# Electron Kinetics with Attachment and Ionization from Higher Order Solutions of Boltzmann's Equation.

код для вставкиСкачатьAnnalen der Physik. 7. Folge, Band 46, Heft 1, 1989, S. 21-40 VEB J. $. Barth, Leipzig Electron Kinetics with Attachment and Ionization from Higher Order Solutions of Boltzmann's Equation By R. WINKLER*,G. L. BRAGLIA**, J. WILHELM* *) Zentralinstitut fur Elektronenphysik, Greifswald, DDR **) Dipartimento di Fisica, Universith di Parma, Itnlia A b s t r a c t . An appropriate approach is presented for solving the Boltzmann equation for electron swarms and nonstationary weakly ionized plasmas in the hydrodynamic stage, including ionization and attachment processes. Using a Legendre-polynomial expansion of the electron velocity distribution function the resulting eigenvalue problem has been solved a t any even truncation-order. The technique has been used to study velocity distribution, mean collision frequencies, energy transfer rates, nonstationary behaviour and power balance in hydrodynamic stage, of electrons in a model plasma. and a plasma of pure SF,. The calculations have been performed for increasing approximation-orders, up to the converged solution of the problem. I n particular, the transition from dominant attachment to prevailing ionization when increasing the field strength has been studied. Finally tho establishment of the hydrodynamic stage for a selected case in the model plasma has been investigated by solving the nonstationary, spatially homogeneous Boltzmann equation in twoterm approximation. Elektronenkinetik mit Anlagerung und Ionisation aus Losungen der Boltamann-Gleiehung in hoherer Ordnung I n h a l t s u b e r s i c h t . Ein geeignetes Verfahren zur Losung der Boltzmann-Gleichung fur Elektrononschwarme und fur nichtstationare, schwachionisierte Plasmen im hydrodynamischen Zustand wird vorgestellt, welches 1onisa.tions- und Anlagerungsprozesse mit enthalt. Unter Verwendung einer Legendre-Polynom Entwicklung der Elektronen-Geschwindigkeitsverteilungsfunktionwurde das resultierende Eigenwertproblem in beliebiger gerader Abbruchordnung gelost. Diese Technik fand zur Untersuchung der Geschwindigkeitsverteilung, der mittleren StoOfrequenzen, der Energieubertragungsraten? des nichtstationaren Verhaltens und der Energiebilanz im hydrodynamischen Zustand von Elektronen in einem Modellplasma und in reinem SF, Verwendung. Die Berechnhgen erfolgten fur zunehmende Approximationsordnungen bis zur konvergenten Losung des Problems. Insbesondere wurdc der mit wachsender Feldstarke erfolgende Ubergang von dominierender Anlagorung zu vorherrschender Ionisation studiert. SchlieSlich erfolgte die Untersuchung der Einstellung des hydrodynamischen Zustandes fur einen ausgewahlten Fall im Pllodellplasma, indem die nichtstationare, raumlich homogene Boltzmanngleichung in 2-Termnaherung gelost wurde. 22 Ann. Physik Leipzig 46 (1989) 1 1. Int8rodnction Recently a new approach has been developed t o determine the solution of the Boltzmann equation and relevant macroscopic quantities for electron swarms and stationary weakly ionized plasma,s a t high orders of approximation [l,21. This approach has been applied to study the electron behaviour in different model, inert and molecular gases. These investigations were limited t o conservative inelastic collisions [I- 61. A still open problem concerns the study of the additional impact of non-conservative processes, such as ionization and attachment. Some efforts in this direction have recently been made for model gases [7, 81 and molecular nitrogen [9], considering mainly the isolated effect of attachment a t very low fields or ionization a t relat'ively high values of EIN. The present paper deals with the extension of our just mentioned t,echniclue t o include non-conservative inelastic processes, i. e. ionization and attachment. Using this extended approach, the electron kinetics in a model gas (with attachment and ionization) and in pure SF, are studied and the results properly interpreted. The main interest of our investigation is for spatially integrated properties of electron swarms and electron kinetic properties of nonstationary spatially homogeneous plasmas in hydrodxnamic stage. I n particular, results relevant to mean collision frequencies and corresponding energy transfer rates are reported and discussed. This is made for increasing approximation-orders, beyond the conventional two-term approximation, and for a large range of (time-independent) reduced-field strength BIN. However, the nonstationary behaviour and the power balance of the electrons is also considered in detail. I n addition, the relaxation of the velocity distribution function to the hydrodynamic stage is studied for the mentioned same model plasma, having recourse to the solution of the nonstationary, spatially homogeneous Boltzrnann equation in two-term approximation. Starting from different initial distributions, the same steady-state velocity distribution and corresponding macroscopic quantities are obtained as found when solving the kinetic equations in hydrodynamic stage in two-term approximation which, in turn, also confirms the validity of our new, more general and complex solution-technique relative to the hydrodynamic stage. 