HAL Id: jpa-00218401
https://hal.archives-ouvertes.fr/jpa-00218401
Submitted on 1 Jan 1979
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
RELATIVISTIC EFFECTS IN THE CENTRAL FIELD APPROXIMATION : THE RELATIVISTIC
PARAMETRIC POTENTIAL METHOD.
E. Luc-Koenig
To cite this version:
E. Luc-Koenig. RELATIVISTIC EFFECTS IN THE CENTRAL FIELD APPROXIMATION : THE
RELATIVISTIC PARAMETRIC POTENTIAL METHOD.. Journal de Physique Colloques, 1979, 40
(C1), pp.C1-115-C1-121. �10.1051/jphyscol:1979121�. �jpa-00218401�
JOURNAL DE PHYSIQUE Colloque C 1 , suppldment au n 2 , Tome 40, fdvrier 1979, page C 1 - 1
IS
RELATIVISTIC EFFECTS I N THE CENTRAL FIELD APPROXIlviATION : THE: RELATIVISTIC PARAMETRIC POTENTIAL PETHOD.
E. Luc-Koenig
L a b o r a t o i r e Aim6 Cotton, C.N.R.S. 11, G t i m e n t 505, 91405-Orsay, France.
Resume : Grgce B s a s i m p l i c i t 6 e t s a r a p i d i t 6 , l a M6thode du P o t e n t i e l Parametrique R e l a t i v i s t e (RYLAC) r e n d p o s s i b l e des c a l c u l s e x t e n s i f s p o r t a n t s o i t s u r t o u t e une sequence i s o 8 l e c t r o n i q u e , s o i t sur un s p e c t r e p a r t i c u l i e r . Les r 6 s u l t a t s obtenus peuvent Q t r e a n a l y s e s B l ' a i d e du d6veloppement r e l a t i v i s t e , s u i v a n t l e s p u i s s a n c e s de Z-I e t Z2a2
,
i n t r o d u i t par Layzer e t Bahcall. C e t t e a n a l y s e montre que p a r f o i s l e s c o r r e c t i o n s r e l a t i v i s t e s s u r l e s f o n c t i o n s r a d i a l e s ne peuvent d t r e n e g l i g e e s e t q u ' a l o r s c e s e f f e t s peuvent d t r e i n c l u s simplement dans un c a l c u l a u premier o r d r e de RELAC.A b s t r a c t : Owing t o i t s simplicity and i t s r a p i d i t y , t h e R e l a t i v i s t i c Parametric P o t e n t i a l Kethod (RELAC) makes i t p o s s i b l e t o perform e x t e n s i v e c a l c u l a t i o n s e i t h e r on a complete i s o e l e c t r o n i c sequence o r on a g i v e n spectrum. The c o r r e s p o n d i n g r e s u l t s can be analyzed through t h e use of t h e relativistic -dependent
2
3
theory of I a y z e r and Bahcall which i n t r o d u c e s a double power-serles expansion i n 2-I and Z a
.
It i s shown t h a t sometimes t h e r e l a t i v i s t i c corrections t o t h e r a d i a l wavefunctions cannot be n e g l e c t e d . Then t h e corresponding e f f e c t s can be i n t r o d u c e d e a s i l y i n a f i r s t o r d e r c a l c u l a t i o n of RELAC.INTRODUCTION.
The i n c r e a s i n g i n t e r e s t i n t r a n s i t i o n proba- b i l i t i e s f o r h i g h l y i o n i z e d atoms makes i t impor- tan: t o s t u d y c a r e f u l l y t h e s y s t e m a t i c t r e n d s of wavelengths and o s c i l l a t o r s t r e n g t h s a l o n g iso- e l e c t r o n i c sequences. Then, t h e p r i n c i p a l emphasis i s not on an a c c u r a t e p r e d i c t i o n f o r a given t r a n - s i t i o n but on a n o v e r a l l study of t h e r e g i o n s where r e l a t i v i s t i c e f f e c t s a r e important and on a n analy- s i s of t h e c h a r a c t e r i s t i c s of t h e s e e f f e c t s . A s Z i n c r e a s e s i n an i s o e l e c t r o n i c sequence, t h e r e l a t i v i s t i c e f f e c t s become s o important t h a t t h e u s u a l p e r t u r b a t i v e t r e a t m e n t i n t h e P a u l i approxi- mation [ I ] i s no more v a l i d . Moreover a r e l a t i v i s - t i c t r e a t m e n t which does not account c o r r e c t l y f o r
I
t h e i n t e r e l e c t r o n i c i n t e r a c t i o n cannot g i v e r e l i
-
a b l e r e s u l t s s p e c i a l l y f o r low and moderate s t a g e s of i o n i z a t i o n . Consequently one h a s t o d e a l simul-
l a u l i approximation which i n t r o d u c e s r e l d t i v l s t i c c o r r e c t i o n s only t o f i r s t o r d e r i n a* ( a 1s t h e f m e - s t r u c t u r e c o n s t a n t ) t o t h e one-electron opera- t o r s (mass, Darwin and s p i n - o r b i t terms) and t o t h e two-electron o p e r a t o r s ( s p i n - o t h e r - o r b l t and spin- s p i n i n t e r a c t i o n s ) . Owing t o t h e b i e l e c t r o n i c terms; t h e e x a c t s o l u t i o n s of t h e P a u l i e q u a t i o n a r e not o b t a i n e d d i r e c t l y and one g e n e r a l l y u s e s n o n - r e l a t i v i s t i c wavefunctions o b t a i n e d f r o s a z e r o o r d e r h a m i l t o n i a n i n t r o d u c i n g a s w e l l a s p o s s i b l e t h e e l e c t r o s t a t i c i n t e r a c t i o n and one c a l c u l a t e s t h e r e l a t i v i s t i c e f f e c t s t o f i r s t o r d e r of p e r t u r - b a t i o n t h e o r y . This t r e a t m e n t i n t r o d u c e s only t h e i n t e r m e d i a t e c o u p l i n g and t h e r e l a t i v i s t i c c o r r e c - t i o n s on t h e energy l e v e l s . For h i g h 2-values t h i s approach i s inadequate s i n c e i t i s necessary t o i n t r o d u c e t h e r e l a t i v i s t i c c o r r e c t i o n s t o t h e r a - d i a l p a r t of t h e wavefunctions. The most a c c u r a t e method of c a l c u l a t i n g r e l a t i v i s t i c r a d i a l wave- t a n e o u s l y with r e l a t i v i s t i c and' correlation ef-
f u n c t i o n s i s t h e Dirac-Hairtree-Fock method [2,3] ; f e c t s ; t o a v o i d complicated and r a t h e r huge calcu-
however t h i s method i s very time-consuming and i s l a t i o n s a compromise must be found i n t h e t r e a t m e n t
not e a s y t o apply in t h e c a s e of multi-valence- of t h e two e f f e c t g .
