SOME PROPERTIES OF ONE ELECTRON ATOMS IN INTENSE MAGNETIC FIELDS

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SOME PROPERTIES OF ONE ELECTRON ATOMS IN INTENSE MAGNETIC FIELDS

M. Mcdowell

To cite this version:

M. Mcdowell. SOME PROPERTIES OF ONE ELECTRON ATOMS IN INTENSE MAGNETIC FIELDS. Journal de Physique Colloques, 1982, 43 (C2), pp.C2-387-C2-396.

�10.1051/jphyscol:1982229�. �jpa-00221841�

JOURNAL DE PHYSIQUE

Colloque C2, supplément au n°21, Tome 43, novembre 1982 page C2-387

SOME PROPERTIES OF ONE ELECTRON ATOMS IN INTENSE MAGNETIC FIELDS

M . R . C . McDowell

Mathematics Department, Royal Holloway College,(University of London),Egham Hill, Egham, Surrey TW20 OEX, U.K.

Résumé. - Les propriétés des systèmes à un électron dans un champ magnétique intense sont tout d'abord passées en revue. Un travail récent sur les états liés et sur les transitions entre états liés est ensuite discuté ainsi que le problème de l'espacement des niveaux d'énergie associés au mouvement transverse, dans le con- tinuum. De nouveaux résultats sur la variation de cet espacement avec z sont présentés dans le cas d'un champ d'intensité 107G, pour m = - 1 . On montre que les états de nombre quantique principal n â 5 deviennent "liés" pour z petit. Ceci est interprété en termes d'ionisation par champ. Des lois d'échelle pour la diffu- sion d'électrons par des cibles à un électron sont ensuite rapide- ment discutées.

Abstract. - The properties of one electron systems in intense magnetic fields are reviewed. Recent work on bound states and bound-bound transitions is discussed, as is the problem of the energy level spacing for the perpendicular motion in the

continuum. New results are presented for the variation of this spacing with z in the case of a field of 107G, for m = - 1 . It is shown that states with Landau quantum number n <. 5 become "bound" at small z . This is interpreted in terms of field ionisation. Scaling laws for electron scattering by one electron targets are briefly discussed.

1. Introduction. - We consider some properties of one-electron atoms in intense magnetic fields. Relativistic effects are neglected.

We first discuss the bound states and their scaling. This leads on to a discussion of the bound-bound transition probabilities, and their scaling. The main new results concern an adiabatic approx- imation to the energies and wave functions of the quasi-Landau levels embedded in the continuum. Some remarks are made on photoionisation, and recent exact and approximate scaling laws for this process

reviewed. The problem of electron scattering by one electron targets in a magnetic field is discussed, and it is shown that the first Born approximation is valid for all energies at sufficiently high nuclear charge.

2. Bound states and energy levels. - We consider a uniform static magnetic field B along the z-axis, and choose cylindrical polar co- ordinates (p,e,z). We write

(1)

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1982229

C2-388

w i t h

JOURNAL DE PHYSIQUE

and suppose t h a t t h e mass o f t h e n u c l e u s may be t a k e n a s i n f i n i t e . F i n i t e mass e f f e c t s a r e s i g n i f i c a n t a t v e r y h i g h f i e l d s (y >> 1) and have been c o n s i d e r e d b y , among o t h e r s , Herold e t a l ( l ) . We t a k e t h e n u c l e a r c h a r g e a s a o e . The SchrBdinger e q u a t i o n i s

w i t h

and t h e e i g e n f u n c t i o n s Y have a conserved v a l u e m o f a z i m u t h a l quantum number and p a r i t y ^{T .} We may p u t

t o o b t a i n

w i t h

and

The boundary c o n d i t i o n s a r e

and i n g e n e r a l we c o n s i d e r q u a d r a t i c a l l y i n t e g r a b l e s o l u t i o n s . The d i f f e r e n t i a l e q u a t i o n ( 6 ) i s o f c o u r s e n o n - s e p a r a b l e , and r e c o u r s e must b e had t o approximate methods. Work up t o 1975 h a s been reviewed by ~ a r s t a n g ( ~ ) , and we w i l l comment o n l y on some r e c e n t r e s u l t s . For weak f i e l d s t h e q u a d r a t i c Zeeman t e r m f y 2 p 2 may be t r e a t e d a s a p e r t u r b a t i o n . For v e r y h i g h f i e l d s (y >> 1) t h e Coulomb t e r m -F 2a0 may b e c o n s i d e r e d a s t h e p e r t u r b a t i o n and an expansion made i n

