Mathematical modelling of a microwave plasma generator

U.S.S.R. Comput.f#aths.Mith.Phys., Vo1.25,No.3,pp.149-157,1985 Printed

0041-5553/85 $10.00+0.00 Pergamon Journals Ltd.

in Great Britain

MATHEMATICAL MODELLING OF A MICROWAVE PLASMA GENERATOR* A.R. MAIKOV,

A.G. SVESHNIKOV

and S.A. YAKUNIN

Using the example of the design of a plasma microwave generator, a nonstationary conservative numerical mathematical model is constructed of Maxwell's equations and the collision-free Vlasov equation, for which, the integral laws of conservation-variation of mass, energy and momentum are satisfied at the discrete level. In this paper we develop and prove effective numerical algorithms for solving the problem of the evolution of a collisionless plasma, described by the Vlasov system of kinetic equations The problem of constructing appropriate and Maxwell's equations for the electromagnetic field. difference schemes will be considered using the example of a specific physical system, namely, a plasma microwave generator /l/. When constructing conservative difference schemes for this selfconsistent problem, we will essentially use theideaofrepresentingthedifference analogucs of differential operators developed in /2-4/. The plasma microwave generator is a section of cylindrical waveguide, hollow for O
particles. 1. We will consider the plasma in the the evolution

of the electron

In this case, selfconsistent field approximation. density function is described by Vlasov's equation

distribution

(1.1) where v-v(u)-u(I+u*jc*)-'., F=F(x, U, t)=E+c-'[v, H], Maxwell's equations 1 aE

and the fields E and II are solutions

4n. cl,

rotH=cdt+

rot+_-laH

(1.2)

c at’

(1.3)

tliv H=O,

div E=4n(p+p,).

of

j(x.t)=e5 f(x,u,t)v(u)d'u,

(1.4)

p(x.~)=ejj(x,u.t)d’a; R’

and

p0 is the density

of the fixed

ions.

The initial

conditions

are

f(x,u,0,=i,(x,u).

while

the boundary

conditions

include

(1.5) H(x.O)=H,(x)

E(x,O)=E,(x).

the formulation

-

*Zh.vychisl.llat.mat.riz.,25,6,883-095,1905

14'1

of the problem.

(1.6)

150 2. The so-called method of macroparticles /5/ has become widely used to solve Vlasov's equation with a selfconsistent field. In this method, the initial conditions for (1.1) are chosen in the form x Mx.u)-z

8(X-x.)6(u-II"). n-l

Then the distribution density function IN(x, u, t) - the generalized solution of (1.1) with these initial conditions - corresponds to the pattern of motion of N model macroparticles with charges e..,and masses mv such taht esle=mn/m. while e,N is the total charge of all the electrons of the actual plasma. The solution of Vlasov's equation with initial conditions (1.5) is sought as the weak limit of /x(x,u,t) as N-cm and fFP converges weakly to f,h The problem of the convergence of the method of macroparticles for systems with different types of interaction between the particles has been discussed in a number of papers (see, for ~. example, /6, 7/). In particular, it is shown in /7/, that convergence occurs for a system with electrostatic interaction, with certain assumptions. It is usually assumed that a particle is not a point particle, but is "smeared", and its natural density is specified by a certain fairly smooth "smearing function" 8(z). We will henceforth assume that the carrier 1 is compact, and that

3. It is only possible to calculate the field in practice in a finite region of space. It seems natural to choose V, as this region. However, the formulation of the boundary conditions for the fields at the open end of the system meets with serious difficulties. One way of getting round this is to use a certain version of the principle of limit absorption Consider a system which differs from that described above solely in the fact that it /9/. consists of the cavity V,,L>Lo, with the same geometry as V,, but We will closed at the end a-L with an ideal conductor (Fig.2). assume that the whole of the space inside V, is filled with a o-_eonst>O and e-p-i, which has no effect medium of conductivity The solution of the initial on the motion of the electrons. problem will be sought as the limit of the solutions for the system in VL as L+a and o-0. The equations of the method of macroparticles for thissystem Fig.2 in VL have the form -

au, er E = at mB

rotE=-Teat,

1

mrc’

dz

k--1,2 ,...,

II,

N,

dt---KUU’

an

rotH_LaE

c at

4n +e(J+aE),

where (1.9)

E.&= j

E,(r,z)l(r-r,,z-z~)drdz,

(1.10)

with the initial conditions for the fields (1.5) and the macroparticles, and with the conditions that the tangential components of E should be zero on av, - the surface of 1“. 4. As is well-known I

d.“Jf~ 1

a‘ “1

-9

0

p=en

z

Af(r-rh,

z-z,).

