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ELECTRICAL FLUCTUATIONS AND THE LIMIT OF SENSITIVITY OF ELECTRICAL INSTRUMENTS*
II. THE SHOT EFFECT AND RELATED PHENOMENA
V. L. Granovskii, Moscow
Contents
- Theory of the shot effect in the absence of space charge. — 2. Experimental investigations of the shot effect in a cathode tube in the absence of space charge. — 3. The shot effect in photocells. — 4. The influence of space charge on fluctuations of the electron current. a) Depression of the shot effect. b) Anomalous fluctuations in the presence of positive ions. c) The thermal effect. — 5. “Age-related” variations of electron emission (flicker effect). — 6. The shot effect in the current of positive ions. — 7. The threshold of sensitivity of electronic instruments.
1. Theory of the shot effect in the absence of space charge
In the first part of the present review, electrical fluctuations occurring in conductors as a consequence of thermal motion were considered. The conditions under which these phenomena occur are characterized essentially by the following: between the carriers of electricity, the atoms of the substance of the conductor, and the electrical circuit as a whole, there is a continuous mutual exchange of energy. Thanks to this, the mean energy per one degree of freedom of the electrical circuit in thermal motion is the same and is equal to \(\frac{kT}{2}\) (the entire theory of the thermal effect is constructed on the basis of the hypothesis of an equipartition of energy over degrees of freedom)**. Therefore the energy of the thermal effect at a given temperature is the same in any conductor. It is not affected even by the presence of current in the latter, since the total ordered displacement in an insignificantly
* See Uspekhi fizicheskikh nauk, XIII, no. 6, 805, 1933.
** One may think that even if the distribution of electron velocities according to Fermi statistics is taken into account, the fluctuation energy of the circuit as a whole will still be equal to \(\frac{kT}{2}\), since the whole circuit represents a “particle” with a relatively enormous mass, for which Fermi statistics must pass over into the classical one. However, an exact calculation according to the new statistics has not been carried out.
to a lesser degree changes the kinetic energy of the electrons; the velocity acquired by them under the action of the electric field in the conductor is small compared with the mean velocity of thermal motion. But here an important reservation must be made: this is so only in good conductors, where the field strength \(E\) cannot assume large values. If the field is so strong that the kinetic energy acquired by an electron during one mean free path is comparable with the mean energy of thermal motion, or even greater than it, then the theorem of the equipartition of energy becomes inapplicable. In this case, all conclusions based on it concerning the magnitude of electrical fluctuations likewise lose their force; they must be replaced by other, generally more complicated calculations, which would take into account both the thermal and the drift velocity of the electrons. We may pass to another limiting case, when collisions between elementary particles are so rare that the exchange of energy occurring in them may be altogether neglected. In this case the motion of each individual electron becomes completely independent of all the other electrons and atoms. Such a case occurs in a tube with a highly rarefied ionized gas, or in an electron tube at low current density. Under these conditions, as we shall see, fluctuations must also occur, but the laws governing them will be entirely different from those for the thermal effect. The existence of these phenomena was first pointed out by Schottky[^1] in 1918; in the same work Schottky also gave the foundations of a quantitative theory of the phenomenon he discovered, in which all its most important features were already contained. As the simplest case Schottky considered the motion of an electron stream in a vacuum; we now turn to an exposition of his theory, freed from a number of initial inaccuracies and somewhat generalized.
Fig. 1. Diagram for observing the shot effect.
Let \(A\) (Fig. 1) be an electron tube evacuated to a high vacuum; between the cathode and the anode there flows an electron current \(i_0\). This current cannot be pictured as a continuous uniform stream of electricity. On the contrary, it must be likened to a hail of individual small shot striking the anode. In such a process there can be no question of continuity and uniformity. If the emission and motion in the tube of each electron are regarded as completely independent events, then the sequence of impacts of these shot will be determined by the laws of chance. On the average, during a short interval of time \(\tau\), the cathode will be reached by
\[ \bar{n} = \frac{i_0}{e}\tau \tag{1} \]
electrons (\(e\)—the charge of the electron). However, in separate intervals of time \(\tau_1\), \(\tau_2\), etc., the number of electrons that have reached the cathode will be either greater or less than this mean. Consequently, here too the current intensity, independently of thermal motion, must be subject to fluctuations. These fluctuations Schottky called the shot effect (Schroteffekt; in English, shot-effect).
The probability that the number of electrons which have struck the anode lies within the limits from \(n\) to \(n + dn\) can be expressed by the following formula of probability theory:
\[ W(n)=\sqrt{\frac{n}{2\pi}}\,e^{-\frac{(n-\bar n)^2}{2\bar n}}\,dn . \tag{2} \]
If we introduce into consideration the quantity \(\delta=\dfrac{n-\bar n}{\bar n}\), which shows the relative deviation from the mean, then formula (2) may be rewritten as follows:
\[ W(\delta)\,d\delta=\sqrt{\frac{\bar n}{2\pi}}\,e^{-\frac{\bar n\delta^2}{2}}\,d\delta . \tag{2′} \]
This formula, expressing the probability of a given deviation \(\delta\), is very close to formula (3″) of the first part of the present survey. Using these formulas, we can calculate the mean magnitude of the fluctuations of the current intensity in the tube. The deviation of the current intensity \(i\) from the mean value \(i_0\) may be written as:
\[ j=i-i_0=\frac{ne}{\tau}-\frac{\bar n e}{\tau}=(n-\bar n)\frac{e}{\tau}, \]
and its mean square:
\[ \overline{j^2}=\overline{(i-i_0)^2}=\overline{(n-\bar n)^2}\frac{e^2}{\tau^2} =\bar{\delta^2}\,\bar n^{\,2}\frac{e^2}{\tau^2}. \tag{3} \]
We calculate \(\overline{\delta^2}\) from formula (2′):
\[ \overline{\delta^2}=\int_{-\infty}^{+\infty}\delta^2 W(\delta)\,d\delta=\frac{1}{\bar n}. \tag{4} \]
Substituting this value into (3), we obtain:
\[ \overline{j^2}=\frac{1}{\bar n}\,\bar n^{\,2}\frac{e^2}{\tau^2} =\frac{\bar n e^2}{\tau^2} =\frac{i_0 e}{\tau}. \tag{3′} \]
The quantity \(j\) may be regarded as a certain very irregularly varying current that is superposed on the constant current \(i_0\). Its effective value is:
\[ j_{\mathrm{eff}}=\sqrt{\overline{j^2}}=\sqrt{\frac{i_0 e}{\tau}}. \tag{3″} \]
This formula contains all the basic features of the shot effect. We see that its magnitude depends on factors entirely different from those determining the thermal effect. The shot effect pro—
proportional to the square root of the electron current intensity; at very small currents it falls to zero. We shall see below that, at large current intensities, when a considerable space charge is formed between the electrodes, the shot effect again decreases, but this phenomenon lies outside the scope of the elementary theory now under consideration. Further, the shot effect proves to depend on the magnitude of the elementary charge; if electricity were a continuous fluid, this effect would not exist. Finally, it is inversely proportional to the square root of those intervals of time over the course of which the current intensity is determined each time.
Despite its great fundamental significance, formula \((3')\) can hardly be verified directly. To obtain measurable values of \(j_{\mathrm{eff}}\), one must take \(\tau\) so small that during it it is impossible to measure the current intensity by ordinary methods. However, one can indicate an experimental arrangement that permits observation and measurement of the magnitude of the shot effect. It is necessary to connect, in series with a tube, an electrical resistance of some kind and observe the potential difference at its ends caused by the current passing through the tube. This voltage will have a variable component arising from the shot effect; if the latter quantity is too small for direct measurement, it may be amplified by means of a multi-tube amplifier. It was precisely in this way that the shot effect was studied experimentally; the differences in the methods of the various authors consisted only in the choice of the resistance inserted in the anode circuit, in the amplifier, in the instrument measuring the current at the amplifier output, and in the methods of calibrating the apparatus.
Let us begin with consideration of the most general case. Suppose that the anode load of the tube is some resistance \(Z\) (Fig. 1). This quantity, generally speaking, is complex and depends on the frequency \(f\). To determine the voltage fluctuations in it, one must resolve the current fluctuations \(j\) into a spectrum, determine the voltage at the terminals of the resistance \(Z\) caused by each of the current components \(j\), and sum all these partial voltages.
In fact, the shot effect is a very rapidly and irregularly varying current, having no definite frequency. We may, however, represent it, on the basis of Fourier’s theorem, in the form of an infinite sum of sinusoidal currents of different frequencies and different phases:
\[ j=\sum_{k=1}^{\infty} j_k=\sum_{k=0}^{\infty} A_k \cos \omega_k t+ \]
\[ +\sum_{k=0}^{\infty} B_k \sin \omega_k t=\sum_{k=0}^{\infty} C_k \sin \left(\frac{2\pi k}{T}t+\varphi_k\right) \tag{5} \]
where \(T\) is some large (in comparison with the proper perio-
...of the circuit) time interval, and the coefficients \(A_k\), \(B_k\), and \(C_k\) are related to one another by the relation:
\[ A_k^2+B_k^2=C_k^2. \tag{5'} \]
These currents will produce in the circuit alternating voltages of the corresponding frequencies. To each partial current \(f_k\) there will correspond its own partial voltage \(v_k\), defined by the equality:
\[ v_k=j_k Z_k, \tag{6} \]
where \(Z_k\) is the value of the resistance \(Z\) at the frequency \(f_k=\dfrac{k}{T}\)—a quantity, generally speaking, complex and different for different frequencies. The square of the effective voltage of a given frequency is:
\[ \overline{v_k^2} = Z_k^2 \overline{j_k^2} = Z_k^2 C_k^2 \overline{\sin^2(\omega_k t+\varphi)} = \frac{1}{2} Z_k^2 C_k^2 = \]
\[ = \frac{1}{2} Z_k^2\left(A_k^2+B_k^2\right) \tag{7} \]
The total voltage at the terminals of the resistance \(Z\) will be equal to the sum of all \(v_k\). Since these separate voltages are statistically independent, we can add only their squares, and for the voltage fluctuations in the circuit \(v_s\) we find the expression:
\[ \overline{v_s^2} = \sum \overline{v_k^2} = \frac{1}{2}\sum Z_k^2 C_k^2. \tag{8} \]
We see that, in order to calculate \(\overline{v_s^2}\), it is necessary to know the quantities \(A_k^2\) and \(B_k^2\), i.e., to determine the intensity of the individual terms of the expansion of the shot effect in a Fourier series. This analysis, carried out by Schottky, constitutes one of the most important elements of the whole theory. In it the basic features of the phenomenon itself appear very clearly. First of all it must be noted that exact knowledge of the individual coefficients \(A_k\) and \(B_k\) is impossible. Owing to the completely irregular, random course in time of the function \(j(t)\), there are no data sufficient for calculating these coefficients; one can only note that their magnitude may depend to a large degree on the choice of the time interval for which the expansion is made, and may also vary strongly and irregularly from one coefficient to another. However, if we take a series of such coefficients, for example from \(A_k\) to \(A_{k+\Delta k}\), where \(\Delta k\) is sufficiently large in absolute value but small in comparison with \(k\), then the mean value of the square of these coefficients will prove to be quite well defined. For this it is necessary to take in the series (5) only terms with sufficiently high values of \(k\), so that \(k \gg \Delta k\) and \(\Delta k \gg 1\). This is always feasible; it is only necessary to choose the fundamental period of the expansion \(T\) sufficiently large. Then, in the expression (8) for \(\overline{v_s^2}\), in all cases actually encountered, the terms corresponding to small values of \(k\) will play no role for one of the following two reasons: either 1) the magnitude \(Z_k\) for these terms is vanishingly small, for example if the load \(Z\) is a resonant circuit, or 2) these terms...
will not be further amplified if a resonant amplifier is used for observation; one of these two circumstances is always present in experiments. The expression for the coefficient \(A_k\), according to Fourier’s theorem, is:
\[ A_k=\frac{2}{T}\int_{0}^{T} j\cos\omega_k t\,dt \tag{9} \]
and for \(B_k\):
\[ B_k=\frac{2}{T}\int_{0}^{T} j\sin\omega_k t\,dt, \tag{9'} \]
for their squares:
\[ A_k^2=\frac{4}{T^2}\left\{\int_{0}^{T} j\sin\omega_k t\,dt\right\}^{2} =\frac{4}{T^2}\int_{0}^{T}\int_{0}^{T} jj'\cos\omega_k t\cos\omega_k t'\,dt\,dt' * \tag{10} \]
and correspondingly for \(B_k\).
Let us note that the product \(j(t)\cdot j(t')\) is certainly a positive quantity only for very close moments of time \(t\) and \(t'\). For more remote \(t\) and \(t'\) this product may have either sign.**
We shall use this circumstance in calculating \(\overline{A_k^2}\) and \(\overline{B_k^2}\) as follows. Divide the period \(T\) into intervals \(\Delta t\), so small that during each of them \(j(t)\), \(\cos\omega_k t\), and \(\sin\omega_k t\) may be regarded as constant, and at the same time sufficiently large
* The transformation of the square of a definite integral into a double integral may be explained as follows: the square of a definite integral is the product of some sum by a second similar sum; in this case each term of the first sum must be multiplied by each term of the second, and the products added; this sequence of operations is represented by the double integral of the product of two identical functions of different variables, with both integrations taken over identical limits.
