NEW CONCEPTS OF GAS DISCHARGE DEVELOPMENT

G. V. Spivak, Moscow

INTRODUCTION

The basic idea of Townsend’s theory of discharge, as is well known, consists in the fact that, in order to maintain a self-sustained discharge or breakdown of a discharge gap at high pressure, two avalanches moving toward one another are necessary: an avalanche of electrons and an avalanche of positive ions, the latter taking, alongside the electrons, an active part in the ionization of the gas. When both avalanches begin to reinforce each other, a self-sustained discharge or breakdown takes place, i.e., a discharge whose course does not depend on an external ionizer. That precisely this is characteristic of Townsend’s theory is seen from the so-called ignition condition for the discharge, which, for the case of a field between plane electrodes, has the form:

\[ \alpha = \beta e^{(\alpha-\beta)d}, \tag{1} \]

where \(\alpha\) and \(\beta\) are coefficients characterizing ionization of the gas by electrons and by positive ions (usually \(\alpha \gg \beta\)), and \(d\) is the distance between the electrodes. The physical meaning of relation (1) is that the number of pairs of charges created by a positive ion over 1 cm of path, i.e., the numerical value of \(\beta\), must reach a certain magnitude for equation (1) to hold; otherwise no self-sustained discharge will occur. At atmospheric pressures \(\beta\) does not depend on the material of the cathode, which is precisely what led Townsend to regard such an ionic avalanche as essentially important for ig...

…of the discharge. Condition (1) can also be generalized to the case of a nonuniform field. The fact that experiments at low pressures show a dependence of \(\beta\) on the material of the cathode presents no difficulty for a theoretical interpretation; it is only necessary, alongside \(\alpha\) and \(\beta\), to introduce a new factor \(\gamma\), characterizing the knocking out of electrons when a positive ion strikes the cathode. The breakdown condition then has the form:

\[ \alpha(1+\gamma)=(\alpha\gamma+\beta)e^{(\alpha-\beta)d}. \tag{2} \]

The conception of gas ionization by positive ions was introduced by Townsend in order to explain the deviations of the observed values of the current through the discharge gap from the theory that originally took into account only ionization by electron impacts. Townsend’s theory, which explained quite well not only qualitatively but also quantitatively a whole series of regularities, could be regarded as essentially corresponding to reality; however, in recent times, in connection with the work of Rogowski, doubts have arisen concerning some of its fundamental propositions. The remarkable experimental investigations of Rogowski and his pupils,\(^{1}\) who studied the development of a discharge through a gas in time with the aid of a cathode oscillograph, which made it possible to follow how the phenomenon proceeds over the course of \(10^{-9}\) sec, showed that breakdown develops in \(10^{-7}\)—\(10^{-8}\) sec, which is entirely unexpected for Townsend’s theory. With a discharge-gap length of \(1\) cm and a field strength of \(30\ \mathrm{kV/cm}\), which at atmospheric pressure causes breakdown of this gap, the velocities of positive ions are of the order of \(10^{5}\ \mathrm{cm/sec}\), and in a time of \(10^{-8}\) sec the positive ions will practically not move from their place, will not be able to create new electrons, and consequently it is difficult to expect the emergence of the ion avalanche required by Townsend’s theory. It must be noted that the time needed for the complete development of a discharge through a gas is made up of two parts. First, of the time that elapses from the moment the voltage is applied, but no current passes through the discharge gap. This first pro-

the interval of time ends when a current arises. The magnitude of this interval is determined by statistical fluctuations: the discharge can begin only when, somewhere between the electrodes, initial ionization arises at random as a result of a radioactive or other process. Secondly—from the interval of time needed for the complete development of one or another type of discharge. The first interval, with the aid of a sufficiently strong ionizer, can be made practically equal to zero. The region of contradiction between theory and experiment is precisely the second interval of time. The breakdown time which, according to Townsend, should be expected in the present case ought to have been of the order of \(10^{-5}\) sec. Townsend’s theory in general is not beyond reproach, and one can also point to a number of phenomena that do not fit within its framework; nevertheless, until recently there could be no doubt about the principal aspect of the theory, about the reality of the coefficient \(\beta\). If, for the statistical case, the velocities of electrons and positive ions are of little significance, then for explaining such rapid ignition of the discharge they are very important. In subsequent works by Rogowski[^2] and his collaborators, who repeatedly photographed the state of the discharge at very small intervals of time, it was found that the main focus of ionization (intense glow) is located at the cathode, and not in the anode region, as should have been expected, where the avalanche of electrons, increasing according to an exponential law, would, according to Townsend, produce the greatest excitation of the gas. If one also takes into account that, according to Loeb’s data,[^3] ionization by positive ions at field gradients of \(30\ \mathrm{kV/cm}\) is too small for an independent discharge to be able to arise, then the presence of an ion avalanche within the framework of Townsend’s theory becomes doubtful. Only in fields of \(1.5\cdot 10^{6}\ \mathrm{V/cm}\) can sufficiently large values of \(\beta\) be expected. Evidently it is necessary to introduce certain essential changes into the mechanism of discharge development at atmospheric pressure in order to understand the contradictions that have arisen. Rogowski was the first to propose that the cause of the increase of current be attributed not only to ionization by positive ions