t e r m s o f Landau f u n c t i o n s . A t i n t e r m e d i a t e f i e l d s t r e n g t h s , y % I, v a r i a t i o n a l methods a r e a v a i l a b l e , e i t h e r i n t e r m s o f s p e c i f i e d b a s i s f u n c t i o n s , o r i n a Hartree-Fock approach. Here y = y ( a o = 1) and we w i l l s e e l a t e r t h a t i t must be s c a l e d f o r a. f I. Most c a l c u -

l a t i o n s a r e c o n f i n e d t o t h e l o w e s t f o u r t e e n l e v e l s , t h a t i s t h o s e which go t o t h e n t h l e v e l , n

5

3. However P a t i l ( l 7 ) h a s r e c e n t l y g i v e n a n a l y t i c e x p r e s s i o n s f o r t h e h i g h f i e l d e n e r g i e s o f s t a t e s w i t h n 2 3 . A t low f i e l d s we may l a b e l t h e e n e r g i e s by t h e quantum numbers

( n R m ) a s i n t h e f i e l d f r e e c a s e . The r e s u l t s ( S c h i f f and s n y d e r ( 3 ) ) have r e c e n t l y been compared w i t h v a r i a t i o n a l c a l c u l a t i o n s by Ruder e t a 1 ( 4 ) . The p e r t u r b a t i o n c a l c u l a t i o n i s a c c u r a t e f o r t h e ground s t a t e t o b e t t e r t h a n 1% f o r y l < 0.43 and t o b e t t e r t h a n 0.7% f o r a l l t h e s e s t a t e s a t y l = 4 . 3 when compared w i t h t h e v a r i a t i o n a l c a l c u l a t i o n s o f Kara and ~ c ~ o w e l l ( 5 ) . The ( 3 s o > and 13do> s t a t e s

combine a s

The t a b u l a t e d v a l u e s omit t h e f i n i t e mass c o r r e c t i o n term which i s o f o r d e r 1 . 5 ym(eV)

-

^{ym Ry.}

A mixed v a r i a t i o n - p e r t u r b a t i o n c a l c u l a t i o n which u s e s o n l y two v a r i a t i o n a l p a r a m e t e r s by Cohen and ~ e r m a n ( 6 ) f o r

I

l s o > ,

1

2 s o > ,

12p0>, 1 2 p ? l > f a i l s b a d l y f o r 12so> a t y , = 0.43.

The a d i a b a t i c approximation h a s been used by Simola and V e r t a m ~ ( ~ ) and by ~ u n n e r ( 8 ) and polynomial a p p r o x i m a t i o n s t o t h e energy l e v ~ l s g i v e n by Wunner and ~ u d e r ( 9 ) . They g i v e r e s u l t s f o r y < 2 x 1 0

.

The s t a t e s a r e l a b e l l e d by t h e Landau quantum numbers (n,m) and t h e number v o f nodes i n t h e l o n g i t u d i n a l wave f u n c t i o n . For odd, b u t n o t f o r e v e n , v t h e e n e r g i e s a proach t h e i n f i n i t e - f i e l d l i m i t by y s 2 x 10'. Wunner and Ruder(97 u s e t h e s e r e s u l t s t o compute t h e wave l e n g t h s o f t h e Balmer s e r i e s l i n e s i n f i e l d s a p p r o p r i a t e t o n e u t r o n s t a r s . An a l t e r n a t i v e approach h a s been used by Bender e t a l ( 1 8 ) f o r t h e l o w e s t s t a t e o f e a c h m. They expand

w i t h k = 2 ( I m l

+

^{1) and rl} i s a d i m e n s i o n l e s s p a r a m e t e r r e l a t e d t o t h e minimum o f ,V ( p , z ) . They o b t a i n a n a l y t i c e x p r e s s i o n s f o r E-, t h r o u g h E,.