(1.111

L-1

The satisfaction of (1.3) for the solutions of (1.8) at any instant of time follows from the fact that these equations are satisfied for the initial conditions (1.6). We will assume that It is also known that for a "particle + field" system, desecribed by Eqs. the latter holds. For the solutions of the (1.7)-_(l.lO), the laws of variation of momentum and energy hold. difference scheme proposed below, their discrete analogues are satisfied.

2.

Definitions

and auxiliary

results.

BY constructing discrete analogues of the fundamentalintegralrelations of vector analysis (Stokes theorem, Gauss's theorem etc.) we can obtain conservative difference schemes for the equations of mathematical physics /3, 4/. For example, the satisfaction of the law of variation of energy of the electromagentic field +$

is based

formally

I (Ez+Hz)rdrdz=- s ((J,E>+oE*)2nrdrdz VL Y‘

on the identity

151

(n[A,B]Ms.r

(2.2)

(-
If the discrete analogue (2.1) issatisfied forthedifference operator ROT, the law of variation of energy for the soltuions of Maxwell's difference equations is automatically satisfied. We will henceforth use the basic ideas in /3, 4/ for constructing conservative difference schemes for the selfconsistent problem (1.7)-(1.10). Certain features of the method are connected with the fact that the components of each of the fields E and R are specified on nets that are shifted with respect to one another. 1. For the most compact description of the material it is best to introduce the following notation. We will call the set of similar cell-parallelepipeds, the vertices of which (the nodes of the net) make up the set

nQ~~$=(((i+t)ah,. (j+q)j%., the

net

Q,:P;y, in

everywhere,

R"m(r, z, t) where

h,, h., TAO,

e, n, i3 are any of the numbers

cross-section"

(n+O)r);

are constants,

0, ‘I?, 1; (L, B=i

(2.3)

i, i, tv+Z, i%).

or

fixed

‘I.

for this net.

We mean by

Henceforth,

Q,c;T’ the "spatial

of this net, i.e. the net with nodes nP,:::'-( ((i+e)ah.,

(j+q)i3hz)4C

i, F-Z, iSO),

and we will mean by (n+e)r, a=Z).

Q,e, the "time cross-section" Q,:r$, (the nodes n,,, are the points ,*.01 The net &,n, consists of the same cell-rectangles. We will denote by o(P)

the cell of this net with centre at the point P. We will denote the r and z coordinates of the point M by r(M) and z(M). Everywhere, where the specific values of a, p, 5, 11,0 are unimportant, instead of O,:f,?, we will use the notation

52, instead

5?,.and the space and time periods We will denote by

of

Q,:$

we will use &instead

of these nets will be denoted

5r(Q)-~~~J~:, the space of the functions

by

of

Q,,, we will

use

Ar=ah,, 4z=fih,.Jt=r.

specifiedonthenet

Q$,!:,=G!.

Suppose v is a certain subset of Rz and the net g is specified. We will denote by a,(V) that part of the net OS where any of its cell m is contained in V, and by Q(V) the net Q,(l)XOr. If the specific form of V is unimportant or is clear from the context, we will also use the notation Q-Q(V). 30, - the Theinternalnormal n=h, n.); n,‘+n,‘=l, where n,, n,=O, *I, is defined on boundary of Bs, everywhere, apart from a finite number of points. The closure of that part of the boundary where n,-0 (or np=O), will be denoted by as,. (or as,). The nodes of the net Q, not lying on aQ.XQ,, will be called internal. 2. Suppose F,Es(ai) and RE$(a2) are two scalar net functions, defined on parts of two different, generally speaking, nets: Q1--521(V,), Q2=R2(V,). We will put Ql@Q2=QlUQ2, PEiI1,

F,(P), S(SiW2)=(F,@‘F.)

