On the Ignition of a Gas Discharge*

V. Rogovskii, Aachen*

1. Statement of the Problem

The ignition of a gas discharge is, as its name indicates, an optical phenomenon: at first the dark discharge gap suddenly becomes filled with light. Experience shows, however, that an electrical process is associated with this optical phenomenon: the gas gap, which previously did not conduct, immediately becomes a conductor (the phenomenon of breakdown). We shall henceforth take the view that the essential process in ignition is the electrical process, not the optical one. We assume that the high electron density arising during breakdown chiefly causes the visible glow. We also pose the question of how it happens that, when a sufficiently high voltage is applied, a good conductor is obtained from an insulator.

It is usually accepted that in one limiting case (an insulator) small, non-self-sustaining currents are attributed to the substance; these, with sufficiently weak external ionization and sufficiently high voltages, flow through the gas gap to a noticeable, though still weak, degree. In the second limiting case (a conductor), extraordinarily large currents flow through the gas gap at low voltages. The properties of the gas gap in this case will be governed by the laws of the arc or glow discharge. If the arc is regarded, as is usually done, as a degenerate glow discharge, then the second limiting case may be identified with the glow discharge.

A satisfactory explanation of ignition must account both for the weak non-self-sustaining currents before the ignition process and for the properties of the glow discharge after ignition of the discharge, and must also satisfactorily describe the transition from one form of discharge to the other.

2. Townsend’s Theory

Townsend was the first successfully to develop this problem. In order that, at ordinary voltages, it should in general be possible

* On the ignition of a gas discharge, see also Kurchatov, UFN, 9, 685, 1929; Spivak, UFN, 11, 726, 1931.
* Physik. Zs.*, 33, 797, 1932. Translation by G. V. Spivak.

for ignition to occur, according to Townsend it is necessary to have at least one single electron or one ion in the field in the gas gap. Thus, we assume the presence of one initial electron that has been released from the cathode. If the voltage is sufficiently small, then this initial electron will wander in a zigzag fashion between atoms and molecules from cathode to anode, and nothing special happens thereby. If the voltage increases, then this initial electron can acquire on its free paths so much energy that, upon colliding with atoms or molecules, it destroys them and thus releases, by impact ionization, an electron from atomic bonds. The old and new electrons, continuing their path to the anode, can, at sufficiently high voltage, again release new electrons, etc. An electron avalanche is formed1. Then, as is known from Townsend, for one initial electron there arrive at the anode

\[ e^{\alpha L} \tag{1} \]

electrons (\(\alpha\) is the ionization coefficient of electrons, \(L\) is the distance between the electrodes, \(e = 2.718\ldots\)).

If not a single electron, but new electrons are emitted from the cathode uniformly at definite intervals (current density \(i_0\)), then, as a consequence of impact ionization, a constant current density is obtained,

\[ i = i_0 e^{\alpha L}. \tag{2} \]

However, this is not yet an independent current and by no means ignition.

What is important for ignition, according to Townsend, occurs only at higher voltage.

If the voltage increases further, then the positive ions that arise under impact ionization as the residues of atoms can also ionize atoms by impact. We shall take into account only the so-called surface ionization by positive ions, in which the ions release electrons from the metal of the cathode. Let one ion release, by surface ionization, \(\gamma\) electrons. Then the dark current becomes greater than in (2) and, according to Townsend, has the magnitude

\[ i = \frac{i_0 e^{\alpha L}}{1 - \gamma(e^{\alpha L} - 1)} . \tag{3} \]

Let the distance \(L\) between the electrodes be given. If the voltage \(U\) increases, the field strength \(E\) also increases, and with it the values of the ionization coefficients of electrons \(\alpha\) and positive ions \(\gamma\). Finally, at a sufficiently high voltage \(U_{\text{п}}\), the denominator in (3) becomes zero and the current \(i\) becomes infinitely large. If this occurs, then, according to Townsend, breakdown has taken place. Accordingly, Townsend’s condition for breakdown (ignition) reads:

\[ \gamma(e^{\alpha L} - 1) = 1. \tag{4} \]

This formula and the experiments lying at its basis constitute, properly speaking, the content of the Townsend theory of breakdown.

3. Is the Townsend theory sufficient?

The Townsend theory, as the works of Townsend and Schumann have shown, has yielded many good results. It therefore contains, without doubt, much that is correct. On the other hand, it is not the final word on the process of ignition of a gas discharge. It gives an account of the increase of the dark current up to ignition. It contains a special voltage \(U_p\) (see above), which must not be crossed (the denominator in (3) becomes zero). As for the second limiting case of a glow discharge after ignition has occurred, Townsend’s theory says nothing. According to the theory, everything proceeds continuously. It contains no indications whatever of a jump-like process, which, according to experiment, ignition is. The theory, one might say, brings us to the boundary of the catastrophe, but as to how the catastrophe itself proceeds and how it ends, it leaves us in uncertainty.

