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CALCULATION OF THE PHOTOIONIZATION CROSS SECTIONS OF EXCITED LEVELS OF RARE
GAS ATOMS
P. Ranson, J. Chapelle
To cite this version:
P. Ranson, J. Chapelle. CALCULATION OF THE PHOTOIONIZATION CROSS SECTIONS OF
EXCITED LEVELS OF RARE GAS ATOMS. Journal de Physique Colloques, 1979, 40 (C7), pp.C7-
25-C7-26. �10.1051/jphyscol:1979712�. �jpa-00219091�
JOURNAL DE PHYSIQUE CoZZoque C7, suppl6ment au n07, Tome 40, JuiZZet 2979, page C7- 25
CALCULATION OF THE PHOTOIONIZATION CROSS SECTIONS OF =CITED LEVELS OF RARE GAS ATOMS
P. Ranson and J. Chapelle.
C.B. P. H. T. -C. I. R.S. Orle'ans, France
I - INTRODUCTION : The c a l c u l a t i o n o f t h e photoio- n i z a t i o n c r e s s s e c t i o n s of e x c i t e d l e v e l s o f r a r e g a s atoms i s of g r e a t i n t e r e s t f o r many problems l i k e emiss,ion and a b s o r p t i o n of l i g h t by plasmas o r hot g a s e s o r l i k e t h e d e t e r m i n a t i o n o f o p t i m a l experimental c o n d i t i o n s f o r l a s e r d i s c h a r g e s . Ab i n i t i o c a l c u l a t i o n s a r e g e n e r a l l y complex and it is e a s i e r t o use semi-empirical methods. One of them i s t h e quantum d e f e c t method which p e r m i t s t o c a l c u l a t e t h e r a d i a l p a r t c o n t r i b u t i o n on t h e c r o s s s e c t i o n s . I n t e r m e d i a t e c o u p l i n g i s used f o r t h e a n g u l a r p a r t .
11- CALCULATION OF THE CROSS SECTION /1/
11%- General form : Wfth t h e assumption o f t h e c e n t r a l f i e l d approximation, t h e photoioniza- t i o n c r o s s s e c t i o n of an atom i n a s t a t e i by a photon of energy hv i s given by t h e formulae :
-
-..,
d = , 90 Bohr r a d i u s , R, Rydberg c o n s t a n t , a i s t A ? ? s t i c a l weight of t h e atom.
except o f a f a c t o r r , a r e t h e r a d i a l wave f u n c t i o n s f o r bound s t a t e i = (n,Q) and for f r e e s t a t e j = (
E
= k 2 ,e')
and where C i j comes from t h e i n t e g r a t i o n of a n g u l a r p a r t .II-b- Calculation of radial wave function : The quantum d e f e c t method /2/ ( p = n-nx,, nX ef-.
f e c t i v e quantum number) based 0% t h e ~ o u l o m b ap- proximation v a l i d f o r g r e a t d i s t a n c e s between t h e c o r e and t h e e l e c t r o n a l l o w s t h e e v a l u a t i o n o f r a d i a l wave f u n c t i o n . Vith t h i s approxfmation, f o r t h e bound s t a t e s and p a r t f c u l a r l y f o r t h e high
e
s t a t e s (
c >
21, t h e s o l u t i o n o f t h e r a d i a l SchrB- d i n g e r e q u a t i o n d f v e r g e s f o r s m a l l r. I t is t h e n n e c e s s a r y t o i n t r o d u c e a c t-off f a c t o r with t h e form f c ( ( , r ) = (1-e- 'er)a4+1 which g i v e s , i n t h e c a s e ,)A< 1, a good v a r i ' a t i o n of t h e wave f u n c t i o n n e a r t h e o r f g f n e and ?n t h e g e n e r a l c a s e which p e r m i t s t o keep a f i n i t e wave f u n c t i o n . The c o e f f i c i e n t t@ i s a d j u s t e d by t h e n o r m a l i s a t i o n c o n d i t f o n .For t h e f r e e s t a t e s , t h e r a d f a l wave f u n c t i o n f s a l i n e a r c o m b i n a t i c n o f r e g u l a r and i r r e g u l a r so- l u t i o n o f t h e r a d i a l Schradinger equation which g i v e s an asymptotic from
where t h e phase s h i f t &'(k2) i s c a l c u l a t e d by e x t r a p o l a t i o n of t h e quantum d e f e c t ) t o p o s f t i - ve e n e r g i e s . It i s a l s o n e c e s s a r y t o u s e a c u t - o f f f a c t o r o f t h e same form a s f o r t h e bound s t a t e s f o r t h e i r r e g u l a r p a r t . It i s a d j u s t e d by f i t t i n g t h e a f i r s t extremum i n t h e i r r e g u l a r p a r t with t h e
o t h e r s and w i t h t h o s e o f r e g u l a r p a r t .
