- Email: [email protected]
On stark broadening functions for nonhydrogenic ions
Volume 33A, number 7
ON S T A R K
PHYSICS LETTERS
BROADENING
FUNCTIONS
14 December 1970
FOR
NONHYDROGENIC
IONS
S. K L A R S F E L D Institut de Physique Nucl~aire, Division de Fhysique Th$orique *, 91 Orsay, France Received 12 October 1970
Rapidly convergent integral representations are given for several functions used in calculating profiles of isolated ion lines.
S e m i c l a s s i c a l b r o a d e n i n g c a l c u l a t i o n s of s p e c t r a l l i n e s e m i t t e d by p o s i t i v e nonhydrogenic ions in a p l a s m a have been r e p o r t e d by many a u t h o r s [1-6]. The n e c e s s i t y of using h y p e r b o l i c t r a j e c t o r i e s for the p e r t u r b i n g e l e c t r o n s m a k e s the evaluation of the s e c o n d - o r d e r t i m e i n t e g r a l , which defines the b r o a d e n i n g functions A and B, much h a r d e r than for n e u t r a l l i n e s . A b e t t e r u n d e r s t a n d i n g of the a n a l y t i c s t r u c t u r e of the new functions m a y be r e a c h e d again on a p p e a l i n g to contour i n t e g r a t i o n techniques. However, the o r i g i n a l method, which p r o v e d so efficient in the n e u t r a l c a s e [7], now m e e t s with s e r i o u s d i f f i c u l t i e s and does not s e e m to be w o r k a b l e . We have u s e d i n s t e a d a m o d i f i e d v e r s i o n , which s t a r t s by c o n s i d e r i n g d i r e c t l y the ' d i a g o n a l ' functions a s s o c i a t e d with a Debye s h i e l d e d e l e c t r o n - i o n i n t e r a c t i o n [8]. While this does not p r o v i d e c l o s e d f o r m e x p r e s s i o n s for A and B, it y i e l d s for both s i m p l e and f a s t c o n v e r g e n t i n t e g r a l r e p r e s e n t a t i o n s , which a r e v e r y convenient for n u m e r i c a l computation. In the n o n - s h i e l d i n g l i m i t t h e s e r e a d (definitions and notations a r e the s a m e a s in ref. [6]):
f
dt c o s 2 ~ t c o s h 2 t K 0 ( 2 ~ cosh t), A(~,E) = 4 ~2e2exp(Tre) ~/2 0 dO exp(2~O) sin2~ K0(2~e s i n 0 ) B(~, ¢) = 4 ~ 2 ~ 2 f 0
(1)
oo
- 4 ~2 ~2exp(Tr~)
f
d t s i n 2 ~ t cosh 2 t K0(2~e cosh t ) .
(2)
0 The subsequent i n t e g r a t i o n o v e r the i m p a c t p a r a m e t e r is e x p r e s s i b l e in t e r m s of oo
a(~,E)
oo
= f de'A(~,C)/~',
(3)
b(~,e) = f d ~ ' S ( ~ , e ' ) / E ' .
£
E
T a k i n g account of eq. (1), (2) one e a s i l y gets i n t e g r a l r e p r e s e n t a t i o n s for a and b, e.g., 7r/2
b(~,e)
=
2~ef
d O e x p ( 2 ~ O ) s i n O K l ( 2 ~ e s i n 0 ) - 2~e
exp(~)f
dt s i n 2 ~ t c o s h t K l ( 2 ~ e c o s h t ) ,
(4)
dt c o s 2 ~ t sinh 2t K0(2~ e c o s h t ) .
