \(M\)) and, on the other, by the requirements on the noise level \(\rho\) and the desired value of \(a\). Taking, for example, \(M=1\), \(a=\rho=3\), we obtain \(\gamma_{\max}=10\,000\). This magnitude is therefore sufficient if the purpose of using the electron multiplier consists only in lowering the noise level. In many cases it is possible to limit oneself to still smaller values of \(\gamma\)—of the order of 1,000.

In practical circuits with photoelectric devices, in addition to the elimination of noise, the elimination of other interferences having the character of pickup is also an essential question. In this case the magnitude of the output signal also gives an advantage to electron multipliers. Finally, another argument in their favor is the possibility of reducing the subsequent amplification. These last two advantages are mainly economic (shielding, extra tubes, power consumption) and operational in character and constitute an argument in favor of the use of multipliers, for example—

measure, in sound cinema. In an objective assessment, however, here it is also necessary to take into account another aspect of the matter—the comparative high cost of the multiplier, the power consumption for its supply, etc.

1. P. A. Snitsyna^5 carried out an investigation of noise in electron multipliers of various designs and at different values of the total amplification. This work, in general, showed good agreement with the theory set forth above. As for the threshold sensitivity, in these experiments it was determined as the luminous flux producing noise equal to the dark-current noise, and proved to lie within the limits \(10^{-8}\)—\(10^{-9}\) lm at room temperature.

2. Everything stated above undoubtedly indicates that, when operating with small primary electron currents, electron multipliers have certain advantages over simple photocells with an external photoelectric effect, consisting in a considerable increase of the signal-to-noise ratio, which on the average various authors^3,4, in agreement with one another, estimate as two-hundredfold. Since this advantage, being based on the fact that an electron multiplier is a current amplifier, is an advantage of a fundamental character, it provides electron multipliers with a quite definite field of application in which these devices have no competitors. This is the field of weak modulated light and electron fluxes over a wide frequency band, i.e. the field of high-quality (mechanical and electronic) television. And indeed, electron multipliers are successfully used (for example, at the Kiev television center) as the photosensitive element of television transmitters. Another method of using them consists in employing an electron multiplier as part of a television tube of the iconoscope type, in which the signal is obtained at the expense of secondary electrons knocked out of the mosaic by the scanning beam.^6 These elementary electron currents are amplified in the multiplier, the output of which is connected to the input of a tube amplifier.

B. Dark currents in electron multipliers

1. Among other applications of photoelectron multipliers in present-day practice, a prominent place belongs to their use as instruments for measuring and recording very small light fluxes (astro- and spectrophotometric measurements, recording of stellar occultations, etc.). The use of modulated light fluxes is, of course, entirely possible here. In this case, regarding the question of the use of electron multipliers one can say the same as was set forth above, with the corresponding corrections for a narrower frequency band. As a rule, however, it is preferred to dispense with light modulation and without subsequent tube amplification, feeding the output current of the multiplier to a sufficiently sensitive measuring instrument. In this case the sensitivity threshold of the multiplier is determined by its own

interference—the “dark current” of the multiplier, i.e., the current that exists at the output of the multiplier in the absence of illumination of the photocathode.

Dark currents in multipliers may be produced by at least six phenomena: 1) ohmic leakage in the collector circuit; 2) optical feedback—a phenomenon consisting in the fact that radiation (visible, ultraviolet, or soft X-ray) which sometimes arises in the last stages of the multiplier under the action of electron bombardment causes photoelectrons to be torn out of the photocathode; 3) ionic feedback—a phenomenon consisting in the fact that positive ions formed (as a consequence of insufficient vacuum) in the region of the last stages, where the density of the ionizing electron flux is especially high, bombard the first stages of the multiplier and the photocathode, knocking out electrons there; 4) thermoelectronic currents of the photocathode and of the first multiplication stages; 5) cosmic radiation; 6) cold emission of the electrodes or of conductors connected with them.

Whereas factors 1)—3) and 6) are not in the nature of fundamental limitations and can be eliminated by comparatively simple methods (the design of the multiplier; see below), the two other factors constitute fundamental limitations, and suppressing them requires rather complicated measures. In addition, it should be noted that factors 2) and 3), unlike the others, are secondary in the sense that they are capable only of maintaining the dark current, not of producing it. For a detailed investigation of dark currents in electron multipliers we are indebted to D. Reichman,^7 who considers 1), 3), 4), and 6) to be the most significant of the factors listed above.

Taking into account that the only fundamentally ineradicable one of these four factors is thermoelectronic emission of the photocathode, and also that the others may, if not be eliminated completely, then be substantially suppressed by corresponding changes in the design of the multiplier—considerable work having been done in this direction for one type of multiplier—we shall postpone their consideration until the discussion of design questions, confining ourselves in this section only to thermoelectronic dark currents.

1. Generally speaking, the dark current of thermoelectronic origin is small and constant in comparison with the currents arising as a result of ionic feedback, cold emission, and ohmic leakage (in the case where these factors have not been eliminated). Because of this, it does not impose such serious limitations on the use of multipliers when working with small luminous fluxes as do dark currents caused by other reasons. Owing to its constancy it may even (provided the temperature is sufficiently constant) be neutralized by means of an appropriate compensation circuit, which, however, is an undesirable complication. Another, and more serious, objection to the thermoelectronic current arises when working with modulated light.

In this case the signal-to-noise ratio proves to be reduced owing to the noise produced by the dark current.

Suppression of the dark current of thermoelectronic origin can be accomplished only by cooling the multiplier. It then turns out that the dark current can be reduced so much that the multiplier becomes capable of measuring much smaller (approximately by two orders of magnitude) light fluxes. The great interest of experiments of this kind lies not only in the fact that they provide a means for creating a very valuable instrument, but also in the fact that such measurements provide, to a very high degree, a convenient method for measuring the thermoelectronic constants of photocathodes, since for these surfaces the temperature interval in which one may assume constancy of their properties has so low an upper limit that direct measurement of thermoelectronic currents is associated with great difficulties even for large areas of the emitting surfaces. Measurements by this method have so far been carried out only for oxygen-silver-cesium and oxygen-silver-rubidium surfaces.

The thermoelectronic emission of any surface may be expressed by Richardson’s formula, which has the form

\[ I = AT^2 e^{-\frac{B}{T}} = AT^2 e^{-\frac{eW}{kT}}, \tag{16} \]

where \(I\) is the emission current in amperes per square centimeter, \(A\) is a constant having the dimension \(\mathrm{A}/\mathrm{cm}^2\), \(T\) is the absolute temperature of the emitting surface, and \(B\) is another constant, equal to

\[ B = 1.16 \cdot 10^4 W, \tag{17} \]

if the electron charge \(e\) is expressed in coulombs, Boltzmann’s constant \(k\) in joules per degree, and the work function \(W\) in volts. The constant \(A\), as was shown theoretically, is a universal constant, i.e., its magnitude does not depend on the nature of the surface and is equal to:

\[ A = 120\ \mathrm{A}/\mathrm{cm}^2 \mathrm{deg}^2 . \tag{18} \]

This value of \(A\) was established experimentally for many pure metals. However, for complex surfaces, in particular for the oxygen-silver-cesium cathode, the measured values of \(A\) may differ from the indicated value by an enormous factor (up to \(10^{30}\)). This discrepancy can be explained by assuming that in the present case the work function varies with temperature. Under this assumption we arrive at a more complicated form of Richardson’s equation, namely one in which the quantity \(B\) (or the work function) is a power series in \(T\). Nevertheless, because of the great difficulties in determining the coefficients of this series with even the minimally significant degree of accuracy, one usually uses not the general expression but its simple form (16).

Thus, even in the case of complex surfaces their emission properties are characterized by two constants, \(A\) and \(W\). In temperature studies the experimental data are represented in the form of

ELECTRON MULTIPLIERS

of the graph of the dependence \(\lg \dfrac{I}{T^2}\) on \(\dfrac{1}{T}\), which gives a straight line. The slope of this straight line determines the work function \(W\), and its position relative to the axis determines the constant \(A\).

A substantial role in measurements of this kind is played by the method of changing the temperature of the emitting surface and by measuring this temperature. It is best to place the electron multiplier in a vessel with an insulating liquid circulating between this vessel and a coil that is heated or, correspondingly, cooled. Such a method ensures equality of the temperature of the walls of the bulb at all points and, consequently, after a sufficiently long interval of time has elapsed, a uniform temperature of all the internal parts of the multiplier. The temperature of the emitting surface can be monitored by observing the temperature of the liquid. In doing so it is necessary to make sure that the emitting surface has also acquired this temperature. This is indicated by the cessation of changes in the dark current. Another, more direct method consists in introducing a thermocouple inside the bulb, the junction of which is brought into contact with the metal of the emitting surface.