** The product \(\overline{j(t)j(t')}\), averaged over a large number of separate moments of time, shows how strongly the value \(j(t)\), which occurred at some moment of time \(t\), affects subsequent values of \(j\). This quantity is a function of the time interval \(t-t'\), \(\overline{j(t)\cdot j(t')}=\Phi(t-t')\), and it is natural to regard this function as symmetric, having a maximum at \(t-t'=0\), and decreasing monotonically on both sides from the maximum. It is usually called the correlation function. The faster the influence of a given current value \(j\) on subsequent values disappears, the faster the function \(\Phi\) decreases on both sides from zero, and the smaller the width of the correlation band. In the case we are considering, the shot effect without space charge, the motions of the electrons occur quite independently of the preceding ones; therefore the width of the correlation band is extremely small. By this we shall express the fact that the term \(M_k\) in the form (10″), upon averaging, must vanish. Using the theory of correlation, the subsequent calculations could have been somewhat shortened; however, in order to present more clearly the physical premises of the derivation, we give it in a simpler and more expanded form. We shall encounter the meaning of correlation again below in § 4.
for a significant number of electrons to have passed to the anode of the tube during each of them (otherwise the concept of \(j\) will lose its meaning, and it will be impossible to use the formulas of statistics). This can be done in all practically encountered cases. Indeed, if we take even a very high natural frequency of the circuit \(\omega_0\), for example \(\omega_0=1\cdot10^{10}\ \mathrm{sec.}\), corresponding to a wavelength of \(20\ \mathrm{cm}\), and an electron current in the tube of only \(1\ \mathrm{mA}\), then even then the mean time interval between the passages of two electrons, equal to \(1.6\cdot10^{-16}\ \mathrm{sec.}\), will be so much smaller than the period of any term of the series (5) having a real value (here the remark concerning terms corresponding to very low frequencies may be repeated) that the condition of constancy of \(\cos\omega_k t\) and \(\sin\omega_k t\) during the interval \(\Delta t\) can easily be satisfied. We can then replace the double integral (10) by a double sum:
\[ A_k^2=\frac{4}{T^2}\sum_0^T\sum_0^T j\Delta t\cdot j'\Delta t'\cos\omega_k t\cos\omega_k t'. \tag{10′} \]
We split this sum into two parts: collecting, on the one hand, the terms for which \(t=t'\), and, on the other hand, all the remaining ones \((t\ne t')\),
\[ A_k^2=\frac{4}{T^2}\left[\sum_0^T j^2(\Delta t)^2\cos^2\omega_k t+M_k\right], \tag{10″} \]
where \(M_k\) denotes the sum of all terms in which \(t\ne t'\).
Since \(T\gg T_k\) (\(T_k\) is the period of the term of the Fourier series under consideration), each value of \(\cos\) will be repeated many times under the sign of the first sum. Therefore the summation may be carried out in the following sequence: first add all terms corresponding to one value of \(\cos\), and then add all these sums corresponding to different \(\cos\). Each of these sums is equal to the product of \(\cos^2\omega_k t\), the number of such terms \(\left(\dfrac{T}{\Delta t}\right)\), and the mean value \(\overline{j^2}(\Delta t)^2\). The latter may be calculated by formula (3), where \(\Delta t\) must be substituted for \(\tau\):
\[ \overline{j^2}\Delta t^2=\frac{i_0 e}{\Delta t}\cdot\Delta t^2=i_0 e\Delta t. \]
This value of \(\overline{j^2}\Delta t^2\) proves to be the same for all terms of the first sum in expression (10″); therefore this sum may be rewritten as
\[ \sum_0^T j^2(\Delta t)^2\cos^2\omega_k t = i_0e\sum_0^T \cos^2\omega_k t\Delta t = i_0e\cdot\frac{T}{2}, \tag{11} \]
since the mean value of \(\cos\omega_k t\) over a period is equal to \(1/2\).
As for the second sum, which we denoted by \(A\), it cannot be calculated exactly, but the order of magnitude of its probable values can be estimated
of values can be, using the known statistical theory of the sum of \(p\) oscillations of equal amplitude \(a\) with random phases; if \(p\) is large, then the order of magnitude of the probable values of this sum will be \(a\sqrt{p}\). The number of terms in the first sum we have computed was \(\left(\dfrac{T}{\Delta t}\right)\), the number of terms in the second is \(\left(\dfrac{T}{\Delta t}\right)^2-\dfrac{T}{\Delta t}\). The order of magnitude of the terms of both sums is the same; if for it we retain for the time being the notation \(a\), then for the first sum we obtain \(\dfrac{T}{\Delta t}a\), for the second:
\[
\sqrt{\left(\frac{T}{\Delta t}\right)^2-\frac{T}{\Delta t}}\cdot a \cong \frac{T}{\Delta t}a.
\]
Consequently, in formula (10″) both terms have the same order of magnitude. Since the first term is positive, while the second may be either \(>0\) or \(<0\), the quantity \(A_k^2\) may have very different values, both large and small. With an increase of the fundamental period \(T\) one cannot indicate any definite limit toward which the value of the individual coefficient \(A_k\) would tend.
Let us now form the sum of the squares of the coefficients \(A\) from \(k\) to \(k+\Delta k\):
\[
\sum_{k}^{k+\Delta k} A_k^2
=
\frac{4}{T^2}\sum_{k}^{k+\Delta k}\left[i_0 e\frac{T}{2}+M_k\right]
=
\frac{2i_0\Delta k}{T}
+
\frac{4}{T^2}\sum_{k}^{k+\Delta k} M_k.
\]
The last sum, consisting of \(\Delta k\) terms of arbitrary magnitude and sign, by the theorem already mentioned, will be a quantity of the order
\[
\frac{4\sqrt{\Delta k}}{T^2}\,|M_k|,
\]
whereas the first sum has magnitude of the order
\[
\frac{4}{T^2}\Delta k\,|M_k|.
\]
If \(\Delta k\) is sufficiently large, then the second sum may be neglected. Finally we find:
\[
\sum_{k}^{k+\Delta k} A_k^2=\frac{2i_0 e\Delta k}{T}.
\tag{12}
\]
The mean value of \(A_k^2\) is determined from this:
\[
\overline{A_k^2}
=
\frac{\displaystyle\sum_{k}^{k+\Delta k} A_k^2}{\Delta k}
=
\frac{2i_0 e}{T}.
\tag{12′}
\]
In an entirely analogous way we shall find that
\[
\overline{B_k^2}=\frac{2i_0 e}{T},
\]
the same value as for \(\overline{A_k^2}\).
Hence, finally, by formula (5):
\[
\overline{C_k^2}=\overline{A_k^2}+\overline{B_k^2}=\frac{4i_0 e}{T}.
\tag{12″}
\]
These very important formulas show that \(\overline{C_k^{\,2}}, \overline{A_k^{\,2}}\), and \(\overline{B_k^{\,2}}\) are the same for all \(k\). In other words, in the spectrum of the shot effect all frequencies within the limits indicated above are represented with the same intensity.
Regarding the spectrum of the shot effect, some authors (for example Thornton Fry \(^{12}\)) have expressed themselves in the sense of denying its very existence. Such a judgment is based on the indeterminacy of the values of the individual coefficients \(A_k\), \(B_k\), and \(C_k\) and on the absence of definite limits for them. However, as we have seen, \(\overline{C_k^{\,2}}\) has a perfectly definite value, and this is precisely the quantity that we determine experimentally. However selective the apparatus with which the measurement is made may be, it always receives not one frequency but a certain interval of frequencies, and by means of it one can determine only the mean value of the spectral coefficient \(\overline{C_k^{\,2}}\) in this interval. This applies not only to the spectrum of an electric current but also to any other spectrum. Rayleigh, studying the structure of white light, established that here too one can determine the energy only for some frequency interval \(d\omega\). The spectrum of the shot effect has the same reality as the spectrum of white light.
Let us return to the calculation of \(\overline{v_s^{\,2}}\). Substituting \((12'')\) into (8), we find
\[ \overline{v_s^{\,2}}=\frac{2i_0e}{T}\sum |Z_k|^2. \]
To pass to integration, let us introduce, instead of the argument \(k\), the frequency \(f\):
\[ f=\frac{k}{T}; \quad \text{then } \Delta f=\frac{\Delta k}{T}. \]
We may write (putting \(\Delta k=1\))
\[ \overline{v_s^{\,2}}=\frac{2i_0e}{T}\sum_k |Z_k|^2 \Delta k =2i_0e\sum_f |Z_f|^2 \Delta f \]
and, replacing the summation sign by an integral:
\[ \overline{v_s^{\,2}}=2i_0e\int_0^\infty |Z(f)|^2\,df. \tag{13} \]
Formula (13) is the general expression for the voltage fluctuations caused by the shot effect for an arbitrary load in the tube circuit. We see that here too the features of the phenomenon that we have already noted in formula \((3'')\) are preserved: the magnitude of the effect is proportional to the mean strength of the anode current and to the magnitude of the elementary charge.
Let us now turn to the consideration of individual special cases. Let the anode load be a circuit consisting of a resistance coil \(R\) and a self-inductance \(L\) and of a capacitance \(C\) connected in parallel with it (Fig. 2). Concerning the relative magnitude of
of these parameters ($R$, $L$, and $C$) we as yet make no assumptions, so that the circuit may be either periodic or aperiodic. For such a circuit $Z_k$ is expressed by the formula:
\[ Z_k^2=\frac{R^2+\omega_k^2L^2}{\omega_k^2R^2C^2+(1-\omega_k^2CL)^2}, \tag{a} \]
where $\omega_k=2\pi f_k$.
Let us introduce the notation:
\[ \omega_0=\frac{1}{\sqrt{LC}},\quad x=\frac{\omega_k}{\omega_0}\quad \text{and}\quad r=\frac{R}{L\omega_0}. \]
Then formula (a) can be transformed:
\[ Z_k^2= \frac{\dfrac{R^2}{L^2\omega_0^2}+\dfrac{\omega_k^2}{\omega_0^2}} {C^2\omega_0^2\left[\dfrac{\omega_k^2}{\omega_0^2}\dfrac{R^2}{L^2\omega_0^2}+ \left(\dfrac{1}{LC\omega_0^2}-\dfrac{\omega_k^2}{\omega_0^2}\right)^2\right]} = \frac{r^2+x^2}{C^2\omega_0^2\left[x^2r^2+(1+x^2)^2\right]}. \tag{b} \]
Fig. 2.
Substitute (b) into formula (13), in which we transform the integral from the variable $f_k$ to the variable $x$ ($x=\frac{2\pi}{\omega_0}f$); we find:
\[ \overline{v_s^2}= \frac{2i_0e}{2\pi C^2\omega_0^2} \int_0^\infty \frac{r^2+x^2}{x^2r^2+(1-x^2)^2}\,dx. \]
The integral in formula (13′) can be evaluated as follows: first note that the integrals
\[ J_1=\int_0^\infty \frac{dx}{(1+x^2)^2+x^2r^2} \quad \text{and} \quad J_2=\int_0^\infty \frac{x^2dx}{(1-x^2)^2+x^2r^2} \]
have one and the same value, since the second can be obtained from the first by substituting into it $x=\frac{1}{z}$ and then replacing, after the transformation, the letter $z$ by $x$. Therefore each of them is equal to half the integral:
* This expression is most simply obtained by using the complex representation of resistances under alternating current. The impedance of one branch is
\[ \overline{Z}_L=R+j\omega_kL; \]
the impedance of the other is
\[ \overline{Z}_C=\frac{1}{j\omega_kC}; \]
for a parallel connection
\[ \frac{1}{\overline{Z}}= \frac{1}{\overline{Z}_L}+\frac{1}{\overline{Z}_C} = \frac{1}{R+j\omega_kL}+j\omega_kC = \frac{1-\omega_k^2LC+j\omega_kRC}{R+j\omega_kL}, \]
or
\[ \overline{Z}= \frac{R+j\omega_kL}{(1-\omega_k^2LC)+j\omega_kRC}, \]
whence
\[ Z^2=\left|\overline{Z}\right|^2= \frac{R^2+\omega_k^2L^2}{(1-\omega_k^2CL)^2+\omega_k^2R^2C^2}. \]
\[ J_3=\int_0^\infty \frac{(1+x^2)\,dx}{(1-x^2)^2+x^2r^2}, \]
which represents their sum. Introducing into it the substitution \(x-\dfrac{1}{x}=ry\), we find:
\[ J_3=\int_{-\infty}^{\infty}\frac{dy}{y^2+1}=\frac{\pi}{r}. \]
Consequently:
\[ J_1=J_2=\frac{\pi}{2r}. \]
Now the integral in formula (13) can easily be calculated:
\[ \int_0^\infty \frac{r^2+x^2}{x^2r^2+(1-x^2)^2}\,dx = r^2J_1+J_2=\frac{\pi}{2r}(r^2+1), \]
and for \(\overline{v_s^2}\) we find:
\[ \overline{v_s^2} =\frac{2i_0e}{C^2\omega_0\,2\pi}\cdot\frac{\pi}{2r}(r^2+1) =\frac{i_0eL}{2C^2R}(r^2+1). \tag{14} \]
Formula (14), more general than Schottky’s original result,* was first obtained (by an entirely different method) by Thornton Fry\(^{12}\). One may examine its separate special cases. If the damping of the circuit is small, then \(r\ll 1\), and
\[ \overline{v_s^2} =\frac{i_0eL}{2C^2R} =\frac{i_0eR}{2L}\,Z_{\mathrm{res}}^2 =\frac{i_0e}{\theta}\,Z_{\mathrm{res}}^2, \tag{14′} \]
where \(\theta\) is the settling time of the circuit. Formula (14′), found by Schottky, was later derived by Führth\(^{10}\) in another, considerably simpler way (see Appendix I). When the damping is reduced to very small values, \(\overline{v_s^2}\), as we see, increases very strongly. Expression (14′) permits the following formulation: the shot effect produces in a tuned circuit an alternating voltage of the same magnitude as a sinusoidal current of frequency \(\omega_0\) and effective strength \(i_e=\sqrt{\dfrac{i_0e}{\theta}}\).