A s was f i r s t shown by Surmelian and O ' C ~ n n e l l ( ~ O ) t h e e n e r g i e s s c a l e a s

and t h e wave f u n c t i o n s a s

a s may r e a d i l y be proved from (1) and ( 2 ) . Thus n o n - r e l a t i v i s t i c e n e r g i e s a r e known f o r t h e l o w e s t f o u r t e e n s t a t e s f o r t h e e n t i r e i s o - e l e c t r o n i c sequence. F i g . 1 shows t h e v a r i a t i o n o f t h e l o w e s t

energy l e v e l s w i t h f i e l d s t r e n g t h (a, = 1 ) .

1

; o

-

-

- W 1-

Fog 1 - 35

25 20 0 0 3

F i g . 1

I$ The s i x l o w e s t l y i n g l e v e l s

o f atomic hydrogen a s a f u n c t i o n of magnetic f i e l d s t r e n g t h . B (Gauss) ( a f t e r Ruder e t a l ( 4 ) ) .

6 7 8 9 10 11 12 13

Logl0B (GI

c2-390 JOURNAL DE PHYSIQUE

3. Dipole m a t r i x elements and o s c i l l a t o r s t r e n g t h s .

-

E i n s t e i n A i j c o e f f i c i e n t s and o s c i l . l a t o r s t r e n g t h s of t r a n s i t i o n s connecting t h e lowest f o u r t e e n l e v e l s have been given by a number of a u t h o r s (Brandi e t a 1 (ll)

,

Smith e t a 1 (12)

,

Kara and ~ c ~ o w e l l ( 5 )

,

Wunner e t a 1 (13)

,

~ a d e h r a ( 1 4 ) )

.

The r e s u l t s a r e i n g e n e r a l i n c l o s e agreement, and t h o s e f o r s i x t r a n s i t i o n s a r e shown i n Fig. 2 , adopted from r e f . ( 1 3 ) .

Fig. 2

O s c i l l a t o r s t r e n g t h s f o r t r a n s i t i o n s among t h e low l y i n g l e v e l s of atomic hydrogen a s a f u n c t i o n of f i e l d s t r e n t h ( a f t e r Ruder e t a l 7 4 ) .

I t i s important t o n o t e t h a t f n n l can vary by almost f i v e o r d e r s of magnitude when y changes by a : f a c t o r of 100' This i s , i n p a r t i c u l a r , t h e c a s e f o r t h e ( 2 s 0 > ⁺ 12po> t r a n s i t i o n . The d i s c o r d a n t r e s u l t s of r e f . (11) and r e f . ( 8 ) f o r t h e very weak 13d

+

^{l > +}

1

3p

+

^{l >} t r a n - s i t i o n s have been r e s o l v e d by Ruder e t a l ( 4 ) i n favour of Brandi e t a l ( l 1 ) . Wunner e t a l ( 1 3 ) show t h a t t h e d i p o l e m a t r i x element s c a l e s as

s o t h a t t h e o s c i l l a t o r s t r e n g t h s c a l e s a s ( 1 5 )

a s might have been expected i n view of t h e sum r u l e .

These r e s u l t s have been used by Ruder e t a 1 ( 1 5 ) t o d i s c u s s t h e spectrum of H-like i r o n (Fe X X V I ) i n neutron s t a r a c c r e t i o n l a y e r s where f i e l d s o f o r d e r 10' t o 1013 G may be p r e s e n t . This corresponds t o e f f e c t i v e f i e l d s y i n t h e range 0 . 1

2

y 5 10. The Ly l i n e has a photon energy i n c r e a s i n g from 7 keV a t 1 0 " ~ t o 21 keV a t 1013G and a l i f e t i m e of o r d e r 1 0 - ' ~ s .

