ENERGY LEVELS AND ELECTROMAGNETIC TRANSITIONS OF ATOMS IN SUPERSTRONG MAGNETIC FIELDS

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ENERGY LEVELS AND ELECTROMAGNETIC TRANSITIONS OF ATOMS IN SUPERSTRONG

MAGNETIC FIELDS

G. Wunner, H. Ruder

To cite this version:

G. Wunner, H. Ruder. ENERGY LEVELS AND ELECTROMAGNETIC TRANSITIONS OF

ATOMS IN SUPERSTRONG MAGNETIC FIELDS. Journal de Physique Colloques, 1982, 43 (C2),

pp.C2-137-C2-152. �10.1051/jphyscol:1982211�. �jpa-00221821�

JOURNAL DE PHYSIQUE

Colloque C2, supplément au n°ll, Tome 43, novembre 1982 page C2-137

ENERGY LEVELS AND ELECTROMAGNETIC TRANSITIONS OF A T O M S I N SUPERSTRONG MAGNETIC FIELDS

G. Wunner and H. Ruder

Institut fttr Theoretisehe Physik, Universitât Erlangen-Nù'rnberg, D-8520 Erlangen, F.R.G.

Lehrstuhl fur Theoretisehe Astrophysvk, Universitât Tubingen, D-7400 Tiïbingen, F.R.G.

Résumé. - Lorsque les champs magnétiques appliqués à l'atome deviennent si intenses que la force de Lorentz devient comparable ou plus grande que la force de Coulomb, pour des champs B s- 4,7-lCr T*Z2 (Z la charge du noyau), la structure atomique doit être réanalysée. Ces conditions sont celles des étoiles à neutrons. Nous présentons les résultats obtenus au cours de cette tentative, pour les systèmes à un électron (niveaux d'énergie, transitions électromagné- tiques), pour les systèmes à deux électrons et l ' i o n Ht.

Abstract. - The paper reviews the results obtained so far in the endeavour to recalculate the whole of atomic physics for magnetic field strengths so strong that the Lorentz forces are comparable with or larger than the Cou- lomb forces, i.e. B>A.7'105 T'Z2(1 = nuclear charge). Investigations of this kind are stimulated by the existence of such huge fields in the vicinity of neutron stars. Detailed results are presented for one-electron systems

(including energy levels, electromagnetic bound-bound, bound-free, and free- free transitions), for two-electron systems, and for the Hi-ion.

1. Introduction.- The interpretation of the spectra of white dwarfs and accreting neutron stars confirms the existence of the huge magnetic fields associated with these compact cosmic objects. Magnetic fields of the order of 1 02 - 105 T have been observed in white dwarfs (cf. Angel et al.El]), and of the order of 1 07 - 1 09 T in neutron stars (cf. Trumper[2]). On account of the prevailing physical conditions the atomic spectra of these objects should be dominated by the lines of the hydrogen- like and helium-like ions. Therefore, the accurate knowledge of the atomic proper- ties of these ions in strong magnetic fields constitutes an essential ingredient for the detailed calculation of the ionisation equilibria, radiative processes, etc. in the atmospheres of white dwarfs and the accretion columns of X-ray pulsars.In recent years numerous investigations have been undertaken to determine the enerqy levels and electromagnetic transitions of hydrogen-like ions (cf.[3-ll]), whilefor ions with more than one electron only limited results are available for energy levels, and electromagnetic transitions have not been discussed to date.The purpose of this paper is to summarize the results that have been achieved so far in the endeavour to recalculate the whole of atomic physics for magnetic fields so strong that the atomic structure is dominated by the Lorentz forces rather than by the Coulomb for- ces. We shall use hydrogen-like systems as a paradigm for demonstrating in detail the influence on all atomic properties and shall then consider helium-like systems.

Furthermore the changes of molecular structure caused by strong magnetic fields will be discussed for the H^-ion.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1982211

C2-138 JOURNAL DE PHYSIQUE

2. Energy l e v e l s f o r hydrogenic i o n s i n a r b i t r a r y magnetic f i e l d s a. The general two-body problem i n a homoqeneous maqnetic f i e l d

For a system o f charged p a r t i c l e s t h e separation o f t h e motion o f t h e c e n t r e o f mass by no means c o n s t i t u t e s a problem as t r i v i a l as i n t h e f i e l d - f r e e case, C l a s s i c a l l y speaking every charged p a r t i c l e gyrates ( n e g l e c t i n g t h e mutual i n t e r a c t i o n ) about t h e magnetic f i e l d , which i m p l i e s t h a t t h e c e n t r e o f mass does n o t move u n i f o r m l y . This p i c t u r e has l e d several authors t o t h e i n c o r r e c t conclusion t h a t i n t h e pre- sence o f a magnetic f i e l d t h e r e e x i s t s no conserved q u a n t i t y f o r t h e t r a n s l a t i o n a l motion.[l2

,

131. For n e u t r a l systems however, i t immediately f o l l o w s from t h e t r a n s - l a t i o n a l i n v a r i a n c e o f t h e problem t h a t t h e generalized momentum o p e r a t o r Po, de- f i n e d as t h e generator o f an i n f i n i t e s i m a l t r a n s l a t i o n , i s conserved. Taking i n t o account t h a t t h e t r a n s l a t i o n must be accompanied by a gauge t r a n s f o r m a t i o n o f t h e v e c t o r p o t e n t i a l

A

one o b t a i n s Po f o r a n e u t r a l two-body system (charges e+=e,e-=-e, masses m,, m-):

where R = (m-r-+ m+r )/(m-+m ), r = ( C

- s ) ,

and t h e v e c t o r o o t e n t i a l has been chosen-in t h e form

~f!)

⁼ l / $ ( ~ x ~ ) . Txe o p e r a t o r Po commutes w i t h t h e Hamil t o n i a n H, and simultaneous eigenfunctions 9 o f H and Po ( w i tTi eigenvalues E and h s ) read

which leads t o t h e eigenvalue equation f o r t h e wave f u n c t i o n b ( 1 ) :

where M = m-+ m+, p = (m-.m+)/M,

p

= - i h a / a ~ , and Vc(r) = -e / ( 4 r s o r ) i s t h e Coulomb 2 i n t e r a c t i o n .

The eigenfunctionsand t h e eigenvalues o f (3) depend on K,B,M, and u ,i .e.

,

4 = +(K,B,MYp,t-) and E = E(K,B,M,u). A c l o s e r i n s p e c t i o n o f t h e s t r u c t u r e o f t h e Schrodinger equation ( 3 ) y i e l d s t h e s c a l i n g laws

Thus, i n o r d e r t o e x p l o i t the s c a l i n g freedom i n t h e quantum number

K,

t h e s o l u t i o n s o f ( 3 ) have t o be known, f o r some g i v e n v a l u e of K

+

0, as continuous functions o f t h e magnetic f i e l d s t r e n g t h and o f t h e masses involved.