e l e c t r o n c o n f i g u r a t i o n s . For t h e s e r e a s o n s d i f f e -
PAUL1 APPROXIMATION. r e n t a u t h o r s use much simpler methods. For example
Cowan and G r i f f i n [4] add t h e mass and Darwin terms For low s t a g e s of i o n i z a t i o n ( I ( 8 ) , t h e
i n t o t h e r a d i a l Hartree-Fock e q u a t i o n s , and n e g l e c t e l e c t r o s t a t i c i n t e r a c t i o n betweer. t h e e l e c t r o n s i s
t h e s p i n - o r b i t term which depends on t h e t o t a l more important t h a n t h e r e l a t i v i s t i c e f f e c t s , s o
a n g u l a r momentum j of t h e e l e c t r o n . This procedure t h e c a l c u l a t i o n s c a n be c a r r i e d o u t u s i n g t h e
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979121
c1-116 JOURNAL DE PHYSIQUE
i n t r o d u c e s -,he major r e l a t i v i s t i c e f f e c t s i n t o t h e wavefunctions w h i l e r e t a i n i n g a n o n - r e l a t i v i s t i c formalism i n t h e c a l c u l a t i o n of atomic s t r u c t u r e .
r e 1
Thc r a d i a l wavefunction R o b t a i n e d in t h i s way nd
g i v e s a good approximation t o t h e weighted average of t h e l a r g e component of t h e r e l a t i v i s t i c wave- f u n c t i o n s G
n.e J
Consequently t h e d i r e c t r e l a t i v i s t i c e f f e c t [5]
( c o n t r a c t i o n of t h e o r b i t a l s towards t h e n u c l e u s coming from t h e i n c r e a s e of t h e e l e c t r o n mass) a s w e l l a s t h e i n d i r e c t r e l a t i v i s t i c e f f e c t [5] (ex- pansion of t h e o u t e r n o n - p e n e t r a t i n g o r b i t a l s P >, 2 owing t o t h e g r e a t e r s h i e l d i n g of t h e nucleus by t h e r e l a t i v i s t i c s and p o r b i t a l s ) a r e introdu- ced i n agreement w i t h t h e r e s u l t s of a r e l a t i v i s t i c s e l f - c o n s i s t e n t - f i e l d c a l c u l a t i o n . On t h e o t h e r hand t h e r a d i a l wavefunctions a r e not j-dependent ; consequently t h i s method does n o t always a l l o w us t o c a l c u l a t e c o r r e c t l y J-dependent q u a n t i t i e s such a s branching r a t i o s w i t h i n a m u l t i p l e t o r f i n e - s t r u c t u r e i n t e r v a l s . To o b t a i n j-dependent correc- t i o n s i t i s necessary t o include the s p i n - o r b i t i n t e r a c t i o n in t h e z e r o o r d e r h a m i l t o n i a n . Care must be taken i n t h e form of t h e s p i n - o r b i t term n e a r t h e n u c l e u s t o a v o i d a r t i f i c i a l divergence i n t h e c a l c u l a t i o n s [6]. The n a t u r a l scheme i s t h e n t h e j - j c o u p l i n g which c o m p l i c a t e s g r e a t l y t h e c a l c v l a t i o n s ; consequently t h i s method i s used mainly t o s t u d y a l k a l i s p e c t r a [7].
RELATIVISTIC 2-EXPANSION.
For high s t a g e s of i o n i z a t i o n ( I )_ 18 ) t h e i n t e r a c t i o n between t h e n u c l e u s and t h e e l e c t r o n s becomes more i m p o r t a n t t h a n t h e i n t e r e l e c t r o n i c i n t e r a c t i o n and t h e r e l a t i v i s t i c e f f e c t s a r e predo- minant. In t h i s c a s e t h e r e l a t i v i s t i c Z-dependent t h e o r y of I a y z e r and Bahcall [8] i s very s u i t a b l e , s i n c e t h e z e r o o r d e r h a m i l t o n i a n Ho i s a sum of hydrogenic Dirac h a m i l t o n i a n hD
.