2. Theoretical Background Our starting point is the Boltzmann equation aF - + v . V,F at e + Ck C i p ( F )+ Ci(F) + C u ( F ) , - A E . V,F = Cez(F) m for electron swarms, or its spatially uniform version for nonstationary plasmas. Here, F ( r ,u,t ) is the electron velocity distribution function, E a constant electric field, fie(., t ) the electron density, e, and m the charge magnitude and the mass of the electrons Moreover, Cel,C i p , Ciand Ca are the collision integrals for elastic collisions, conservative inelastic collisions, ionization and attachment, respectively. Substitution of the expansion [ 101 F(r, V , t ) = F(")(v)n, + F("(v) . (-V,.) n, + .. . (2) of the velocity distribution function (for the hydrodynamic stage) with respect to the gradient of the electron density into the Boltzmann equation, after integration over the R. WINKLERe t al.. Electron Kinetics with Sttnchment and Ionization velocity space yields the continuity equation [ 101 - arL, _ - vn, - [jwFf")dv at + J C'(F(l))dv + jC'"(F('')dv] + ... = 0 . X(-V,) n, - y'", yc = J C'(F( 0 )) dw, Here - y IVL - Fa == - J C'&(F(")) dw , where G' ancl P are the mean collisioii frequencies for ionization and attachment. i\'hen substituting (2) antl (3) into ( I ) , the following equation - ' O E. V,F(O)= Cel(F(0))+ 2 c y ( ~ ( 0 ) ) m (5) ! i ! + Ci(j7CO))+ C'U(J'(0') - FpfO) for F('), with the natural normalization jFC0)dv = 1, is obtained. Ff")is the velocity tlistribritioii function of spatially integrated s ~ a r i n sor nonstationary, spatially homogeneous plasmas. For an electric field E = 73e, the (Iistribution fiinctioii B""' lias the particular velocity depentlence F(")(v,vJv) (cf. [ 21) and can thus br given in the Legeiitlre-polyiiornjals espaiision form 21-1 F(O)(v,% / V ) = - 2: f n ( v ) P,(v,/v), n-0 (ti) which IS written here for an even number 21 of terms. This representation of F("),as a generalization of eqs. (18) antl (19) of [a]due t o the inclusion of ionizatioii antl attachment, leads to the ordinary differential equation system m with the resultant normalization condition 4n v2fo 0 dv = 1 for the fimction f,. This differential equation system contains additional difference terms. It is of order 21 a i d involves the eigen-value GIN, Here, &el, &!", Qt and &" are the total collisioii cross sections relewnt to the different collision processes, whereas mv;/2 and mv:/2 are the energy losses In conservattve-inelastic antl ionizing collisions. SnO is the Kroneckei- symbol. The neutrals, of mass M and density N , are assumed t o be a t rest. Furthermore, hierarchy (7) is obtained by an additional espansion of the collision integrals with respect t o the mass ratio mlM antl by truncating the expansion after the leading term, for each collision process and each equation of the hierarchy. 24 Anh. Physik Leipzig 46 (1989) 1 Isotropic scattering in all of the collision processes has been assumed. Moreover, for each ionization event, it is supposed that the two electroils after the collision have the same kinetic energy. A transformation from speed 71 t o energy U = ( m / 2 ) u 2 in (7), finally yields the hierarchy of equatiorls 3 5 n 5 21 - 1, fel = 0, with the normalization condition I’ Lpi2 f o( u) d U = 1, 0 where The reduced quantities in (8) are Kote that detailed expressions of the mean frequencies P / N and ;“IN .for ionization and attachment can be obtained from (4) using expansion (6). They are given by and ill determined by the isotropic part f o ( U ) of bhe velocity distribution function F(”’by averaging over the energy space. Note also that 6 = 2m/M while 1J;’)= ( m / 2 ) $ , r“ = (m/2)vf are the energy losses in coiisrrvative-iaelastic and ionization processes. To determine the velocity distri1)ntion P(O)(w)at a given (even) order of approrimatioii with 22 terms, hierarchy (8) (inrolving 21 equations) has t o be solved for given cross sections &ez, Qi7’, &“ and &“ and given values of BIN, taking into account the normalization condition (9). However, the solution requires the simultaneous calculation of the R. WIKKLERet al., Electron Kinetics with Attachment and Ionization 25 reclucect frequency GIN of nonconservative inelastic processes, which is involved in (8). Since GIN depends on the same solution fo, strictly spiking, the solution of hierarchy (8) is a nonlinear problem. Furthermore, appropriate averaging of kinetic equation ( 5 ) for F(O)over the velocity space yields the particle-number and power-balance equations for electroD swarms, or nonstationary plasmas, with the normalized total mean power loss @IN = l?/N 2 @IN f k + $IN + @ I N . (13) Here, Ge is the spatially integrated electron density for swarms or the electron density for homogeneous plasmas. U is the mean electron energy, P the mean power input from the electric field and U"l, @,?l and? l are the mean energy losses produced by the different collision processes. All the mean quantities in (12) and (13) are determined by the first two terms f o and f l of the Legendre polynomial expansion (6) according t o the equations u do = j- fo(U)d U , u3/2 0 lj2 P N -= - -L(A) 3 m 2 Uel N N - Ua eoE F, 0 00 M o Ui1)= q I J -, N - U f l ( U )d U . 