Since f o r s t r o n g magnetic f i e l d s these s o l u t i o n s can be obtained n u m e r i c a l l y only, t h e p r a c t i c a l treatment o f t h e two-body problem i n t h e general case

K

c 0 c o n s t i - t u t e s a l a b o r i o u s task. I n t h e f o l l o w i n g , we s h a l l concentrate on t h e case

K

= 0, which represents, a l s o i n magnetic f i e l d s , t h e p h y s i c a l l y reasonable s t a r t i n g o o i n t f o r i n v e s t i g a t i n g t h e i n t e r n a l p r o p e r t i e s of two-body systems. Here t h e Hamiltonian i n ( 3 ) becomes a x i a l l y symmetric,and 4 can always be chosen as an e i g e n f u n c t i o n o f 1, =(I x p), w i t h eigenvalue hm. Then the M-dependence of t h e energy i s simply con- t a i n e d in-a diagonal term, and t h e s c a l i n g laws can be formulated most c o n v e n i e n t l y i n terms o f t h e s o l u t i o n s o f t h e problem f o r M - a ( w i t h o u t l o s s o f g e n e r a l i t y we t a k e mqco and abbreviate m+/(m+

+

m-) = A )

( c f . Pavloy-Verevkin and Z h i l i n s k i i [ I 4 1 ). I t i s seen t h a t , f o r a given m- and K=O, once t h e problem has been solved i n t h e l i m i t m++m f o r every value o f

B,

^{t h e}

-

s o l u t i o n s o f t h e two-body problem w i t h mass m- and a r b i t r a r y f i n i t e values o f m+

a r e a l s o known. From ( 5 ) t h e s c a l i n g law f o r t h e d i p o l e strengths, d , . , " = ( ( ~ ' ( r ( q ) / a , ( r ) ( ~ , q = O , + l ,

can be d e r i v e d ( T ' ,T denote s e t s o f quantum numbers c h a r a c t e r i z i n g t h e i n i t i a l and the f i n a l s t a t e , r e s p e c t i v e l y , r ( q ) a r e t h e s p h e r i c a l components o f t h e r e l a t i v e vector, and a, i s the Bohr r a d i u s ) :

d,. ,(q)(m-, m+ , B) = I.-24.,(Q)(m-, m+ -. m, B/12) , I. = m+ /(m+

+

^{m - ) .}

( 6 ) From (5b) and ( 6 ) t h e s c a l i n g behavior o f o s c i l l a t o r strengths, t r a n s i t i o n probabi- 1 it i e s , etc., i s immediately c a l c u l a t e d . Important a p p l i c a t i o n s o f t h e mass s c a l i n g laws are t h e hydrogen atom w i t h f i n i t e p r o t o n mass, and p o s i t r o n i u m ( c f . Wunner e t a1. [15] ). I t should be noted t h a t , c o n t r a r y t o w i d e l y h e l d opinion, t h e e f f e c t of t h e f i n i t e p r o t o n mass i s n o t always o f t h e a n t i c i p a t e d order o f maqnitude me/mproton' The l a s t term i n (5b) causes a s h i f t between s t a t e s w i t h d i f f e r e n t

m

by ,im.heB/mDroto,,a6..29.6 eV-B/4.7

.

^{10 T}

,

which, i n h i g h f i e l d s , becomes comparable h t h - t h e Coulomb b i n d i n g energies, and thus i s by no means n e g l i g i b l e ( c f

.

Wunner e t a1

.

^{[ I 6 1 ).}

I n c o n t r a s t t o t h e f i e l d - f r e e case, t h e well-known Z - s c a l i n g i s possible, i n t h e presence o f a magnetic f i e l d , o n l y i n t h e l i m i t m,-w. For t h e sake o f completeness we l i s t t h e s c a l i n g laws f o r t h e wave f u n c t i o n s and energies:

( c f . Surmelian and O'Connell [17] ) . From t h i s we can d e r i v e t h e s c a l i n g laws f o r t h e d i p o l e s t r e n g t h s

d,.,"yZ, B) =

z-

24.,"'(z = 1, BIZ^) ,

( 8 ) and analogously from (7b) and ( 8 ) those f o r the o s c i l l a t o r strengths, e t c .

The p r a c t i c a l a p p l i c a t i o n o f t h e s c a l i n g f o r t h e energies and d i p o l e s t r e n g t h s r e - q u i r e s t h e knowledge o f t h e continuous B-dependence o f these q u a n t i t i e s f o r m+- w

.

A r i g o r o u s a n a l y t i c a l s o l u t i o n t o t h i s problem i s n o t possible, and a reasonable way i s t o solve t h e Schrodinger equation n u m e r i c a l l y accurate f o r a s u f f i c i e n t l y dense sequence o f d i s c r e t e B values.

b. S o l u t i o n o f t h e Schrodinqer equation f o r i n f i n i t e proton mass

I n s o l v i n g t h e Schrodinger equation, q u i t e g e n e r a l l y two asymptotic ranges o f t h e magnetic f i e l d s t r e n g t h have t o be d i s t i n g u i s h e d

,

namely t h e l o w - f i e l d r e g i o n BKBZ = 2a2m,2 ~ ~ / ( e h ) . ~ ~ = 4 . 7 . 1 0 ~ T . Z ~ and t h e h i g h - f i e l d r e g i o n B>BZ, where e i t h e r t h e coulomb f o r c e s o r t h e Lorentz f o r c e s are dominant.

Our procedure f o r determining as a c c u r a t e l y as p o s s i b l e hydrogenic wave f u n c t i o n s i n magnetic f i e l d s o f a r b i t r a r y s t r e n g t h s t a r t s from expanding t h e wave f u n c t i o n s i n terms o f s p h e r i c a l harmonics f o r low f i e l d s

am =

Zf

( r ) Ylm(d

,

^{( 9 )}

w h i l e f o r h i g h magnetic f i e l d s , where t h e s p e r i c a l symmetry o f t h e Coulomb p o t e n t i a l i s destroyed by t h e magnetic f i e l d t o an ever i n c r e a s i n g e x t e n t , t h e expansion i n terms o f Landau s t a t e s

oL:l(r,)