ions, but also to space charges that increase the field at the cathode, without as yet providing a detailed mechanism for this influence. According to Rogowski’s initial conception, the space charge as the cause of discharge instability nevertheless still does not give a sufficiently rapid growth of the breakdown process, and the development of a large space charge proceeds in a time much longer than the observed breakdown time. In general, in Townsend’s theory and experiments, if ionization by positive ions is real, a mixed action with the influence of the space charge should take place, which is especially manifested as one approaches the onset of a self-sustained discharge. In the two years that have elapsed since the appearance in this journal of Kurchatov’s article,^4 the question has undergone intensive further development. The present article is a review of a number of recent works devoted to the mechanism of breakdown of a gas gap and to a modification of Townsend’s theory.

Theoretical Investigations

In Schumann’s opinion,^5 the only cause capable of playing a role in so rapid a creation of breakdown conditions is positive space charges; it is not necessary to think only, as Rogowski (1926) imagined, that in Townsend’s theory positive ions must traverse the whole discharge gap, 1 cm long, for which they require \(10^{-5}\) sec, for if at some point the field has become sufficiently large, then the positive ions need traverse only very small distances for the discharge to become self-sustained. It is known directly from measurements that at a field strength of \(95\ \mathrm{kV/cm}\) a discharge gap of \(0.1\ \mathrm{mm}\) is broken down, and at \(400\ \mathrm{kV/cm}\) a gap of \(0.01\ \mathrm{mm}\). If such fields can arise at the cathode, then an insignificant displacement of the positive ions already gives breakdown. Distortions of the field of this kind can be produced by space charges arising sufficiently rapidly under the action of intense electron bombardment of the gas molecules, i.e., in the presence of sufficiently

large \(\alpha\). Measurements recently made by Pawlow\(^6\) show that the coefficient \(\alpha\) for atmospheric pressure is sufficiently large and indicate the possibility of the occurrence of such considerable fields. The distortion of a uniform field, leading to its increase, is very significant for a discharge if it occurs at the cathode, for here the appearance of additional electrons leads to a considerable increase in the avalanche; an electron arising somewhere along the path of the discharge is not multiplied to the same extent as an electron arriving from regions close to the cathode. It is also necessary to note the insufficiency of Rogowski’s analysis (1926), since, while taking into account the growth of positive space charge, it does not give values for the distorted force field, which changes continuously with the change in the density of the space charges acting upon the growth of these charges. Rogowski’s analysis does not fully take into account the role of space charge and is inapplicable for cases of high fields and large pressures, and corresponds much more to the case of weak fields and low pressures. In this respect a substantial step forward was made by Hippel and Franck\(^7\), who pointed out that, although Rogowski’s basic remark concerning the contradiction between the breakdown time and the time required by a positive ion to traverse the discharge gap remains valid, nevertheless the basic ideas of Townsend’s theory can be preserved. The paradox is resolved by the fact that, owing to large differences in the velocities of the electron and the positive ion, the electrons go to the anode practically instantaneously, while the positive ions remain as a space charge; moreover, the electrons following the first electron avalanche already move in a field distorted by the positive space charge; in this case the coefficient \(\alpha\) must become larger and larger for each subsequent avalanche, and we have an interaction between the immobile positive space charge and the electron avalanche, mutually reinforcing one another, instead of the interaction between two avalanches of electrons and ions that we had in Townsend’s ...

Townsend theory. Further, one can imagine that in sufficiently strong fields the positive ions situated near the cathode strike it and knock out additional electrons in a quantity sufficient to satisfy the breakdown condition, which again lies within the framework of Townsend’s theory. For the case in which the distance between plane electrodes is equal to 1 cm at atmospheric pressure and the field strength at breakdown is 31.7 kV/cm and \(\alpha\) is therefore equal to 18.5, Hippel and Frank calculate what potential arises as a consequence of the positive space charge produced by the first electron avalanche. The calculation is made in order to verify how quickly a high field strength arises that could lead to breakdown. Assuming that one electron moves in a channel with cross-sectional area \(df\), we have Poisson’s equation for our case in the form:

\[ df\frac{d^2 V_1}{dx^2}=4\pi\varepsilon\alpha e^{\alpha x}, \tag{3} \]

where \(\varepsilon\) is the charge of the electron, \(x\) is the path in centimeters. Integrating (3), we obtain:

\[ V_1=\frac{4\pi\varepsilon}{\alpha df}\left(e^{\alpha x}-1\right), \tag{4} \]

where the first constant of integration is taken equal to zero, which physically means neglecting the decrease of the field near the anode, and the second constant is determined from the condition that at \(x=0\), \(V_1=0\).