11-c- Radial matrix element : For bound s t a t e s r a d i a l wave f u n c t i o n s a r e c a l c u l a t e d from t h e a s y m p t o t i c expansion with t h e c o r r e c t i o n f o r s m a l l r a d i u s . For f r e e s t a t e s , a s e r i e s expansion /3/ is used. With t h i s method, t h e m a t r i x element can b e c a l c u l a t e d i n a n a l y t i c form. We o b t a i n a n a l t e r - n a t e s e r i e s expansion which converges only i f t h e c o n d i t i o n k n x < 1 i s f q l f i l l e d . The c o n d i t i o n o f convergence can be l e s s r i g i d i f t h e i n t e g r a t i o n i s made from 0 t o R where R i s c a l c u l a t e d s o a s t o i n t r o d u c e only a very s m a l l e r r o r ( l o b 4 ) . Then, t h e v a l i d i t y o f t h e c a l c u l a t i o n i s l i m i t e d t o t h e c o n d i t i o n knX,( 2 by t h e i n c r e a s e of t h e term of t h e s e r i e s .
11-d- The intermediate coupling for t h e rare gas The term C i j of t h e formulae ( 1 ) g i v e s t h e a n g u l a r p a r t of t h e c r o s s s.ections. The e x c i t e d s t a t e s of r a r e g a s e s < j i He) a r e uor c e l e v d n t of
.
t~lly U S U ~c o u p l i n g schemes o f d i f f e r e n t s o r b i t a l and s p i n mo- menta. I n d e e d , t h e a p p r o p r i a t e c o u p l i n g i s interme- d i a t e beyween t h e LS coupling and t h e J j o r J e c o u p l i n g . For t h i s purpose, t h e observed energy l e - v e l s a r e used f o r t h e decomposition o f r e a l s t a t e s on t h e J j s t a t e s b a s i s . For t h e continuum s t a t e s , t h e computed c o e f f i c f e n t s a r e e x t r a p o l a t e d from high l y i n g l e v e l s . Then, t h e v a l u e of C i j can be c a l c u l a t e d v i a m a t r l x a l g e b r a .
11-c-The continuum s t a t e s phase s h i f t s The quantum d e f e c t method g i v e s t h e r e l a t i o n b e t - ween e x t r a p o l a t e d quantum d e f e c t ( i n t h e c o n t i - nuum) and phase s h i f t s of f r e e s t a t e s f o r a s e r i e s np5 2 ~ 3 / 2
,1/2-j
n1 P(\K,J].