(n: Landau quantum number, m: magnetic quantum number, See, e.g., Canuto and Ventura [18]) i s more a p p r o p r i a t e

am = xgn(z)@nm Lan

(5 1

n

C2- 140 JOURNAL DE PHYSIQUE

I n s e r t i n g (9), o r ( l o ) , i n t o t h e Hamiltonian o f t h e problem, and p r o j e c t i n g on t h e d i f f e r e n t s p h e r i c a l harmonics, o r Landau states, y i e l d s a system o f i n t e g r o -

d i f f e r e n t i a l - e q u a t i o n s f o r t h e expansion f u n c t i o n s fl ( r ) , o r gn(z). which i s coupled v i a t h e m a t r i x elements o f t h e diamagnetic term i n t h e Hamiltonian,

o r those o f t h e Coulomb p o t e n t i a l ,

v,!$)(z) = <o::~(> ) IZe / ( 4 ~ ~ ~ r ) l @ ~ ? ~ 2

(5 I>,

r e s p e c t i v e l y . Both systems o f equations possess t h e mathematical s t r u c t u r e o f t h e Hartree-Fock equations encountered i n f i e l d - f r e e atomic physics. We t h e r e f o r e took t h e well-known Froese-Fischer code (Froese-Fischer [19]), which has proved so successful i n f i e l d - f r e e Hartree-Fock c a l c u l a t i o n s , and adapted i t t o t h e r e q u i r e - ments o f t h e two d i f f e r e n t ranges o f the magnetic f i e l d s t r e n g t h (Proschel [20]

,

Rosner [ 2 1 ] ) . By v i r t u e o f i t s a p t choice o f t h e i n t e g r a t i o n mesh and i t s r e f i n e d i n t e g r a t i o n methods, t h e code i s n o t o n l y n u m e r i c a l l y v e r y stable, b u t a l s o v e r y f a s t , and a l l o w s an i n c l u s i o n o f more than 10 components irl (9) and (10) w i t h o u t any s u b s t a n t i a l expenditure o f computer t i m e o r memory. The convergence o f t h e method can be seen by successively i n c r e a s i n g t h e maximum number o f components i n - cluded i n (9) o r (10). A d e t a i l e d d e s c r i p t i o n o f our numerical orocedure i s found i n r e f s . [20,21]

.

c Results and d i s c u s s i o n

As a r e p r e s e n t a t i v e example f o r t h e c a l c u l a t i o n s performed by us, i n Table 1 we l i s t t h e energy values ( i n u n i t s o f t h e Rydberg energy E,, =2m,c2 /2 ? 13.6 eV) o f t h e

" F

^{0 , ~ 0,2 II~,}

" ¹⁰

^{0 ) 0,2 1111,}

L

n i n e lowest m=O s t a t e s o f t h e H atom f o r v a r i o u s magnetic f i e l d s t r e n g t h s i n t h e range 1 0 - ~ < p < 1 0 ~ (P=Pl= B/B1 ). The s t a t e s a r e l a b e l l e d by b o t h t h e i r f i e l d - f r e e quantum numbers and t h e i r asymptotic (p-m) quantum numbers n,m,u ( u i s t h e number o f nodes of t h e dominant l o n g i t u d i n a l wave f u n c t i o n i n (10) For p - + m ) , which a t t h e same time expresses t h e correspondence between t h e d i f f e r e n t asymptotic quantum num- bers. For small f i e l d s , a s u b s c r i p t o f t h e type Nl/N2 i n d i c a t e s t h a t t h e energy va- l u e was computed i n c l u d i n g up t o N2components i n (9), and t h a t t h e s i g n i f i c a n t d i -

C Z

+

t

(b) s 1

I I

I ^/

.+'

I+,+@ ^A+'c ,+++

&I.+

,++'

-3,go-

-

^3,91-

-3,92-

-**+ -+-**

F i g u r e 1.- Convergence o f t h e r e s u l t s f o r t h e energy values i n dependence on t h e maximum number nk o f c o n f i g u r a t i o n s i n c l u d e d i n (9) and (10). The curves r e f e r t o t h e s t a t e w i t h asymptotic quantum numbers ls0/000 (a:P=7, b:P= 20). I t i s recog- n i z e d t h a t i n general t h e s p h e r i c a l computation (curves marked by " s " ) converges more r a p i d 1 y than does the c y l i n d r i c a l one (curves marked by " c " ) . F o r i n c r e a s i n g

p ,

however, t h e c y l i n d r i c a l curves f l a t t e n , and more c o n f i g u r a t i o n s a r e needed i n t h e s p h e r i c a l c a l c u l a t i o n t o reach s a t u r a t i o n . The behaviour shown can be used t o e x t r a p o l a t e t h e energy values t o "+a.

I* I

(a) +

I ,

;,+' -

5 3 9

-

'/

c +,' /

/+' /

/+

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^5,60-

+.J+

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r *oo-mn-ram . , N . . - - - " 7 - -

.,.. . - - -

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2

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w . r Z ' - , ³ s s m u -

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a m . 7 z

I s . - c , u m w r E m . - c l o

n

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nn nnn on n n n - - n w o --n*,.-,,.., .s

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d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d

C2-142 JOURNAL DE PHYSIQUE

g i t s given were a l r e a d y obtained f o r N, components. For i n t e r m e d i a t e and s t r o n g f i e l d s , t h e s i n g l e s u b s c r i p t i n d i c a t e s t h e maximum number o f c o n f i g u r a t i o n s used i n t h e computations i n ( 9 ) o r ( l o ) , and t h e number f o l l o w i n g t h e s l a s h gives t h e values towards which t h e l a s t d i g i t s tend i f use i s made o f t h e e x t r a p o l a t i o n method discussed i n Fig. 1. The s o l i d l i n e s below c e r t a i n energy values mark t h e boundaries from whereon t h e Landau expansion produces b e t t e r r e s u l t s than t h e expansion i n terms o f s p h e r i c a l harmonics. The mesh o f p - p o i n t s i n Table 1 i s chosen i n such a way t h a t i n a q u a d r a t i c i n t e r p o l a t i o n f o r i n t e r m e d i a t e p-values one t o two s i g n i - f i c a n t d i g i t s a t most a r e l o s t .

