Static breakdown in gas. Part III

In the first part (https://www.dhirubhai.net/pulse/static-breakdown-gas-part-i-sergei-vasiliev-ib1bf/?trackingId=p4i%2Bx%2BTPTMe0YyAthbbLrw%3D%3D) we considered the essence of the phenomenon of electric current generation in gases. In its normal state, air is a dielectric, otherwise there would be no power lines. But sometimes, for example during lightning, the air becomes a conductor. This means that free charge carriers have appeared in the air. Why? How can a neutral air molecule be turned into a charged particle? It is necessary to tear an electron from it, more often they say "ionize". As experience shows, an initially non-conductive gas can be ionized in several ways – by heating the gas, irradiating it with ultraviolet light, X-rays, radioactive radiation (alpha-rays, beta-particles, gamma-radiation). If there is no electric field in the external space (for example, in a gas tube), where charged particles have appeared, then these particles do not move anywhere in a direction. Recombination may occur – the electron will return to the ion, turning it back into a neutral atom.

So, two competing processes are taking place – ionization and recombination. If now an electric field is created in the studied space (gas tube), then the ions will start moving towards the cathode (negatively charged electrode), and the electrons – towards the anode (positively charged electrode), an electric current will arise in the external circuit. Since recombination has not disappeared, and some of the charges recombine, then with an increase in voltage, an increase in current will be observed, since an increasing part of the charged particles will reach the electrodes. At a sufficiently high voltage on the AC gap, all charged particles will reach the electrodes, and the current will reach saturation – with a further increase in voltage, no further increase in current will be observed. The magnitude of this saturation current depends on the degree of ionization of the gas.

If we now remove the cause of ionization, and the discharge stops, such a discharge is called non-self-sustaining, it can exist only in the presence of an external ionizer. If the discharge continues in the absence of an external ionizer, such a discharge is called self-sustaining, it is self-sustaining. Due to what? Because the free electrons that are formed, accelerating in an electric field, have time to accelerate and acquire energy sufficient for the impact ionization of neutral gas atoms. That is, the condition for the ignition of a self-sustaining discharge is Wkin?>?Wion. Each gas has such a characteristic as the ionization energy Wion (we provided a table of such values in the first part).

In the second part (https://www.dhirubhai.net/pulse/static-breakdown-gas-part-ii-sergei-vasiliev-gfgpf/?trackingId=p4i%2Bx%2BTPTMe0YyAthbbLrw%3D%3D), we obtained relationship of exponential increase in the number of charge carriers in the gap from the cathode to the anode under the influence of an electric field due to the impact ionization of molecules by electrons (this is how an avalanche is formed)

where the coefficient α is called the impact ionization coefficient (or the first Townsend ionization coefficient), it is equal to the number of ionization events performed by an electron per 1 cm of path along the field E. Knowledge of the ionization coefficients α is so important that the experimental determination of this characteristic of the ionization process served as one of the most pressing problems in the development of gas discharge physics as an exact science. To measure this value, J. Townsend proposed an experiment in 1902, from which an empirical formula was obtained for the analytical description of the coefficient α:

where P is the gas pressure; E is the electric field strength; the constants A and B are determined experimentally.

Let's continue now. The occurrence of an avalanche is not yet a breakdown. Starting from a certain distance d between the electrodes, the increase in the electron concentration according to the formula Ne(x) = N0e^(αx) ceases to be valid. The current grows faster than exponentially with increasing d. To explain this effect, Townsend suggested that additional electrons arise when the cathode is bombarded by positive ions of the avalanche. To evaluate the process of occurrence of these secondary electrons, the coefficient γ is introduced – the so-called coefficient of secondary emission of electrons from the cathode (or the second ionization coefficient of Townsend), which characterizes the probability of the appearance of a new electron on the cathode after the passage of one avalanche; it is determined by the type of gas, the cathode material and its surface condition, the gas pressure P and the field strength E. The data on γ are imperfect and often contradictory, but this is mitigated by the fact that in most cases γ enters the formulas under the logarithm sign, and therefore the results are not very sensitive to the choice of this value. Approximate values of the coefficient γ for some gas/cathode material combinations are given in Table 1.

Let N(x) be the number of electrons in section x. Then, after passing the path dx, their number will increase due to ionization collisions by the value dN = N(x)αdx. If initially N0 electrons leave the cathode, then because of impact ionization of the anode N0(d) = N0e^(αd) particles will be reached, where d is the distance between the cathode and the anode. The number of secondary electrons knocked out of the cathode Nsecond is equal to the number of ions that reached it, multiplied by γ. From the point of view of the probability of an increase in the number of electrons due to secondary emission from the cathode, this is equivalent to the fact that each electron of the avalanche, passing the path dx, releases ωdx secondary electrons from the cathode, where ω = αγ.

Then

Secondary electrons Nsecond with probability ω are produced by all electrons, both primary and secondary (since secondary electrons create a new avalanche, with new ions, which again tear electrons from the cathode, and so on). Then the total number of electrons Ne leaving the cathode is equal to

consequently

where do we get it from

Considering the primary (impact) and secondary ionization (from the cathode), the anode is reached by N(d) = Ne*e^(αd) electrons, therefore

or for electron current

Here (e^(αd) – 1) is the number of ions in one avalanche initiated by one electron (subtraction of one is performed to consider only the created ions, without considering the initiating electron). Each of these ions rushes to the cathode and causes the appearance of γ(e^(αd) – 1) secondary electrons, which rush to the anode.