It is remarkable that criticism of the Townsend theory of breakdown did not come from this side. It was hardly felt that, in essence, Townsend, with his formal condition (4), gives the completion of the region of non-self-sustained discharge, while he remains completely silent about the ignition process itself. Criticism of the Townsend theory of breakdown followed only by way of comparing the theory with observations at impulse voltages. Experiment had long since shown that ignition takes place in an extremely short time, of the order of \(10^{-7}\) sec. The increase of the dark current according to the Townsend theory from small to infinitely large values proceeds, at static voltages, noticeably more slowly, and in other cases even more slowly by an order of magnitude. Only after this disagreement between theory and experiment did discussion of the question of ignition and of the Townsend theory of breakdown begin anew.

4. Ionization and space charge

(Ignition of the discharge and the glow discharge)

In connection with this discussion, the problem became topical in a direction that especially concerns us: in extending Townsend’s theory by bringing in space charge. This problem was not new, but owing to the difficulties it presented, it had made little progress. The inadequacy of the Townsend theory, revealed by oscillographic and optical methods, led to the problem acquiring great attractiveness. As a result of a number of investigations, it became evident that the description of ignition processes had earlier been unsatisfactory.

It is curious that at the same time we encountered the necessity of uniting the Townsend theory with space—

by charge in another region of the gas discharge; the explanation of phenomena in the cathode parts of the glow discharge all demanded their solution. But here too, for the solution of common difficulties, which had been pointed out more than once by authorities, work in this field led the investigator to elucidations of important propositions—though ones different from those found for the ignition of the discharge. The problem was thus worked out for two extreme cases, without arriving at a definitively satisfactory completion. Naturally, both problems—ignition as well as the glow discharge—had to merge together. And not only because both seek to establish a connection between ionization and space charge, but also because the glow discharge physically represents the completion of the ignition process. Thus the theory of ignition had to include within itself the theory of the glow discharge as well. This is what happened.^9 Alongside the theory of ignition, the theory of the glow discharge continued to develop simultaneously, at first in the form of an approximation, and then in a more rigorous calculation.^10 Part of what crystallized in this process will be briefly described by me below.

5. Idealization for the Approximation

Let us first assume that there is an ideal voltage source. It has zero resistance and is infinitely powerful.

The voltage is applied directly to the electrodes of the gas gap. Whatever may occur in the discharge gap, let the voltage at the electrodes remain unchanged. We shall then be able easily to reduce phenomena with a voltage source of finite power to this ideal case. Owing to the high velocity of the electrons and the low velocity of the positive ions, only the positive space charge need be taken into account. The magnitude of the latter depends, of course, on the magnitude and distribution of the ionization currents. In our first approximation we wish to abstract from this close connection between space charge and ionization phenomena. Therefore we allow the space charge all possible values and consider, in this way, the action of the whole scale of space charges on ionization phenomena.

At first the positive space charge changes the field in the discharge gap, as is known, in such a way that the field at the cathode is strengthened, while at the anode it is weakened. The field in doing so may have the most varied distribution. In our idealization we shall choose the limiting case: the field grows in jumps on the cathode side and falls to zero on the anode side. This is true for the case where the space charge is distributed not through the volume, but over the surface. Under this far-reaching idealization, the space charge leads to step-like ...

distortions of the field. For the region of propagation of the field, i.e. for the region in which there actually is a field, the mechanical arrangement of the electrodes has a secondary significance. The mechanical distance is only an upper limit of the possible regions of propagation of the field. In this idealization, the place and magnitude of the volume (surface) charge determine the region of propagation of the field. If \(U\) is the voltage, then all regions in which there is a field satisfy the condition:

\[ U \equiv E_0L_0 \equiv E_1L_1 \equiv E_2L_2 \equiv \ldots \tag{5} \]

If the voltage \(U\) is plotted as the ordinate, and \(L\) as the abscissa, then the totality of all regions of propagation of the field is represented by the straight line \(PQRS\) (Fig. 1), passing at a distance \(U\) from the axis of abscissae.

Fig. 1. Voltage and the region of propagation of the field.

6. Step-like distortion of the field and ionization

The question of the influence of space charge on ionization will now be replaced by the question: how does ionization change with distortion of the field at constant voltage.

a) Ionization by electrons

The region of propagation of the field is at first sufficiently large (point \(P\) in Fig. 1). At a constant voltage \(U\) we have a weak average field strength. The ionization coefficient of the electrons \(\alpha\) falls exponentially with the field strength \(E\):

\[ \alpha = c_1 e^{-\frac{c_2}{E}};\quad c_1 \text{ and } c_2 \text{ are constants} \tag{5a} \]

to zero. The product \(\alpha L\), entering into equation (1), becomes extremely small. The number of newly formed \((e^{\alpha L}-1)\) electrons and ions also falls to zero with increasing \(L\).

Let the field be contracted (point \(Q\), Fig. 1), which may occur upon displacement of the electrodes or as a result of space charge. The region of propagation of the field \(L\) then decreases. The average value of the field strength \(E\) increases and, with it, at least at first, the electron ionization coefficient \(\alpha\) increases according to the exponential law [equation (5)]. The number of electrons and positive ions formed, \((e^{\alpha L}-1)\), increases very strongly. With suitable values of \(U\) and \(L\), it may increase to tens of thousands, millions, and hundreds of millions.

