QUICK INTERPRETATION OF UNRESOLVED HYPERFINE AND/OR ZEEMAN STRUCTURES IN STELLAR SPECTRA

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QUICK INTERPRETATION OF UNRESOLVED HYPERFINE AND/OR ZEEMAN STRUCTURES IN

STELLAR SPECTRA

J. Bauche, J. Oreg

To cite this version:

J. Bauche, J. Oreg. QUICK INTERPRETATION OF UNRESOLVED HYPERFINE AND/OR ZEE-

MAN STRUCTURES IN STELLAR SPECTRA. Journal de Physique Colloques, 1988, 49 (C1),

pp.C1-263-C1-265. �10.1051/jphyscol:1988156�. �jpa-00227476�

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JOURNAL DE PHYSIQUE

C o l l o q u e C 1 , Supplt5ment au n 0 3 , Tome 49, Mars 1988

QUICK INTERPRETATION OF UNRESOLVED HYPERFINE AND/OR ZEEMAN STRUCTURES IN STELLAR SPECTRA

J . BAUCHE and J . OREG*

Laboratoire Aim6 Cotton, Bat. 505, F-91405 Orsay Cedex, France

" ~ u c l e a r Research Centre of Negev, POB 9001, Beer-Sheva, Israel

ABSTRACT

Some a t o m i c o r i o n i c l i n e s e m i t t e d by t h e s t e l l a r atmospheres a r e v e r y u s e f u l c l u e s f o r e s t i m a t i n g t h e l o c a l m a g n e t i c f i e l d s . However, t h e i r u s e i s o f t e n c o m p l i c a t e d by i n s u f f i c i e n t

e x p e r i m e n t a l r e s o l u t i o n and by t h e o c c u r e n c e o f h y p e r f i n e s t r u c t u r e . But t h e a v e r a g e wavelength and t h e w i d t h o f t h e

component set c o r r e s p o n d i n g t o e a c h t y p e o f p o l a r i s a t i o n a r e g i v e n by compact f o r m u l a e , i n t e r m s of t h e m a g n e t i c f i e l d , a n d of t h e Land6 f a c t o r s a n d h y p e r f i n e m a g n e t i c and e l e c t r i c c o n s t a n t s o f b o t h i n v o l v e d l e v e l s .

I . INTRODUCTION

Many d i f f e r e n t phenomena c o n t r i b u t e t o t h e b r o a d e n i n g o f t h e a t o m i c l i n e s i n s t e l l a r s p e c t r a . I n a d d i t i o n t o t h e i n s t r u m e n t a l w i d t h , which c a n b e c o n s i d e r e d n e g l i g i b l e i n many c a s e s , t h e r e may o c c u r w i d t h s due t o , e . g . , s e l f - a b s o r p t i o n , a t o m i c c o l l i s i o n s , D o p p l e r e f f e c t , S t a r k s p l i t t i n g , Zeeman s p l i t t i n g , and h y p e r f i n e s t r u c t u r e . Among t h e l a t t e r , o n l y t h e h y p e r f i n e s t r u c t u r e ( h f s ) i s c h a r a c t e r i s t i c o f t h e a t o m i c l i n e c o n s i d e r e d , a n d c a n o f t e n b e m e a s u r e d i n l a b o r a t o r y e x p e r i m e n t s . The o t h e r s b r i n g i n f o r m a t i o n a b o u t t h e s t e l l a r medium.

I n t h e f o l l o w i n g , i t i s d e a l t w i t h t h e t y p i c a l c a s e of an a t o m i c l i n e which e x h i b i t s Zeeman e f f e c t a n d / o r m a g n e t i c - d i p o l a r a n d / o r e l e c t r i c - q u a d r u p o l a r h f s , b u t which i s u n r e s o l v e d , due t o t h e c o a l e s c e n c e o f c l o s e components t h r o u g h one o t h e r b r o a d e n i n g phenomenon w i t h a s m a l l w i d t h . For e a c h of t h e t h r e e Zeeman p o l a r i s a t i o n s , A, C + , and 0 - , t h e w e i g h t e d a v e r a g e wavenumber of t h e components and t h e i r r m s d e v i a t i o n from t h i s a v e r a g e a r e d e r i v e d a s s i m p l e f u n c t i o n s o f t h e m a g n e t i c f i e l d and o f t h e Zeeman a n d h f s c o n s t a n t s o f t h e l e v e l s . They may p r o v e c o n v e n i e n t t o o l s f o r e v a l u a t i n g t h e p a r a m e t e r s o f a n u n r e s o l v e d l i n e .

