Electrical Fluctuations and the Sensitivity Limit of Electrical Instruments*

V. L. Granovskii, Moscow

5. “Secular” Variations of Electron Emission

All the experiments described above for detecting and measuring the shot effect (except for Hartmann’s experiments, which gave poor results) were carried out at radio frequencies. They established the existence of the effect predicted by Schottky, and the conditions under which it appears in its pure form. The study of voltage fluctuations in the circuit of an electron tube at low frequencies, undertaken by Johnson,⁹ led to the discovery of a new fluctuation effect.

Johnson worked with two tubes: one had a pure tungsten cathode, the other an oxide cathode. The load was a tuned circuit whose natural frequency could be varied from 6000 to 8 hertz. This was achieved by changing coils (coils of 0.025 henry, 0.1 and 38 henries were used) and capacitors (a set of capacitances from 0.001 to 7.5 μF). The circuit was connected to the input of a five-stage resistance amplifier, giving a total voltage gain of \(3 \cdot 10^5\) times and a power gain of \(10^{14}\) times; the current at the output was measured with a thermocouple. The entire apparatus—the measuring circuit, amplifier, and thermocouple—was calibrated with a sinusoidal alternating current from a tube generator. Unfortunately, Johnson does not give the frequency characteristic of his amplifier, indicating only a comparatively small value¹ of distortion. The results were processed according to the formula:

\[ \overline{v^2} = \frac{e i_0}{2C^2} \frac{L}{R\left(1+\dfrac{L}{R R_1 C}\right)} \left[ 1+\frac{C}{L}\frac{R^2}{1+\dfrac{R}{R_1}} \right], \tag{25} \]

where \(R_1\) is the total shunting resistance, including the resistance of the tubes. Formula (25) was derived by Frey¹² and represents a generalization of formula (14) of § 1. This formula gives the voltage fluctuations directly in the measuring circuit.

* Continuation; see Uspekhi Fizicheskikh Nauk 13, 805, 1933; 15, 435, 1935.

Apparently, Johnson took the gain of all frequencies of significance to be the same; Johnson does not discuss the magnitude of the error arising from this.

All the experiments he carried out showed that the observed magnitude of the fluctuations does not correspond to that calculated by formula (25). The discrepancy is systematic, namely: 1) the observed fluctuations prove to depend on the frequency, which should not be the case for the shot effect; 2) when the current strength \(i_0\) is increased, they grow not proportionally to its first power, as Schottky’s theory requires, but rather proportionally to the square. These results are clearly seen in Figs. 23a and 23b. Fig. 23a shows the ratio of the observed fluctuations to those expected according to

Fig. 23a. Electron-current fluctuations at low frequencies (Johnson).

Schottky’s theory at various frequencies for a tube with a tungsten cathode. We see that at frequencies exceeding 1000 hertz the experiment agrees satisfactorily with the theory; however, at lower frequencies, and especially below 100 hertz, the fluctuations increase strongly; at 10 hertz they exceed the theoretical value by a factor of 50. With an oxide cathode the magnitude of the fluctuations proves to be still considerably greater, and deviations from the theoretical value begin at higher frequencies. The dependence on current strength is shown in Fig. 23b, where along the abscissa are plotted the calculated values of the fluctuations for a current strength varied from 0.1 to 5 mA and constant frequency, and along the ordinate—the observed values. These observations were made at different frequencies; at all frequencies a more rapid growth of the fluctuations with current strength was obtained than is required by Schottky’s theory. In addition, Fig. 23b once again shows that at the lowest frequencies (curves \(D_1\) and \(D_2\)) the fluctuations are especially large.

ELECTRICAL FLUCTUATIONS

The enormous magnitude of the effect observed by Johnson and the predominance in it of low frequencies compel one to think that here we are dealing not with the chaotic emission of individual electrons, but with comparatively slow, though considerable, oscillations in the emission of whole regions of the cathode. According to Johnson, the cause of such “secular” variations* in the emissive capacity of the cathode may be any processes affecting the surface of the cathode: evaporation and reverse condensation, diffusion, recrystallization, or any other structural changes, bombardment by gas ions, chemical processes, etc. Schottky^4 found that the cause of the phenomenon discovered by Johnson is fluctuations in the density of the layer of foreign atoms on the cathode surface. These atoms may settle on the surface of the cathode from outside—for example, be adsorbed from the residual gas in the lamp—or, conversely, diffuse to it from within. The first mechanism is more probable in the case of a tungsten cathode, the second in the case of an oxide cathode. The elementary event here is the residence of one foreign atom on the surface of the cathode. Each such atom changes the emissive capacity of some adjacent region of the cathode and causes an increase in electron emission throughout the entire time of its residence on the surface.

Fig. 23b. Fluctuations of the electron current at low frequency as a function of current intensity. Curve \(A_1\) was taken at \(0.1\ \mathrm{H}\) and \(0.8\ \mu\mathrm{F}\), \(A_2\)—at \(0.1\ \mathrm{H}\) and \(0.1\ \mu\mathrm{F}\), \(A_3\)—\(0.1\ \mathrm{H}\) and \(0.01\ \mu\mathrm{F}\), \(B_1\)—\(0.025\ \mathrm{H}\) and \(0.2\ \mu\mathrm{F}\), \(B_2\)—\(0.025\ \mathrm{H}\) and \(0.02\ \mu\mathrm{F}\), \(D_1\)—\(38\ \mathrm{H}\) and \(1.00\ \mu\mathrm{F}\), \(D_2\)—\(38\ \mathrm{H}\) and \(0.10\ \mu\mathrm{F}\), \(D_3\)—\(38\ \mathrm{H}\) and \(0.01\ \mu\mathrm{F}\), and \(D_4\)—\(38\ \mathrm{H}\) and \(0.0022\ \mu\mathrm{F}\).

Schottky compares this phenomenon with random fluctuations of the surface brightness of a luminous body and, by analogy, calls it the flicker effect (English flicker, German flackern—flickering).

The essential difference between this kind of fluctuation and the shot effect consists not only in the fact that each elementary event produces a significantly larger current impulse than the emission of an individual electron, but also in the fact that the magnitude of the impulse depends on the duration of the individual event. Therefore the mean residence time of a foreign atom on the surface \(\tau\) (or the reciprocal

* We call them “secular” because their mean period is large in comparison with the time intervals between the emission of individual electrons.

its magnitude \(a=\dfrac{1}{\tau}\) must enter into the expression for the magnitude of the effect. Schottky gave a theory of the phenomenon analogous to his theory of the shot effect, i.e. proceeding from consideration of the spectrum of the flicker effect. The electron current is expanded in a Fourier series:

\[ i=i_0+j=i_0+\sum_{k=1}^{\infty} A_k \cos \omega_k t+\sum_{k=1}^{\infty} B_k \sin \omega_k t, \]

whose fundamental period \(T\) must be taken very large in comparison with \(\tau\).

