Progress in the kinetic description of non-stationary behaviour of the electron ensemble in non-isothermal plasmas

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Progress in the kinetic description of non-stationary behaviour of the electron ensemble in non-isothermal

plasmas

J. Wilhelm, R. Winkler

To cite this version:

J. Wilhelm, R. Winkler. Progress in the kinetic description of non-stationary behaviour of the electron ensemble in non-isothermal plasmas. Journal de Physique Colloques, 1979, 40 (C7), pp.C7-251-C7- 267. �10.1051/jphyscol:19797441�. �jpa-00219445�

JOURNAL DE PHYSIQUE Colloque C7, supplkment au no 7, Tome 40, Juillet 1979, page C7-251

Progress in the kinetic description of non-stationary behaviour of the electron ensemble in non-isothermal plasmas

J. Wilhelm and R. Winkler

Zentralinstitut fiir Elektronenphysik, Institutsteil V, Gasentladungsphysik ⁽⁽ R. Seeliger ^D, 22 GreifswaldIDDR

Abstract. - The understanding of the temporal behaviour of a plasma at several time varying conditions is of great interest under general physical aspects as well as from the point of view of very different applications. After an short survey concerning the investigations related to such problems in the following we discuss some more comprehensive results which were recently obtained in the investigations of the electron ensemble both at the field free collision dominated relaxation in the afterglow during the first period of temporal decay and at the collision dominated relaxation in a plasma with additional heating by an electric field. These studies take into account all main collision processes using realistic functions for the energy dependence of the correlated collision cross sections.

The results were attained by the numerical deterinination of the isotropic velocity distribution function of the electrons and resulting macroscopic quantities. In this way on the basis of kinetic theory a deeper insigth into the temporal relaxation mechanism of the electron ensemble in weakly ionized collision dominated plasmas can be gained for many different models of the temporal behaviour of the electric field as the jump-like change and the continuous aperiodical as well as periodical alteration of the field. Thus with the developed sure and widely appli- cable numerical methods a firm basis has been established to perform further investigations of more general problems grouped around the relaxation models presented here.

1. Introduction. - There is a lot of problems whose solution requires a realistic description of non- stationary states of a homogeneous non-isothermal collision dominated plasma. Illustrating examples are applications of ac-discharges up to some 10 kHz for plasma light generation, the specific techniques of high frequency discharges, the afterglow behaviour with its special relations to the study of several ele- mentary processes and so on. On the other hand there is a widespread interest in the knowledge of relaxation under the aspect of gas laser physics, optical transient processes and from the plasma chemical point of view.

For the physical understanding of the temporal behaviour of plasmas in the course of their decay or under the action of time-variable fields, or in general terms during the temporal transition between several states of the system, it is of great interest to investigate the dominant relaxation processes by which the time dependence of the macroscopic plasma properties is determined. A detailed microphysical knowledge of these processes is of general importance, and, more specifically, it makes a decision possible which degrees of freedom of the components can be treated as being in an approximate equilibrium.

In a first step a qualitative classification can be given for bounded plasmas with the aid of some basic time constants as z, : typical diffusion deter- mined life times (for instance by ambipolar diffusion of charged particles and by heavy particle diffusion of the main excited atoms or molecules), z, : the

mean life times of the excited atoms and/or molecules (for the most important metastables and atoms in the resonant states) and z, : the electron relaxation times for impulse and energy transfer to the heavy particles.

The values of these time constants are especially dependent on the pressure of the heavy plasma components. If we presume a pressure of about 1 torr then the diffusion times z, will possess values in a range of roughly one order of magnitude around one millisecond ; those of the life times z, of excited states have a similar variance around one hundred microseconds and those of z, around some tenths of microseconds.

Furthermore, if z designates the characteristic time constant of a process which causes the temporal changing of the plasma - for example the temporal alteration of an electric field - the resulting very different situations depend primarily on the relation of z to the other time constants mentioned before.

For instance in the case of the action of a periodical electric field the resulting degree of modulation of the macroscopic quantities as the electron density,- the mean electron energy, the concentration of excited atoms etc. will be very widely due to the different orders of magnitude of z.

In this connection some additional remarks appear to be necessary : the classification scheme given is only a rough one. For the investigation of the plasma in each specific time range a further sub-classification of the basic time constants mentioned is necessary

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19797441

C7-252 J . WILHELM AND R. WINKLER

beqause several processes with their own characte- ristic time constants are often involved. As repre- sentative examples suffice it to mention here the dissociation process via the vibrational states of molecules by eV-, VV- and VT-exchange [l, 21 and the dynamic behaviour of column plasmas as res- ponse to external disturbances [3].

Though there are still other examples, from the plasma chemical point of view the chemical relaxation time z, of a chemical reaction can also play an impor- tant role; its magnitude compared to the other characteristic times mentioned is essential e.g. for answering the question if the classical Arrhenius

Table I.

Problem

-

Field'free decay

Transition to a sta- tionary state with non-vanishing elec- tric field

High frequency plasma

Authors

Bayet, Delcroix, Denisse [Ill

Schuler a.0. [12]

Holway, Jr. [ I 31 Kahalas, Kashian [14]

Osipov [15]

Eaton, Holway, Jr. [16]

Wright [17]

Stenflo [18]

Ghatak a.0. [19]

Winkler [20]

Koch [21]

Shoda, Ghatak [22]

Braglia, Ferrari [23]

Polman [24]

Englert [25]

M iiller, Miiller 1261

Carleton, Megill [27]

Naidis 1281 Wagner a.0. [29]

chemistry is applicable or if the non-equilibrium kinetics has to be used [4, 51.

