52
IZVESTIYA VUZ. FIZIKA
ON THE PARAMETRIC DESCRIPTION O F AN INHOMOGENEOUS, OPTICALLY DENSE PLASMA I. The D i s p e r s i o n Shape of a S p e c t r a l Line N. G. P r e o b r a z h e n s k i i and G. A. Kolobova I z v e s t i y a VUZ. F i z i k a , No. 4, pp. 8 8 - 9 2 , 1966 The source inhomogeneity parameter is calculated for various distributions of excited atoms and variations of the line width along the discharge cross section, for a "poxely" dispersive shape of the spectral liFle.
The frequency dependence of the inhomogeneity parameter, which varies with increase of the radial dependence of the line width, is determined. It is shownto be necessary to take this fact into account in calculating the shape of a self-reversed line. In c a l c u l a t i n g the i n t e g r a l i n t e n s i t y , the c o n c e n t r a tion of e x c i t e d a t o m s or ions, the o s c i l l a t o r s t r e n g t h s , e t c . , for an i n h o m o g e n e o u s , o p t i c a l l y d e n s e p l a s m a , it is n e c e s s a r y to know the s o - c a l l e d d e g r e e of i n h o m o g e n e i t y of the s o u r c e . This p a r a m e t e r , i n t r o duced in the m o d e l s of Cowan and Dieke [1] and B a r r e l s [2, 3], c h a r a c t e r i z e s the v a r i a t i o n along the d i s c h a r g e c r o s s s e c t i o n of the s o u r c e function, defined by Chand r a s e k h a r a s the r a t i o of the coefficient of e m i s s i o n e u to the a b s o r p t i o n c o e f f i c i e n t x v ; thus one l i m i t i n g c a s e is that of a r a d i a t i n g f i l a m e n t at the c e n t e r of the s o u r c e , s u r r o u n d e d b y a "shell" of a b s o r b i n g a t o m s , while the o t h e r is a h o m o g e n e o u s l a y e r , in which r a d i a t i n g and a b s o r b i n g a t o m s a r e u n i f o r m l y " m i x e d . " B a r t e l s [2,3] n o t e s that the d e g r e e of irdaomogeneity is not c o n s t a n t over the e n t i r e line, but v a r i e s with f r e q u e n c y , p a r t i c u l a r l y in the r e g i o n s of s m a l l Av, which is due to the v a r i a t i o n of the shape of a n o n r e a b s o r p t i v e l i n e along the p l a s m a c r o s s s e c t i o n . In the p r e s e n t work we c o n s i d e r the f r e q u e n c y d e p e n d e n c e of the i n h o m o g e n e i t y p a r a m e t e r for v a r i o u s g r a d i e n t s of the line width and the s o u r c e function toward the axis of the d i s c h a r g e , for " p u r e l y " d i s p e r s i v e line shape. We p r e s e r v e the b a s i c p h y s i c a l p r e m i s e s of the t h e o r y of B a r t e l s [2], which a r e a p p l i c a b l e to m o s t s o u r c e s , p a r t i c u l a r l y to a r c s : 1) the p l a s m a l a y e r is a x i a l l y s y m m e t r i c , and the point of o b s e r v a t i o n l i e s in the p l a n e of a p e r p e n d i c u l a r c r o s s s e c t i o n of the l a y e r ; 2) the s o u r c e function Jr(x) = = ~v(X)/~v(X) is a s s u m e d to d e c r e a s e m o n o t o n i c a l l y f r o m the a x i s of the p l a s m a to the p e r i p h e r y ; 3) within the l i m i t s of the s p e c t r a l l i n e Jr(x) is p r e s u m e d i n d e p e n d e n t of f r e q u e n c y (Kirchhoff law); 4) the n o n r e a b s o r p t i v e l i n e has the shape of a s y m m e t r i c c u r v e with a m a x i m u m , so that the i n t e n s i t y on each side d e c r e a s e s m o n o t o n i c a l l y to zero. F o r this c a l c u l a t i o n we u s e the c o o r d i n a t e s y s t e m t a k e n by B a r r e l s , i . e . , we i n t r o d u c e the r e l a t i v e c o -
w h e r e t and to a r e , r e s p e c t i v e l y , the g e o m e t r i c a l coo r d i n a t e and the g e o m e t r i c a l t h i c k n e s s of the l a y e r if the zero point is outside the s o u r c e , and the r e l a t i v e optical length x
- = f ~, (~') dx',
w h e r e x* (x) = , "~(x)
the a b s o r p t i o n coefficient. F u r t h e r , a c c o r d i n g to B a r t e l s the shape of the line x ' ( u , x ) can be w r i t t e n in the following f o r m :
~' (u, x)
1
x . . . . , to 2
(1)
"~(u, x) fn (x)
(a)
'
f -,/(,, x) d~ = -~d- ,2
(4/
mc {}
w h e r e f is the o s c i l l a t o r s t r e n g t h of the l i n e , n(x) is the c o n c e n t r a t i o n of a b s o r b i n g a t o m s , and u = A v / 6 is the r e l a t i v e f r e q u e n c y . F o r a "purely" d i s p e r s i v e l i n e ~e 2
.z (u)
mc (~, + ~r
(5)
'
w h e r e 5 is the h a l f - w i d t h of the n o n r e a b s o r p t i v e line. Using Eq. (5), we obtain an e x p r e s s i o n f o r x*(x),
n* (x)y (x) u ~ + y'-' (x) ~* (x) =
(6)
"": ,r~ (x) y (x! d x '
2 ! ~ + y2 (x)
where
y (x) = ~.(X)_
n* (x)
(0) '
n (x) "~
j n (x) ax
A s s u m i n g for s i m p l i c i t y n*(x) = 1, y(x) = 1 - 2ax, a = [6(0) - 6(1/2)]/6(0), we obtain f r o m Eq. (2) the following e x p r e s s i o n for ~(x):
1 t
is the n o r m a l i z e d value of
j ~ (~) a~
In
o r d i n a t e x,
(e)
0
"(~)=~-"
u2+l
u~+ 11 - 2 (1 - ~ ) x ] 2 In u~+-----L u ~ + ~z
a(l/2) '
~=
~(o)"
(7)
SOVIET PHYSICS JOURNAL
53
0.7
,FZ=I
0,5
o.t
!"