We now pass directly to the limiting case, when the field is concentrated in a small region at high values near the cathode (point \(S\), Fig. 1). It is noticeable that for the forma-

of an electron avalanche, since for the formation of a full avalanche not only the length of the path is important, but also the entire free path. If \(E\) is very large, then \(\alpha\) remains finite [expression (5)]; \(\alpha\) cannot exceed the value \(c_1\). If \(L\) approaches zero, then \(\alpha L\) and \((e^{\alpha L}-1)\) likewise approach it. This means that the number of newly formed electrons and ions will again be small. The contraction of the field leads to an interesting result in the formation of electrons and positive ions, which we must note.

If, at constant voltage \(U\), the region of propagation of the field \(L\), going from large values, becomes small, then the number of electrons and ions \((e^{\alpha L}-1)\) arising from one initial electron increases, beginning from arbitrarily small values for a large region of propagation of the field to large values as the field is contracted. This number of newly formed charge carriers then reaches a maximum and, finally, with a further decrease of \(L\), again falls to zero (Fig. 2).

Fig. 2. Number of secondary electrons and ions formed by one initial electron as a function of \(L\). The voltage \(U\) is constant. Approximate form of the dependence.

b) Ionization by positive ions

How do matters stand with positive ionization, especially with the ionization coefficient of positive ions, when the distortion of the field is appreciable?

To the quantity \(\gamma\) one must assign a value depending on the energy of the striking positive ions. For very large \(L\) and constant voltage \(U\), and hence for a weak field \(E\), the energy of the positive ions is small, and \(\gamma\) is likewise small. With contraction of the field the field strength increases and the energy of the positive ions also increases continuously. The contraction of the field thus leads to an increase of the values of \(\gamma\).

As experiment has shown, the value of \(\gamma\) cannot exceed four[^11], i.e., even at extremely high voltages a single positive ion cannot, upon impact with the cathode, liberate more than four electrons. In the general case this will be less than four. Therefore we have noted: the coefficient \(\gamma\) of surface ionization increases with contraction of the field, but can in the best case—and only at very high voltages and with considerable contraction of the field—reach the value four.

7. Increase of ionization

For ignition and for a glow discharge, what is important is not only ionization by electrons in itself, and also not only the ionization coefficient \(\gamma\), but also the known relation between both, which

we shall call the “ionization increase” \(\mu\). We arrive at this quantity as a consequence of the following physical consideration.

Let \(N_0\) electrons fall at once onto the cathode. In the region of propagation of the field \(L\), along their path they produce, in total, \(N_0(e^{\gamma L}-1)\) new electrons and positive ions. If the ionization coefficient is equal to \(\gamma\), then \(N_0(e^{\gamma L}-1)\) positive ions, after traversing the path \(L\), liberate at the cathode \(N_0\gamma(e^{\gamma L}-1)\) additional electrons. This phenomenon, consisting of ionization by electrons and by positive ions, we shall call the ionization process. In this ionization process just described there is, in nature, at once included at least a second process, etc. In reality, each positive ion need not traverse the entire region of the field. The whole region of propagation of the field forms, for the path of the positive ions, the upper boundary. In our idealization we have assigned this upper boundary to all positive ions. This leads to the result that all ionization acts are superposed upon one another stepwise. In reality these steps are smoothed out. No appreciable deficiencies arise in our treatment as a consequence of this.* If we wish to know how strongly the intensity of the ionization currents increases from step to step, then we must form the ratio \(\mu\) for the initial electrons of two successive ionization processes. We obtain:

\[ \mu=\frac{N_0\gamma(e^{\gamma L}-1)}{N_0}=\gamma(e^{\gamma L}-1). \]

We have called this ratio \(\mu\) the ionization increase. It is clear that if the ionization increase is less than unity, then a current once arisen falls to zero; if the ionization increase is greater than unity, then a current once arisen grows unstably; but if the ionization increase is exactly equal to unity, then the current that arises remains stable and should no longer change. Here we arrive at the Townsend breakdown condition, which, under our method of treatment, is interpreted as the equality \(\mu=1\). Since Townsend assumes in his formula (3) the continuing liberation of electrons from the cathode by extraneous ionization, at \(\mu=1\) the current does not remain constant, but grows more and more, to infinity.

Of the decrease, constancy, or increase of the current we can learn something only by observing the ionization increase \(\mu\).

8. Distortion of the field and ionization increase

Let us see what effect the space charge and the distortion of the field have on the ionization increase \(\mu\), moreover

* We have something similar when, instead of operating with distributed capacitance and self-inductance, we consider them as concentrated. With distributed self-inductance and capacitance the current grows stepwise. With concentrated conductivity these steps are smoothed out.

we again assume that \(U=\mathrm{const}\) (a sufficiently powerful voltage source).

If the region of propagation of the field \(L\) is very large, the field strength \(E\) is small, so that, by spreading the electrodes apart, we can attain a small number of electrons and positive ions formed \((e^{\alpha L}-1)\), and likewise a small ionization coefficient of the positive ions \(\gamma\). Correspondingly, the ionization increase remains sufficiently small: \(\mu=\gamma(e^{\alpha L}-1)\). The ionization increase for sufficiently large \(L\) lies sufficiently close to zero.