Next, for the energy input in the plasma by any electric field the electron component is of basic importance because the energy transfer takes place via this ensemble to the heavy components, which are the source of light or the initial material of special chemical reactions etc. From this point of view we can expect that in the first stage each relaxation response of the plasma after a perturbation in the electric field is dominantly determined by temporal changes in the electron component because of certain high speed processes resulting from the small inertia

Method

-

Representation by the eigen- functions of the collision operator

Polynomial expansion with a Maxwellian weight factor Special case of [13]

Special case of [I 31

random-Walk analysis and collision integral expansion Legendre polynomial ex- pansion

Laguerre polynomial ex- pansion

Legendre polynomial ex- pansion and additional La- guerre expansion

Legendre polynomial ex- pansion

Legendre polynomial ex- pansion

Legendre polynomial ex- pansion

Representation by eigen- functions

Difference-method Monte Carlo Monte Carlo

Approximation of first or- der in the electric and ma- gnetic field

Laplace transformation

Gas

-

Model systems with repul- sive central force law

-

r - "

(5

<

^s

<

co) for the colli- sion process

Model system

Model system Afterglow in Ar

Response to a pulsed ac field and relaxation in model systems

General relaxation proper- ties

Model plasma

Response to a pulsed field for a model of elastic colli- sions (similar to Cs, He, N,) He, Ne, Ar, Kr, Xe Model plasma He

Relaxation times in Ar after strong increase of electric field

Air

Model plasma Ne

PROGRESS IN KINETIC DESCRIPTION O F NON-STATIONARY BEHAVIOUR OF ELECTRON ENSEMBLE C7-253

of the electrons. In many cases alterations primarily connected with the greater inertia of the heavy particles will not become noticeable before this point.

The subsequent remarks are devoted to the temporal behaviour of the electron component within the discussed short first time period on the basis of the non-stationary Boltzmann equ;ion. Some efforts were undertaken after World War I1 to attack perti- nent problems under several points of view and with different methods.

First, we may mention here the well known work of Margenau [6], S. C. Brown [7], their coworkers and others who investigated the behaviour of high frequency discharges by means of kinetic theory and/or measurements. These investigations are dealing primarily with such specific aspects of the plasma properties in microwave fields as the electric conduc- tivity, the break down behaviour etc. For this purpose the electron distribution function is developed not only in Legendre polynomials related to the velocity space but also in a Fourier series in time. Especially a frequency range is coxered in which only a non- stationary evolution of impulse transfer and there- fore also only such one of the anisotropic part of the distribution function occur whilst there is a nearly time independent energy transfer due to the colli- sions and as a result practically no alteration of the isotropic part during one cycle because of the high frequencies considered. Furthermore some research has also been done to describe more in detail the generation of higher harmonics of the distribution function - particularly of its anisotropic part - in wide frequency ranges of field and collisions, using the same kinetic basis [S].

Moreover, we would like to mention the research work done by Parker and Lowke, Lucas, Skullerud as well as by others [9] on the space-time development of an electron swarm based on the Boltzmann equa- tion or Monte Carlo simulations. Finally the investi- gation of the time evolution of the distribution function in a beam-excited field free Xe and Ar laser discharge by Elliot and Greene [lo] is worth mentioning in this context.

In the following we refer to collision dominated plasmas with low and medium degrees of ionization.

In these the relaxation of the electron component is determined by energy and impulse transfer from the electrons to the atoms and/or molecules which is realized by collisions. The main processes under this aspect are the elastic and the inelastic collisions

- such as excitation and dissociation - between electrons and the heavy particles. In addition, with increasing ionization degree also the Coulomb inter- action, especially among the electrons, gains a remar- kable influence particulary on the interchange of energy between the electrons of the different energy ranges and thus indirectly on the energy transfer from the electrons to the atoms. Furthermore, in certain gases there are also other kinds of collisions

which can win an additional influence on the relaxa- tion processes, for example attachment in the case of electronegative gases or the recombination of electrons and ions with growing degree of ionization.

Because of the numerous kinds of possible collision processes we will confine ourselves to plasmas with sufficiently low ionization degrees and therefore, besides the influence of an electric field, t o the action of elastic and inelastic collisions. Also the main properties of the temporal development and the relaxation mechanism of the electron component can be understood already from studies of inert gas- plasmas.

Under these aspects those former works are of interest in which particular attention is given to the temporal energy relaxation by collisions. Special problems are the decay or in broader terms the temporal transitions between different states as well as the periodic behaviour and also its adjustment.

We may mention here a number of papers based on the kinetic approach (*), being roughly classified in the table I.

Most of the cited publications with numerical results are restricted to elastic collisions or to special models of inelastic collisions with the exception of the papers [25,26] and [27], in which exciting collisions

are also included.

In the following we will discuss some more compre- hensive results which were recently obtained in the investigations both of the field free collision domi- nated relaxation in the afterglow during the first period of temporal decay and of the collision domi- nated relaxation of the electron ensemble in a plasma with additional heating by an electric field. These calculations take into account all main collision processes, using realistic functions for the energy dependence of the correlated collision cross sections.