Another case is an aperiodic circuit containing only \(R\) and \(C\) (\(L=0\)). From formula (14) we find for this case (\(r\gg 2\)):
\[ \overline{v_s^2}=\frac{i_eR}{2C}. \tag{15} \]
* In the 1918 paper, where the concept of the shot effect was first set forth and the most essential elements of its theory were given, Schottky made two mathematical errors: 1) the integral denoted by us as \(J_2\) was calculated incorrectly; 2) for the quantity \(C_k^2\) a value was found which in fact is obtained for \(A_k^2\) and \(B_k^2\). One of these errors was discovered and corrected by Johnson\(^{8}\), the other by Schottky himself\(^{3}\). Schottky’s 1922 paper already contains the correct results.
This case actually occurs when “purely ohmic” resistance is connected into the anode circuit, since one must always take into account the capacitance of the leads parallel to it, the internal capacitance of the tube, the input capacitance of the subsequent amplifier, etc. Note, however, that if the capacitance \(C\) tends to zero, then according to formula (15) \(\overline{v_s^2}\to\infty\).
The latter result can also be obtained directly from formula (13), since for \(L=C=0\) we have \(Z(f)=R\)—a quantity constant for all frequencies; then
\[ \overline{v_s^2}=2i_0 e R^2 \int_0^\infty df=\infty . \]
This result is, of course, incorrect. The reason is the circumstance that formula (13) and the subsequent ones are inapplicable in the present case, when very low and very high frequencies play the same role as the intermediate frequencies; for these limiting frequencies, as we have already noted above, formula (12′), expressing the distribution of the shot effect over the spectrum, is incorrect. The case of “purely ohmic” resistance has been studied experimentally in a number of works; however, as we shall see below, in doing so some limited frequency interval was always taken.
In order to obtain formulas that can be checked by direct experimental data, it is necessary to take one further step. The voltage fluctuations in the anode circuit are too small to be measured directly with a pointer instrument, or even a mirror instrument, for alternating current. Indeed, let us take, for example, the case of a tuned circuit; let us put in formula (14′): \(i_0=4\ \mathrm{mA}\), \(Z_{\mathrm{res}}=100\,000\ \Omega\), \(\dfrac{1}{\Theta}=10^3\ \mathrm{sec.}\); we find: \(\overline{v_s^2}=4\cdot 10^{-3}\cdot 1.6\cdot 10^{-19}\cdot 10^3\cdot 10^{10}=6.4\cdot 10^{-9}\), or \(v_s=8\cdot 10^{-5}\ \mathrm{V}\). An alternating voltage of such magnitude is difficult to measure directly, but it is quite possible to amplify it with a cathode amplifier to a magnitude allowing measurement by various methods. In doing so one must take into account that the amplification coefficient \(\mu\), generally speaking, is a function of frequency and, consequently, will be different for different \(v_{ks}\). A partial voltage \(v_s(f)\) in the anode circuit will correspond, at the output of the amplifier, to the voltage:
\[ V_s(f)=\mu(f)v_s(f). \]
The total alternating voltage at the amplifier output is determined as follows:
\[ \overline{V^2}=\int_0^\infty \mu^2(f)v_s^2(f)\,df =2i_0e\int_0^\infty \mu^2(f)Z^2(f)\,df . \tag{16} \]
Formula (16) is the basis for the experimental verification of the theory of the shot effect. Two special cases deserve attention here. If the amplifier possesses a sharply resonant frequency-
characteristic, so that \(\mu(f)\) has appreciable values only in a narrow frequency interval near the frequency \(f_0\), for which \(\mu(f_0)\) is maximal, then in this interval \(Z(f)\) may be regarded as constant and one may put
\[ Z(f)=Z(f_0). \]
Then formula (16) will take the form:
\[ \overline{v_e^{\,2}}=2i_0 e Z^2(f_0)\int_0^\infty \mu^2(f)\,df. \tag{16'} \]
In this expression \(\int_0^\infty \mu^2(f)\,df\) represents the area of the frequency characteristic of the amplifier plotted on a quadratic scale. Its value is necessary for determining the shot effect. We obtain an analogous formula also in another particular case, when the anode load \(Z\) represents a “purely ohmic” resistance \(R\). Then
\[ \overline{v_e^{\,2}}=2i_0 e R^2\int_0^\infty \mu^2(f)\,df \tag{16''} \]
for any form of the frequency characteristic of the amplifier.
We have set forth here the theory of the shot effect following Schottky’s method. A whole series of attempts was made to construct the theory of this phenomenon by other methods. Fort’s work \(^{10}\) has already been mentioned; in it a special case of a tuned circuit is considered at once, but in such a brief and elementary (though not rigorous) form that for many purposes (for example, pedagogical ones) it is preferable to Schottky’s long and exhaustive analysis; this derivation is given in Appendix 1. Ornstein and Burger \(^{11}\) gave a theory of the phenomenon in a very compressed form, leading to the final result in the form of equation (14) or (14′); at the same time the question of the spectrum of the shot effect, which is of enormous importance for experiment, is completely bypassed. This theory is based on consideration, by statistical methods, of the differential equation of a circuit connected into the tube circuit; the approach somewhat resembles Ornstein’s treatment of problems of Brownian motion or of fluctuations of thermal origin. T. Fry \(^{12}\) constructed a theory of the shot effect by considering the power delivered by the tube current to any load connected in series with it. Fry refuses to consider the spectrum of the phenomenon and tries to prove its nonexistence. Great attention is devoted to clarifying those physical hypotheses which are laid at the foundation of the theory of this phenomenon.
Finally, S. Ballantine \(^{16}\) derived the formulas for the shot effect from the Rayleigh–Schuster theorem, and in this case a generalization of the theory is obtained that is also valid for very high frequencies.
2. Experimental investigations of the shot effect in a cathode tube in the absence of space charge
The first experimental study of the shot effect was undertaken by Hartmann5, 6, 7 in the laboratory of the Siemens concern, with Schottky’s direct participation. To obtain the shot effect in as pure a form as possible, a special tube (diode) was made with a tungsten cathode, careful insulation of both electrodes, and an additional branch with charcoal, which could be cooled with liquid air during each series of experiments and thereby ensured a good vacuum in the tube. As the anode load an oscillatory circuit was used, attention being paid to obtaining the smallest possible damping. The condenser was mica, with capacitance from \(10^{-6}\) to \(10^{-8}\) F; for precise adjustment an air condenser of capacitance \(2\cdot 10^{-9}\) F was connected in parallel. The coil had an iron core, \(L=0.16\text{—}0.121\) henry (depending on the frequency). The resistance of the circuit is composed not only of the resistance of the coil and the leakage of the condenser, but also of the parallel-connected internal resistance of the tube \(R_i\) and the input resistance of the amplifier. In order to obtain as high an \(R_i\) as possible, the work was carried out at sufficiently high anode voltages, already falling in the region of saturation current; here \(R_i\) varied within the limits from \(23.8\cdot 10^5\) to \(1.2\cdot 10^5\ \Omega\), depending on \(i_0\), which was varied from 0.88 to 20 mA. As for the damping action of the amplifier, in order to reduce it the amplifier input was made up of a capacitance and a resistance; the most advantageous values proved to be \(R_b=0.5\ \mathrm{M}\Omega\) and \(C_b=5\cdot 10^{-9}\) F. As a result, for the watt resistance of the circuit values from 2.4 to 13.9 \(\Omega\) were obtained, depending on the frequency; for \(Z_0\), respectively, from \(1.26\cdot 10^4\) to \(2.37\cdot 10^{-5}\ \Omega\).
From a comparison of the data for \(Z_0\) and \(R_i\) it is clear that the internal resistance of the tube cannot in this case be considered infinitely large. In view of the absence at that time (1921) of a rigorous theory for this case, Hartmann took \(R_i\) into account in the following way. He writes the strength of the sinusoidal current equivalent to the shot effect not as \(j_e=\sqrt{\dfrac{i_0 e}{\theta}}\), but
\[ j'_e=\sqrt{\frac{i_0 e}{\theta}}\frac{R_i}{R_i+Z'_0}, \tag{17} \]
where \(Z'_0\) is the impedance of the circuit with the parallel-connected input of the amplifier. Hartmann assumes the effective value of the voltage on the circuit to be equal to:
\[ v_s=j'_e Z'_0=j'_e\frac{R_i Z_0}{R_i+Z'_0}. \tag{17′} \]
It is easy to see that such a method of allowing for \(R_i\) contains an error, since it is assumed that \(R_i\) proves to be the same for all frequencies, which in reality is not the case (see below, § 4a). Most strongly \(R_i\)
is manifested at frequencies close to the resonant one, and more weakly at more distant ones. Neither Hartmann nor the other physicists who criticized his work analyzed the magnitude of the error introduced in this way. Concerning the construction of the amplifier, apart from the fact that it was four-stage, Hartmann gives no details. At the output of the amplifier a telephone was connected, which served for an aural comparison of the magnitude of the shot effect with the magnitude of a certain sinusoidal voltage. Hartmann adopted the following experimental procedure. First the amplifier was connected to the circuit in which the shot effect was to be measured, and the latter was observed by means of the telephone. Then, instead of the circuit under test, a generator of sinusoidal current, tuned to the same frequency as the circuit, was connected to the amplifier through a potentiometer.
Fig. 3. Diagram of Hartmann’s apparatus for measuring the shot effect.
By adjusting with the potentiometer the voltage \(v_1\) applied to the input of the amplifier, the experimenter obtained in the telephone a sound of the same loudness as that produced by the shot effect. Under this condition Hartmann considered that the effective value of the voltage from the generator was equal to the effective voltage of the shot effect:
\[ v_s = v_1. \]
Hartmann did not take into account the possibility of different amplification at different frequencies; on the basis of special experiments carried out by him, he considered that the amplification of all frequencies of significance in his apparatus was the same. In reality this, of course, is impossible, and ignorance of the exact frequency characteristic leads only to ignorance of the error in determining the effect being measured.
The general scheme of the experiment is shown in Fig. 3. The diode \(R\), the filament ammeter \(A\), the rheostat, the milliammeter \(a\), the coil \(L\), and the capacitor \(C\) are enclosed in one metallic grounded box \(K\). From the circuit the voltage is fed by a cable, enclosed in a grounded sheath, to the switch \(U\), which by the same shielded cable is connected with the amplifier \(V_k\), placed
in another iron box \(K_3\). The sinusoidal-current generator \(E\)—Hartmann reports nothing about its construction—is connected through the transformer \(T_r\) to the first potentiometer (resistance box) \(W\); in series with them is included the measuring thermoelement \(Th\), connected to a direct-current galvanometer. The voltage from the resistance \(r_1\) is applied to the second box \(r_2\); from part of the latter (\(r_3\)) wires go to the switch \(U\). The resistances are chosen so that \(r_2 \gg r_1\) and \(r_2 \gg r_3\); then, if the current measured by the thermoelement \(Th\) is equal to \(i\), the voltage \(v_1\) supplied to the amplifier will be equal to:
\[ v_1 = i r_1 \frac{r_3}{r_2}. \]
To avoid the possibility of feedback between the amplifier and the generator, the latter also, together with all its apparatus and almost all the batteries, is enclosed in the iron box \(K_2\); the generator is moved 9 m away from the other boxes.