(5)
0
0
or alternatively, after a partial integration,
~/2 b(~,e) = - 1 / 2 ~ + ~e 2 f
d 0 e x p ( 2 4 0 ) sin 20K0(2~e s i n 0 ) 0 oo
+ ~e 2 exp ( ~ )
f 0
* Laboratoire associ~ au C. N. R.S. 437
Volume 33A, n u m b e r 7
PHYSICS
LETTERS
14 D e c e m b e r 1970
In t h e l i m i t ~ ~ 0 o n e f i n d s B ( 0 , ~) = 0, b(0, ~) = ~ / 2 . On t h e o t h e r h a n d , f r o m t h e i n t e g r a l r e p r e s e n t a t i o n s g i v e n a b o v e o n e c a n e a s i l y e x t r a c t v a r i o u s a s y m p t o t i c e x p a n s i o n s f o r ~ -~ oo. In p a r t i c u l a r f o r > 1 we f o u n d B ( ~ , e ) ~ ( 1 / 2 ~ e) h 2 ( 1 / e ) + ( 1 / 1 6 43 e 3) [ h 4 ( 1 / e ) + ( 1 / 3 e ) h h ( 1 , / e ) ] + . . .
(6)
b(~,~) ~ ( 1 / 2 ~ ) [ h l ( 1 / e ) - 1] + ( 1 / 4 8 ~ 3 e 3 ) h 4 ( 1 / e ) + . . . .
(7)
w h e r e hn(cz)
= (d/dc~)nh(~)
and
oo
h(e~) = f du e x p ( e t u ) K o ( u ) = (1 - o~2)-1/2(~ - a r c c o s
c~),
(c~ < 1).
(8)
0 Eq. (6) c o i n c i d e s w i t h t h e e x p a n s i o n g i v e n p r e v i o u s l y by S a h a l - B r O c h o t [6]. T h e l a s t f o r m u l a e a r e c l e a r l y i n a d e q u a t e w h e n ~ i s v e r y c l o s e to u n i t y . A s a m a t t e r of f a c t t h e a s y m p t o t i c b e h a v i o u r f o r ~ ~ oo a n d E = 1 h a s a c o m p l e t e l y d i f f e r e n t c h a r a c t e r (~ = 1 i s a s o - c a l l e d t r a n s i t i o n p o i n t ) . In t h i s c a s e we o b t a i n e d B(~, 1) ~ - 2 . 8 8 6 42/3 + . . .
(9)
b(~, 1) ~ 0.684 ~-2/3 + . . .
(10)
It i s a l s o p o s s i b l e to w o r k out u n i f o r m a s y m p t o t i c e x p a n s i o n s . A + i B ~ 2~ 1/2 e x p ( - i ~ / / 3 ) e 3''~ G(~') ~2/~ + . . .
For instance:
,
(tl)
where O9
G(~) :
. f l d ! l 3/2 e x p ( ~ t - 1 3 , / 3 ) .
[ : 2 exp(2i~/3)(e-1)~2/3.
(12)
0 If !~[ i s s m a l l o n e c a n u s e G(~) ~ 0 . 9 4 0 + 1.114~ + 0.767~ 2 . Eq. (11) e s t a b l i s h e s
(13)
t h e l i n k b e t w e e n e x p a n s i o n s of t h e t y p e (6) a n d (9).
It i s a p l e a s u r e to a c k n o w l e d g e s t i m u l a t i n g d i s c u s s i o n s B r ~ c h o t a n d H. v a n R e g e m o r t e r .
w i t h D r s . J. C o o p e r , H. G r i e m ,
S. S a h a l -
Refere~wes [lj H.R. Griem, Phys. Rev. L e t t e r s 17 (1966) 509: Phys. Rev. 165 (1968) 258. [21 S. Brdchot, Phys. L e t t e r s 24A (1967) 476. [3}J. Cooper and G.K. Oertel. Phys. Rev. L e t t e r s 18 (1967) 985. [4] J. Davis and D . E . R o b e r t s , Proc. Phys. Soc. 92 {1967) 889; J. Phys. (B) 1 (1968) 48. [5] V. A. Alekseev and E. A. Yukov. Optics and Spectr. (USA) 25 (1968) 363. [6] S. Sahal-Brdchot Astr, Astrophys. 1 (1969) 91; 2 (1969) 322. [7] S. Klarsfeld, Phys. L e t t e r s 32A (1970) 26, and t P N O R e p o r t TH/177(1970). [8] This modified v e r s i o n has been applied previously to neutral line broadening by J. Cooper and S. Klarsfeld (unpublished).