The results were attained by the determination of the isotropic velocity distribution function of the electrons and resulting macroscopic parameters. In this way a physical understanding of the macroscopic properties is attained on the basis of kinetic theory.

It is also possible to gaid an illustrating qualitative insight into the temporal occurrences with the aid of defined and numerically determined characteristic relaxation times, though we are not in a positioq here to present such a concept in detail because of the limited time.

2. The theoretical background. - Basis of the electron kinetics of a non-stationary anisothermal weakly ionized plasma under the above mentioned aspects described by a kinetic equation is the Boltz- mann equation, in which we consider elastic and

(*) The papers use the Boltzmann equation or Monte Carlo methods. Besides this there are investigations applying balance equations; also relaxation processes due to Coulomb interaction only are not within the scope of our considerations.

C7-254 J. WILHELM A N D R. WlNKLER

inelastic collisions. In the Lorentz approximation the tion in the Legendre polynomials leads to the equation usual development o f t h e velocity distribution func- system

in which the transformation to the volt equivalent of the momentary energy was performed in conformity with eo U = mv2/2.

Here f and fA are according to

the normalized isotropic and the first anisotropic part of the distribution function.

From the normalized function f the electron con- centration n e e ) is determined by the expression

Furthermore

%/PO = (2 eo/m)1'2 (E~Po) (llU1l2)

,

Vd/pO = (2 eo/m)112 U1l2 ng Qd(U) , (4a, b, c)

Q/po = (2 eo/m)112 u1I2 ng Qp(U)

represent the characteristic frequency for the action of the electric field and the collision frequencies for impulse transfer and-for the inelastic processes res- pectively. Finally E(t) is the homogeneous electric field, Qd(U) and QF(U) are the total cross sections for impulse transfer - and for the different inelastic collision processes, t = p o t the normalized time, p o the time-independent pressure of the neutral atoms related to 0 OC, ng the atom concentration at 1 torr and O O C , Uk the threshold potentials of the several inelastic processes. Because of the special binary collision processes between electrons and atoms considered here the introduction of such normalized time scale 7 becomes possible and in the same way we thus obtain normalized relaxation times.

In (lb) we have as usual neglected the unimportant terms of the inelastic collisions.

2.1 THE FIELD FREE CASE. - Under the condition E = 0 the eqs. (la, b) for f and f, are not connected with one another and can be handled separately.

Eq. (lb) for the anisotropic part ^fA is reduced to a simple usual differential equation of first order with the momentary energy U as parameter 1301 ; thus the solution can be found only if an initial function f, at

7

⁼ 0 is given. On the other hand (la) is simplified to a partial differential equation of first order for f with additional difference-terms due to the ine- lastic collisions. The temporal behaviour o f f has a unique solution in the ranges of

7 >

^{0 and U 2 0 if}

an initial distribution at the moment T = 0 and the boundary condition

lim f(U,

t)

= 0

U+ co ( 5 )

for all

7

are given ; the latter condition follows from the fact that the distribution function can in prin- ciple be normalized. The concentration and energy balances, consistent with (la), are obtained by inte- gration over U from 0 to co and by such integration after multiplication with U respectively in the form

means - the volt equivalent of the mean electron energy, Ue' and

pF

the volt equivalents for the energy losses by elastic and the different inelastic collisions normalized per one electron and time unit and accordingly

uc

designates the total energy loss by collisions.

Especially for these quantities there are the rela- tions

PROGRESS IN KINETIC DESCRIPTION OF NON-STATIONARY BEHAVIOUR OF ELECTRON ENSEMBLE C7-255

with

v p

as the frequency for the k-th inelastic col- In order to solve this system of partial differential lision process. equations it is useful to estimate the rapidity of the

temporal alteration of the isotropic and the aniso- 2 . 2 NON-VANISHING ELECTRIC FIELD. - In this tropic part f and f ~ .

case the eqs. (la, b) are connected with one another. Next there are the relations

which are realized nearly for all gases. A detailed qualitative discussion of the equation system (la, b) shows that using the supplementary condition

(i.e. the characteristic frequency for the temporal alteration of the electric field is much smaller than the fre- quency of impulse transfer due to elastic collisions) a rapidity of the temporal changing of the anisotropic part follows which is by some orders of magnitude greater than that of the isotropic part. Then the formal solution of (1 h) is reduced under the given conditions (8) to

and we can see that the adjustment of the anisotropic part proceeds exponentially with the frequency for impulse transfer to a state which is determined by the momentary electric field

~ ( i )

and by the isotropic distribution

f

(U,

7).

Therefore, instead of (1 b) its quasi-stationary approximation can be used.

If we suppose for the further investigation that the characteristic frequency for the field alteration is always small in comparison with pd/p0, i.e. excluding ac-discharges in the'range ofvery high frequencies, the system (la, b) is simplified to the following parabolic partial differential equation of second order with additional difference- terms

In (10) ~ ( t ) / p , is the only plasma parameter deter- A detailed discussion suggests to apply the condi- mining the temporal behaviour. tion [31]

Now, in the same ranget

>

0 and U 2 0 the unique

solution of (10) requires an additional boundary

a

lim

-

^{f (U,}

7)

= 0 for

7

2 0

condition at U = 0 in relation to the field free case. u-o

au

^{(1 1)}

C7-256 J. WILHELM A N D R. WINKLER i.e. that for a nearly arbitrary behaviour of Q , for _web

U -, 0 the field term does not represent a source

/E/h!, a 5 v em-'rorr-'

of electrons in the concentration balance (6a).