2 - j" U2QT U)f&U ) F- = ( G ) 3 _ - w j" N 3 U&iP(U ) fo( U ) d U , 0 ;i Ui-, N 7= ):( 112 m U2Qa(U)f,,(U)d U . (14) 0 As one can see from ( l a ) , larger deviations from the stationary behaviours of particle number and power balance in hydrodynamic stage can be expected if the mean collision frequency of ionization becomes remarkly different from that of attachment or if only one of these processes occurs. 3. Blain Features of the Solution Technique The detailed solution procedure of hierarchy (S), without the inclusion of nonconservative inelastic processes, has been presented in [2 , I]. Now, this procedure must be extended to include ionization and attachment. Of course important features of the former approach will remain unchanged. The basic ideas of our former procedure for solving the simplified version of hierarchy (8) at an arbitrary even order 21 of approximation are the following. The linear system of d l ordinary differential equations constitutes a weakly singular system a t small electron energies and a strongly singular system a t large electron energies, with U = 0 Ann. Pliysik Leipzig 46 (1989) 1 26 and U = 00 as singular points. This can be shown with appropriate power series representations of @ ( U ) a t small U , and QiP(U) immediately above the thresholds U;lr of (conservative) inelastic processes and with asymptotic series presentations of these same cross sections a t large U . Separate considerations on the structure of the general solution of the hierarchy in the region of small and large energies lead to conclude that its general solution contains I nonsingular and 1 singular fundamental solutions in both energy regions. The physically relevant solution can be (uniquely) determined by the nonsingular part of the general solution (NSPGS) a t low and high energies, by continuous connection of these NSPGS's a t a n appropriate connection point U,: and by the additional normalization according to condition (9). Finally, the distribution function 3"") a t a n even-order of approximation (2Z) is numerically fourid by an Z-fold backward integration of the appropriate simplified version of hierarchy (8) down t o U, (starting from a sufficiently large U,) and by a 22-fold forward integration up to U , (starting from U = 0). Special measures must be undertaken to preserve the linear independence of the contributions t o the NSPGS, particularly during the 2-fold backward integration, and to start the 21 forward integrations directly a t the singular point U = 0. Now, in the generalized case of interest here, the solution of hierarchy (8) is a nonlinear problem. However, it can be dealt with as an eigen-value problem. The proper eigen-value i;/N has to be self-consistently found, which can be done in an interative and solve hierarchy (8), with normaliway. To this'end, we start, with an estimate zation (9), using this estimated value. This approximate solution is used to improve ( ; / N ) Pand then a further approximate solution of (8) and (9), belonging to this latter After estimate, is determined, which in turn yields a further improvcment of having repeated a sufficient number of these st,eps the solution of (8) and (Y), and the self-consistent eigen-value ?;/AT,can be obtained. However, t o be able to make such steps, an extension of our previous procedure of solution is necessary, in order to include nonconservative processes. Special aspects of this extension, with respect to the procedure detailed in [2, I], are the following. As obvious from hierarchy (8), the term containing Y/N involves the square root of the energy, which leads to a somewhat different structure of (8) when compared to the former hierarchy (18) of [2, I]. Despite this fact, if hierarchy (8) is properly transformed, i.e. we pass from the electron energy U to the square root U1ls as independent variable (i.e. to the speed v), for a given value of (Y/N)p,it is found t o be again a weakly singular system a t small energies and a strongly singular system a t large energies, with U = 0 and U = co as singular points, if appropriate power and asymptotic series presentation of &"(U) and &i(U)are assumed. Thus it is further found that the general solution of (8) in the regions of small and large energies contains I nonsingular and 1 singular fundamental solutions as in the simpler case. The nonsingular solutions must be isolated to construct the NSPGS in both energy regions and, finally to determine the hy the same approximate iionsingular solution of (a), for any given estimate of procedure detailed in [2] for the simpler case. In order t o determine the 1 nonsingular fundamental solutions f ( m,j ) , 1 5 j 5 I, a t large energies by the mentioned backward integration from U,, a t this particular enwgy and, repeatedly, a t appropriate lower energies U* down to U , an approximate representation f (') of the fundamental solutions of (8) must be calculated using ansatz (42) of [2]. To t,his end, the nontrivial solutions of the