If \(L\) becomes smaller (\(U=\mathrm{const}\)), which can occur by bringing the electrodes closer together, then the number of electrons and positive ions formed \((e^{\alpha L}-1)\) increases, as does the coefficient \(\gamma\). The ionization increase \(\mu\) also increases.

In the special case, with the proper shifting of the electrodes, the ionization increase reaches the value one. If the region of propagation of the field \((L)\) is reduced still more, which can now occur as a consequence of the space charge, then the ionization increase becomes greater than one and, in some cases, at sufficiently high voltage reaches values \(10^2, 10^4, 10^6, 10^8\). If the region \(L\) becomes very small and the field strength is large, then the ionization coefficient \(\gamma\) remains less than four. The number \((e^{\alpha L}-1)\)—of the electrons and positive ions formed—falls close to zero; the same happens with the ionization increase \(\mu=\gamma(e^{\alpha L}-1)\). We thus come, in contracting the field, again to that position where the ionization increase becomes equal to one, and then less than one. Thus, if we start from large \(L\), then in contracting the field the ionization increase \(\mu\) grows from values less than one to values equal to one, then exceeds one, reaches a considerable maximum* and then again passes to one, and then again falls to values less than one (Fig. 3).

Fig. 3. Ionization increase \((\mu)\) as a function of the magnitude of the region of propagation of the field. Voltage \(U=\mathrm{const}\). Shape of the dependence.

For each voltage \(U\), new curves for \(\mu\) are obtained, but they all have the form of a hump, and as the voltage rises the maximum also rises strongly.

The ionization increase is, therefore, a function of the voltage \(U\) and of the region \(L\) over which the field propagates.

If we plot the value \(\mu\) on an axis perpendicular to the plane of Fig. 4, then at the end points of the entire set

* This maximum is shifted, in comparison with the maximum in Fig. 2, toward smaller regions of propagation of the field and toward larger field strengths.

we obtain a surface similar to a humpbacked one. On the zero curve we have \(\mu = 1\). This curve is shown in Fig. 4, where it is indicated by a thick line, and in Fig. 5 it is repeated. We can approach it if we move a small distance toward a side perpendicular to the ordinate. We expect that this curve is close to the coordinate axes, which can well be represented by a V-shaped curve. In the region surrounding the surface of the level \(\mu = 1\), the ionization increase becomes greater than unity and, as Fig. 4 shows, by a sufficiently large amount (see, for example, the plotted levels). Between the coordinate axes and the level line \(\mu = 1\) lies the region with values \(\mu < 1\).

9. Ignition as a jump from one equilibrium position to another

We have prepared everything for explaining ignition. We take the voltage \(U\) as constant and decrease \(L\) from large to ever smaller values. At first there occurs a decrease—

Fig. 4. The totality of ionization increases \(\mu\), represented by the level lines.

Fig. 5. Level line \(\mu = 1\). Breakdown curve.

of \(L\) mechanically, by bringing the electrodes closer together. So long as, at large \(L\), we have an ionization increase \(\mu < 1\), we are in the region of Townsend dark currents. An avalanche arising from a single initial electron will die out here. With prolonged external ionization the current remains stable and constant. This is the initial equilibrium position before ignition occurs.

The boundary of these static currents is reached when the electrodes have been moved together so much that, at the assumed constant voltage, we reach the level line \(\mu = 1\). This curve (Figs. 4 and 5) indicates the boundary of the stable discharge region. The level line \(\mu = 1\) is therefore identical with the usually experimentally established breakdown curve. The current that has arisen can here still be held in equilibrium. If the field region is compressed still more, which occurs because of the increasing space charge,* then we arrive in the region of values \(\mu > 1\). Once arisen, the current then becomes unstable. Then there grow

* For this only a small accidental increase of \(\mu\) is needed.

current, the space charge, the distortion of the field, and the growth of ionization reaches very high values. Finally, we arrive at the summit of our curve (approximately point \(R\) in Fig. 4) and then descend along the other side of the same curve.

Now the action of the space charge and the distortion of the field act in the opposite direction. If earlier the action of both these factors led to an accelerated growth of the current, now they act in a retarding manner. The current, the space charge, and the field distortion still grow, but at an ever-decreasing rate (from \(R\) to \(S\) in Fig. 4). Finally, the discharge passes to those values of the “growth of ionization” \((\mu)\) which correspond to the level line \(\mu = 1\), and here acquires a new and stable equilibrium position.

Our consideration thus gives a complete picture of stable small, non-self-sustaining dark currents in the Townsend region, of the unstable growth of currents when leaving this region, and of the new equilibrium position with large self-sustaining current densities. The discontinuity in ignition—the transition from one equilibrium position to another, from small current densities to large ones, from a non-self-sustaining discharge to a self-sustaining one—is well represented in our treatment.

10. The Second Equilibrium Position Is Identical with the Glow Discharge

If we start with a single electron, and at constant voltage the region over which the field is distributed has become so small that the growth of ionization \(\mu\) is exactly equal to unity, then the discharge will be self-sustaining and stable. The current itself remains small in this case. At the beginning of every ionization process we always have only one single electron.