For \(\overline{A_k^2}\) and \(\overline{B_k^2}\) we have expressions analogous to (10):

\[ \overline{A_k^2}=\frac{4}{T^2}\int_0^T j(t)\cos \omega_k t\, j(t')\cos \omega_k t'\,dt\,dt' \]

and, correspondingly, for \(B_k\), with cosines replaced by sines.

The mean current \(i_0\) corresponds to some mean number of foreign atoms on the cathode surface \(N_0\); if, however, the number of these atoms \(N\) differs from the mean by \(n=N-N_0\), then the current also deviates by the corresponding amount \(j\). Schottky assumes that each foreign atom causes a change in the current by one and the same fraction \(F\); the changes in the current caused by individual atoms add additively. Such an assumption may be justified if the individual atoms are, on the average, located sufficiently far from one another and, consequently, together occupy only a small part of the cathode surface. Then

\[ j=nFi_0 \tag{26} \]

and

\[ \overline{A_k^2}=\frac{4F^2 i_0^2}{T^2}\int_0^T\int_0^T n(t)n(t')\cos \omega_k t\cos \omega_k t'\,dt\,dt' . \]

We can repeat here the same reasoning as in the consideration of the shot effect, and we find that in order to calculate \(\overline{A_k^2}\) we must establish the correlation between the values of \(n\) for the times \(t\) and \(t'\), i.e. determine \(\overline{n(t)n(t')}\). This quantity is a function of \(t-t'=\Delta t\). Schottky believes that, in its determination, an essential role is played only by those molecules which were on the surface both at the beginning and at the end of the interval \(\Delta t\), since the remaining molecules have no statistical connection with one another and can affect \(n(t)\) and \(n(t')\) both in the same and in the opposite direction. It is here that \(\tau\)—the mean residence time of a molecule on the surface—enters into the consideration. Namely, if

if at the beginning of the interval there were \(N_1\) of them, then at the end of it there remains

\[ N' = N_1 e^{-\frac{\Delta t}{\tau}} = N_1 e^{-\alpha \Delta t} \left(\alpha=\frac{1}{\tau}\right). \tag{27} \]

Further, according to what was said above,

\[ \overline{n(t)n(t')}= \overline{(N_1-N_0)(N'-N_0)} = \overline{n\left[(N_0+n)e^{-\alpha\Delta t}-N_0\right]} = \]

\[ =\overline{n^2}e^{-\alpha\Delta t} -\left(1-e^{-\alpha\Delta t}\right)\overline{n}N_0 = \overline{n^2}e^{-\alpha\Delta t} = N_0e^{-\alpha\Delta t}, \]

since, by the general theorems of statistics, \(\overline{n}=0\) and \(\overline{n^2}=N_0\) [see § 1, formulas (3) and (3′)].

Now we may write, replacing \(t'\) under the integral by \(t+\Delta t\) and passing to the variable \(\Delta t\) (the limits, for given \(t\), being from 0 to \(\infty\)):

\[ \overline{A_k^2} = \frac{4F^2 i_0^2}{T^2}N_0 \int_0^T \cos \omega_k t\,dt \int_0^\infty e^{-\alpha\Delta t}\cos\omega_k(t+\Delta t)\,d(\Delta t) = \]

\[ = \frac{4F^2 i_0^2}{T^2}N_0 \int_0^T \cos^2\omega_k \frac{\alpha}{\alpha^2+\omega_k^2}\,dt = \frac{4F^2 i_0^2}{T^2}N_0 \frac{\alpha}{\alpha^2+\omega_k^2} \frac{T}{2} = \]

\[ = \frac{2}{T}\, \frac{F^2 i_0^2 \alpha N_0}{\alpha^2+\omega_k^2}. \]

We shall find an analogous expression for \(B^2\) as well. The mean sum of the squares of both coefficients is

\[ \overline{C_k^2} = \overline{A_k^2} + \overline{B_k^2} = \frac{4}{T}\, \frac{F^2 i_0^2\alpha N_0}{\alpha^2+\omega_k^2}. \tag{28} \]

At this point one may stop in the exposition of the theory of the flicker effect. It is sufficient to compare formula (28) with the corresponding formula (12″) for the shot effect from § 1 in order to see all the characteristic features of Johnson’s results. We note, first of all, that whereas the intensity of all components of the shot effect increases proportionally to \(i_0\), here it is proportional to \(i_0^2\). Further, we see that the spectrum of the flicker effect, in contrast to the shot effect, exhibits a definite frequency dependence; namely, \(\overline{C_k^2}\) tends to a constant limiting value at very low frequencies, and decreases to zero at high frequencies. The fluctuations observed experimentally represent the sum of both effects. Therefore at high frequencies practically only the shot effect remains; at low frequencies, such that \(\omega_k^2>\alpha^2\), the flicker effect rapidly begins to increase. This is what Johnson observed in his experiments. The circumstance that the increase of the observed effect begins at frequencies of the same order as \(\alpha\) can be used to determine \(\alpha\) and, consequently, the mean residence time of a foreign atom on the surface

cathode. Of course, for this one cannot directly use formula (28). It is necessary to calculate the voltage fluctuations in the anode circuit from the formula:

\[ \overline{v^2}=\int_0^\infty Z^2(\omega)\,C^2(\omega)\,d\omega \]

and to compare this quantity with the experimental data. Schottky made the calculation and found, from Johnson’s data, that in the case of an oxide cathode \(\tau \simeq 10^{-3}\) sec., while in the case of a tungsten cathode \(\tau \simeq \frac{1}{20}\) sec. Further, having calculated the factor \(F\) by the theory of electrical images (for numerical determination it is necessary to use experimental data on the influence of adsorption of gas by a monomolecular layer on electron emission), Schottky obtained the possibility, from Johnson’s measurement results, of determining \(N_0\), i.e. the density of the atoms located on the cathode surface.

Fig. 24. Depression of the shot effect and flicker effect (Pearson). Curve \(B\)—thoriated cathode, \(A\)—oxide cathode.

Thus, if the shot effect is a method that makes it possible to study the conditions under which the electron current flows in a tube, then the flicker effect makes it possible to study processes on the cathode surface in a new and distinctive way. In general, electrical fluctuations are beginning to find more and more application as a method for investigating various physical phenomena. We shall encounter some examples of this kind at the end of the article.