Putting \(x=1\) cm in equation (4), Hippel and Frank find the increase of potential at the anode which would take place if the part of the space in front of the anode did not have a reduced field strength owing to the presence of positive space charge, decreasing the field strength in some parts of the space between the electrodes and increasing it near the cathode. The distribution of potential is approximately represented by a straight line (instead of a curve), which begins at the origin of coordinates and goes to the left of the straight line of potential distribution in a plane condenser, intersecting the ordinate axis higher than

31.7 kV/cm (Fig. 1, straight line 2). From equation (4), for \(x=1\) cm and passing to practical units, we have:

\[ V_1=\frac{10.6}{df}\ \mathrm{V/cm^2}. \tag{5} \]

The quantity \(df\) is found from the consideration that, with a photocurrent from the cathode of \(1.5\cdot 10^{-10}\) A, the delay time of the discharge from Rogowski oscillograms will be \(10^{-7}\) sec, and during this time 100 electrons will fly out from the cathode. Each is assigned an area \(df=1\ \mathrm{mm^2}\). Substituting in equation (5), we obtain the distortion:

\[ V_1=1060\ \mathrm{V}. \]

Fig. 1.

For the next batch of electrons, which have flown out in the subsequent \(10^{-8}\) sec, the field will no longer be 31.7 kV/cm, but, with the addition of the distortion obtained, 32.7 kV/cm, and \(\alpha\) will be 22.7; this will give a new additional voltage \((df=10\ \mathrm{mm^2})\).

\[ V_2=2500\ \mathrm{V}. \]

Now already \(\alpha=30.5\). These field increments are very sensitive to an increase in \(\alpha\), for \(\alpha\) enters into the exponent (straight line 3). If one more avalanche passes in a time of \(10^{-8}\) sec, then the additional voltage at the anode, and approximately also the field strength at the cathode, will be increased by 37 kV/cm, i.e. the new field strength will exceed the former by more than a factor of two, and naturally at this time breakdown is to be expected. The fact that at overvoltages, i.e. at voltages above the breakdown potential difference, breakdown, as oscillograms show, proceeds faster is understandable, for \(\alpha\) becomes considerably larger. The calculation of Crumpel and Frank indeed shows that such a short breakdown delay time can be qualitatively explained by the rapid accumulation of positive charges. Nevertheless

one may point out a number of weak spots in the theory of Hippel and Frank. From the calculations of the authors mentioned it is clear that the distribution of potential, owing to their failure to take into account the boundary condition at the anode, leads to values of the potential and of the field strength higher than in reality; for if there is an increase of the field strength at the cathode, then there must also be a decrease of the field strength at the anode. Moreover, in reality we do not have the simultaneous motion of 100 electrons during \(10^{-7}\) sec., and the electrons are emitted continuously, not in stages; thus, if the head of the first avalanche moves in the field without positive space charges, then the electrons already following it move in a distorted field. In other words, the time interval \(10^{-7}\) sec. for the first avalanche is excessively large. In addition, in Hippel and Frank the distortion of the field depends on how many electrons fly out from the cathode in 1 sec.; however, the experiments of Rogowski and Tamm\(^8\) show that, at a certain illumination of the cathode, further illumination does not lead to a decrease in the lag time of breakdown. Moreover, Loeb\(^9\) believes that velocities of \(10^8\ \text{cm/sec}\), adopted for the electrons, have been taken too large (the time from one avalanche to another). These velocities are of the order of \(10^7\ \text{cm/sec}\). Thus, in \(10^{-8}\) sec. the subsequent avalanches will not reach the anode. On the other hand, \(\alpha\) in Hippel and Frank, extrapolated from Townsend’s data for low pressures, must be much larger, which will lead to still more powerful space charges. A refinement of the calculations of Hippel and Frank was carried out by Kaptzov, who, taking sufficiently small time intervals, took into account the change of \(\alpha\) with each subsequent sufficiently small interval of time, while observing the boundary conditions at the electrodes. These more rigorous calculations also showed a rapid increase of the gradient at the cathode due to space charges, leading to a breakdown time of \(10^{-7}\) sec. Recently Shuman\(^ {10}\) also took up this problem. We shall dwell somewhat on his calculations, for they will give us the possibility of understanding the nature of the distribution of luminosity in the photographs of Rogowski\(^2\) and Torok,\(^ {11}\)

does not agree with the notions of the original Townsend theory. The first electron avalanche moves toward the anode. The electron current grows according to the law:

\[ nv=(nv)_0 e^{\alpha x}. \tag{6} \]