Applying this additional boundary condition, we parameter: [ ~ o r r

SI

obtain the same concentration balance (6a) but instead of (6b) the generalized form of the energy balance

d -

-

---

^{U =} U,/po -

F7"/Po .

dt (12)

Here

-

-

UF(~)~PO = PO be(El~,)2 (1 3) is the normalized energy input from the electric field per electron and per time unit with the electron

*.,-

mobility be due to

po b,c) = - (2 eo/m)"2 - omJ&+.du (14)

3 n, Fig. ^/ la.

3. Review of the temporal relaxation of the electron distribution function and of characteristic macroscopic parameters. - 3 . 1 RELAXATION IN A TEMPORALLY DECAYING PLASMA WITHOUT FIELD HEATING. - With E ⁼ 0 the solution of (lb) gives

w i t h L ( ~ ) as an arbitrary initial distribution for the anisotropic part. We observe that the anisotropy decays exponentially with the frequency rd(U)/po in the normalized time

2.

Starting for instance from a stationary state in the column plasma of a usual glow discharge and assuming typical values for the diffusion cross section of 10-l6 to 10-l5 cm2, the anisotropic part f, is markedly damped down within some l o p 9 torrs. Because of the relation

for the particle current density of the electrons practically the same short relaxation time results also for the electric current of the glow discharge plasma.

Such situation is realized for instance after a jump- like switching off of the electric field in a stationary plasma.

For the same kind of decay the temporal evolution of the isotropic part f was investigated [30, 321 in the inert gas neon and the molecular gas hydrogen in order to illustrate the general behaviour. (la) is reduced to a partial differential equation of first order with difference terms.

As initial condition we choose the stationary iso- tropic distribution

Fig. Ib.

existing at

7

= 0 which characterizes the moment of switching off the field. The numerical solution was performed with the aid of a finite difference method using centered second-order-correct difference ana- logs. The concentration and energy balance (6a, b), consistent with the applied kinetic equation, were used to control the precision of the calculated iso- tropic distribution. Figures l a , b show the temporal development of the isotropic distribution as function of U with ?as parameter performed for a neon plasma, where we described the exciting collisions by a lumped state with a threshold energy of Ue = 16.6 V. The figures are related to the two different states deter- mined by (Elp,),, = 5 and 0.3 V/(cm torr). These initial stationary states are characterized by pre- vailing energy losses due to exciting or elastic colli- sions. It can be seen that the first part of the decay is determined first of all by the energetic situation in

PROGRESS IN KINETIC DESCRIPTION O F NON-STATIONARY BEHAVIOUR O F ELECTRON ENSEMBLE C7-257

the initial state where the field is switched off. If this state is dominantly determined by energy losses ^{m o}

owing to exciting collisions (Fig. la) at first only a H 2

quick damping of the tail takes place during a charac- ^{- 1} (&)st*1o Z K ' ~

teristic time of x torr s for a noticeable tem- paromster:i [ ~ o r r s l

poral alteration. The depopulation of the high- ,,., ^- energy range is finished within x 3 x 1 0-7 torr s.

Then the relaxation is continued via the further _lo.3 thermalization of the distribution only by elastic collisions with a characteristic time of x 10- torr s for a noticeable change. In the second case (Fig. lb) the initial distribution is essentially restricted to the energy range where only elastic collisions occur; ^{l o - '} therefore a noticeable alteration of the distribution is found in the characteristic time z l o p 6 torr s because now exciting collisions are insignificant.

Because of the very different atomic data of mole- ^10-7, cular hydrogen compared to those of the inert gas neon also a very different relaxation behaviour of ,,.s

the distribution function in hydrogen results. In ^{0 5 10 15}

figures 2a, b the time development of this function is _{Fig. 2b.}

presented for the initial field strengths (Elp,), = 23 and 10 V/(cm torr).

Performing the calculation, we used the main collision cross sections of vibrational excitation, of electronic excitation for singulet and triplet levels and of ionization, moreover the diffusion cross section for elastic collisions [33]. As it can be seen from figure 2a there exists a varied relaxation beha- viour in the energy range from 0.5 to x 9 V of marked vibrational excitation and in the range greater than x 9 V of electronic excitation and

range. This characteristic time of lo-' torrs can be observed also from figure 26 where for the decay particularly the vibrational excitation is significant.

Furthermore we would like to emphasize that the characteristic decay times obtained for hydrogen are shorter by orders of magnitude compared to those in the neon plasma, which is primarily due to the differences in the atomic data.

ionization. Whereas the characteristic time for noti- 3 . ^IN A PLASMA WITH FIELD

ceable decay of the distribution amounts to ^- In the cases considered in the previous section the torr in the latter range we find such a relaxation processes proceed without energy input by time Of 10-8 torr for the first an electric field so that the temporal development of the isotropic distribution and thus of the macroscopic behaviour always leads to a monotonous energy loss of the electron component and therefore causes a thermalization effect. A more complicated situation will result if in addition an electric field acts on the electrons. Then the temporally successive states will

2 be determined by two competing processes, the

23 Vcm-91orr.l

momentary energy input from the electric field and

mrameter: i TO,^.] the energy losses by the different kinds of collisions.

