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C. I. CALCULATIONS IN THE SILICON SEQUENCE
A. Hibbert
To cite this version:
A. Hibbert. C. I. CALCULATIONS IN THE SILICON SEQUENCE. Journal de Physique Colloques,
1979, 40 (C1), pp.C1-122-C1-124. �10.1051/jphyscol:1979122�. �jpa-00218402�
JOURNAL DE PHYSIQUE
Colloque
Cl, suppl6ment au n
O 2,Tome
40,f6vrier
1979,page
C1-122C. I. CALCULATIONS I N THE SILICON SEQUENCE
A . Hibbert
Department of Applied Mathematics and T h e o r e t i c a l Physics, The Queen's U n i v e r s i t y of B e l f a s t , B e l f a s t BT7 INN,
Northern I r e l a n d .
~ e s u m 6
-
Nous avons u t i l i s e l a methode de 1 ' i n t e r a c t i o n c o n f i g u r a t i o n n e l l e pour c a l c u l e r l e s f o r c e s d t o s c i l l a t e u r d e s t r a n s i t i o n s 3 ~ e-
3 ~ 0 : 3 ~-
3 . ~ 3 ~ 3 ~ 3e t ~ ~ - 3 ~ 2 3 ~ 4 s ; 3 ~ e-
3 ~ 0 : 3 ~-
3~ ~ 3e t - 3 ~ ~ 3 ~ 3 d 3 ~~ ~~ de S i I, P I1 e t S 111. Les v a l e u r s u t i l i s a n t l e couplage L-S s o n t en accordavecl'exp6rience,mais on d o i t a j o u t e r l e s i f f e t s r e l a t i v i s t e s pour l e s E t a t s 3p3d 3 ~ 0 e t 3 ~ 0 .A b s t r a c t
-
Lar e c o n f i g u r a t i o n i n t e r a c t i o n expansions a r e used t o e v a l u a t e o s c i l l a t o r s t r e n g t h s of 4Pe-
3 ~ 0 : 3 ~-
3~ ~ 3and 3 ~~- 3 ~ ~ 3 ~ 4 s ; ~~-
'Do: 3 ~ 2 3 ~ ~-
3 ~ 3 ~ ) and - 3 ~ ~ 3 ~ 3 d t r a n s i t i o n s i n S i I , P I1 and S 111. Values o b t a i n e d i n L-S coupling a r e s a t i s f a c t o r y f o r a l l t h e t r a n s i t i o n s except t h o s e i n v o l v i n g t h e 3p3d s t a t e s where r e l a t i v i s t i c e f f e c t s a r e expected t o be important.1. INTRODUCTION 2 . WAVE FUNCTIONS AND ENERGY LEVELS
There have been a number of r e c e n t experimental s t u d i e s of t r a n s i t i o n s i n members of t h e s i l i c o n i s o - e l e c t r o n i c sequence 11-72.
corresponding t h e o r e t i c a l s t u d i e s have concentra-
t e d on t r a n s i t i o n s of t h e form 3 ~
-
3s3p3. ~ 3 ~ ~ Aymar 1 81
used a p a r a m e t r i c p o t e n t i a l i n c l u d i n gs p i n - o r b i t i n t e r a c t i o n s t o i n v e s t i g a t e t h e 3 ~ e
-
3 ~ 0 , 3 ~ 0 t r a n s i t i o n s f o r S i I through Ca V I I . Even though t h e number of c o n f i g u r a t i o n s was s m a l l , t h e same c a n c e l l a t i o n e f f e c t s were e v i d e n t a s were found by Beck and Sinanoglu 1 9 , 1 0 1 w i t h t h e i r more e l a b o r a t e NCMET p r e s c r i p t i o n . The c a n c e l l a t i o n i s due t o major c o n f i g u r a t i o n i n t e r a -2 2
c r i o n (C.I.) between 3s3p3 a n d 3 s 3p3d (and 3s 3p 4 s f o r 3 ~ 0 , though t o a Lesser e x t e n t ) .