When the head of the avalanche has reached the anode, for the density of positive ions we have:

\[ p=\alpha(nv)_0 e^{\alpha x}\frac{d-x}{v}, \tag{7} \]

where \(v\) and \(n\) are the velocity and density of the electrons, \(p\) is the density of positive ions. The meaning of equation (7) is that during the time \(\frac{d-x}{v}\), while the head of the avalanche travels from the point \(x\) to the anode, a positive space charge will accumulate at the point \(x\). The resultant charge is given by:

\[ p-n=n_0 e^{\alpha x}(\alpha d-\alpha x-1)=\rho. \]

Hence, for the field distribution after the passage of the first electron avalanche we have:

\[ \frac{dE}{dx}=\frac{n_0}{\Delta}e^{\alpha x}(\alpha d-\alpha x-1), \tag{8} \]

where \(\alpha\) is assumed constant throughout the passage of the first avalanche. \(\Delta=1/4\pi\cdot 9\cdot 10^{11}\). Integrating equation (8), we find:

\[ E_x-E_0=\frac{n_0}{\Delta}\left[e^{\alpha x}(d-x)-d\right], \tag{9} \]

where \(E_0\) is the value of the field strength at \(x=0\), i.e. at the cathode. Having specified the potential difference between anode and cathode \(V_0-V_a=-V\), it is not difficult to find the integration constant \(E_0\). Since \(E_x=-\frac{dV}{dx}\), integrating equation (9), we find:

\[ V_0-V_x=\frac{n_0}{\Delta\alpha}\left[\left(d-x+\frac{1}{\alpha}\right)(e^{\alpha x}-1)-x(\alpha d+1)\right]+E_0x; \]

for \(x=d\):

\[ E_0=-\frac{V}{d}-\frac{\frac{n_0}{\Delta\alpha^2}\left[e^{\alpha d}-1-\alpha d(\alpha d+1)\right]}{d}. \tag{10} \]

The second term on the right-hand side of equation (10) gives the increase of the field at the cathode due to the space charges. The magnitude of this distortion, as we see from equation (10), de-

depends on \(n_0\), \(\alpha\), and \(d\), increasing as these quantities increase. From equation (9) the field strength can be determined for any point of the discharge gap during the passage of the first electron avalanche. The distribution of the field strengths and of the density of space charges after the passage of the first avalanche is shown in Fig. 2. As \(x\) increases, the field strength gradually decreases from the cathode side; just before the anode it drops sharply, then rises somewhat at the anode itself. Where the field strength is large, electrons move rapidly and intense luminosity arises. At both electrodes there appear layers where the discharge takes place. Photographs of Toepler taken at very high voltages acting for a short time have precisely this character. Under the action of subsequent avalanches, the density \(p'\) of positive space charges continues to increase in proportion to the time elapsed after the passage of the first avalanche, since the positive ions are regarded as immobile.

Fig. 2.

\[ p'=\alpha n_0 v e^{\alpha x} t', \]

where

\[ t'=t-\frac{d}{v}. \]

The field distribution is then given by:

\[ \frac{dE'}{dx}=\frac{1}{\Delta}\,\alpha n_0 v t' e^{\alpha x}. \]

Integrating, we have:

\[ E_x'-E_0'=\frac{n_0}{\Delta}vt'\left(e^{\alpha x}-1\right), \]

and the corresponding distribution of the potential is given by:

\[ V_0'-V_x'=\frac{n_0}{\alpha\Delta}vt'\left(e^{\alpha x}-\alpha x+1\right)+E_0'x. \]

G. V. Spivak

Since for \(x=d\)

\[ V_0' - V_\alpha' = -V, \]

then approximately

\[ E_0' = -\frac{V}{d}\cdot \frac{n_0}{\alpha \Delta}\cdot \frac{vt'e^{\alpha d}}{d}. \]

From equation (10) we have:

\[ E_0 = -\frac{V}{d}\cdot \frac{n_0}{\Delta x^2}\cdot \frac{e^{\alpha d}}{d}. \]

The field distortions after the first avalanche and the following ones are related to one another as \(1/\alpha vt'\), if \(vt'=d\), i.e., by the time the second avalanche arrives at the anode the field distortion will be \(\alpha d\) times greater than when the first avalanche arrived. It must be noted that the field distortion should turn out to be considerably greater, for each subsequent avalanche encounters a field strength increased as a result of the passage of the preceding avalanche, and it becomes larger and larger with the growth of the number of avalanches, as we have already seen from the arguments of Tippel and Frank.

