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THEORETICAL SIMULATION OF BEAM FOIL DECAY CURVES FOR RESONANCE TRANSITIONS
OF HEAVY IONS
W. Wiese, S. Younger
To cite this version:
W. Wiese, S. Younger. THEORETICAL SIMULATION OF BEAM FOIL DECAY CURVES FOR
RESONANCE TRANSITIONS OF HEAVY IONS. Journal de Physique Colloques, 1979, 40 (C1),
pp.C1-146-C1-148. �10.1051/jphyscol:1979125�. �jpa-00218405�
JOURNAL DE PHYSIQUE Colloque C1, suppl6ment au n O 2, Tome 40, fkvrier 1979, page C1-146
THEORETICAL SIMULATION OF BEAM FOIL DECAY CURVES FOR RESONANCE TRANSITIONS OF HEAVY IONS
W. L. Wiese and S. W . Younger
National Bureau o f Standards, Washington, D.C. 20234, U .S.A.
Resume.--Nous avons e f f e c t u e une etude t h e o r i q u e de l l i n f l u e n c e des cascades s u r des mesures de duree de v i e q u i u t i l 7 s e n t l a technique "beam-foil
.
Nous nous somnes penches s u r t o u t s u r l e s ions lourds, e t en p a r t i c u l i e r s u r l a r a i e de rdsonance de Kr V I I I . A p a r t i r de c a l c u l s thdoriques pour l e s durees de v i e e t l e s populations i n i t i a l e s , nous avons p r o d u i t des courbes de d g c l l n q u i simulent l e s c o n d i t i o n s d ' e x c i t a t i o n de l a technique "beam-foil". La courbe de d g c l i n t h g o r i q u e q u i r e p r o d u i t l e mieux p o s s i b l e l e s courbes exp6rimentales a donn6 l a m@me dur6e de v i e que l e s expgriences, l o r s q u e ces d e r n i e r e s o n t subi l a m@me analyse de cascades.Cependant, l e s techniques conventionnel l e s d'analyse en e x p o n e n t i e l l e s n ' o n t pas pu e x t r a i r e l a d u d e de v i e p r i m a i r e u t i l i s d e pour l e c a l c u l l a courbe thdorique.
Abstract.--We have made a t h e o r e t i c a l study o f t h e i n f l u e n c e o f cascades on heavy i o n beam- f o i l l i f e t i m e s , c o n c e n t r a t i n g on t h e resonance l i n e o f Kr V I I I . Using t h e o r e t i c a l data f o r t h e l i f e t i m e s and i n i t i a l populations o f e x c i t e d s t a t e s , we have c o n s t r u c t e d decay curves s i m u l a t i n g beam f o i l e x c i t a t i o n c o n d i t i o n s
.
The t h e o r e t i c a l decay curve producing t h e b e s t f i t w i t h experimental decays y i e l d e d t h e same l i f e t i m e as t h e experfments when subjected t o t h e same cascade a n a l y s i s , b u t customary exponential f i t t i n g techniques were n o t a b l e t o e x t r a c t the t h e o r e t i c a l primary l i f e t i m e a c t u a l l y used i n i t s c o n s t r u c t i o n .1. I n t r o d u c t i o n . - - A t t h e l a s t i n t e r n a t i o n a l Table I. Experimental, simulated and t h e o r e t i c a l beam-foil conference, we discussed t h e o s c i l l a t o r l i f e t i m e s ( i n n s ) f o r t h e 4p l e v e l o f Kr V I I I s t r e n g t h s o f resonance t r a n s i t i o n s i n some heavy Beam f o i 1 experiments Theory Simulation ions t h a t a r e o f s p e c i a l importance t o magnetic
0 . 3 6 ~ , 0.39 5 0.26' 0.35'
f u s i o n research [I]. A l a r g e discrepancy between
0 . 3 5 ~ . 0.35'~ 0.36 8
t h e o r e t i c a l and beam-foil data was found f o r t h e 0 . 2 5 ~
4s-4p t r a n s i t i o n i n t h e C u - l i k e i o n s which r e - mained approximately constant along t h e sequence.
On t h e basis o f t h e i n f o r m a t i o n a v a i l a b l e a t t h a t time, we concluded t h a t i t was probably due m a i n l y t o t h e n e g l e c t o f core p o l a r i z a t i o n e f f e c t s
i n the c a l c u l a t i o n s .
I n t h e meantime, advanced c a l c u l a t i o n s f o r t h e Cu sequence u s i n g the m u l t i - c o n f i g u r a t i o n Hartree-Fock method 121 and many-body p e r t u r b a t i o n t h e o r y [3] have y i e l d e d i d e n t i c a l f-values which a r e s t i l l 20-30% above those d e r i v e d from beam- f o i l data. There can be l i t t l e doubt now t h a t these advanced t h e o r e t i c a l r e s u l t s f o r compara- t i v e l y simple atomic systems a r e c l o s e t o t h e t r u e numbers. The puzzle remains, however, as t o why t h e beam-foil data should be so d i f f e r e n t .