Measurements of this kind were carried out with multipliers of various designs (see below). With one of them (a less advanced one, where, along with emission from the photocathode, emission from the multiplying electrodes could also play some role), Reichmann measured the constants for an oxygen-rubidium surface, obtaining for \(W\) the value \(1.03\ \mathrm{V}\) and for \(A\) values from \(5 \cdot 10^{-9}\) to \(50 \cdot 10^{-9}\ \mathrm{A/cm^2\,deg^2}\). With a more advanced design, the corresponding measurements for an oxygen-silver-cesium surface gave: for the work function the value \(W = 0.75\ \mathrm{V}\) \((\pm 0.05\ \mathrm{V})\) and for the constant \(A\)

\[ A = S \cdot 2.5 \cdot 10^{-7}\ \mathrm{A/cm^2\,deg^2} \qquad (0.1 < S < 10). \]

These values of the thermionic constants for oxygen-silver-cesium cathodes imply a dark-current value at the output at room temperature (\(\sim 25^\circ\)) of about \(1\ \mu\mathrm{A}\), which, with a photocathode sensitivity of about \(10\ \mu\mathrm{A/lm}\) and a total amplification of \(10^7\) times, corresponds to a threshold sensitivity (by which we mean the luminous flux producing at the output a current equal to the dark current) of approximately \(10^{-8}\ \mathrm{lm}\). In the most successful multipliers this quantity turns out to be somewhat lower, reaching \(2.5 \cdot 10^{-9}\ \mathrm{lm}\).

It is interesting to note that these data are in complete agreement with the above-mentioned data of Sinitsyn\(^5\), since there too, from the very method of measuring the noise, it is clear that threshold sensitivity is understood to mean equality of the output currents from the light signal and from the dark current.

1. Since the measurements of Reichmann and Sinitsyn were made on multipliers of completely different designs (which, however, were similar in that both systems ensured impeccable focusing, so that any “electron losses” during passage from cascade to cascade were absent) and under different conditions, this com-

comparison of the results shows that the threshold sensitivity of a multiplier with an oxygen–silver–cesium photocathode actually lies within the limits \(10^{-8}—10^{-9}\ \mathrm{lm}\) (at room temperature). This quantity is small enough to make the electron multiplier an indispensable instrument in the region of such small light fluxes, but at the same time it is still so large as to make its reduction desirable.

Taking into account that the thermionic constants of the given surface cannot be controlled at will, there are no effective ways of further reducing the dark current of multipliers with this type of cathode. Nevertheless, some improvement can still be achieved along three lines.

1) By reducing the area of the photocathode to the minimum required by the operating conditions (operation with concentrated light fluxes is meant).

2) By increasing the integral sensitivity of the photocathode (as a result, the ratio of the thermionic current to the photoelectric current will decrease, especially since cathodes with high photosensitivity possess relatively small thermionic emission, and vice versa\(^{8}\)).

3) By maintaining the multiplier at a low temperature.

The last method is undoubtedly the most effective, provided only that it is feasible under the given practical operating conditions. According to measurements by the same Reichman, lowering the temperature of the photocathode to \(-28^\circ\) made it possible to reduce the sensitivity threshold to \(2\cdot 10^{-11}\ \mathrm{lm}\).

Finally, especially valuable results may be expected when the oxygen–silver–cesium cathode is replaced by another cathode possessing a larger work function. The use of oxygen–silver–rubidium cathodes already gives a perceptible improvement. Still better results will undoubtedly be given by an antimony–cesium photocathode, both because of its high work function\(^{9}\) (\(\sim 1.7\ \mathrm{V}\)) and because of its exceptionally large quantum yield, reaching about \(30\%\)\(^{10}\) in the region of the selective maximum. This cathode is applicable, however, exclusively for work in the visible region.

C. Output powers of electron multipliers

1. Above we established two regions of application of electron multipliers in which they provide undoubted fundamental advantages. These are the regions of small modulated and unmodulated light and electron fluxes. A third region, related to these, is only just emerging—namely, the use of multipliers for recording individual elementary particles. In this section we shall consider the question of the loads permissible for multipliers and establish the reasons that hinder the broad use of multipliers as applied to large and medium light fluxes.

2. As experience shows, it does not appear practically possible to make the output currents of modern electron multipliers greater than \(1—5\ \mathrm{mA}\), which, with a potential difference of \(50—200\ \mathrm{V}\) at

cascade corresponds to 0.05 to 1 W. These figures refer to the oxygen–silver–cesium emitters that are almost exclusively in use at the present time. It is quite clear that such small powers and currents at the output are insufficient for directly actuating ordinary devices (reliable relays, reproducing units), so that subsequent amplification by means of vacuum tubes becomes necessary. Thus a modern multiplier proves unable to replace the entire amplifying system, and this circumstance, together with the problem of power supply, which we shall discuss below, is one of the substantial obstacles limiting the possibilities of using these devices. At the same time, this limitation is not of a fundamental nature, since it is a consequence of the physical properties of modern materials possessing high coefficients of secondary emission, as will now be explained. Therefore any progress in this field will immediately lead to an expansion of the field of application of multipliers in technology.

1. The figures indicated above for the limiting loads on the emitters (which fluctuate, generally speaking, depending on the dimensions of the emitting electrodes) cannot be exceeded because of two circumstances. The chief of these is the irreversible change (decrease) of the secondary-emission coefficient of the last cathodes, and together with it of the overall gain of the multiplier, which assumes a catastrophic character when the output is increased beyond the indicated limits, reducing the service life of the device to several hours—several tens of hours. The second and less essential circumstance is the decrease of \(\sigma\) (also in the last cascades), which is reversible in character. The reason for these latter changes lies in two factors that are inherent properties of modern efficient emitters of the complex-surface type and rooted in their semiconductor nature: namely, the decrease of \(\sigma\) under the influence of an increase in the density of the primary current, and under the influence of a rise in temperature. The difficulty of this latter kind can be circumvented by increasing the number of multiplication cascades.

2. We see, therefore, that the most essential primary factor determining the behavior of the emitter, and along with it of the multiplier, is the temperature of the emitting surface acquired by it during operation. Since the rise in temperature is a consequence of the bombardment of the emitting surface by electrons, we must turn to a consideration of the energy balance of the emitter.

The dynamic temperature equilibrium in which the emitter finds itself, for any value of the load, is determined by the difference between the energy acquired and lost by it per unit time. The energy arriving at the emitter is exhausted by the kinetic energy of the primary electron stream, and therefore the input in our balance may be expressed [without taking into account the distribution of primary electrons over energies, for which there are grounds (see below)] as

\[ Q_p = I_p V_p, \tag{19} \]

where \(I_p\) is the primary-current strength and \(V_p\) is the potential difference between the given stage and the preceding one. The expenditure of energy takes place in two ways: a) through the energy carried away by secondary electrons, which in general form can be expressed as

\[ Q_s=e\int_{-aV_p}^{V_p} N_s(V_s)\,dV_s, \tag{20} \]

where \(e\) is the electron charge, \(N_s(V_s)\) is the distribution function of the secondary electrons over energies \((V_s)\), and \(a\) is a certain coefficient taking into account the fact that part of the secondary electrons, as follows from the experiments of Pyatnitskii\({}^{11}\) and Vudynskii\({}^{12}\), leaves the emitter only in the presence of an accelerating field. Taking into account that no analytical expression exists for the function \(N_s(V_s)\), the last equality, for practical purposes, is conveniently replaced by

\[ Q_s=I_s \overline{V}_s, \tag{21} \]

where \(\overline{V}_s\) is the mean value of the energy of the secondary electrons, which can be determined by graphical integration of the electron velocity-distribution curve, and \(I_s=eN_s\), where \(N_s\) here denotes the total number of secondary electrons.

In addition to the energy carried away by secondary electrons, the emitter loses energy through heat transfer, which, if one disregards negligible losses through convection due to residual gases and vapors of the alkali metal, can occur in two ways—by removal of heat through the leads on which the electrode \(Q_T\) is mounted, and through radiation \(Q_R\). Thus,

\[ Q_p-(Q_s+Q_T+Q_R)=0. \tag{22} \]

It is not our aim to investigate this equation in detail; we intend to draw from it only certain qualitative conclusions for which there are sufficient experimental data, and which are of interest as indications of the circumstances that must be taken into account in designing multipliers.