Comparing w i t h t h e b e s t values f o r t h e energies o f these s t a t e s known t o d a t e (Prad- daude [ 2 2 ]

,

Cabib e t a l . [ 2 3 ] f o r p(2.5, Simola and Virtamo

14)

f o r p > / 2 . 5 ) , i t i s found t h a t our r e s u l t s e i t h e r reproduce t h e i r values, o r even l i e s l i g h t l y lower on account of t h e i r h i g h e r number o f s i g n i f i c a n t d i g i t s . I t i s e v i d e n t from Table 1 t h a t t h e number o f s i g n i f i c a n t d i g i t s o b t a i n a b l e i n t h e present c a l c u l a t i o n s i s l a r - gest i n t h e l o w - f i e l d regime ( p < 1); t h i s i s o f course due t o t h e f a c t t h a t here t h e diamagnetic term i n t h e Hamiltonian represents o n l y a small p e r t u r b a t i o n t o t h e f i e l d - f r e e s p h e r i c a l problem. I n p a r t i c u l a r , i t can be v e r i f i e d from Table 1 t h a t , as p o i n t e d o u t by Ruder e t a l . [ 241

,

t h e simple formulae o f a standard p e r t u r b a t i o n c a l c u l a t i o n ,

where n designates t h e p r i n c i p a l quantum number, aB i s t h e Bohr r a d i u s and P

w i t h

are capable of producin t h e c o r r e c t energy values w i t h a remarkable accuracy up t o magnetic f i e l d s p

- l o 8 - l o 2

f o r t h e s t a t e s under consideration. I n t h e regime o f i n t e r m e d i a t e f i e l d strengths, 0.1 6

P 4

10, where t h e dominance o f t h e Coulomb po- t e n t i a l i s g r a d u a l l y superseded by t h a t of t h e magnetic f i e l d , the number o f necess- a r y angular momentum components r a p i d l y increases. I t i s

,

however, o n l y f o r p21-10 t h a t t h e Landau expansions y i e l d more accurate r e s u l t s u s i n g f e w e r c o n f i g u r a t i o n s than would be r e q u i r e d i n t h e corresponding angular momentum expansion.

For i l l u s t r a t i o n a l purposes, i n F i g . 2 t h e energies o f several l o w - l y i n g s t a t e s w i t h m = 0, -1, -2 a r e p l o t t e d as continuous f u n c t i o n s o f t h e magnetic f i e l d s t r e n g t h . Fig. 2 e x h i b i t s t h e w e l l known f a c t t h a t f o r e v e r y value o f m t h e r e e x i s t s one (asym- p t o t i c a l l y node-less) s t a t e which i s s t r o n g l y lowered i n energy ( " t i g h t l y bound s t a t e s " ) , w h i l e t h e energies o f a l l o t h e r s t a t e s ("hydrogen-like s t a t e s " ) a r e l e s s than one Rydberg, and i n the l i m i t p - co converge t o t h e f i e l d - f r e e eigenvalue spec- trum o f t h e H atom (Loudon [ 2 5 ] ) . I n

fat:,

w r i t i n g t h e energy values f o r f i n i t e

P

and f i x e d m i n t h e f o r m E, = -EH/( ^ij

+

^&), where B = u/2 f o r even v and ^ij =(u+1)/2 f o r odd v

,

i t i s found t h a t t h e ( f o r m a l ) quantum d e f e c t 6,- r a p i d l y tends t o some constant value f o r even and odd u

,

r e s p e c t i v e l y , as u goes t o i n f i n i t y . This f a c t can be used t o c a l c u l a t e w i t h s a t i s f a c t o r y accuracy, f o r given p and m, t h e energies o f a l l h i g h l y e x c i t e d bound s t a t e s o f t h e H atom. I t may be noted, however, t h a t t h e k n o G d g e o f t h e energy values f o r u

>>

1 i s more o r l e s s o f academic i n t e r e s t , s i n c e t h e widths o f these l e v e l s due t o t h e i n t e r a c t i o n w i t h t h e r a d i a t i o n f i e l d , o r w i t h o t h e r atoms, e a s i l y become comparable w i t h t h e energies themselves.

Because o f t h e e x t r a o r d i n a r y accuracy o f t h e i r energy values, t h e wave f u n c t i o n s (9) and (10) a r e p a r t i c u l a r l y w e l l - s u i t e d f o r c a l c u l a t i n g m a t r i x elements and e x p e c t a t i o n

Figure 2.- Energies of low-lying m=O, -1, -2 states of the H atom in units of the Rydberg energy as a function of the magnetic field parameter

p .

The states are labelled by their asymptotic field-free and adiabatic approximation quantum numbers.

E upper continuum

2

Figure

3.-

Magnetic-field de-

+ m _ c

oendence of the qround state

r

energy of hydrogenic uranium

(non-relativistic and relativi-

5 0 0

stic calculation). At

B z

, I l l

4.5.10" T the ground state

B in l o 9 T

energy reaches the lower con- tinuum, which may result in the spontaneous production

-m c

of positrons.

- - -

⁴

m - 0 n = -1

Table 2.- Mean square radius and value

of the wave function at the brigin of

B ^<rz /a;> a a ; l p i o ~ I' <r2 /a:>

the lowest m=O state, and mean square radius of the lowest m=-1 state of the H atom as a function of the magnetic field strength. (For m=-1,$(0)=0 for a1

l

values of P .

⁾

The subscripts indicate the number of configurations and the method used in computing the values ("a": eq.

(9),

" b" :eq. (10). For (-1) read 10" , ^etc.

;

a, denotes the Bohr radius.

C2-144 JOURNAL DE PHYSIQUE

values, which a r e more sensitive t o approximation methods than are energies. Table 2 provides, f o r various magnetic f i e l d s streng.ths in the range 0

< p <

l o 3 , the mean square r a d i i of the lowest m = 0 and

m

= -1 s t a t e s of the H atom, and, f o r m = 0, a l s o the value of the wave function a t the o r i g i n , calculated using our multi-confi- guration wave functions (9) and (10). The subscripts attached t o the r e s u l t s indi- cate the type of expansion ("a" f o r ( 9 ) , "b" f o r (10))and the number of configura- t i o n s necessary t o guarantee the numerical s t a b i l i t y of the r e s u l t s with respect t o a l l the d i g i t s given. The dramatic decline of t h e mean square radius, and

the

strong

increase in the value of the

m

= 0 wave function

a t

the nucleus, which a r e evident from t h e tabulated r e s u l t s , a r e altogether c l e a r l y indicative of the t o t a l rear- rangement of atomic s t r u c t u r e which occurs a s t h e magnetic f i e l d increases beyond some c r i t i c a l value of

p .

Concluding t h i s section we note t h a t the solution of the Dirac equation f o r the H atom in strong magnetic f i e l d s (Lindgren and Virtamo 1261) reveals t h a t r e l a t i v i s t i c e f f e c t s a r e negligible even a t magnetic f i e l d strengths & - ~ ~ ~ = 4 . 4 1 . 1 0 ~ ~ ,where the level distance in the Landau spectrum becomes equal t o the r e s t mass of t h e electron.