Fig. 4. Results of Hartmann’s experiments
Hartmann’s experiments showed, first of all, that after eliminating all mechanical and electrical disturbances and the noise from one tube of the amplifier (the latter was achieved simply by selecting the tube), only the noise produced by the diode and by the circuit connected with it remained in the telephone. In this noise a tone corresponding to the natural frequency of the circuit was quite noticeable; however, its intensity fluctuated all the time. It was therefore not easy to compare it exactly in intensity with the generator current; Hartmann estimates the observational errors at 10–20% at lower frequencies and at 30% at high ones. Having measured \(v_1\) for various frequencies from \(\omega_0 = 1500\ \mathrm{sec}^{-1}\) to \(\omega_0 = 15\,000\ \mathrm{sec}^{-1}\) and equating it to \(v_s\), Hartmann could, by formula (17), determine \(j'_e\), and from it in turn the value of the elementary charge \(e\). The correct value of \(e\) is as much a touchstone for the theory of the shot effect as the correct value of the Boltzmann constant \(k\) is for the theory of the thermal effect. The results obtained by Hartmann* at different frequencies and two
* In Hartmann’s original 1931 paper the measurement results were treated according to an incorrect formula of Schottky (see the note on p. 445), as a result of which values for \(e\) were obtained a thousand times smaller than those found by Millikan. Hartmann and Schottky concluded from this that the elementary theory of the shot effect was inapplicable to the actual phenomena occurring in a cathode tube, and even attempted to give an explanation of such a large discrepancy based on an idea of the mutual influence of individual acts of electron emission.
at different emission currents in the diode \((i_0 = 2\text{ mA}\) and \(i_0 = 20\text{ mA})\), are presented graphically in Fig. 4*. From this graph it is seen that the observed values of \(e'\) fluctuate about the correct mean value. At the same time, a definitely pronounced course is obtained for the dependence of \(e\) on the natural frequency of the circuit \(\omega_0\); moreover, it turns out that the electron charge depends somewhat on the emission current. Thus it may be established that Hartmann’s experiments in any case prove the existence of the shot effect. But from the quantitative point of view the agreement with theory is unsatisfactory. The reason must be sought either in the incompleteness of the theory or in some errors in the experimental method.
We have already indicated above that one source of errors may lie in formula (17); unfortunately, no analysis of the magnitude of this error was made. R. Fürth\(^{10}\) revealed another and, apparently, very important source of errors, lying in one of the links of the experimental setup, namely, in the receiving apparatus—the telephone and the observer’s ear. From the theory of the phenomenon it is seen that the quantity \(\overline{v_s^{\,2}}\) is subject to measurement; it is made up of the squares of the voltages of partial oscillations of different frequencies [see formula (8)]. For this it is necessary that the measuring apparatus give readings proportional to the square of the voltage and at the same time independent of frequency; then it will indeed add all the partial \(\overline{v_k^{\,2}}\). Neither condition is fulfilled in the present case. The loudness of sounds perceived by the ear, in the first approximation according to the Weber–Fechner law, is proportional to the logarithm of the intensity. If the intensity of the sound emitted by the telephone is taken as proportional to the square of the voltage \(V\), then it turns out that the loudness
\[ s = c \lg \frac{V^2}{V_0^2} = 2c \lg \frac{V}{V_0}, \]
where \(V_0\) is the voltage corresponding to the threshold of sensitivity of the ear. Further, \(V_0\) depends strongly on frequency; the acoustic output of the telephone also depends strongly on frequency. Thus the receiving apparatus in Hartmann’s setup is completely unsuited for measuring such a phenomenon as statistical voltage fluctuations, or for comparing it with a sinusoidal voltage. Fürth, however, succeeded, in a very ingenious way, in establishing how Hartmann’s results should be corrected in order to eliminate the error arising from the properties of the telephone and the ear. After introducing the corresponding corrections into Hartmann’s numbers, Fürth obtained the following table of values for \(e\) (Table 1).
These numbers fluctuate much less than Hartmann’s original data, and their mean is close to the accepted value for \(e\). Some influence of the frequency on the result is still noticeable, but already from these data one may judge that Schottky’s theory is correct and that an improvement in the agreement with experiment may be expected with an improvement of the method.
A real confirmation of the correctness of Schottky’s theory not only from the qualitative but also from the quantitative side was given
TABLE 1
| \(\omega_n\) | At \(i_0 = 2\) mA, \(10^{-10}\) CGS | At \(i_0 = 20\) mA, \(10^{-10}\) CGS |
|---|---|---|
| 2,000 | 4.0 | 3.8 |
| 3,000 | 8.2 | 6.2 |
| 4,000 | 7.3 | 5.7 |
| 5,000 | 8.1 | 6.2 |
| 6,000 | 6.6 | 5.4 |
| 7,000 | 4.5 | 5.5 |
| 8,000 | 3.3 | 5.0 |
| 9,000 | 2.5 | 3.5 |
| 10,000 | 2.2 | 4.0 |
| 11,000 | 2.2 | 4.0 |
| 12,000 | 5.3 | 2.8 |
| 13,000 | 4.4 | 3.3 |
| Average | 4.9 | 4.6 |
In the work of Hull and Williams\(^{13,14}\). Like Hartmann, these investigators observed voltage fluctuations in the tuned anode circuit of a tube; however, a whole series of changes were introduced into the method and technique of the experiment, ensuring the attainment of a quantitatively correct result. First of all, the method of observation was changed. Hull and Williams abandoned the use of a telephone and of subjective observation, in view of their failure to meet the requirement of proportionality to the square of the voltage and independence of frequency. Instead, they used a vacuum-tube detector; the current rectified by the latter was measured by a galvanometer and potentiometer by the null method. Special attention was paid to selecting the proper characteristic of the detecting tube; by changing the bias and the resistance in the grid circuit, it was possible to ensure that the anode current was proportional to the square of the voltage within the limits of 1.5 V. Correspondingly, the amplification factor of the entire amplifier had to be adjusted so that the voltage amplitude on the detector grid did not exceed 0.75 V. The importance of exact observance of the condition \(\Delta i \sim \Delta V^2\) is shown by the authors’ preliminary experiments, in which \(e\) was obtained 50 times (!) smaller than the true value if the amplification was too large and individual voltage peaks entered the saturation-current region.
Further, in contrast to Hartmann’s method, which used nonselective amplification, Hull and Williams employed a sharply resonant amplifier tuned to the natural frequency of the oscillatory circuit. The frequency characteristic of the amplifier was carefully measured at each new tuning, and the amplified voltage was calculated by formula (16). Such a method, of course, gives a result of greater accuracy than the use of formula (14′) under the assumption—obviously not fulfilled in reality—that all frequencies are amplified uniformly. The very frequency region in which the measurements were made,
was also changed by the experimenters. Instead of audio frequencies, which Hartmann used, Hull and Williams worked at frequencies from 600 to 1000 kilocycles. As we shall see below in the description of the Johnson phenomenon, this transition to high frequencies is necessary in order to obtain the shot effect in pure form. Finally, the American investigators identified and eliminated from their experiments one more factor that could distort the magnitude of the shot effect—namely, the space charge in the tube. The presence of an appreciable space charge means such an electron density in the tube that an interaction arises between them, influencing their flight in the tube. In such a case the motions of electrons from cathode to anode are no longer entirely independent events; but then the possibility of applying formula (2), which is based precisely on the assumption of the independence of the transitions of individual electrons, disappears, and with it the whole further theory collapses. Special experiments carried out by Hull and Williams convincingly showed that with the appearance of a space charge in the tube the magnitude of the shot effect falls sharply (see below, p. 467). Therefore, in order to obtain the phenomenon in pure form, they replaced the diode by a three-electrode tube, whose grid, connected to the anode, was to eliminate the possibility of formation of a space charge. For this purpose a U.V. 199 Radiotron tube was chosen, having comparatively small diameters of the filament (0.015 mm) and of the grid (2.5 mm); with a voltage on the grid and anode of \(+120\ \mathrm{V}\), an electric field of such strength was produced at the surface of the cathode \(\left(E = 30\,000\ \frac{\mathrm{V}}{\mathrm{cm}}\right)\) that the occurrence of any considerable space charge was excluded. Operation in this regime (“temperature-limited current”) has the further advantage that under this condition the current in the tube is saturated and its internal resistance \(R_i\) is close to infinity; this brings the experimental conditions closer to the elementary theory.
The method of measurement consisted, as in Hartmann’s work, in comparing the shot effect with a sinusoidal electromotive force of resonant frequency, the magnitude of the latter being chosen so that the detector gave one and the same current. If the effective value of the sinusoidal voltage equivalent to the shot effect is denoted by \(v_1\), and the amplification factor at the resonant frequency by \(\mu_0\), then the following equality must hold:
\[ 2 i_0 e \int_0^\infty Z^2(f)\,\mu^2(f)\,df = v_1\mu_0^2 \]
(see formula [16]), whence:
\[ v_1^2 = 2 i_0 e \int_0^\infty Z^2(f)\left[\frac{\mu(f)}{\mu_0(f)}\right]^2 df. \tag{18} \]
Substituting the expression for \(Z(f)\) into formula (18) and passing from the variable \(f\) to \(x\), the authors transform (18) to the form:
\[ v_1^{\,2}=\frac{e i_0 L F}{2C^2R}, \tag{18,a} \]
where
\[ F= \frac{ \displaystyle \int_{0}^{\infty} \frac{x^2\psi(x)\,dx}{(1-x^2)^2+r^2x^2} }{ \displaystyle \frac{\pi}{2r} } \]
—the “amplification factor”—represents the ratio of the square of the voltage obtained at the output of the given amplifier from the shot effect to that which would be obtained with “uniform” amplification of all frequencies (the denominator of the expression for \(F\) is equal to the value which the numerator would have for \(\psi(x)=1\));
\[ \psi(x)=\frac{\mu^2(x)}{\mu_0^2(x)}. \]
From formula (18′) one can determine the elementary charge:
\[ e=\frac{2C^2Rv_1^{\,2}}{Li_0F}. \tag{19} \]
The general circuit of the setup is given in Fig. 5. \(S\) is the triode in which the shot effect is measured; \(L, C, R\) are the anode circuit. The amplifier, placed in a separate copper box, consists of four plistrons; the first three tubes operate in the amplifying regime and are loaded with tuned circuits. The amplification coefficient of each stage is 40–45; the total amplification of the whole amplifier is \(73\,000 \simeq (42)^3\); the agreement of the last two numbers serves as evidence of the absence of regeneration, which could have distorted the result of the measurements. A sample of the frequency characteristic of the amplifier is shown in Fig. 6. The 4th tube of the amplifier operates in the detector regime; its current is compensated by a potentiometer according to the readings of the galvanometer \(G\). The sinusoidal voltage \(v_1\) is applied through a special induction divider of a very original design, consisting of two coaxial cylinders, the self-induction per 1 cm of length of which can be calculated; the current strength in the loop \(i_1\) is measured with the aid of a thermocouple; then the voltage \(v_1\) can be calculated: \(v_1=i_1\omega L\).
The principal measurements were made at a frequency of 725 kilocycles. The data obtained are presented in Table 2.
We see that for different values of the emission current strength, very close values are obtained for \(e\). Individual measurements differ from the mean by no more than 3%, and if the first two measurements, made at weak current, are discarded, then by less than 2%. The mean value itself differs from the value determined by Millikan (\(1.591\cdot 10^{-9}\) coulomb) by only 0.3%. The agreement of theory with the experimental data is excellent. The authors report that they
measurements were also made at other frequencies in the interval from 600 to 1,000 kilocycles, and for \(e\) the same values were obtained at all frequencies; however, the experimental figures found at other frequencies are not given in the article.
Fig. 5. Diagram of the Hall and Williams apparatus.
An even more brilliant confirmation of Schottky’s theory was provided by the work of Williams and Vincent\(^{15}\). In this work, instead of a tuned circuit, a simple ohmic resistance \(R = 48\,300\ \Omega\) was inserted in the tube circuit to obtain voltage fluctuations. At the comparatively high frequencies used by the experimenters, it turned out that the capacitance and leakage of the amplifier input, which must be regarded as connected in parallel with \(R\), could not be neglected. The total impedance of this whole system
\[ Z = \frac{R}{\sqrt{1+\omega^{2}R^{2}C^{2}}} \]
proved to be considerably smaller than \(R\). Therefore it was measured each time in the experiment. The amplifier was of the same
Fig. 6. Example of a partial characteristic of the Hall and Williams amplifier.
of the same type as in the work of Hull and Williams, but contained five amplification stages instead of 3. The overall amplification factor was not measured, since its absolute magnitude when using the “substitution method” (comparison of the shot effect with a sinusoidal voltage) is of no importance; according to the authors’ estimate it was of the order of \(10^6\). Rectification was carried out not by a diode tube, as in the preceding work, but by a thermoelement connected through a transformer to the amplifier output; the thermoelement current was measured by a galvanometer,
TABLE 2
| \(J\) (mA) | \(v_1\) (μV) | \(F\) | \(\sqrt{\overline{v_s^2}}\) (μV) | \(\sqrt{\overline{j^2}}\) | \(e\) (coulomb) |
|---|---|---|---|---|---|
| 1 | 65.0 | 0.776 | 73.8 | 0.204 | \(1.541\cdot 10^{-10}\) |
| 2 | 89.4 | 0.763 | 102.2 | 0.282 | \(1.640\cdot 10^{-10}\) |
| 2 | 88.3 | 0.763 | 101.0 | 0.279 | \(1.603\cdot 10^{-10}\) |
| 3 | 102.7 | 0.750 | 118.5 | 0.327 | \(1.595\cdot 10^{-10}\) |
| 3 | 102.7 | 0.750 | 118.5 | 0.327 | \(1.595\cdot 10^{-10}\) |
| 4 | 113.4 | 0.740 | 131.8 | 0.364 | \(1.570\cdot 10^{-10}\) |
| 4 | 114.4 | 0.740 | 133.1 | 0.367 | \(1.595\cdot 10^{-10}\) |
| 5 | 122.5 | 0.727 | 143.8 | 0.397 | \(1.566\cdot 10^{-10}\) |
| 5 | 122.5 | 0.727 | 143.5 | 0.396 | \(1.556\cdot 10^{-10}\) |
| Average . . | \(1.583\cdot 10^{-10}\) |
shunted by a large capacitor. Despite this latter measure, when working at a frequency of 50 kilocycles the galvanometer deflections underwent very large fluctuations, which made observations difficult. On going over to higher frequencies these fluctuations disappeared; apparently, their origin is connected with the Johnson phenomenon (flicker effect). As in the investigations of Hull and Williams, the method of comparison with a sinusoidal voltage of resonant frequency was used. In view of the sharp resonance of the amplifier and the comparatively slow variation of the dependence on frequency, in the present case the voltage at the amplifier output can be determined from formula (16′), which we rewrite as follows:
\[ \overline{V_e^2}=2i_0 e Z^2 A\mu_0^2, \]
where
\[ A=\int_0^\infty \frac{\mu^2(f)}{\mu_0^2}\,df. \]
The equivalent voltage \(v_1\) gives at the amplifier output the voltage:
\[ V_1=\mu_0 v_1. \]
Since
\[ \overline{V_e^2}=V_1^2, \]
therefore
\[ v_1^2=2i_0 e Z^2 A, \]
whence
\[ e=\frac{v_1^2}{2i_0 A Z^2}. \]
\(A\) was determined graphically, as the area of the quadratic resonance curve; \(Z\) was measured at that frequency for which the ordinate of the resonance curve passed through the center of gravity of the area under this curve. We give the results of these measurements, remarkable in their accuracy (Table 3).