Measurement e r r o r s appear t o be excluded since, as 1 Table I shows, the f i v e experimental r e s u l t s 14-83 agree v e r y c l o s e l y . Cascade a n a l y s i s o f t h e experimental decay curves seemed t o i n d i c a t e o n l y t h s presence o f a second l o n g - l i v e d exponential.
Nevertheless, i n o r d e r t o g a i n a b e t t e r understand- i n g o f cascade e f f e c t s we decided t o undertake a comprehensl/ve a n a l y s i s o f them by way o f a theo- r e t i c a l s i m u l a t i o n o f beam-foil decay curves [9].
2. S i m u l a t i o n
of
Beam-Foil Decay Curves.-- To perform a beam-foil s i m u l a t i o n one must have a v a i l a b l e t h e r e l e v a n t t h e o r e t i c a l t r a n s i t i o n p r o b a b i l i t i e s o r l i f e t i m e s , as w e l l as t h e r e l a t i v e i n i t i a l populations o f t h e cascade s t a t e s w i t h respect t o t h e primary l e v e l[lo].
A. T r a n s i t i o n p r o b a b i l i t i e s . Our s i m u l a t i o n s t u d i e s have shown t h a t t h e most c r i t i c a l p i e c e of data i s an accurate l i f e t i m e f o r t h e primary l e v e l . For the cascading t r a n s i t i o n s , l i f e t i m e s obtained from approximate t h e o r e t i c a l methods, such as t h e Coulomb approximation, appear t o be s a t i s f a c t o r y .
B. I n i t i a l populations. D e t a i l e d i n f o r m a t i o n concerning t h e i n i t i a l d i s t r i b u t i o n o f e x c i t e d s t a t e s a f t e r f o i l e x c i t a t i o n i s v e r y sketchy. This i s e s p e c i a l l y so f o r heavy ions which may show l e v e l d i s t r i b u t i o n s q u i t e d i f f e r e n t from l i g h t i o n s [9]. Considering t h i s r a t h e r u n s a t i s f a c t o r y s t a t e o f a f f a i r s , we have based our s i m u l a t i o n study n o t on u n c e r t a i n assumptions about t h e populations, b u t i n s t e a d on comparisons w i t h a c t u a l decay curves. A number o f r e l a t i v e l y simp1 e p o p u l a t i o n models were s e l e c t e d and t e s t e d by f i t t i n g t h e r e s u l t a n t decay curves t o an a c t u a l beam-foil decay curve f o r t h e 4s-4p m u l t i p l e t o f Kr V I I I . One o f these p o p u l a t i o n
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models achieved a v e r y c l o s e fit, f a r s u p e r i o r t o t h e others. I n t h i s model, t h e r e l a t i v e p o p u l a t i o n o f a l e v e l w i t h e f f e c t i v e p r i n c i p a l quantum number n* and o r b i t a l angular momentum quantum number L, compared t o t h e primary l e v e l , i s
As another comparison, we show i n Fig. 2 a s i m u l a t i o n o f t h e decay o f t h e 4p PI/* 2 l e v e l ( t h e
0
596 A l i n e ) o f Kr V I I I , along w i t h t h e experimental
I
I I I I1
A g r a p h i c a l i l l u s t r a t i o n o f t h i s f i t i s provided by Fig. 1, which i s reproduced from o u r r e c e n t paper 191. The c i r c l e s a r e a d i r e c t f i t t o the
Fig. 1. F i t o f t h e experimental decay curve o f D r u e t t a and Buchet ( c i r c l e s ) t o t h e com- p u t e r s i m u l a t i o n . The i n s e r t i s an en- largement o f t h e boxed region, showing slopes o f t h e p r i m a r y decay (P) and v a r i o u s cascade c o n t r i b u t i o n s due t o Rydberg (R1, R2, R3) and y r a s t
(Y)
cascades.
Fig. 2. F i t of t h e experimental decay curve o f Knystautas [8] ( d o t s ) t o t h e computer s i m u l a t i o n .
p o i n t s o f Knystautas ( d o t s ) [8]. Again we a r e a b l e t o produce a c l o s e f i t t o t h e observed decay, p a r t i c u l a r l y i n t h e i m p o r t a n t "1 i n e a r " r e g i o n where t h e primary l i f e t i m e i s u s u a l l y determined.
Considering d i p o l e s e l e c t i o n r u l e s i n t h e cascade c h a i n t h e r e l a t i v e i n i t i a l populations o f t h e e x c i t e d s t a t e s f o r t h i s case a r e
experimental data; t h e t r i a n g l e s assume a s h i f t o f .5 nun i n t h e absolute f o i l p o s i t i a n and i n d i c a t e t h a t u n c e r t a i n t i e s i n t h e f o i l p o s i t i o n may be o f some s i g n i f i c a n c e . Most o f t h e cascading occurs through t h e " y r a s t " 1 evel s, which have been i n - cluded up t o n = 22. By a p p l y i n g t o t h e simulated decay a customary cascade a n a l y s i s , i . e . a two- exponential fit, we obtained e x c e l l e n t agreement w i t h t h e experimental r e s u l t s , b u t n o t w i t h t h e t h e o r e t i c a l l i f e t i m e f o r t h e 4p s t a t e a c t u a l l y used i n c o n s t r u c t i n g t h e simulated curve (see Table I).