linear and homogeneous equation system (43) of [2] for the expansion coefficients of the approximate funtlamenta.1 solution f(')has to be determined after having raplaced in (43) the quantity q( U*) by q( U*) $( U*) @( U*) ( U*)-1/2 .i;/lv.These approximate fundamental solutions are used as initial values for the backward integra- + + + 27 R. WINKLERe t nl., Electron Kinctics with Attachment and Ionization tion and in order to preserve the linear independence in the calculation of the nonsinguU ) a t large energies can be lar fundamental solutions f(”,j). Then, the NSPUS f(“)( obtained by a superposition of these fundamental solutions according to (45) of [2]. I n order to start the forward integration a t the singular point U = 0, power series presentations of the Z nonsingular fundamental solutions f(‘)( U ) and of the I particular solutions f ( ” j ) ( U ) near U = 0 are used. The particular solutions result from the difference terms in the lowest equation of hierarchy (8), containing fo(U U i p ) and fo(2U Ui). These difference terms are already known when performing the calculat’ion of the NSPGS a t high energies, i.e. from U , down t o the connection point U , 5 Nin(U;[?’,Ui), and are determined by the functions fO(“*i)(U Ui7’) and fc,(”*7)(2U Ui). Thus, they can be dealt with as known inhomogeneities of the differential equation system. Instead of (52) and ( 5 3 ) of [a] the initial part of the nonsingular fundamental solutions near U = 0, now can be represented as + + + + +f ( r ) W +p u + O(U)], f ‘ ” ( U ) == Ur[@ (15) with r = 0, . . . , 1 - 1, where the components of the coefficients are recursively given by the equations gd+l = O forO<nI:l- jp = 1, f g = 4n 1, + 2n - 1 4n 1 2 r ~- 2r - 2 4 ~. 3 272 2% 1 2r ~ ~ +3 + + 2 jg_.1 = 2 n + 1 2 r + 2 n + 3 - 2n ~ for n = I, - Pcr),f ( r )and f(‘) for n (r - n = I, ..., I - I, + 1) . . ., 1 - 1, f$~=OforO<n<Z- - 1 1, for n = 1, ..., I - I , if)= o for r = 0, ..., z - 2, for r = 0, ..., I - 2 and n = I., ..., 1 - 1, 2 n 2r + 2 n - t jgo2= !?I ! 2n--12 4n 2n-Ir-n+2 for r = I - 1 and n = 1 - 1, -.., 1. +1 3 (16) Ann. Physik Leipzig 46 (1989) 1 28 I n (16) the expansions Q“(U) = 5@U., m - Q“(TJ) = v=o 3 QZU. v=o have been used for small energies so that the quantities p , and q$ are given by p , = @/(eoE/N) and & = @ / ( e o E / N ) . Concerning the power series representation of the particular solutions at small U , the expansions of the cross sections C Q - Qp( U )= 2 Qiy,,(U - lJif’)”,Q ( U ) = 2 Qf( U - Ui)”, v=l ”=l + + as well as the expansions of the functions fo(*yi)(U UiP) and f,,(”J)(2U U i ) are necessary, in analogy to (56) of [ 2 ] . Then, the initial part of the particular solutions can be given the forin f (Pd( u)= j%?b7) + j(~,i)u1/2 + J’(P,~)U + j%,i)U3/2 + O( ,573/2), I l j l l , with j ~ ~ v= i ) - =0 fp,j) = (18) for 0 4n+3 n 2 r ~ - 3 - 2n+14n-l for YL = < n -< 21 - 1 , _ n+2 1 - 1, ..., 1. ~ for n = I, .. ., I - 1, (19) Prolongation of the initial parts o f f ” ) and f(I’J), by forward integration of ( 8 ) , u p to the connection point U, yields the NSPUS f (O)( U ) a t small energies by superposition of the nonsingular fundamental solutions and the particular solutions, according to (60) of [2]. The physically relevant solution in the simpler case detailed in [ 2 ] is found by a continuous connection of the NSPGS’s f P) and f (O)at U , and a concluding normalization. Because of a special property of the lowest equation of hierarchy (18) of [ 2 ] the requirement of continuous connection a t U , of the components of the NSPGS’s, with the excepautomatically ellforces the continuous connection tion of t’hat belonging to f‘,oo) and also of the latter. Thus, by this procedure, one of the 21 superposition constants remains free and can be determined when applying the normalization condition (21) of [ a ] , which is identical with condition (9) of this paper. I n the generalized case considered here, the connection off (w) and f (O)mustbe made in the same way, however with functions f(”)and f ( O ) obtained with the estimated ( G / N ) 2 , , which does not lead t o an automatical continuity of f{”)(U,) with f{o)(U,), fy), R. WINKLFRet al., Electron Kinetics with Attachment and Ionization 29 and thus t o the desired physical solution. This can be seen when integrating (analytically) the lowest equation of hierarchy (8) over both NSPG’s, i.e. from 0 to U, and from Uc t o 00, and when taking the sum of the two resulting relations. This leads to write that When performing the continuous connection and the normalization in the mentioned way, the resultant function systems, which are uniquely determined by special superpositions from both NSPGS’s, satisfy the reduced relation + j- Uqif6”’ dU Ufjb”) dU );( 1 Do 3uc [fP’(Uc)- f!”’tUc)l - Ui 7 UqajiO) dU - 0 (21) = 0. P UC These continuously connected and normalized function systems, in (21) are denoted by the same symbols as before for simplicity. Equations (10) and (11) make obvious that, when the estimated (G/N)pconverges to its self-consistent value i / N , again the automatiU,) and flW)( U,) is established and the physically relevant solution cal