EXCITATION AND DECAY OF BOUND STATES NEAR THE IONIZATION LIMIT

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EXCITATION AND DECAY OF BOUND STATES NEAR THE IONIZATION LIMIT

David Ellis

To cite this version:

David Ellis. EXCITATION AND DECAY OF BOUND STATES NEAR THE IONIZATION LIMIT.

Journal de Physique Colloques, 1979, 40 (C1), pp.C1-152-C1-154. �10.1051/jphyscol:1979127�. �jpa- 00218407�

J O U R N A L DE PHYSIQUE Colloque C1, suppldment au n " 2, Tome 40, fdvrier 1979, page C1-152

EXCITATION AND DECAY OF BOUND STATES NEAR THE IONIZATION LIMIT

David G. E l l i s

U n i v e r s i t y o f Toledo, Toledo, Ohio 43606, U.S.A.

RLsumd. Un modele c l a s s i q u e s i m p l e de l a c a p t u r e d ' u n d l e c t r o n , l o r s d e l ' e x - c i t a t i o n par f a i s c e a u - l a m e , s u g g e r e que l e s n i v e a u x de Rydberg d e n d l e v d e t (1

l i m i t d , p e u v e n t B t r e p e u p l d s d ' u n e manihre s u b s t a n t i e l l e . En i n c l u a n t l e s c a s c a d e s d e t o u s l e s o r d r e s , o n d e r i v e une r e l a t i o n s i m p l e e n t r e une t e l l e p o p u l a t i o n i n i t - i a l e e t l a queue n o n - e x p o n e n t i e l l e de l a d i s t r i b u t i o n .

A b s t r a c t . A s i m p l e c l a s s i c a l model of e l e c t r o n p i c k u p i n beam-foil e x c i t a t i o n s u g g e s t s t h a t Rydberg s t a t e s of h i g h n and l i m i t e d (1 may b e s u b s t a n t i a l l y pop- u l a t e d . I n c l u d i n g c a s c a d e s o f a l l o r d e r s , a s i m p l e r e l a t i o n i s d e r i v e d between s u c h a n i n i t i a l p o p u l a t i o n and t h e r e s u l t i n g power-law t a i l o f a d e c a y c u r v e .

INTRODUCTION

I n L i n e e m i s s i o n from h i g h l y c h a r g e d i o n s f o l l o w i n g beam-foil e x c i t a t i o n , e x p e r i m e n t a l decay c u r v e s may show a power-law t i m e dependence f o r l o n g t i m e s , r a t h e r t h a n t h e more f a m i l i a r e x p o n e n t i a l d e c r e a s e . [ I ] These "power-law t a i l s "

have been e x p l a i n e d a s produced by c a s c a d i n g from h i g h l y e x c i t e d s t a t e s , o n t h e a s s u m p t i o n t h a t t h e i n i t i a l p o p u l a t i o n s of t h e s e s t a t e s d e c r e a s e a s some power o f t h e p r i n c i p a l quantum number. 121 However, many q u e s t i o n s r e m a i n , s u c h a s t h e im- p o r t a n c e o f h i g h e r - o r d e r v s d i r e c t c a s c a d e s , y r a s t v s Rydberg c a s c a d e s , t h e c o n n e c t i o n between t h e e x p o n e n t of t h e d e c a y c u r v e and t h a t of t h e pop- u l a t i o n s , and t h e c o n n e c t i o n between t h e s e and t h e p h y s i c s o f t h e b e a m f o i l e x c i t a t i o n p r o c e s s .

T h a t a power-law t a i l d o e s r e s u l t n a t - u r a l l y from a power-law p o p u l a t i o n d i s t r i b u t i o n c a n of c o u r s e be s e e n v e r y e a s i l y . Assume t h e l e v e l o f i n t e r e s t ( n l , ( l l ) i s f e d by a s e r i e s of f i r s t - o r d e r c a s c a d e s from l e v e l s (n,(1) w i t h n = 2 , 3 , 4 , " ' . Then i f h i g h e r - o r d e r c a s c a d e s a r e i g n o r e d t h e p o p u l a t i o n o f l e v e l (nl,!Ll) is g i v e n by 131:

c o n v e r t e d t o a n i n t e g r a l and i t s a s y m p t o t i c be- h a v i o r e s t a b l i s h e d by s t e e p e s t d e s c e n t . We f i n d i m m e d i a t e l y a s t / ~ -> :

p l ( t ) Q c o n s t x ( t / ~ ) - ~ (2) w i t h m = ( b + c - l ) / a ,

where we h a v e assumed

Beyond t h i s o b v i o u s c r u d e r e l a t i o n s h i p we would l i k e t o s u g g e s t two s e p a r a t e p o i n t s which may b e r e l e v a n t t o some o f t h e open ques-

t i o n s a b o u t t h i s phenomenon. F i r s t , i t can b e shown t h a t ( 2 ) r e m a i n s t r u e f o r a c a s c a d e p r o c e s s o f a r b i t r a r i l y h i g h o r d e r . Second, e l e c t r o n p i c k u p from t h e f o i l s u r f a c e might i n some c i r - c u m s t a n c e s p o p u l a t e s t a t e s o f h i g h n b u t l i m i t e d