u ?
;/
0.3
]
o,j 0
0.5
,,0
/5
2.O
~.5
0
0.5
I~
Z.5 --.------ t/
Z.O
Fig. 2
Fig. 1
0.8
0.0
t
O.7
-----
~ --
~g,2 ~33 ..Z
0,8
L I
t6
f
1 /
a2 o
~s
1.o Fig. 3
15
o
Fig. 4
e5
54
IZVESTIYA VUZ. FIZIKA
The e x p r e s s i o n for the i n h o m o g e n e i t y p a r a m e t e r q(u) will have the following f o r m :
'z~ j(~) 12. j' ~3 -d~ _,/, Y~m
q(u) =
K (u)
j'~ ]j(~ d~
- M (u)"
(8)
After integrating K(u) and M (u) by parts, taking into account the symmetry of the integral functions and the normalization condition, and using expression (2), we obtain
~3. f (x) dx q
=
4
(9)
.I r.]" (x) dx o
w h e r e f(x) = J u ( x ) / J u m is a s s u m e d to be m o n o t o n i c a l l y d e c r e a s i n g f r o m 1 on the axis to zero on the s o u r c e b o u n d a r i e s a c c o r d i n g to the law f(x) = 1 - (2x) n. The c a l c u l a t i o n s w e r e c a r r i e d out for the v a l u e s n = 1/2; 1;2. In the s i m p l e s t c a s e , for n = 1, the f o r m u l a for c a l c u l a t i n g M(u) has the f o r m
•(u, 4)=
, 2 (1--~)
• L
Ill
u - ~ + I -] 9 "I
u2 4- ~1-J
u'~+l +
•
(10)
]:i
Because of the complexity of the expressions for integer n and the untabulated integrals for half-integer n, the quantities q(u) and M(u) were calculated using Gauss's method of numerical integration for each n with the following v a l u e s of ~7= 6(1/2)/6(0): 0.1; 0.2; 0.33; 0.5; 0.8. The p r e c i s i o n of the i n t e g r a l c a l c u l a t i o n s was found to be s u f f i c i e n t upon u s i n g four k t e r m s in G a u s s ' s f o r m u l a . The r e s u l t s of the c a l c u l a t i o n s a r e p r e s e n t e d g r a p h i c a l l y . The function q(u) is g i v e n in F i g s . 1, 2, and 3, for n = 1/2, 1, and 2, r e s p e c t i v e l y . The v a l u e s of
were varied in each case from 0.i to 0.8. In Fig. 4 are presented the functions q(u) and M(u) at constant i?=0.2 forn= 1/2, i, and 2. From analysis of the curves it is possible to conclude: I. The inhomogeneity parameter q(u) is determined, on the one hand, by the distribution of excited atoms in the source (which in the given case is characterized by the value of n), and on the other hand, by the radial dependence of the line width, specified by the value of 7. 2. With the variation of the line width along the discharge cross section, q(u) varies from the center of the line toward the sides in such a way that for all values of ~ there is a minimum at u = 0 and a monotonic increase, as u ~ ~o, to the boundary value specified by the magnitude of ~? (Figs. i, 2, 3). The more rapidly the line width in the discharge decreases, i.e., the smaller is ~?, the larger will be qffo), and the lower will be the minimum of q(u). As ~? increases, i.e., as the decrease in the line width becomes small, q(u) approaches the constant value q0 = (n + l)/(n + 3). The rise from the minimum to the boundary value lies in the region of maximum line width (-2 < u < 2). 3. The behavior of the function M(u), calculated at the same values of the parameters, is analogous to the behavior of q(u); since as n increases, i.e., as the distribution of excited atoms becomes more uniform, the growth of M(u) outstrips the growth of q(u) (Fig. 4). 4. Clearly, the dependence of the inhomogeneity parameter q and of M on frequency must, above all, deform the region of the dip in a self-reversed line. It is of interest in all plasma diagnostics to study the influence of the frequency dependence of q(u) and M(u) on the line shape in this region, for various gradients of the variation of the concentration of excited atoms and the line width along the discharge cross section; this will be carried out in the future. REFERENCES i. 418, 2. 3.
R. Cowan and G. Dieke, Rev. Mod. Phys., 1948. H. Bartels, J. Phys., 125, 597, 1949. H. Bartels, J. Phys., 126, 108, 1949.
30 D e c e m b e r 1964
20,
N o v o s i b i r s k State U n i v e r s i t y Kirov Tomsk Polytechnic Institute