However t h e s e r i e s with same p a r i t y and same t o t a l momentum J a r e mu- t u a l l y perturbed and i t i s n e c e s s a r y t o t a k e t h i s p e r t u r b a t i o n s i n t o account i n t h e c a l c u l a t i o n of e x t r a p o l a t e d quantum d e f e c t s . The method proposed by Edlen / 4 / i s used. The quantum d e f e c t o f a s t a t e i belonging t o t h e s e r i e s 1 and p e r t u r b e d by t h e s t a t e s k of t h e s e r i e s 2 i s g i v e n byT
+
s p e c t r a l term. The unperturbed p a r t np' can be approximated by a q u a d r a t i c e x p a n s i o L f t h e Pnergy.111- APPLICATION TO THE LEVELS
s
np51
2~7
(n+l)s ;c.2, T
is5
and-
s4 %h PASCHEN ,VOT%?ONIWe have c a l c u l a t e d t h e p h o t o i o n i z a t i o n c r o s s sec- t l o n s o f t h i s two l e v e l ; f o r Ne, AT, Kr ,.Xe atoms.
The t r a n s i t i o n s i s p o s s i b l e t o f r e e s t a t e s w i t h
9' = 1. For t h e l e v e l < s 5 ( J = 2 only t r a n s i t i o n s t o t h e ?on?c c o r e 2 ~ 3 / 2 a r e allowed and t h e se- l e c t f o n r u l e s l i m i t t h e t r a n s i t i o n t o 5 l e v e l s (pIO, pg. pg, 9, p6). For t h e l e v e l i ~ ~ ( ~ = 11,
t r a n s i t i o n s t o t h e two i o n i c c o r e s 2 ~ 3 / 2 , 2 ~ 1 2 a r e p o s s f b l e and o n l y t h e t r a n s i t i o n s t o l e v e i pg
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979712
a r e f o r b i d d e n .
The f i g u r e s l a - d show t h e r e s u l t w i t h comparison t o o t h e r t h e o r e t i c a l works (Elc Carm-Flawry / 5 / , Hyman / 6 / , Hazi-Rescigno /7/ and H a r t q u i s /9/).
F o r Ne, o u r r e s u l t s a r e i n r e l a t i v e l y good a g r e e - ment w i t h o t h e r v a l u e s ( e x c e p t /9/), b u t f o r A r ,
i m p o r t a n t d i s c r e p a n c i e s e x i s t and f o r Ky and Xe, o n l y t h e r e s u l t s o f H a r t q u i s t a r e i n good a g r e e - ment. Few e x p e r i m e n t a l r e s u l t s a r e a v a i l a b l e . F o r Xe, t h e v a l u e s o f Rundel and a 1 /8/ a r e i n - t e r m e d i a t e between d i f f e r e n t ~ t h e o r e t i c a l r e s u l t s .
Our method g i v e s w i t h a minimum o f c a l c u l a t i o n s t h e p h o t o i o n i z a t i o n c r o s s s e c t i o n i n c l u d i n g s e v e r a l phenomena (summation on a l l f i n a l s t a t e s , t r a n s i - t i o n s between d i f f e r e n t i o n i c c o r e , m u t u a l p e r t u r - b a t i o n o f s e r i e s ) .
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F i g . 1 : P h o t o i o n i z a t i o n c r o s s s e c t i o n s f q r l s 5 and ls4 l e v e l s . S o l i d c u r v e : o u r r e s u l t s ; d o t - d a s h e d c u r v e : Mc Cann.-Flammery;
dashed c u r v e : H a r t q u i s t , ( 0 : l s 4 ) , ( A : l s 5 ) . F i g . l a : Neon, d o t t e d c u r v e : Hazi-Rescagno ( d i p o -
l e velocity);-..-.. Hazi-Yescagno ( d i - p o l e l e n g h t 1.
F i g . l b : A r g o n , d o t t e d c u r v e : Hyman ( n o t e t h e s c a l e change f o r H a r t q u i s t r e s u l t s and f o r o u r r e s u l t s n e a r l y t h e t h r e s h o l d ) .
F i g . l c : Krypton, d o t t e d c u r v e : Hyman.
F i g . I d : Xenon, c r o s s e s : e x p e r i m e n t a l r e s u l t s of Xundel and a l .
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