11. PRINCIPLES

The t h e o r y o f t h e Zeeman e f f e c t of h y p e r f i n e s t r u c t u r e i s w e l l e s t a b l i s h e d . F o r a g i v e n l e v e l , i t g o e s t h r o u g h t h e d i a g o n a l i s a t i o n o f t h e sum

H = H z + H m + H *

o f t h e Zeeman and h y p e r f i n e o p e r a t o r s i n t h e s u b s p a c e c o r r e s p o n d i n g t o t h e r e l e v a n t n u c l e u s a n d a t o m i c l e v e l . These o p e r a t o r s c a n be w r i t t e n [I], f o r a l e v e l d e n o t e d J

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1988156

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C1-264 JOURNAL

DE

PHYSIQUE

where pB B i s t h e s o - c a l l e d normal Zeeman i n t e r v a l , g, i s t h e Lande f a c t o r , I i s t h e n u c l e a r - s p i n a n g u l a r momentum, and A, and B, a r e t h e magnetic and e l e c t r i c h f s c o n s t a n t s of t h e l e v e l . When both quantum numbers I and J a r e l a r g e r t h a n 1 / 2 , some e i g e n v a l u e s EFM o f H ( w i t h

F

=

f + 3)

a r e v e r y complicated f u n c t i o n s o f pBB g,, A, and B,. The same i s t r u e f o r t h e expansions o f t h e e i g e n v e c t o r s c v e r t h e b a s i s s t a t e s I ( I , J ) F M ) , and f o r t h e e l e c t r i c - d i p o l e s t r e n g t h s

S F M , F W * , ~ ⁼ 1 ( ( I , J ) FM I D , ' ~ ' 1 ( I , J 1 ) F'M') 1 ' (1) o f t h e components o f a J

-

J' l i n e f o r any p o l a r i s a t i o n q = 0, 51, where

L(')

i s t h e e l e c t r i c - d i p o l e moment o f t h e system.

T h e r e f o r e , L t i s remarkable t h a t t h e weighted moments

FMF'M' FMF'M'

o f t h e component e n e r g i e s f o r a g i v e n p o l a r i s a t i o n a r e simple f u n c t i o n s o f p , ~ g,, pBB g,., A,, A,,, B, and B J , , when t h e weight i s t h e e l e c t r i c - d i p o l e s t r e n g t h SF,,,,,. ,,

.

T h i s can be understood b y means o f t h e second- q u a n t i s a t i o n method [ 2 1 . More p r e c i s e l y , one can g e t r i d of t h e c o m p e t i t i o n between t h e Zeeman and h f s c o u p l i n g s i n t h e same way a s t h e i n t e r m e d i a t e c o u p l i n g can be d e a l t with f o r u n r e s o l v e d t r a n s i t i o n a r r a y s [ 3 1

.

The weighted a v e r a g e energy o f t h e components i s p, and t h e c o r r e s p o n d i n g r m s d e v i a t i o n i s t h e s q u a r e r o o t of t h e v a r i a n c e v ⁼

p, -

(pl)

'.

For each of t h e t h r e e p o l a r i s a t i o n s X , 6+ and o-

( w i t h MI- M = 0 , l and - I ) , p, and v a r e polynomials of t h e f i r s t and second o r d e r i n t e r m s of pBB g,, p,B g,., A,, A,., B, and B,.

.

The a v e r a g e energy r e a d s

p, = E, + & [ p , ~ g , { 2 + ( J - J ' ) ( J + J t + l ) )

+

p,Bg,, {2- ( J - J ' ) ( J + J 1 + l )

) I

^{/ 4}

( 3 ) where E = 0 , + 1 -1 f o r t h e t r a n s i t i o n s

n,

_Q+,Q-

The v a r i a n c e i s t h e sum o f t h r e e p a r t s .

( i ) For x t r a n s i t i o n s , t h e sum v,

+

v,,

+

v,,., where

w i t h Z = J ' ( J 1 + l )

-

J ( J + l )

-

2 .

v,. i s deduced from v, by t h e exchange of J and J ' .

