CONTINUOUS ULTRAVIOLET ABSORPTION BY Si+(4Pe) : AN EXPLORATORY R-MATRIX CALCULATION

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CONTINUOUS ULTRAVIOLET ABSORPTION BY Si+(4Pe) : AN EXPLORATORY R-MATRIX

CALCULATION

J. Tully

To cite this version:

J. Tully. CONTINUOUS ULTRAVIOLET ABSORPTION BY Si+(4Pe) : AN EXPLORATORY R-MATRIX CALCULATION. Journal de Physique Colloques, 1988, 49 (C1), pp.C1-185-C1-188.

�10.1051/jphyscol:1988135�. �jpa-00227455�

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JOURNAL DE PHYSIQUE

Colloque C1, Suppl6ment au n03, Tome 49, Mars 1988

CONTINUOUS ULTRAVIOLET ABSORPTION BY s ~ ' ( ~ P ~ ) : AN EXPLORATORY R-MATRIX CALCULATION

J.A. TULLY

Observatoire de Nice, BP 139, F-06003 Nice Cedex, France

RESUME: La methode de l a m a t r i c e R e s t u t i l i s e e pour c a l c u l e r l a p h o t o i o n i s a t i o n % p a r t i s du prem-er e t a t e x c i t e 3 ~ 3 ~ 2 4Pe du

s i t .

Les deux termer t r i p l e t s l e s p l u s bas 3s3p PO e t 3$ 3 ~ e de 1 ' i o n residue1 Sit* s o n t i n c l u s dans l e c a l c u l . 11s sont repre- sent& par des f o n c t i o n s d'onde, avec i n t e r a c t i o n de c o n f i g u r a t i o n , c o n s t r u i t e s avec s i x o r b i t a l e s spectroscopiques: lsY2s,2p,3s,3p,3d. Des s e c t i o n s e f f i c a c e s p a r t i e l l e s e t t o t a l e s sont tabulees pour des energies photoniques q u i v o n t du s e u i l (705

9)

jusqul% 2.5 Ryd (365 A). Quelques resonances dues Ei des @ t a t s a u t o i o n i s a n t s 3p np sont delinees e t portees s u r des graphiques.

ABSTRACT: The R-matrix method i s used t o ca!culate p h o t o i o n i s a t i o n from the f i r s t e x c i t e d s t a t e 3 ~ 3 ~ 2 3 ~ e o f

sit.

The lowest two t r i p l e t terms 3s3p 3 ~ 0 and 3p2 3 ~ e o f t h e r e s i d u a l i o n

sit+

are i n c l u d e d i n the c a l c u l a t i o n . They a r e approximated by c o n f i g u r a t i o n i n t e r a c t i o n wavefunctions which a r e constructed from s i x spectroscopic o r b i t a l s : ls,2~,2p,3~,3p,3d. P a r t i a l and t o t a l cross-sections a r e t a b u l a t e d f o r phgton energies ranging from t h r e s h o l d (705 A) up t o 2.5 Ryd (365 A). Some o f the 3p np a u t o i o n i s i n g resonances a r e d e l i n e a t e d and shown g r a p h i c a l l y .

1.. INTRODUCTION

The aluminium-like i o n S i I I possesses a r i c h l i n e spectrum i n t h e v i s i b l e and u l t r a - v i o l e t and, owing t o t h e h i g h cosmic abundance o f s i l i c o n , i s detected i n many a s t r o - nomical o b j e c t s . It i s t h e r e f o r e d e s i r a b l e t o have r e l i a b l e i n f o r m a t i o n about the atomic p r o p e r t i e s o f t h i s i o n and i n p a r t i c u l a r i t s photoabsorption spectrum.

Although two d e t a i l e d c a l c u l a t i o n s (1,2) have been c a r r i e d o u t f o r p h o t o i o n i s a t i o n from the ground s t a t e , no serious attempt has y e t been made t o o b t a i n the continuous a b s o r p t i o n c o e f f i c i e n t f o r e x c i t e d s t a t e s . Her one presents some p r e l i m i n a r y r e s u l t s f o r photoabsorption b y t h e metastable $pe s t a t e .

The R-matrix approach ( 3 ) i s a powerful method, based on t h e close c o u p l i n g approxi- m t i o n ( 4 ) , f o r o b t a i n i n g t h e i n i t i a l bound and f i n a l s c a t t e r i n g s t a t e s w i t h which t o compute the c r o s s - s e c t i o n by means o f Fermi's "Golden Rule". One has used a m o d i f i e d v e r s i o n o f t h e w e l l known R-matrix package RMATRX (5) t o compute the cross- s e c t i o n s f o r t h e processes shown s c h e m a t i c a l l y i n f i n u r e 1.

2. DETAILS OF THE PRESENT R-MATRIX CALCULATION

The program RMATRX r e q u i r e s as i n p u t wavefunctions f o r s t a t e s o f the r e s i d u a l ion, i n t h i s case

s i f t .