From the point of view presented at the beginning of the paper there exist several problems which are of general interest; for instance the question how the transition is performed to a stationary state which is determined by a time independent value of the electric field and produced for example by a sudden jump in the field. It is further of some impor- tance to understand the behaviour in continuous time dependent fields. In particular it is useful to explain the state behaviour in aperiodical and perio- dical electric fields in different situations according to diverse conditions of parameters.

Besides the practical aspects of such questions it

Fig. 212. seems of physical interest to identify the dominant

C7-258 J. WILHELM A N D R. WINKLER

relaxation processes under varied parameter condi- tions and thus to provide a detailed understanding on a kinetic basis including the different time constants which characterize them. In addition to the field free decay, where we used characteristic times for noti- ceably altering the distribution, now a characteristic time for the adjustment of the final state gains increasing interest, for instance the adjustment of a stationary state or of the periodical behaviour.

Moreover, it is important to determine the limits of the characteristic time for noticeable alterations of the electric field at which the instationary behaviour of the electron component changes to a quasi-statio- nary variation in the given field or to the other extreme case of a slow time dependent behaviour of the quan- tities in rapidly varying fields compared to the field alteration.

Many of the questions mentioned here concerning the relaxation of the electron component in a weakly ionized plasma can already be clarified via the inves- tigation of an inert gas plasma with its more simple structure according to the collision processes. In the last years a sure and a widely applicable method was developed [31] to obtain a very precise solution of eq. (10) as an initial-boundary value problem for different parameters. In order to perform the cal- culation of the isotropic distribution a generalization of the Crank-Nicolson difference-method has been used including an iterative treatment of the supple- mentary difference terms resulting from the inelastic collisions.

In the following sections we will discuss results

state. For this we present besides the distribution f(U,

7)

first of all the time development of two quan- tities, the mean energy

U6J

described by (7a), which characterizes the electrons in the bulk of their dis- tribution and the normalized lumped frequency of exciting collisions v,fi)/p,. The latter is a special case of (7d), because we want to confine ourselves primarily to inert gas plasmas, where in certain cases we can work with such a summarized excitation level, v, characterizing the high energy tail of the distribution. Besides this, two other quantities are very helpful for physical understanding, the energy transfer relation RF and the energy loss quotient R, which in conformity with (13) and (7b, c) are des- cribed by

The first term compared the energy input from the electric field with the total energy loss by collisions and is therefore, according to the energy balance (12), representative for the degree of the non-stationarity.

The second quantity describes the ratio of the energy loss by elastic collisions to the total energy loss by both types of collisions.

Our investigations were performed in neon plasma using the well known data on the cross sections. We always started with the stationary distribution deter- mined by (E/P,)~ which represents the electric field before momentary switching to the new value (E/P,)~.

As representative examples we consider the transi- tions in table 11.

obtained in solving some of the mentioned problems. Table 11.

3 . 2 . 1 Relaxation after a jump-like change of the (Elp,), [V/(cm torr)]

- ( E ~ P , ) ~ [V/(cm torr)]

electricfield. - At first we discuss some results [31,34] ^-

concerning this case, which allows a physical under- 10 (0.0092) standing of the relaxation mechanism and the main 2 (0.116)

reasons which determine the characteristic times for 0.6 (0.866) 1 (0.408) the adjustment

5

= p, z, of the final new stationary 0.3 (1.000)

PROGRESS IN KINETIC DESCRIPTION O F NON-STATIONARY BEHAVlOUR O F ELECTRON ENSEMBLE C7-759

The numbers in brackets are the values of R, for the stationary initial and final states, in each case determined by (Elp,). It can be seen that the initial states vary from such ones determined dominantly by energy losses due to exciting collisions to those determined by elastic losses. On the other hand the final state is characterized by nearly equal elastic and exciting energy losses.

In figure 3 we can see the time alteration of the distribution function determined by the chosen tran- sition from (Elp,), = 10 V/(cm torr) to

(E/P,)~ = 1 V/(cm torr)

.

Due to the great field heating in the initial stationary state immediately after the jump-like decrease of the field a marked depopulation of the high energy tail takes place, noticeable already at normalized times of x torr s. This behaviour is caused mainly by the action of the exciting collisions. The whole relaxation process is finished within some l o p 6 torr s.

Figures 4a, b show the time development of

U

and vJpo and figures 5a, b that of the energy transfer relation R, a_nd the energy loss quotient R, as function of the time t. It should be emphasized that despite very diverse stationary initial distributions all tran- sitions to the new state are realized in nearly the same time of 4-7. l o p 6 torr s. Generalizing this result we have verified that, starting from different sta- tionary initial distributions, the characteristic time for each adjustment of a new stationary'3tate is dominantly determined by the special situation .in the energy loss due the different kinds of collisions in the final state and is nearly independent of the specific initial stationary distribution.

In order to overlook the range of alterations of the adjustment times, which is rather wide due to broad changes in the parameter (E/P,)~ of the final states, we discuss some results which are presented

Fig. 4h.

Fig. 50.

Fig. 4u. Fig. 5h.

C7-260 J. WILHELM AND R. WINKLER

Fig. 6.

in figure 6 and describe several transitions to new' stationary states. The values of (E/P,)~ used for the final states cover a range of field strengths which are of interest for many column plasmas. We can observe that the adjustment time

<

changes by nearly four orders of magnitude due to the very different energy kansfer situations in the final states caused by the collision processes. These values of

<

can be taken for example from the adjustment of the energy transfer relation R$) towards the value 1 of the stationary state for the different transitions presented in the figure 7.

parameter. a-b

Additionally, in figure 6 the values of the total energy losses per sec (Uc/p,),, in the final stationary states are given, which illustrate that the decrease of the adjustment time is related to an increase in (Uc/p0), of nearly the same order.

