ELECTRICAL BREAKDOWN OF GASES.

(A Critique of the Ionization Theory of Breakdown.)

I. Kurchatov, Leningrad.

The ionization theory of discharge in gases, in many respects, may be regarded as a model of the mathematical treatment of physical processes.

Townsend and then Schumann succeeded, not only qualitatively but also quantitatively, in describing the entire variety of discharge phenomena both in a uniform and in a nonuniform electric field.

The numerous experimental data invariably fitted into the framework of conceptions about ionization by positive and negative ions, which form the basis of the theory. However, despite the full perfection of this grandiose construction, it is now already quite clear that Townsend’s theory cannot describe the breakdown of a gas at atmospheric pressure and normal temperatures, and also, apparently, is incorrect for other regions of pressure. The contradictions between the theory and experiment were revealed by Rogowski, to whose exposition of the work this article is devoted.

Townsend’s Theory.

Its principal content must be taken to be the requirements of ionization of the gas by positive ions. When the gas is ionized only by electrons, the current increases strongly, but breakdown cannot occur; only in the case when positive ions ...

ions will cause the appearance of new carriers of electricity when the negative and positive avalanches begin to reinforce one another; the current reaches values characteristic of breakdown.

“Let us suppose that a positive ion forms \(\beta\), and an electron \(\alpha\), carriers of electricity in one centimeter of its path; if \(u\) and \(v\) are respectively the mobilities of both carriers, then for \(\dfrac{dn_+}{dt}\) and \(\dfrac{dn_-}{dt}\) we have the following relations:

\[ \frac{dn_-}{dt}=-\frac{d}{dt}(n-v)+\alpha(n\cdot v)+\beta(n+u) \tag{1} \]

\[ \frac{dn_+}{dt}=-\frac{d}{dt}(n+v)+\alpha(n\cdot u)+\beta(n+v). \tag{2} \]

Noting that

\[ n-v+n+u=i, \]

we obtain, for the stationary process, the equation:

\[ \frac{d}{d}(n-v)-(\alpha-\beta)(n-v)-\beta i=0, \]

whose integral, in the general case, when \(\alpha\) and \(\beta\) are functions of the field gradient, gives:

\[ n-v=e^{\int_{0}^{x}(\alpha-\beta)\,dx} \left\{ C_1+i\int_{0}^{x}\beta e^{\int_{0}^{x}(\alpha-\beta)\,dx}\,dx \right\} \]

\[ n+u=e^{\int_{0}^{x}(\alpha-\beta)\,dx} \left\{ C_2-i\int_{0}^{x}\alpha e^{\int_{0}^{x}(\alpha-\beta)\,dx}\,dx \right\} \]

Taking as boundary conditions \(n-v=0\) at \(x=0\) (cathode), \(n-v=i\) at \(x=d\), \(n+u=0\) at \(x=d\), and \(n+u=i\) at \(x=0\), we determine the constants \(C_1\) and \(C_2\), which turn out to be respectively equal to \(1\) and \(i\).

Using the values of \(C_1\) and \(C_2\), we obtain the relation

\[ \int\limits_{0}^{a} d e^{\int\limits_{0}^{x}(\beta-\alpha)^1\,dx}\,dx=1 \tag{3} \]

as the condition for a stationary current and, at the same time, for breakdown. In equation (3) there is implicitly specified a relation between the gradient of the electric field and the length of the discharge space. Schumann assumed \(\alpha=k\beta\) and, using empirical dependences of \(\alpha\) on the field gradient, explained a number of regularities of the breakdown of gases in fields of various forms. The construction set forth, however, does not touch upon the question of the time during which the process required by the equations written above can become established.

Fig. 1.

The question of this time is precisely that region of the contradictions between theory and experiment with which, along with others, Rogowski has been especially much occupied in recent years.

Investigations of gas breakdown under voltage impulses.