The i n t e r e l e c - t r o n i c i n t e r a c t l o n V and t h e R r e i t o p e r a t o r B [ I ] a r e t r e a t e d a s p e r t u r b a t i o n s . The e i g e n v a l u e s of Ho a r e g i v e n by t h e S o m e r f e l d formula [ I ] and2 2
c a n be expanded in powers of Z a ( Z i s t h e n u c l e a r c h a r g e ) . A l l s t a t e s w i t h t h e same p a r i t y
and with t h e same p r i n c i p a l quantum numbers form a
" complex [g]. I f a l l d i s t a n c e s a r e s c a l e d by
t h e n u c l e a r charge
-
i. e . r-. p = Z r-
t h e f i r s t o r d e r Hamiltonian can be w r i t t e n a sH =
[vl
( z 2 a 2 )+
2'a2 E~ (z2a2)] where1
z
V1 andB, depend on Z only through z2a2 and where Z- 1 a p p e a r s a s t h e n a t u r a l parameter i n t h e perturba- t i o n expansion, l i k e i n t h e n o n - r e l a t i v i s t i c Z- dependent t h e o r y [g]. Consequently t h e r e l a t i v i s t i c energy c a n be w r i t t e n a s a double power s e r i e s ex- pansion i n z2a2 and Z-' a s s o c i a t e d r e s p e c t i v e l y w i t h r e l a t i v i s t i c and e l e c t r o s t a t i c i n t e r a c t i o n s : E =
z2 Y '5
E nm ( ~ az - ~
) ~;z
a p p e a r s simul-n=O m=O
t a n e o u s l y i n t h e two expansions reminding u s t h a t relativistic and e l e c t r o s t a t i c i n t e r a c t i o n s cannot b e s e p a r a t e d . The f i r s t terms i n t h e expansion a r e t h e f o l l o w i n g :
t h e y correspond t o t h e n o n - r e l a t i v i s t i c hydrogenic energy ( E ~ ~ )
,
t h e e l e c t r o s t a t i c i n t e r a c t i o n ( E ),
t h e one- and t w o - e l e c t r o n r e l a t i v i s t i c ope-0 1
r a t o r s ( E , ~ and E r e s p e c t i v e l y ) ; these f o u r 1 1
terms a r i s e from t h e intra-complex i n t e r a c t i o n s . The n o n - r e l a t i v i s t i c extra-complex i n t e r a c t l o n cor- responds t o EO2
.
When t h e r e a r e more t h a n one l e v e l w i t h a g i v e n t o t a l a n g u l a r momentum J i n t h e complex, t h e above expansion i s not unique, b u t t h e c o e f f i c i e n t s can be expressed i n terms of z2a2/2-' [8]. The predominant terms in t h e expan- s i o n a r e n o t t h e same a c c o r d i n g t o t h e s t u d i e d range of Z.
For low. Z-values t h e n o n - r e l a t i v i s t i c e l e c t r o s t a t i c i n t e r a c t i o n w i t h t h e extra-complex c o n t r i b u t i o n (EO,,
EO2) i s predominant ; f o r h i g h 2-values t h e r e l a t i v i s t i c one- and two-elec- t r o n o p e r a t o r s (E10,
E ) g i v e r i s e t o t h e main1 1
c o n t r i b u t i o n s . The r e l a t i v i s t i c Z-expansion method h a s been used t o study i o n s w i t h u p t o t e n e l e c - t r o n s i n t h e f i r s t two s h e l l s [lo-141
,
b u t i t cannot be used e a s i l y t o s t u d y e x c i t e d complexes f o r many-electron i o n s , s i n c e t h e r e i s a g r e a t d e a l of s t a t e s i n a g i v e n complex. Consequently i n s t e a d of the a n a l y t i c a l expansion of t h e r e l a t i v i s t i c hydrogenic wavefunctions i t i s b e t t e r t o use nume- r i c a l r e l a t i v i s t i c wavefunctions. Then a c e n t r a l p o t e n t i a l more r e a l i s t i c t h a n t h e hydrogenic onecan be used, t o t a k e account of t h e mean i n t e r a c - t i o n between t h e e l e c t r o n s .
RELAC AND SOXE TYPICAL RESULTS I N HIGHLY IONIZED ATOMS.
-
The R e l a t i v i s t i c Parametric P o t e n t i a l Method, RELAC, [ 151 p r o v i d e s a n example of such a treatment.