If, however, we have not one single electron, but billions of such electrons, and the voltage and the region of field distribution again correspond to \(\mu = 1\), then this current too will be stable. But the current is billions of times greater than before. We see that the condition \(\mu = 1\) may be conditioned by the presence of any large currents. The same holds for large currents in the glow discharge; for it too we must require \(\mu = 1\). Our Fig. 4, in which the level line \(\mu = 1\), corresponding to the values at breakdown, is drawn with a heavy line, is also suitable for the glow discharge. In doing so, however, a definite assumption is obviously made: we must have the right to replace the field in the glow discharge by a stepwise field distribution. In the region of normal and anomalous cathode fall this approximation can indeed be made. For the stationary glow discharge, of the entire breakdown curve \(\mu = 1\) the part lying near the axis of ordinates (near breakdown) is relevant. Only here do the increased space charge and the contraction of the field give values of \(\mu\) less than unity;

only here can the discharge be in a stationary state, whereas in another part of the curve (with very similar conditions) the spatial charge intensified at \(\mu>1\) leads only to the transition of the discharge indicated above. The nearest part of the field distribution upon breakdown should, approximately, correspond to the initial regions of field propagation in the anomalous and normal cathode fall of the glow discharge (dark cathode space).

Just as the level line \(\mu=1\) (the breakdown curve) has a minimum in voltage, so this must also be the case in a glow discharge, as is known for the case of normal cathode fall.

How can it happen that the curve \(\mu=1\) (the breakdown curve) is suitable both for small currents at breakdown and for large currents of the glow discharge?

We can reach the limiting curve \(\mu=1\) in two ways. One time—from the region \(\mu<1\). If we do this, then we find ourselves in the region of small currents immediately before breakdown. It is otherwise if we reach the region \(\mu=1\) from regions with increasing ionization, \(\mu>1\), and if the current previously belonged to the region \(\mu>1\) and was for a long time unstable; then the large currents, the currents of the glow discharge, also correspond to the curve \(\mu=1\).

Reviewing what has been said in §§ 9 and 10, we come to the conclusion that the second final state, into which the current passes, owing to the stabilizing action of the space charge after an unstable increase, can be nothing other than the glow discharge. We have thus fulfilled the entire program outlined in § 1 for an ideal, powerful voltage source.

11. A voltage source of finite power

(Voltage drop and current rise. The falling characteristic due to space charge. Arc discharge)

How will our consideration change if the power of the current source is of finite magnitude and some resistances are unavoidable?

This question will become clear if we take into account certain properties of the glow discharge, which can just as easily be obtained as consequences of our consideration[^14]. Here they will be given without proof.

1. To each value of the “cathode fall” of the glow discharge there corresponds a definite current density; the current densities increase strongly with voltage.

2. The current density increases with pressure quadratically.

3. In an anomalous cathode fall a stable discharge is possible when the cathode is entirely covered by the discharge. Only in normal cathode fall, i.e. at minimum voltage—

or a glow discharge; the discharge can be maintained with a partially covered discharge.

The table below gives values for voltages, current densities, and pressures for air in certain cases. Also given here are data of interest for what follows on density and power. A detailed examination of these figures gives an idea of the conditions that must be satisfied by the ideal powerful voltage source which we used in the preceding sections.

TABLE 1*

Ignition occurs, for example, at a voltage of 2 thousand V. At a pressure of 1 mm Hg ignition ends, with an ideal current source, at a current density of 0.3 A/cm².

At pressures of the order of 1 atm, after the ignition occurring at 2 thousand V, we must supply no less than 180 thousand A/cm².

At 100 atm we arrive at almost actual current densities of 2 billion A/cm². At high pressures and large electrode dimensions, even at such low voltages as 2 thousand V the discharge gap turns into a current-devouring monster. The current strengths (current density × size of the electrodes) may become so great that no terrestrial source can maintain them.

The current source frees itself from such a large demand for current by means of a fall in voltage. As the current increases during ignition, when the current has become sufficiently large and unstable, the voltage across the discharge gap falls for a current source of finite power. The set of ionization growths will be intersected not on the straight line \(U=\mathrm{const}\), but on a curve displaced toward lower voltages (approximately \(RI\), Fig. 4). Nevertheless, throughout the entire course of this curve the ionization increase is greater than unity. Despite the fall of voltage on the electrodes, the current continues to grow until it reaches the level of the line \(\mu=1\). We, thus

* Recalculated according to the similarity law. Temperature effects are not taken into account.

Thus, we obtain a falling characteristic, which is characteristic for ignition when ordinary sources of energy are used. It is also immediately clear that, for sufficiently large electrode surfaces and for a finite power of the current source, the completion of ignition will result in the minimum possible current density on the partially covered cathode, i.e., the “normal cathode fall.”

Also instructive is an examination of the power densities in our table; at \(1\ \mathrm{mm}\ \mathrm{Hg}\) it gives values of \(0.6\ \mathrm{kW}/\mathrm{cm}^2\), at \(1\ \mathrm{atm}\)—values of \(3.6\cdot 10^5\ \mathrm{kW}/\mathrm{cm}^2\), and at \(100\ \mathrm{atm}\)—even the fantastic value \(3.6\cdot 10^9\ \mathrm{kW}/\mathrm{cm}^2\).

