DIFFUSION

OF

ATMOSPHERE L.

SLOW WHEN

A.

Gava!las

ELECTRONS

IN

INELASTIC and

Yu.

AN

INERT

COLLISIONS M.

GAS OCCUR

UDC 539.186.2

Kagan

We have p r e v i o u s l y considered the problem of finding the diffusion electron c u r r e n t s which a r i s e in a gas which fills the space between a s o u r c e electrode and a receiving electrode [1]. It was assumed that the emitting s u r f a c e emits a uniform single-velocity electron beam n o r m a I to the surface, and that the i n t e r action of the electrons with the gas atoms has the form of isotropic elastic scattering. Eddington's method, empIoyed in the t h e o r y of radiant energy t r a n s f e r [2], was used to solve the probIem. In this paper the p r o b l e m of obtaining the diffusion c u r r e n t s is solved on the assumption that the e l e c t r o n energy E 0 is f a i r l y large, so tllat in t h e i r collisions with atoms atomic excitation as well as elastic scattering is possible. 1. The Plane Symmetry C ase. As p r e v i o u s l y we will introduce the "distribution function of the s c a t t e r e d e l e c t r o n s " f so that fd~2dv will define the number of elastically s c a t t e r e d electrons in a volume dv, whose velocities lie within the solid angle d a . F o r two infinite plane e l e c t r o d e s a distance L apart,the kinetic equation for determining the d i s t r i bution function of the elastically s c a t t e r e d electrons will have the following f o r m :

COS

~ df .!_f= 1 (f -" dx + 1,o ~ - I ~ sin~'dl)' + 4~,v},J--s176e--~7.

(1)

0

Here h 0 is the mean f r e e path of an e l e c t r o n with energy E 0 with r e s p e c t to total scattering (we will denote the total c r o s s section by Q0); k is the mean f r e e path of an electron with energy E 0 with r e s p e c t to elastic s c a t t e r i n g (we will denote the elastic s c a t t e r i n g c r o s s section by Q); J0 is the density of the emitted e l e c tron current; x is the distance f r o m the emitting surface; ~ is the angle between the direction of the velocity and the axis; and Q 0 - Q =Q1 is the total c r o s s section of all f o r m s of inelastic p r o c e s s e s which a r e possible for electron energies equal to E 0. We introduce the variable r = x ~ and the p a r a m e t e r ~ =Q/Q0 = )t0/X, which gives the fraction of elastic collisions. Equation (1) can then be written in the f o r m t~cos~

+ f = -~ .

fsin{}'d~' + Jo~ .e 4~v

'~ "

(2)

0

We seek the solution in the f o r m of a s e r i e s in Legendre polynomials, confining o u r s e l v e s to the f i r s t two t e r m s , i.e., f(% 0) = A (~) + B (~) cos ~

(3)

We obtain the two equations required to find A(r) and B(r) in the following w a y : first, by integrating (2) over the solid angle and substituting (3); second, by multiplying (2) by cos ~ and then integrating over the solid angle and substituting (3). As a result we obtain Leningrad Institute of P r e c i s i o n Mechanics and Optics. T r a n s l a t e d f r o m I z v e s t i y a VUZ. Fizika, Vol. 11, No. 8, pp. 64-70, August, 1968. Original article submitted D e c e m b e r 6, 1967.

9 Consultants Bureau, a division of Plenum Publishing Corporation, 227 g/est 17th Street, New York, N. Y. 10011. All rights reserved. This article cannot be reproduced for any purpose whatsoever without permission of the publisher. A copy of this article is available from the publisher for $15.00.