We have already pointed out above that, in determining the energy arriving at the emitter with the primary current, there is no need to take into account the energy distribution of the primary electrons. This follows from the form of the electron energy-distribution curves for effective emitters, a characteristic feature of which is the position of the maximum at small values of \(V_s\) (of the order of 1–3 V) and its sharpness. This is evident, for example, from the curve of Fig. 2, which refers to an oxygen–silver–cesium emitter 11. Thus, without appreciable error it may be assumed that all primary electrons have one energy \(V_p\).

In determining \(Q_s\), on the basis of the same properties of the distribution curves, we may assume that the energy of all secondary electrons is the same and equal to \(V_s\) (since we do not have in mind точ-

…calculation). Using Pyatnitskii’s data\(^{11}\) for an oxygen–silver–cesium cathode, we find for \(V_p = 50\ \mathrm{V}\):

\[ Q_p = 50 I_p; \qquad Q_s = 3 I_p \sigma; \qquad \eta_{50}=\frac{3\sigma}{50}\approx \frac{1}{17}\sigma \]

and for \(V_p = 300\ \mathrm{V}\):

\[ Q_p = 300 I_p; \qquad Q_s = 2 I_p \sigma; \qquad \eta_{300}=\frac{2}{300}\sigma=\frac{1}{150}\sigma. \]

Unfortunately, it is not known exactly what the dependence of \(\sigma\) on \(V_p\) is in the case under consideration. It will not, however, be an understatement to assume that \(\sigma_{50}\leqslant 2.5\) and \(\sigma_{300}\leqslant 6\). In this case the energy efficiency for the two cases will be, respectively,

\[ \eta_{50}\simeq 14\% \quad \text{and} \quad \eta_{300}=4\%. \]

We see, therefore, first of all, that the energy losses by the emitter due to the emission of secondary electrons are negligible, and that the greater part of the energy of the primary beam is expended on heating the emitter, so that the principal term in the energy expenditure is its heat transfer. Moreover, it is clear that with increasing \(V_p\) the efficiency decreases, since the increase of \(\sigma\) is outweighed, on the one hand, by the increase of \(V_p\) and, on the other, by the decrease of \(V_{s\max}\). This circumstance is one of the essential arguments in favor of so-called “low-voltage” multipliers, where the voltage per cascade is made as small as possible.

Fig. 2

Taking into account that the emitters of a multiplier have total surfaces of the order of several square centimeters, and also that the leads on which they are mounted have a diameter not greater than \(1\ \mathrm{mm}\) for a length of several centimeters, it should be thought that the principal of the two paths of heat loss is radiation. In particular, radiation begins to play a large role at high temperatures, since the total amount of radiated heat is proportional to the fourth power of the temperature difference between the emitter and the surrounding medium, whereas heat losses through thermal conductivity are proportional only to the first power of this quantity. Therefore, raising the upper limit of the emitter’s operating temperatures offers substantial advantages with respect to increasing the output powers of multipliers.

1. Summarizing the above, we arrive at the conclusion that the limitations on the applicability of modern multipliers for large primary

flows is due to the insufficient thermal stability of the emitting materials. At the basis of this property of theirs lies their low energy efficiency. Also an unfavorable factor is the heat-transfer conditions in which the electrodes of the multiplier are found.

An increase in the permissible output powers of multipliers may be accomplished in the following ways:

1) Improvement of the energy balance of the emitter. Since at high \(\sigma\), apparently, it is in principle impossible to have large values of \(V_{s\max}\), the only practically realizable method here is as large a reduction as possible of the cascade voltage (low-voltage multipliers). A supplement to this may be the construction of electrodes with the largest possible radiating surfaces.

2) Use of emitting materials capable of operating at high temperatures.

D. Power supply of electron multipliers

1. Further limitations with regard to the possibility of using electron multipliers arise in connection with the problem of their power supply.

This limitation is not of a fundamental nature, but is extremely important in the technical sense. Since the operating voltage for each cascade must amount to from 50 to 200 V, this means a total voltage of the order of 1000—3000 V. Such voltages can easily be obtained only from rectifiers, i.e., in the presence of an alternating-current mains supply. All possible other devices (batteries, high-voltage machines, voltage converters) are not sufficiently reliable. In addition, at such high voltages the question of insulation becomes more complicated, in connection with which external conditions (for example, humidity) begin to play a role. Therefore electron multipliers may be regarded as devices suitable only in various stationary installations, where weight, dimensions, etc., do not play a decisive role and where, in general, the most favorable conditions can be created for the multiplier.

In any case, it always seems desirable to use the supply voltage most effectively, i.e., to have as much amplification as possible for each volt of the total voltage applied to the multiplier. It is perfectly clear that the possibilities presented here are determined by the dependence of \(\sigma\) on \(V\). Without specifying an analytical expression for this function, one may simply use the experimental curves \(\sigma=f(V_p)\).

In order to find the most advantageous value of the voltage per cascade, it is sufficient to find the conditions for the maximum of the expression\(^2\)

\[ \Sigma=\sigma^n, \tag{23} \]

where \(\Sigma\) is the total amplification of the multiplier, \(\sigma\) is the amplification per cascade, and \(n\) is the number of cascades. Here it is assumed that \(\sigma\) does not change from cascade to cascade, i.e., the dependence of \(\sigma\) on tem-

temperatures and densities of the primary beam; in other words, what is involved here is small specific loading of the emitters.

Since \(n=\dfrac{V}{V_0}\), where \(V\) is the total voltage and \(V_0\) is the cascade voltage, then

\[ \Sigma=\sigma^{\frac{V}{V_0}} . \tag{24} \]

Differentiating this equality and setting the first derivative equal to zero, we have:

\[ \frac{d\Sigma}{dV_0} = \sigma^{\frac{V}{V_0}-1} \left( \frac{V}{V_0}\frac{d\sigma}{dV_0} - \frac{V}{V_0^2}\sigma\ln\sigma \right) =0, \tag{25} \]

whence we find the condition for the maximum

\[ \frac{d\sigma}{dV_0} = \frac{\sigma}{V_0}\ln\sigma . \tag{26} \]

Luk’yanov\(^{13}\) solved the same problem in a somewhat different way, by introducing an analytical expression for the function \(\sigma=f(V_p)\), which he obtained on the basis of a definite physical picture of the phenomenon in the following form:

\[ \sigma=AV_0e^{-\mu V_0}, \tag{27} \]

where \(A\) and \(\mu\) are constants to be determined from the empirical dependence of \(\sigma\) on \(V_p\), and \(V_0\) is the energy of the primary electrons. In this case the expression for the total amplification \(\Sigma\) has the form:

\[ \Sigma=\sigma^n=(AV_0)^n e^{-\mu V_0 n}. \tag{28} \]

Assuming, as before, the total voltage to be \(V=nV_0\), we have:

\[ \Sigma=\left(\frac{AV}{n}\right)^n e^{-\mu V}. \tag{29} \]

Differentiating this equality with respect to \(n\), we find, by setting the first derivative equal to zero, the most advantageous number of cascades \(n_{\mathrm{opt}}\) for the given \(V\):

\[ n_{\mathrm{opt}}=\frac{AV_0}{e}. \tag{30} \]

Thus we see that for each specified value of \(V\) and for each type of emitter (the dependence of \(\sigma\) on \(V_p\)) there exists a definite number of cascades most advantageous in the sense of the magnitude of the total amplification. This circumstance is taken as the basis for the design of so-called “low-voltage multipliers,” already mentioned by us.

III. CLASSIFICATION OF PHOTOELECTRON MULTIPLIERS

1. Electron multipliers used for the amplification of weak (of the order of microamperes and below) electron currents, i.e., primarily photoelectron multipliers, are at present represented by a rather considerable number of designs, so that there arises a natural need for their classification. Although such a classification can, generally speaking, be carried out in several different ways,

taking as the basis, for example, the number of multiplication stages, the most expedient system will undoubtedly be one based on a minimum number of secondary features. Such a classification must obviously be based above all on the most fundamental physical characteristics of the designs. We believe these to be the following: 1) the group of applications, 2) the number of control fields, 3) the presence or absence of focusing of the electron beam.

All existing types of multipliers fit into such a system of basic features without any difficulty, and there is no reason to think that there will be no place in it for new designs.