continuity of is found. Returning to the remark given above on the iterative improvement of a n estimate (Y/N), of G/N, eq. (21) indicates the possibility of an iterative control of (G/N)P.When the parameter ( G / L V )used ~ in (8) approaches the self-consistent eigen-value G/N2the simple equation /I0)( I G/N - (Y/N)p= 0 (22) must be satisfied. Solving hierarchy (8) for the estimated (Y/N), and calculating the corresponding value of G/N with the resulting solution according to (10) and (ll),the left-hand part of (22) will be non-zero. However, the calculated value of G/N can be considered to be a function of the estimate (Y/N), used for the solution. Therefore, an improved estimate (;/N)JJcan be found locking (for example with the Newton method) for a n improved value of the zero point of equation (22) which is then used for the next iterative step, i.e. the next solution of (8),as mentioned above. I n this way, after some steps, the physical relevant solution together with the self-consistent eigen-value can be obtained. 4. Results and Discussion The extended technique has been applied to investigate velocity distribution and resulting macroscopic quantities of electrons in an appropriate model gas and in pure SB’, . A range of field values has been considered for which either ionization or attachment, if not both of these non-conservative processes, contribute to the eigen-value GIN. The results are calculated for increasing approximation orders (2Z). 30 Ann. Physik Leipzig 46 (1989) 1 4.1. Model Gas For this first analysis a former model gas, already treated in [I, 21, is extended to take into account of ionization and attachment. The model includes elastic collisions and single excitation, ionization and attachment processes. The relevant cross sections &ez(U)=6.10-16cm2, U > O e V , 0, u 5 uy, Q?'(U) = &(U - U ; p ) / A U , i &', U U51) 2 Uip + A U , U(;p= 1 eV, A U = 2U5 Uc,l, +AU, - 0.2 eV, Q;??= 4 . 10-l6 em2, are presented in Big. 1. Furthermore, a mass of four atomic mass units (S = 2 m/M = 2.744. lo-*) is used for the gas particles. Such a model gas shows the typical features of a n electronegative gas, namely large attachment at low electron energies, excitation a t intermediate energies and ionization a t the higher energies. First results for this model have already been reported in [ 1t]. Table 1reports the mean electron energy 6and the normalized collision frequencies for excitation, ionization and attachment for different EIN values and increasing approximation orders 21. As obvious, a t low EIN attachment and a t high E / N ionization dominate in the particle balance (12) and cause a large temporal decrease or increase of the total electron number in the swarm or of the electron density in the homogeneous plasnia The change of the dominance of the two non-conservative processes occurs for a field strength E / N between 200 and 300 Td. 10 7 model Ir/Qa 0 5 I ;I b'ig. 1. Collision cross sections for the model gas dy Pig. 2. Distributions fo and f, for the model gas as obtained with %term and 8-term approximation at 50 Td model EIN =300 Td - 8-term 10” Fig. 3. Distributionsf,, andfi for the model gas as obtained with 2-term and 8-term approximation a t 300 Td Ann. Physik Leipzig 46 (1989) 1 32 - Table 1. Mean electron energy U (in eV) and mean collision frequencies (in cm3/s) for excitation (Y;I>/N),ionization (?/N) and attachment in the modcl gas a t different E/N's (in Td) and different approximation orders 21 ~____ 20 2 4 6 8 50 200 300 500 ~ ~ - 1.746-8 1.6 7 9 8 1.663-8 1.657-8 4.680-16 2.731-15 2.756-15 2.728-16 l.214-8 1.209-8 6.481-12 1.126-11 1.122-11 1.119-11 6.067-9 6.230-9 6.218-9 6.206-9 6.140-10 6.@29-10 li.6"-10 6.627-10 2.219-9 2.416-9 2.418-9 2.415-9 6.715-8 5.640-8 3.233-9 2.261-9 2.260-9 1.324-9 1.460-9 1.464-9 1.462-9 6.559-9 6.612-9 6.514-9 c1.514-9 7.520-10 8.214-10 8.225-10 5.213-10 6.997-9 6.718-9 6.676-9 6.667-9 1.570-8 6 8 1.236 1.245 1.243 1 .242 2 4 6 8 2.354 2.396 2.395 2.393 2.918-8 2.921-8 2.917-8 2 4 6 4.761-8 8 4.978 4.936 4.937 4.936 2 6.873 4 6 8 6.770 6.771 6.771 5.640-8 2 4 9.636 9.476 6.852-8 6 9.473 Y 9.473 3 1 100 0.7473 0.7365 0.7377 0.7388 1.672-8 1.566-8 1.5V8-8 3.916-8 4.iO7-8 4.706-8 4.706-8 5.639-8 6.i67-8 6.766-8 6.766-8 2.260-9 ~ 1.204-8 1.201-8 When passing froin the two-term to the eight-term approximation, the normalized attachment frequency iP/X changes up t o about 10% in the EIN range we hare considered. The normalized ionization frequency PIN, however, changes froin less than 1 94 a t 500 Td to more than 500% at 50 Ttl when increasing the approsirnation order. All the quantities show a good convergence a t high EIN when increasing the value of 21 and a satisfactory convergence a t the lowest values of EIN. The modification of these average quantities result from the alteration of the isotropic distribution function f o with the (increasing) approximation order a t different EIN. Pigs. 2 and 3 show the behaviour of f o and f l as a kuunction the electron energy U , for the 2- and 8-term approximation a t 50 and 300 Td. I n correspondence of the lower field strength, the ionization region belongs