TABLE 3
| \(f_0=146\) kilocycles; \(Z=34510\,\Omega\) \(A=12880\) | \(f_0=114.25\) kilocycles; \(Z=36870\,\Omega;\ A=6231\) | ||||
|---|---|---|---|---|---|
| \(i_0\) (mA) | \(v_1\) (μV) | \(e\) (coulomb) | \(i_0\) (mA) | \(v_1\) (μV) | \(e\) (coulomb) |
| 0.206 | 31.7 | \(1.587\cdot10^{-19}\) | 0.206 | 23.6 | \(1.595\cdot10^{-19}\) |
| 0.310 | 39.1 | \(1.605\cdot10^{-19}\) | 0.310 | 28.9 | \(1.590\cdot10^{-19}\) |
| 0.367 | 42.4 | \(1.595\cdot10^{-19}\) | 0.404 | 33.8 | \(1.590\cdot10^{-19}\) |
| 0.404 | 44.4 | \(1.590\cdot10^{-19}\) | 0.417 | 34.7 | \(1.593\cdot10^{-19}\) |
| 0.447 | 46.7 | \(1.590\cdot10^{-19}\) | 0.508 | 37.9 | \(1.589\cdot10^{-19}\) |
| 0.508 | 49.7 | \(1.586\cdot10^{-19}\) | |||
| 0.614 | 54.7 | \(1.590\cdot10^{-19}\) | |||
| 0.715 | 59.1 | \(1.591\cdot10^{-19}\) | |||
| 0.811 | 62.8 | \(1.587\cdot10^{-19}\) | |||
| Average . . | \(1.5912\cdot10^{-19}\) | Average . | \(1.5914\cdot10^{-19}\) |
Only one measurement gives a figure differing by 1% from the mean; all the remaining figures give deviations smaller than 1%. For measurements of a statistical phenomenon such as the shot effect, this is quite extraordinary accuracy. The mean value from both series of observations agrees with the value found by Millikan to the last digit. It may be asserted that, alongside the classical experiments by the falling-drop method, the investigations of the American physicists Hull, Williams, and Vincent are among the most reliable proofs of the discreteness of electric charge.
It is curious to note that Johnson,\(^9\) on the basis of his observations at low frequencies, which led to the discovery of a new fluctuation effect, comes to the conclusion that experiments with the shot effect cannot serve as material for determining the elementary charge. At low frequencies, incorrect values are obtained for \(e\)—
...values, and therefore, on the contrary, one should use the value of \(e\) known from other investigations in order to determine the limits of applicability of Schottky’s theory. Such a point of view, in our opinion, is incorrect. Every method of measuring any physical constant is based on the study of some definite phenomenon and has definite limits of application, beyond which the given phenomenon is complicated by others.
These limits are established by analyzing the results of the experiment itself, and there is no need at all for this to know the sought quantity from other sources. This was also the case with the falling-drop method, which, as is known, gives correct results only within a certain range of drop radii. The use of drops of too small a radius led Ehrenhaft and his school to systematic errors in the determination of \(e\). In exactly the same way, by analyzing the results of observations of the shot effect, one can ascertain the conditions for obtaining the effect in pure form and obtain the correct value of the elementary charge. We shall see, when considering Johnson’s experiments, that they make it possible to establish that range of frequencies at which the pure shot effect is obtained. Similarly, the experiments of Williams and his collaborators show the necessary restrictions on the anode voltage. Taken together, they give a complete picture of the conditions under which the shot effect has a constant magnitude and can serve as a fully valid material for determining the charge of the electron.
A number of works in which the shot effect was also observed in a cathode tube at temperature-limited current will be touched upon in §§ 4 and 5.
3. Shot Effect in Photoelectric Cells
The concept of fluctuations of the electron current, confirmed in the study of thermoelectron emission in a cathode tube, is also applicable to photocurrent. The emission of each photoelectron constitutes a separate elementary act associated with the absorption of one quantum of light. The departure of individual photoelectrons occurs independently of one another; therefore Schottky’s theory can be applied here completely. Experimentally, the shot effect in photoelectric cells was studied by Orban\(^{20,21}\) and Kingsbury\(^{22}\). Orban studied photoelectric cells with cesium and potassium cathodes, both evacuated and gas-filled. The aim of the investigation was to establish both the general magnitude of the effect and its course in time as a function of the composition of the circuit. The experimental arrangement is shown in Fig. 7. The photoelectric cell \(Ph\) was illuminated by the lamp \(L\) through the variable diaphragm \(Bl\). The photocurrent passed through the resistance \(R\); the voltage at the ends of the latter was fed to the input of the three-stage amplifier \(I\), then filtered by the circuit \(S\) and further amplified by a second amplifier, also three-stage. The voltage at the output was measured by a vacuum-tube voltmeter with a square-law characteristic. The amplification coefficient of the apparatus at different frequencies was determined
by experiment; the resistance of the photoelectric circuit was calculated from the measured resistances and capacitances of the various sections; the value of \(e\) was computed from the experimental data by formula (16). Experiments with vacuum photocells were carried out with different amplifier adjustments, different resistance in the photocell circuit, and different illumination, i.e., different photocurrent.
Fig. 7. Measurement of the shot effect in a photocurrent (Orban).
Figure 8 shows the dependence of \(\overline{V_s^2}\) on the photocurrent at various resistances. The values calculated by formula (16″) are shown by straight lines, and those found experimentally by points. We see that satisfactory agreement is obtained between experiment and theory; \(\overline{v_s^2}\) is proportional to \(i_0\) and \(R^2\). To check the independence of the magnitude of the effect from frequency, the value of \(e\) was determined at different frequencies: 800, 2,000, 3,000, and 4,000 hertz; the results are shown in Fig. 9; each value of \(e\) is the average of 30 measurements. From these data it is evident that, within the limits of the accuracy of the measurements, there is no dependence of \(e\) on frequency; it should, however, be noted that the accuracy is not high and that there are too few points. The coincidence of the theoretical and experimental values of the effect indicates
Fig. 8. Dependence of the shot effect in a photocell on the photocurrent strength (Orban).
Fig. 9. Frequency dependence of the shot effect in a photocell (Orban): —·—·— adopted value of \(e\); — — — limits of experimental accuracy.
that in the photoelement there is no space charge. This is explained not only by the small current strength, but also by the cathode area, large in comparison with a cathode tube, so that the emission density, in comparison with the latter, is in fact negligible.
Of interest are the curves recording the shot effect, obtained by Orban with the aid of an oscillograph. These curves, shown in Fig. 10, were obtained with different tuning of the amplifier. Beside each curve is given the frequency characteristic of the amplifier. The first pair of curves refers to very selective tuning to the frequency \(f_0 = 800\) cycles; to obtain such sharp tuning, the experimenter used coupling through a tuning fork instead of an oscillatory circuit (mechanical resonance). In this case it is seen that the oscillations
Fig. 10. Recording of the shot effect by an oscillograph.
1) Very selective tuning:
a) natural frequency of the string — 1000 hertz
b) ” ” ” — 6000 ”
2) Weakly selective tuning
3) Nonselective tuning.
Visible labels in the figure:
a); b); \( \frac{1}{800}\,\mathrm{sec} \); \(f_{\text{string}} = 1000\,\mathrm{Hz}\); \(f_s = 6000\,\mathrm{Hz}\); time scale; \( \frac{1}{800}\,\mathrm{sec} \); \(f_s = 1000\,\mathrm{Hz}\); time scale; \( \frac{1}{800}\,\mathrm{sec} \); b); \(f_s = 6000\,\mathrm{Hz}\); time scale; \( \frac{1}{400}\,\mathrm{sec} \).
On the frequency-characteristic plots: voltage in arbitrary units; 400, 800, 1200; amplification; 10, 20, 30; amplification; 10, 20; 800, 1200, 1600 Hz.
the voltages at the output are almost pure tuning-fork oscillations, the initial phase of which remains unchanged for a long time. The curve in Fig. 10,2 was obtained with weak tuning sharpness; the coupling is no longer through a tuning fork, but through an electrical circuit. In this case we also see series of regular oscillations; however, these series are much shorter, and the initial phase changes much more often. Finally, in Fig. 10,3 a recording with nonselective amplification is shown; the curve represents completely irregular changes of current intensity. Thus, the considerations concerning the dependence of the character of fluctuations on damping, which were made in Part I of the present review as applied to the thermal effect, are fully applicable to the shot effect.
Orban also carried out experiments whose purpose was to establish the possible existence of a dependence of the shot effect in the photocurrent on the type of light flux. For this purpose the photocell
Fig. 11. Kingsbury’s experimental circuit (shot effect with photocurrent).
was subjected, in addition to illumination by a projection lamp, also to illumination by a gas-filled lamp and a gas burner. In all cases the luminous flux was regulated by a diaphragm so that the photocurrent \(I\) remained unchanged. Owing to the different sensitivity of the cathode to light of different spectral composition, in these three cases we shall have three different luminous fluxes; in 1 sec., in each of these cases, a different number of quanta will be absorbed. However, the number of electrons emitted per second is one and the same, and therefore from the general theory of the shot effect it follows that the magnitude of the latter must also be the same in all three cases. The experiment confirmed this. It must be admitted that this result is essentially trivial; there are no grounds for seeing why anything else could have been obtained.
All the more surprising is it that in Kingsbury’s work\(^{22}\) an apparently different result was obtained on this question. His method differed somewhat from that adopted by Orban. To avoid calibrating the apparatus, in particular the amplifier, Kingsbury compared the shot effect of the photocurrent with the shot effect of the thermionic current in one and the same circuit. The latter, under conditions of temperature-limited current, has been well studied and can indeed serve as a certain standard for comparison. The experimental circuit is shown in Fig. 11. The photocell \(Ph\) and the cathode tube \(S\) are connected in parallel with one another and in series with a resistance \(R_1 = 10\) megohms. The voltage fluctuations at the terminals of the latter
are produced either by a photocurrent, if the cathode of the lamp \(S\) is not heated, or by a thermionic current, if the photoelement is not illuminated. These fluctuations are transmitted through a \(0.006\,\mu F\) capacitor to the grid of the first tube of the amplifier (9 stages, with resistances; \(\mu_0 \simeq 3\cdot 10^5\)); at the output there is a rectifier and a direct-current instrument. The shot effect was measured in both devices at the same current strength \(i\). It turned out that, in agreement with the theory, \(\overline{v_s^2}\) for the photocurrent is also proportional to \(i_0\); however, in absolute magnitude, \(\overline{v_s^2}\) for the photocurrent was 12% greater than for the thermionic current of the same strength. Theoretically, there should have been no such discrepancy; Kingsbury did not establish its cause.
The study of the dependence of the shot effect on the nature of the radiation also led this experimenter to strange results.
Fig. 12. Influence of the lamp filament heating on the shot effect of the photocurrent (Kingsbury).
Experiments carried out with a series of filters showed that there is no such dependence. However, experiments carried out with one and the same lamp at different values of the filament current gave the following result (Fig. 12). Along the abscissa is plotted the logarithm of the current strength; along the ordinate, \(\lg v_s\); the parameter of the curves is the current strength heating the light source. Experimental data are given for three values of this quantity: 14, 15, and 16 A. With increasing heating, the entire spectrum of the lamp shifted toward shorter wavelengths. It is evident from the figure that, in this case, the magnitude of the current fluctuations also changes; namely, at one and the same photocurrent strength, \(\overline{v_s^2}\) is the greater, the weaker the heating of the source. Kingsbury, to be sure, attempted to conclude that the composition of the spectrum does not affect the shot effect, since the curves for different filament-current strengths run parallel. However, this conclusion is unfounded, since constancy of the shot effect requires not parallelism, but complete coincidence of the curves.