1. By groups of applications we mean not all possible specific applications, but only two main groups. One of them is the one that uses the fundamentally new possibilities opened up by the electron multiplier, i.e., reduction all the way to the limit of the noise level and the threshold of sensitivity. This group may be called the group of indicators of extremely small light and electron fluxes. This does not mean, of course, that every design belonging to this group is actually used in precisely this way, but means only that the properties of the designs in principle permit gains sufficient for this purpose.

The second group, which may be called the group of sensitive photocells, includes all multipliers in which secondary emission is used only for some increase in the overall sensitivity of the device (for increasing the output current).

The second main feature is the number of independent control fields that ensure the passage of the electron beam from the \(n\)-th to the \((n+1)\)-th stage. This number may be either one or two.

As for the third main feature, it must be stipulated that here and everywhere below we attach to the term “focusing” only the meaning that the electrode system of the multiplier, at the proper values of the electrode potentials, ensures that every electron emitted by the \(n\)-th stage reaches the \((n+1)\)-th. In other words, this means that the system at each stage ensures complete utilization of the electron beam (provided, of course, that the external field is sufficiently strong compared with the space-charge field).

We shall adhere henceforth, in describing the designs of electron multipliers, to this classification system, which is presented visually in the attached diagram.

IV. DESIGNS OF PHOTOELECTRON MULTIPLIERS

A. Indicators of Extremely Small Fluxes

1. The principal requirement imposed on a multiplier of this class is high overall gain (from \(10^{4}\) to \(10^{6}\)—\(10^{7}\)). This circumstance determines their belonging to the group of multi-electrode devices.

The most essential task in designing such multipliers is the implementation of focusing of the electron beam,

Classification of Photoelectronic Multipliers

• Photoelectronic multiplier
• Indicator of extremely small fluxes
  • Multiplier with one control field
  • System with focusing
    (crossed electric and magnetic fields)
    • Kubetsky tube
    • Zvorykin multiplier
  • System without focusing
    (crossed electric fields)
    • Farnsworth multiplier
    • “Nonmagnetic” Kubetsky tube
    • Timofeev multiplier
• Photoelement with increased sensitivity
  • Multiplier with one control field
  • System without focusing
    (crossed electric fields)
    • Farnsworth multiplier
    • “Nonmagnetic” Kubetsky tube
    • Timofeev multiplier
  • Multiplier with two control fields
  • Systems without focusing
    (emitters of through action)
    • Multiplier with funnel-shaped electrodes (Timofeev)
    • Multiplier with electrodes in the form of grids (Weiss)
    • Multiplier with electrodes in the form of troughs or spoons (Kubetsky)
    • Multiplier with electrodes in the form of thin continuous layers
  • Systems with focusing
    • Multiplier of the $L$ and $T$ types (Zworykin)
    • Multiplier of the checkerboard type (Zworykin and Rajchman)
• Single-stage multiplier
  • Yamsa and Salzberg
  • Gerlich
  • Timofeeva

necessary for using each of the multiplying electrodes to the full extent, without scattering of the electron stream and without reducing the total gain as a result of the transition of electrons from the \(n\)-th to the \((n+m)\)-th cascade instead of the \((n+1)\)-st.

Fig. 3

1. The most obvious way of carrying out such focusing is the use of crossed electric and magnetic fields. For this reason the first multipliers were those first implemented by Kubetsky with mixed control (known under the name of “Kubetsky tubes”). Subsequently the same principle was used by Zvorykin, who also gave an approximate theory of the operation of such devices\(^{3}\).

The construction of Kubetsky’s tube is shown in Fig. 3. This multiplier is a glass tube, the inner surface of which is coated with a layer of emitting material divided into inclined rings \((E_1, E_2, \ldots)\). Each of these rings, which are emitters, as well as the photocathode \(K\) and the collector \(C\), has platinum leads through the wall of the tube. Beginning with \(K\) and ending with \(C\), the electrodes have uniformly increasing positive potentials. The magnetic field is perpendicular to the plane of the drawing and is directed so that an electron torn out of an emitter and accelerated by the field of the portion of the next ring situated above it is deflected toward the emitting surface of the next ring. Another variant of this method of control, implemented by Zvorykin, is shown in Fig. 4, where the arrangement of the electrodes with respect to the magnetic field and with respect to one another is the same as in Kubetsky’s tube. The upper row of electrodes is intended only to create a field gradient at the surface of the corresponding emitter; for this purpose each electrode of this row is connected to the following emitter.

Fig. 4

Fig. 5

These electrode systems possess very good focusing properties with a proper choice of the strengths of the magnetic and electric fields, as is easily seen from the simplified consideration, given below, of the problem of electron motion in the corresponding system of fields.

Let the \(X\)-axis of the coordinate system (Fig. 5) represent the direction from the cathode \(K\) to the collector \(C\) of electrons in the electron multiplier; the \(Y\)-axis coincides with the direction of the (assumed homogeneous one—

... electric field between each pair of plates and the \(Z\)-axis—with the direction of the magnetic field (also homogeneous). Then, under the simplifications mentioned above, which consist in the following: a) the potential of each upper electrode is assumed to be considerably higher than the potential of the following emitter (which, as is clear from Fig. 4, is not actually the case, since these potentials are equal), and b) the initial velocities of the emitted electrons are equal to zero (which in general is close to real conditions in the case of complex surfaces), we can describe the motion of the electron by the following system of equations:

\[ \left. \begin{aligned} m\,\frac{d^{2}x}{dt^{2}} &= eH\,\frac{dy}{dt},\\ m\,\frac{d^{2}y}{dt^{2}} &= eE - eH\,\frac{dx}{dt},\\ m\,\frac{d^{2}z}{dt^{2}} &= 0, \end{aligned} \right\} \tag{31} \]

the solution of which gives

\[ \left. \begin{aligned} x &= \frac{E}{H^{2}}\,\frac{e}{m}\left(\frac{eHt}{m}-\sin\frac{eHt}{m}\right),\\ y &= \frac{E}{H^{2}}\,\frac{e}{m}\left(1-\cos\frac{eHt}{m}\right),\\ z &= 0. \end{aligned} \right\} \tag{32} \]

As is easy to see, these equations are the equations of a cycloid, so that the paths of the electrons will be represented by the curves in Fig. 5. From these equations it is also easy to determine the distance between the points at which the electrons fall on neighboring emitters, \(x_{0}\), and the maximum height to which they rise above the emitters, \(y_{0}\), namely:

\[ x_{0}=2\pi\,\frac{E}{H^{2}}\,\frac{m}{e} \tag{33} \]

and

\[ y_{0}=\frac{2E}{H^{2}}\,\frac{m}{e}. \tag{34} \]

All the formulas obtained are strictly valid only in the case where we have the arrangement of fields indicated in Fig. 5. In actual multipliers this is, of course, not so, since the potentials of the emitters increase from the cathode to the collector, as a result of which the potentials of the upper plates also increase. All this leads to the appearance of an electric-field gradient in the \(X\) direction. In addition, neither the electric field in the \(X\) and \(Y\) directions nor the magnetic field is homogeneous. This complicates the conditions to such an extent that it makes a more exact analytical treatment impossible.

If the distribution of the initial velocities of the electrons (in magnitude and direction) is taken into account, then for the electron paths, instead of cycloids, one obtains a family of trochoids, degenerating into a cycloid when the initial velocity is equal to zero. In this case it turns out that de...

focusing of the beam in the direction of the \(X\) axis constitutes a negligible fraction of the distance between the points at which the electrons strike the emitters, as determined without taking into account the velocity distribution. In the design of Fig. 4 there is observed a rather considerable defocusing in the direction of the \(Z\) axis (Fig. 5), caused by the absence of orienting fields in this direction, and also by the space charge of the beam (especially in the last stages, where its density is high). Matters may even go so far that the beam goes beyond the limits of the emitter. The design of Fig. 3 is freer from this defect, since, first, here, when moving in the direction \(Z\) along the emitter, the direction of the normal changes and, second, because the electron flux is embraced on all sides by the emitting surface.

Fig. 6

Despite the good focusing qualities of this design of multipliers (especially the design of Fig. 3), they are not particularly convenient from the standpoint of applications. The reason for this is the presence of two focusing fields, between the intensities of which, for the best focusing, there must exist a rather strictly defined ratio. This can be seen from Fig. 6, which shows the output current of the multiplier (at unchanged light flux and total voltage) as a function of the magnetic-field intensity. It is evident from the figure that the conditions of maximum sensitivity are expressed very sharply.