Our description of the secular variations of the electron current would be incomplete if we did not mention the influence of space charge on them. As for the shot effect, the space charge here has a smoothing action, ordering the motion of the electrons and reducing the amplitude of current oscillations. Therefore, with an increase in the density of the electron current without an increase in their velocity, beginning from a certain moment one should observe a cessation of the growth of the flicker effect, and then its decrease. This fact had already been established in Johnson’s work, who, for his investigations, had to restrict himself to currents not exceeding 5 mA. An increase in the anode current strength by increasing the heating, without changing the anode voltage, up to 5 mA caused an increase in the fluctuations; however, at larger current strengths their magnitude fell sharply. The same phenomenon was also observed by Pearson\(^{37}\), from whose work we borrow Fig. 24. This figure shows

two curves representing the dependence of the magnitude of the fluctuations on the current strength at constant anode voltage (the current is varied by means of the filament heating). Curve B refers to the thorated cathode; it shows a pure shot effect, in which, beginning with $i_0 = 3$ mA, depression sets in. Curve A was taken with a cathode coated with BaO; the fluctuations, many times larger than those of the first cathode, undoubtedly represent secular oscillations of the emission. At $i_0 > 4$ mA they also fall sharply and, as is seen from the figure, at large current strengths become even smaller than the fluctuations with the thorated cathode. This latter circumstance cannot be explained, since the author, unfortunately, gives no other data on the conditions of the experiment, in particular on the construction of the tubes. The very fact of depression of the flicker effect at high charge density is, however, presented extremely clearly.

6. Shot effect in a current of positive ions

The theory of the shot effect set forth in § 1 was tested for the electron current in cathode tubes (§ 2) and photoelectric cells (§ 3). It is easy, however, to convince oneself that it is also applicable to a current made up of any carriers of electricity whatever, provided only that their motions are statistically independent of one another and that their distribution in time is completely random. Neither the sign of the charge nor the mass of the carriers of electricity plays any role in deriving the formulas of § 1. Therefore one may expect that in a current of positive ions a shot effect should likewise occur; moreover, its magnitude should be determined by the same formulas as for the electron current, i.e., at very small space charge—by formulas (13) and (16), and at sufficiently large space charge—by formula (21).

This question was studied experimentally by Williams and Haxford.^23 The principal experimental difficulty consisted in obtaining a source of positive ions satisfying the requirements of the theory of the shot effect; above all, it had to give as pure as possible an emission of ions of one sign. After a number of experiments, the authors settled on a thermionic emitter developed by Kunsman,^33 emitting $K^+$ ions. It consisted of a tungsten spiral wound on a quartz tube 8 mm in diameter and 6 cm long. A paste made of a mixture of $K_2O$ and $Fe_2O_3$ in paraffin oil was applied to the spiral; after ignition this paste bound to the spiral in a single solid layer. Such an emitter, used as the anode, gave fairly constant currents of $K^+$ ions up to 400 μA; the current strength decreased by no more than 5% per hour. The electron emission was insignificant; even at a much higher heating, which was not used in the measurements (900°), at which $i_+ = 1200$ μA, the electron current amounted to no more than 12 μA, i.e. 1%. At lower temperatures the electron emission decreased much more rapidly than the ionic emission; therefore, with ionic emission up to 400 μA, the electron emission could be neglected.

A substantial role in the ionic current is played by space charge. Since the mobility of ions is much smaller than that of electrons, for the same current strength the space charge in the former case will be many times larger. Strong fields are necessary in order to prevent the accumulation of ions. Therefore the cathode was made in the form of a cylinder coaxial with the anode, and the gap between them was left only 1 mm; the voltage on the cathode was increased as the emission was intensified, for the same purpose. During the measurements the tube was connected to a pump and evacuated to \(p \ll 10^{-5}\) mm. The method of measurement adopted was the same as in Williams’s other work—the “substitution” of a sinusoidal EMF; the amplifier, the method of calibration, and the measurement of the voltage at the output were also borrowed from earlier works.

Fig. 25. Depression of the shot effect in the current of \(K^+\) ions (Williams and Haxford); upper curve at 200 μA, lower curve at 400 μA.

The frequency band passed by the amplifier was chosen from 112 to 121 kilocycles. To check the entire apparatus, the charge of the electron was first determined from the shot effect in the electron current; from a series of measurements it was found that \(e = 1.585 \cdot 10^{-19}\) coulomb (arithmetic mean) and \(e = 1.589 \cdot 10^{-19}\) coulomb (weighted mean). Then the shot effect in the current of \(K^+\) ions was measured. We reproduce the results of the measurements in Table 6; the first column gives the values of the cathode voltage, the second the ionic-current strength, the third the equivalent sinusoidal voltage \(v_1\) (see § 2), and the fourth the value of the charge of the \(K^+\) ion calculated from these data.

The value of \(e_+\) is obtained of the correct order and, at small current strengths, the error is less than 1%. However, with increasing current strength a small but systematic trend in the value of \(e_+\) is observed; the decrease of the latter indicates that the space charge of the \(K^+\) ions is beginning to manifest itself.

Williams and Haxford studied the depression of the shot effect in the current of positive ions by recording curves of the dependence of the shot effect on the cathode voltage at constant current strength to the cathode (analogously to the method of Kazanovsky and Williams described above). A very distinct picture of the depression is visible in the curves of Fig. 25. Along the abscissa axis is plotted the voltage between cathode and anode; along the ordinate axis, the value of \(e_+\) calculated from the experiment. At high voltages and relatively small emission, \(e_+\) tends toward a definite, and moreover correct, limit; for \(i_0 = 400\) μA, \(e_+\) approaches the limit more slowly than for \(i_0 = 200\) μA, since the ion density is greater in the former case. At small voltages and, consequently, large emission (so

as \(i_0 = \mathrm{const}\)) the shot effect decreases; the drop becomes especially sharp at \(V_k < 100\ \mathrm{V}\). Over the whole voltage range

TABLE 6

Shot effect of the current of positive ions
(Williams and Haxford)

from 50 to 800 V the depression at \(i_0 = 400\ \mu\mathrm{A}\) is greater than at \(i_0 = 200\ \mu\mathrm{A}\). Thus it has been proved that in the current of positive ions the shot effect exists and is expressed by the same equations as in electron current.

7. Sensitivity threshold of electronic instruments

Electrical fluctuations are of interest not only as one of the characteristic chapters of physical statistics. We have already seen that in a number of cases they can serve as a method for studying the microstructure of electric current: the mechanism of emission, space charge, etc. However, the greatest practical significance of the problem of fluctuations lies in the fact that they limit the sensitivity threshold of physical instruments, including electrical ones. In Part I of the review we already touched upon this question in connection with the thermal effect in conductors. We shall now analyze the influence of all kinds of electrical fluctuations on the sensitivity threshold of two basic electronic instruments: the cathode tube and the photocell.