This method, which i s t h e r e l a t i v i s t i c v e r s i o n of t h e Parametric P o t e n t i a l Method of K l a p i s c h [ 1 6 ] , has been d e s c r i b e d e x t e n s i v e l y elsewhere [IS-171 ; consequently we g i v e h e r e only i t s main c h a r a c t e - r i s t i c s . The c e n t r a l p o t e n t i a l which o c c u r s i n t h e z e r o o r d e r h a m i l t o n i a n i s w r i t t e n a s an a n a l y t i c a l f u n c t i o n depending on a s e t of parameters. Each parameter d e s c r i b e s t h e r a d i a l d e n s i t y of charge i n a complete s h e l l of t h e i o n i c c o r e ; i t is propor- t i o n a l t o t h e i n v e r s e of t h e mean r a d i u s of t h e s h e l l , and i n c r e a s e s a s Z i n an i s o e l e c t r o n i c sequence. Even f o r many-electron i o n s no more t h a n 4 o r 5 parameters a r e needed. The o p t i m a l v a l u e s of t h e parameters a r e o b t a i n e d by minimizing t h e t o t a l f i r s t o r d e r energy of e l t h e r t h e ground l e v e l o r t h e ground complex of t h e s p e c t r u q . When t h e o p t i m a l p o t e n t i a l i s known, t h e i n t e r m e d i a t e cou- p l i n g and t h e c o n f i g u r a t i o n mixing a r e o b t a i n e d by d i a g o n a l i z i n g t h e m a t r i x of t h e t o t a l h a m i l t o n i a n ; c o r r e l a t i o n e f f e c t s can be p a r t i a l l y i n t r o d u c e d by i n c l u d i n g s e v e r a l complexes i n t h e m a t r i x . From t h e wavefunctions o b t a i n e d i n t h i s way, i t i s pos- s i b l e t o study f o r example Land4 f a c t o r s , f i n e o r h y p e r f i n e s t r u c t u r e s and o s c i l l a t o r s t r e n g t h s ; i n t h i s c a s e t h e l e n g t h o r v e l o c i t y f o r m u l a s given by Grant [2] can be used and t h e comparison of the two corresponding v a l u e s g i v e s a t e s t of t h e t r e a t m e n t
0
of c o r r e l a t i o n s . For wavelengths g r e a t e r t h a n 3 A , r e t a r d a t i o n i s n e g l i g i b l e i n t h e study of o s c i l l a t o r s t r e n g t h s [ l a ] . The f i r s t advantage of RELAC i s t h a t it i s p o s s i b l e t o p r f o r m .'lab i n i t i o " s t u d i e s f o r any i s o e l e c t r o n i c sequence. Moreover t h e r e i s a g r e a t s a v i n g i n computer time w i t h r e s p e c t t o r e l a t i v i s t i c S e l f - C o n s i s t e n t - F i e l d methoas ; indeed a g i v e n o r b i t a l
is
d e s c r i b e d f o r any atomic s t a t e by only one r a d i a l wavefunction which i s o b t a i n e d by s o l v i n g an homogeneous d i f f e r e n t i a l e q u a t i o n . Ii,oreover a l l s t a t e s of g i v e n p a r i t y and t o t a l angu- l a r momentum J a r e s t u d i e d s i m u l t a n e o u s l y . ~ F i n a l - l y t h e o r t h o g o n a l i t y of t h e wavefunctions i s auto-m a t i c a l l y ensured and t h e r e a r e no o v e r l a p i n t e - g r a l s in t h e s t u d y of non-diagonal q u a n t i t i e s which g r e a t l y s i m p l i f i e s t h e c a l c u l a t i o n
[
191. Conse- q u e n t l y RELAC i s a very s u i t a b l e t o o l t o perform e x t e n s i v e c a l c u l a t i o n s consuming not t o o much COT-p u t e r time e i t h e r on a given spectrum o r on a com- p l e t e i s o e l e c t r o n i c sequence.
RELAC h a s been used by K l a p i s c h e t a l . [20,21]
t o s t u d y molybdenum i o n s
-
from No XV ( ~ i - l i k e ) t o Mo XL ( ~ 1 - l i k e )-
which occur i n t h e Tokamak d i s c h a r g e ; i n o r d e r t o be a b l e t o i d e n t i f y t h e i o n i z a t i o n s t a g e s i n t h e d i s c h a r g e , one h a s no c h o i c e but t o compare t h e e x p e r i m e n t a l spectrum with calculations of wavelengths and i n t e n s i t i e s f o r many transitions and f o r v a r i o u s s t a g e s of I o n i z a t i o n . An example of an e x t e n s i v e s t u d y i n a g i v e n s p c t r u m i s provided by t h e i d e n t i f i c a t i o n of f o r b i d d e n l i n e s i n t h e Ho X V spectrum [22].
Two i n t e n s e l i n e s seen a t 58.832 and 57.927
1
i n t h e spectrum of t h e TFR Tokamak a r e a t t r i b u t e d t o9 10
t h e E2 t r a n s i t i o n s (3d 4 s ) J = 2 -. (3d )J=O
.
This a n a l y s i s is s u p p o r t e d by a n "ab i n i t i o " conpu- t a t l o n of t h e wavelengths, which reproduces t h e experimental d a t a w i t h a r e l a t i v e accuracy b e t t e r than 5 x However t h e t r a n s i t i o n p r o b a b i l i t i e s f o r t h e s e l l n e s a r e of 4 o r d e r of magnitude weaker t h a n t h e t r a n s i t i o n p r o b a b i l i t i e s f o r t h e E l r e s o - nance l i n e s coning from t h e e x c i t e d l e v e l s
(3d 4 p ) ~ = 1 9
.
Consequently t h e E2 l i n e s a r e n o t observed i n l a b o r a t o r y plasmas approaching l o c a l thermodynamic e q u i l i b r i u m , o r where the e l e c t r o n d e n s i t y i s s u f f i c i e n t l y high l i k e i n the vacuum s p a r k . On t h e c o n t r a r y in a low d e n s i t y plasma, such a s i n t h e Tokamak, t h e c o l l i s i o n a l depopula- t l o n is weak ; s o t h e m e t a s t a b l e l e v e l s of t h e lowest e x c i t e d c o n f i g u r a t i o n 3d 4 s may be h i g h l y 9 depopulated and t h e E 2 l i n e s may appear with a n i n t e n s i t y comparable t o t h a t of t h e allowed r e s o - nance l i n e s . The computed i n t e n s i t i e s t a k e i n t o account a l l p o s s i b l e e x c i t a t i o n and d e - e x c i t a t i o n t r a n s i t i o n s between t h e 91 lowest energy l e v e l s due t o electron-impact and r a d i a t i v e decays ; t h e n t h e t h e o r e t i c a l spectrum o b t a i n e d w i t h t h e plasma para- m e t e r s ( e l e c t r o n d e n s i t y n and temperatureTe ) r e p r o d u c i n g t h e Tokamak regime r e p r o d u c e s very w e l l t h e e x p e r i m e n t a l spectrum a s can be s e e n in F i g . 1.