This may be explained by the f a c t t h a t even in the r e l a t i v i s t i c treatment the spa- t i a l dependence of the spinors as regards the motion perpendicular t o the magnetic f i e l d i s given by the Landau functions Lan

( 2 ) ,

and thus f o r the motion paprallel t o the f i e l d the electron ultimately f e e l s t h e same e f f e c t i v e potentials as in the n o n - r e l a t i v i s t i c treatment. R e l a t i v i s t i c corrections a r e therefore only of the order of the r a t i o of t h e Coulomb binding energies (a few hundreds eV f o r the t i g h t - l y bound s t a t e s ) and t h e electronic r e s t mass, and thus negligible. However, r e l a - t i v i s t i c e f f e c t s can no longer be neglected f o r higher nuclear charges, where, by the Z-scaling law ( 7 b ) , f o r t i g h t l y bound s t a t e s binding energies of hundreds of keV may be obtained. For hydrogenic uranium ( Z = 92) i n Fig. 3 ) the r e l a t i v i s t i c energy of the ground s t a t e i s plotted as a function of the magnetic f i e l d strength.

I t i s seen t h a t f o r B 2

lo9

T r e l a t i v i s t i c corrections indeed become sizeable, and f o r B 4.8 x 10" T the ground s t a t e plunges i n t o t h e f i l l e d Dirac sea

,

which entail s the spontaneous production of positrons [ 271 (decay of the vaccuum)

.

^Magne-

t i c f i e l d strengths of t h i s magnitude a r e expected

t o

occur f o r very short times in heavy-ion c o l l i s i o n s [28].

3. Electromagnetic t r a n s i t i o n s of hydrogenic ions in strong magnetic f i e l d s

a. Bound-bound t r a n s i t i o n s

The calculation of electromagnetic t r a n s i t i o n s i s a standard chapter of quantum mechanics, the e s s e n t i a l formula being the famous "golden rule" ( s e e , e.g., Heitler [29]). However the resulting f i n a l expressions f o r f i e l d - f r e e t r a n s i t i o n s which may be found in atomic physics l i t e r a t u r e cannot simply be applied t o atoms in strong magnetic f i e l d s . This i s mainly because of the completely altered s t r u c t u r e of atomic wave functions. Thus the recalculation of the cross sections and t r a n s i - tion p r o b a b i l i t i e s of a1 1 relevant processes i s r e q u i s i t e .

Bound-bound t r a n s i t i o n s of hydrogenic ions in strong magnetic f i e l d s have been d i s - cussed in considerable d e t a i l by Wunner and Ruder and Wunner e t a l . [8]within the framework of the "adiabatic approximation". This approximation amounts t o including in t h e Landau expansion ( 9 ) s o l e l y the contribution of

n

⁼ 0, which i s j u s t i f i e d when the Landau level distance 4P EH i s large compared with the Coulomb binding energies, i . e . f o r p>>1. In p a r t i c u l a r , the adiabatic approximation becomes exact in the l i m i t

p -

c o . The i n t u i t i v e meaning of the adiabatic approximation i s t h a t in strong magnetic f i e l d s the motion of t h e electron perpendicular t o the magnetic f i e l d in (undisturbed) Landau o r b i t s i s much f a s t e r than t h a t parallel t o the f i e l d , whence f o r t h i s motion the electron e f f e c t i v e l y f e e l s the Coulomb potential aver- aged oher the Landau o r b i t . In f a c t , t h e longitudinal p a r t s of the adiabatic

I o6

1 0' 1

oa

^10' 10'' 1

o1$ 1 o f 2

^{1 0 13 B i n G}

,

1 0 ⁿ

zoni

^v 8 8 z 8 r t t i t t 8 n m j - 3 ~ + ~ r r r r n . t 7 7 ~ ~ ~ m n r - r - P E R T U R B A T I O N A D I A B A T I C A P P R O X I M A T I O N

7

^{l o}

1 o8 1

o9

B i n G

F i g u r e 4.- 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 H atom as a f u n c t i o n o f t h e magnetic f i e l d strength. Top: r e s u l t s obtained u s i n g p e r t u r b a t i o n theory and a d i a b a t i c approxi- mation wave f u n c t i o n s , crosses r e f e r t o refs.[7,30,31]. Bottom: r e s u l t s f o r i n t e r - mediate f i e l d s t r e n g t h s obtained u s i n g m u l t i - c o n f i g u r a t i o n wave f u n c t i o n s (9) and (10).

C2-146 JOURNAL DE PHYSIQUE

approximation wave f u n c t i o n s a r e obtained by s o l v i n g one-dimensional Schrodinger equations w i t h t h e e f f e c t i v e p o t e n t i a l s V::)(Z) (see (12)). The comparison o f t h e energy values c a l c u l a t e d i n a d i a b a t i c approximation w i t h t h e h i g h l y accurate ones determined w i t h t h e h e l p o f t h e mu1 t i - c o n f i g u r a t i o n wave f u n c t i o n s (10) shows t h a t f o r t h e t i g h t l y bound s t a t e s t h e d e v i a t i o n s a r e about 1% f o r

p

= 50, and about 0.2%

f o r b =

l o 3 .

w h i l e t h e enerqies o f hydroqen-like s t a t e s a r e even more accurate by -

-

a f a c t o r o f 5 t o 10.

I n F i g . 4a we have p l o t t e d t h e o s c i l l a t o r strengths fT(2) o f (om = q ) - t r a n s i t i o n s between l o w - l y i n g s t a t e s

,

r 1 o f t h e H atom as a f u n c t i o n o f t h e magnetic f i e l d strength. The values f o r p

>

1 have been c a l c u l a t e d using a d i a b a t i c approximation wave f u n c t i o n s , those f o r

p < l o - *

u s i n g o r d i n a r y Schrodinger o e r t u r b a t i o n t h e o r y wave f u n c t i o n s . Because o f t h e stronger s e n s i t i v i t y o f m a t r i x elements t o approxi- mation methods, t h e a d i a b a t i c approximation curves have been dashed f o r p

<

50, where t h e accuracy i s expected t o become worse than-5%. A d d i t i o n a l l y we have i n - cluded i n Fig. 5a t h e few values f o r the o s c i l l a t o r s t r e n g t h s e x i s t i n g i n t h e l i t e - r a t u r e f o r B

3 l o 3

T (Smith e t a1. [30]; Brandi, Santos, and Miranda[31] ; Kara and McDowell [ 7 ] ) . I t i s seen t h a t i n t h e r e g i o n o f small

P

t h e o s c i l l a t o r s t r e n g t h s o f those t r a n s i t i o n s which a l s o occur a t

p

= 0 d i f f e r o n l y s l i g h t l y from t h e f i e l d - f r e e values, w h i l e t h e o s c i l l a t o r s t r e n g t h s o f t h e o t h e r t r a n s i t i o n s increase w i t h a simple power law i n p according t o t h e removal o f t h e energy degeneracy o f t h e r e s p e c t i v e states. I n t h e h i g h - f i e l d r e g i o n a l l o s c i l l a t o r s t r e n g t h s decrease due t o t h e s h r i n k i n g s p a t i a l extension o f t h e atomic wave f u n c t i o n s . The decrease i n Am

+

0 t r a n s i t i o n s i s more pronounced since here t h e extension perpendicular t o t h e magnetic f i e l d i s e s s e n t i a l , and a s y m p t o t i c a l l y these o s c i l l a t o r s t r e n g t h s behave as I / p . I n order t o b r i d g e t h e gap i n Fig. 4a where n e i t h e r u e r t u r b a t i o n theory n o r t h e a d i a b a t i c approximation i s a p o l i c a b l e , we used m u l t i - c o n f i g u r a t i o n wave f u n c t i o n ( 9 ) and (10) t o c a l c u l a t e t h e r e l e v a n t m a t r i x elements. The r e s u l t s obtained [32]

f o r t h e o s c i l l a t o r strengths i n t h e range

l o 3

G < T ( 5 x

l o 6

T a r e shown i n F i g . 4b.