438
ON S T A R K
PHYSICS LETTERS
BROADENING
FUNCTIONS
14 December 1970
FOR
NONHYDROGENIC
IONS
S. K L A R S F E L D Institut de Physique Nucl~aire, Division de Fhysique Th$orique *, 91 Orsay, France Received 12 October 1970
Rapidly convergent integral representations are given for several functions used in calculating profiles of isolated ion lines.
S e m i c l a s s i c a l b r o a d e n i n g c a l c u l a t i o n s of s p e c t r a l l i n e s e m i t t e d by p o s i t i v e nonhydrogenic ions in a p l a s m a have been r e p o r t e d by many a u t h o r s [1-6]. The n e c e s s i t y of using h y p e r b o l i c t r a j e c t o r i e s for the p e r t u r b i n g e l e c t r o n s m a k e s the evaluation of the s e c o n d - o r d e r t i m e i n t e g r a l , which defines the b r o a d e n i n g functions A and B, much h a r d e r than for n e u t r a l l i n e s . A b e t t e r u n d e r s t a n d i n g of the a n a l y t i c s t r u c t u r e of the new functions m a y be r e a c h e d again on a p p e a l i n g to contour i n t e g r a t i o n techniques. However, the o r i g i n a l method, which p r o v e d so efficient in the n e u t r a l c a s e [7], now m e e t s with s e r i o u s d i f f i c u l t i e s and does not s e e m to be w o r k a b l e . We have u s e d i n s t e a d a m o d i f i e d v e r s i o n , which s t a r t s by c o n s i d e r i n g d i r e c t l y the ' d i a g o n a l ' functions a s s o c i a t e d with a Debye s h i e l d e d e l e c t r o n - i o n i n t e r a c t i o n [8]. While this does not p r o v i d e c l o s e d f o r m e x p r e s s i o n s for A and B, it y i e l d s for both s i m p l e and f a s t c o n v e r g e n t i n t e g r a l r e p r e s e n t a t i o n s , which a r e v e r y convenient for n u m e r i c a l computation. In the n o n - s h i e l d i n g l i m i t t h e s e r e a d (definitions and notations a r e the s a m e a s in ref. [6]):
f
dt c o s 2 ~ t c o s h 2 t K 0 ( 2 ~ cosh t), A(~,E) = 4 ~2e2exp(Tre) ~/2 0 dO exp(2~O) sin2~ K0(2~e s i n 0 ) B(~, ¢) = 4 ~ 2 ~ 2 f 0
(1)
oo
- 4 ~2 ~2exp(Tr~)
f
d t s i n 2 ~ t cosh 2 t K0(2~e cosh t ) .
(2)
0 The subsequent i n t e g r a t i o n o v e r the i m p a c t p a r a m e t e r is e x p r e s s i b l e in t e r m s of oo
a(~,E)
oo
= f de'A(~,C)/~',
(3)
b(~,e) = f d ~ ' S ( ~ , e ' ) / E ' .
£
E
T a k i n g account of eq. (1), (2) one e a s i l y gets i n t e g r a l r e p r e s e n t a t i o n s for a and b, e.g., 7r/2
b(~,e)
=
2~ef
d O e x p ( 2 ~ O ) s i n O K l ( 2 ~ e s i n 0 ) - 2~e
exp(~)f
dt s i n 2 ~ t c o s h t K l ( 2 ~ e c o s h t ) ,
(4)
dt c o s 2 ~ t sinh 2t K0(2~ e c o s h t ) .