t o the high energy tail of the distribution, where the largest increase of f o occurs when passing to higher orders of approximation. This leads to the large increase of Y,JLVwe have just mentioned. At higher fields, the ionization mainly occurs in the body of the distribution. This part of the distribution, however, remains nearly unchanged when increasing 21. Then only small changes of the mean ionization frequency will result a t high E / X . Since, generally, the electrons in the body of the distribution mainly contribute to the mean collision frequencies of attachment and excitation and to the R. WINKLERe t a]., Electron Kinetics with Attachment and Ionization 33 mean electron energy, the smaller variations observed for these quantities are immediately understood. The alteration of the first contribution ( f l ) to the anisotropy of the velocity distribution, is very similar to that of fo, a t not too small electron energies, when passing from 2-term to 8-term approximation. A comparison of these results with those relevant to the model gas with only conservative collision processes [ 1, 21, indicates that the aclditional presence of ionization and attachment does not change substantially the convergence behaviour of the distributions fo and fl and the resulting macroscopic quantities. This is due to two main reasons. J n the ionization region the exciting collisions are also acting so that an additional ionization only enlarges, to some extent, the effectiveness of the inelastic collision processes in this energy region. On the other hand, attachment presents a n intense inelastic process in the model a t small energies which, however, occurs only in a very limited energy range and with a relatively small energy loss per collision. Therefore, for any approximation order, this process has a certain impact on the structure of the distribution a t small energies (particularly for small fields) but does not lead to remarkable corrections to the distribution when passing from the %term approximation t o higher order treatments, which can be immediately seen from Figs. 2 and 3. Table 2. Mean power input PIN, total power loss in collisions U'IN and individual power losses +'IN, *IN, $/A' and S I N (in eV cn13/s) for the model gas at different E / Y s (in Td) and different approximation orders 2Z ElX 10 50 PIN $IN $IN Vy/N +/s 7 3 9 4 - 12 7.278-12 7.289-12 7.300-12 6.997-9 6.718-9 - 6.67 6-9 - (i.667-9 - 200 300 500 - cap 7.634- 9 7.211-9 7.1.51-9 7.134-9 2 1.604-9 4 1.576-9 (i 1.673-9 8 1.571-9 1.464-8 1.394-8 1.383-8 1.381-8 1 4 6 6.853-9 6.669-9 6.671-9 6.67 1-9 2.185-8 2.171-8 2.163-8 2.157-5 1.570-11 1.604-11 1.599-11 1.697-11 1.579-8 1.573-8 1.566-8 1.562-8 4.f;80-15 2.721-14 2.756-14 2.728-14 1.8194 1.769-8 1.770-8 1.770-8 3.2433-8 3.269-8 3.254:8 3.261-5 4.21'1-11 4.378-11 4.373-11 4.368-11 "918-8 2.921-8 3.917-8 "916-8 6.481-11 1.125-10 1.122-10 1.119-10 3.147-9 3 ".)0-9 3.2149 4.708-8 6.605-8 6.510-8 8 4.646-8 4.644-8 4.645-8 5.509-8 1.285-10 1.281-10 1.281-10 1.281-10 4.761-8 4.707-8 4.706-8 4.706-8 6.140-9 6.619-9 6.G29-9 6.627-9 1.174-9 1.275-9 1.277-9 1.276-9 > 4 6 8 8.647-8 8.640-8 8.538-8 8.539-8 8.029-8 7.998-8 7.999-8 7.997-8 2.069-10 2.040-10 2.041-10 2.041-10 5.716-8 3 _._& "93-8 6.640-8 7.047-10 7.756-10 6.640-8 5.639-8 2.260-8 2.261-8 2.260-8 7.781-10 2 1.908-7 1.875-7 1.875-7 1.875-7 1.349-7 1.336-7 1.336-7 1.336-7 3.415-10 3.358-10 3.358-10 3.368-10 6.852-8 6.767-8 6.766-8 6.766-8 6.559-8 6.512-8 6.514-8 6.514-8 4.017-10 4.385-10 4.395-10 4.392-10 s 100 - 21 2 4 6 8 > 4 6 i 4 6 8 5.509-8 6.044-9 3.97 7 - 9 5.950-9 5.937-9 1 .id 3.208-9 7.784-10 34 Ann. Physik Leipzig 46 (1989) 1 The energetic aspect is analysed in Table 2 where the (normalized) power input PIN from the electric field and the (normalized) power loss in elastic (i5?/N), exciting ( P I N ) ,ionizing ( @ I N ) and attaching ( P I N ) collisions are reported together with the total power loss in collisions @/AT given by (13). As one can see, a t low EIN the power loss in collisions is remarkably larger than the power input from the electric field whilst, a t high BIN, the situation is reversed. This clearly indicates that, due t o the action of attachment and ionization, in the power balance (12) for the hydrodynamic stage a pronounced nonstationary behaviour, i.e. large deviations from the stationary situation with PIN = @IN, occurs. The power, a t low BIN, is mainly dissipated in attachment and excitation processes. When increasing EIN the loss by excitation and ionization becomes dominant and the powei input becomes larger than that lost in collisions. In addition, it is worth noting that the normalized power losses @IN and @IN clue to attachment and ionization depend, as for Ga/N aiitl ?IN, on the approximation order and the corresponding changes of power input PIN and total power loss @IN in collisions amounts to some per cent. 4.2. Pure SF6 \Ve will report now the results of some calciilations for electrons in SF,, a gas which is of great importance from the technological point of view. This gas presents a pronounced energy dependence of some cross sections