F o r

a+

t r a n s i t i o n s , t h e sum v',

+

v',,,

+

v',,,

,

where

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v',, i s deduced from v', by t h e exchange of J and J ' .

( i i ) For t h e magnetic h f s , t h e q u a n t i t y

where X = 2

-

J ( J + l ) - J ' ( J 1 + l )

.

(iii) F o r t h e e l e c t r i c h f s , t h e q u a n t i t y

I V . DISCUSSION AND CONCLUSION

Two p o i n t s a r e noteworthy i n Eqs. ( 3 )

-

( 9 ) : ( i ) The average energy pl does not depend on t h e h f s

( i i ) No cross-product of any p a i r of parameters of d i f f e r e n t o r i g i n s , i . e . , Zeeman, d i p o l a r o r quadrupolar h f s , appears i n t h e v a r i a n c e s . This i s l i n k e d with t h e d i f f e r e n t t e n s o r i a l behaviours o f t h e t h r e e o p e r a t o r s i n t h e e l e c t r o n i c and n u c l e a r s p a c e s .

P r o p e r t y ( i ) means t h a t t h e v a l u e of B can be deduced d i r e c t l y from t h e d i s t a n c e between t h e average wavelengths of t h e a + and

a-

components, p r o v i d e d t h e Land6 f a c t o r s of t h e l e v e l s a r e known.

T h i s h a s been known ( i n t h e absence of h f s ) s i n c e t h e paper by Shenstone and B l a i r [ 4 ]

.

I n a r e c e n t work 151, Garstang has c a l c u l a t e d e x p l i c i t l y t h e p o s i t i o n s and s t r e n g t h s of a l l t h e components of t h e l i n e a 6 ~ l / z

-

z 6 ~ 1 / 2 i n Mn I, f o r i n c r e a s i n g values of t h e magnetic f i e l d . He has chosen t o use, f o r t h e a n a l y s i s , e i t h e r t h e average wavelengths of t h e 6 s t r o n g e s t 6+ and o f t h e 6 s t r o n g e s t

o-

components, o r t h e wavelengths of t h e outermost components of b o t h t y p e s . This way of d o i n g i s supported by t h e marked asymmetry of t h e <T+ and

a-

p a r t of a p u r e Zeeman s t r u c t u r e , o f t e n r e f e r r e d t o a s

shade

o r

shade

a.

Indeed, it would o f t e n be d i f f i c u l t t o a s s i g n a Gaussian shape t o an u n r e s o l v e d a + o r 0- p r o f i l e . The c a s e of f o r b i d d e n E2 o r MI l i n e s i s not more f a v o u r a b l e [ 6 ] .

I n t h e absence of magnetic f i e l d , t h e width measured f o r an u n r e s o l v e d h f s p a t t e r n b r i n g s q u a n t i t a t i v e i n f o r m a t i o n on t h e h f s s p l i t t i n g s of t h e l e v e l s . Thus, it may be a t o o l f o r t h e assignment o f t h e l i n e .

The simple formulae d e r i v e d above f o r t h e average wavelength and r m s d e v i a t i o n of u n r e s o l v e d Zeeman components i n t h e presence of h f s avoid t h e u s e o f any d i a g o n a l i s a t i o n . They can a l s o be used i f t h e s t r u c t u r e i s p a r t i a l l y resolved, provided t h a t t h e s p e c t r a l c o n t o u r i s n e a t and unblended with o t h e r l i n e s .

REFERENCES

[ I ] B . R . JUDD, Operator Techniques i n Atomic Spectroscopy (McGraw-Hil1,New York, 1963)

.

[ 2 ] B . R . J U D D , Second Q u a n t i z a t i o n and Atomic Spectroscopy (Johns Hopkins U n i v e r s i t y P r e s s , Baltimore, 1967)

.

[ 3 ] C . BAUCHE-ARNOULT, J . BAUCHE and M. KLAPISCH, Phys. Rev.

m,

2424 ( 1 9 7 9 ) .

[ 4 ] A . G . SHENSTONE and H . A . BLAIR, P h i l . Mag. 8, 765 ( 1 9 2 9 ) . [ 5 ] R . H . GARSTANG, J. Opt. Soc. Am. B;L, 311 ( 1 9 8 4 ) .

[ 6 ] J.CZERWINSKA, J. Opt. Soc. Am. B4, 1349 ( 1 9 8 7 ) .