For a breakdown to form, at least one new initiating electron must appear again after the avalanche has passed on the cathode. Only if this condition is met does a repeated avalanche occur, then another avalanche, and so on: this leads to the formation of an independent (self-sustaining) multi-avalanche discharge. Mathematically, this condition for the discharge to be independent is written as 1 – γ(e^(αd) – 1) = 0. Indeed, one initiating electron leads to the formation of (e^(αd) – 1) pairs of charged particles. Each of the resulting ions drifts toward the cathode and, with probability γ, knocks an electron out of it. If all the ions together knock out at least one electron, i.e. γ(e^(αd) – 1) = 1, then the discharge will become independent, since in this case each initiating electron creates, through a certain number of secondary processes, a new electron at a place equivalent from the standpoint of discharge development. Mathematically, this is equivalent to the current in the gap becoming infinite (since it is self-sustaining). The resulting condition γ(e^(αd)?–?1) ?=?1 is called the Townsend condition for the ignition of a self-sustaining discharge; in this case, the insulating layer in the gap between the anode and cathode breaks down and a self-sustaining discharge develops.

Since the coefficients α and ω depend on the field strength (and therefore on the voltage across the gap U), as well as on the mean free path (and, therefore, on the pressure P), that is, α/P = f(E/P) and ω/P = g(E/P), then from the condition of self-sufficiency of the discharge, we can obtain the dependence of the discharge voltage on P and d.

Indeed, from the expression

where do we get it from

If the mean free path of an electron is λ, then the total number of electron collisions per unit path is Q = 1/λ. Then the number of ionizing collisions will be α = Qi*ηi = ηi/λi, where λi is the mean free path for the ionization reaction, and ηi is the fraction of ionizing collisions from the total number of collisions. Note that if an electron over a mean free path λi has gained energy equal to the ionization potential Ui, then e*E*λi = e*Ui is valid, whence λi = Ui/E. The probability that, with a mean free path λ, an electron will travel a path greater than or equal to λi is determined by the relation e^(-λi/λ), the same relation is equal to the fraction ηi of ionizing collisions from the total number of collisions. Consequently,

Since the mean free path λ is inversely proportional to the pressure P, i.e. λ ～ 1/P, we can write the equality 1/λ = KP, where K is a certain proportionality coefficient. Then the last expression can be written as

and comparing the resulting expression with the expression

we see that K = A, and B = Ui*A. Then expression

can be rewritten in the following form

Considering the expression for the mean free path obtained in Part I,

we can get

then A = σ/(kT). Expression

is known as Paschen's law, formulated by Friedrich Paschen in 1889. This expression describes a curve with a minimum. Let us determine the value of (Pd)min at which the breakdown voltage Us is minimal. To do this, let us denote Pd = x, and A/(ln(1+1/γ)) = C. Then Paschen's law can be written in the following form

To find the minimum, we differentiate this expression with respect to x and set the result equal to zero:

As you can see, here we need to calculate the differential of the product of two functions. Using the formula (UV)’ = U’V + V’U, where is denoted (considering that the derivative of the logarithm of a complex function (lnU)’ = U’/U)

we can write the following

Substituting Paschen's law into this expression, we obtain UsMIN:

that is, finally

At a gap voltage less than UsMIN, it is impossible to cause a breakdown (provided that the field distribution between the electrodes is uniform). In this case, the gas pressure P and the distance between the electrodes d do not matter.

As the avalanche lengthens and the number of carriers in it increases, the charge near the front of the developing avalanche increases, and the electric field strength of the avalanche also increases more and more. At a certain strength, the discharge can spread practically without the participation of electrodes, due to the high strength of the avalanche itself. The so-called avalanche-streamer transition occurs, the transition of the discharge from the multi-avalanche form to the streamer form. The criterion for such a transition is the fulfillment of the condition αd?=?20.

A streamer is a set of thin branched channels through which electrons and ionized gas atoms move in peculiar streams. A streamer is a precursor to a breakdown during a corona or spark discharge under conditions of relatively high pressure in a gas and a relatively large distance between the electrodes. The branched glowing channels of the streamer lengthen and, eventually, close the gap between the electrodes – continuous conductive threads and spark channels are formed.

The formation of a spark channel is accompanied by an increase in current in it, a sharp increase in pressure, the emergence of a shock wave at the boundary of the channel, which we hear as the crackling of sparks (thunder and lightning in miniature). The head of the streamer, located in the front part of the channel thread, glows the brightest. Depending on the nature of the gaseous medium between the electrodes, the direction of movement of the streamer head can be one of two: this is how anode and cathode streamers are distinguished.

As the interval lengthens, for long intervals, the appearance of repeated streamers in the wake of the first streamer is possible. This occurs because where the streamer passed, the gas heats up, the density of the gas decreases, its electric strength decreases, and new streamers with their additional heating can appear and spread in the streamer wake, and so on. As a result of the local increase in temperature, thermal ionization begins in it, and the electrical conductivity increases.

Unlike the electron avalanche, the streamer is characterized by a high speed (about 0.3% of the speed of light) of propagation of the streamer head to the anode or cathode, which is many times higher than the electron drift speed simply in an external electric field. At atmospheric pressure and with a distance between the electrodes of 1 cm, the speed of propagation of the cathode streamer head is 100 times higher than the speed of the electron avalanche. For this reason, the streamer is considered as a separate pre-breakdown stage of an electric discharge in a gas.

Part I - https://www.dhirubhai.net/pulse/static-breakdown-gas-part-i-sergei-vasiliev-ib1bf/?trackingId=p4i%2Bx%2BTPTMe0YyAthbbLrw%3D%3D

Part II - https://www.dhirubhai.net/pulse/static-breakdown-gas-part-ii-sergei-vasiliev-gfgpf/?trackingId=p4i%2Bx%2BTPTMe0YyAthbbLrw%3D%3D

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