and, particularly, a very intense attachment process a t low electron energy. Fig. 1 reports several collision cross sections, Fig. 4. Collision cross scctions for SF, R. WINKLER et al., Electron Kinetics with Attachment and Ionization 35 as proposed in [12]. The figure reports the cross section for momentum transfer in elastic collisions (&“I), the total cross sections for lumped vibrational excitation (Q:P), lumped electronic excitation (QgP), ionization (@) and for the two main attachment processes (QT, Qg), leading t o the production of SF, and SF, ions. The energy losses per vibrational excitation, electronic excitation and ionization are 0.0954, 9.8 and 15.8 eV, respectively. A further attachment process leading to production of F- ions is neglected since the corresponding cross section (Qf) is very small. Tables 3 and 4 report some results relevant t o mean energy, collision frequencies, power input and power-loss rates for SF, a t different field strengths and for increasing approximation orders. Table 3. Mean electron energy (in eV) and mean collision frequencies (in cm3/s) for electrons in SF, at different E / N s (in Td) and different approximation orders 21 - Yy/N E/N 21 U 70 2 4 6 5.07 5.05 5.05 1.47-8 1.48-8 1.48-8 5.64-10 5.60-10 5.60-10 1.29-13 1.50-13 1.49-13 5.74-10 5.67-10 5.66-10 4.77-11 5.10- 11 5.14- 11 100 2 4 6 5.35 5.32 5.31 1.35-8 1.38-8 1.38-8 9.80-10 9.67-10 9.65-10 1.55-12 1.74-12 1.73-12 5.76-10 5.53-10 5.50-10 3.74-11 4.06-11 4.11- 11 300 2 4 6 8 7.53 7.47 7.46 7.46 8.60-9 8.78-9 8.82-9 8.83-9 5.27-9 5.14-9 5.13-9 5.13-9 3.98-10 4.15-10 4.14-10 4.14-10 6.80-10 5.95-10 5.72-10 5.63-10 2.19-11 2.18-11 2.21-11 2.22-11 500 2 4 6 8 9.45 9.35 9.34 9.34 6.52-9 6.63-9 6.64-9 6.66-9 9.13-9 8.85-9 8.84-9 8.84-9 2.05-9 2.09-9 2.09-9 2.09-9 6.08-10 5.50-10 5.23-10 5.11-10 1.69-11 1.65-11 1.63-11 1.64-11 The specific effects found in the model case in the particle and power balance, occur also in SF,. So, a t low EIN, the first attachment process dominates in the particle balance and the corresponding total power loss in collisions is remarkably larger than the power input from the electric field. However, a t large fields the ionization process dominates in the particle balance and the power input becomes larger than the total power lost in collisions. The transition between these two situations occurs now a t somewhat larger fields. The corrections to the collision and power transfer rates produced by increasing approximation orders are relatively small and amounts t o less than 20% in the EIN range considered here. Our previous studies [ 111 with only conservative collisions have shown that larger corrections to the two-term approximation are mainly observed when the total cross section for inelastic collisions becomes comparable or even larger than that for elastic collisions over larger energy regions. As one can deduce from Fig. 4, in SF, such a situation occurs in the region of SF- roduction by attachment and of vibrational excitation, 6 p. but only in a narrow energy region below 1eV. This explains the relatively small corrections introduced by higher order calculations in SF,. I n addition, it should be mentiolied that, due t o the large maximum value and the strong energy dependence of some cross sections, the accurate numerical solution of the eigen-value problem in pure SF, is not a t all simple. 36 Ann. Physik Leipzig 46 (1989) 1 Table 4. Power input, total power loss in collisions and individual power losses (in eV cm3/s)for the different collision processes in S F , a t various E / N (in Td) and different approximation orders 21 - - - UgPIN $IN UYIN UZ/N 3.72-12 3.71-12 3.71-12 1.40-9 1.41-9 1.42-9 5.53-9 5.49-9 5.49-9 2.04-12 2.36-12 2.36-12 1.25-10 1.24-10 1.24-10 2.58-11 2.75-11 2.76-11 1.11-8 1.10-8 1.10-8 4.06-12 4.03-12 4.03-12 1.29-9 1.31-9 1.32-9 9.60-9 9.47-9 9.46-9 2.45-11 2.74-11 2.74-11 1.22-10 1.18-10 1.17-10 2.02-11 2.20-11 2.22-11 8 5.66-8 5.64-8 5.65-8 5.65-8 5.89-8 5.79-8 5.78-8 5.78-8 7.19-12 7.11-12 7.10-12 7.10-12 8.21-10 8.37-10 8.41-10 8.43-10 5.16-8 5.04-8 5.03-8 5.03-8 6.29-9 6.56-9 6.54-9 6.54-9 1.34-10 1.18-10 1.14-10 1.12-10 1.16-11 1.15-11 1.17-11 1.18-11 2 4 6 8 1.36-7 1.35-7 1.35-7 1.35-7 1.23-7 1.20-7 1.20-7 1.20-7 1.04-11 1.03-11 1.03-11 1.03-11 6.22-10 6.33-10 6.34-10 6.35-10 8.94-8 8.67-8 8.66-8 8.66-8 3.24-8 3.30-8 3.30-8 3.30-8 1.18-10 1.07-10 1.02-10 9.98-11 8.72-12 8.59-12 8.56-12 8.60-12 21 PIN ?IN 70 2 4 6 3.94-9 3.95-9 3.95-9 7.09-9 7.06-9 7.07-9 100 2 4 6 7.79-9 7.81-9 7.82-9 300 2 4 6 500 - U!PIN E/N 5. Relaxation to the Hydrodynamic Stage To give a n example of tendency t o the hydrodynamic stage and to assess the validity of the results obtained by solving the kinetic equation relevant to the hydrodynamic stage, we also have solved the spatially homogeneous version of the nonstationary Boltzmann equation (1)in two-term approximation. I n this approximation the Legendre polynomial expansion (which we write here in terms of electron energy U and normalized time N t ) leads (after some approximation to the collision integrals as detailed in sect. 2) to two partial differential equations with