A q u i t e d i f f e r e n t approach i s r e q u i r e d f o r t h e Rydberg l e v e l s , and has been d i s c u s s e d by Clark and Taylor i n a number of p a p e r s ( 1 6 ) . Since t h e y w i l l be d i s c u s s i n g t h e i r work i n d e t a i l a t t h i s meeting we merely n o t e h e r e t h a t t h e y c o n s i d e r s t a t e s n e a r t h e i o n i s a t i o n l i m i t , where t h e i o n i s a t i o n p o t e n t i a l i s of t h e same o r d e r a s t h e c y c l o t r o n energy. They use a very l a r g e Sturmian f u n c t i o n b a s i s and o b t a i n t h e eigenvalues f o r n

2

23, and corresponding o s c i l l a t o r s t r e n g t h s , t o a r e s o l u t i o n of 0.1 cm-'at 4.7T. No s t u d y has y e t been made of t h e v a r i a t i o n of o s c i l l a t o r s t r e n g t h w i t h f i e l d , but we would a n t i c i p a t e s i g n i f i c a n t v a r i a t i o n when Ay s 0 6 ( A E n n , ) . A t n = 23, A E n I n + l % 1.5

lo-'

Ry and a t 4.7 T I y = 4.25 10' Ry, s o one would n o t expect much change between n manifolds f o r s m a l l f i e l d changes. However, we would expect s i g n i f i c a n t v a r i a t i o n among t h e Balmer emission l i n e s

from a given n .

4 . Continuum l e v e l s .

-

Much l e s s i s known about t h e continuum l e v e l s of one-electron systems i n magnetic f i e l d s . The photoabsorption experiments of Garton and T o m k i n s ( l g ) , Garton e t a 1 ( 2 0 ) , C a s t r o e t a 1 (21)

,

Gay e t a 1 (22) and Delande and Gay (23) appear t o show a l e v e l spacing j u s t above t h e i o n i s a t i o n t h r e s h o l d o f AEn,,+l = 3y Ry. The spacing expected i n t h e absence o f t h e Coulomb i n t e r a c t i o n i s of course 2y R 3 , t h e Landau s p a c i n g . The e x p l a n a t i o n was f i r s t given by Edmonds(2 1. He argued t h a t n e a r E = 0 t h e s e m i - c l a s s i c a l frequency a s s o c i a t e d w i t h motion along t h e z-axis was s m a l l compared w i t h t h a t of t h e c y c l o t r o n motion, and e v a l u a t e d t h e energy i n a plane p e r p e n d i c u l a r t o z using a JWKB approximation,

where n i s t h e number of nodes i n t h e r a d i a l wave f u n c t i o n ; p i t h e c l a s s i c a l t u r n i n g p o i n t s . The Landau spacing i s recovered when z >> p

,

b u t f o r small z he found ( d ~ / d n )

-'

⁼ 3.16 y Ry.

This work was l a t e r r e f i n e d by tara ace(^^) and ~ a u ( 2 6 ) . Very r e c e n t l y G a l l a s and otconne11(27) have shown, following S t a r a c e and Rau, t h a t f o r t h e motion i n t h e z = 0 p l a n e , t h e energy spacing can be

expressed a n a l y t i c a l l y i n terms of complete e l l i p t i c i n t e g r a l s , o f t h e f i r s t and t h i r d kinds. A p p l i c a t i o n s t o t h e experimental spacings i n Rb have been r e p o r t e d by Economou e t a l ( 2 8 ) and t o B a and S r by Fonck e t a l ( 2 9 ) . There i s no need t o make t h e JWKB approximation:Kara and ~ c ~ o w e l l ( ~ ~ ) have solved ( 6 ) d i r e c t l y f o r z = 0. T h e i r r e s u l t s f o r two c a s e s a r e compared i n Table I. The s p a c i n g s found a r e appreciably d i f f e r e n t from t h e JWKB Coulomb-Landau l i m i t . I t i s w e l l known t h a t

Table 1

Exact s o l u t i o n s of eqn. ( 6 ) i n t h e z = 0 p l a n e compared w i t h JWKB v a l u e s . AE i s t h e l e v e l s e p a r a t i o n . A l l q u a n t i t i e s a r e i n u n i t s of y Ry.

A E A ~ ~ + " ]

⁺ 2y Ry a s n becomes l a r g e . What had n o t f u l l y been apprec- i a t e d b e f o r e t h e Kara and McDowell c a l c u l a t i o n s was t h a t t h e r e a r e no s o l u t i o n s f o r n < nmin (y) f o r t h e motion p e r p e n d i c u l a r t o t h e Z-axis. Thus Kara and McDowell found (Table 1) t h a t many q u a s i - Landau l e v e l s were missing i n t h e z ⁼ 0 p l a n e a t f i e l d s below

l o 9

^G.