(P)=

Pd2,

F,(P), i F,(P)+F,(P),

P&lilsIS.

this definition obviously gives the usual point-by-point when ni--82 functions. We will require in addition that B1--9,::,$I,, Q2=Q,::,$;,,. We will

sum of the scalar choose

some nodes

net PEQZ

and consider the nodes P,‘~a2 such that jr(P,‘)-r(P)/Gh./2, Iz(P<‘)-z(P)(
If QJ=Q2, we put Q1@5/2--81. In any case, as can easily be seen, (2.3) is satisfied for rr((rl@152). We will put (F,@F,)(P/)-F,(P)F,(Pl(P)). r.F(P)== 3. Suppose FeF(gl). We will define a multiplication operator r..by putting If Ql does not contain points lying on the axis of symmetry, we can similarly r(P)F(P). define a multiplication operator (i/r).. 4. The r-semisum operators (.),s,,,,, and the difference derivative A/Ar which transform (0) S:r,,"\, into Sr:r$.;,, , will be defined as follows: P, and P1 are nodes Q,E,n,s,,such that r(P?)-r(P,)-Ar, O.S[F(P,)+F(P,)]

and P' is the middle

of the section

y$P’)== The z- and

P,P1; P'E!&$~,\~~.

We will put

F,,,,,,(P')=

and

t-half-sum

operators

+F(P,)-F(P,)

and (.)(",,,,,

I.

( .) (‘Id and the difference

derivatives

1,1~3

152 and A/At are defined the operator (.)$,,

similarly. the product

All these operators commute with one another. of the operators of the=-,=-, and t-half-sums,

value of the each of the indices 'g, 9. 0 all the multiplied operators

is equal to the sum of the corresponding

We will call when the indices

of

5. We will mean by a vector net function the triple scalar net function F=(FR. Fe, Fz), where FR, FB, FZ, possibly, are defined on different nets. We determine the action of the operators A/At, AjAr. A/As, (.)$!, component-by-component. We define the difference operators DIV and ROT by the equations DT"F=iA(rFR) Ar r

eAE AZ

ROrF = _ _AF8

AFR --AZ

1

We will confine ourselves [.,.] for pairs of vector net periods

AZ

’

’ AFZ Ar

1 A(rF8) --. Ar

’ r

1

to determining the scalar product (., .) and the vector product functions, whose components are specified on nets with integer


6.

Suppose

Q=S&XQ,

and

Fe(Q).

Consider I&s=

We will

(-FR@GZ)@

(FfPJGB) e (-Fe*GZ)

(FZ@GR) (

).

the new net

u o(P). F-znns

call the sum 1s k‘kAz= Iris

2 I'(P)&Az P&i*

A surface net integral of the function F with respect

a volume net integral. be the sum

;P(P')Azs

to

a,n, will

s_ FAZ dP.3

of all the sections over the middles P integral with respect to a,Q,. Henceforth we will use the notation

We can obtain Lemma 1. AFG -= At Lemma

2.

Lemma 3.

the following

&a,.

results directly

For any scalar net function, AF F”.’ AG _ +G”“’ _ At ’ At

Similarly,

from the definitions

specified

given

AF AG AFG = F,v .O)- +G,%.o, -, Ar ’ Ar Ar

1s

net

above.

aF AG AFG = F,a,v,, - +G @m K AZ AZ

.

DI\'ROTF=O.

then

,.,,,,,@lQZ. In particular, and I,=Ini~(i82),,,,,,,~(Il!i) where I,=(rnl),,,,,,~,~(lrr2),a ,:., we have -&(F~.E.,)&A~=

c%kl. ',J

the surface

on a common net, we have

For any vector net function F we will have Suppose F,,+T(C,,,);

we can determine

if

F,,FIEF(Q~),

153

~enmra 4

(theanalogueoftheNewton-Leibnitz I,saii,,,+r~~= I

The following

formula). 1 a2(‘QS

*

kl.

equation

holds:

aJAr. ‘i,)

We recall that in paragraph 3 of Sect.1 we denoted 7. the cavity of the auxiliary system by V,. Wewill assume the following (see Fig.3):

Q+Q,‘,:::,,

se=se(v,),

522=52~!~;1,~,,nz=az(rne.), QR=Q,‘;;;o,,

.-._.__ Fig.3

QP=Q:;:.‘d.O,,

v- for ne,. n - for PZS, X-- forQ&, and 0 - for QQS

PR=RR(KB.). Izp=QIP(lQeS),

QI=Q;:,:;;*.!,:,.

for next to (lying on) the axis of symmetry of The part of the boundary &QZ (or &rle,&QR), r(P)=/&/2 (or r(P) =O) the system, i.e. the part such that for any of its points the equation is satisfied, will be denoted by a,.ciz (or a.,.ne, a,,,Cm). We will define the "smearing net function":

-& j j

i%(P;r,z)=

a(r-r’,

z-z’)&’

dz’,

m2es;

’ ’ U,P,

For any fixed pair

(m,z,)

we have

TO,+swe.~.