Thus, in order to obtain maximum sensitivity it is necessary to have the possibility of regulating the magnetic field, which is easily realizable only when using an electromagnet or when providing special devices for moving a permanent magnet. Both are cumbersome; an electromagnet requires a power source, while permanent magnets are not distinguished by any particular constancy. The dependence of sensitivity on the total voltage at unchanged magnetic field is of an equally sharp character.

Fig. 7

1. The first multistage multiplier with purely electrostatic control was proposed by P. T. Farnsworth\(^2\). A diagram of its construction is shown in Fig. 7. In this type of multiplier, multiple bombardment of the surface producing secondary emission (this is the inner surface of the walls of the tube between the leads \(K_1\) and \(K_2\), representing an oxygen–cesium cathode) is achieved by having every emitted electron come under the action of two fields—one radial, created by the potential difference between the layer on the

of the bulb surface and by a wire stretched along the axis of the device; the other—longitudinal—field arises owing to the potential difference applied between \(K_1\) and \(K_2\) (for this purpose the silver layer on which the cesium cathode is formed is applied not directly to the glass, but to a thin layer of nickel or platinum, nonoxidizing under treatment with oxygen and having a resistance of \(0.5\)—\(2\ \mathrm{M}\Omega\); as a result, a uniform potential drop is produced along the entire tube). Under the action of these fields the primary electron, torn out by the light entering the tube through the window \(O\), strikes the opposite wall at a distance determined by the diameter of the tube and by the ratio of the two fields to the energy determined by the potential difference between the points of emission and impact, and there produces secondary emission. The secondary electrons move on in the same manner, and the process continues until the electrons are collected by the funnel-shaped collector located in the lower part of the tube (some of the electrons fall directly on the filament and do not take part in the subsequent multiplication).

Thus, crossed electrostatic fields are used here for control. This method of controlling the electron stream was further developed, as will be discussed below.

This construction represents a very elegant solution to the problem of electrostatic control. In particular, what is practically important is that it is not a multi-lead construction. This makes it possible to give the multiplier comparatively small dimensions. At the same time, it also has a number of essential disadvantages, rooted in the circumstance that the radial field decreases as one approaches the output end of the tube. The consequence of this is the impossibility of dispersing considerable space charges, which leads to a limitation of achievable output currents, to a reduction in gain, and to a shortening of the working section of the light characteristic.

Another variant of control by means of two electrostatic fields was realized by Kubetskii in a multiplier design differing from that just described in that, instead of creating a field gradient along the tube by means of the potential drop across a high-resistance layer, it was created by means of an external voltage divider; the emitting surface itself (also a layer on glass) was divided into separate rings (Fig. 8). This construction, along with all the disadvantages of the preceding one, also has all the defects of a multi-lead system.

Fig. 8

The method of two electrostatic fields found further application in the design proposed by P. V. Timofeev. This construction, the diagram of which is shown in Fig. 9, is also multi-lead and differs in that the electrodes are sep-

individual plates placed inside an evacuated cylinder; the leads from the plates are brought out through an ordinary leg. Otherwise this arrangement can be obtained from the preceding one if the cylindrical tube is cut by a plane passing through its axis, the half-cylinders are unfolded onto a plane, and the filament \(F\) is replaced by a sparse grid \(G\). The presence of a grid instead of a filament improves the conditions for the dissipation of space charges, but, on the other hand, the open form of the electrodes facilitates losses of the electron stream, since focusing is absent here.

Fig. 9

The absence of focusing is the principal shortcoming of all three of the last constructions. It leads to the fact that the mean gain per stage proves to be considerably lower than in systems with focusing, for the same values of the interstage voltage. Thus, for example, whereas the construction of Fig. 10, with 100 V per stage, gives \(\sigma\) about 2.5 (with an oxygen–silver–cesium emitter), systems with focusing give \(\sigma\) about 4.7–5.

1. The next large group of multipliers is characterized by the presence of a single controlling (electrostatic) field, created by applying the corresponding potentials to the electrodes. They are divided into two subgroups, to one of which belong systems with focusing and to the other—systems without it.

For the second subgroup the use of perforated electrodes is characteristic. There exist three constructions, differing in the arrangement of the electrodes. The first of them, proposed by Timofeev (shown in Fig. 10), has electrodes in the form of hollow truncated cones, the inner surfaces of which are coated with an emitting layer. Fig. 11 shows the scheme of another construction, first proposed by Kubetsky, where the emitting electrodes have the form of a Venetian blind. Finally, the third construction, with electrodes in the form of grids, was described by Weiss\(^{14}\).

Fig. 10

Fig. 11

All these three constructions possess the fundamental shortcoming that in them the complete utilization of the electrons liberated at each of the electrodes is impossible. This circumstance is rooted in the structure of the electrodes, on the one hand, and in the arrangement of the fields, on the other. The fact that the electrodes have apertures leads to the passage of part of the electrons from the \(n\)-th electrode through the \((n+1)\)-st without knocking out secondary ones—this is one reason for the decrease in efficiency. Another consists in the fact that the accelerating field, which should draw electrons from each electrode, is screened by that electrode itself.

and does not pull out all the electrons. It is obvious that the degree of utilization of each stage is determined by two factors: 1) the filling factor of the electrode cross section and 2) its transparency. Since improvement of one of these parameters automatically leads to deterioration of the other, the optimum solution must be a compromise. The greatest transparency at the greatest filling factor can be achieved with the smallest possible openings, with the narrowest and shortest possible solid intervals. The third of the designs comes closest to these conditions, and the first is farthest from them. Weiss multipliers had grids with tens of thousands of loops per \(1 \text{ cm}^2\).

The ideal electrode for multipliers of this design would be one semitransparent to electrons and to the electric field which, under the action of the primary stream falling on one side, would give appreciable secondary emission from the opposite side. This principle of the construction of emitting electrodes was patented by Kubetsky; however, all attempts at its practical realization have so far yielded no positive results.

Fig. 12                 Fig. 13

1. The first multipliers with a single controlling field and with focusing were described as early as 1936 by Zworykin and his co-workers\(^3\). Fig. 12 shows the scheme of the so-called type-\(L\) multiplier. Each electrode here consists of two joined cylinders, the outer surface of the joint being cut by a plane inclined at an angle of \(45^\circ\) to the axes of both cylinders. These cuts are the surfaces emitting secondary electrons. The photocathode may be made in the form of a grid or a louver. The focusing properties of this system are due to the presence of gaps between the cylinders forming neighboring electrodes. These gaps play the role of electric lenses. Fig. 13 shows the scheme of another multiplier design based on the same principle. This is the so-called type-\(T\) multiplier. In this case there must be the same relation between the dimensions of the electrodes as before (see Fig. 12), namely:

\[ u + v = 2D, \]

\[ u = \frac{2D}{3}; \qquad v = \frac{4D}{3}; \]

the focusing properties of these lenses do not depend on the potential of the electrodes.

However, despite the fact that focusing is provided in these designs, their use is possible only at very small currents,

since the scattering of space charges is hindered by the shielding action of the cylinders themselves. Therefore such designs are usually used only in those cases where the use of only a small number of cascades is contemplated.

1. Of special interest is a recently developed design of a multiplier with a single control field and focusing, based on a detailed study of the fields and electron paths in an electrostatic multiplier1. We shall dwell on this design in more detail, especially in connection with the fact that, in creating it, various sources of dark currents in electron multipliers were carefully investigated, and also because new general experimental methods were used in connection with it, which may have wide application in the creation of new designs of electronic devices.

Analytical consideration. Let us consider a certain number of electrodes at known constant potentials, and the trajectory described by a charged particle under the influence of electrostatic fields produced only by these charged electrodes. This means, in particular, that we shall not take into account the fields produced by the charged particles themselves, i.e., space charges.

In such an electrode system the potential \(\varphi\) at any point of space satisfies Laplace’s equation:

\[ \Delta^2 \varphi = 0 \tag{35} \]

and is determined by the corresponding boundary conditions, i.e., it assumes prescribed values on the surfaces of the metallic electrodes. Let the charged particle have mass \(m\) and charge \(e\), and represent a charged sphere of vanishingly small radius, which is an admissible approximation as applied to electrons or ions.

