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>
HAL Id: jpa-00221821
https://hal.archives-ouvertes.fr/jpa-00221821
Submitted on 1 Jan 1982
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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
Erlangen,
Lehrstuhl
Tiïbingen,
fttr Theoretisehe
F.R.G.
fur Theoretisehe
F.R.G.
Physik,
Universitât
Astrophysvk,
Erlangen-Nù'rnberg,
Universitât
Tubingen,
D-8520
D-7400
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 électromagnétiques), 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 Coulomb 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 freefree 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 10 2 - 10 5 T have been
observed in white dwarfs (cf. Angel et al.El]), and of the order of 10 7 - 1 0 9 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 hydrogenlike and helium-like ions. Therefore, the accurate knowledge of the atomic properties 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 forces. 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 presence 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, def 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-):
s),
where R = (m-r-+ m+r )/(m-+m ), r = ( C and t h e v e c t o r o o t e n t i a l has been
.
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,
chosen-in t h e form ~f!) = l / $ ( ~ x ~ )Txe
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 2 / ( 4 r s o r ) i s t h e Coulomb
interaction.
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 closer inspection o f the structure o f the
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 cont 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 probabi1 i t 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
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
me/mproton'
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
the dipole strengths
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 .T1 .0Z~~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
oL:l(r,)
terms o f Landau s t a t e s
(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
Lan
am= xgn(z)@nm (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)
2
) IZe / ( 4 ~ ~ ~ r ) l(5
@ I>,
~ ? ~
= <o:~(>
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]
.
"F
0,~
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II~,
I
,
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-5 3 9-
I
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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 recogn 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 1y 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.
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
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
~
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
range 1 0 - ~ < p < 1 0 (P=Pl=
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 numbers. 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 val 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 -
\~.....-.-.,\,..
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2
2
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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 (Praddaude [ 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
calculation,
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
with
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 pot 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 necessa 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 (asymp 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 spectrum 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
+m_c
r
500
,
I
l
l
B in lo9T
-m c
--n = -1
m - 0
B
<rz/a;>
aa;lpio~
I'
<r2/a:>
Figure 3.- Magnetic-field deoendence of the qround state
energy of hydrogenic uranium
(non-relativistic and relativistic calculation). At B z
4.5.10" T the ground state
energy reaches the lower continuum, which may result in
the spontaneous production
of positrons.
4
Table 2.- Mean square radius and value
of the wave function at the brigin of
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.
JOURNAL DE PHYSIQUE
C2-144
values, which a r e more sensitive to 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 radii of the lowest m = 0 and m = -1 s t a t e s of the H atom, and, f o r m = 0,
also the value of the wave function a t the origin, calculated using our multi-configuration wave functions (9) and (10). The subscripts attached to the r e s u l t s indicate the type of expansion ("a" f o r ( 9 ) , "b" f o r (10))and the number of configurations 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 the 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 the tabulated results, are altogether clearly indicative of the t o t a l rearrangement of atomic structure which occurs as the 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
. 1 0 ~ ~the
e f f e c t s are negligible even a t magnetic f i e l d strengths & - ~ ~ ~ = 4 . 4 1 ,where
level distance in the Landau spectrum becomes equal t o the r e s t mass of the electron.
This may be explained by the f a c t that even in the r e l a t i v i s t i c treatment the spat i a l dependence of the spinors as regards the motion perpendicular to the magnetic
Lan
f i e l d i s given by the Landau functions
( 2 ) , and thus f o r the motion paprallel
to the f i e l d the electron ultimately f e e l s the same effective potentials as in the
non-relativistic treatment. Relativistic corrections are therefore only of the
order of the r a t i o of the Coulomb binding energies (a few hundreds eV f o r the tightly bound s t a t e s ) and the electronic r e s t mass, and thus negligible. However, relat 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 (7b), 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) in 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 into the f i l l e d Dirac sea , which
entail s the spontaneous production of positrons [ 271 (decay of the vaccuum) . Magnet i c f i e l d strengths of t h i s magnitude are expected t o occur f o r very short times
in heavy-ion collisions [28].
3. Electromagnetic transitions of hydrogenic ions in strong magnetic f i e l d s
a. Bound-bound transitions
The calculation of electromagnetic transitions i s a standard chapter of quantum
mechanics, the essential formula being the famous "golden rule" (see, e.g., Heitler
[29]). However the resulting final expressions f o r field-free transitions 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 structure
of atomic wave functions. Thus the recalculation of the cross sections and transition probabilities of a1 1 relevant processes i s requisite.
Bound-bound transitions of hydrogenic ions in strong magnetic f i e l d s have been discussed in considerable detail 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 the Landau expansion ( 9 ) solely 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 particular, the adiabatic approximation becomes exact
in the limit p - c o . The intuitive meaning of the adiabatic approximation i s that
in strong magnetic f i e l d s the motion of the 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 effectively f e e l s the Coulomb potential averaged oher the Landau o r b i t . In f a c t , the longitudinal parts of the adiabatic
10
n
Io6
zoniv
1 oa
1 0'
8
8
z 8 r t ti
t
t
8
PERTURBATION
nm
10'
10''
j - 3 ~ + ~ r r r r n . t 7 7
1of2
1o1$
~
~
~
10
m
n
13
r
ADIABATIC APPROXIMATION
,
BinG
7
-
r
-
lo
1o8
1o9
Bin G
Figure 4.- O s c i l l a t o r strengths f o r the 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 using p e r t u r b a t i o n theory and a d i a b a t i c approximation wave functions, 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 strengths obtained using 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).