From t h e convergence o f t h e r e s u l t s w i t h i n c r e a s i n g maximum number o f c o n f i g u r a t i o n s i t may be concluded t h a t a t l e a s t t h r e e s i g n i f i c a n t d i g i t s can be guaranteed f o r t h e values o f t h e o s c i l l a t o r s t r e n g t h s over t h e whole range o f B shown i n F i g . 4b. Figs.

4a and 4b c l e a r l y e x h i b i t t h e t o t a l rearrangement, n o t o n l y o f l e v e l s t r u c t u r e , b u t a l s o o f emission spectra t h a t takes place when p becomes o f order u n i t y .

With t h e h e l p o f t h e s c a l i n g laws (7b) and (8), t h e r e s u l t s o f Fig.4 can immediately be t r a n s f e r r e d t o hydrogenic ions. With regard t o a p p l i c a t i o n s t o s t r o n g l y magne- t i z e d a c c r e t i ng neutron s t a r s w i t h source temperatures kT- 10 keV, hydrogen-1 i ke i r o n (Fe XXVI) i s o f p a r t i c u l a r i n t e r e s t . A c a r e f u l estimate o f t h e r e l e v a n t physi- c a l parameters has r e c e n t l y l e d Ruder e t a l . [ 3 3 ] t o t h e p r e d i c t i o n t h a t photon f l u x e s i n t h e Lyman-a l i n e which should be observable w i t h spectrometers a t present o r i n t h e near f u t u r e can be expected from a c c r e t i n g neutron s t a r s o f n o t t o o h i g h X-ray l u m i n o s i t y . The fundamental importance o f t h e a c t u a l observation o f magneti- c a l l y s t r o n g l y s h i f t e d i r o n l i n e s t o t h e physics o f neutron s t a r s as w e l l as t o atomic physics i n s t r o n g f i e l d s i s evident,

b. Bound-free t r a n s i t i o n s

To a r r i v e a t t h e general i o n i s a t i o n formula o f a plasma t h e accurate knowledge o f t h e cross s e c t i o n s o f t h e f o l l o w i n g elementary bound-free processes a r e a necessary i n p u t from atomic physics: 1 ) p h o t o i o n i s a t i o n and photorecombination, 2) c o l l i s i o n a l i o n i s a t i o n and three-body recombination. Once t h e cross s e c t i o n s a r e known i t can be decided under which physical c o n d i t i o n s t h e e l e c t r o n i c , o r uhotonic, i o n i s a t i o n - recombination mechanism c o n t r i b u t e s dominantly t o e s t a b l i s h i n g t h e i o n i s a t i o n eaui- l i b r i a . The knowledge o f t h e i o n i s a t i o n e q u i l i b r i a , i n t u r n , i s p r e r e q u i s i t e t o q u a n t i t a t i v e l y c a l c u l a t i n g atomic l i n e spectra ( b o t h i n emission and absorption) from cosmic X-ray sources. We now r e p o r t t h e r e s u l t s o f c a l c u l a t i o n s f o r p h o t o i o n i - s a t i o n s [34] and impact i o n i s a t i o n [35] performed f o r magnetic f i e l d s B

- l o 7

-109T w i t h i n t h e a d i a b a t i c approximation u s i n g accurate numerical l o n g i t u d i n a l wave func- t i o n s f o r s t a t e s both i n t h e d i s c r e t e and t h e continuous spectrum.

Fig. 5a shows t h e computed p h o t o i o n i s a t i o n cross s e c t i o n o f t h e

H

atom f o r B = 4.7 x 1 0 * ~ f o r photons i n c i d e n t perpendicular, and w i t h p o l a r i s a t i o n p a r a l l e l , t o the magnetic f i e l d . The cross s e c t i o n e x h i b i t s

-

c h a r a c t e r i s t i c of t h e presence of a magnetic f i e l d

-

spikes a t photon energies which correspond t o Landau e x c i t a t i o n s .

B

0

5 5 109 163

4.10~ 8 . 1 0 ~ 12.10~ lh104 2.10~ 2;.104 E in ERy,

0.3 2 18 2 72 326 h w in KeV

Figure 5a.-Total photoionisation cross section of the H atom for

8=4.70.10"

T as a function of the photon energy (or the energy distance from the ionisation threshold) for photons incident perpendicular, and polarisation parallel, to the field (lower curve:

B=O).

The spikes in the cross section correspond to the exci- tation of the electron into higher Landau levels.

Figure

5b.-

Same as

Fig.

5a for hydrogenic iron.

-A

"gl""

& 1

B = 6.7-lof2 G , 3=9O0. g l h

6 . 6 .

Fe

XB!7

2

-

0 - - 2

-

-6

.

-6.

-8

+

^P

0 2.103 8.103 12.104 16-106 2.106 E i n E R y ,

20 74 12 8 183 23 7 292 Rw in KeV

C2- 148 JOURNAL DE PHYSIQUE

The comparison w i t h t h e f i e l d - f r e e r e s u l t shows t h a t i n a s t r o n g magnetic f i e l d t h e cross s e c t i o n i s increased by several orders o f magnitude. From F i g . 5b, where t h e same s i t u a t i o n i s presented f o r Fe XXVI, i t can be seen t h a t a l s o f o r l a r g e r n u c l e a r charges jumps a r e present a t the Landau e x c i t a t i o n s ; however, f o r a given magnetic f i e l d s t r e n g t h t h e d i f f e r e n c e w i t h respect t o t h e f i e l d - f r e e cross s e c t i o n i s l e s s pronounced, caused by t h e stronger Coulomb f o r c e s .

Fig. 6 shows the cross s e c t i o n f o r impact i o n i s a t i o n o f t h e H atom as a f u n c t i o n o f t h e energy o f t h e incoming e l e c t r o n f o r

P

= 50 and p =

lo3.