Since one's aim was t o c a r r y o u t an ex 1 r a t o r y c a l c u l a t i o n , one chose t o i n c l u d e o n l y two s t a t e s , v i r 3s3p 3 ~ 0 and 3p4 QPe. Furthermore, one repre- sented these by f a i r l y simple c o n f i g u r a t i o n i n t e r a c t i o n wavefunctions b u i l t up from o r b i t a l s ls,2s,2pY3s,3p ( 6 and 3d ( 7 ) . The t h e o r e t i c a l ( x erimental ) i'onisation t h r e s h o l d energies are: I ( l P 0 ) = 1.298(1.292) Ryd and = 2.013(1.994) Ryd. I n t h e t h i r d stage o f RMATRX t h e thresholds were a d j u s t e d i n t h e standard f a s h i o n (5) so as t o agree ui t h observation.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1988135

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t

F i g u r e 1. Schematic r e p r e s e n t a t i o n o f t h e photoabsorption process

The R-matrix boundary was s e t a t RA = 15.15625 a.u., which was deemed s u f f i c i e n t l y l a r g e t o ensure convergence of t h e d i p o l e l e n g t h and v e l o c i t y m a t r i x elements. A r e l a t i v e l y l a r g e boundary was necessary since t h e v e r s i o n o f RMATRX t h a t one used ignores t h e c o n t r i b u t i o n t o t h e d i p o l e i n t e g r a l s from the o u t e r region.

One chose NRANG2 = 15 "continuum" basis o r b i t a l s t o span t h e i n n e r region. This enables one t o use t h e present model f o r photon energies up t o about 2.5 Ryd.

3. RESULTS

F i u r e 1 shows t h e s i x continuum channels based on t h e two lowest t r i p l e t s t a t e s of Si'. With each o f these channels i s associated a p r o b a b i l i t y ( i . e . p a r t i a l cross- s e c t i o n ) f o r p h o t o i o n i s i n g S i + and l e a v i n g the r e s i d u a l i o n i n t h e corresponding channel s t a t e , w h i l e a t the same time e j e c t i n g a photoelectron o f given wave number k at&i o r b i t a l angular momentum 1. The p a r t i a l cross-sections i n megabarn u n i t s ( i . e . 10- cm2) a r e given i n Table 1 as f u n c t i o n s o f t h e photon energy i n Rydberg u n i t s . The t o t a l cross-section, which i s the q u a n t i t y o f i n t e r e s t t o astronomers, i s a l s o tabu1 ated.

The cross-sections f o r absorption i n t o t h e kd 4 DO and kd 4 ~ 0 channels e x h i b i t broad minima o f t h e type f i r s t r e p o r t e d by Rudkj$bing (8) i n connection w i t h t h e absorption spectrum o f sodium. T h i s behaviour i s caused y a change i n s i g n o f t h e r a d i a l m a t r i x

B

elements as t h e photon energy increases. The Do l e n g t h cross-section, f o r example, drop r a p i d l y from t h e value 1.14Mb a t t h r e s h o l d (1.292 Ryd) t o zero a t about 1.75Ry.

The Po c r o s s - s e c t i o n vanishes a t a s l i g h t l y h i g h e r energy.

2

Between t h e 3 ~ 0 and 3 ~ e thresholds, i . e . f o r 705A > h > 365A, t h e smooth v a r i a t i o n o f t h e c r o s s - s e c t i o n i s i n t e r r u p t e d by a u t o i o n i s i n g resonances. These a r e caused by l h e 3p2np terms o f S i + which are embedded i n t h e 3s3pks and 3s3pkd continua. The two

Po channels i n t e r a c t s t r o n g l y and w h i l e the resonance i n ks produces an i n d e n t a t i o n i n t h e p a r t i a l cross-section, t h a t i n t h e kd channel gives r i s e t o a complementary peak r e s u l t i n g i n o n l y a s l i g h t trough when t h e two a r e summed. F i g u r e 2 i l l u s t r a t e s t h i s behaviour f o r t h e n=5 resonance which i s l o c a t e d a t V-1.74255Ryd and has a w i d t h (FWHM) o f 2.3%-03 Ryd. The f i r s t resonance i n t h e $0; channel occurs a t about 1.52Ryd and i s o f the Beutler-Fano type w i t h a negative q c o e f f i c i e n t . The l e n g t h and v e l o c i t y cross-sections are compared i n f i g u r e 3 and a r e seen t o d i f f e r by a l a r g e amount. The n e x t resonance (n=5) occurs c l o s e t o where t h e background cross-section passes through zero and consequently i t s p r o f i l e i s approximately Lorentzian. This resonance i s shown i n f i g u r e 4 f o r t h e geometric mean o f t h e l e n g t h and v e l o c i t y p a r t i a l cross s e c t i o n s . A curious f e a t u r e i n t h e form o f a narrow crevasse occurs on the s h o r t wavelength s i d e o f t h e p r o f i l e and i s associated w i t h a small b l i p i n the phase s h i f t . The c r o s s - s e c t i o n f a l l s suddenly t o zero a t t h e p o i n t where t h e energy d e r i v a t i v e o f t h e phase s h i f t passes through zero. One assumes t h a t t h i s has no physical s i g n i f i c a n c e and i s simply due t o t h e p r o x i m i t y o f an R-matrix p o l e a t 1.7376 Ryd.