(5)
0
0
or alternatively, after a partial integration,
~/2 b(~,e) = - 1 / 2 ~ + ~e 2 f
d 0 e x p ( 2 4 0 ) sin 20K0(2~e s i n 0 ) 0 oo
+ ~e 2 exp ( ~ )
f 0
* Laboratoire associ~ au C. N. R.S. 437
Volume 33A, n u m b e r 7
PHYSICS
LETTERS
14 D e c e m b e r 1970
In t h e l i m i t ~ ~ 0 o n e f i n d s B ( 0 , ~) = 0, b(0, ~) = ~ / 2 . On t h e o t h e r h a n d , f r o m t h e i n t e g r a l r e p r e s e n t a t i o n s g i v e n a b o v e o n e c a n e a s i l y e x t r a c t v a r i o u s a s y m p t o t i c e x p a n s i o n s f o r ~ -~ oo. In p a r t i c u l a r f o r > 1 we f o u n d B ( ~ , e ) ~ ( 1 / 2 ~ e) h 2 ( 1 / e ) + ( 1 / 1 6 43 e 3) [ h 4 ( 1 / e ) + ( 1 / 3 e ) h h ( 1 , / e ) ] + . . .
(6)
b(~,~) ~ ( 1 / 2 ~ ) [ h l ( 1 / e ) - 1] + ( 1 / 4 8 ~ 3 e 3 ) h 4 ( 1 / e ) + . . . .
(7)
w h e r e hn(cz)
= (d/dc~)nh(~)
and
oo
h(e~) = f du e x p ( e t u ) K o ( u ) = (1 - o~2)-1/2(~ - a r c c o s
c~),
(c~ < 1).
(8)
0 Eq. (6) c o i n c i d e s w i t h t h e e x p a n s i o n g i v e n p r e v i o u s l y by S a h a l - B r O c h o t [6]. T h e l a s t f o r m u l a e a r e c l e a r l y i n a d e q u a t e w h e n ~ i s v e r y c l o s e to u n i t y . A s a m a t t e r of f a c t t h e a s y m p t o t i c b e h a v i o u r f o r ~ ~ oo a n d E = 1 h a s a c o m p l e t e l y d i f f e r e n t c h a r a c t e r (~ = 1 i s a s o - c a l l e d t r a n s i t i o n p o i n t ) . In t h i s c a s e we o b t a i n e d B(~, 1) ~ - 2 . 8 8 6 42/3 + . . .
(9)
b(~, 1) ~ 0.684 ~-2/3 + . . .
(10)
It i s a l s o p o s s i b l e to w o r k out u n i f o r m a s y m p t o t i c e x p a n s i o n s . A + i B ~ 2~ 1/2 e x p ( - i ~ / / 3 ) e 3''~ G(~') ~2/~ + . . .
For instance:
,
(tl)
where O9
G(~) :
. f l d ! l 3/2 e x p ( ~ t - 1 3 , / 3 ) .
[ : 2 exp(2i~/3)(e-1)~2/3.
(12)
0 If !~[ i s s m a l l o n e c a n u s e G(~) ~ 0 . 9 4 0 + 1.114~ + 0.767~ 2 . Eq. (11) e s t a b l i s h e s
(13)
t h e l i n k b e t w e e n e x p a n s i o n s of t h e t y p e (6) a n d (9).
It i s a p l e a s u r e to a c k n o w l e d g e s t i m u l a t i n g d i s c u s s i o n s B r ~ c h o t a n d H. v a n R e g e m o r t e r .
w i t h D r s . J. C o o p e r , H. G r i e m ,
S. S a h a l -
Refere~wes [lj H.R. Griem, Phys. Rev. L e t t e r s 17 (1966) 509: Phys. Rev. 165 (1968) 258. [21 S. Brdchot, Phys. L e t t e r s 24A (1967) 476. [3}J. Cooper and G.K. Oertel. Phys. Rev. L e t t e r s 18 (1967) 985. [4] J. Davis and D . E . R o b e r t s , Proc. Phys. Soc. 92 {1967) 889; J. Phys. (B) 1 (1968) 48. [5] V. A. Alekseev and E. A. Yukov. Optics and Spectr. (USA) 25 (1968) 363. [6] S. Sahal-Brdchot Astr, Astrophys. 1 (1969) 91; 2 (1969) 322. [7] S. Klarsfeld, Phys. L e t t e r s 32A (1970) 26, and t P N O R e p o r t TH/177(1970). [8] This modified v e r s i o n has been applied previously to neutral line broadening by J. Cooper and S. Klarsfeld (unpublished).
438