(1, f a v o r i n g Rydberg o v e r y r a s t c a s c a d e s . HIGHER ORDER CASCADES

t h

C o n s i d e r a n o r d e r c a s c a d e p r o c e s s o r i g i n a t i n g from a Rydberg s e r i e s (n,L) a s be- f o r e , and t e r m i n a t i n g o n t h e l e v e l o f i n t e r e s t ( n l , " ) . T h e r e w i l l b e N-1 i n t e r m e d i a t e s t a t e s i n v o l v e d , l a b e l l e d by quantum numbers ( n 2 , R 2 ) ,

( n 3 , E 3 ) , - - - ( % , ! L N ) . The c o n t r i b u t i o n t o P l ( t ) from t h i s p r o c e s s w i l l b e [ 3 ] :

Here a i s t h e decay c o n s t a n t , A t h e t r a n s i t i o n

e - a n ~

X [ ( a -a

)...

...).

p r o b a b i l i t y and P t h e p o p u l a t i o n . Now i t seems ( 4 )

nNeN ne (anla,-ane)

c l e a r t h a t b e c a u s e a d e c r e a s e s w i t h n , t h e t e r m s

n The N m i s s i n g t e r m s i n t h e b r a c k e t a l l t e n d t o

i n e-alt w i l l b e n e g l i g i b l e f o r l a r g e t and i n -

have l a r g e r d e c a y c o n s t a n t s and s o we would l i k e d e e d tha's i s e a s y t o show. F o r t h e same r e a s o n ,

t o n e g l e c t them. I n f a c t i t c a n b e shown t h a t t h e n-t.. t e r m s dominate t h e sum, s o i t c a n b e

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979127

t h e y l e a d t o power-law t e r m s a l s o , b u t ' d e c r e a s - i n g more s t e e p l y w i t h t t h a n t h a t from t h e f i r s t term.

We f i r s t c o n s i d e r a f i x e d set o f v a l u e s f o r L2,...LN, s o t h a t we a r e c o n s i d e r i n g

t h

o n l y one p a r t i c u l a r scheme o f N-- o r d e r c a s c a d e , i n v o l v i n g s t a t e s i n t h e g i v e n s e q u e n c e o f Rydberg s e r i e s : L + L N + L N - l + - . . ~ . Having t h i s r e s u l t we must t h e n add i n a s i m i l a r c o n t r i b u - t i o n from e a c h p o s s i b l e s u c h s e t of L ' s . I t

k

t u r n s o u t t o b e s u r p r i s i n g l y e a s y t o p r o v e t h a t , f o r a g i v e n sequence ( L 2 , - - ' kN), t h e sums o v e r n , n ~ , - - - 3 c a n be done i n t h e l a r g e t l i m i t and l e a d t o t h e same r e s u l t a s i n t h e f i r s t - o r d e r c a s e , namely e q u a t i o n ( 2 ) . The i m p o r t a n t p o i n t i s t h a t e a c h s e q u e n c e (Lg,..-P, ) may pro-

N

duce a d i f f e r e n t c o n s t a n t , b u t t h e exponent m i s t h e same. Thus when a l l s e q u e n c e s a r e summed, and even when t h e sum o v e r o r d e r s N i s done, t h e a s y m p t o t i c r e s u l t i s s t i l l t h e same b e c a u s e m depends o n l y o n t h e e x p o n e n t s per- t a i n i n g t o t h e s e r i e s (n,L) and n o t a t a l l on any p r o p e r t y o f t h e i n t e r m e d i a t e s t a t e s . P h y s i c a l l y of c o u r s e t h e p o i n t i s t h a t t h e l A n g e s t p e r i o d s a r e t h o s e g o v e r n i n g t h e f i r s t d e c a y i n t h e c a s c a d e p r o c e s s , and s o i t i s t h e p r o p e r t i e s of t h e o r i g i n a t i n g Rydberg s e r i e s t h a t d e t e r m i n e t h e t- b e h a v i o r of t h e decay c u r v e .