difference terms describing the temporal evolution of the two distribution parts fo( U , N t ) and fl( U , N t ) [ 131. Using the quasistationary approximation for fl and the solution technique discussed in [14- 161, the resultant equation for fo(U,N t ) is numerically solved up t o the hydrodynamic stage, starting from appropriate initial distribution, f,,( U , 0). For the latter distribution we have assumed that Thus the energy UM coxresponding to the maximum and the energy width A U M are parameters of the initial distribution. The calculations have been performed a t EIN = 300 Td for the model gas specified in sect. 4.1, starting from two different initial distributions fo( U , 0), relevant t o the parameter combinations V M= 1eV, A U M = 1 2 eV and UM = A U M = 9 eV. Fig. 5 illustrates these initial distributions and the resultant distribution of hydrodynamic stage (which is established a t a normalized time of about N t = 3 lo8 s) and which completely agrees with the function fo( U ) reported in Fig. 3 and obtained in two-term approximation. Figs. 6 and 7 present the relaxation to the hydrodynamic stage of the ~ - lo-4 rmode( EIN =300 Td IO-~ 2-term F _ _ _ UM=AUM=9 eV I Fig. 6. Initial distributions (-, hydrodynamic stage I 2- / I , - - - ) and resultant distribution (- , . -) from calculations up t o the ,,Yo/" __--,, / - --+Nt[108cm~3s] 38 Ann. Physik Leipzig 46 (1989) 1 collision frequencies for excitation ($')IN, ionization (?IN) and attachment (Ga/N), and also of the mean electron energy 0, the power input (PIN)and the total power loss (@/hr).Also from the macroscopic point of view, the establishment of hydrodynamic values for the macroscopic quantities occurs in the mentioned time and yields in any case, the same values reported in Tables 1and 2, for EIN = 300 Td and two-term approximation. Therefore, in agreement with expansion (2) relevant to the hydrodynamic stage in a homogeneous plasma, a time independence of mean collision frequencies for ionization and attachment, mean electron energy, power input and power loss per electron, is obtained after the relaxation period, however, in the presence of a pronounced nonstationarity of particle number and power balance, because of the remaining temporal evolution of the electron density. This clearly also agrees with Fig. 8 where the growth of electron density is presented up t o times which are remarkably exceeding that requi16 14 model EIN = 300 Td ?-term I 12 eV, AU,=12 eV ---Uu,=l I \ I - - -UM=AUM=9 eV 1 \ 10 a I I I I - 6 N$lO~cm%] 0.5 1 1.5 2 2.5 Fig, 7. Temporal relaxation of mean energy, power input and total power loss in collisions R. WINKLERet al., Electron Kinetics with Attachment and Ionization 39 red to establish the hydrodynamic stage. Note that, after the initial relaxation period, a pure esponential growth of the density is observed in agreement with the hyclrodynamic balance equation (12). A similar behaviour of the electron velocity distribution function has also been observed in previous studies on the temporal evolution of distribution function and macroscopic quantities, following the disturbations of the electric field strength produced by rectangular impulses of appropriate period [ 161. 2c 1c 8 6 4 E/N=300Td 2-term __ U,=l eV, A UM=12 eV _ _ _ _ U,=AUM=9 2 - eV Nt[lO9cm3s] 1 I0 I I 1 L . I I I 2 3 Fig. 8. Temporal growth of the electron density during the relaxation to, and after the establishment of, the hydrodynamic stage References [l] WINKLER,R.; BRAGLIA, G. L.; HESS,A.; WILHELM,J.: Beitr. Plasmaphys. 24 (1984) 657. WINKLER,R.; WILHELM, J.; BRACLIA, G. L.: I1 Nuovo Cimento 7 D (1986) 641. WINKLER,R.; BRAGLIA, G. L.; HESS,A.; WILHELM, J.: Beitr. Plasmaphys. 25 (1985) 351. BR~GLIA, G. L.; WILHELM, J.; WINKLER,R.: Lettere a1 Nuovo Cimento 44 (1985) 257. B~acLr.4,G. L. ; W~LIIELM, J.;J~INKLER, R. : Lettere a1 Nuovo Cimento 44 (1985) 365. BRAGLIA, G. L.; WINKLER,R.; WILHELM, J.: I1 Nuovo Cimento 7 D (1986) 681. BLEVIN,H. A.; FLETCHER, J.; HUNTER, S. R.: Phys. Rev. A 31 (1985) 2215. [2] [3] [4] [5] [6] [7] Ann. Physik Leipzig 46 (1989) 1 40 [8] NESS,K. F.; ROBSON, R. E.: Phys. Rev. A 34 (19%) 2185. [9] PHELFS, A. V.; PITCHFORD, L. C.: Phys. Rev. A 3 1 (1985) 2932. [lo] KUMAR, K.; SKULLERUD, H. R.; ROBSON, R. E.: Aust. J. Phys. 33 (1980) 343. [Ill WINKLER, R.; BRAQLIA, G. L.; WILHELM, J.: XVII th ICPIG, Budapest 1985, Invited Papers, pp. 22. [12] YOSHIZAWA, T.; SAKAI,Y.; TACASHIRA, H.; SAKAMOTO, S.: J. Phys. D: Appl. Phys. 12 (1979) 1839. [13] WINKLER,R. ; DILONARDO, M.; CAPITELLI, M. ; WILIIFLM,J.: Plasma Chem. Plasma Process. 7 (198i) 245. [I 41 WILHELM, J.;WINKLER, R. : Ann. Phys. (7) 37 (1980) 35. [16] WINKLER, R.; WILHELM, J.; HESS,A.: Ann. Phys. (7) 42 (1985) 53i. [lG] WINKLER, R.; WILHELM,J.: Comput. Phys. Commun. 20 (1980) 113. Bei der Redaktion eingegangen am 6. Juni 1988. Anschr. d. Verf. : Dr. R. WINKLERund Prof. Dr. J. WILHELM Zentralinstitut fur Elektronenphpsik-Bereich 6, Robert-Blum-Str. 8- 10 Greifswald DDR-2200 Prof. Dr. G. L. BRAGLIA Dipartimento di Fisica, Universita di Parma, CIEQP - GNEQP, 43100 Parma, Itnlia

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