48

dB(z)+ 3 ( I - - z ) d ~:

A(,:)--

3/0 4r

tc

dA (-:)

e

':

(4)

'

~ B (~) ~- 0.

d~

(5)

The s y s t e m can be r e d u c e d to the single s e c o n d - o r d e r equation

d2A(,)

3(1--~)

A(z)

3/' o

e

-

'~

(6)

4~r vtc

Equation (6) can be solved f o r ~-< 1. The following form of the general solution satisfies this condition: K

z

21/3(--1--

A (~) -

~c~)C,

e V:~('--~)~ -- e -V3{~-'~ ~-] + 3jo

+ 2-C~ I ev 8(,.-~)~- .§ e-K32v-~T~-]

4rcv(3~'--2)

e

-~

(7)

T h e n B ( r ) can be e x p r e s s e d as

_~_C2V~I-K)2x

[ e Va~

:

T -- e - g ~

-:]

+ 4~vK (3K--2)

e

q

(s,

.

The c o n s t a n t s C l and C 2 a r e d e t e r m i n e d f r o m the b o u n d a r y conditions which e x p r e s s the fact that t h e r e is no flux of s c a t t e r e d e l e c t r o n s f r o m the r e g i o n x-< 0 and x >- L =/2

Here

r 2=

L/;t.

~:/2

S f (0, ~) cosl} sin 0 e,) = o;

,If(%, o) cos {}sin ~}dl} = 0 '

0

0

(9)

Substituting (3) in (9) we obtain the r e l a t i o n s h i p b e t w e e n A ( r ) and B ( r ) f o r r =0 and T = r 2.

Using (9), C 1 and C 2 can be e x p r e s s e d as

C~ = / o

31s

,,

4~v(3K--2)

% + 5]/3{1--~) sh l/~-(1--K) % _ _1 e _ z ~~ 3 tc 2

5/,2 ch l / 3 ( ~ X ~

2 ~ ch V ~ ( ~ -

~! ~,~+ ~ ~ I5 2

a C~ = J 0 4~v(:3/c--2)

X

Dcch V3(1 --~}

(10)

sh I / 3 ( ~ - ~ ) -

2V3(1-K)

E

5itch V~3(I--K)

7 _-- _4~ ~

K

]fl~(1--t<)

K31-.)

~

7--4~

% +

_.:z (11)

sh F 5/1 - ~

Substituting (10) and (11) in (7) and (8) we obtain A ( r ) and B ( r ) . The flux d e n s i t y of e l a s t i c a l l y s c a t t e r e d p a r t i c l e s incident on the c o l l e c t o r e l e c t r o d e (r = r 2) is d e t e r m i n e d f r o m the r e l a t i o n H2 = 2~ v j f ( %

(12)

~) cos ~ sin~d0.

0

Using the e x p r e s s i o n f o r A ( r ) and B ( r ) this flux can finally be e x p r e s s e d as

N~

=

Jo 2 ~ c h ]/3(1--K)

~

I

/'2

~ 3 5 l / 3 ( 1 - ~ ) shlf3-(1-- /c) -Ce ~ - - - ~1e

-~] J

' 2]/3(I--K)

shV3(I--Kj'--K

49

V3(1 - i~) R:

.shV3(1

-

-

K)

e ~ }- 2 V 3 ( I - ~)

tc

2~c c h ] / 3(--f--~ ~ ) % - tK

(13) 7 .---_4,~s h V 3 ( - - i ~ - _ ~ T ~ i L 2V 3(l--K)

-

It is interesting to c o m p a r e the r e s u l t s obtained with the data given in [3], where an expression was derived for the flux of elastically s c a t t e r e d electrons at the collector, obtained by solving a s i m i l a r p r o b lem by another approximate method ( C h a n d r a s e k h a r ' s method [4]). F i g u r e 1 c o m p a r e s the r e s u l t s obtained in [3] (the continuous lines) with the flux calculated f r o m f o r m u l a (13) (shown by the points). We h a v e p l o t ted the r a t i o H2, g/H2, 1 along the ordinate axis; H2, t~ iS the flux at the collector for a given value of the p a r a m e t e r ~; and H2, i is the flux at the collector when ~=10 Along the a b s c i s s a axis we have plotted the values of the p a r a m e t e r ( l - x ) , which gives the fraction of inelastic collisions. We have plotted curves for t h r e e values of ~-2 (~"2 =3, T 2= 5, and T 2 = 10). Note that f o r K~ 1 Eq. (13) in the limit takes the same form as Eq. (8) of [1], which e x p r e s s e s the flux of electrons at the collector calculated on the assumption that the interaction of the electrons with the gas atoms is in the form of elastic scattering. 2. Cylindrical Symmetry. The problem of the diffusion of electrons in the case of two infinite coaxial cylindrical eIeetrodes is of much g r e a t e r p r a c t i c a l interest, since for this g e o m e t r y the r e sults of the calculations can be c o m p a r e d with experiment, as was done in the case of elastic scattering in [5]. Suppose the s o u r c e of electrons is an infinite cylinder of radius r 1 and the r e c e i v e r is a cylinder c o axial with it of radius r 2. F o r this case the kinetic equation for elastically scattered electrons has t h e f o r m (r-r,)