I n Fig. 3 we present a d e t a i l e d decomposition o f t h e y r a s t c o n t r i b u t i o n s t o t h e 4p Pl2 I2 decay.
The sum o f t h e i n d i v i d u a l cascade curves (each o f which represents t h e f r a c t i o n a l p o p u l a t i o n o f t h e 4p Pl2 I2 1 evel as produced by t h e i n i t i a l popu- . l a t i o n o f t h e p a r t i c u l a r cascading l e v e l ) and t h e primary equals t h e simulated r e s u l t o f F i g . 2.
The f i r s t several cascades have f i n a l slopes very near t h a t o f t h e primary, and a l l have the appear- ance o f "growing-in" cascades. T h i s i s due t o t h e
c1-148 JOURNAL DE PHYSIQUE
m u l t i - e x p o n e n t i a l c h a r a c t e r o f t h e i n d i v i d u a l cascade c o n t r i b u t i o n s . For example, as seen a t t h e p r i m a r y l e v e l p, t h e t w o - l e v e l cascade c h a i n j+k+p i n c l u d e s n o t o n l y t h e e f f e c t s o f r e p o p u l a t i n g k and p by e l e c t r o n s decaying f r o m j, b u t a l s o t h e i r d e p o p u l a t i o n due t o downward t r a n s i t i o n s . F o r l o w l y i n g " f a s t " cascades, t h i s d e p o p u l a t i o n determines t h e t i m e dependence o f t h e cascade.
Thus even though t h e cascading l e v e l s m i g h t have v e r y much s h o r t e r l i f e t i m e s t h a n t h e p r i m a r y , t h e n e t cascade c o n t r i b u t i o n s t o t h e decay c u r v e may d i s p l a y v e r y s i m i l a r t i m e dependences, b u t o f f s e t i n time, t h u s g r e a t l y c o m p l i c a t i n g t h e problem o f e x t r a c t i n g a c c u r a t e p r i m a r y l i f e t i m e s .
F i g . 3. I n d i v i d u a l y r a s t cascade c o n t r i b u t i o n s t o t h e 4p PI,* 2 s i m u l a t e d decay curve.
3. Conclusions.--The p r i n c i p a l r e s u l t s o f o u r s i m u l a t i o n o f t h e An = 0 resonance t r a n s i t i o n o f Kr V I I I may be summarized as f o l l o w s :
( a ) The s i m u l a t i o n s show t h a t most o f t h e cascading o c c u r s t h r o u g h t h e y r a s t l e v e l s . U s i n g t h e o r e t i c a l d a t a we c o u l d c l o s e l y reproduce e x p e r i m e n t a l decay curves. We were, however, u n a b l e t o r e c o v e r t h e t h e o r e t i c a l p r i m a r y l i f e t i m e s used f o r t h e i r c o n s t r u c t i o n [see Table I) when we a p p l i e d t h e customary e x p o n e n t i a l cascade a n a l y s i s .
The r e p l e n i s h m e n t r a t i o , o f t e n used as a measure o f t h e importance o f cascades i n a decay, was always q u i t e small i n o u r s i m u l a t i o n s , d e s p i t e heavy cascading; ( b ) For p r i m a r y l e v e l s w i t h l a r g e p r i n c i p a l quantum number ( n
>
4) t h e d r o p - o f f i n p o p u l a t i o n w i t h n a c c o r d i n g t o t h e p o p u l a t i o n d i s t r i b u t i o n g i v e n by Eq. ( 3 ) i s r e l a t i v e l y slow, l e a d i n g t o f a i r l y s t r o n g l y p o p u l a t e d cascading l e v e l s ; ( c ) W i t h i n an i s o e l e c t r o n i c sequence cascade e f f e c t s f o r An = 0 t r a n s i t i o n s s h o u l d become p r o g r e s s i v e l y more s e r i o u s w i t h i n c r e a s i n g z, because o f t h e d i f f e r e n t z s c a l i n g o f t h e A-values f o r t h e p r i m a r y and cascade t r a n s i t i o n s ;( d ) The energy l e v e l schemes o f most heavy i o n s a r e q u i t e complex and o f t e n i n c l u d e d o u b l y e x c i t e d d i s c r e t e l e v e l s below t h e i o n i z a t i o n l i m i t which d i r e c t l y f e e d i n t o t h e upper l e v e l s o f t h e p r i n - c i p a l resonance l i n e . Thus s i g n i f i c a n t c o n t r i - b u t i o n s from many h i g h e r l y i n g l e v e l s a r e t o be expected which should general1 y 1 ead t o complex decay curves f o r heavy i o n s .
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