From the energy loss quotient RIt in the final state also presented in figure 6 we can see that the greatest alteration in the adjustment time takes place in the range where the transition is perfo;med from the states dominantly determined by elastic collisions to such determined by exciting collisions. In neon this region is approximately given by

0.5 V/(cm torr) ,< (Elp,) d 3 V/(cm torr) as illustrated in figure 6 by the course of the energy loss quotient RESt in dependence of (E/P,)~ However at lower as well as higher (E/P,)~ values there are still further but smaller changes in the adjustment time. At these values the energy losses are realized almost exclusively via one kind of collision process.

The strong alteration of the adjustment time can be explained qualitatively with the fact that the energy transfer per electron becomes increasingly more efficient with growing (E/pJf. On the one hand with increasing (E/P,)~ the electrons transfer a higher amount of energy to the neutral atoms in elastic collisions. On the other hand in neon above

(Elp,) x 0.5 V/(cm torr)

the exciting collisions, which are much more effective than the elastic collisions because at least the threshold energy is lost per one collision, gain more and more importance. This can be distinctly recognized from figure 8, where some selected stationary distribution functions are shown which illustrate the extraordi-

Fig. 7. Fig. 8.

PROGRESS IN KINETIC DESCRIPTION OF NON-STATIONARY BEHAVIOUR OF ELECTRON ENSEMBLE C7-261

(*) In [25] a similar sensitive dependence of this adjustment time on the normalized electric field strength (Elp,,), in the final stationary state was found by Monte Carlo investigations in a He plasma.

narily different population of electrons in their 8~

-"

^I

, ^{, , , ,} , ^{, , , ,} ,

^,

^,

^{, i}

energy space for different electric field strengths k-5

-

Fig. IOU.

(E/po)st

(*I .

The observed dependence of the adjustment time will be valid even if we do not start from stationary states correlated to (E/p.), but from another initial distribution according to a special phygical situa- tion [32]. To illustrate this with a representative 6

example we applied initial distribution functions of the form

Finally, we will make a short remark concerning the dependence of the adjustment time on atomic data such as the different magnitude of cross sections and atommass in different gas plasmas. In figures 10a,b you can see the evolution of the mean energy

0

and the lumped excitation frequency vJp, for the same transitions

-

¹ -

-

^'\

-

hlr: 1-3

-

7- -

-

-

-

-b-, ^7'

--____

^{1 1- hlrzs-1}

-

---- _--- -- -

i 'f ^{--. -.}

^{\-- '\,, hlr.1-o,z 'b,} \,

^{- - - -}

-

Fig. 10h.

18

( 2 n

+

3)n+'312'

u n

e-(2n+3)"/(2u)

h ( U ) = 9 5-

f i ( 2 n

+

1) ! !

cn

^+(3/2)

^-

^-

^-

n 2 O (18)

- -

-

which becomes a Maxwellian distribution for n = 0 ;

for greater n more beam-like distributions around L

-.

^---_____

_{--- --}

the mean energy result. In figure 9, besides the

--

^',

initial stationary distribution at

( E / p J i = 0.5 V/(cm torr) ; ₃

distributions according to (18) for n = 0.3 and 10 with the same mean energy are shown. Despite the very diverse initial distributions we found in each case nearly the same adjustment time in which the same final state was reached.

.-..Ar:l-QZ I,,

-

-

.,. '. -

- ^'\

'\ ',

-

- parameter:o-b '\

'\

'\

-

- O*E/PO)I , b=(E/po)f ,

',

-

- [v/(cm b r r ) ] ^\ \,

-

-

'.

- "\

---_a

-

2-

_- -

^{fCTorrs1 '\}

^\. ----______ ^-

1 / 1 1 1 1 1 1 1 1 1 1 l 1 1 1 1 l I 1 1 1

o 10q7 10-5 lo4

C7-262 J . WILHELM AND R. WINKLER

Table 111.

for Ne as well as Ar [35]. Indeed there are conside- rable changes in the values of the adjustment time, especially for the first two transitions where diffe- rences of about one order of magnitude for the two considered gases result. This dependence of the adjustment time can be explained first and foremost by the great differences in the energy transfer situa- tion for the two gases at the same field strength

( E / P ~ ) ~ in the final stationary state and finally by the different atomic data.

3 . 2 . 2 Relaxation in aperiodically altering jields. - If the electric field is acting during the relaxation process, in general a more complex situation results because of the rivalry of the continuously altering energy input by the field and the energy loss in colli- sions. Limiting cases are on the one hand the nearly jump-like change of the electric field - as treated above - and on the other hand the quasi-stationary change simultaneously with the field. Apart from other questions it is of interest to find relations which characterize these limiting cases. In order to investi- gate these questions we considered an aperiodical continuous field alteration between the initial and final values (Efp,), and ( E / P , ) ~ according to the expression

during the normalized time

7,

[36]. Next we discuss Table IV.

the results of the calculations performed for the

parameters of the following table. In figure l l a the (EIPoX (EIPO), -

temporal development of the summarized frequency [V/(crn torr)] [V/(cm torr)] t , [torr s]

- - -

v,F)/p, for exciting collisions is presented for Ne as _{I 0.35} 2 ~ 1 0 - ' ; 1 ~ 1 0 - ~ ; 5 ~ 1 0 - ~ ;

full lines. Curve parameter is the normalized time

- 1 ~ 1 0 - ~ ; 2 . 5 ~ 1 0 - ~ ; ~ X I O - ~ ;

tE for adjustment of the final value of the electric 1 x l o - 4 ; 4 x l o - 4 field.