Z(P;

3. A difference scheme for the equations of the method of macroparticles. Consider

the net functions Zr. L’Z,. 81EF,(I,.

EZC=Sr,,,,.

k=l,

2,.

, N.

Jz,fz&-(Qr),

P*P,%=w2p), EZ. HZEF(RZ),

ER, HR=F(PR),

Ee, He=s(ne).

Here (3.1)

13.2b)

(3.3a)

(3.3b)

(3.4a) (3.4bl These

functions

are connected

by the set of equations +$AxEz*.

(3.5)

rnn

AZ. xand (at Internal

points

rn& -epyl uzp

C2R@QeW2Z@Qp) . zt=-DIV(J.+oE"'),

(3.6)

of the net

13.7) (3.81

Moreover,

they satisfy

the boundary

conditions

154

.

p-p-o El?==0

on

anp, a.mua....m?.

EZ-0

on

.3,QZ\&,.QZ,

on

3.10) 3.11) E0-0

on

r3CH3

3.12)

and the initial conditions, defining the values of the functions on those layers of the corresponding nets which have coordinate t-o. The satisfaction of Poisson's equation (see (1.3)) and the law of variation of mom of the particle + field system for solutions of the analytical equations of the method of macroparticles (1.7)-(1.10) is formally based on the equation of continuity (1.11). The net functions JZ (see (3.2b)) and p (see (3.2a)), however, are not related by any difference analogueofthisequation. The new "corrected" current densities J' and charge p are introduced to get round this difficulty. We can obtain IZ and i for specified net functions JZ and p by solving Eq.(3.7) as an equation in p (3.13) and then using (3.11). In order that the value of l/p in (3.7) should be limited from zero at least as long as there is at least one particle inside the cavity, it is sufficient to

requirethat

4.

inf a(r,z)Se>O. ,W
To solve the difference scheme (3.1)-(3.12) we carried out discrete analoguesofthelaws of variation of energy and momentum, of the equation divE=-bnp,and divH-O. We will note the scheme without giving the complete proof. 1. We denote by ePthe total mechanical energy of the macroparticles

On the basis of Lemma 1, Eq.(3.5), and the definition variation of energy of the macroparticles

&ZI,

of

we can

obtain

the law of

(4.1)

From Maxwell's difference equations of the energy of electromagnetic field

(3.8) and (3.9) we can obtain

the law of variation

(4.2)

M- ~~ir((E,E)+
(4.3)

jj o(E("',E"'>2xrArAz,

(4.4) (4.5)

where

l--lQR.@lQ0~@lC2Zs. The statements of Lemmas 3 and 4 in fact denote that for the operator ROT constructed

above, the discrete analogueof (2.2) is satisfied. converted into a surface integral K=K:,+K:. Here K, = j n,[ (rEe)

(‘1) 8 HZ”“‘-

Due to this, the volume

(rH8)““’

@

integral

EZL"]Az,

K can be

(4.6)

6‘

where equal

a,-alpe.@rnz,), al=a,(m3,~rm,). A direct check shows that the net integrals on the right-sides of (4.1) and (4.21, are to one another.

This leads to the following

result.

Proposition 1 (law of variation of energy of a particle + field system in the discrete To solve the difference scheme (3.1)-(3.12) we must satisfy the equation problem). N

;

(Mf x8,)=-D+K,+K,, L.-l

where M and D are defined by (4.3) and (4.4), while K,+K,, as can be seen (4.71, istheanalogueofthePoynting vector flux through the boundary. 2. It follows from (3.7), (3.81, and Lemma 2 that

from

(4.3) and

155

$ Hence,

in order

DIV E-c DIV ROT H’“‘-4n

for the following

equation

DIV(J’+oE’“‘)

to be satisfied

=4+. on any time layer: (4.8)

DIVE=4x;, it is sufficient for it to be satisfied the equation DIV H=O. The net function

for the initial conditions.

This is also true for

N

will be called the total momentum of the macroparticles in the discrete model. equations of motion of the macroparticles (3.5) we obtain the law of variation the macroparticles

The integral on the right-side can be rewritten in the form

of this equation,

taking Poisson's

equation

From the of momentum

of

(4.8) into account,

&$!,,, DIVE Ar AZ = C.