-2--v f=~),0

vvf+

~-v f

f ( r , O , ~ ) T 2 -~de - +--

Here, as above K=Q/Q0 is the fraction of elastic collisions. tem of coordinates, where the z - a x i s is parallel to the axes that the velocity v at the given point is in the xz-plane. Let z - a x i s , and let ~o be the angle between x - a x i s and the v e c t o r write Eq, (14) in the following f o r m :

io 4~},

. -r~ e

r2

-

~x--?-

(14)

We introduce at the point r a r e c t a n g u l a r s y s of the cylinders, and the x - a x i s is chosen so | be the angle between the velocity v and the r. Introducing the variable ~- = r/X, we can 1

sin 0

COS? O~

9

~

-~ K

~-~

4=v--~- e

.

(15)

F o r r e a s o n s given in [1] we seek the solution in the f o r m of the following expansion of the distribution function: f(~, 0, ~) = A (~) + B(~)cos~ sin O.

(16)

Using the method described above we obtain equations for determining A(~') and B (T)" (~-xl)

B, + I _ B +

3(I--g) Ic

A:

3]o ~_L' e 4zV 9

'

,4' -[- l _ B = 0.

(17)

K xt

Eliminating B and putting 3(1 - - ~ ) M ; R"2

L,

we obtain an inhomogeneous B e s s e l equation:

47~ ~)

~A"

50

3]o~1 .e r =

+ ~ A ' - - ~2M~-A - -

~ Le

/r

~

(18)

T h e g e n e r a l s o l u t i o n of t h e h o m o g e n e o u s e q u a t i o n c a n b e w r i t t e n in t h e f o l l o w i n g f o r m :

H2, 1

A=

( C~ -- C~ ln ~

) Io ( M~) -- C,Ko ( M , ) ,

w h e r e I o and K o a r e m o d i f i e d B e s s e l f u n c t i o n s . W e s e e k a s o l u t i o n of Eq. (18) b y t h e m e t h o d of v a r i a tion of c o n s t a n t s ; we e x p r e s s CI(T) and C20-) in t h e f o l lowing form:

t-

-

e

'~ 9 l o ( M ' ~ ) d ' c

+

R,

,

h;

C'(;):--L-(/ e-~-(ln~'lo(M=)+Ko( MOd;+R~.)) . ge/

{of

(19)

a=

Fig~ 1 T h e b o u n d a r y c o n d i t i o n s , s i m i l a r to (9), c a n b e w r i t t e n a s

for ~ 1 , 0

3/2=

0

"~

3/2=

,t"si.,o o fz.cos 0

=o

for

'~ ~ "~2 -

(20)

~/2

T h e s e c o n d i t i o n s r e d u c e to A (r

~

2 _ B (r 3

= O,

A0:2)-- 2 B(,~,)=0. 3

U s i n g t h e b o u n d a r y c o n d i t i o n s we o b t a i n e q u a t i o n s which e n a b l e us to d e t e r m i n e R 1 and R 2. F o r c o n v e n i e n c e we u s e t h e f o l l o w i n g n o t a t i o n :

,z, (~) = In m___2~o (M~) + ~o (m~), ~-

~,

(~) = in ~- ],(M~) -- K, (M~),

-~

~

"eL

a=

2 -3

tr

__2"

":-i

2

('~1) -- ", (z,),

= ' / 0 (M~,,) +

~ = Io(M%) -- - - t~M[, (M%), 3

2 u m [ , (M~.), 3

p= -

ah.