Let us consider the schematic circuit of the input of a tube amplifier (Fig. 26). Let the emf generator \(E\) be connected to the grid of the first tube of the amplifier \(A\); between the grid and the filament of the tube a load \(Z_g\), in general complex, is connected; in this quantity we also include the internal capacitance and resistance between cathode and grid. In the anode circuit of the tube there is also a complex load \(Z_a\); voltage oscillations across the latter are transmitted to the subsequent stages of amplification. The generator \(E\) itself has some internal resistance \(z\). In the diagram it is shown as galvanically connected to the tube; in reality, the coupling is often inductive

(for example, the coupling of the antenna with the first tube of the receiver) or capacitance. For these cases Fig. 26 should be regarded as an equivalent circuit, sufficient to give us a general orientation in the subject; a more detailed consideration is not now part of our task. In practice the question is usually posed as follows: for a given generator resistance \(z\), one must choose the operating regime of tube \(A\) and of the load \(Z_g\) and \(Z_a\) so as to be able to make out the smallest possible emf \(E\), and what is the attainable threshold in this case?* Let us first determine what fluctuation phenomena we shall have to reckon with. The following are to be considered: 1) the thermal effect in the input circuit, 2) the shot effect of the grid current in this same circuit, 3) the thermal effect in the anode circuit, 4) the shot effect of the anode current in this circuit, 5) the flicker effect in the same circuit, 6) the thermal effect in the internal resistance

[In the block of the diagram: “Subsequent amplification stages.”]

Fig. 26. Circuit of the input of a tube amplifier.

of the tube. Fluctuations caused by positive ions in the region of space charge we discard: with a good vacuum and a cathode with a metallic surface their magnitude is negligible. We shall also not speak of other sources of noise: leakage between electrodes, static charges on the glass, poor contacts, etc., assuming that they have been eliminated in the manufacture of the tubes. That this can in fact be achieved to a very considerable degree we shall see below.

Of the fluctuation effects listed above, some occur in the grid circuit, and some in the anode circuit. Therefore their role in the total noise will not be the same; the first group of noises is amplified by the whole amplifier, including the first tube, whereas amplification of the second group occurs only in the subsequent amplification stages. In order to be able to compare them, it is necessary to reduce all the phenomena under consideration to one

* For example, in the case of an inductively coupled antenna, the emf \(e_1\) arising in it can be transferred to the secondary circuit with the factor

\[ m=\frac{n_2}{n_1} \]

(the ratio of the numbers of turns in the secondary and primary coils), and the component resistances of the antenna \(r_1\) and \(x_1\)—with the factor \(m^2\).

circuit; we shall do this for the input circuit. For simplicity of calculation, let us assume that the amplifier passes the frequency band \(f_2—f_1\) and amplifies uniformly within this band. Let us agree that the shot effect will be denoted by the subscript \(s\), the thermal effect by the subscript \(t\), the external anode circuit by \(a\), the internal one by \(i\), and the grid circuit by \(g\); the effects of the anode circuit, recalculated to the grid circuit, will be denoted by the subscript \(g\) on the left.

The thermal effect in the grid circuit is caused by the watt components of the impedances \(z\) and \(Z_g\), which must be regarded as connected in parallel. Denoting these components by \(r\) and \(R_g\), we write:

\[ \overline{v_{tg}^{2}}=4kT\int_{f^1}^{f_2}\frac{rR_g}{r+R_g}\,df, \tag{29} \]

If the pass band of the amplifier is so narrow that within its limits \(r\) and \(R_g\) may be considered constant, then

\[ \overline{v_{tg}^{2}}=4kT\frac{rR_g}{r+R_g}(f_2-f_1)=4kTR_1\Delta f, \tag{29'} \]

where \(R_1\) is the total watt resistance of the input, and \(\Delta f=f_2-f_1\).

At \(T=300^\circ\mathrm{K}\) we obtain:

\[ \overline{v_{tg}^{2}}=1.64\cdot10^{-20}R_1\Delta f. \tag{29''} \]

The magnitude of the voltage from the generator applied to the grid, which we shall call the “signal,” is equal to

\[ v_{\mathrm{sign}}=\frac{EZ_g}{z+Z_g}. \tag{30} \]

The solution of the problem of the greatest sensitivity of a circuit is sometimes sought by making, by any means whatever, the magnitude \(\overline{v_t^2}\) as small as possible. This, of course, is wrong, since in doing so it may easily turn out that the magnitude of the signal is also greatly reduced. Only those methods of reducing noise are acceptable as solutions which do not at the same time weaken the signal or, in any case, increase the ratio of signal to noise (“signal-to-noise ratio” of American authors). This ratio, which we shall denote by \(\dfrac{c}{w}\), in the presence of a single thermal effect is expressed as follows:

\[ \left(\frac{c}{w}\right)^2= \frac{v_{\mathrm{sign}}^2}{V_{tg}^{2}} = \frac{E^2Z_g^2}{4kT(Z_g+z)^2R_1\Delta f}. \tag{31} \]

From formula (31) we directly see two indications: 1) it is desirable to lower the temperature of the input circuit—a measure of little practical applicability; 2) the frequency band \(\Delta f\) must be made as narrow as possible; the limit to reducing \(\Delta f\) is imposed only by the character and duration of the signals.* Further, if the resistance \(Z_g\) com-

* With a very small \(\Delta f\), the setting time of the apparatus increases greatly, and if the latter becomes comparable with the duration of the signals, then the latter will be reproduced with distortions.

if it is complex, it is advantageous to decrease its wattful component \(R_g\) and to increase as much as possible the wattless component \(X_g\). If, however, the entire load in the grid circuit is wattful, then the conclusion will be somewhat different. Put in formula (31) \(Z_g=R_g\) and \(z=r\). We find

\[ \left(\frac{c}{u}\right)^2=\frac{E^2R_g}{4kT[R_g+r]r}, \tag{31′} \]

whence it is seen that it is desirable to make \(R_g\) as large as possible. The usually applied condition \(R_g=r\) does not give the best result; for \(R\to\infty\) the ratio \(\frac{c}{u}\) proves to be \(\sqrt{2}\) times larger. In this case we simply find:

\[ \left(\frac{c}{u}\right)^2=\frac{E^2}{4kTr\Delta f}. \]

The next source of noise in the same circuit is the shot effect of the grid current. The latter may be composed of the electron current, the current of positive ions, and also secondary emission, photoeffect, etc. The magnitude of the shot effect of the grid current will be the sum of the effects caused by each of these currents, and therefore will be expressed by the equality:

\[ \overline{v}=2e\sum i_g\int_{f_1}^{f_2}\left(\frac{Z_\sigma z}{Z_g+z}\right)^2\,df \tag{32} \]

or, under the assumption of sharp tuning of the amplifier:

\[ \overline{v_{sg}^{\,2}}=2e\sum i_g Z_1^2\Delta f, \tag{32′} \]

where \(\sum i_g\) is the arithmetic (not algebraic) sum of all currents to the grid, \(Z_1=\dfrac{Z_g z}{Z_g+z}\). Since \(Z_1\) cannot be decreased, as we saw above, it is necessary to reduce as far as possible all grid currents; for this a negative potential on the grid, a good vacuum, etc., are required. If the grid is left “freely suspended” at a potential established as a result of equilibrium between currents of both directions, then the quantities \(\sum i_g\) and \(Z_g\) will be large; numerous observations show that in this case the noise is considerably intensified (see, for example, \(^{37}\)). Conversely, with a sufficiently negative potential this whole effect may be less than the thermal one. Indeed, let the input resistance be wattful, so that \(Z_1=R_1\). The ratio of the thermal effect in the grid circuit to the shot effect is:

\[ \frac{\overline{v_{tg}^{\,2}}}{\overline{v_{sg}^{\,2}}} = \frac{4kTR_1\Delta f}{2e\sum i_g R_1^2\Delta f} = \frac{2kT}{eR_1\sum i_g}. \tag{33} \]