Nussbaumer [ll], i n a s t u d y of t h e i s o - e l e c t r o n i c Fe X I I 1 , h a s p o i n t e d o u t t h e dangers of l i m i t e d C. I. i n such s i t u a t i o n s . For example, t h e i n c l u s - i o n of t h e c o n f i g u r a t i o n 3 ~ 3 ~ 3 d ' r e s u l t s i n a s u b s t a n t i a l change (even of s i g n ) i n t h e expansion c o e f f i c i e n t s of t h e dominant c o n f i g u r a t i o n s , even though i t s own c o e f f i c i e n t i s a n o r d e r of
magnitude s m a l l e r . The purpose of t h i s paper i s t o extend t h e NCMET c a l c u l a t i o n by t h e i n c l u s i o n
Because of t h e a n t i c i p a t e d C . I . between t h e low-lying e x c i t e d s t a t e l e v e l s , we considered i t n e c e s s a r y t o o b t a i n a s good a r e p r e s e n t a t i o n a s p o s ~ i b l e of t h e s e a s w e l l a s of the ground s t a t e . The type of c a l c u l a t i o n i s s i m i l a r t o t h a t employed e a r l i e r [12,13] i n which t h e r a d i a l f u n c t i o n s
P a r e a n a l y t i c f i n i t e sums of S l a t e r - t y p e n
t
o r b i t a l s . The ls,Zs,2p,3s,3p f u n c t i o n s were chosen a s t h e HF f u n c t i o n s of t h e ground 3 ~ e s t a t e
[14], w h i l e t h e remaining f u n c t i o n s c o n t a i n p r e c i s - e l y (n-1) b a s i s f u n c t i o n s s o t h a t t h e orthonorm- a l i t y i n t e g r a l s determine t h e b a s i s c o e f f i c i e n t s i n each c a s e . This e s s e n t i a l l y l e a v e s t h e exponents of t h e S l a t e r f u n c t i o n s a s parameters which may be v a r i e d .
The 3d f u n c t i o n was v a r i e d t o optimize t h e important 3s -3d c o r r e l a t i o n i n t h e ground s t a t e . The 3s 3p3d s t a t e was then r e p r e s e n t e d by 2
2 2
a combination of 3s 3p3d and 3s 3p4d, w i t h t h e 4d exponents being optimized on t h e c o r r e c t eigen- v a l u e i n a C I expansion, i n c l u d i n g where a p p r o p r i - a t e 3 ~ and 3s 3p4s. 3 2 ~ ~ The 4 s f u n c t i o n was optimized on t h e eigenvalue dominated by t h e c o n f i g u r a t i o n 3s 3 ~ 4 s . The n-values of our r a d i a l 2 of a l l - e x t e r n a l c o r r e l a t i o n , w i t h i n an L-S frame- f u n c t i o n s t h e r e f o r e do n o t n e c e s s a r i l y have work, and t o p r e s e n t t h e o r e t i c a l v a l u e s f o r s p e c t r o s c o p i c s i g n i f i c a n c e .
t r a n s i t i o n s p r e v i o u s l y c o n s i d e r e d only by experim- I n a d d i t i o n , we determined a 5d f u n c t i o n
e n t a l techniques. t o account f o r t h e s e m i - i n t e r n a l e f f e c t 3 ~
+
3 ~ ~3 s 3pnd, w h i l e t h e 4 f f u n c t i o n was determined by 2 o p t i m i z i n g 3s3p3
+
3s3p 4 f . 2 J u s t a s t h e c o r r e l a -Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979122
t i o n 3d o f t h e ground s t a t e i s d i f f e r e n t from t h e e x c i t e d s t a t e ' s s p e c t r o s c o p i c 3d ( r e p r e s e n t e d i n o u r c a l c u l a t i o n s by a c o m b i n a t i o n o f o u r 3d and 4d f u n c t i o n s ) , s o t h e 3p f u n c t i o n s o f t h e s t a t e s a r e n o t i d e n t i c a l . We d e t e r m i n e d t h e 4p f u n c t i o n a s a c o r r e c t i o n t o t h e 3p f o r t h e e x c i t e d s t a t e s . F i n a l l y , we d e t e r m i n e d 5 s a n d 5p f u n c t i o n s t o o p t i m i z e t h e M-shell c o r r e l a t i o n of t h e ground s t a t e . We found l i t t l e d i f f e r e n c e between t h e s e t s o f r a d i a l f u n c t i o n s f o r t h e 3 ~ 0 and 3 ~ 0 s t a t e s , and f o r t h e f i n a l c a l c u l a t i o n s we used t h e same s e t o f f u n c t i o n s f o r a l l s t a t e s .