The first systematic experiments on the breakdown of air with a very short duration of applied voltage were described by Altmesser, who used damped oscillations of high frequency; his results are given in Fig. 1. Along the ordinate axis is given, in percent of the static voltage, the magnitude of the difference between the static and the short-time breakdown potential; along the abscissa axis is the length of the spark gap; on the curves are marked the frequencies of the oscillations used. The experiments were carried out with spheres of diameter \(2.5\ \mathrm{cm}\). At a frequency of \(0.5\cdot 10^6\), the static and short-time potentials are practically equal to each other

one another; only at \(2.5 \cdot 10^6\) does a difference between the two values become noticeable, reaching \(60\%\) at \(0.5 \cdot 10^7\). The field between spheres of \(2.5\ \mathrm{cm}\), at the distances used in the investigation, may be regarded as more or less uniform. In a nonuniform field with spheres of \(0.25\ \mathrm{cm}\), the divergence between the static and the short-duration breakdown potentials becomes noticeable already at \(0.5 \cdot 10^6\), reaching \(100\%\) at \(0.5 \cdot 10^8\). The data were obtained for the electrodes of the discharger illuminated by ultraviolet rays; with “dark” electrodes the divergence is considerably greater.

Leont’eva, using the same method of damped oscillations, investigated the breakdown of air at very small distances between the electrodes; the results of these experiments are in good agreement with Al’tgemisofer’s data. The works of Peek, who worked with pulses of very short duration, must be regarded as very authoritative. With spheres of \(12\ \mathrm{cm}\) radius, for sparks up to \(10\ \mathrm{cm}\) long, Peek, with pulses of duration up to \(10\) sec., found no difference between the breakdown potentials and the static ones. A number of other investigators dealt with these questions with approximately the same experimental results; however, all the material listed cannot be considered convincing, since the voltage in the circuits was not measured directly, but was calculated from the circuit data. Especially valuable, in view of what has been said, are the investigations of breakdown under voltage pulses carried out by Rogowski, Flegler, and Tamm with the cathode oscillograph of the electrotechnical laboratory of the Aachen Polytechnic Institute. In Fig. 2 is shown one of the photographs representing the breakdown of air at atmospheric pressure; in Fig. 3—a oscillogram of a spark discharge in air under the same

Fig. 2.

conditions; for each oscillogram the scale of the abscissae (time) is given below.

Tamm took oscillograms of discharges with electrodes of various materials, at different temperatures and different gas pressures. The first two factors do not noticeably affect the character of the oscillogram. At low pressures (1 mm) in air, as well as in hydrogen at atmospheric pressure, the oscillograms clearly show two stages of the discharge greatly extended in time. This latter circumstance is extremely important, and we shall return to it below. Analysis of the data shows that at

Fig. 3.

static voltages breakdown occurs after \(\sim 10^{-7}\) seconds. After the voltage is applied, increasing the potential by 30% relative to the static value reduces the “delay” to \(10^{-1}\) seconds. It should be noted that the current at breakdown rises very sharply, by a jump, a rise that is very difficult to explain from Townsend’s point of view.

Calculation of the breakdown time according to Townsend’s theory.

Remaining faithful to the theory, we must assume that, \(10^{-7}\) seconds after the voltage is switched on, the avalanche of positive ions already begins to act upon the electron avalanche. Starting from the data for the mobility of positive ions and constructing the picture of the motion of the electron

and ionic avalanches, Rogowski showed that this time must have an entirely different order; under the most extreme assumptions it cannot be less than \(10^{-5}\) seconds.

The velocity of ions (electrons) with charge \(e\) in a field with gradient \(E\) is usually represented by the formula:

\[ u=E\frac{e}{m}\cdot\frac{l}{v_m}, \tag{A} \]

where \(m\) is the mass of the ion, \(v_m\) is the mean velocity of thermal motion at the given temperature, and \(l\) is the mean free path. Formula \((A)\) is valid only for those cases in which the velocity of thermal motion is considerably greater than the velocity acquired by the ion in the electric field. At gradients close to breakdown this condition is not fulfilled. For the velocity of a positive nitrogen ion in a field with a gradient of \(30\,000\ \mathrm{V/cm}\) we obtain the value \(2.2\cdot10^{6}\ \mathrm{cm/sec}\), whereas the velocity of thermal motion for it will be only \(4.6\cdot10^{4}\ \mathrm{cm/sec}\).

In large fields at atmospheric pressure it is much more correct to use the formula

\[ u=\sqrt{E\frac{e}{m}\cdot\frac{2}{\pi}}\ \mathrm{cm/sec}, \tag{A'} \]

derived under the assumption that at each collision the ion completely loses its energy. If the calculation is carried out by this formula, then in a field with a gradient of \(30\,000\ \mathrm{V/cm}\) the electron and the ion will move with velocities respectively of \(4.3\cdot10^{7}\ \mathrm{cm/sec}\) and \(0.96\cdot10^{6}\ \mathrm{cm/sec}\). It cannot be considered that the assumption underlying formula \((A')\) is admissible; however, as analysis of formulas in which elastic and inelastic impacts of molecules are taken into account shows, formula \((A')\) gives sufficiently good results under conditions of strongly developed ionization.