(4.9)

The law of variation of momentum of the particle + field system can now be obtained in the discrete model in the same way as in classical electrodynamics. We add the following terms to the right-side of (4.9):

Ar 1\.z,

1

It follows from Maxwell's equation (3.8) and (3.9) that A'=B'-C'--A=B=O. Applying 1 and 3 to the sum of the net inteqrals A+B+C+A'+B'-i-C', we prove the following. Prcqosition 2 (the law of variation solve the difference scheme (3.11-(3.121

of the momentum of the particle we must satisfy the equation

Lemmas

+ field system).

To

$?.+8.)=-b-4naS*. where r[-HR@E0+H8@ER]Arh

8=& ,Ci&ies $- = +j

r[ (HZ’“‘)Z-

(,,““‘)‘-

(HR""')z]Ati+

+aj r:R,!:;:,HZ~"JIAz. a,=a,InRs~a,IQe,ea,lnp.,

a,=a,Ine,~a,Inp,.

Hence, for the discrete model we also have laws of variation of the total particle + field system.

5.

of the energy

and momentum

An algorithm for solving the problem. The difference equations of the motion of the macroparticles (3.5) and (3.h) are implicit Below we will consider one We can use iterational methods to solve them. and non-linear. of the possible versions of an iterational algorithm for solving the difference problem (3.1'(3.12).

156

Suppose we know

F”={Z,“,

D‘ZA”. J”, “J”, p”. “p’. E”,H”)

- the values of the net functions on the

n-th time layer, and F"=(Zk", lJZ*‘. J‘,f”,p’, $, E’. H’}, which are the s-th iterations of the values of these functions on the (n-i-1) -th time layer (if s==o, then for all the above functions we put F’=F”). We write the transition F’++P+‘. Stage 1. We denote by EZ,"(Z,',U&', p",EZ") the function EZ,, calculated with respect to ZI", VZ,“, p”* EZ”, Z,‘, UZ,‘, p; EZ”. Consider the equation

(from 0.3b))

(5.1) where A” :

y+uz,”

z’=Z,‘fs ~~;ll-~~;:~~,

y’=UZ~“+rEZ,‘(t’.

(l+y’lc~)‘“+[l+(UZ,‘)‘/c~]~’

y, p”, EZ”).

It turns out that the operator A” is a compressing operator, of successive approximations for Eq.cS.1) reduces to solving (z:" --Z~")=rm.& (UZ, (vz:” Stage

2.

From

-lJZ;)=r~EZk(Z:+‘,

II

UZ:”

+Lp,q, ,p',EZ*).

(3.2) we calculate P'+'-p(z,",z:'),

and we then solve

d+’ +uZ,“)/(DP:+’

and ~~A"~~=rconst, and the method the equations

JZa+‘-JZ(Z,n.

UZ,“, Z:” , UZ:+‘);

(3.13) with these net functions.

Stage 3. Using the functions iZlr' obtained after Stage 2, we solve Maxwell's We denote the fields obtained in this way by E"+'. H”‘.

equations.

we have obtained a set Z;+', UZ;", p"+',;*'I,J’+‘, ia+‘, E’+‘. H’-‘: repeating the whole Hence, process we can obtain the (s+2)-th iteration. The proof of the convergence of the whole iterational process is carried out in the same way as was done in /9/, on the basis of the laws of variation of energy for the field and the p-F'+* particles (see Sect.4, paragraph l), which enable us to establish that the mapping constructed above is a compressing mapping for fairly small ?(A.-z+h.-'). The solution of the equation of continuity (3.13) with boundary conditions (3.10) at stage 2 can be obtained, for example, by the pivotal condensation method. From Maxwell's difference equations we can obtain two equation-corollaries

(5.2)