Fig. 3.

In Fig. 3 we have the space charge and the field after the passage of the second avalanche. The field strength at the cathode has greatly increased and at first falls slowly, and then more rapidly near the anode. The discharge begins in a large thickness near the cathode and shifts toward the anode, where only at the end, when the whole breakdown gap becomes highly conducting, do large fields arise. The photographs of Rogovsky and Tamm\(^{2}\) seem to correspond quite well to this distribution of the field strength. In these calculations of the field distribution in the presence of space charges it is also necessary to bear in mind the circumstance that, at very great ionization, the resulting increase of potential at some point leads to the fact that in it the potential becomes higher than at the anode; at a certain point the field turns to zero and may become

even negative. This leads to an accumulation of electrons in those places where the field strength has sharply fallen, and the method of calculation with the usual boundary conditions is inapplicable. Also of interest are the recent considerations of Rogowski^12 on the connection between the usual breakdown condition and that which may be adopted in the presence of space charges as the cause of discharge instability. It turns out that this condition has, approximately, the same form as that obtained from Townsend’s theory. It is natural to expect that the instability leading to breakdown occurs at some definite ratio of the total positive space charge \(Q_R\), located between the electrodes, to the charge on the electrodes \(Q_e\), i.e., for breakdown it must be

\[ \frac{Q_R}{Q_e}=k. \tag{11} \]

If one assumes that the positive charge is produced during the time \(\tau\), which is proportional to the time required for the ion to traverse the distance between the electrodes:

\[ \tau=\mathrm{const}\,\frac{d}{\sqrt{E_0}}, \]

where \(E_0\) is the breakdown field strength. For \(Q_e\) we have:

\[ Q_e=\frac{E_0}{4\pi} \]

and for \(Q_R\):

\[ Q_R=N_0 e^{\alpha_0 d}\tau, \]

where \(N_0\) is the electron current from the cathode. From equation (11) we have:

\[ N_0 e^{\alpha_0 d}\tau=\frac{E_0}{4\pi}\,k' \]

or

\[ e^{\alpha_0 d}=\frac{E_0^{3/2}k'}{N_0d}. \tag{12} \]

This is a transcendental equation for determining the field strength \(E_0\) required for breakdown. Although \(N_0d\) and \(E_0\) vary, nevertheless the right-hand side of equation (12) may be set equal to a constant quantity; then we have:

\[ e^{\alpha_0 d}=\mathrm{const} \]

or

\[ \alpha_0 d=\mathrm{const}. \tag{13} \]

For each value of \(d\) we have, from the approximate relation (13), a corresponding value of the field strength \(E_1\). That the field strength determined from equation (13) differs only slightly from \(E_0\) is evident from the fact that, if the right-hand side of equation (12) changes by a factor of \(1000\), then the difference between \(E_1\) and \(E_0\) at \(d=1\ \mathrm{cm}\) is only \(4\%\). The reason for the good approximation given by expression (13) is that \(e^{\alpha d}\) increases extremely rapidly even for small changes in the field strength \(E_0\). Condition (13) is the typical Townsend–Schumann condition for a self-sustained discharge, not taking space charges into account. Expression (13) may be given the physical meaning that the electron current \((e^{\alpha d})\) must have, near the breakdown voltage, a definite magnitude. The fact, strange at first sight, that both discharge conditions coincide is explained by the circumstance that the current increases sharply only immediately before the breakdown voltage calculated without taking space charges into account, becoming sufficiently large and creating space charges, which immediately lead to breakdown. Thus the field strengths for breakdown “without space charges” and for breakdown due to “field distortion” are experimentally inseparable from one another.

In Rogowski’s opinion\({}^{18}\), space charge cannot always produce the instability that leads to breakdown. Indeed, let a current pass through the discharge gap in an undistorted field:

\[ i_1=N_0 e^{\alpha_0 d}. \]

Under the action of this current a distortion of the field is formed, which leads to another \(\alpha\) and to another value of the current:

\[ i_2=N_0 e^{\int_0^d \alpha\,dx}, \]

where

\[ i_2/i_1=e^{\int_0^d(\alpha-\alpha_0)\,dx}, \]