Put \(T=300^\circ\mathrm{K}\), \(R_1=1\,\mathrm{M}\Omega\), \(\sum i_g=1\cdot10^{-8}\,\mathrm{A}\); we find:

\[ \frac{\overline{v_{tg}^{\,2}}}{\overline{v_{sg}^{\,2}}} = \frac{2\cdot1.36\cdot10^{-23}\cdot300}{1.6\cdot10^{-19}\cdot10^6\cdot10^{-8}} \approx 5, \]

Therefore, with a sufficiently small grid current and input watt resistance, it may be assumed that the fluctuations in the grid circuit are caused chiefly by thermal motion, and therefore one may be guided by the conclusions from formulas (31) and (31′).

The situation is different in the anode circuit, where, usually, owing to the considerably greater current, the shot effect predominates over the thermal one. The shot effect in this circuit is expressed by Llewellyn’s formula (21), which in our case may be written as

\[ \overline{v_{sa}^{2}}=2J_{0}e\left(\frac{\partial i}{\partial J}\right)^{2} Z_{a}^{\prime 2}\,\Delta f, \]

where \(Z_a'=\dfrac{Z_a Z_i}{Z_a+Z_i}\), and \(Z_i\) is the internal resistance of the tube between cathode and grid. For the thermal effect of the external and internal parts of the circuit taken together we may write formula (24):

\[ \overline{v_{ta}^{2}}= 4k\,\frac{T_i R_i Z_a^{2}+T_a R_a Z_i^{2}}{(Z_i+Z_a)^2}\,\Delta f. \]

\[ * \]

The ratio of these effects is

\[ \frac{\overline{v_{sa}^{2}}}{\overline{v_{ta}^{2}}} = \frac{ J_0 e\left(\dfrac{\partial i}{\partial J}\right)^{2} Z_a^{2} Z_i^{2} }{ 2k\left[T_i R_i Z_a^{2}+T_a R_a Z_i^{2}\right] }. \]

Under the usual operating conditions of the tube, the first term of the sum in the denominator is considerably greater than the second, since \(Z_a>Z_i\) and \(T_i>T_a\). This means that the thermal motion inside the tube creates more noise than the thermal effect of the external circuit. Discarding the term corresponding to the latter, we find:

\[ \frac{\overline{v_{sa}^{2}}}{\overline{v_{ta}^{2}}} = \frac{ J_0 e\left(\dfrac{\partial i}{\partial J}\right)^{2} Z_i^{2} }{ 2kT_i R } \]

or, since the internal resistance of the tube at not very high frequencies has mainly a watt component, one may put \(Z_i\simeq R_i\), which gives:

\[ \frac{\overline{v_{sa}^{2}}}{\overline{v_{ta}^{2}}} = \frac{ J_0 e\left(\dfrac{\partial i}{\partial J}\right)^{2} R_i }{ 2kT_i }. \tag{34} \]

It is desirable, of course, that \(\dfrac{\partial i}{\partial J}\) be close to zero, since then, owing to depression, the shot effect will become vanishingly small. Under these conditions, on the anode side of the tube there will remain only one source of fluctuations—thermal motion. However, in practice it is difficult to reduce \(\left(\dfrac{\partial i}{\partial J}\right)^2\) below several de—

\[ \text{* It should be borne in mind that this formula exaggerates the thermal effect of the tube; see § 4b.} \]

ones. Taking \(\left(\dfrac{\partial i}{\partial J}\right)^2 = 0.2\), \(J_0 = 5\ \mathrm{mA}\), \(R_i = 2 \cdot 10^4\ \Omega\), \(T_i = 1100^\circ\mathrm{K}\) (oxide cathode), we find:

\[ \frac{\overline{v_{sa}^{2}}}{\overline{v_{ta}^{2}}} \approx 100. \]

Therefore, at not very low frequencies, we must take chiefly the shot effect into account in the anode circuit. To determine its significance for the threshold sensitivity of the tube, it is necessary to reduce it to the equivalent voltage in the grid circuit. The amplification factor of the tube under operating conditions is:

\[ \mu' = \frac{\mu Z_a}{Z_a + Z_i}. \]

We shall denote the shot effect referred to the grid circuit by \(\overline{g v_{sa}^{2}}\).

\[ \overline{g v_{sa}^{2}} = \frac{\overline{v_{sa}^{2}}}{\mu'^2} = \frac{2 J_0 e \left(\dfrac{\partial i}{\partial J}\right)^2 Z_i^2 \Delta f}{\mu^2} = \frac{2 J_0 e \left(\dfrac{\partial i}{\partial J}\right)^2}{S^2}\,\Delta f, \tag{35} \]

where \(S = \dfrac{\mu}{R_i}\) is the slope of the static characteristic of the tube. It is interesting to note that the quantity \(\overline{g v_{sa}^{2}}\) is entirely independent of the load in the anode circuit. There is no need to calculate the ratio \(\dfrac{c}{w}\) for the shot effect, since comparison of formulas (30) and (35) shows directly that the magnitudes of the signal and of the shot effect depend on entirely different factors, and one may operate with quantities affecting \(\overline{g v_{sa}^{2}}\) without fear of any decrease in the signal. From formula (35) it is clear that it is desirable to work with a small emission current, the strongest possible depression, and a large slope; one of the conditions for satisfying these (in general poorly compatible) requirements is a low anode voltage, ensuring the development of a sufficiently dense space charge. Let us now compare the magnitude \(\overline{g v_{sa}^{2}}\) with the thermal effect of the grid circuit \(v_{tg}\). From (29″) and (35) we find:

\[ \frac{\overline{g v_{sa}^{2}}}{\overline{v_{tg}^{2}}} = \frac{2 J_0 e \left(\dfrac{\partial i}{\partial J}\right)^2}{4 k T R_1 S^2}; \]

substituting \(J_0 = 5\ \mathrm{mA}\), \(\left(\dfrac{\partial i}{\partial J}\right)^2 = 0.2\), \(T = 300^\circ\mathrm{K}\), \(R_1 = 1\ \mathrm{M}\Omega\), \(S = 2\,\dfrac{\mathrm{mA}}{\mathrm{V}}\), we find:

\[ \frac{\overline{g v_{sa}^{2}}}{\overline{v_{tg}^{2}}} = \frac{1}{200}. \]

With a smaller wattless input resistance this ratio becomes larger, but only with a very low-resistance input

the shot effect begins to play a role comparable with the thermal one. Thus, as a result of comparing the various kinds of fluctuations with one another, we arrive at the following conclusion: in amplifiers of medium and high frequency the sensitivity threshold is determined practically by the thermal motion in the input circuit, if the watt resistance of the latter is not very small.