If they are solved, then using He and E8 we can immediately and a similar equation for E8. calculate from Maxwell's equations themselves the fields ER,EZ,HR, and HZ. It can be shown by a direct check that the fields E and H constructed in this way are solutions of the initial Maxwell's equations. Thus, instead of (3.8) and (3.9) at stage 3 we can solve an equation of the form (5.2). Methods of solving such equations are considered in /2/. Notes. 1. If we eliminate and 1-J (3.7) from the scheme (3.1)-(3.12) and put c-p everywhere, then for the simpler difference scheme obtained the law of variation of momentum is not satisfied, but the law of variation of energy is. Note that this scheme, and the initial scheme (3.1)-(3.12) can be modified for a system with non-relativistic electrons. 2. We have assumed from the very beginning that the electrons are completely magnetized (Sect.1, paragraph 1). If this is not so, instead of (1.7) we will have 6 scalar equations describing the motion of the electron due to the action of the Lorentz force in cylindrical coordinates. It turns out that for this system we can construct a conservative difference scheme, the conservatism of which is based on the same principles as above. REFERENCES 1. BOGDANKEVICH L.S., KUZELEV M.B. and RUKHADZE A.A., plasma microwave electronics Usp. Fir. 133, 1, 3-32, 1983. Nauk, 2. SAMARSKII A.A. et al., Theory of Difference Schemes (Teoriya rasnostnykh skhem), Nauka, Moscow, 1980. of difference schemes of mathematical physics in 3. SAMARSKII A.A. et al., Representation operators form, Dokl. Akad. Nauk SSSR, 258, 5, 1092-1095, 1981. 4. SAMARSKII A.A. et al., The use of the method of reference operators to construct difference analoguesoftheoperationsof tensor analysis, Diff. Uravneniya, 18, 7, 1251-1256, 1982. 5. BEREZIN YU.A. and VSHIVKOV V.A., The Method of Particles in the Dynamics of Rarefied Nauka, Novosibirsk, 1980. Plasma (Meted chastits v dinamike razrezhennoi plamy), l/N-limit of inter6. BROWN W. and HEPP K., The Vlasov dynamics and its flutuations in the acting classical particles. Communs. Math. Phys. Vol.Sh, No.2, 101-113, 1977.

157 7. ARSEN'YEV A.A., Removal of smoothing on passing to the liquid limit for systems with Coulomb interaction, Preprint IPMatem, Akad. Nauk SSSR, No.82, 1982. Akad. Nauk SSSR, 8. SVESHNIKOV A.G., The principle of limit absorption forwaveguides,Dokl. 80, 3, 57-64, 1951. 9. SVESHNIKOV A.G. and YAKUNIN S-A., Mathematical modelling of non-stationary processes in plasma-optical systems, Zh. vychisl. Mat. mat. Fiz., 23, 5, 1141-1157, 1983. Translated

U.S.S.R. Comput.Maths.Math.Phys.,Vo1.25,No.3,pp.157-167,1985 Printed in Great Britain

by R.C.G.

0041-5553/85 $lO.OO+O.OO Pergamon Journals Ltd.

EXISTENCE AND UNIQUENESS OF THE SOLUTION OF THE LINEARIZED BOLTZMANN EQUATION IN THE CASE OF INELASTIC COLLISIONS*

B.V. ALEKSEYEV

and I.T. GRUSHIN

The kernel of the linearized Boltzmann inteqro-differential equation with inelastic collisions is isolated. For a wide class of elastic and inelastic collision cross-sections, the square integrability of the third iterated kernel of the equation is proved, and the existence and solution, orthogonal to the invariants of the inelastic uniquenessofa collisions, are proved. The problem of the perturbation of the Maxwell distribution function in bimolecular chemical reactions is solved numerically. 1. Isolation of the kernel. We consider a gas mixture, in which, apart from paired elastic collisions inelastic collisions are possible, leading to bimolecular chemical reactions

of particles,

A,+A,=A,+Aa. We write Boltsmann's

equation

(1.1)

as

++L+=

OL

EJ ,

(1.‘f,‘-f”f,)P,,“‘g,,bdbfledv,+

(1.2)

The generally accepted system of notation is used in Eq.(1.2), see /l, pp.63-68, 134/. When the number of inelastic collisions is of the same order as the number of elastic collisions, the linearized Boltzmann equation has the form /l, p.261/

jL”[ (\‘, .zy )(w,2-+)+“(v,.d,)+

(1.3)

11,

2w,ow,

:;

vg I=iiJ

6-1

rd”‘r:“’ (ha’+!/,‘-h,-h,)X

b sin8 db d0 dE dqadv,. ha)~.rPm’? On dividing term corresponding

Eq.cl.31

by

I:!, we isolate

to reaction L;;=J

In (1.4) we have introduced

in the integral

operator

of the equation

the

number r: jd"'(hl’+h~‘-h,--hp)

the differential

gabo,‘:‘sin

effectiveness

8 d8 dq dv,. cross-section

(1.4) of reaction

(1.1):

(1.5)

*Zh.vychis.Z.Mat.mat.Fiz.,25,6,896-911,1985