These s t a t e s c e r t a i n l y e x i s t a s z + : t h e y a r e t h e f i e l d - f r e e

Landau l e v e l s . F u r t h e r , n i s h e r e t h e number of r a d i a l nodes i n t h e p e r p e n d i c u l a r motion, and i s conserved a s we vary z. We can attempt t o t r a c e t h e s e s t a t e s a d i a b a t i c a l l y , and w r i t e ( 6 ) a s

JOURNAL DE PHYSIQUE

Table 2

Minimum v a l u e of n f o r t h e p e r p e n d i c u l a r motion i n t h e z = 0 p l a n e a t d i f f e r e n t f i e l d s .

f o r f i x e d z = z,. We assume t h e r a d i a l wave f u n c t i o n F z o ( p ) i s a s l o w l y v a r y i n g f u n c t i o n o f z and n e g l e c t Fz = aFzo ( p ) a z and Fzz =

a 2 ~ Z o

( p ) / a z 2 . T h a t i s , we assume < F Z > and < F Z z > < < kz2 where < > i n d i c a t e a v e r a g e s o v e r p f o r f i x e d z = z,. T h i s

remains t o b e confirmed. I n t e g r a t i n g outwards from t h e o r i g i n we can o b t a i n a s t a r t i n g s e r i e s by p u t t i n g

To f i n d t h e e i g e n v a l u e s f o r f i x e d Landau quantum number n , we e i t h e r s t a r t a t z = 0 w i t h Kara and McDowell's v a l u e s , o r a t v e r y l a r g e z w i t h t h e Landau v a l u e . Nuzzo and ~ c ~ o w e l l ( 3 1 ) have c a r r i e d o u t

c a l c u l a t i o n s i n i t i a l l y f o r B =

l o 7

G and m =

-

1. F u r t h e r work i s i n hand. The r e s u l t s t o d a t e a r e shown i n F i g . 3 . For n = 7 and n = 6 t h e e n e r g y o f t h e p e r p e n d i c u l a r motion d e c r e a s e s smoothly from t h e Landau v a l u e w i t h d e c r e a s i n g z t o t h e z = 0 l i m i t . However, f o r n = 5 f o r which no z = 0 e i g e n v a l u e w i t h E > 0 was found, E s ( z ) changes s i g n n e a r z, = 4.0 and goes t o a z, = 0 l i m i t i n g v a l u e o f

-

^{2.095 y Ry.} Note t h a t t h i s l i e s a l m o s t e x a c t l y 3 y Ry below t h e f i r s t continuum s t a t e ( n = 6 ) : h e r e y = 0.0043, s o a t z, = 0 t h e n = 5 s t a t e l i e s j u s t o v e r 100 meV below t h e i o n i s a t i o n t h r e s h o l d . I t f o l l o w s t h a t t h e s t a t e s w i t h n = 0 , 1 , 2 , 3 , 4 l i e below t h i s a t s p a c i n g s o f a b o u t 0.17 e V , and t h u s we p r e d i c t t h e n = 0 s t a t e t o b e bound by a b o u t 1 e V a t zo = 0 . W i t h i n t h e l i m i t s of o u r a d i a b a t i c a p p r o x i m a t i o n , which remains t o be t e s t e d , t h i s a p p e a r s t o mean t h a t t h e v e r y h i g h l y i n g Rydberg s t a t e s a r e i n e f f e c t f i e l d i o n i s e d by t h e m a g n e t i c f i e l d . T h i s might b e d i r e c t l y t e s t a b l e a t f i e l d s o f 100 T

( l o 6

^{G )} where t h e b i n d i n g o f t h e n = 0 l e v e l s h o u l d be a b o u t 0 . 1 eV, p r o v i d e d t h i s i s much g r e a t e r t h a n koT.

/ / 1

The a d i a b a t i c e n e r g i e s o f

1

motion p e r p e n d i c u l a r t o t h e f i e l d i n t h e n = 5 , 6 , 7 s t a t e s o f m = -1 o f t h e continuum a t a f i e l d y = 0.0043 ( 1 0 7 ~ ) .