There are no indications in the literature of direct measurements of the velocities of ions and electrons in large fields close to breakdown. If the mobility of an ion in large fields is taken to be the same as in small fields, then the velocity of an ion in a field of \(30\,000\ \mathrm{V/cm}\) turns out to be \(42\,000\ \mathrm{cm/sec}\). It may, however, be stated with confidence that the magnitude

in this way the velocity calculated will be too small. At high pressures and small fields the positive ion must be regarded as a certain complex of molecules; in high fields this complex should break up, and at the same time the velocity of motion should increase. On the other hand, in large fields the velocity grows proportionally not to the first power of the field gradient, but only to its half, and the value obtained above for the velocity of the positive ion, \(1\cdot 10^5\ \mathrm{cm/sec}\), must be considered the upper limit of the truly possible velocity.

The magnitude of the velocity of the positive ion enables Rogowski to calculate the time necessary for the electron avalanche to begin to change under the action of the avalanche of positive ions. Starting from the general equations, Rogowski found the value of the numbers of electrons and positive ions as functions of \(x\) (the distance between the electrodes), \(X\)—the field gradient, and the time of application of the voltage. Three intervals of time were investigated separately.

1) \(0 \leq r=\dfrac{d}{v}\). During this interval of time the positive ions may be considered immobile and \(\beta=0\). The equation for the electrons will be:

\[ \frac{1}{v}\frac{\partial(n_-v)}{\partial t} = -\frac{\partial}{\partial x}(n\cdot v)+\alpha(n\cdot v), \]

and after integration

\[ (n_-v)=e^{\alpha x}f(x-vt). \]

The function \(f(x-vt)\) is determined from the following conditions: 1) at \(t=0\) and at \(x=0\), \(n_-v=0\); 2) for any \(t\), at \(x=0\), \(n\cdot v=(n\cdot v)_0\). The function \(f(x-vt)\) thus represents a rectangular wave (Fig. 4), the front of which reaches the anode in the time \(\tau=\dfrac{d}{v}\). For positive ions, the equation, under the same assumptions as above, may be written in the following form:

Fig. 4.

\[ \frac{1}{u}\frac{\partial(n_{+}u)}{\partial t}=(n_{-}v)\alpha, \]

which, after integration, gives:

\[ n_{+}=\alpha(n_{-}v)_0 e^{\alpha x} t_1, \]

where \(t_1\) is the time elapsed after the electron avalanche from the cathode has passed together with the abscissa \(x\). At the moment

\[ t=\tau=\frac{d}{v} \]

the number of positive ions as a function of \(x\) will be expressed by the formula:

\[ n_{+}=\alpha(n_{-}v)_0 e^{\alpha x}\frac{d-x}{v}. \]

The admissibility of the simplifications made above obviously depends on the value of the quantity \(\eta\), which is given by the equality:

\[ \eta=\frac{(n_{+}u)\beta}{(n_{-}v)\alpha}. \]

By the end of this first interval, \(\eta\) is equal to:

\[ \eta=\frac{u}{v}\frac{\beta}{\alpha}(d-x)\alpha. \]

It is not difficult, by calculation, to verify that \(\eta\), depending on \(x\), varies between \(\frac{1}{100}\) and \(\frac{1}{10000}\); our solution, therefore, remains quite legitimate for these time intervals.

2) \(\frac{d}{v}<t<\frac{d}{v}+\frac{d}{u}\). If for this second interval of time we retain our assumptions and consider the positive ions immobile and \(\beta=0\), then the electron avalanche will undergo no changes; the same positive ion cloud will remain immobile, but will grow according to the equation

\[ n_{+}=\alpha(n_{-}v)_0 e^{\alpha x}\left\{t' + \frac{d-x}{v}\right\}. \]

If, however, we assume that only \(\beta\) is equal to zero, then the electron avalanche will not change in this case either, while the ion cloud will move toward the cathode. Let us turn to

equations for positive ions; in the present case they will have the following form:

\[ \frac{1}{u}\frac{\partial(n_{+}u)}{\partial t'} = -\frac{\partial(n_{+}u)}{\partial x} +\alpha(n-v)_0 e^{\alpha x}. \]

Integration gives:

\[ (n_{+}u)=F(x+ut')-(n-v)_0e^{\alpha x} \]

and leads to the function \(F(x+ut')\), which must satisfy the following conditions:

1) at the instant \(t'=0\) \((t=\tau)\), \(n_{+}\) is equal to:

\[ n_{+}=\alpha(n-v)_0 e^{\alpha x}\frac{d-x}{v}; \]

2) for all values of \(t'\), at \(x=d\), \(n_{+}=0\).