The e n e r g y s p l i t t i n g s between t h e ground s t a t e and t h e 3 ~ 3 ~ 0 , 3 3 ~ 0 ~ and 3 ~ ~ 3 ~ 4 s ~ 3 ~ 0 s t a t e s w e r e i n g e n e r a l l y good agreement w i t h e x p e r i m e n t a l s p l i t t i n g s a s may be s e e n from T a b l e I. The 3 s 3p3d s t a t e s were c o n s i s t e n t l y h i g h e r 2 t h a n e x p e r i m e n t , by s e v e r a l thousand cm-l,
a l t h o u g h t h e o r d e r of t h e m u l t i p l e t s a r i s i n g from t h e c o n f i g u r a t i o n was c o r r e c t . I t i s n o t c l e a r t o u s why t h i s s h o u l d be. R e l a t i v i s t i c e f f e c t s a r e l i k e l y t o b e i m p o r t a n t h e r e ( s e e l a t e r ) a l t h o u g h t h e f i n e s t r u c t u r e s p l i t t i n g o f t h e s t a t e s i n v o l v - ed i s a t most a few hundred cm-I. It i s p o s s i b l e t h a t a n i m p o r t a n t c o n f i g u r a t i o n h a s been o m i t t e d ( t h o u g h we have b e e n c a r e f u l t o c h e c k t h e s e ) o r t h a t f u r t h e r r e f i n e m e n t s i n t h e r a d i a l f u n c t i o n s a r e needed. For S i I and P 11, t h e d i f f e r e n t s t a t e s a r e w e l l s e p a r a t e d . However, f o r S 111, t h e 3p4s 3 ~ 0 and 3p3d 3 ~ 0 s t a t e s l i e c l o s e t o g e t h e r , and t h e e r r o r i n t h e l a t t e r r e s u l t s i n o u r c a l c u l a - t i o n p l a c i n g them i n t h e wrong o r d e r . The
i n t e r a c t i o n between them, which i s s u b s t a n t i a l , c o u l d w e l l be wrong.
3 . OSCILLATOR STRENGTHS
One o f t h e m o t i v a t i o n s o f t h i s work was t h e f - v a l u e o b t a i n e d by L i v i n g s t o n e t a l .
(11
f o rt h e 3s23p2 3~
-
3 ~ 3~ t r a n s i t i o n i n P 11. The 3 ~ ~ l i f e t i m e i s t w i c e a s l o n g a s t h a t o b t a i n e dp r e v i o u s l y 1 2 , 1 5 3 . The e a r l i e r v a l u e s had. been c o n f i r m e d by NCMET c a l c u l a t i o n s 191; we wished t o s e e i f a l l - e x t e r n a l c o r r e l a t i o n had t h e e f f e c t o f r e d u c i n g t h e f - v a l u e . The r e s u l t shown i n T a b l e I r e p r e s e n t s c o r r e l a t i o n o n l y w i t h i n t h e M - s h e l l , b u t t h e v a l u e i s below t h a t o f NCMET, and i s , i n f a c t , i n v e r y good agreement w i t h t h e r e l a t i v e l y s i m p l e c a l c u l a t i o n of Aymar (81. T h a t t h i s i s t r u e a l s o f o r S I11 s u g g e s t s t h a t h e r v a l u e s f o r more h i g h l y i o n i z e d s y s t e m s a r e a l s o r e l i a b l e . It
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i s , however, d i f f i c u l t t o p l a c e a n y r e l i a n c e on more t h a n t h e f i r s t s i g n i f i c a n t f i g u r e f o r s u c h t r a n s i t i o n s . We performed a c a l c u l a t i o n which was e s s e n t i a l l y NCMET i n s t y l e b u t i n c l u d i n g 4 s i n t h e HF s e a . The l e n g t h and v e l o c i t y v a l u e s f o r P I1 were 0.017 and 0 . 0 2 1 f o r t h e l o w e s t t r a n s i t i o n , w h i l e t h e v a l u e s f o r t h e t r a n s i t i o n i n v o l v i n g 3 s 3p4s were h a r d l y changed. 2 The c a u s e of t h e change was a m o d i f i c a t i o n i n t h e w e i g h t i n g s o f t h e c o n f i g u r a t i o n s d o m i n a t i n g t h e u p p e r s t a t e and t h e c a n c e l l a t i o n .
S e r i o u s c a n c e l l a t i o n s do n o t o c c u r i n t h e 3p2
-
3p4s t r a n s i t i o n f o r S i I and P 11, and agreement w i t h e x p e r i m e n t i s v e r y good. For S i I, work by Garz 1 51
s u g g e s t s a n e x p e r i m e n t a l v a l u e up t o 10% h i g h e r t h a n t h a t of Savage and Lawrence i 1 5 ],
a l t h o u g h t h e l i f e t i m e of t h e u p p e r l e v e l s v a r i e s somewhat w i t h J. Even though we have d o u b t s a b o u t o u r c a l c u l a t i o n f o r S 111, b e c a u s e o f t h e p o s s i b l e i n c o r r e c t i n t e r a c t i o n4 . RELATIVISTIC EFFECTS
Our main p u r p o s e h a s been t o
c a l c u l a t e m u l t i p l e t o s c i l l a t o r s t r e n g t h s . But f o r medium s i z e d atoms, i t is* i m p o r t a n t t o l o o k a t how r e l a t i v i s t i c e f f e c t s ( s p i n - o r b i t i n t e r a c t i o n s e t c . ) change s u c h r e s u l t s . We have made some p r e l i m i n - a r y i n v e s t i g a t i o n s i n t h i s d i r e c t i o n f o r P 11.