5. P h o t o i o n i s a t i o n .

-

The p h o t o i o n i s a t i o n c r o s s s e c t i o n i s g i v e n by (30)

where we a d o p t a q u a n t i s a t i o n l e n g t h o f Lz a l o n g t h e f i e l d . The d e n s i t y o f s t a t e s i s

L Z 1

p ( E Z ) ⁼

-

'

-

27~ 2kZ

where E, = kZ.2 i s t h e energy o f t h e e j e c t e d e l e c t r o n i n Rydbergs, and S i f i s t h e d l p o l e m a t r i x element. C a l c u l a t i o n s have been made by

~ u d e r (32) and h i s c o l l e a g u e s f o r Fe XXVI a t v e r y h i g h f i e l d s (y >> 1) u s i n g an a d i a b a t i c approximation f o r t h e continuum wave f u n c t i o n

where t h e z-component i s o b t a i n e d by s o l v i n g

w i t h t h e boundary c o n d i t i o n o f no o u t g o i n g wave a t z =

--

^and

r e g u l a r and i r r e g u l a r s o l u t i o n s

They u s e t h e f u l l e l e c t r o m a g n e t i c Harniltonian and do n o t make t h e d i p o l e a p p r o x i m a t i o n , b u t r e s t r i c t t h e m s e l v e s t o p h o t o i o n i s a t i o n i n t o t h e f i r s t Landau l e v e l . A s c a l i n g l a w ( 3 7 ) t h e n a l l o w s them t o w r i t e down t h e r e s u l t f o r any one e l e c t r o n system

w i t h ⁰ (aO,B,E,E'

,k)

⁼ a i l 0 ( 1 , ~ a o ~

, E ~ G ~

, E ' ~ Z ~ I k a ; l ) (21) E + h k c = E l . -

Kara and ~ c ~ o w e l l ( 30) have made s i m i l a r c a l c u l a t i o n s a t i n t e r m e - d i a t e f i e l d s t r e n g t h s

l o 7

t o

lo9

^G ( 0 . 0 0 4 3 1 y 1 0 . 4 3 ) f o r p h o t o i o n i s - a t i o n from b o t h t h e Iso> and 12po> s t a t e s , u s i n g a c c u r a t e ground s t a t e wave f u n c t i o n s . F o r t h e continuum wave f u n c t i o n s t h e y choose

Because o f t h e c h o i c e o f z = z o t h e y found t h e main r e s o n a n t behaviour a t t h e t h r e s h o l d s o f t h e bound s t a t e s of p e r p e n d i c u l a r motion i n t h e z = 0 p l a n e , r a t h e r t h a n a t t h e Landau e n e r g i e s ( z ^{+ a).} The d i s - c u s s i o n i n s e c t i o n 4 above s u g g e s t s t h a t s i n c e t h e s e e n e r g i e s v a r y with z some s o r t o f a v e r a g i n g p r o c e d u r e w i l l b e needed t o o b t a i n a c c u r a t e r e s u l t s . Kara and McDowell found, f o l l o w i n g Blumberg e t a l ( 3 3 ) t h a t t h e c r o s s s e c t i o n behaved n e a r t h r e s h o l d a s kz f o r I ~ m l = 1 t r a n - s i t i o n s b u t a s k g 1 f o r

l a m 1

= O odd p a r i t y bound s t a t e s , and t h e r e v e r s e f o r even p a r i t y bound s t a t e s . They used t h e d i p o l e approximate f o r which t h e s i m p l e s c a l i n g l a w ( 3 7 )

a v ( a o , B ) = a i 2 a v , ( l , ~ / a ~ ~ ) w i t h

a p p l i e s . There a r e no a v a i l a b l e measurements e x c e p t t h o s e of Blumberg e t a l ( 3 3 ) on S-, b u t b o t h e x p e r i m e n t and t h e o r y f o r H- a r e i n hand.