The power density at high pressure after ignition may reach such values that the electrodes will quickly burn up and the gas temperature will become so great that strong thermal ionization will arise. Our table leads us directly to the conclusion that at high pressure the natural completion of the discharge is not a glow discharge, but an arc discharge.

12. Results of a More Rigorous Consideration*

(Lowering of the ignition voltage by irradiation. Repeated ignitions. Extinction voltage)

We wish briefly to indicate what a more rigorous calculation can add to the basic data. In the approximation, the ionization growth was taken in the form \(\mu=\gamma(e^{\alpha L}-1)\). A more rigorous calculation, which does not assume a step-like field but seeks to relate the form of the field to the ionization phenomena, takes the ionization growth in the form:

\[ \mu=e^{\int_0^L \alpha dx}(\gamma-1). \]

In the approximation we sought all discharges, briefly speaking, stable discharges, which satisfy the condition \(\gamma(e^{\alpha L}-1)=1\) (line 1, Figs. 4 and 5). Accordingly, in a more rigorous calculation we shall have in mind those stable discharges which satisfy the condition:

\[ \gamma\left(e^{\int_0^L \alpha dx}-1\right)=1. \tag{1} \]

In the approximate consideration the totality of stable discharges lay on a curve. It formed a singly infinite set. With a more rigorous calculation all stable discharges form not a singly, but a doubly infinite set, and this constitutes our first generalization of the question.

This can be easily justified. Let the field strength at the cathode be given, \(E_k=\mathrm{const}=E_{k0}\). Then by this the ionization coefficient [[unclear: continuation cut off]]

* Exact justification—see Rogowski, Arch. F. Elektrotechn., 26, 613, 1932.

\(\gamma=\gamma_0=\mathrm{const}\), since we can set them in correspondence with the field strength at the cathode. The condition \(E_k=\mathrm{const}\), together with those according to (6), leads to another condition, namely that the integral

\[ \int_0^L \alpha\,dx=\mathrm{const}. \]

The latter condition (Fig. 6) can be satisfied: 1) for large values of \(\alpha\), a sufficiently small decrease in the field strength and small distances \(L\); it can also be satisfied: 2) for a large decrease in the field strength, small values of \(\alpha\), and large distances \(L\). Large field strengths correspond to small current densities \(i\). A considerably weakened field strength corresponds to large current densities (Fig. 6). Thus with the condition \(E_k=\mathrm{const}\) there is associated yet another infinite set of stable discharges, which differ from one another by a different drop of the field strength and by the magnitude of the current density. Also the field strength at the cathode \(E_k\) can run through an infinite scale of values. The set of stable discharges given by (6) is therefore doubly infinite.

Fig. 6. Field distribution for different current density \(i\), but at constant field strength at the cathode, \(i_1<i_2<i_3\).

Thus there may be an infinitely large number of times more stable discharges than could have been supposed from the first approximation.

Condition (6) alone is still insufficient for a rigorous calculation. We must compare it with the known current equation

\[ i=\varepsilon(pu+nv), \tag{7} \]

where \(\varepsilon\) is the elementary charge, \(p\) is the density, \(u\) the velocity of the ions, \(n, v\) the density and velocity of the electrons; with the space-charge equation

\[ \frac{dE}{dx}=-4\pi\varepsilon(p-n), \tag{8} \]

where \(x\) is the distance from the cathode; with the ionization equation

\[ \frac{d(nv)}{dx}=\alpha(nv) \tag{9} \]

and with the velocity equations

\[ \begin{aligned} u&=k\sqrt{E},\\ v&=l\sqrt{E}, \end{aligned} \tag{10} \]

where \(k\) and \(l\) represent the “mobilities.” This system of equations, the solution of which is necessary for full clarity in the ...

…the value \(n\) for various discharges. It is clear that, on the basis of this calculation, the characteristic properties of the positive column, cathode and anode drops, and the so-called hindered discharge can be explained. An important result, which was not obtained from the first approximation, was also found for ignition. In our first approximation, for a given distance between the electrodes \(L=L_0\) there corresponded a definite ignition voltage \(N=N_d\). At ignition, the jump proceeded from a definite small field strength at the cathode \(E_{k n}\) to a definite large field strength at the cathode \(E_{k m}\). The current density then jumps (for an ideally powerful current source) from certain extremely small values to very large values (Fig. 4, course of the lines \(QS\)). In the approximate treatment we had a jump at two definite limiting positions. In a more rigorous calculation all this is infinitely more varied. For \(L=\mathrm{const}\) there is not one voltage at which ignition can occur, but a whole scale of such voltages. The voltage \(U_D\) of the approximation will be only the upper limit of this scale of voltages. In the jump corresponding to ignition, we have a transition between field-strength values lying close to one another. The same is true also for the current densities passed through in jumps. These relationships are visible in Fig. 7, where the field strength at the cathode is plotted as the independent variable, and the corresponding voltages and current densities as dependent functions.