The i o n i s a t i o n t h r e s - holds are marked by v e r t i c a l dashed 1 ines. Comparing w i t h t h e f i e l d - f r e e r e s u l t

( P

= 0) we see t h a t t h e c r o s s s e c t i o n i s decreased as t h e magnetic f i e l d increases.

The c a l c u l a t i o n s have n o t y e t been extended beyond t h e f i r s t Landau t h r e s h o l d where again spikes i n t h e cross s e c t i o n w i l l occur.

As f a r as t h e time-reversed processes a r e concerned i t must be noted t h a t , as a con- sequence o f t h e d i f f e r e n t d i m e n s i o n a l i t y o f t h e d e n s i t y o f f i n a l s t a t e s o f t h e e l e c - t r o n (three-dimensional f o r B = 0, one-dimensional i n t h e s t r o n g l y magnetised case), t h e f i e l d - f r e e t r a n s f o r m a t i o n laws can no l o n g e r be applied. For example, t h e r e l a - t i o n between t h e cross s e c t i o n o f photorecombination o f an e l e c t r o n w i t h Landau quantum number n ' = 0, and k i n e t i c energy E ' > 0 t o a bound s t a t e w i t h n = 0, and energy Ec 0

,

accompanied by t h e emission o f a photon o f wave v e c t o r (.Kkc=E1+

JEI

) and t h e corresponding i o n i s a t i o n cross s e c t i o n i s obtained as

d o r e c - ( k a L ) * i o n

- -

-

d n 4 n O t o t a l

where aL = (ZK/~B)"* denotes Larmor quantum length, and a s i m i l a r law can be d e r i v e d f o r three-body recombination.

-

The question as t o t h e predominance o f one o f t h e two i o n i sation-recombination mechanisms depends, of course, on t h e p a r t i c u l a r physi- c a l s i t u a t i o n ( e l e c t r o n d e n s i t y , photon d e n s i t y , temperature, etc.), and w i l l n o t be pursued here.

c. Free-free t r a n s i o n s (bremsstrahlung

The bremsstrahlunq process i s t h e dominant mechanism f o r t h e ~ r o d u c t i o n o f t h e y-quanta emitted-from t h e a c c r e t i o n columns o f neutron s t a r s : The elementary brems- s t r a h l u n g cross s e c t i o n t h e r e f o r e e n t e r s as a l o c a l source term i n t o every s e l f - c o n s i s t e n t r a d i a t i v e t r a n s f e r c a l c u l a t i o n which aims a t t h e c o r r e c t d e s c r i p t i o n o f the e m i t t e d continuous X-ray spectrum.,Starting p o i n t o f our own numerical compu- t a t i o n s [36] f o r magnetic f i e l d s B-10

-

10' T were t h e attempts undertaken pre- v i o u s l y by Canuto e t a l . [37]

,

Virtamo and Jauho [38]

,

and Akopyan and T s y t o v i c h [39]

.

Our r e s u l t s were obtained f o r e l e c t r o n s moving f r e e l y p a r a l l e l t o t h e magnetic f i e l d (Born's approximation), and i n Landau o r b i t s perpendicular t o t h e f i e l d . Furthermore i t was assumed t h a t t h e e l e c t r o n s occupy t h e lowest Landau l e v e l (n = n r = 0) b o t h i n t h e i n i t i a l and t h e f i n a l s t a t e . For a magnetic f i e l d o f 4.4 x 10' T F i g . 7 shows, i n an exemplary way, t h e d i f f e r e n t i a l cross s e c t i o n per energy i n t e r v a l o f t h e e m i t t e d photon f o r t h e energy Ee of t h e e l e c t r o n i n t h e i n i t i a l s t a t e o f

10%,

o r 90%, o f t h e Landau l e v e l d i s t a n c e

'fiwg,

r e s p e c t i v e l y . F o r every v a l u e o f Ee 4 curves are shown, corresponding t o t h e 2 p o l a r i s a t i o n s o f t h e hoto on, and t o t h e 2 possi- b i l i t i e s o f whether o r n o t t h e d i r e c t i o n o f t h e z-momentum o f t h e e l e c t r o n i s r e - versed. I t i s seen t h a t i n general t h e backward process i s suppressed w i t h respect t o t h e forward process; furthermore, f o r small e l e c t r o n energies p o l a r i s a t i o n o f t h e photon i n t h e k-B-plane i s predominant, w h i l e f o r e l e c t r o n energies c l o s e r t o t h e Landau threshoTd-pol a r i s a t i o n perpendicular t o t h e k B - p l ane becomes more and more favourable as t h e photon energy increases. F o l d i n g the cross s e c t i o n w i t h a given e l e c t r o n d i s t r i b u t i o n f u n c t i o n leads t o t h e photon spectrum produced by t h e elemen- t a r y bremsstrahlung process. It must be stressed, however, t h a t , i n an o p t i c a l l y t h i c k plasma, t h e u l t i m a t e l y e m i t t e d spectrum may s t i l l be reprocessed by t h e i n t e r - a c t i o n o f t h e bremsstrahlung quanta w i t h o t h e r e l e c t r o n s (Comptonisation).

4. Two-electron systems i n s t r o n q maqnetic f i e l d s

The s t a r t i n g p o i n t f o r t r e a t i n g h i g h e r atoms i n s t r o n g magnetic f i e l d s i s t h e l e v e l

Figure

6.-

Total cross section for impact ionisation of the H atom in strong magnetic fields as a func- tion of the energy of the incoming electron. For comparison, the field-free cross section is also shown. The dashed lines mark the ionisation thresholds.

Fisure 7.- Energy distribution of the bremsstrahlung quanta in a mag- netic field ~ = 4 . 4 . 1 0 ~

T

for two va- lues of the energy Ee of the elec- tron in the initial state (wB=eB/me).

On account of the one-dimensionality enforced by the magnetic field, in the process only the component of the electron momentum para1 lel to the field is altered (solid lines:

no change of direction, dashed lines:

change of direction; 1: polarisa- tion of the photon in the k-B plane, 2: polarisation perpendicuTaF to the k-8 plane).

-

-

o 02 a& a6 08

w/w,

l o - ' 1 oO 1 o f l P~ 30+3

Figure

8.-

Ground state

0 - - l o

- 5 0

E?EH

- 1 0 0 . -

I I 1 1 1 l l l r n 1 1 1 1 1 1 1 1 1 I 1 1 1 1 1 1

-

- - - .

energy of He as a func-

1

^MCHF

^- _ tion of the magnetic

field parameter bZ. So-

-

-

lid curve: SCHF calcula-

--

tion in adiabatic appro-

-

ximation 1401, dashed

-

curve: spherical MCHF

-

calculation (ls2p-,

3 ~ - 1

-

+ ls4f-,

3F_, )

with in-

- clusion of the diamag-

--I--

^{I I I I}

netic term [42].