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T a b l e 1. P a r t i a l a n d t o t a l c r o s s s e c t i o n s i n M b units. + l e n g t h ,

++

v e l o c i t y .

kp 4 ~ 0 kp 4 ~ 0 k p 4 ~ 0 T o t a l

0.01 ⁸ ^b L

,

⁸ , , , ,

1

1.760 1,765

photon energy lRyd)

F i g u r e 2. P r o f i l e s of t h e l e n g t h p a r t i a l c r o s s s e c t i o n s f o r t h e 3 p 2 5 p 4 ~ 0 r e s o n a n c e . k s c h a n n e l _{( o ) ,} k d c h a n n e l ( a ) , s u m (x).

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F i g u r e 3. L e n g t h ( o ) and v e l o c i t y ( + ) p a r t i a l c r o s s s e c t i o n s n e a r t h e 3 p 2 4 p & D O r e s o n a n c e .

photon energy (Ryd)

F i g u r e 4. P r o f i l e o f t h e g e o m e t r i c m e a n c r o s s s e c t i o n f o r t h e 3 p 2 5 p 4 ~ 0 r e s o n a n c e .

4. A NOTE ON THE OUTER REGION SOLUTIONS

The i n n e r and o u t e r r e g i o n s o l u t i o n s a r e matched on t h e R-matrix boundary. I n t h e present case one assumed t h a t the value o f RA was s u f f i c i e n t l y l a r g e t o j u s t i f y the use o f a Coulomb p o t e n t i a l i n the o u t e r region. The asymptotic s o l u t i o n s were there- f o r e obtained by i n t e r f a c i n g t h e t h i r d stage o f RMATRX t o a program f o r generating Coulomb f u n c t i o n s . The v a l i d i t y o f t h i s approximation was t e s t e d by comparing some o f the present r e s u l t s w i t h those obtained by i n t e r f a c i n g RMATRX t o t h e AYSPCK ( 9 ) package. I n some cases d i f f e r e n c e s o f more than a few percent were encountered and so f u r t h e r work i s c l e a r l y r e q u i r e d f o r t h i s p h o t o i o n i s a t i o n process i n o r d e r t o have complete confidence i n the r e s u l t s .

A f i n a l note concerning the use o f ASYPCK (9) i s o f relevance here. It i s i m p o r t a n t t o take account o f t h e charge dependent n o r m a l i s a t i o n used by Crees f o r t h e r a d i a l continuum f u n c t i o n s ( c . f . (9), s e c t i o n 2.4.1). T h e s e ' o s c i l l a t e a s y m p t o t i c a l l y w i t h amplitude d(z/k), where z i s t h e charge o f t h e r e s i d u a l i o n . This f a c t was overlooked i n t h e previous R-matrix c a l c u l a t i o n devoted t o S i I I and consequently the cross- sections given i n reference ( 2 ) should be d i v i d e d by 2 ( t h e charge o f s i t + ) . ACKNOWLEDGEMENTS

I thank Dr. K.A. B e r r i n g t o n f o r p r o v i d i n g me w i t h copies o f t h e programs used i n t h e present study and a l s o f o r a c l a r i f y i n g discussion. Dr. C.J. Zeippen helped i n l o c a l i s i n g the z f a c t o r discrepancy associated w i t h the use o f RMATRXtASYPCK.

REFERENCES

1. Daum G.R. and K e ? l y H.P. 1976 Phys.Rev. A13, 715.

2. T a y l o r K.T., Zeippen C.J. and Le Dourneuf M. 1984 J.Phys.6 17, L157.

3. Burke P.G. and T a y l o r K.T. 1975 J.Phys.B 8, 2620.

4. Burke P.G. and Seaton M.J. 1971 Methods Comput. Phys. 10, 1.

5. B e r r i n g t o n K.A., Burke P.G., Le Dourneuf M., Robb W.D., T a y l o r K.T. and Vo Ky Lan.

1978 Comput. Phys. Commun. 14, 367.

6. Clementi E. and R o e t t i C. 1974 Atomic Data and Nucl. Data Tables 14, 177.

7. B a l u j a K.L. and t l i b b e r t A. 1980 J.Phys.6 13, L327.

8. Rudkjbbing

N.

1940 Publ. Kbh. Obs. 18, 1.

9. Crees M.A. 1980 Comput. Phys. Commun. 19, 103.