JOURNAL DE PHYSIQUE
C2-146
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 mu1t 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%
s t a t e s a r e even more accurate by
f o r b = l o 3 w h i l e t h e enerqies o f hydroqen-like
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
p < lo-* 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 , those f o r
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 approximation 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 magnet 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 ike
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 physic 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 magnetic 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 eauil 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 funct 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
4.10~
55
0
0.3
8.10~
109
12.10~
163
lh104
2 18
2.10~
2 72
2;.104 E in ERy,
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 excitation of the electron into higher Landau levels.
-A
"gl""
&
6.
1
B = 6.7-lof2G , 3=9O0.g l h
Fe
XB!7
6 .
2
-
0-2
-
-6
.
-6.
-8 +
0
20
P
2.103
74
8.103
128
12.104
183
16-106
23 7
Figure 5b.- Same as Fig. 5a for hydrogenic iron.
2.106 E i n E R y ,
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 consequence 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
dorec - (kaL)*
ion
-dn
4n
Ototal
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 physic 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 no 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 bremss 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 comput 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 prev 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 possib 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 element 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 function 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 magnetic field ~=4.4.10~T for two values of the energy Ee of the electron 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: polarisation of the photon in the k-B plane,
2: polarisation perpendicuTaF to the
k-8
- plane).
o
a&
02
08
a6
w/w,
lo-'
- - - - . -I
0-
1 MCHF
-lo
_
1 oO
I
1
1 1
1 ofl
lllrn
P~
1
1 1 111111
I
30+3
1 1 1 111
- 5 0 --
-
E?EH -
-
-
-100.-
--I--
I
I
I
I
Figure 8.- Ground state
energy of He as a function of the magnetic
field parameter bZ. Solid curve: SCHF calculation in adiabatic approximation 1401, dashed
curve: spherical MCHF
calculation (ls2p-, 3 ~ - 1
+ ls4f-, 3F_, ) with inclusion of the diamagnetic 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 can 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
(B>B,,
i.e. D, = B/B,>l)
a r e f u l f i l l e d ; then t h e
t h e a d i a b a t i c approximation
..
I.
L
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 ,
4i(i)
Lan
= on.=o,m
1
(1~1 ) ' g m .
i
i
('i)
u
'
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 , t h 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 > 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 .
z
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 extens i o n o f t h e method t o multi-Landau-configuration forms o f t h e N-electron wave funct 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 presence 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
B/4.7 x I O ~ T
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
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
.
s t r o n g magnetic f i e l d , s t a t e s t h a t do n o t b i n d i n t h e f i e l d - f r e e case, such as t h e
lowest m = -1 s t a t e , become b i n d i n g f o r p 0.
+
0
"H;
B=O. 05
(in Ry)
(in R Y )
-1.0
-1 0
0
1
2
R
(in a.u.)
3
t
F i g u r e 9.-Energy o f t h e lowest m=O and m=-1 s t a t e o f t h e H, i o n as a
of t h e i n t e r n u c l e a r separation R ( i n atomic u n i t s ) f o r v a r i o u s values
magnetic f i e l d parameterp. As p increases, t h e e q u i l i b r i u m d i s t a n c e
t e d t o s m a l l e r values, and t h e b i n d i n g energy s t r o n g l y increases; i n
l a r , t h e a n t i - b i n d i n g m=-1 s t a t e becomes a b i n d i n g s t a t e .
function
o f the
i s shifparticu-
I t i s easy t o see t h a t t h e c o n f i g u r a t i o n o f t h e :H i o n w i t h t h e molecular a x i s par a l l e l t o t h e magnetic f i e l d represents an e n e r g e t i c minimum. A p e r t u r b a t i o n t h e o r y
estimate shows, however, t h a t t h e z e r o - p o i n t energies associated w i t h t h e v i b r a t i o n s
of t h e n u c l e i perpendicular t o t h e magnetic f i e l d f o r s u f f i c i e n t l y l a r g e j may become o f the same order o f magnitude as t h e Coulomb b i n d i n g energies. Obviously t h e
Born-Oppenheimer approximation breaks down i n such a s i t u a t i o n . Furthermore, the eff e c t s associated w i t h t h e f i n i t e mass o f t h e protons and t h e n e t s i n g l y p o s i t i v e
charge o f t h e system are estimated t o be o f the o r d e r o f t h e proton c y c l o t r o n energy, and thus a r e expected t o l e a d t o n o n n e g l i g i b l e c o r r e c t i o n s t o t h e b i n d i n g ene r g y i n t h e h i g h - f i e l d regime. This once again i l l u s t r a t e s t h e f a c t t h a t t h e t r e a t ment o f atomic and molecular systems i n s t r o n g magnetic f i e l d s , m o t i v a t e d by t h e d i s covery o f such f i e l d s i n compact cosmic objects, s t i l l r a i s e s a number o f unanswered,
though f a s c i n a t i n g , quantum mechanical problems - problems t h a t should be t a c k l e d
i n t h e near f u t u r e .
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