Such a particle, moving in a conservative force field determined by the potential \(\varphi\), will obey Newton’s equation of motion

\[ m \frac{d^2 \overline{R}}{dt^2} = - e \operatorname{grad} \varphi \tag{36} \]

(where \(\overline{R}\) is the radius vector of the particle), as well as the initial conditions. In order to obtain the equation of the trajectory, generally speaking, it is sufficient to integrate Laplace’s equation (35) exactly, substitute the found values of the potential into equation (36), and then eliminate time from it. Instead of this one may use the principle of least action (which sometimes proves much simpler). In this case time is eliminated from the very beginning. In the case considered by us, the principle of least action states that in a conservative system of forces the trajectory between two points \(A\) and \(B\) is such that the integral

\[ S = \int_A^B 2W\,dt \quad \text{or} \quad S = \int_A^B mv\,ds \tag{3[[unclear: remainder of formula number]]} \]

(where \(W\) is the kinetic energy, \(v\) the velocity, \(ds\) an element of the trajectory, and \(dt\) an element of time) retains a constant value.

With a proper choice of the zero of potential \(\varphi\), such that the total energy of the particle becomes zero, i.e., such that \(|e\varphi|=|W|\), the second of the integrals (37) becomes, up to constant factors,

\[ S=\int_A^B \sqrt{\varphi}\,ds . \tag{38} \]

It is assumed here that the velocity of the particle is so small that relativistic corrections may be neglected.

Considering the two-dimensional problem and introducing Cartesian coordinates \(x\) and \(y\), we obtain:

\[ S=\int_A^B \sqrt{\varphi}\sqrt{1+\left(\frac{dy}{dx}\right)^2}\,dx . \tag{39} \]

This integral assumes a stationary value in the case where the Euler equation is satisfied:

\[ \frac{d}{dx}F_{y'}-F_y=0, \tag{40} \]

where

\[ F(x,y,y')=\sqrt{\varphi(x,y)}\cdot \sqrt{1+y'^2} \tag{41} \]

and \(y'\) denotes the derivative of \(y\) with respect to \(x\).

This last equation reduces in our case to the so-called ray equation

\[ \frac{d^2y}{dx^2}=\frac{1}{2\varphi}\left(\frac{d\varphi}{dy}-\frac{dy}{dx}\frac{d\varphi}{dx}\right)\left[1+\left(\frac{dy}{dx}\right)^2\right]. \tag{42} \]

From equation (42) it is evident that the trajectory of a charged particle in a given electrode configuration does not depend on its charge or mass and remains unchanged under a proportional change of the potential of all electrodes. Consideration of the equation of motion of the particle (36), with \(\vec R\) replaced by \(\lambda \vec R\), further shows that increasing the dimensions of the system leads only to a corresponding increase in the dimensions of the trajectory itself. These circumstances are of essential importance for what follows.

Generally speaking, analytical integration of these differential equations is not possible in the case of most electrode systems, even the simplest ones. As for numerical integration, this very lengthy and laborious process can be successfully replaced by experimental techniques, which we shall consider below.

Electrolytic bath and graphical methods. The method of the electrolytic bath consists in placing a model of the electrode system in an electrolyte and assigning to all electrodes potentials proportional to the potentials of the electrodes of the device. Under these conditions, the potential of any point of the electrolyte proves to be proportional to the potential of the corresponding point

of the field of the electrode system under actual conditions, so that it becomes possible to construct the potential distribution in the entire system. The potential distribution on the surface of the electrolyte corresponds to that in the plane of symmetry of the electrode system, which separates the immersed part of the model and its mirror image with respect to the surface. The two most frequently encountered types of electrode systems, possessing axial symmetry and consisting of cylindrical electrodes with generators normal to a certain plane, do in fact have such planes of symmetry¹⁶.

The problem of finding electron trajectories from the potential distribution can be solved by numerical integration, as was already noted above. It is more convenient, however, to use graphical methods, which mainly consist in replacing the trajectory by segments of various simple curves, such as straight lines, circles, parabolas, etc. One such method, known as the “circle method,” is based on a very simple relation between the radius of curvature \(R\) at the point \(P\) of the trajectory, the value of the potential \(V\) at this point, and the component \(E_r\) of the electric field normal to the trajectory at the same point \(P\).

For a particle with initial energy \(eV_0\), by the law of conservation of energy we have

\[ \frac{mv^2}{2}=e(V+V_0). \tag{43} \]

Since the centripetal acceleration is proportional to the radial force, one may write:

\[ \frac{mv^2}{R}=eE_r. \tag{44} \]

Eliminating \(mv^2\) from (43) and (44), we find the relation

\[ R=2\frac{V+V_0}{E_r}, \tag{45} \]

which is the basis of the “circle method.”

Suppose that we have a system of equipotential lines (a plane problem), and that a particle arrives at the equipotential line \(V-\Delta V\) with velocity \(v\) (Fig. 14). The center of the circular arc, which is an approximation to the segment of the trajectory between the lines \(V-\Delta V\) and \(V\), will lie on the line \(PTC\), normal to the trajectory at the point \(P\). The distance \(Y=PQ\) is approximately the shortest distance between the lines \(V\) and \(V-\Delta V\), so that

\[ E=\frac{\Delta V}{Y}, \tag{46} \]

Fig. 14

where \(E\) is the field strength at the point \(P\), expressed with the accuracy with which one may regard the change in potential from \(V-\Delta\)

to \(V\) linear with the distance along the trajectory. The field component in the direction \(PC\) will then be

\[ E_r = E \cos a, \tag{47} \]

where \(a = \angle QPC\) (\(QT\) is drawn perpendicular to \(QP\)).

Taking into account that

\[ \cos a = \frac{Y}{Z}, \tag{48} \]

where \(Z = PT\), from equations (45) and (48) we find that

\[ R = 2 \frac{V + V_0}{\Delta V} Z. \tag{49} \]

Thus, in order to find the position of the center it is sufficient to multiply the distance \(PT\) by the known coefficient \(2(V+V_0)/\Delta V\). The arc of the circle \(PP_1\) represents the required approximation of the trajectory segment. Repeating this operation a sufficient number of times, one can construct the entire trajectory. In doing so there is no need to construct \(V\) each time, since the point \(C\) must necessarily lie on the line \(P_1C\) (because the trajectory segments must pass smoothly one into another without violating the continuity of direction). Thus, each time the point \(Q\), the tangent \(QT\), and the arc \(PP_1\) are to be determined.

The properties of equation (45) have been used in the construction of very ingenious machines for automatic plotting of trajectories \(^{17,18}\). This equation has also been applied to a graphical method similar to that described \(^{19}\).

Mechanical model. An extremely convenient experimental method, which makes it possible simultaneously to find the potential distribution and the electron trajectories, is based on the analogy between the motion of a small ball rolling over a properly stretched elastic (rubber) membrane under the action of gravity and the motion of a charged particle in an electric field. Such a technique has been used by many authors \(^{16,20–22}\) to demonstrate the action of electron-optical systems. This method proves applicable to the case of the motion of charged particles in a plane normal to the generating lines of electrode surfaces, which are assumed to be cylindrical and to have infinite extent in the direction of the generatrix or, at any rate, to be sufficiently long that the edge effect has no influence.

The mechanical model of an electrode system is constructed by installing on a horizontal board rigid cylindrical surfaces whose directrices are similar to the directrices of the electrodes under investigation and whose heights are proportional to their potentials. If the trajectory of a negatively charged particle, for example an electron, is being investigated, then the electrode possessing the smallest potential must be the highest of all; for a positive ion, the reverse is true.

Rubber sheeting, previously stretched over a rectangular or circular frame, is then pressed against the model in such a way that it is in contact with the cylinders along the entire length of their upper edges. This means applying pressure to the rubber downward toward the electrodes at those places where its tension does not otherwise allow contact to be obtained. The horizontal projection of the trajectory of the center of a heavy solid ball rolling over this stretched sheet under the action of gravity proves to be similar to the trajectory of a charged particle moving under the corresponding electrical conditions, provided only that the initial conditions are identical in both cases. The validity of this can easily be shown.

Let us first determine what form the surface of the rubber sheet will assume under the indicated conditions. To this end, let us suppose that the rubber was stretched over the frame in such a way that its tension was uniform and large in comparison with the additional tension arising as a result of bringing it into contact with the cylindrical electrodes. Analytically, this may be expressed as the lifting of an initially undisturbed plane surface along a certain contour to a height \(z(x,y)\) relative to the plane of the surface, in which the coordinate axes \(X\) and \(Y\) of a rectangular Cartesian system are located. We shall assume that the deformable surface is an ideal two-dimensional body, offering no resistance to bending; moreover, its potential energy is proportional to its area. Such a body, in striving to assume the form corresponding to the minimum of potential energy, must reduce its surface to a minimum.