We thus obtain for \(F(z)=F(x+ut')\), for \(z\le d\),

\[ F(z)=(n-v)_0 e^{\alpha z}\left\{1+\alpha(d-z)\frac{u}{v}\right\}, \]

for \(z\ge d\),

\[ F(z)=(n-v)_0 e^{\alpha d}. \]

Neglecting the formation of positive ions in the interval between \(t=0\) and \(t=\tau\) [omitting the term \(\alpha(d-z)\frac{u}{v}\), which is smaller than unity], we obtain \((n_{+}u)\) as a wave \(F(x+ut')\) moving from the cathode with velocity \(u\), and the space function \((n-v)_0e^{\alpha x}\) (Fig. 5). Figs. 6a and 6b show the displacement of the positive volume charge. Analysis of \(\eta_{+}\) shows that, toward the end of the second interval, this quantity reaches large values close to unity, and here

Fig. 5.

it is already possible for lowering by positive ions. In the following, third interval of time, breakdown occurs.

3) \(\dfrac{d}{v}+\dfrac{d}{u}\leq t\leq \dfrac{2d}{v}+\dfrac{d}{u}\). Taking \(\beta\) to be nonzero, analogously to the preceding case it is easy to show that the number of electrons—

Fig. 6a

Fig. 6b

Fig. 6c

Fig. 6d

on the anode by the end of this interval of time must be given by the equation:

\[ (n-\nu)=(n-\nu)_0 e^{\alpha d}\left\{1+\frac{\beta}{\alpha}e^{\alpha d}-\frac{\beta}{\alpha}-\beta x\right\}, \]

and since \(\alpha\) is always greater than \(\beta\), in this interval of time there will occur a strong increase of the electronic

avalanches—here the interaction of avalanches begins, leading to breakdown. Assuming that breakdown occurs in a time equal to \(\frac{d}{u}\), for a distance between the electrodes of \(1\) cm we obtain, for the time of avalanche formation and breakdown, a value, using the above values of the ion velocity, of \(1 \cdot 10^{-5}\) cm/sec for a field with a gradient of \(30\,000\) V/cm.

Comparison of Townsend’s Theory with Experiment

The data of the theory are thus in sharp contradiction with experiment; experiment shows that breakdown at normal static voltages follows in \(\sim 10^{-7}\) seconds after the application of the voltage; the theory requires a time hundreds and thousands of times greater for sufficient development of the ionization processes. The discrepancy arises because of the low mobility of positive ions. One might think—as Rogowski himself indicates—that the value adopted for the mobility of the ions requires some correction. As the experiments of Dempster show, for the mean free path of an ion one must take larger values than is usually done. According to the experiments of this investigator it turned out that the mobility of hydrogen ions in helium is 9 times greater than that calculated by the formulas of the kinetic theory of gases.

It is not difficult, however, to see that, even if we allow the same effect for oxygen and nitrogen ions, we shall not appreciably change the upper limit of their velocities, since the latter increases in proportion to the square root of the mean free path. In order to explain the discrepancy between experiment and theory by such an increase, it would be necessary to assume that the mean free path of the ion is \(10^4\) and even \(10^6\) times greater than the molecular one, which appears completely impossible, however imperfect we may consider the present experimental material to be.

Analysis of oscillograms also leads to other difficulties. We have already pointed out the step-like character of the discharge, very clearly noticeable in the case of hydrogen and air at relatively low pressures. According to Townsend we

must allow ionization throughout the whole spark gap, which leads to a number of impossible conclusions. From the oscillograms it is easy to calculate that the quantities of electricity flowing in the circuit for one of the stages of the discharge are equal to \(2 \cdot 10^{-6}\ \mathrm{C}\); a positive charge of approximately the same magnitude must remain in the space between the electrodes. The volume positive charge will be 200 times greater than the charges on the electrodes in a uniform field, and by virtue of this this volume charge must cause a “monstrous,” as Rogovskii writes, distortion of the field. In a capacitor with a potential difference \(v\) on the plates, a distance \(d\) between them, and a uniformly distributed positive volume charge, the gradient of the electric field as a function of distance is expressed by the formula:

\[ E=\frac{v}{d}-2\pi qd\left\{1-\frac{2x}{d}\right\}=4\pi\sigma-2\pi q\left\{1-\frac{2x}{d}\right\}, \]

where \(\sigma\) is the surface charge of the capacitor plates in the case of a uniform distribution of potential, and \(q\) is the total volume charge in the capacitor. If we assume that \(q\) is only 2 times greater than \(\sigma\), then at \(x=d\) the field gradient will already be 2 times greater than the average; at \(q=200\sigma\) the field strength at the point \(x=d\) will be 100 times greater than the average, while for \(x<\frac{d}{2}\) the field will reverse its direction, so as to become again, at the other electrode at \(x=0\), 10 times greater than the average. Under such conditions it is quite impossible to imagine constancy of current and voltage in the spark, if one recalls the sensitivity of ionization phenomena to changes in the field. Ionization throughout the whole spark gap thus becomes doubtful.

To clarify the mechanism of breakdown, at the end of 1928 Rogovskii carried out an investigation of the structure of the glow in a spark discharge.

The investigation was carried out photographically during breakdowns under impulse loading. Flat electrodes were placed in a glass vessel, which could be evacuated and then filled with the gas under investigation; by rotating one of the ground joints it was possible to change the distance between the electrodes.

In the vessel, on a special glass joint, tubes with a radium preparation were placed in order to excite the initial ionization of the gas.

The voltage was applied to the discharge gap from a special circuit for obtaining rectangular pulses, the duration of which varied from several units to hundreds of \(10^{-7}\) sec.

In view of the fact that the intensity of a single discharge proved too small and did not produce a noticeable blackening of the plate, for one and the same photograph from 50 to 100 breakdowns were produced under, of course, the same experimental conditions.

a  b  c

Fig. 7.

As was shown in the oscillographic investigation of breakdown, two stages must be distinguished in the discharge; the first stage of the discharge at atmospheric pressure is very short, while at very low pressure, on the contrary, its length is “infinitely great”; in the discharge space a “glow” is observed by the eye.

The photographs taken by Rogowski show that, with short applications of voltage, when it is possible to record the glow corresponding to the first stage of the discharge, even at pressures of 40 mm of mercury it has the same character as in a glow discharge; upon transition to pulses of greater duration the glow assumes the form that until now alone had been known—the form of a continuous luminous band. In photographs a, b, c and d (Fig. 7) Rogowski’s data are presented for the breakdown of air at a pressure of 40 mm and a distance between

electrodes of 3 mm with pulses of different duration. In the photographs Kruykovo’s dark space and the strongly developed negative glow of the glow discharge are clearly visible.

This work of Rogowski’s further deepens the contradictions in Townsend’s theory; the complete analogy between the structure of the glow during the breakdown of air in the region of already comparatively high pressures and the structure of the glow of a glow discharge, where the passage of current develops in a field strongly distorted by space charges, speaks directly against the conception of a spark discharge as a discharge in a uniform field.

Ionization by positive ions is insignificant in a glow discharge and does not always take place here; it occurs only in long tubes in the column of positive glow; neither this column nor the second Faraday dark space is present in Rogowski’s photographs. A continuous structure of the glow has always been very weighty evidence of ionization throughout the whole space by both carriers of electricity; according to Rogowski it arises as a result of secondary processes connected, apparently, with the (local) heating of the electrodes and the tearing out from them of positive metal ions. Even if the experiments described cannot yet be considered decisive, in any case they are of considerable interest and constitute valuable material for any new theory of the phenomena under consideration.

Conclusion.

Rogowski’s works show that there are very deep cracks in Townsend’s theory.

Rogowski does not put forward any new complete theory. In his opinion, it is certain only that breakdown begins with ionization by electrons; subsequently a number of secondary processes are possible which maintain the discharge. One may think that the positive ions, formed sufficiently close to the cathode, bombard it and tear out new electrons; one may think that they recombine with radiation,

which gives a photoelectric effect from the electrodes, etc. All these possibilities had already been foreseen by Townsend, but in their time they were set aside as not satisfying experiment. It may be that an analysis of these processes from the standpoint of the new conceptions will now lead to different conclusions than in Townsend’s time.

In conclusion, we should like, together with Rogowski, to note the cardinal role of the oscillographic study of the breakdown process, which enabled this investigator to pose anew for physics the problem of discharge in gases.

Literature

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