Representing \(\alpha-\alpha_0\) in the form of a series:

\[ \alpha-\alpha_0=\frac{d\alpha}{dE}\,dE+\frac{1}{2}\frac{d^2\alpha}{dE^2}(dE)^2+\ldots \]

on the other hand assuming that \(dE\) is nothing other than the additional field \(F\), produced by space charges, we have:

\[ \int_{0}^{d}(\alpha-\alpha_0)\,dx = \frac{d\alpha}{dE}\int_{0}^{d}F\,dx + \frac{1}{2}\cdot \frac{d^{2}\alpha}{dE^{2}}\int_{0}^{d}F^{2}\,dx . \]

But

\[ \int_{0}^{d}F\,dx=0, \]

for however the field may change, the boundary conditions remain unchanged. Then we have:

\[ i_2/i_1 = e^{\frac{1}{2}\cdot\frac{d^{2}\alpha}{dE^{2}}\int_{0}^{d}F^{2}\,dx} \tag{14} \]

Since the curvature of the curve giving the dependence of \(\frac{\alpha}{p}\) on \(\frac{E}{p}\) (\(p\) is the gas pressure) is such, it follows from equation (14) that when the curvature is positive, the space charge amplifies the subsequent electron avalanches. If the curvature is negative, then the space charge has the opposite effect, and the cause of the instability must be sought in other phenomena. Thus in helium there seems to be no region of positive curvature; but for it the increase of current due to space charges falls away. It is true that in helium the space charge strengthens the action of the positive ion avalanche, which is not taken into account in formula (14). Undoubtedly, also, besides the action of the space charge, phenomena such as the liberation of electrons from the cathode under the action of bombardment by positive ions play a substantial role in breakdown. Explaining the rapid breakdown of small discharge gaps by the presence of arising space charges encounters the difficulty that, although \(\alpha\) is large for small \(d\), \(e^{\alpha d}\) rapidly decreases with decreasing \(d\), so that it is difficult to expect the necessary distortions of the field. It is possible that the sufficiently large local fields arising somewhere in the discharge gap lead to the positive ions being set in motion, and the discharge then proceeds by the Townsend mechanism.

This point of view of Rogowski is already quite close to the opinion of Loeb,^9 who considers that electrostatic fields alone, applied to the electrodes, of the order of 30 kV/cm, are far from sufficient to impart to $\beta$ a value leading to breakdown. If, under the action of space charges, fields of $1.5 \cdot 10^5$ V/cm arise at the cathode, then only then is a noticeable ionic avalanche possible. According to Loeb, a breakdown-development time of $10^{-8}$ sec., if comparatively small velocities are assumed, can be explained by the fact that the electrons traverse a small interval, producing large space charges over this section. It is true that the mechanism of such a breakdown requires sufficiently large initial values of $\alpha$, for which there are as yet no experimental indications.

Slepian^11 also made an attempt to explain breakdown in $10^{-8}$ sec. by space charges and by a thermal ionizing wave developing as a result of heating the gas by powerful currents. Slepian sees the necessity of introducing a new ionizing agent, namely thermal ionization, in the fact that space charges alone are insufficient to explain the colossal conductivity that develops during breakdown, when the final current reaches values of thousands of amperes within a fraction of a microsecond. All the more must one also bear in mind that space charges, intensifying the field in some places, weaken it in others, which affects the conductivity. For plane electrodes the number of electrons at the point $x$, $N = e^{\alpha x}$, changes as a result of radial diffusion. On the basis of the data of Townsend and Rogowski, the radius of the beam as a function of the field strength $E$ under the action of diffusion becomes:

\[ R^2 = 8.18 \cdot 10^5\, E^{-1/2}. \tag{15} \]

Since the electron density at the point $x$ also depends on transverse diffusion, for the electron density $n$ one must take the quantity:

\[ n = \frac{3N}{2\pi R^2}, \]

where $R$ is given by equation (15).

These electrons, having moved by \(1\ \text{cm}\), will give the body \(neE\). To calculate the temperature of the air, one must know its heat capacity. Taking for the quantity of heat in \(1\ \text{cm}^3\) of air the value \(1.2\cdot 10^{-3}T\), where \(T\) is its absolute temperature, we have the relation for determining \(T\):

\[ neE = 1.2\cdot 10^{-3}T \]

or

\[ T = 7.82\cdot 10^{-15}E^{3/2} e^{\alpha x}, \tag{16} \]