At very low frequencies (below 1000 hertz) one has to take into account the secular variations of emission. We saw in § 5 that the flicker effect in this range of frequencies is tens of times greater than the shot effect, especially with oxide cathodes. Consequently, its magnitude must be comparable with the thermal effect also for a high-ohmic input. To reduce the flicker effect, in addition to increasing the sharpness of tuning of the amplifier, one should make use of depression at a high space-charge density; for this, a low anode voltage and a considerable emission density are necessary.

It is very convenient to represent the resultant noise of the tube in the following form. We single out, on the one hand, the thermal effect of the input circuit, which does not directly depend on the tube, and express it by formula (29) or (29′); on the other hand, we collect all the noises that depend on the tube (the shot effect in both circuits, the flicker effect, the thermal motion in the tube), and represent them as the thermal effect of some equivalent resistance \(R_L\), placed in the input circuit.* Then the entire noise of the input and of the first stage of the amplifier will be expressed as follows:

\[ \overline{v^2}=4kT(R_1+R_L)\Delta f. \tag{36} \]

The signal-to-noise ratio is:

\[ \frac{c}{ш}=\frac{E^2 Z_g^2}{(z+Z_g)^2\,4kT(R_1+R_L)\Delta f}. \tag{37} \]

We again arrive at the conclusions which we already made when considering the thermal effect alone in the input circuit, namely, that it is desirable to increase as much as possible the reactive component of the input resistance and to decrease \(R_i\); in addition, it is necessary by all means to reduce \(R_L\). If the entire input resistance is watt resistance (\(Z_g=R_g\)), then, on the contrary, \(R_g\) must be increased. In this latter case the thermal effect of the input circuit must exceed all the other sources of noise, and it alone must determine the threshold of a sensitive amplifier.

Now we can answer the question posed at the beginning of this paragraph. It is necessary to make the input circuit as wattless as possible, to eliminate the grid current, to suppress the shot and flicker effects by the space charge, and to tune the amplifier sharply

* We neglect the thermal effect in the anode circuit, since its magnitude is small in comparison with the other effects.

** We are entitled to write such a formula because all the effects considered are proportional to \(\Delta f\).

and (if this depends on the experimenter) work at high frequencies. Then—in the ideal case—we shall have only one fundamentally unavoidable source of noise left: thermal motion inside the tube. Let us calculate its value, referred to the input circuit. First find the voltage in the anode circuit:

\[ \overline{v_{ti}^{2}}=4kT_iR_i\frac{Z_a^2}{(Z_a+R_i)^2}\Delta f, \]

divide by \(\mu^2\):

\[ g\overline{v_{ti}^{2}}=\frac{4kT_iR_i}{\mu^2}\Delta f =\frac{4kT_i}{\mu S}\Delta f =\frac{4kT_i}{G}\Delta f =\frac{\mu^2 Z_a^2}{Z_a+R_i} \tag{38} \]

\(G\)—the merit factor of the tube.

Let us put \(T_i=1100^\circ\mathrm{K},\ R_i=2\cdot10^4\ \Omega,\ \mu=20,\ \Delta f=5000\) hertz; then \(_g\overline{v_{ti}^{2}}=1.5\cdot10^{-14}\ \mathrm{V}\), or

\[ (_gv_{ti})_{\mathrm{eff}}\approx 1.2\cdot10^{-7}\ \mathrm{V}. \]

We find a value of this order of magnitude for a triode; in a two-grid screened tube both \(R_i\) and \(\mu\) are one order greater, and therefore, according to formula (38), the noise threshold should be still lower by a factor of 3–4.

Thus, the theoretical sensitivity threshold of a tube amplifier is determined by thermal motion in the internal resistance of the first tube. In the literature one may often encounter references to the shot effect as such a threshold,\(^{1,18}\) and its magnitude is sometimes calculated from the Schottky formula.\(^{38}\) As we have seen, this is incorrect, especially the latter. Schottky’s theory is applicable only in the absence of space charge; but then the tube does not act as an amplifier at all. The clearest, though not exhaustive, discussion of the question may be found in the papers of Pearson\(^{37}\) and of Johnson and Llewellyn.\(^{32}\)

In reality, of course, it is very difficult to approach the limit just calculated. First, it is impossible to eliminate completely the other kinds of fluctuations; and second, in ordinary tubes there are noises arising from other causes as well. Metcalf and Dickinson,\(^{30}\) who investigated this question in the laboratory of the General Electric Co. (USA), studied noises in the low-frequency region. They found, on a large number of specimens, that an important role is played by charges accumulating on the walls of the bulb and on the insulating supports inside the tube and then discharging at random; leakage through the tube base is also significant if its resistance is less than \(10^{10}\ \Omega\). On the basis of their work they constructed a tube in which measures were taken to eliminate these phenomena and the noises of fluctuation origin were considerably reduced. In particular, under the obvious influence of the work of Zscher and Williams (see § 46), they introduced an artificial space charge at the control grid, producing a strong depression of the shot and flicker effects and at the same time

not allowing positive ions to exert their harmful action. This tube was put on the market by the General Electric Co. under the designation PJ—11 and is characterized by the following data: \(V_a = 135\ \mathrm{V}\), \(V_g = -1.5\ \mathrm{V}\), \(i_a = 0.45\ \mathrm{mA}\), \(\mu = 30\), \(R_i = 100\,000\ \Omega\). In this case the noise in the frequency band from 3 to 250 cycles is equivalent to \(0.3\)—\(0.7\ \mu\mathrm{V}\) at the input. The other least noisy tubes of the same firm give noise from \(2.5\ \mu\mathrm{V}\) and higher.

Pearson\({}^{37}\) investigated a whole series of tubes of the Western Electric Co. The investigation was carried out by a method representing a direct realization of formula (36): the tube under study was placed in the first stage of an amplifier; first the noise at the output of the amplifier was measured with the grid connected to the cathode through a very small resistance, then resistance was gradually introduced into the grid circuit and selected of such a magnitude that the noise at the output doubled. Obviously in this case \(R_g = R_L\), and, consequently, the noise of the tube is equal to \(4kTR_g\Delta f\); the latter quantity was calculated from the known \(T_1, R_g\), and \(\Delta f\). By this method Pearson found that the smallest noise is given by the triode 102 G.