C2-394 JOURNAL DE PHYSIQUE

6. S c a t t e r i n g o f e l e c t r o n s by one e l e c t r o n t a r g e t s .

-

~ c ~ o w e l l ( 3 4 ) has r e c e n t l y reviewed t h e work t o d a t e on t h e s c a t t e r i n a wroblem. and attempted

t o

d e r i v e s c a l i n g laws. Using t h e uni-dimensional d e n s i t y of f i n a l s t a t e s f o r t h e s c a t t e r e d e l e c t r o n t h e c r o s s s e c t i o n f o r t h e t r a n s i t i o n i n which t h e i n c i d e n t e l e c t r o n makes a t r a n s i t i o n from a continuum s t a t e Inimiki> t o a f i n a l continuum s t a t e Infmfkf> while t h e t a r g e t goes from i n i t i a l bound s t a t e In$b)Rimgi> t o

Inib)aimk > t o l n i b ) e p l f > i s ( 3 5 ) i

The only d e t a i l e d c a l c u l a t i o n s t o d a t e a r e by ~ h s a k i ( ~ ~ ) who c o n s i d e r s

a t f i e l d s of 10' and

lo9

Gauss and impact e n e r g i e s up t o 90 eV.

Unfortunately he uses unperturbed wave f u n c t i o n s , which a r e of d o u b t f u l value f o r t h e ground s t a t e a t such f i e l d s , and t o t a l l y u n r e l i a b l e f o r t h e e x c i t e d s t a t e s . I n a d d i t i o n he c a r r i e s o u t t h e c a l c u l a t i o n i n t h e F i r s t Born Approximation (FBA) which i s u n l i k e l y t o be v a l i d a t such e n e r g i e s . H i s q u a n t i t a t i v e r e s u l t s a r e t h u s u n l i k e l y t o be c o r r e c t b u t t h e main q u a l i t a t i v e f e a t u r e may be. The i n c i d e n t and s c a t t e r e d e l e c t r o n a r e , i n FBA, r e p r e s e n t e d by unperturbed Landau f u n c t i o n s s o he f i n d s l a r g e r e s o n a n t enhancements of t h e e x c i t a t i o n c r o s s when t h e k f 2 = (AEif

+

2 j y ) Ry, j = 1, 2 , 3

... .

Consider an e l e c t r o n s c a t t e r e d by a o n e - e l e c t r o n t a r g e t of n u c l e a r charge a o . The bound e l e c t r o n s e e s t h e f u l l n u c l e a r charge, b u t t h e i n c i d e n t e l e c t r o n s e e s only t h e r e s i d u a l Coulomb f i e l d of charge

a

-

¹ We can t h e r e f o r e p a r t i t i o n t h e Hamiltonian a s

where IIi i s t h e f u l l momentum ( i = 1, 2 ) . I n FBA t h e T-matrix i s , n e g l e c t i n g exchange,

where Y (

- -

^{; 1 )} s a t i s f i e s ( 3 ) w i t h charge a

-

¹ and

Y ( - .

;r,)

w i t h charge a o . I t i s c l e a r t h a t no simple s c a l i n g law e x i s t s . An approximate r e s u l t may be obtained when a, >> 1. The FBA m a t r i x element s c a l e s a s a o , s o f o r t h e n u c l e a r charges a 1 6 a 2 >> 1, hence f o r i n c i d e n t e n e r g i e s k$ = a 2 k2 and f i e l d s B i = a i B

( i = 1 , 2 )

,

and

S O

The second Born term < i l ~

G:Iv>

scales-independent of charge, since the Green's function G$ scales as a o 2 ; the third and higher terms go as a;',

...

etc. That is, provided a. >> 1, the Born series is an expansion in a . Thus for sufficient large charge (28) is true for the full cross section to a good approximation, at all enerqies

In the case of atomic hydrogen alone, where the continuum electron see the magnetic field modified by a short range field it may be a satis- factory approximation to represent this electron by a Landau function.

This is not true when the residual charge a

-

¹ is non-zero:

then the solutions of (3) must be used for both electrons. Detailed calculations will be very difficult, as cross sections for the atomic processes must be summed over all allowable final quasi-Landau states and averaged over all initial quasi-Landau states. Presumably for transitions in, e.g. hydrogen-like iron, which are of interest for astrophysical purposes, one could assume a Boltzmann distribution of electron energies, but it is not clear how these should be partitioned among quasi-Landau levels.

Acknowledgements

I am indebted to S. Nuzzo (Mrs. M. Zarcone) for permission to use joint results. Some clarification of my ideas was achieved in con- versations with Professor M. Mittleman and Dr. C. Clark.

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