Fig. 7. Ignition and burning voltages for a discharge interval at \(L=\mathrm{const}=1/100\ \mathrm{cm}\). Atmospheric pressure. Above the ionization voltage increase \(\mu>1\), below it \(\mu<1\).

The curves (Fig. 7) are suitable for all stable discharges that satisfy condition (6), equations (7), (8), (9), (10), and the requirement \(L=\mathrm{const}\). The jump-like change at ignition is evident from the fact that two stable discharges correspond to one and the same voltage, differing from one another by different field strengths at the cathode and by different values of current density, so that it may occur—

* Rogowski, loc. cit., and also Schumann, loc. cit. The equations indicated above are the simplest basis on which it is possible to transform the theory of the spatial discharge into Townsend theory. The object of further investigation will be to establish how great a change in one or another proposition must be for there to be agreement not with the idealized glow discharge, but with the real one.

proceeds from one form of discharge with a low current density to another form with a higher current density. Below our voltage curve (Fig. 7) \(\mu < 1\), above it \(\mu > 1\), and this is the region of unstable states. As a consequence of the rigorous calculation we have from Fig. 7:

1. For a definite distance between the electrodes there exists not one single ignition voltage, but an entire totality of such voltages.

2. The various ignition voltages differ from one another in the magnitude of the initial field strength at the cathode and in the initial current density.

3. There exists a least ignition voltage, which coincides with the extinction voltage. These results of rigorous calculation give a complete explanation of the dependence of the ignition voltage on irradiation and on preliminary lowering of the discharge gap (point \(M_3\), Fig. 7). Strong external ionization and residual charges after extinction of the discharge can reduce the ignition voltage to the theoretical value of the extinction voltage. This is in agreement with experiment. The fact that we observe a quite definite ignition voltage is caused by the weakness of the usual preliminary ionization.

Thus we come to the end of our theoretical consideration. Historical development shows that the problem of ignition of a gas discharge was not simple. That which was essential could be clarified by simple reasoning; this is the merit of our method of consideration, which, owing to the overwhelming number of satisfactorily explained phenomena, becomes sufficiently convincing.

13. Some experimental additions

To the account set forth above are added certain experimental results. We subject the gas gap to a short (\(10^{-7}\) sec.), sufficiently high, shock voltage, having a rectangular form. We choose a low pressure and photograph the luminous phenomenon that arises after \(10^{-7}\) sec. and immediately disappears. Evidently, after \(10^{-7}\) sec. we have the characteristic features of an ordinary glow discharge. Likewise the dependence on pressure, as shown by the contraction of the dark space, corresponds well to the glow discharge usually observed in experiment. Our experiments may serve as clear proof that upon completion of ignition the distribution of light, field, and space charges is exactly the same as in an ordinary glow discharge, which is in full agreement with the theoretical ideas developed here.

I show in Fig. 8 the voltage drop, recorded with the aid of an oscillograph, for an extremely slowly increasing static voltage. We know from the preceding that the voltage drop is not a single property of the gas gap,

ON THE IGNITION OF A GAS DISCHARGE

and is due to the weakness of our current source. In the oscillogram the step-like breakdown interval attracts attention—it corresponds to the transition of the discharge form from corona discharge, and then from glow to arc. The oscillogram must be interpreted as follows*: first, in the interval \(AB\), there develops, as can be shown by an exact numerical calculation, the unstable rise of the current described by us. Already at point \(A\) it is in full swing. But the current density is still too small,

Fig. 8. Step-like voltage drop during breakdown. Static breakdown. Pressure \(p = 500\) mm Hg.

Fig. 9. After ignition has occurred—the voltage is equal to the arc voltage. Pressure \(p = 400\) mm Hg. An oscillation is superposed on the step. The reason for this is noticeable to be the insufficiency of the decoupling capacitors, which were taken large for increasing the sensitivity.

Fig. 10. The length of the step decreases with increasing pressure. \(p = 740\) mm Hg.

Fig. 11. The step has disappeared. Pressure \(p = 1500\) mm Hg.

Fig. 12. Ignition under impulse voltage. \(p = 760\) mm Hg.

in order to produce a noticeable voltage drop in the oscillogram. Only at point \(B\) is the current density sufficiently large, and it becomes

* I shall return to this question in detail elsewhere. The above explanation is based on the assumption that ignition is caused by the single initial electron.