At \(i_a = 1.2\ \mathrm{mA}\), \(V_a = 130\ \mathrm{V}\), \(R_i = 45\,000\ \Omega\), \(R_a = 50\,000\ \Omega\), \(\mu = 30\), and a frequency band from 10 to 15,000 cycles, this tube exhibited a noise equal to \(\overline{v^2}/\Delta f = 0.64 \cdot 10^{-16}\ \mathrm{V^2/cycle}\). At low frequencies—from 5 to 205 cycles—this quantity increases to \(2.2 \cdot 10^{-16}\ \mathrm{V^2/cycle}\) (flicker effect), while above 1500 cycles it decreases to \(0.58 \cdot 10^{-16}\). Consequently, the noise of this tube is less than that of the PJ—11 tube, since the latter gives, per unit frequency band in the region from 3 to 250 cycles, a noise

\[ \frac{\overline{v^2}}{\Delta f} \simeq 8 \cdot 10^{-16}\ \mathrm{V^2/cycle}. \]

Noise of the same order as PJ—11 is given by the tubes 264-B (triode) and 259-B (screened, indirectly heated); in the region of higher frequencies their noise is considerably lower. Pearson calculated the theoretical minimum noise for the tubes he investigated and found that the actual noise in the 102-G tube is 3.8 times, and in the 264-B tube 2.1 times, higher than the theoretical. These tubes, therefore, are already not far from the ideal.

The tube 38 (pentode) also has a very low threshold. E. Johnson and Neffert\({}^{39}\) found that, with a band of 10,000 cycles and the proper operating conditions, it gives a noise of \(1.5\ \mu\mathrm{V}\), which corresponds to

\[ \frac{\overline{v^2}}{\Delta f}=2.25\cdot 10^{-16}\ \mathrm{V^2/cycle}; \]

its operating conditions in this case were as follows: \(V_g = -1.5\ \mathrm{V}\), \(V_{\mathrm{scr}} = 6\ \mathrm{V}\), \(V_a = 12\ \mathrm{V}(!)\), \(i_a' = 6 \cdot 10^{-5}\ \mathrm{A}(!)\), \(R_i = 1.2 \cdot 10^6\ \Omega\), and \(\mu = 160\). These authors set themselves the aim of constructing an amplifier for measuring very weak alternating EMF’s, for which purpose they used the tube 38. The circuit of one stage is shown in Fig. 27. Four such stages give, at 1000 cycles, an amplification of 400,000 times. In order to lower the noise threshold, the authors resorted to an unusually sharp tuning of the amplifier. For this purpose, in the third stage, instead of a resistance, a tuned circuit with a parallel-connected dynatron was introduced. This achieved

bandwidth of 1 hertz at \(f=200\) hertz and of 10 hertz at \(f=10\,000\) hertz.

Thanks to such sharp tuning it proved possible to measure a voltage down to \(10^{-8}\,V\) at 400 hertz and to detect voltages down to \(10^{-9}\,V\). Such extraordinary sensitivity is bought at the price of an enormous time required for establishing the apparatus; already with a frequency band \(\Delta f=5\) hertz it was necessary to use at the output an instrument with a period of 1 min. The threshold of sensitivity of cathode tubes was also studied by Hafstad,\(^{34}\) who used an FP-54 tube for recording pulses from \(\alpha\)-particles in an ionization chamber, by Mollin and Ellis,\(^{18}\) who investigated a number of tubes of the Mazda firm, etc., and by Brenschede and Fiman,\(^{35}\) who studied German tubes.

We now turn to the threshold of sensitivity of photoelectric cells. If, as usual, the photoelectric cell is closed through a resistance \(R\) and the latter is connected to the grid of an amplifier tube (Fig. 28), then noises may arise from the following causes: 1) the shot effect

Fig. 27. One stage of the amplifier (E. Johnson and K. Neitzert).

Fig. 28. Circuit for connecting a photoelectric cell.

of the photocurrent, 2) thermal motion in the resistance \(R\), 3) the shot effect of the grid current of the cathode tube. Since the space charge in the photoelectric cell has negligible density, all phenomena connected with it, including the internal thermal effect, are absent; the magnitude of the shot effect can be estimated by the formulas of § 1. Of the three effects present, the last can be reduced by decreasing the grid currents in the tube; then the principal role will be played by the shot effect of the photocurrent. Indeed, its magnitude (under the same assumptions that we made in considering the cathode tube) is:

\[ \overline{v_s^2}=2i_0 e R^2 \Delta f, \]

whereas the thermal effect is:

\[ \overline{v_t^2}=4kTR\Delta f. \]

Their ratio is

\[ \frac{\overline{v_s^2}}{\overline{v_t^2}}=\frac{i_0 e R}{2kT}=19.4\, i_0 R \]

(in practical units); since usually \(i_0 R>1\), it follows that \(\overline{v_s^2}>\overline{v_t^2}\).

The signal-to-noise ratio will be calculated taking both effects into account. Let, when the luminous flux changes, the photocurrent change by

\[ V_{\text{sig}}=\Delta i R. \]

Then

\[ \frac{c}{w}= \frac{\Delta i^{2}R^{2}}{2i_{0}eR^{2}\Delta f+4kTR\Delta f} = \frac{\Delta i^{2}R}{2\Delta f\,(i_{0}eR+2kT)} . \tag{39} \]

This formula shows that it is desirable to increase the modulation of the photocurrent and the resistance \(R\), and to decrease the pass band of the amplifier. If \(i_{0}R \gg 1\), then the noise is determined almost exclusively by the shot effect, and formula (39) reduces to

\[ \frac{c}{w}=\frac{\Delta i}{2i_{0}e\Delta f}. \tag{39′} \]

For sufficiently large \(R\), the signal-to-noise ratio no longer depends on the resistance. Let us denote the “current” sensitivity of the photocell \(\frac{\Delta i}{\Delta\Phi}\) by \(S\) (\(\Phi\) is the luminous flux). Then the smallest change in flux that can be measured with certainty is determined as follows:

\[ \Phi_{\min}= \frac{\sqrt{\overline{v^{2}}}}{RS} = \frac{\sqrt{2i_{0}e\Delta f}}{S}. \tag{40} \]

Let, for example, \(i_{0}=10^{-8}\,\mathrm{A}\), \(\Delta f=100\) hertz, \(S=10^{-4}\,\frac{\mathrm{A}}{\mathrm{lm}}\); then \(\Phi_{\min}=5.6\cdot 10^{-8}\) lumens.*

In the present review we have considered various kinds of electrical fluctuations and their role in limiting the sensitivity threshold of electrical instruments. We have already had occasion to point out, in passing, the significance of electrical fluctuations as a means of penetrating into the mechanism of electric current under various conditions. A very interesting attempt in this direction was made by Gaeuors and Bosors, \(^{41}\) who studied fluctuations of electric current in solid dielectrics with the aim of deciding in this way the dispute about ionic or electronic conductivity in them. An entirely different field of application of fluctuations was opened by Hessler’s work, \(^{40}\) which showed the role of electrical fluctuations as initial impulses causing oscillations in tube generators. Andronov, Witt, and Pontryagin \(^{42}\) considered the question of fluctuations as an apparatus permitting the study of oscillations of electrical systems. An analysis of these interesting investigations, however, lies outside the scope of our article.