where \(e\) is the charge of the electron. This relation gives the increase of temperature along the path of the ionizing electron. If the temperature reaches \(4000^\circ\), then from Saha’s equation it is seen that the path becomes strongly ionized as a result of thermal ionization. Equation (16) shows that, for a given external field (at the electrodes), the temperature rise along the gradient increases very rapidly with the coordinate. In equation (16) the space charges have not yet been taken into account in any way. If, as an estimate, one assumes that the critical gradient for breakdown is determined by the fact that at the end of the discharge gap the temperature reaches \(4000^\circ\), then from equation (16) the length of the breakdown gap can be calculated. Comparison of these data with Schumann’s data leads to breakdown distances greater than is observed experimentally. Evidently there is one more effect at work, which causes thermal ionization to occur more rapidly—namely, the influence of the space charge. The development of the discharge may also be represented as the propagation of an ionization column, the head of which consists predominantly of electrons. The large field forces arising at the head impart considerable velocities to the electrons and lead to the column, piercing the gas, moving rapidly forward.\(^{15}\) The positive ions remain immobile. The coordinate of the center of gravity of the positive ions \(\bar{x}\) is then related to the coordinate of the center of the sphere of the electron beam \(x\) by the relation:

\[ x = \bar{x} + \frac{1}{\alpha}\left(1 - e^{-\alpha x}\right). \tag{17} \]

Then, for the increase of the field \(\Delta E\) at the head of the electron avalanche, we have:

\[ \Delta E = e^{\alpha x} e \left[\frac{1}{R^{2}}-\frac{1}{(R+\Delta x)^{2}}\right], \tag{18} \]

where \(\Delta x = x-\bar{x}\), and \(R\) is taken from equation (15). With increasing \(x\), the additional field grows extraordinarily rapidly. In Fig. 4 are shown the field strengths at the head of the ionization column, drawn with the aid of relation (18), with \(E\) determined by the initial field. The steep rise of the field also corresponds to an increase of the temperature up to \(4000^\circ\) and higher.

Fig. 4.

As the length of the breakdown gap one takes that value of \(x\) from which \(E\) begins to rise steeply. Comparison of these lengths with experimental data already gives good agreement. As for the breakdown time, it is calculated as the time required for the positive ion to travel from the place where thermal ionization began to the cathode. This gives breakdown times of the order of \(10^{-7}\)—\(10^{-8}\) sec., i.e. in agreement with the observed values.

Loeb sees a weak point of Slepian’s theory in the assumption that electrons heat the gas directly by impacts [equation (18)]. Meanwhile, rapidly moving electrons to a considerable extent transfer their energy to atoms in the form of excitation and ionization of the latter, and not in the form of imparting to them thermal velocity, owing to the difference in masses. In Loeb’s opinion, the gas is heated only after breakdown has occurred, at the expense of collisions of positive ions with gas molecules. Moreover, Shah’s theorem, which underlies Slepian’s calculation, can be applied only to established processes. In the arising

discussion, Slepyan¹⁶ points out that in his calculation he used relatively small values of \(\alpha=13\) (for \(E=30\,kV/cm\)). If one takes the larger value \(\alpha=22\), then, despite the small share of energy transferred in a collision by each electron to an atom of the gas, this nevertheless gives a significant value in connection with the increase in the number of electrons. Further, in Slepyan’s opinion, although Shah’s equation, strictly speaking, applies to steady-state phenomena, nevertheless thermal ionization is present. It must nevertheless be added that Slepyan’s calculations are very qualitative and not very convincing. On the other hand, there is the opinion¹⁷ that it is unnecessary to go into the details of the breakdown mechanism and to seek which of the two points of view (the original one of Rogowski and Schumann) better corresponds to reality. It is better to content oneself with a formal, phenomenological theory, treating breakdown as a statistical phenomenon. But it is of course well known from the history of our science that, for successful progress, alongside phenomenological theory there must also be developed a theory that enters into the details of the phenomena.

Experiments

Among recent experimental works devoted to oscillographic recording of breakdown times, of great interest is the work of Stenbeck,¹⁸ who investigated the ignition of a glow discharge as a function of time by means of a Braun tube at pressures of several millimeters, with the tube filled with Ar, He, Hg, \(N_2\), and \(H_2\). The voltage applied to the tube was about 400 V. The oscillograms obtained gave, for the development time of the maximum current, in hydrogen \(1\cdot10^{-6}\), in helium \(5\cdot10^{-6}\), in nitrogen \(4\cdot10^{-6}\), and in argon \(6\cdot10^{-6}\) sec. As we see, these times are of the same order of magnitude as required by Townsend’s theory, and Stenbeck’s experiments can thus be regarded as a complete confirmation of the Townsend conceptions connected with the motion of the positive ion, especially since it is clearly seen that the breakdown time increases with increasing atomic weight. The apparent discrepancy with Rogowski’s experimental data, who, as we know, observed times