In t h i s n o t a t i o n t h e e q u a t i o n s f o r d e t e r m i n i n g t h e c o n s t a n t s R 1 and R 2 b e c o m e ~R1 + [~R~ = ~',

~R, 4 ~R~ = ~'.

The constants themselves can be expressed as R, = ~;a - - ~

.

R2 = ~P -- ;'~

(21)

T h e c u r r e n t which a r r i v e s p e r unit l e n g t h of t h e c o l l e c t o r e l e c t r o d e (~- = ~'2) is

0

0

3~2,~

51

Taking (20) and (21) into account, 12 can be written as

H2, ~t. H~, 1

io=4~2r~A('72)=jo3~r2 "

L_t e ~-

R2--1n N/ 9 I1 + In

tc

, Io(M'72)

2

The expression for the c u r r e n t which r e t u r n s per unit length of the e m i t t e r electrode has the following f o r m : 712".:

I, = --2~rw ; sin2Od0 J'f('Tl) COS,~d, 0

:]o3v.rx ~J- e z-

/$

~J2

I,+I.,+

In

R2--1nM---.R~

Io(Mz,)--R/~o(M% ) .

As in the case of planar s y m m e t r y , the expressions f o r the fluxes in the limit as ~ 1 and M ~ 0 reduce the f o r m u l a s obtained in [1]. We can v e r i f y this by the example of the e x p r e s s i o n for the flux density which is r e t u r n e d to the e m i t t e r electrode. The following relations hold:

r2

Fig. 2

Lima = x-.'l M~O

2

1

3

"Tt

+ In ~ = a',

Lim ~= 1, .'r M~O

Lim~,~ = In %~e. . . .

In ~,e . . . .

f e~- _'71d= = ~l',

M~0

Lim g =

2

1

-t- tn z2 = ~',

M..-,.0

Lim s = 1, M-:-0

Lim~=(e:~--e-~,) M~0

7'

-- }~' In

-?

a'p'__Tt~ , ]

/r 1

M-r,-O

The flux density at the collector H2 = - 2 w A ( ~"1). - - 2-~-e-('-~-'-o+ 3In ~ + 3e", [Ei(~,) -- Ei(~2)] Lira H~ = J0 "72 b:-~l M-+O

2

2

_ _ 4 _ _1_ "71 "72

+31r,

~2

-"71

which a g r e e s with f o r m u l a (24) of [1]. In the n u m e r i c a l calculations the integrals I 1 and 12 were evaluated on a computer. F i g u r e 2 illustrates the effect of the probability of inelastic p r o c e s s e s (1-~) on the electron flux density a r r i v i n g at the c o l l e c t o r . Along the ordinate axis we have plotted the ratio of the flux density at the collector f o r a given value of the p a r a m e t e r (l--K), namely, H2, to, to the flux density when t h e r e a r e no inelastic collisions, namely, H2, 1. Along the a b s c i s s a axis we have plotted the quantity ~-2=r2/X. The five c u r v e s c o r r e s p o n d to a change in the probability of inelastic collisions in the range from 1 to 5%. LITERATURE 1. 2. 3. 4. 5.

52

CITED

L . A . Gavallas and Yu. M. Kagan, I z v e s t i y a VUZ. Fizika [Soviet P h y s i c s Journal], no. 2, 1965. Sobolev, Radiant E n e r g y T r a n s f e r in the A t m o s p h e r e s of Stars and Planets [in Russian]. P~ I. Chantry, A. V. Phelps, and G. I. Schulz, Phys. Rev., vol. 152, no. 1, 1966. S. C h a n d r a s e k h a r , Radiative T r a n s f e r [Russian translation], 1953. L . A . Gavallas and Yu. M. Kagan, I z v e s t i y a VUZ. F i z i k a [Soviet P h y s i c s Journal], no. 4, 1967.