we shall notice a corresponding voltage drop. The current in \(b\), for example, is equal to \(10^{-2}\) A. The distortion of the field in the discharge gap, as an exact calculation shows, has become so great that we practically have a region over which a field of the order of the normal cathode fall is distributed. In Figs. 4 and 7 we are still, practically at the initial voltage, in the region of ionization increase \(\mu > 1\). The increase of the current continues, and with it the voltage drop as well. In Fig. 7 the corresponding point of our discharge gap traverses the path \(M_0, M_0'', M_2\). Falling voltages correspond to them, but still increasing currents. Point \(C\) of the oscillogram corresponds exactly to \(M_2\) of Fig. 7. The increase of the current here, at \(\mu = 1\), stops. Nevertheless, on the oscillogram the increase of the current (section \(CD\)) continues. This occurs because the discharge arises in a small channel (a single initial electron is assumed) and the cathode surface at point \(C\) is not yet covered by current. Diffusion of the electrons enlarges the discharge channel, and then, finally, the point \(M_2\), corresponding to the discharge in Fig. 7, begins slowly to move toward point \(M_3\) (for a large electrode surface). As a result of the enlargement of the discharge channel the current increases, and because the current source has finite power, the voltage drop slowly increases. Meanwhile, in the discharge gap something new is being prepared, the cause of which is an increase in the temperature of the electrodes and of the gas column (§ 11). This new circumstance brings about a jump into the arc discharge, which arises just as unexpectedly, according to the oscillogram of Fig. 8, as the glow discharge did. With a still more greatly reduced voltage and a corresponding increase of the current, the breakdown ends at low voltages and high densities, as an arc discharge. Proof of this is provided by the oscillogram of Fig. 9, which indicates a small residual voltage, much less than the minimum burning voltage of the glow discharge. We saw earlier that temperature effects increase very strongly (to the third power) with pressure. In agreement with this, the width of the step decreases with pressure (Fig. 10). Then, still during the first voltage drop and without any noticeable step, the transition from glow to arc discharge occurs (Fig. 11).

Fig. 13. Reignition. Zero voltage and voltage of static breakdown.

14. Oscillograms under impulse voltages and repeated ignitions

For oscillograms 8–11 there were definite static voltages. We can noticeably, and even extremely strongly, in-

increase the rate of the ignition process and the rate of voltage fall at breakdown. An illustration of all this is furnished by oscillogram 12, which differs by an extraordinarily short preparatory time for breakdown and by the absence of a step, from Figs. 8–10.

Finally, let us also dwell on Fig. 13, which points to certain characteristic repeated ignitions. If one looks at Fig. 7, they are understandable without further explanation. Let us take, for example, the transition \(M_1 M_2\). Such a steep fall after the second ignition has occurred is not unexpected according to our theory (see Fig. 7); it must proceed exactly as rapidly as the corresponding part of the main ignition.^14

LITERATURE

1. Reviews: Townsend, Hand. d. Radiologie 1, 1920; Schumann, El. Durchbruchfeldstärke von Gasen; Seeliger, Gasentladungen; Mierdel, Hdb. d. Experimentalphysik, XIII, 3.

2. Rogowski, Arch. f. Elektrotechn. 381, 16, 761, 1926.

3. Townsend, loc. cit., 381.

4. Rogowski, Flegler u. Tamm, Arch. f. Elektrotechn. 18, 479, [[unclear: year cut off]]; Rogowski u. Tamm 20, 625, 1928; v. Hamos, Ann. d. Phys. 7, 857, [[unclear: year cut off]]; Dunnington, Phys. Rev. 38, 1535, 1931; Snoody, Phys. Rev. 40, 409, [[unclear: year cut off]]; Holzer, Zs. Physik 77, 676, 1932.

5. Rogowski, Arch. f. Elektrotechn. 20, 194, 1928; Slepian, El. W[[unclear]], 1928, 761; Loeb, Science, 509, 1929; Franck u. Hippel, Zs. Physik 57, [[unclear]], 1929; Schumann, Zs. f. phys. Tech. 11, 194, 1930; Kapzov, Zs. Ph[[unclear]] 95, 380, 1932.

6. Rogowski, Arch. f. Elektrotechn. 24, 689, 1930.

7. Seeliger, Naturwiss. 16, 655, 1928.

8. Döllenbach, Physik. Zs. 26, 483, 1925; Günterschulze, Zs. Physik 20, 1, 1925; Seeliger, Gasentladungen, S. 347; Compton u. Morse, Phys. Rev. 30, 305, 1927; Steenbeck, Zs. Physik 53, 192, 1929; Rogowski, Arch. f. Elektrotechn. 24, 689, 1930; v. Hippel, Zs. Physik 76, 1, 1932; Holzer, Physik. Zs. 25, 157, 1924; Mierdel, Hdb. d. Exp. Physik 13, 3, 412.

9. Rogowski, Arch. f. Elektrotechn. 25, 551, 1931.

10. Rogowski, Arch. f. Elektrotechn. 26, 643, 1932.

11. Baerwald, Ann. d. Phys. 67, 167, 1921; Scheider, Ann. d. Phys. [[unclear]], 381, 1931.

12. Rogowski, Arch. f. Elektrotechn. 25, 525, 1931.

13. Oscillograms of this kind in Rogowski u. Klemperer, Arch. f. Elektrotechn. 24, 127, 258, 1931; Buss, Arch. f. Elektrotechn. 26, 206, 1932.

14. Further oscillograms: Rogowski, Flegler u. Tamm, Arch. f. Elektrotechn. 19, 235, 1928; in addition, works of the Electrotechnical Institute in Aachen II–V; Krug, Zs. f. tech. Physik 13, 377, 1932; figures 9–[[unclear]] taken from Dr. Russ; Viehman, Arch. f. Elektrotechn. 25, 259, 1931.

1. Avalanche. ↩