Kingsbury’s article provides no material for judging the possible source of this phenomenon, which contradicts both the theory and the experimental data of Ornstein and of Kingsbury himself (the experiments with filters). Apparently, the later work of Ornstein was technically carried out better and deserves greater confidence; the causes of Kingsbury’s erroneous, in all probability, results still await discovery.
Both authors also carried out experiments on fluctuations of the photocurrent in gas-filled photoelements. The magnitude of the fluctuations here turns out to be much greater than in a vacuum photoelement, since
as the effect of ionization of the gas is also superposed (see below). According to Orban, the amplification of the “noise” in a gas-filled photoelement may be from 4 to 50 times; at the same time the proportionality of \(V_s^2\) to \(R\) and \(i_0\) is preserved; the latter, however, is not confirmed by Kingsbury’s experiments. Here too, evidently, new investigations must be awaited.
4. Influence of Space Charge on Fluctuations of the Electron Current
a) Depression of the Shot Effect
In the preceding paragraphs, in discussing experimental investigations of the shot effect, it has already been emphasized that, in order to obtain it in pure form, the absence of space charge is necessary. The shot effect is, in essence, fluctuations of the emission current. The condition of statistical independence of the individual electrons reaching the anode reduces to two separate requirements:
-
Statistical independence of the emission of individual electrons by the cathode; complete randomness of the distribution of emission both over the surface of the cathode and in time.
-
Exact reproduction in the anode current of the fluctuations of emission.
Fig. 13. Space charge in a cathode lamp.
For the realization of the latter requirement it is necessary that, on the way to the anode, the electrons be subjected to no other influences except the accelerating field, the same for all; in particular, that there be no appreciable interaction between them. Only when these requirements are fulfilled can the electron current be regarded as corresponding to the distribution formula (2), and Schottky’s theory is based on its application. Phenomena that occur when the first of these conditions is not observed will be considered in the next paragraph. We now turn to the analysis of deviations from the second requirement. If the density of electrons at some point on the path between cathode and anode is so great that it begins to influence the distribution of potential in space, then the motions of the individual electrons cease to be independent. The electrostatic interaction between them can no longer be neglected; it must be taken into account that each new electron emitted by the cathode is repelled by the preceding ones, emitted before it. When an electron enters the space-charge cloud (Fig. 13), its motion is retarded (up to its reaching the surface of minimum potential); moreover, if many electrons were emitted immediately before it, it is retarded more strongly, and if few, the retardation is weaker. Consequently, the space charge hinders the occurrence of appreciable deviations of the electron current from its mean value; it smooths the fluctuations, diminishes their magnitude. Electrons emerging from the cloud
of the space charge in the direction toward the anode, move in a more ordered fashion than those emitted directly by the cathode. In the limit, at a sufficient electron density, their motion toward the anode will become completely uniform; the instants of their arrival at the anode will be separated by exactly equal time intervals. Under these conditions, the phenomenon that we observe as the shot effect disappears altogether. These considerations agree well with the data of numerous experiments.
Fig. 14. Anode current under complete suppression of the shot effect.
A careful consideration of the process taking place in the limiting case of complete limitation of the current by space charge is very useful for clarifying what we actually call the shot effect. In the case now under consideration the strength of the anode current will likewise not be strictly constant in time; it may be represented by a curve of the kind shown in Fig. 14. Each hump represents the arrival of one electron; the time intervals between them $\tau$ are strictly identical. However, this will not be the shot effect, and in practice we cannot observe such a phenomenon. Let us expand this curve in a Fourier series; since it is strictly periodic (period—
Fig. 15. Anode current in the presence of the shot effect.
$\tau$), we obtain a constant term and a series of harmonic components with frequencies $\omega_k = \dfrac{2\pi}{\tau} k$ ($k$ an integer). Let the mean current be $i_0 = 0.1\ mA$; then
\[ \tau = \frac{e}{i_0} = \frac{1.59 \cdot 10^{-19}}{1 \cdot 10^{-4}} \simeq 1.6 \cdot 10^{-15}\ \text{sec}. \]
Consequently, the frequency of the fundamental tone in our expansion will be $n_1 = \dfrac{1}{\tau} = 0.6 \cdot 10^{15}\ \text{sec}^{-1}$, and the corresponding wavelength $\lambda = 3 \cdot 10^{10} \cdot 1.6 \cdot 10^{-15} = 4.8 \cdot 10^{-5}\ cm$—of the order of the wavelength of blue light. The overtone frequencies will be still higher; it is clear that there is no question of observing current fluctuations of such a frequency.
In the absence of a space charge, i.e., with temperature-limited current, the strength of the anode current will be expressed by a curve of the kind shown in Fig. 15. Individual electrons reach the anode in disorder; the function $i(t)$ can no longer be expanded in a series, but can be represented only by a Fourier integral, in which,
As was shown above, all frequencies, except very high ones, are represented uniformly. For the shot effect, therefore, what is characteristic is not simply the fact that charge arrives at the anode in portions, but the chaotic nature of this arrival.
Thus, the presence of a space charge of appreciable magnitude diminishes the shot effect. If one measures the voltage fluctuations in the anode circuit in the absence of space charge \(\left(\overline{v_s^2}\right)\) and in its presence \(\left(\overline{v_s^{\prime 2}}\right)\), then the latter quantity proves to be smaller*; the ratio \(\dfrac{\overline{v_s^{\prime 2}}}{\overline{v_s^2}}=D\) is called in the American literature the depression of the shot effect. However, besides ordering the motion of the electrons, the transition to the space-charge regime also affects the magnitude of the observed effect in another respect. Namely, instead of a saturation current we now have to deal with the inclined part of the tube’s volt-ampere characteristic, where \(i_a\) depends not only on emission, but also on the anode voltage. This means that the fluctuations of anode voltage caused by the shot effect must exert a feedback influence on the current in the tube. Llewellyn\(^{26}\) derived, by a completely formal method, an expression that takes into account both influences of space charge on the magnitude of the shot effect. The strength of the anode current \(i\) is in general a function of \(e_a\), \(e_g\), and \(J\)—the current emitted by the cathode. If we assume that \(e_g=\mathrm{const}\), then the change in the anode current is expressed as follows:
\[ \delta i=\frac{\partial i}{\partial e_a}\delta e_a+\frac{\partial i}{\partial J}\delta J =\frac{1}{R_i}\delta e_a+\frac{\partial i}{\partial J}\delta J \]
\[ \left(R_i=\frac{\partial e_a}{\partial i}\ \text{— the internal resistance of the tube}\right). \]
Let a load \(Z\) be connected in series with the tube. Then a change in current strength by \(\delta i\) will produce at its terminals a voltage change \(v=Z\delta i\); owing to this, the voltage on the tube will decrease by the same amount, and consequently:
\[ \delta e_a'=-Z\delta i \quad \text{or} \quad \delta i=-\frac{1}{Z}\delta e_a. \]
Substituting this expression into the preceding formula, we find:
\[ -\frac{1}{Z}\delta e_a=\frac{1}{R_i}\delta e_a+\frac{\partial i}{\partial J}\delta J \]
or
\[ \delta e_a=-\frac{R_iZ}{R_i+Z}\cdot\frac{\partial i}{\partial J}\delta J =-Z'\frac{\partial i}{\partial J}\delta J, \]
where \(Z'=\dfrac{R_iZ}{R_i+Z}\) is the resistance of the tube and the load in parallel connection. Consequently:
\[ v_s'=Z'\frac{\partial i}{\partial J}\delta J. \tag{20} \]
\[ \text{* Assuming that other sources of fluctuations, besides the shot effect, are absent.} \]
If \(\delta J\) represents spontaneous fluctuations of the emission current, which in § 1 we denoted by \(j\), then we may make use of its Fourier-series expansion (5). For each component of this series one may write the mean square of the corresponding partial voltage:
\[ \overline{(v_s')_k^2}=\frac{1}{2}C_k^2\left(\frac{\partial i}{\partial J}\right)^2 Z_k^2 . \]
As was shown in § 1:
\[ \overline{C_k^2}=\frac{4J_0e}{T}. \]
Passing from the sum to an integral, we find, by analogy with the derivation of formula (13):
\[ \overline{(v_s')^2}=2J_0e\left(\frac{\partial i}{\partial J}\right)^2 \int_0^\infty Z'^2(f)\,df . \tag{21} \]
In this formula, which at high space-charge density replaces formula (13) and serves as its generalization, both of the above-mentioned effects of the space charge are plainly visible. The appearance of a finite resistance of the tube is taken into account by introducing \(Z'\) instead of \(Z\), the internal resistance of the tube being connected in parallel with the load and damping it. The ordering of the motion of the electrons is expressed by the appearance of the factor \(\left(\dfrac{\partial i}{\partial J}\right)^2\). In the case of temperature-limited current (no space charge), \(\dfrac{\partial i}{\partial J}=1,\ R_i=\infty,\ J_0=i\), and we return to formula (13). On the contrary, for a current completely limited by space charge, \(\dfrac{\partial i}{\partial J}=0\), and the shot effect must disappear. To determine the depression \(D\), we must stipulate that measurements of the shot effect both in the presence and in the absence of space charge are made with the same \(Z'\); for this purpose the change in the tube resistance \(R_i\) must be compensated by a corresponding change in the wattless component of the load \(Z\). Then we immediately find that the depression is
\[ D=\left(\frac{\partial i}{\partial J}\right)^2 . \tag{22} \]
Experimentally, the depression of the shot effect was discovered by Hull and Williams in the work already cited.\(^{14}\) Observing the magnitude of the fluctuations at different cathode heats and at constant anode voltage, they established that only at comparatively weak heating, when there is no appreciable space charge, is the “theoretically correct” magnitude of the shot effect obtained. If, however, without changing \(V_a\), the cathode heating is increased and thereby the density of the space charge is increased, then the observed shot effect becomes smaller than the “theoretical” one, i.e., that calculated by Schottky’s formula. We give the results of the meas—
measurements obtained with a UV 199 Radiotron tube at \(V_a=130\text{ V}\) and \(V_g=-6\text{ V}\); the normal filament heating of the tube is \(0.20\text{ A}\) (see Table 4).
From this table it is evident that even at a weaker heating than that recommended for normal use of the tube, the depression reaches 0.2. Similar results were also obtained with other tubes.
Llewellyn, \(^{26}\) Moullin and Ellis, \(^{18}\) Kosanowsky and Williams \(^{24}\), and other authors likewise observed a strong depression of the shot effect when space charge appeared. However, obtaining exact, reproducible figures and verifying Llewellyn’s formula
TABLE 4
Dependence of the shot effect on filament heating (Hull and Williams)
| Filament current | \(T_k^\circ\text{K}\) | Emission current (mA) | Shot effect \(\mu\text{V}\), observed | Shot effect \(\mu\text{V}\), calculated | Shot effect \(\mu\text{V}\), depression |
|---|---|---|---|---|---|
| 0.140 | 1 675 | 1.0 | 67 | 71.7 | 0.93 |
| 0.150 | 1 750 | 2.0 | 71 | 87.7 | 0.82 |
| 0.152 | 1 765 | 2.5 | 51 | 83.8 | 0.61 |
| 0.160 | 1 805 | 3.0 | 38 | 77.2 | 0.49 |
| 0.167 | 1 850 | 3.5 | 28 | 73.0 | 0.39 |
| 0.170 | 1 867 | 4.0 | 13.6 | 75 | 0.18 |
| 0.182 | 1 940 | 5.0 | 15.9 | 80 | 0.20 |
(21) proved to be a much more difficult task than obtaining the normal effect with temperature-limited current and verifying Schottky’s theory.
b) Anomalous fluctuations in the presence of positive ions
The principal obstacle to achieving a pure depression effect proved to be the instability of the space charge, the slightest fluctuations of which are reflected in an increase of the fluctuations in the anode current. A particularly harmful role is played by positive ions, if they appear in regions of high electron concentration. Indeed, owing to their considerable mass and, consequently, much lower mobility, they exert a very great influence on the electronic space charge. As Langmuir showed, one positive ion, entering the region of the electronic space charge, neutralizes the action of \(\sqrt{\dfrac{M}{m}}\) electrons (\(M\)—mass of the ion, \(m\)—mass of the electron), i.e., of several hundred electrons. Correspondingly, a large batch of electrons is at once released from the cloud and flies to the anode.
therefore each positive ion causes a sharp increase in the electron current; the presence of even a small number of such ions entails anomalously large fluctuations of the anode current. The source of the positive ions may be the gas present in the tube, or the electrodes. Both cases have been studied experimentally.
The influence of ionization of the gas present in the tube on the fluctuations of the current in it was studied by Ballantine^17 both theoretically and experimentally. His attempt at a theoretical analysis of the question is based on the assumption that each positive ion appearing in the tube as a result of impact ionization and then moving toward the cathode causes a certain brief rise of the anode current—“current impulse” \(i(t)\); the form of this function depends on the operating regime of the tube and on the place where the given ion arose.