Such a n e x t e n s i v e s t u d y of a g i v e n spectrum i s
JOURNAL DE PHYSIQUE
T F R Spectrum
" E i 3dm-3d94p
Theoretlcol Spectrum
1
;:=lli;m-sEi Lr:
n
X
Fig. 1
-
Xo XV Experimental TFR Spectrum compared w i t h t h e o r e t i c a l spectrum.p o s s i b l e only through t h e s i m p l i c i t y and t h e r a p i - d i t y of the c a l c u l a t i o n s u s i n g RELAC.
From RELAC it i s a l s o p o s s i b l e t o perform c a l c u l a t i o n s f o r complete i s o e l e c t r o n i c sequences.
Elsewhere a d e t a i l e d s t u d y of t h e resonance l i n e s of t h e Kg sequence i s p r e s e n t e d [23]. The reso- nance l i n e s correspond t o t h e t r a n s i t i o n s s + ( 3 s )J=O 2 -. p l ( 3 s 3 p ) J = l
.
The lowest exci-0
t e d l e v e l p; c o r r e s p ~ n d s t o t h e term 'P and t o t h e o r b i t a l 3p i n t h e IS and j-J c o u p l i n g
1/2
n e g l i g i b l e ; f o r t h e two-electron o p e r a t o r s t h e r e l a t i v i s t i c term l a 2 A3
I
remains s m a l l e r t h a n t h e n o n - r e l a t i v i s t i c c o n t r i b u t i o n / AzG1 1 .
F i g . 2
-
2-dependence of t h e f v a l u e s f o r t h e resonance l i n e s of t h e Mg I sequence ; a ' and a " correspond t o t h e so-. pi and so-+ pq t r a n s i t i o n s r e s p e c t i v e l y ( f and f ").
r e s p e c t i v e l y and t h e second e x c i t e d l e v e l p" t o 1 t h e 'P term and 3p o r b i t a l . I n t h e non-
.
3/2
r e l a t i v i s t i c Z-dependent t h e o r y [g]
,
t h e s p i n - o r b i t i n t e r a c t i o n i s t r e a t e d t o f i r s t o r d e r of p r t u r b a - t i o n t h e o r y , s o t h e o s c i l l a t o r s t r e n g t h f ' cor- responding t o p1 i n c r e a s e s e sz5
and f " cor-1
r e s p o n d i n g t o pt' d e c r e a s e s a s 2-I
.
The r e s u l t s 1o b t a i n e d from RELAC a r e p r e s e n t e d on Fig. 2
.
Forlow 2-values they a r e i n agreement w i t h t h e predic- t i o n s of t h e n o n - r e l a t i v i s t i c 2-dependent t h e o r y , but d i f f e r f o r high 2-values ; indeed f 1 p r e s e n t s a s i g n i f i c a n t maximum and f " a f l a t minimum
.
These r e s u l t s can be i n t e r p r e t e d i n t h e framework of t h e r e l a t i v i s t i c 2-dependent t h e o r y [8] and t h r e e d i f f e r e n t r e l a t i v i s t i c e f f e c t s a r e t o be con- s i d e r e d : i n t e r m e d i a t e c o u p l i n g , energy, and r a d i a l m a t r i x element of t h e t r a n s i t i o n o p e r a t o r i n t h e
l e n g t h form. For Z
5
60 t h e r e i s no c o n t r i b u t i o n from i n t e r m e d i a t e c o u p l i n g s i n c e p; and p" cor-1 respond t o pure j-j s t a t e s i n t h e 3s3p configu- r a t i o n . The wavenumber of t h e t r a n s i t i o n a i s given by t h e r e l a t i v i s t i c 2-expansion :
( 3 ) . For intracomplex t r a n s i t i o n s ffi = 0 ; f o r h i g h
00
2-values t h e extra-complex c o e f f i c i e n t hEO2 i s
Moreover t h e o n e - e l e c t r o n r e l a t i v i s t i c term A E10 v a n i s h e s f o r t h e so
-
pi t r a n s i t i o n f o r which AJ = 0 ; consequently t h e corresponding wavenumber o 1 i n c r e a s e s very slowly and does n o t d i f f e r very much from t h e r e s u l t s of t h e n o n - r e l a t i v i s t i c t h e o r y . On t h e c o n t r a r y f o r h i g h Z-values o"i n c r e a s e s v e r y r a p i d l y a s
z4 ,
which e x p l a i n s t h e i n c r e a s e of f l '.