breakdown time of the order of \(10^{-8}\) sec., may be interpreted in such a way that, when the pressure is decreased, the breakdown voltage also decreases; moreover, if \(d\) does not change, then \(E/p\) changes hardly at all. Further, since \(\alpha/p=f(E/p)\), then as the pressure \(p\) decreases \(\alpha\) will decrease, and consequently it is natural to expect an increase in the breakdown time. In fact, for relatively small \(\alpha\) the space charges are small, and the first positive ions, moving in a weakly distorted field, must, by striking the cathode, create new electrons. Then the breakdown time should be of the order of the transit time of the positive ions across the discharge gap. Rogowski’s experiments were repeated recently by Krug,^19 who basically confirmed the results of the former, except that in some cases breakdowns occur in jumps, which Hippel and Frank interpreted as a phase of anomalous cathode fall during the development of the discharge. Krug’s oscillograms did not reveal such steps. Krug sees their presence in Rogowski’s work as due to the influence of oscillatory processes in the measuring circuit. As an essential achievement one may regard the fact that time intervals of \(10^{-9}\) sec. are spread out by Krug sufficiently widely along the abscissa axis. The temporal course of the discharge at atmospheric pressure was investigated by Gamos,^20 using a Kerr condenser with nitrobenzene placed between two nicols. The condenser transmitted the glow of the discharge gap during definite time intervals (\(10^{-9}\) sec.). By repeating a definite stage of the discharge many times, it was possible to record on a photographic plate the propagation of the glow with time, and also to study the spectrum of the glow at different moments of time. The delay of breakdown was likewise found to be of the order of \(10^{-8}\) sec. It is interesting to note that already \(2\cdot10^{-9}\) sec. after the application of the voltage, a glow appears at the anode with strongly broadened lines (Stark effect), which indicates the presence of strong fields and thus is experimental proof of the existence of space charges. The spectrum in the final stage of the discharge is already different. The change

the material of the electrodes had no influence on the glow spectrum. The most recent experimental work of Fimann²¹ in Aachen led to the conclusion that the earlier measurements of Rogowski and his pupils were connected with an inaccuracy in the determination of the breakdown voltage, which was greater than the static breakdown voltage by up to 20%. Thus there was an overvoltage which, to be sure, has little effect on the velocity of the positive zone, but the rates of growth of the positive space charge, owing to the increase of $\alpha$, become greater.

Conclusion

At the present time it may already be considered sufficiently established that the occurrence of breakdown at atmospheric pressure and with large discharge gaps cannot be explained without connection with space charges. The rise of the current begins with a stationary space charge, whose interaction with the avalanche of electrons leads ultimately to sufficiently strong fields and to the possible initiation of an ionic avalanche; moreover, the positive ions do not have to traverse the entire discharge gap—it is sufficient if they ionize near the place of their origin. In addition to the action of the space charge, a number of secondary phenomena begin to act—the emission of electrons from the cathode, excitation of the gas, high temperature, etc.

Oscillographic recording of the discharge at reduced pressure, where the field is little distorted by space charges, showed that the delay time of ignition does not contradict the usual concepts of Townsend’s theory.

Literature

1. Kurchatov, UFI 10, 685, 1929. Literature up to 1928 is given there as well.
2. Rogowski und Tamm, Arch. f. Elektr. 20, 625, 1928.
3. Loeb, Journ. Frankl. Inst. 205, 305, 1928.
4. Rogowski, Arch. f. Elektr. 16, 496, 1926.
5. Schumann, Z. techn. Phys. 11; 58, 1930.
6. Paavola, Arch. f. Elektr. 22, 443, 1929.

References

1. Hippel and Franck, Z. techn. Phys. 57, 636, 1929.
2. Rogowski and Tamm, Arch. f. Elektr. 20, 107, 1928.
3. Loeb, Journ. Frankl. Inst. 210, 15, 1930.
4. Schumann, Z. techn. Phys. 11, 131, 1930.
5. Torok, Journ. Am. Inst. El. Eng. 47, 177, 1928.
6. Rogowski, Z. techn. Phys. 60, 776, 1930.
7. Rogowski, Arch. f. Elektr. 23, 569, 1930; 24, 679, 1930.
8. Slepian, Electrical World. 91, 761, 1923; ETZ 49, 1162, 1928.
9. Holm, Arch. f. Elektr. 18, 80, 1927.
10. Slepian, Journ. Frankl. Inst. 210, 473, 1930.
11. Shaposhnikov, Vestnik elektrotekhniki No. 7/8, 100, 1930; No. 1, 13, 1931.
12. Steenbeck, Z. f. techn. Phys. 10, 480, 1929.
13. Krug, Z. techn. Phys. 11, 153, 1930.
14. Hamos, Ann. d. Phys. 7, 837, 1930.
15. Viehmann, Arch. f. Elektr. 25, 253, 1931.