In any case the dashed lines denote the alterations calculated in a quasi-stationary way whilst the curve marked with 0 is the change after jump-like switching off of the electric field. The figure clearly shows that the non-stationary time behaviour changes between these two limits, an evolution which is close to that after jump-like alteration of the electric field at suf- ficiently small field transition times

&

and a behaviour at sufficiently high

&

which coincides with the quasi- stationary course. This result is self-evident from the physical point of view. It can also be seen that the limit times are in this special case x 2 x and

The change of the time behaviour between these two limits will once again be demonstrated very clearly by the course of the quantity R, in figure 1 I b.

In the range of small field transition times

iE

we notice a strongly non-stationary behaviour - the values of R, are far away from the stationary value 1 -

which becomes smaller and smaller with increasing

&.

The marked disturbance in the stationary state at smaller

iE

results fpm_a quick change in the momen- tary energy input U F ( t ) whilst the momentary energy loss

U,G)

by collisions follows this alteration much

PROGRESS IN KINETIC DESCRIPTION O F NON-STATIONARY BEHAVlOUR O F ELECTRON ENSEMBLE C7-263

more slowly. Thus we observe that, via the rapidity of field alteration, the time dependent electric field exerts in addition a noticeable control on the relaxa- tion process and thus also on the adjustment time to the new stationary state or to the quasi-stationary behaviour of the electron component.

the losses by collisions. In the further statements we will give some examples which we have obtained in investigations of the temporal development of the electron ensemble in a dc-field (Elp,), with a super- imposed ac field with the amplitude (Elp,),. Whilst in [37] the non-stationary behaviour was investigated for small degrees of modulation in connection with the studies on the impedance, in which the linea- rized Boltzmann equation was used, we are now [38]

in a position to examine situations with high degrees of modulation.

To calculate the periodical temporal evolution of the isotropic distribution it would be sufficient to know the real periodical state at one moment and to use it as initial state. But because such a distribution is unknown it is only possible to obtain an insight into the periodical behaviour by calculation of an adjustment process starting from a chosen initial state. Also from this point of view it is of interest not only to establish the periodical behaviour as dependent on different parameter conditions but also the adjustment to this final periodical state.

The following considerations are related to a field course

E(t)/p, = (Elp,), [l

+

M sin 2 n?ip] (20) with the degree of modulation

Fig. I l h.

M = ( E l ~ o ) , l ( ~ l ~ o ) , (21) 3 . 2 . 3 The time evolution in periodical fields. - and the normalized time of a period of the electric In this case also a very complex situation is to be ^, field. As a representative example we consider the expected which results from the competition between case described by the parameters given in table V the momentary energy input by the electric field and for a Ne plasma.

C7-264 J . WILHELM AND R. WINKLER

Table V.

- t, : 5 x 2 x 5 x

1 x 5 x 2 x torrs

To illustrate the behaviour after adjustment to the periodical state in figure 12 the alteration of the energy distribution is exhibited over one period (here des- cribed by the parameter z =

tfiP

counted from the start

- of .a period after the adjustment process) with

t, = 2 x l o p 6 torr s. It can be noticed that the high energy tail of the distribution increases at the beginn- ing of a period until the field has reached nearly its maximum value (which is given at z = 114) and then decreases until the field has almost reached its mini- mum. Then the distribution returns to its initial course again. In the low energy range the distribution shows a reverse behaviour and does not show such sensitive changes.

Next, let us have a look at the figures 13a, b where the temporal change of the mean energy and the summarized frequency v,/p, for exciting collisions is presented as full lines according to the parameters

t , of table V. It should be pointed out that at greater

- -

t,-values the temporal alteration is realized in a nearly quasi-stationary way so that at each moment the non-stationary values are close to the stationary ones also presented in these figures as dashed lines which are at any moment correlated to the electric field. With decreasing timet, of a period the deviation from the quasi-stationary course becomes increasingly greater, an effect which is coupled with an increasing reduction of the modulation of the macroscopic quantities and a growing phase shift between the

Fig. 13h.

macroscopic quantities and the electric field. The continuous transition of the temporal behaviour from a nearly quasi-stationary one to the limiting case of small amplitude modulation can be easily understood under the aspect of the energy balance (12), which is consistent with the Boltzmann equation.

Figure 14 shows the non-stationary course of the

Fig. 13a. Fig. 14.

PROGRESS IN KINETIC DESCRIPTION OF NON-STATIONARY BEHAVIOUR O F ELECTRON ENSEMBLE C7-265

energy input

GIpo

and the total loss

clp0

by colli- sions. There are only small differences between input and loss at the greatest chosen

7,

and thus we find disturbances of only minor importance in the sta- tionary energy balance. With decreasing period time

-

t , the deviations continue to grow. It is obvious that

this is primarily caused by a change of the course of the energy losses via collisions. This latter phenome- non is combined with a continuous diminution of the modulation due to the fact that collision processes can compensate the momentary energy input less and less during larger parts of one period. Finally we would like to illustrate the adjustment behaviour using two examples which show the temporal development of the lumped collision frequency ve/po for the two different cycle times

-

t , = 5 x and

7,

= 5 x lo-* torrs ,

starting in each case from 3 different initial stationary distributions. The initial conditions are correlated to the zero passage of the ac-part as well as to the maximum and the minimum of the electric field.