The r e l a t i v i s t i c expansion f o r t h e r a d i a l m a t r i x element af t h e t r a n s i t i o n opera- t o r i s t h e f o l l o w i n gRoo i s t h e n o n - r e l a t i v i s t i c hydrogenic v a l u e , RG1 g i v e s t h e c o n t r i b u t i o n of t h e n o n - r e l a t i v i s t i c e l e c t r o s t a t i c i n t e r a c t i o n and R: corresponds t o t h e r e l a t i v i s t i c c o r r e c t i o n s t o t h e r a d i a l p a r t of
Fig. 3
-
Zdependence of t h e t r a n s i t i o n r a d i a l i n t e g r a l s .t h e wavefunctions. F i g u r e 3 g i v e s t h e v a r i a t i o n of Z x WS i n terms of
z 2
; t h i s q u a n t i t y does1/2-'j
n o t r e a c h a l i m i t i n g v a l u e a s Z i n c r e a s e s , s o t h e r e l a t i v i s t i c c o r r e c t i o n s g i v e r i s e t o a n e g a t i v e c o n t r i b u t i o n t o t h e r a d i a l i n t e g r a l . For f "
,
t h i s c o n t r a c t i o n of t h e r e l a t i v i s t i c o r b i t a l s toward t h e nucleus i s t h e predominant c o n t r i b u t i o n and f "d e c r e a s e s f o r h i g h Z-values. In c o n c l u s i o n , i n o r d e r t o i n t e r p r e t t h e 2-dependence of t h e o s c i l - l a t o r s t r e n g t h s f o r t h e resonance l i n e s of t h e Ng sequence i t i s necessary t o add e f f e c t i v e o p e r a t o r s t o t h e n o n - r e l a t i v i s t i c h a m l l t o n i a n , which modify t h e c o u p l i n g and the energy of t h e l e v e l s ; t o t h e o r d e r a 2 t h e s e c o r r e c t i o n s a r e i n t r o d u c e d in t h e P a u l i h a n i l t o n i a n . Simultaneously spin-dependent c o r r e c t i o n s t o t h e r a d i a l p a r t of t h e wavefunctions a r e t o be i n t r o d u c e d ; it i s d i f f i c u l t t o account f o r t h e s e c o r r e c t i o n s i n a n o n - r e l a t i v i s t i c scheme a s i t w i l l be shown below and we c a l l t h i s e f f e c t a "purely r e l a t i v i s t i c e f f e c t " [24].
OPERATOR AND WAVEFUNCTION CORRECTIONS.
It is easy t o understand t h e o r i g i n of t h e o p e r a t o r and wavefunction c o r r e c t i o n s i n t h e frame- work of a s i n g l e - e l e c t r o n model [25]. The r e l a t i - v i s t i c wavefunction
IY>
c a n be expanded i n termsThe l e a d i n g term i n t h e expansion of t h e l a r g e component 15) i s t h e n o n - r e l a t i v i s t i c wavef unc- o n 1 ) ; a 2 IF,,) i s t h e f i r s t o r d e r r e l a t i - v i s t i c c o r r e c t i o n t o t h e l a r g e component. For t h e s m a l l component I t ) ) t h e l e a d i n g term i s of o r d e r a and can be w r i t t e n i n terms of
I
5,) :a
l q
0 ) =J-- 2mC ( 6 ) .The m a t r i x element of an even p e r t u r b a t i o n V P e v a l u a t e d between r e l a t i v i s t i c s t a t e s i s t h e sum of two terms coming from t h e l a r g e and s m a l l compo- n e n t s , and can be expanded i n terms of a 2
The f i r s t term g i v e s t h e n o n - r e l a t i v i s t i c l i m i t ; t h e second a r i s e s from t h e r e l a t i v i s t i c c o r r e c t i o n s t o t h e l a r g e components and corresponds t o t h e wavefunction c o r r e c t i o n . The t h i r d g i v e s t h e con- t r i b u t i o n of t h e s m a l l components ; u s i n g t h e r e l a - t i o n ( 6 ) , t h i s term i s e q u a l t o t h e matrix element
(so] ",if 1 cb)
of t h e e f f e c t i v e o p e r a t o re v a l u a t e d between n o n - r e l a t i v i s t i c wavefunctions.
The opezator c o r r e c t i o n i s e a s y t o e v a l u a t e , f o r example t h e e f f e c t i v e o p e r a t o r f o r t h e energy i s t h e B e l t - P a u l i h a m i l t o n i a n ; Drake [26] g i v e s t h e g e n e r a l form of t h e r e l a t i v i s t i c c o r r e c t i o n s t o r a d i a t i v e t r a n s i t i o n p r o b a b i l i t i e s i n t h e v e l o c i t y form. Abragam and van Vleck [27] study the r e l a t i - v i s t i c c o r r e c t i o n s t o Land6 f a c t o r s . The wavefunc- t i o n c o r r e c t i o n cannot be n e g l e c t e d . For example t h e m a t r i x element of t h e o p r a t o r
-.
r can be w r i t t e n a s :For hydrogenic wavefunctions t h e r a t i o of t h e wave- f u n c t i o n c o r r e c t i o n Wf t o t h e o p e r a t o r c o r r e c t i o n
Op i s e q u a l t o -3.9 and 1.2 f o r t h e t r a n s i t i o n s I s l l 2
-
2p3/2 and I s l l 2-
2p1/2 r e s p e c t i v e l y . I n t h i s c a s e t h e wavefunction c o r r e c t i o n s a r e g r e a t e r t h a n t h e o p e r a t o r c o r r e c t i o n s , . e s p e c i a l l y f o r t h e Aj#
0 t r a n s i t i o n . To t h e o r d e r a 2 t h e wavefunction c o r r e c t i o n s can be o b t a i n e d by s o l v i n ga - 1 2 0 JOURNAL DE PHYSIQUE
a n inhomogeneous e q u a t i o n [25] and a r e e q u i v a l e n t t o a f i r s t o r d e r c a l c u l a t i o n of c o n f i g u r a t i o n i n t e - r a c t i o n s [24,25] through t h e B r e i t - P a u l i o p e r a t o r s ; i t i s n o t p o s s i b l e t o include t h e s e c o r r e c t i o n s t h r o w h t h e i n t r o d u c t i o n of an e f f e c t i v e o p e r a t o r , c a l c u l a t e d between n o n - r e l a t i v i s t i c wavefunctions.