Fig. 16. Circuit for measuring fluctuations in a cathode tube according to Ballantine.
Each such impulse constitutes an elementary act for the type of fluctuation that we are now considering. If it were possible to determine the function \(i(t)\), then it would be possible to calculate exactly the magnitude of the fluctuations and their distribution over the spectrum. Ballantine did not succeed in doing this. Nevertheless, it proved possible to determine the dependence of the noise arising as a result of ionization on the current strength, pressure, and kind of gas, without detailing the form of the “current impulse.” Ballantine found that the noise should be proportional to the gas pressure, to the current strength to the power
\[ \frac{5}{3}, \]
to the molecular weight, and should depend in a definite way on the ionization probability for the given gas. The experiments were carried out in a two-electrode tube with argon, mercury vapor, and so-called “residual gas” emitted from the glass of the tube (chiefly a mixture of \(\mathrm{H_2}\), \(\mathrm{N_2}\), and \(\mathrm{CO}\)). Ballantine’s method of work is of some interest (Fig. 16).
Like Kingsbury, he avoided the difficult work of calibrating the apparatus by comparing the measured fluctuations with the shot effect in the tube at temperature-limited current. For this purpose, a diode \(B\) was connected in parallel with the experimental tube \(A\). First, the current fluctuations in tube \(A\) were measured at non-satu-
in a gas-filled diode; then the filament current of the diode was gradually increased until the instrument at the amplifier output no longer showed a double deflection; in this case one may assert that the current fluctuations in both tubes are identical. The magnitude of the shot effect in the diode was calculated simply from the strength of the electron current flowing through it. The tuned circuit was connected not directly into the anode circuit of the tube, but through a transformer; frequencies from 500 to 1500 kilocycles were used. The amplified voltage was not immediately rectified, but was first combined with a sinusoidal e.m.f. of the corresponding frequency from a special generator. The beats obtained in this way were detected and then amplified by a special low-frequency amplifier, after which they were measured with a pointer instrument.
Ballantine established that, for a given gas, the noise in the tube is directly proportional to the gas pressure, as required by theory. Further, he found that the dependence of the noise on the electron current is expressed by the law \(i^{1.5}\), which differs somewhat from the theoretical \(i^{1.66}\). Finally, as regards the magnitude of the fluctuations in different gases, for mercury vapor, argon, and “residual gas” they stand caeteris paribus in the ratio \(1 : 0.13 : 0.07\). Theory gives for Hg, Ar, \(N_2\), and CO the following ratios \(1 : 0.12 : 0.07 : 0.075\). The agreement, as we see, is quite satisfactory. The whole body of results, and especially the increase of the fluctuations with increasing ion mass, makes the assumption concerning the mechanism of action of positive ions quite convincing.
Fig. 17. The theoretical dependence of the shot effect on the anode voltage at constant anode-current strength.
Anomalous fluctuations of the electron current, caused by emission of positive ions from a heated cathode, were investigated by Kozanowski and Williams. Working with cathodes coated with oxides of Ba and Sr, they established that “activation” of the cathode by thermal treatment causes an increase in the fluctuations observed at a given current strength. Using an arrangement analogous to that of Hull and Williams (the only change was in the technique of measuring \(Z\)), they determined the magnitude of the fluctuations at constant anode current but variable anode voltage; the value \(i_0\) was kept constant by regulating the filament current. Under these conditions one might expect: (a) according to the elementary theory, which does not take space charge into account, a constant value of the fluctuations (Fig. 17, curve A); (b) taking account of the influence of space charge, a depression at small values of \(i_a\), when the emission is large and the space charge is strongly developed (Fig. 17, curve B). In reality, something different was obtained (Fig. 18). At the beginning of activation a depression was indeed observed at small \(V_a\)
(curve I; the rise at the end of the curve is explained by the authors as ionization; probably not all the gas had been removed from the tube). With further treatment of the cathode the depression disappears and is replaced, on the contrary, by a maximum of fluctuations (curves II and III); at sufficiently large values of the anode voltage their magnitude remains constant.
Fig. 18. Dependence of fluctuations on activation of an oxide cathode (Kozanowski and Williams).
It turned out that in this case the reverse current from the cathode (the current observed when the anode is at a negative potential) also increases strongly. By the magnetron method it was shown that positive ions emitted by the cathode play a fairly considerable role in it. Kozanowski and Williams showed by an elegant direct experiment that the anomalously large magnitude of the fluctuations in the presence of space charge is explained precisely by the presence of the latter. For this purpose a tube was made containing, in the immediate neighborhood, two cathodes: one of pure tungsten, emitting electrons alone, and the other of the Känsman construction, emitting almost exclusively K ions (for the construction of this cathode, see below, § 6). Both cathodes were surrounded by a common anode. Curves of the same type as in Figs. 17 and 18 were taken, i.e., at variable anode voltage and constant $i_0$. The experiments showed that when a tungsten cathode alone is operating, the normal course of the shot effect is obtained, with a depression at small $V_a$ (Fig. 19, curve I), and likewise when one Känsman cathode is operating (see § 6). When both cathodes act jointly, however, so that positive ions enter the electronic space charge in large numbers, then, at a high space-charge density, there is obtained not a depression but, on the contrary, a maximum of fluctuations (curve II). The latter curve has exactly the same form as curve III in Fig. 18. This proves the correctness of the explanation of the phenomenon given by Kozanowski and Williams.
Fig. 19. Fluctuations of the anode current with pure and mixed emission.
J. Donal^29 observed anomalous fluctuations of the anode current, of analogous origin, under the action of oxygen on a heated tungsten filament. In this case ions \(WO_2^+\) and \(WO_3^+\) are formed; entering the region of space charge, they are captured in the potential minimum, and thereby produce an anomalous effect of considerable magnitude. Donal calculated the total increase of current and the increase of fluctuation that may be expected in this case, and arrived at conclusions in agreement with his observations.
Thus it is beyond doubt that, in the presence of positive ions in the discharge, the normal effect cannot be obtained. If observations do show depression during the development of space charge, its magnitude will in any case be incomplete. In order to test Llewellyn’s theory, it is necessary to provide a purely electronic positive charge. However, with an ordinary cathode and in its vicinity this cannot be done. As has been established by numerous experiments,^* a heated cathode always emits, in addition to electrons, a certain number of positive ions. Usually these are ions of various impurities; but Wahlin and Smith found that ions of the heated metal itself are also emitted. Despite the fact that the percentage of their emission is negligible, they exert a noticeable influence on the space charge. Thus, Zscher and Williams^25, carrying out measurements of the shot effect in the presence of space charge in a number of ready-made factory lamps with tungsten cathodes, found discrepancies in the results of individual experiments of 5–10%, whereas all the experimental conditions were maintained so constant that the discrepancy should not have exceeded 2%. The experimenters found that the cause of the inconstancy of the results was the emission of positive ions by the cathode.
These authors succeeded, however, by an ingenious method, in creating a purely electronic space charge in the tube. For this they used a two-grid tube in which the space charge was formed near the outer grid. To grasp the idea of this method of operation, let us first consider a tube with one grid. Suppose that a certain positive voltage of sufficiently large magnitude is applied to the grid. Then the space charge of electrons between the filament and the grid is absent; the electrons are accelerated to a considerable velocity, and some of them fly beyond the grid. The positive ions emitted by the cathode are repelled by the grid and return back to the cathode. Between the grid and the anode a purely electronic current is obtained, with respect to which the grid may be regarded as a “virtual emitter.” Under these conditions, between the grid and the anode one can observe the pure shot effect, but without space charge. The latter in this region is too small, since the electron velocities are very large;
^ See the article by Compton and Langmuir, Uspekhi Fizicheskikh Nauk*, XI, 33, 1931.
it is smaller than between the cathode and the grid.* Since \(\rho\) near the cathode is already so small that it does not affect the electron current, then beyond the grid \(\rho\) will certainly have no effect at all. To obtain a large \(\rho\) here, it is necessary to slow down the motion of the electrons; this is achieved by introducing a second grid with a retarding field. The distribution of the potential in such a tetrode is explained in Fig. 20. Between the cathode and the first grid \(G_1\) the potential rises; the electrons are accelerated, and the positive ions return back to the cathode. The second grid \(G_2\) has a potential only slightly higher than the cathode; the field between it and the first grid retards the electrons, which, on reaching \(G_2\), lose almost all their velocity. Quite slow electrons pass beyond the grid \(G_2\). Thus in the given case the second grid may be regarded as a “cathode” emitting
![Figure 20 and Figure 21 diagrams]
Fig. 20. Distribution of potential in a tetrode in the experiments of Zscher and Williams.
Fig. 21. Depression of the shot effect in a pure electron current (Zscher and Williams).
pure-electron emission with very small velocities. Depending on the anode voltage we can obtain either a temperature-limited current (large \(V_a\), curve \(I\) in Fig. 20) or a current limited by space charge (small \(V_a\), curve \(II\)). It is extremely important that in the region \(K—G_1\) there is no space charge and that therefore the positive ions emitted from \(K\) cannot produce any appreciable fluctuations of the electron current. The fluctuations of the current passing through \(G_2\) are therefore a reproduction of the fluctuations of the emission of the actual cathode \(K\). This condition, necessary in order that it should be possible to study the shot effect in the electron current between \(G\) and \(A\), is in fact exactly fulfilled. The “emission” of grid \(G_2\) is distributed in time according to the law of chance. Proof of this is furnished by measurements of the fluctuations at high anode voltages, when the pro-
* If \(I\) is the current density, \(\rho\) the volume charge density, and \(v\) the velocity of the electrons, then \(I=\rho v\). Between the grid and the anode \(I\) is smaller than between the cathode and the grid (the radius of the cross section is larger; moreover, part of the current has gone to the grid); on the other hand, in this region (between the grid and the anode) the field is accelerating. Consequently, \(\rho\) here is smaller than near the cathode.
the space charge near \(G_2\) is absent. In this case the normal value of the shot effect is obtained, corresponding to Schottky’s formula; the value of the elementary charge \(e\) derived from these observations proves to be entirely correct. If, however, a smaller potential difference is applied to the anode, then, in parallel with the growth of the space charge, a depression of the shot effect also develops. The result of measuring the depression is presented in Fig. 21. This figure shows two curves: the 1st curve represents the depression of the shot effect
\[ D=\frac{\overline{V_s'^2}}{\overline{V_s^2}}, \]
the 2nd curve—the ratio of the space current (the current to the anode) \(i\) to the current \(J\) “emitted” by the grid \(G_2\); in this case \(V_a\) and \(J\) were varied, while \(i\) was kept constant. We see that the depression changes smoothly with the change of \(\frac{i}{J}\). Zscher and Williams find that from 50 to 200 V the experimental results are satisfactorily represented by the formula
TABLE 5
Depression of the shot effect (according to Zscher)
| \(V_a\) (volts) | \(J\) (microamperes) | \(\dfrac{\partial i}{\partial J}\) | \(\left(\dfrac{\partial i}{\partial J}\right)^2\) | \(\dfrac{\overline{V_s'^2}}{\overline{V_s^2}}\) |
|---|---|---|---|---|
| 200 | 500 | 1.00 | 1.00 | 1.00 |
| 150 | 520 | — | — | 0.941 |
| 100 | 555 | 0.902 | 0.814 | 0.818 |
| 75 | 589 | 0.809 | 0.653 | 0.720 |
| 60 | 625 | 0.771 | 0.595 | 0.637 |
| 50 | 651 | 0.711 | 0.507 | 0.567 |
| 40 | 700 | 0.665 | 0.444 | 0.464 |
\[ D=\left(\frac{i}{J}\right)^2 \]
—the depression is proportional to the square of the decrease in the anode current. There are no direct data in this work for checking Llewellyn’s formula, since the values of the derivative \(\frac{\partial i}{\partial J}\) were not determined. In a later work by Zscher\(^{31}\), formula (22) was checked. It gives data concerning \(\frac{\partial i}{\partial J}\) and compares them with the magnitude of the depression; \(\frac{\partial i}{\partial J}\) varied from 1.00 to 0.666, and the depression correspondingly from 1.00 to 0.464. In this series of experiments: \(i=500\ \mu\mathrm{A}\); \(V_{g1}=22.5\ \mathrm{V}\); \(V_{g2}=1.5\ \mathrm{V}\). The results of comparing the calculations with experiment are given in Table 5.
The last two columns of the table show that the values of \(D\) and \(\left(\frac{\partial i}{\partial J}\right)^2\), even if they do not coincide exactly, are very close; the small discrepancy between them apparently arises from errors
of these very difficult measurements. In this work Llewellyn’s formula received its first and quite satisfactory experimental confirmation.