We f i n d v e r y l i t t l e m u l t i p l e t i n t e r a c t i o n s f o r t h e same J i n t h e 3 ~ s t a t e s ( t h o u g h t h i s s i t u a t i o n 3 ~ ~ d o e s n o t o b t a i n f o r h i g h e r members o f t h e s e q u e n c e
ell]).
There i s o n l y a s m a l l i n t e r a c t i o n between t h e 3p4s 3 ~ 1 and 1 PI s t a t e s , and i n view of t h e l a c k o f l a r g e c a n c e l l a t i o n e f f e c t s i n t r a n s i t i o n s i n v o l v i n g t h i s 3~ s t a t e we do n o t f e e l t h e i n c l u s i o n o f r e l a t i v i s t i c e f f e c t s would b e i m p o r t a n t . A c c o r d i n g l y , f o r t h e f i r s t t h r e e t r a n s i t i o n s d i s p l a y e d i n T a b l e I, we b e l i e v e t h a t L-S c o u p l i n g g i v e s a s a t i s f a c t o r y r e p r e s e n t a t i o n o ft h e s t a t e s i n v o l v e d , e x c e p t p o s s i b l y f o r 3 p 2
-
3p4s i n S 111.
between 3p4s a n d 3p4d, o u r f - v a l u e i s i n agreement Major C.I. a r i s e s i n t h e 3p3d s t a t e s d u e w i t h r e c e n t e x p e r i m e n t 1 7 3 .
3 3 t o r e l a t i v i s t i c e f f e c t s . I n i n i t i a l , r a t h e r
The P
-
D t r a n s i t i o n s were n o t s i m p l e c a l c u l a t i o n s , we ha...) n o t found a n y p u r s u e d much beyond t h e NCMET l e v e l . Our3 improvement i n t h e p o s i t i o n s of t h e 3p3d s t a t e s a s c a l c u l a t i o n of 3 . ~
-
~3s3p c o n f i r m s t h e e a r l i e r 3 ~ ~ a r e s u l t o f r e l a t i v i s t i c e f f e c t s . The major r e s u l t of S i n a n o g l u and Beck [101. The 3p2-
3p3d t h e o r e t i c a l d i f f i c u l t y i s t h a t o f r e p r e s e n t i n g o s c i l l a t o r s t r e n g t h a p p e a r s i n f a i r l y good a c c u r a t e l y t h e s m a l l s p l i t t i n g between t h e 3p3d,3 agreement w i t h t h e e x p e r i m e n t a l work o f L i v i n g s t o n
3 p and levels, which is required f o r the
C 1
e t a l . L l J . c o r r e c t i n t e r a c t i o n between them. More d e t a i l s
w i l l b e g i v e n i n a l a t e r p u b l i c a t i o n .
B IBLIOGRAPHIE
11
A. E. L i v i n g s t o n , J.A. Kernahan, D. J. G. I r w i n and E.H. P i n n i n g t o n , Phys. S c r .12,
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2,
197 ( 1 9 7 1 ) .1 3 1 A.E. ~ i \ ; i n ~ s t o n , P.D. Dumont, Y. B a u d i n e t - R o b i n e t , H.P. G a r n i r , E. ~ i ' e m o n t and N . G r e v e s s e , Beam-Foil S p e c t r o s c o p y ( e d s . I.A. S e l l i n and D . J . Pegg), (Plenum P r e s s : New York) p.339 ( 1 9 7 6 ) .
1 4 3 B. I. Dynefors and I. M a r t i n s o n , Beam F o i l S p e c t r o s c o p y ( e d s . I.A. S e l l i n and D . J . P e g g ) , (Plenum P r e s s : New York) p.223 ( 1 9 7 6 ) .
[ 5 3 T. Garz, A s t r o n . A s t r o p h y s . 2 6 , 4 7 1 ( 1 9 7 3 ) . [ 6
3
D.J.G. I r w i n and A.E. L i v i n g s t o n , Can. J. *Phys. 54, 805 ( 1 9 7 6 ) .
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68,
825 ( 1 9 7 8 ) .[8
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[lo]
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