This " p u r e l y r e l a t i v i s t i c e f f e c t " l e a d s t o a s h i f t of t h e r a d i a l wavefunctions toward t h e nucleus. I n f i r s t approximation t h e s h i f t depends only on t h e j-value of t h e o r b i t a l and d e c r e a s e s w i t h j [24].
Consequently "purely r e l a t i v i s t i c e f f e c t " can be predominant i n t h e study of non-diagonal matrix element f o r which AJ is n o t n u l l and when l a r g e c a n c e l l a t i o n e f f e c t s occur ; indeed i n t h i s c a s e t h e c a l c u l a t e d r a d i a l i n t e g r a l s a r e very s e n s i t i v e t o a s m a l l r e l a t i v e s h i f t of t h e corresponding o r b i t a l s . The s e p a r a t i o n u s u a l l y made between wavefunction and o p e r a t o r e f f e c t s i s n o t unique
[25] ; expansion i n powers of a 2 cannot be d e f i - ned in a n unique way f o r wavefunctlons and opera- t o r s , only t h e expansion of m a t r i x elements i s w e l l defined. The most u s u a l way t o d e r i v e r e l a t i v i s t i c c o r r e c t i o n s i s t h e Foldy-Wouthuysen t r a n s f o r m a t i o n [28]. A t y p l c a l example of "purely r e l a t i v i s t i c e f f e c t " i s t h e anomalous i n t e n s i t y r a t l o i n t h e p r i n c i p a l s e r i e s of a l k a l i . For a l o n g time i t h a s been observed t h a t t h e r a t i o of t h e o s c i l l a t o r s t r e n g t h s f o r t h e two components of t h e doublet d i f f e r s from t h e r a t i o of t h e s t a t i s t i c a l weights of t h e upper l e v e l s . There is no i n t e r m e d i a t e cou- p l i n g c o n t r i b u t i o n s i n c e t h e r e i s only one e l e c t r o n o u t s i d e c l o s e d s h e l l s ; moreover t h e r e i s no wave- number c o r r e c t i o n , s i n c e t h e f i n e s t r u c t u r e of t h e n 2 P e x c i t e d term becomes n e g l i g i b l e a s n increa- s e s ; consequently only t h e r e l a t i v i s t i c c o r r e c t i o n s t o t h e r a d i a l , i n t e g r a l s Q'>ns can e x p l a i n
1/2-n'P. J
t h e anomalies. For cesium, t h i s phenomenon was e x p l a i n e d q u a l i t a t i v e l y by Fermi [29], a s coming from t h 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 through t h e
s p i n - o r b i t i n t e r a c t i o n between t h e n 2 P . l e v e l s . J
For rubidium, we have r e c e n t l y performed r e l a t i v i s - t i c - c a l c u l a t i o n s t o t h e f i r s t o r d e r of RELAC n e g l e c t i n g a l l c o n f i g u r a t i o n i n t e r a c t i o n s
[YO].
The c a l c u l a t i o n s were c a r r i e d o u t up t o t h e p r i n c i - p a l quantum number n=80
,
showing t h a t t h e r a t i o i n c r e a s e s r a p i d l y with n towards a l i m i t i n g v a l u e , which a g r e e s with % r e c e n t experiment [3 I ] . F i n e s t r u c t u r e anomalies i n a l k a l i - l i k e s p e c t r a a r i s e a l s o from important "purely r e l a t i v i s t i c e f f e c t t 1 [24] ; t h e y can be reproduced i n a f i r s t - o r d e r c a l c u l a t i o n of RELAC. I n a n o n - r e l a t i v i s t i c scheme i t i s n e c e s s a r y t o i n t r o d u c e e x p l i c i t e l y c o n f i g u r a t i o n i n t e r a c t i o n s through t h e r e l a t i v i s t i c one- and two-electron o p e r a t o r s . These i n v e r s i o n s can be e x p l a i n e d a s due t o a second-order "cross- i n t e r a c t i o n ' ' between t h e s p i n - o r b i t i n t e r a c t i o n and t h e exchange p a r t of t h e Coulomb i n t e r a c t i o n [32].Here t h e n o n l o c a l exchange p o t e n t i a l i n t r o d u c e s t h e o v e r l a p of two d i f f e r e n t o r b i t a l s and c a n c e l l a t i o n e f f e c t s a r e important due t o t h e s m a l l o v e r l a p of c o r e and valence o r b i t a l s . "Purely r e l a t i v i s t i c e f f e c t 1 ' can a l s o e x p l a i n t h e non z e r o minimum of 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 s of the ground s t a t e s of a l k a l i [33], s p i n f o r b i d d e n t r a n s i t i o n s i n Group I1 d ~ e m e n t s 1341 01 anomalous Land6 f a c - t o r s i n t h e s p c o n f i g u r a t i o n s of Cd and Hg [35] ; i n t h i s l a s t example t h e r e l a t i v i s t i c cor- r e c t i o n s t o t h e o p e r a t o r s [27] a r e almost n e g l i - g i b l e .