The magnitude of the surface area \(z(x,y)\) is given by the integral

\[ S=\iint_A \sqrt{1+\left(\frac{\partial z}{\partial x}\right)^2+\left(\frac{\partial z}{\partial y}\right)^2}\,dx\,dy, \tag{50} \]

where \(A\) is the horizontal projection of the surface inside the boundaries of the contour. The sought function \(z(x,y)\), which brings the integral to a minimum, must satisfy Euler’s differential equation:

\[ \frac{\partial}{\partial x}F_{z_x}+\frac{\partial}{\partial y}F_{z_y}-F_z=0, \tag{51} \]

where

\[ F(x,y,z,z_x,z_y)=\sqrt{1+\left(\frac{\partial z}{\partial x}\right)^2+\left(\frac{\partial z}{\partial y}\right)^2}. \tag{52} \]

This gives

\[ \frac{\partial^2 z}{\partial x^2}\left[1+\left(\frac{\partial z}{\partial y}\right)^2\right] +\frac{\partial^2 z}{\partial y^2}\left[1+\left(\frac{\partial z}{\partial x}\right)^2\right] -2\frac{\partial^2 z}{\partial x\,\partial y}\frac{\partial z}{\partial x}\frac{\partial z}{\partial y}=0. \tag{53} \]

If \(\frac{\partial z}{\partial x}\) and \(\frac{\partial^2 z}{\partial y}\) are small, then (53) becomes

\[ \frac{\partial^2 z}{\partial x^2}+\frac{\partial^2 z}{\partial y^2}=0, \tag{54} \]

that is, the Laplace equation (35), where \(\varphi\) is replaced by \(z\). In other words, for small slopes and similar boundary conditions, the height of any point of the membrane is proportional to the electrostatic potential of the corresponding point of the electrostatic field.

Let now a ball of radius \(R\) roll over the surface \(z(x,y)\) under the action of gravity. We must assume here that there is some friction, since otherwise sliding would occur instead of rolling. We shall assume that “ideal rolling” takes place here, i.e., that the static friction is sufficient for rolling to occur, while dynamic friction is absent. Then our ball will move in a conservative force field, since we neglect energy losses due to friction. If the radius of the ball is sufficiently small in comparison with the radius of curvature of the surface, then the displacement element of the center of the ball and that of the point of its contact with the surface may be taken equal to one another. Therefore, if the center is displaced by \(ds\), then the ball turns through an angle \(d\alpha\), with

\[ d\alpha=\frac{ds}{R}. \tag{55} \]

The total kinetic energy of the ball will be

\[ W=\frac{1}{2}mv^2+\frac{1}{2}I\omega^2 =\frac{1}{2}v^2\left(m+\frac{I}{R}\right), \tag{56} \]

where \(v\) is the velocity of the center of the ball, \(\omega\) is its angular velocity, and \(I\) is the moment of inertia. By the principle of least action, the trajectory described by the ball must be such that the integral

\[ S=\int_A^B 2W\,dt =\int_A^B v^2\left(m+\frac{I}{R^2}\right)dt =\int_A^B\left(m+\frac{I}{R^2}\right)v\,ds \tag{57} \]

(where \(A\) and \(B\) are the beginning and end of the trajectory) has a minimum value.

Since the increase of kinetic energy is equal to the decrease of potential energy, then, with an appropriate choice of the origin on the \(z\)-axis, we shall have:

\[ \frac{1}{2}v^2\left(m+\frac{I}{R^2}\right)=|mgz|. \tag{58} \]

Substituting this expression into the integral \(S\) and neglecting all constant factors, we obtain:

\[ S=\int_A^B \sqrt{z}\,ds =\int_A^B \sqrt{z}\sqrt{1+\left(\frac{dy}{dx}\right)^2+\left(\frac{dz}{dx}\right)^2}\,dx. \tag{59} \]

The integrand here may be regarded as a function only of the arguments \(x, y\), and \(y'\), since \(z\) is a known function of \(x\) and \(y\):

\[ F(x,y,y')=\sqrt{z(x,y)}\sqrt{1+\left(\frac{dy}{dx}\right)^2+\left(\frac{dz}{dx}\right)^2}. \tag{60} \]

If we compare functions (60) and (41) with each other, then, since on the basis of (59) we have that \(z\) is proportional to \(\varphi\), we must regard (60) and (41) as identical, provided only that the quantity \(\left(\dfrac{dz}{dx}\right)^2\) may be neglected, which means small slopes of the stretched rubber sheet. Since these functions are proportional to one another, the Euler equations will be similar, and consequently the trajectories of the charged particle and of the ball will also prove similar to one another, which proves the assertion made above.

The practical realization of the mechanical model is shown in Fig. 15. Its base is a table made of steel tubes, provided with adjusting screws, on which is placed a square glass plate measuring about \(85 \times 85\ \mathrm{cm}^2\). Cylindrical electrodes are conveniently made from aluminum strips about \(1.5\)—\(2\ \mathrm{mm}\) thick, since aluminum, after annealing at \(300\)—\(400^\circ\), can very easily be given any desired shape. These electrodes are fastened in one way or another in the required position on the base glass, on which a diagram of their arrangement is first drawn. The rubber sheet must be homogeneous, with a smooth surface, very elastic, and at the same time sufficiently strong so as not to be punctured by the electrodes and not to deform under the action of the weight of the balls.

Fig. 15

It is stretched over a strong wooden frame, the uniformity of the tension being ensured by drawing on the rubber a figure similar to the frame but of smaller dimensions, after which the sheet is stretched so that the drawing coincides with the edges of the frame. The sheet is pressed against the electrode system by means of auxiliary electrodes of the same shape, placed in the required positions above and held in the desired position by correspondingly arranged holders. Ordinary balls from ball bearings are used as “electrons.” It turns out that balls from \(1.5\) to \(6\ \mathrm{mm}\) in diameter are equally suitable (for the adopted dimensions of the model), so that usually fairly large balls, \(4.5\ \mathrm{mm}\) in diameter, were used. The release of the ball is carried out by means of an electromagnet, whose power supply, when the circuit is opened, is interrupted, accom—

a damped oscillation. This eliminates the possibility that the residual magnetization of the ball will influence its motion. Since, from the point of view of studying the focusing properties of the system, only the initial and final points of the trajectory are of interest, in most cases one can confine oneself simply to visual observation of the motion of the ball, marking scales on the sending and final electrodes (or on the base glass and projecting the image of the scales onto the rubber by means of a light source located under the glass—Fig. 18).

It is also not difficult to record the entire trajectory photographically. If the model is illuminated with intermittent light, for example from a neon lamp operating on alternating current, and the exposure is made during the entire run of the ball, the photograph will show a dotted line representing the trajectory. The distances between successive images of the ball will indicate its relative velocity and, consequently, the values of the potential.

Fig. 16

Fig. 17

Since the ideal conditions that were assumed in the mathematical derivation cannot be realized in practice because of dynamic friction, sagging of the rubber under the weight of the ball, large inclinations of the surface, etc., the applicability of the method was checked directly on an electron device. The diagram of one such device is shown in Fig. 16. It is a model of two adjacent electrodes \(E_1\) and \(E_2\), provided with guard screens \(S\) in order to avoid distortion of the field. The negative electrode \((E_1)\) is made photosensitive, so that the beginning of the electron trajectories is determined by the point to which the light beam is directed. The positive electrode is provided with one movable or a series of fixed electron receivers.

The correspondence between the points of the first and second electrodes is established from the maximum current to the electron receiver, either when the light spot is displaced or (in the case of a movable receiver) when the receiver itself is displaced. Figure 17 shows the dependence of the coordinate

the point of arrival of the electron \((Y)\) from the point of departure \((X)\) for the final electrode configuration. The numbers along the coordinate axes correspond to those plotted in Fig. 18. The dotted curve gives the focusing curve obtained with the mechanical model, while the solid one gives the curve obtained with the apparatus just described. The good agreement between these results and other results, which was also found in a number of other cases, shows the value of the mechanical-model method and points to the possibility of its broad use. It is necessary to note only that this method is not applicable to electrode structures with axial symmetry and cannot be used for solving questions connected with space charges.

Fig. 18

Multiplier designs and elimination of sources of dark currents. The electrode system shown in Fig. 19 arose from combining the conditions for good focusing and for eliminating two of the above-mentioned sources of dark currents, namely ion feedback and cold emission.