Figures 15a, b make it evident that there are very different adjustment times in which the same perio- dical state is reached. These are determined by the special energy transfer situations during the adjust- ment process and depend on the values of the electric field attained during the adjustement process and on the rapidity of the periodical field alteration.

3 . 2 . 4 An example for the comparison between calculated and experimentally determined relaxation.

behaviour of the electron component. - Using the above results, a special examination of the relaxation behaviour of usual column plasmas in inert gases resulting from collision processes and of the addi- tional action of a temporal decaying electric field -

-

which can't be presented here in detail-shows that noticeable first alterations will occur already after very short characteristic normalized times

-

t z torrs

if the energy transfer to the neutral atoms takes place dominantly via exciting collisions in the first period of the development. Time resolved measurements of the isotropic distribution and dependent quantities such as the mean energy, drift velocity, frequency of direct ionization, Townsend coefficient etc. are very complicated and laborious under this condition. On the other hand, when the energy transfer in the first period is realized primarily via elastic collisions, marked variations will appear in times

- t w t o r r s ,

which makes possible a comparison between theore- tical and experimental results. We made such a comparison for the decay of the electron-atom- Bremsstrahlung continuum. This continuum can be measured in a sufficiently short time and permits the registration of the temporal change of the isotropic distribution almost in the whole energy space via the wave length dependence of the Bremsstrahlung inten- sity. Therefore the time behaviour of the Bremsstrah- lung in the range of 200 to 1 000 nm was calculated [39]

in decaying Ne and Ar column plasmas under medium pressure conditions starting from stationary states and using decay profiles for the electric field as they are suggested by experimental measurements ; on the other side time resolved decay measurements of the relative radiation intensity were performed for some ranges of the wavelength 1. To calculate the intensity it is necessary to know the volume-emission-coefficient of this radiation

&(A,:).

We calculated the intensity I per unit length of the column according to the relation

parameter: i , [ ~ o r r s]

I

^lo'

1

/!-3,75*10-8

i

^-

_- 1 -

^{I [10'7Torr r]}

I I L I ^I I ^I I 1 I ,

0 1 2 3 4

C7-266 J . WILHELM AND R. WINKLER assuming the isotropic distribution of the electrons

and the atom concentration N as nearly independent of the radial coordinate r ; N , is the electron concen- tration averaged over the cross section of the column and R the radius of the tube. According to (22) the time dependence of I can be realized via the distribu- tion function and the electron concentration. Under the considered conditions the characteristic time for a noticeable onset of the relaxation process is, as men- tioned already, 2 torr s, whilst marked altera- tions of the electron concentration are to be expected not before 10-4-1'0-3 torrs. Therefore down to a normalized time of w torr s the time depen- dence of the emission is almost exclusively determined by that of the isotropic distribution.

In order to calculate the evolution of the distribu- tion function we must known the temporal change of the electric field. In the ps-range time resolved direct measurements of the internal field are not easy to realize but such of the time course of the external discharge current after switch off are not so difficult.

The approximate course of EG)/~, can be obtained from the decay of the current flowing in the external circuit. Because under the considered conditions the relaxation of the anisotropic part of the distribution function and therefore also of the current occurs in an almost stationary way we can assume in a first step that the electric field is nearly proportional to the external current during the decay. As the measure- ments showed in the majority of cases a nearly linear decay of the discharge current in an absolute decay time td of some ps, we used in the calculations also such linear field decay (the latter indicated by - -

-) and additionally a parabolic decay with the same time t,(-.

. -.

^.) and a jump-like decay (- - - -).

The examples given in figures 16a, b are the decay of the relative intensity I(& t)/I(A, 0) for 1 = 380 nm in neon (j7, = 20 torr, (Elp,), = 0.42 V/(cm torr), td = 1 x s) and for 1 = 330 nm in argon (Po = 8.8 torr

,

(Elp,),, = 0.64 V/(cm torr)

,

that means the agreement of the measurements with the calculations for the linear and parabolic decay profiles is satisfactory. Though the real decay of the electric field in the column plasma is unknown today these and further results we obtained can nevertheless be reasonably explained by the collision dominated relaxation mechanism of the electron component applied here.

The above results show that on the basis of the kinetic theory a deeper insight into the temporal relaxation mechanism of the electron ensemble in collision dominated plasmas can be gained for many

b . 2 0 Torr R=300nrn

0

\ \

0 ; ' .- e ,

,

0 1 2 3

Fig. 16a.

different models of the temporal behaviour of the electric field. In addition, we investigated some more generalized relaxation models of the electron compo- nent which take into account the action of a pure high frequency field [40] as well as ionizing collisions and a particle loss term for the electrons [41], however they cannot be described here in detail. Thus a firm basis has been established to perform further investigations of more general problems grouped around the relaxa- tion models presented here which are of interest under various aspects. Extensions of the kinetic methods to still more generalized conditions, which cover addi- tionally also such processes as the Coulomb interac- tion between the electrons and the attachment of electrons, seem also promising.