To study " p u r e l y r e l a t i v i s t i c e f f e c t " a f i r s t o r d e r c a l c u l a t i o n of t h e r e l a t i v i s t i c c e n t r a l f i e l d i s more complete and s i m p l e r t h a n an e x p l i c i t t r e a t m e n t of c o n f i g u r a t i o n i n t e r a c t i o n s through t h e B r e i t - P a u l i o p e r a t o r s ; moreover c o r r e l a t i o n e f - f e c t s can t h e n be p a r t i a l l y i n t r o d u c e d by t r e a t i n g e x p l i c i t e l y t h e i n t e r a c t i o n between d i f f e r e n t com- plexes. Consequently owing t o i t s g r e a t s i m p l i c i t y and i t s r a p i d i t y t h e RELAC method a p p e a r s t o be a v e r y s u i t a b l e t o o l f o r performing e i t h e r e x t e n s i v e c a l c u l a t i o n s on a complete i s o e l e c t r o n i c sequence o r an e l a b o r a t e s t u d y of g i v e n s p e c t r a . The cor- responding r e s u l t s can e a s i l y be analyzed through t h e use of t h e r e l a t i v i s t i c Z-dependent t h e o r y and i t i s p o s s i b l e t o show t h e o r i g i n of the r e l a t i - v i s t i c c o n t r i b u t i o n s .
BIBLIOGRAPHIE
[I] Bethe H. A. and S a l p e t e r E. E . , Quantum Mecha- n i c s of One- and %o-Electron Atoms, S p r i n g e r Verlag, B e r l i n ( 1957).
[2] Grand I. P., Adv. Phys.
14
(1970) 747.[3] Desclaux J. P., Cornp. Phys. C O ~ ~ X L
2
(1975) 31.[4] Cowan R . D. and G r i f f i n D. C . , J. Opt. Soc.
Am. (1976) 1010.
[5] See f o r example Desclaux J. F. and Kim Y. K., J. Phys. B 8 (1975) 1177.
[6] Condon E. U. and S h o r t l e y G. H . , The Theory of Atomic S p e c t r a , Cambridge, England ( 1964).
[7] See f o r example : Stone P. M . , Phys. Rev.
127
(1962) 1151 ; Weisheit J. C . , Phys. Rev. A
2
(1972) 1621 ; Norcross D. W . , Phys. Rev. A
1
(1973) 606.
[8] Layzer D. and B a h c a l l J.. Annals of Phys. ( N Y )
a
(1962) 177.[g] Layzer D., Annals of P h ~ s . ( N Y )
8
(1959) 271.[ l o ] Doyle H. T., Adv. Atom. Flolec. Phys.
2
(1964) 337.[ I I ] Safronova U. I., J. Quant. S p e c t r o s c . R a d i a t . T r a n s f e r
3
( 1975) 23 1 .[12] Safronova U. I. and Rudzikas 2. B., J. phys. B
2
(1976) 1989.1131
Goldsmith S., J . Phys. B1
(1974) 2315.[14] Ivanov L. N., Ivanova E. P. a n d Safronova U. L, J . Quant. S p e c t r o s c . Radiat. T r a n s f e r
2
(1975) 553.
[15] Koenig E . , P h y s i c a ( u t r . ) ( 1972) 393.
[16] K l a p i s c h M . , Comput. Phys. ~ O I ~ U
2
~(1971) .239.
[17] K l a p i s c h M., Schwob J. L., F r a e n k e l B. S. and Oreg J . , J. Opt. Soc. Am.
fl
(1977) 148.[18] K l a p i s c h M.
,
P r i v a t e communication ( 1978).[ 191 Cheng K. T. and Johnson W. R.
,
Phys. Rev. Aa
(1977) 1326.
[20] K l a p i s c h M . , P e r e l R . , Weil D . , Report EUR CEA FC 827 (1976).
[21] Schwob J . L., Klapisch M., F i n k e n t h a l M . , Schweitzer N., Report EUR CEA FC 887 (1977).
[22] K l a p i s c h H . , Schwob J . L., F i n k e n t h a l M., F r a e n k e l B. S., Egert S., Bar-Shalom A., Breton C., De N i c h e l i s C . and M a t t i o l i M., Phys. Rev. L e t t .
41
(1978) 403.[23] Aymar M.
,
and Luc-Koenig E. ,
Phys. Rev. A3
(1977) 821.
[24] Luc-Koenig E . , Phys. Rev. A Jl (1976) 21 14.
[25] F e n e u i l l e S . , and Luc-Koenig E., Comments Atomic Nol. Phys.
6
(1977) 151.1261 Drake G. W. F., Phys. Rev. A
5-
( 1972) 1979, and Drake G. W. F . , J. Phys. B9
(1976) L169.[27] Abragam A. and v a n Vleck J. B . , Phys. Rev.
(1953) 1448.
[28] Foldy L. L. and Wouthuysen S. A . , Phys. Rev.
(1950) 29.
[29] F e r ~ i E.
,
Z . Physikgq
( 1970) 680.[TO] Luc-Koenig E., a n d B a c h e l i e r A . , J o u r n a l de Physique
2
( I 978)[3 1 1 Liberman S.
,
and P i n a r dJ.,
s u b m i t t e d t o Phys.Rev. A.
[32] Polmgren I,.
,
Lindgren I . , Morrison J . , and# a r t e n s s o n A. M., Z . Phys. A
276
(1976) 179, and Lee T., Rodgers J. Z., Das T. P., and s t e r n h e h e r R . K . , Phys. Rev. A & (1976) 51.[33] S e a t o n M. J., Proc. R. Soc. A 208 ( 195 1 ) 4 18.
[34] Luc-Koenig E . , J. Phys. B
1
( 1974) 1052.[35] Luc-Koenig E.