From the preceding exposition it is clear that there is, in essence, as yet no theory of the depression of the shot effect. Llewellyn’s formula represents only a formal solution of the problem. The task of a deeper theory is to calculate the magnitude of the derivative \(\dfrac{di}{dJ}\) from the data characterizing the emission and the space charge. Zacher and Williams, in the already cited work,²⁵ attempted to take the next step in constructing a theory of this phenomenon, considering the correlation between the values of the current \(i\) at successive instants of time (the idea of this attempt apparently belongs to Uhlenbeck). Assuming that the correlation function between \(j(t_1)\) and \(j(t_2)\) can be represented as:
\[ \overline{f(t_1) f(t_2)} = \varphi(t_1 - t_2) Pe^{-\alpha^2 \frac{(t_1 - t_2)^2}{2}}, \]
where \(\alpha\) is a certain coefficient, reciprocal in magnitude to the “width of the correlation band,”* they found that the observed depression under sharp tuning of the amplifier should be proportional to the factor \(e^{-\frac{\omega_0^2}{2\alpha^2}}\), where \(\omega_0\) is the resonant frequency of the amplifier. However, measurements showed that the depression curves obtained at frequencies of 54,000 hertz and 480,000 hertz are identical to an accuracy of up to 2%. Hence one must conclude that the factor \(e^{-\frac{\omega_0^2}{2\alpha^2}}\) remains practically constant in this case, and this can be only if \(\alpha^2 \gg \omega_0^2\), i.e., if the width of the correlation band is much less than \(\dfrac{1}{480000}\) sec. \(\simeq 2 \cdot 10^{-6}\) sec. Nothing more can be extracted from the theoretical analysis of Zacher and Williams. No other attempts to construct a theory of the depression of the shot effect have been published.
It should be noted that some authors, for example Mollin and Ellis,¹⁸ altogether deny the correctness of the above interpretation of the shot effect and the influence of space charge upon it. According to Mollin and Ellis, the arrival of electric charge at the anode in discrete portions produces a shot effect under any conditions; it cannot disappear even when the current is completely limited by space charge. The role of the latter, from their point of view, reduces only to the finite magnitude of the internal resistance of the tube; they reject entirely the significance of the factor \(\dfrac{di}{dJ}\). As we have seen above, the phenomenon that remains when the space charge is fully developed has little in common with the shot—
* By the latter term we denote that interval of time during which the initial value of the variable quantity is still reflected in subsequent values. In the absence of space charge the width of the correlation band tends to zero, and consequently \(\alpha\) to infinity.
effect and is practically unobservable. Scherer’s experiments confirm the validity of Llewellyn’s formula, and consequently also of those propositions from which it follows. The experiments of Mollin and Ellis themselves show a discrepancy with the formula they derived, differing from Schottky’s theory only by the introduction of \(Z'\) instead of \(Z\); this disagreement between experiment and the theory they propose is especially significant in the presence of a space charge (60%), and the authors acknowledge that they cannot give an explanation for the discrepancy they found. Meanwhile this discrepancy is not difficult to understand: the observed fluctuations are always smaller than the calculated ones because they do not take into account the factor \(\left(\dfrac{di}{dJ}\right)^2\), required by Llewellyn’s theory. Thus neither theoretical considerations nor experimental data speak in favor of the views of Mollin and Ellis.
A new experimental investigation of the question, carried out by Pearson,\(^{43}\) decisively confirms the theory set forth above. This investigator observed the shot effect by means of a triode with a tungsten cathode; the grid and anode, connected together, were at a potential of 9.4 V. The space current was varied within the limits from 1 to 5 mA by changing the filament heating. The emission current \(J_0\) corresponding to each value of the space current \(i_0\) was measured as follows: the heating was kept unchanged, the anode voltage was increased, and the anode current was measured. At large \(V_a\) we are dealing with the “Schottky current” (the influence of the electric-field strength on the emission), which obeys the equation
\[ \lg J'=\lg J + A V_a . \]
Fig. 22. Shot effect in a triode in the presence of a space charge (Pearson); points—experimental data, solid curve—calculated according to Llewellyn, dashed curve—calculated according to Mollin and Ellis.
Plotting the graph of the dependence of \(\lg J'\) on \(V_a^{1/2}\), the author obtained a straight line; its extrapolation back to \(V_a=0\) made it possible to determine \(J_0\). Next, having constructed the curve of the dependence of \(i_0\) on \(J_0\), he determined graphically the derivative \(\dfrac{\partial i}{\partial J}\). The load in the anode circuit was ohmic, \(R_a=1000\,\Omega\). Using these data and Llewellyn’s formula (21), Pearson calculated the expected magnitude of the fluctuations and compared it with experiment. The result is shown in Fig. 22. Both in the saturation-current region (below 3.5 mA) and in the space-charge region up to 4.4 mA, Llewellyn’s formula agrees well with the experimental data; on the contrary, the formula of Mollin and Ellis does not agree with them at all. The rise of the fluctuations at \(i_0>5\) mA is attributed by the author to the action of positive ions emitted by the cathode; the measures that were adopted in Scherer’s work against the latter were not applied here. Nevertheless, the experimental results are sufficiently definite. Ziegler\(^{48}\), in a brief communication, confirms the resul—
Schottky and Pearson. Working with a small diode at \(i=\dfrac{1}{2}J\), he observed a very strong decrease of the shot effect in comparison with the saturation current. Calculating the expected magnitude of the fluctuations according to Moullin and Ellis, i.e., taking into account only the decrease of the internal resistance, he obtained a value 8 times larger than that found experimentally. Thus both the existence of the depression of the shot effect and the generally accepted explanation of it may be regarded as established.
c) Thermal effect of the electron current in a tube
The discovery of the influence of space charge on the shot effect has substantial practical significance. A cathode tube operating as an amplifier or detector is always in the space-charge regime; the conditions required by Schottky’s theory are not the operating regime of the tube.
It is therefore a very important and favorable circumstance that precisely in the operating regime of the tube the shot effect, which creates interfering noise in it, is reduced. However, one must guard against the possible erroneous conclusion that, when the current is completely limited by space charge, all current fluctuations in the tube disappear. The shot effect, i.e., fluctuations of the anode current caused by fluctuations of emission, must indeed disappear. But another phenomenon then appears—the thermal effect, already considered by us in the first part of the present review. The latter is absent if the individual carriers of electricity do not interact and do not exchange energy with one another or with atoms and molecules; this is the case in an electron current at a low density of space charge. One must bear in mind, however, that the velocities of the electrons from the very beginning are not identical either in magnitude or in direction. At a high electron density their motion at first not only does not accelerate, but is even retarded; since electrostatic interaction occurs between them, an exchange of velocities is possible and, as a consequence of this, fluctuations of density. These latter cause fluctuations of voltage and of current in the external circuit; since their cause in the present case is the thermal velocities of the electrons, we must regard them as a thermal effect. Its magnitude can be determined by the Nyquist–Johnson formula:
\[ \overline{E_T^2}=4kT\int_0^\infty R(f)\,df. \tag{23} \]
These fluctuations arise in the internal circuit of the tube; naturally, by the resistance \(R\) here one should understand the internal resistance of the tube \(R_i\). However, what should be taken as its temperature \(T\)? The temperature of the electron cloud is usually taken
equal to the cathode temperature.* This gives grounds for thinking that the temperature of the resistance \(R_i\) should also be considered equal to the cathode temperature. Llewellyn substantiates all these conclusions by considering a certain thought experiment. Let us imagine a tube with an oxide cathode, in whose anode circuit an ohmic resistance \(R_a = R_i\) is included. The anode (and, if necessary, also the grid) voltage is adjusted so that the current is completely limited by space charge, so that there is no shot effect. The tube and the resistance \(R_a\) are placed together in a furnace, where a temperature is maintained sufficient to make the cathode emit; the anode and grid are assumed to emit neither thermionically nor secondarily (a larger work function than that of the cathode). Under these conditions there can be no fluctuations in the tube circuit except thermal ones. In the resistance \(R_a\) fluctuations of electricity occur, producing a current in the whole circuit; half the power developed by it is expended in the internal resistance of the tube \(R_i\). Since the entire circuit is under conditions of thermal equilibrium, on the basis of the second law of thermodynamics we are entitled to assert that the electromotive forces arising inside the tube must develop the same power in the external resistance \(R_a\). The latter, consequently, must be expressed by formula (23), where \(T\) in this case is the temperature of all the electrodes and of the walls of the tube. In Llewellyn’s opinion the temperatures of the anode, grid, and walls play no role whatever; cooling them with water would not change anything in the operation of the tube and of the external resistance. Therefore the temperature entering into formula (23) is the temperature of the cathode.
It is difficult to regard this reasoning as very convincing, especially in the part concerning the influence of the anode temperature. Pearson’s experiments (see below) do indeed call their correctness into question.
Starting from the result he found, Llewellyn further derives a formula expressing the voltage fluctuations on the tube due to the thermal effect in the real case when the temperature of \(R_a\) is not equal to the temperature of the electron stream in the tube. Let the temperature of the latter be \(T_i\), the external temperature \(T_a\); let us denote the fluctuating e.m.f. inside the tube by \(E_i\), the e.m.f. of the external circuit by \(E_a\); finally, the impedance of the external circuit by \(Z_a\) and the internal one by \(Z_i\). The current strength is
\[ J = \frac{E_i + E_a}{Z_i + Z_a}, \]
* When an electron leaves a metal its energy decreases by a definite amount equal to the work function \(\varphi\). Those electrons whose energy was less than \(\varphi\) do not go outside at all; those which had energy \(\varphi\) emerge with zero energy; those which had energy \(\varphi + \varepsilon\) emerge with energy \(\varepsilon\), and so on. The number of electrons of each energy decreases, but the distribution, i.e., the temperature, remains unchanged. This is the same method of reasoning by which it is proved that in different layers of an ideal gas situated in a gravitational field, the temperature is the same.
voltage across the tube
\[ v_T=E_i-Z_iJ=\frac{E_iZ_a-E_aZ_i}{Z_i+Z_a}. \]
The mean square of this voltage is
\[ \overline{v_T^2}=\frac{\overline{E_i^2}Z_a^2+\overline{E_a^2}Z_i^2}{(Z_i+Z_a)^2}, \]
since \(\overline{E_i\cdot E_a}=0\) (see the first part of the review, pp. 814 and 849, formula). In the frequency interval \(df\) we have the following expressions for \(\overline{E_i^2}\) and \(\overline{E_a^2}\):
\[ \overline{E_i^2}(f)\,df=4kT_iR_i\,df \]
and
\[ \overline{E_a^2}(f)\,df=4kT_aR_a\,df. \]
Substituting this into the formula for \(\overline{v_T^2}\), we find:
\[ \overline{v_T^2}(f)\,df = 4k\,\frac{T_iR_iZ_a^2+T_aR_aZ_i^2}{(Z_i+Z_a)^2}\,df; \tag{24} \]
the entire effect is expressed by the integral:
\[ \overline{v_T^2} = 4k\int_0^\infty \frac{T_iR_iZ_a^2+T_aR_aZ_i^2}{(Z_i+Z_a)^2}\,df. \tag{25} \]
If the thermal effect is observed with the aid of an amplifier whose amplification factor is \(\mu(f)\), then at the output we obtain the voltage:
\[ V_T = 4k\int_0^\infty \frac{T_iR_iZ_a^2+T_aR_aZ_i^2}{(Z_i+Z_a)^2}\,\mu^2(f)\,df. \tag{24} \]
These formulas express the thermal effect arising both from the external and from the internal part of the circuit.
The actual fluctuations observed in a tube even with a purely electronic current must be the sum of both effects: the shot effect and the thermal effect. Lævelain made an attempt to verify the results of his theory by a series of experiments; the latter, however, did not yield a clear result. The magnitude of the fluctuations \((\overline{v_s^2}+\overline{v_T^2})\) was calculated for different filament heats; then the same magnitude was measured experimentally. The course of the experimental curve approximately corresponded to the theoretical one; however, the fluctuations found experimentally proved to be smaller than the calculated ones. Apparently, the cause of the discrepancy lies in an insufficiently accurate determination of the quantity \(\frac{di}{dJ}\).
More detailed, although likewise far from sufficiently clear, data are contained in Pearson’s work.\(^{43}\) This author carried out investigations on tubes with oxide and thoriated cathodes; he abandoned tungsten cathodes, since at the high space-charge density required in these experiments, to eliminate the action of positive ions emitted by tungsten,
was impossible. In the tubes selected for the work, a high filament temperature and a low anode voltage were used, owing to which the thermal effect attained considerable development. To what extent the other sources of fluctuations were eliminated and how this was achieved, Pearson does not report. Therefore a complete assessment of his work in this respect is difficult. The experimental method consisted in measuring the noise at constant \(T_i\) (filament), \(R_a\), and \(T_a\), and variable \(R_i\) (varied by changing \(V_a\)). It turned out that the tube noise is less than is required by formula (25), for all values of \(R_i\). However, the course of the experimental points qualitatively corresponds to the course of the theoretical curve; complete agreement between them can be obtained if in formula (25) one substitutes for \(T_i\) not the temperature of the cathode, but another, lower one. Thus, for an oxide cathode (in two cases) it turned out that the effective value \(T_i = 650^\circ\mathrm{K}\), whereas the actual temperature was \(1100^\circ\mathrm{K}\); for a thoriated cathode it was, respectively, 1200 and \(1850^\circ\mathrm{K}\). The reason for such a discrepancy has not been clarified; Johnson and Llewellyn, who discussed the result of Pearson’s work, put forward two hypotheses, between which it is not yet possible to choose. In any case it should be noted that in both experimental investigations published up to the present time (March 1935) a smaller magnitude of the thermal effect was found than is required by formula (25).
(To be continued.)