In Fig. 19, a shows the first of the electrode configurations of an electrostatic multiplier investigated by means of the methods described. It consisted of two rows of flat plates arranged in such a way that the centers of the plates of one row lay opposite the gaps between the plates of the other. As it turned out, such a system has such poor focusing properties that the construction of a multistage multiplier with it is not possible. However, giving the electrodes a semi-cylindrical shape (Fig. 19, b) makes it possible to obtain quite satisfactory focusing. But, on the other hand, multipliers

Fig. 19

with such an arrangement of the electrodes prove to possess very large dark currents, caused primarily by ionic feedback.

The essence of the phenomenon, which we call ionic feedback, consists in the fact that, at considerable densities of the electron current in the region of the last stages of the multiplier, there becomes possible there, despite the low pressure of the residual gases (chiefly vapors of the alkali metal), the formation of a certain number of positive ions. Under the action of the field these ions move toward the first stages and there knock out new electrons, which undergo multiplication and thus create a certain increase in the output current, which in general leads to a further increase in the number of ions, etc. Thus this process has the property of regeneration, which justifies its name. In its physical essence this process is very similar to the processes taking place in gas-filled photocells and, mathematically, can be treated in an analogous way. We shall not dwell on this. Qualitatively, two cases should be distinguished here. The first of them is that in which one positive ion (on the average) knocks out somewhere in the region of the first stages so many secondary electrons that they (after multiplication) give rise to more than one new positive ion. This is the case of the independent existence of the dark current, when it can reach very large values (being limited only by the defocusing action of space charges). In this case the multiplier gives a large current even in the absence of illumination (since the primary electron current always exists in the multiplier owing to thermoelectronic emission and other factors). The second case is when the number of new ions created by one positive ion is less than unity. In this case the dark current exists only as a certain addition to the output current due to photoelectronic or thermoelectronic emission. A characteristic feature of the dark current resulting from the presence of ionic feedback is its dependence on the total voltage on the multiplier. This current grows exponentially.

The most radical method of eliminating ionic feedback is to prevent positive ions from reaching the first stages from the region of the last stages. This can be achieved by giving the electrodes the proper shape and arrangement, as shown in Fig. 19, c, and also by introducing special shields.

However, even in this case, despite the elimination of ionic bombardment of the first stages, a certain dark current remains, depending exponentially on the voltage on the multiplier. This component of the dark current is due to cold electron emission from the edges of the electrodes, especially considerable because of the small value of the work function of compound surfaces. Its elimination is effected, on the one hand, by increasing the gaps between neighboring electrodes and, on the other, by removing all sharp points on the corresponding-

ing edges of the electrodes. The latter is achieved by enclosing the edge of the electrode in a clip (Fig. 20) made of thin sheet material, which is not subjected to welding or to other methods of processing capable of producing sharp protruding parts. As can be seen from Fig. 20, the ends of this clip are bent over and welded to the other edge of the electrode. This ensures, on the one hand, the rigidity of the electrode and, on the other, is favorable in that it weakens the scattering of the electron stream to the sides. In connection with this it should be noted that, to ensure good focusing (absence of field distortions due to the edge effect), it is necessary that the length of the electrode along the generatrix be at least twice the length of the chord.

Fig. 20

The last of the principal sources of dark current is ohmic leakage in the collector circuit. Recognition of this component of the dark current is based above all on its linear dependence on the voltage on the multiplier. Its presence can also easily be verified by switching off several stages of the multiplier while leaving the collector switched on. Finally, the existence of dark current at voltages on the multiplier of several tens of volts also indicates the presence of ohmic leakage, since under these conditions neither cold emission nor the overall amplification has the magnitude required for the existence of other components of the dark current. The method of eliminating leakage reduces to the best possible insulation of the collector. In this way, without using any guard rings, one can obtain an insulation resistance of the collector of about \(10^{13}\)—\(10^{14}\ \Omega\), i.e., have, at a total voltage on the multiplier of \(2\,000\)—\(3\,000\ \mathrm{V}\), a leakage current of about \(10^{-10}\ \mathrm{A}\). With these measures adopted, the design proved fully suitable for measuring, on the one hand, very small luminous fluxes and, on the other, for studying the thermoelectric constants of photocathodes.

Fig. 21

This design also possesses excellent focusing properties. Its special feature is that a series of such cells has a concentrating action on the electron stream. This is easily seen from consideration of Fig. 21, where the focusing curve is given. Suppose that the primary electron is liberated at the point \(x_1\) of the \(n\)-th cascade. Then this electron falls at the point \(y_1\) of the \((n+1)\)-th cascade, which will be, with respect to the \((n+2)\)-nd, the coordinate \(x_2\), and so on. It is easy to see that the position of the electron in passing from cascade to cascade will change

along a rectangular spiral converging to the point \(P\), i.e., to the middle part of the electrode.

This latter design, in all its properties, is the most advanced of the electron-multiplier designs created so far. In it there remain essentially only the quite fundamental shortcomings of multipliers—high supply voltage and thermionic currents of the photocathode. Thus, further improvements of multipliers must proceed along the line of creating more perfect materials for photocathodes and emitters. We believe that the best of the results presently attainable can be obtained by using, on the one hand, sodium–cesium photocathodes and, on the other, emitters made of metallic cesium with dielectric inclusions.

B. Photocells of increased sensitivity

1. Generally speaking, any of the designs of multistage electron multipliers may be used as a photocell with increased sensitivity. The number of stages in this case is determined by the required degree of amplification, convenient values of the supply voltage, and other considerations.

Fig. 22  Fig. 23  Fig. 24  Fig. 25

The use of such multistage photocells is justified in a number of cases by the resulting somewhat lowered noise level and increased output signal of the photocell, which gives, on the one hand, the possibility of reducing the degree of subsequent tube amplification and, on the other, of relaxing the shielding of the photocell and its circuit.

1. In the case where the photocell has only a single stage of amplification, along with any of the multistage designs there may also be used special single-stage devices distinguished by great simplicity, the main feature of which is the placement of the electron collector between the photocathode and the emitter.

In Figs. 22–25 three designs of single-stage multipliers, or, as they are often called, photodynatrons, are shown. The first of them belongs to Iams and Salzberg\(^{23}\); the second (Fig. 23) was developed—

... Botan at VEI, and the third was described by Gerlich. Here, throughout, \(K\) denotes the photocathode, \(E\) the emitter, and \(C\) the electron collector; the arrows show the direction of the light beam. The action of such photocells is clear from the circuit diagram shown in Fig. 25, where the electrode designations are the same as in Figs. 22–24. The primary electrons, moving toward the collector, which has the highest positive potential \(V_1\), partly fly past it and bombard the emitter \(E\), arriving at it with energy \(V_1 - V_2\), where \(V_2\) is the potential of the emitter. The secondary electrons liberated there are directed to the collector, into whose circuit the load is connected.

The magnitude of the current in the collector circuit, along with the magnitude of the photocurrent and the coefficient of secondary emission, depends on the position and configuration of the collector, its potential, and other factors. Consideration of these dependences is of no interest, and we shall not dwell on them.

CONCLUSION

1. The path of development of secondary-electron multipliers—from the idea of using secondary emission to amplify weak electron fluxes and the first devices in which this idea was realized, to devices of practical value—is essentially the path of solving three main problems: a) the problem of secondary-electron emitters, b) the problem of the design of the electrode system, and c) the problem of supplying power to the multiplier.

If the first and second tasks may at present already be considered to some extent solved (the creation of more advanced emitters, the development of general methods for studying multiplier designs and methods for suppressing dark currents), then the third task still awaits its solution. Although the question of power supply does not at first glance appear to be as essential for secondary-electron devices as the first two, in reality its state has a very substantial effect on the possibilities for practical use of multipliers, as was noted above. The most promising direction here, because of the properties of electron multipliers (low powers at high total voltages), is apparently the creation of high-frequency power sources. The possibility of using such power will in all likelihood make it possible to introduce a number of technically important simplifications into multiplier design. All this will undoubtedly broaden the field of application of electron multipliers, just as any improvement in the design, emitting materials, photocathodes, etc., will do.

Thus, for example, if a modern multiplier with an oxygen-silver-cesium cathode has no advantage over other commonly used indicators, then, for example, the creation of new cathodes more sensitive in the region of infrared rays could already pose this question in a new way.

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1. Reference number as printed in the source. ↩