* Addition in proof. Recently a method has been intensively developed for amplifying photocurrents by means of secondary emission produced by accelerated photoelectrons. In this case the strength of the secondary current must be subject to fluctuations for two reasons: a) the chaotic nature of the primary emission, b) the variability of the number of secondary electrons knocked out by one primary electron. Tenning and Kruytgof \(^{49}\), in a work that has just appeared (August 1935), come to the conclusion that amplification by this method worsens this ratio by a factor of 1.3–1.5.

Addendum 1

R. Fürth \(^{10}\) gave a very elementary derivation of formula \((14')\) for the shot effect in a tuned circuit (see Fig. 2). It is based on considering the effect caused by the passage of individual electrons. Each of them, upon reaching the anode, increases the charge of the capacitor by the amount \(e\); the latter is smoothed out by damped oscillations in the circuit. Let us assume that the electrons pass through at equal time intervals \(\tau\); this incorrect assumption will be rendered harmless below. We shall count time from the moment of arrival of the \(n\)-th electron. Let \(R\), \(L\), and \(\omega\) be the resistance, the coefficient of self-induction, and the natural frequency of the circuit, whose damping we assume to be small. We shall compute the result of the superposition of the separate oscillations caused by the electrons from the first to the \(n\)-th. The charge caused on the plates of the capacitor by the first electron will decrease according to the equation:

\[ q_1 = e \varepsilon^{-\frac{R}{2L}(t-n\tau)} \sin \omega (t-n\tau), \]

where \(\varepsilon\) is the base of natural logarithms; analogously we write for the charge caused by the second electron:

\[ q_2 = e \varepsilon^{-\frac{R}{2L}[t-(n-1)\tau]} \sin \omega [t-(n-1)\tau] \]

and so on. In reality the phases of all these oscillations are quite arbitrary with respect to one another. This means that the latter are incoherent; in summing we shall have to add the squares of the amplitudes of the individual oscillations.* At the moment of time \(t=0\), for sufficiently large \(n\), we shall have:

\[ \overline{q^2} = \frac{e}{2}\sum_{k=0}^{\infty} e^{-\frac{R}{L}k\tau} = \frac{e^2}{2\left(1-e^{-\frac{R}{L}\tau}\right)} \]

Since in practice \(\frac{R}{L}\tau\) is always small compared with unity, then

\[ \varepsilon = e^{-\frac{R}{L}\tau} \simeq 1 - \frac{R}{L}\tau; \]

further

\[ \tau = \frac{e}{i_0}, \]

therefore

\[ \overline{q^2} = \frac{e^2}{2\frac{R}{L}\tau} = \frac{eLi_0}{2R}. \]

* By this, in essence, the assumption made at the beginning about passages at equal time intervals is refuted—an assumption that violates the very essence of the conception of the shot effect (see § 4a).

The mean potential difference on the plates of the capacitor is

\[ \overline{v^2}=\frac{\overline{q^2}}{C^2}=\frac{ei_0}{2}\frac{L}{RC^2}. \tag{14′} \]

This is the whole derivation. The assumption that \(t=0\) does not constitute an essential limitation, since in the course of the derivation this instant may be chosen as the instant of arrival of any electron, while the time intervals between them are extremely small. In general, however, the derivation, despite Campbell’s defense of it,\(^{19}\) must be regarded as quite non-rigorous, internally inconsistent, and resting on assumptions that have not been investigated (for example, it has not been clarified what influence is exerted by the assumption of equal time intervals). Moreover, it gives much less than Schottky’s analysis, which establishes the spectrum of the shot effect and makes it possible to arrive at a general formula applicable to any circuit, and also to consider the influence of an amplifier. The interest of this derivation lies in its brevity, because of which, however, its imperfections should not be forgotten. Alongside it there exist several more rigorous derivations, likewise proceeding from a consideration of the influence on the circuit of each individual electron.\(^{12,16,19}\)

Addendum 2 (to the first part of the review)

In the first part of the review* Johnson’s work was described in detail; he had experimentally studied the thermal effect in various conductors. A number of subsequent works confirmed the results he had established with still greater accuracy and over wider ranges both of resistance and of frequencies. Sandeman and Bedford\(^{44}\) made measurements in the frequency band from 30 to 7000 hertz. The input circuit of the amplifier consisted of an ohmic resistance shunted by the internal capacitance of the tube and of the leads. For this case

\[ R(\omega)=\frac{R}{1+R^2C^2\omega^2}, \]

and the general formula

\[ V_T^2=\frac{2kT}{\pi}\int_0^\infty R(\omega)\,d\omega \]

gives, on integration,

\[ V_T^2=\frac{2kT}{\pi C}\left(\operatorname{arctg}RC\omega_2-\operatorname{arctg}RC\omega_1\right). \]

The ohmic resistance \(R\) was taken within the limits \(10^4\)—\(10^6\,\Omega\). The values of \(k\) obtained at room temperature fluctuated only insignificantly; under conditions in which there was no need to fear feedback in the amplifier, one obtained

\[ k=1.39\cdot 10^{-16}\ \text{erg/grad}. \]

* Uspekhi fizicheskikh nauk 13, 805, 1933.

Experiments at the temperature of liquid air gave worse results than those found by Johnson.

Ellis and Moullin45 worked at frequencies from 1700 to 3300 hertz, and each time a small frequency band was taken—from 200 (at 1700 hertz) to 400 hertz (at 3300 hertz). As with Johnson, the frequency characteristic of the amplifier was taken only within known limits; the remaining part of the area under the characteristic was determined from specially derived formulas, and an accuracy of up to \(1/2\%\) was achieved. Measurements were made with only two resistances—\(62\,000\) and \(110\,000\ \Omega\). Several series of observations were carried out; the last, containing 52 separate measurements, gave the mean value \(k = 1.361 \cdot 10^{-16}\) erg/grad. The best results were obtained by Nyquist,47 who extended the range of investigated frequencies from 1 to 10,000 hertz and the range of resistances—from \(1 \cdot 10^{3}\) to \(1.7 \cdot 10^{9}\ \Omega\). He used an amplifier with extremely sharp tuning (for its description see above in § 7, the work of E. Johnson and Nyquist). The results of measurements at various resistances and constant frequency proved to be in complete agreement with the formula \(R(\omega)=\dfrac{R}{1+R^{2}C^{2}\omega^{2}}\). Measurements at various frequencies gave the following results (see Table 7).

TABLE 7

Determination of \(k\) from the thermal effect (Nyquist).

One point clearly fell out of the series (at 7173.5 hertz). The remaining ones lead to the mean value \(1.366 \cdot 10^{-16}\) with a probable error of \(0.016 \cdot 10^{-16}\) erg/grad. This result differs from the accepted value \(1.372 \cdot 10^{-16}\) erg/grad by \(0.4\%\). Such is the accuracy with which the theorem of the equipartition of energy has at present been verified as applied to electrical fluctuations in conductors.

Electrical Fluctuations

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