C2- 150 JOURNAL DE PHYSIQUE

scheme o f t h e H atom, where, i n a s i n g l e - p a r t i c l e p i c t u r e , t h e lowest s t a t e s (m = 0,-1, ...) a r e c o n s e c u t i v e l y occupied by t h e atomic e l e c t r o n s . E v i d e n t l y t h e ca-

n o n i c a l approach t o t h e problem i s a Hartree-Fock (HF) c a l c u l a t i o n i n which f o r every s i n g l e - p a r t i c l e wave f u n c t i o n an expansion (10) i n terms o f Landau o r b i t a l s i s taken, and t h e z-dependent expansion f u n c t i o n s a r e determined s e l f c o n s i s t e n t l y i n t h e course o f a MCHF procedure. A s i m p l e r s i t u a t i o n a r i s e s i f t h e c o n d i t i o n s f o r t h e a d i a b a t i c approximation (B>B,, _{. .} i.e. D, = B / B , > l ) a r e f u l f i l l e d ; then t h e

L I. L

s i n g l e - p a r t i c l e wave f u n c t i o n s can reasonably be described by products o f Landau s t a t e s and l o n g i t u d i n a l f u n c t i o n s ,

L a n

4 i ( i ) = on.=o,m (1~1 ) ' g m . u ( ' i ) '

1 i i i

and t h e many-el e c t r o n wave f u n c t i o n i s represented by a s i n g l e S l a t e r determinant.

The general Hartree-Fock equations i n a d i a b a t i c approximation, t o g e t h e r w i t h a p p l i - c a t i o n s t o t h e i s o e l e c t r o n i c sequence o f He (up t o Fe XXV), have r e c e n t l y been t r e a t e d i n d e t a i l by Proschel e t a1. [40], who employed an adopted v e r s i o n [ 2 0 ] o f t h e successful Froese-Fischer code [ 1 9 ] f o r s o l v i n g t h e HF equations. As one example, t h e m a g n e t i c - f i e l d dependence obtained f o r t h e ground s t a t e energy o f He i s shown i n F i g . 8 ( s o l i d curve f o r PZ>l); t h e energy-lowering e f f e c t o f s t r o n g magnetic f i e l d s on ground s t a t e s i s e v i d e n t again. Note t h a t , i n ~ a r t i c u l a r , th e lowest s t a t e becomes a t r i p l e t s t a t e . Fig. 8 a l s o contains (dashed curve) t h e r e s u l t s o f MCHF c a l c u l a t i o n s i n a s p h e r i c a l b a s i s which s e l f c o n s i s t e n t l y took i n t o account t h e d i a - magnetic Hamiltonian [42] ; i t i s found t h a t t h i s procedure operates remarkably w e l l even up t o PZ-20, and, f o r P

z

> 1 , l i e s below t h e a d i a b a t i c approximation r e s u l t s . T h i s i s o f course due t o t h e f a c t t h a t a s i n g l e S l a t e r determinant cannot account f o r c o r r e l a t i o n s between the e l e c t r o n s i n c l u d e d i n MCHF wave f u n c t i o n s .

Obviously t h e n e x t step w i l l be t o c a l c u l a t e energies and wave f u n c t i o n s o f e x c i t e d s t a t e s o f h e l i u m - l i k e systems i n s t r o n g magnetic f i e l d s w i t h i n Hartree-Fock theory, and, then, t o evaluate t h e m a t r i x elements o f electromagnetic t r a n s i t i o n s between these states. Furthermore a q u a n t i t a t i v e i n v e s t i g a t i o n a l s o o f l i t h i u m - l i k e systems i n s t r o n g magnetic f i e l d s w i l l become n u m e r i c a l l y f e a s i b l e . I n a d d i t i o n , t h e exten- s i o n o f t h e method t o multi-Landau-configuration forms o f t h e N-electron wave func- t i o n can be envisaged.

5. The H: i o n i n s t r o n g magnetic f i e l d s

As a f i n a l p o i n t , + l e t us b r i e f l y discuss t h e behaviour o f t h e simplest m o l e p l a r s t r u c t u r e

-

t h e H, i o n

-

i n s t r o n g magnetic f i e l d s . I n v e s t i g a t i o n s o f t h e H,system i n t h e h i g h - f i e l d regime are o f p a r t i c u l a r i n t e r e s t as they can serve as a u s e f u l guide t o studying q u a n t i t a t i v e l y l i n e a r polyatomic chains i n superstrong magnetic f i e l d s , which are expected t o p l a y an important r o l e f o r t h e s o l i d s t a t e s t r u c t u r e o f neutron s t a r surfaces. References t o t h e e x i s t i n g l i t e r a t u r e may be found i n Wunner e t a l . [43], where energy values, e q u i l i b r i u m i n t e r n u c l e a r separations, and z e r o - p o i n t energies o f n u c l e a r v i b r a t i o n s p a r a l l e l t o t h e f i e l d were c a l c u l a t e d f o r t h e l o w e s t s t a t e s u s i n g t h e a d i a b a t i c approximatjon.

I t i s e v i d e n t t h a t a c o n s i s t e n t d e s c r i p t i o n o f t h e Hp i o n from f i r s t p r i n c i p l e s would r e q u i r e t h e s o l u t i o n o f t h e quantum-mechanical three-body problem i n the pre- sence o f a magnetic f i e l d . Apart from t h e formidable d i f f i c u l t i e s encountered a l - ready i n t h e f i e l d - f r e e case, t h i s t a s k i s a d d i t i o n a l l y complicated by t h e f a c t t h a t f o r char ed systems i n magnetic f i e l d s t h e t o t a l momentum i s no l o n g e r a conserved g u a n t i t y j 4 4 ] . Lacking a s a t i s f a c t o r y s o l u t i o n t o t h i s problem, most o f t h e authors have considered t h e e l e c t r o n moving i n t h e Coulomb f i e l d s o f two protons l o c a t e d a t f i x e d p o s i t i o n s zA = -R/2, zg = +R/2 on t h e z-axis, taken t o p o i n t along t h e d i r e c - t i o n o f t h e magnetic f i e l d .

As an+example Fig. 9 shows the energy curves o f t h e lowest m = 0 and m = - l ! s t a t e of

t h e H2 i o n f o r v a r i o u s values o f t h e magnetic f i e l d parameter B/4.7 x I O ~ T

.

I t i s seen t h a t , as t h e magnetic f i e l d increases, t h e energy curves become s t i f f e r and s t i f f e r , t h e e q u i l i b r i u m i n t e r n u c l e a r d i s t a n c e i s s h i f t e d t o smaller values, and t h e energy a t minimum i s s t r o n g l y lowered. I n p a r t i c u l a r , as an e f f e c t o f t h e