T h i s r e s u l t means t h a t e a c h p o p u l a t e d Rydberg s e r i e s o f l e v e l s (nL) w i l l produce a t e r m i n t h e d e c a y c u r v e which g o e s l i k e t-m(L) a s t-. C l e a r l y t h i s r e s u l t i s much more h e l p - f u l i f o n l y a few s e r i e s a r e i m p o r t a n t , i . e . i f t h e r e i s a r e l a t i v e l y low l i m i t on L. So t h i s a p p r o a c h r e m a i n s o p p o s i t e t o t h a t f o r t h e y r a s t c a s c a d e s , where i t i s assumed t h a t n and L grow l a r g e t o g e t h e r .

CRUDE CLASSICAL PICKUP MODEL

I f we w i s h t o s t u d y ' p o p u l a t i o n s of l e v e l s w i t h h i g h quantum numbers, we s h o u l d b e a b l e t o l e a r n s o m e t h i n g from a c l a s s i c a l model.

We a r e p a r t i c u l a r l y i n t e r e s t e d i n e l e c t r o n p i c k - up from t h e f o i l s u r f a c e which we assume i s l i k e l y t o make a n i m p o r t a n t c o n t r i b u t i o n t o t h e p o p u l a t i o n of t h e s e l e v e l s [ 4 ] . T h i s p r o c e s s w i l l b e s t r o n g l y i n f l u e n c e d by plasma e f f e c t s f o r f a s t h i g h l y c h a r g e d i o n s 151. The s u r f a c e e l e c t r o n s w i l l b e s u b j e c t e d t o a c o m p l i c a t e d

time-dependent f i e l d a s t h e i o n a p p r o a c h e s t h e s u r f a c e a n d t h e n emerges. We want t o g e t a q u a l i t a t i v e i d e a o f how t h e plasma s c r e e n i n g of t h e i o n ' s f i e l d is l i k e l y t o a f f e c t t h e cap- t u r e p r o c e s s -- i n p a r t i c u l a r how t h e r e s u l t i n g p o p u l a t i o n s of h i g h l y e x c i t e d s t a t e s might b e d i f f e r e n t from t h o s e produced i n more f a m i l i a r p r o c e s s e s s u c h a s c h a r g e t r a n s f e r and r a d i a t i v e r e c o m b i n a t i o n . F o r t h i s p u r p o s e we make t h e extreme s i m p l i f y i n g a s s u m p t i o n t h a t t h e s u r f a c e e l e c t r o n is c o m p l e t e l y s h i e l d e d from any know- l e d g e o f t h e i o n ' s a p p r o a c h u n t i l t h e i o n s u d d e n l y emerges w i t h s p e e d v from t h e s u r f a c e a t a t r a n s v e r s e d i s t a n c e r from t h e e l e c t r o n ; a f t e r t h a t we assume t h e e l e c t r o n i s s u b j e c t t o t h e i o n ' s u n s c r e e n e d Coulomb f i e l d . We a l s o assume t h a t t h e e l e c t r o n i s a t r e s t , s i n c e i t s t h e r m a l v e l o c i t y i s l i k e l y t o b e much l e s s t h a n t h e i o n beam v e l o c i t y .

The c o n s e q u e n c e s o f t h i s e x t r e m e model a r e o b v i o u s . F o r a n i o n o f c h a r g e Z ,

t h e e l e c t r o n ' s o r b i t w i l l b e bound i f and o n l y

2 2

i f mv 12 < Ze / r ; t h i s d e f i n e s a c l a s s i c a l c a p t u r e r a d i u s R = 2Z/v 2

.

2 2 2

L=vr and E=-Z /2n ~v 12-Z/r. (5)

T h i s p u t s a n upper l i m i t on t h e o r b i t a l quantum number f o r bound o r b i t s :

a < Lmx w i t h Lmax:vR=2Z/v. ( 6 )

Of c o u r s e t h e s e c l a s s i c a l r e l a t i o n s c a n h a v e no meaning f o r n%1 o r 1-1. Another immediate r e s u l t i s t h a t t h e s e o r b i t s h a v e L =0, which i s a l s o f o u n d i n a n y s i m p l e quantum model b e c a u s e of a x i a l symmetry. I f we g i v e t h e e l e c t r o n some i n i t i a l t h e r m a l v e l o c i t y we would t h e n f i n d some p o p u l a t i o n o f o r b i t s w i t h small LZ, r o u g h l y l i m i t e d by

F i n a l l y we c a n g e t a n e s t i m a t e d p o p u l a t i o n d i s t r i b u t i o n 5 y s i m p l y assuming a u n i f o r m number of s u r f a c e e l e c t r o n s p e r u n i t a r e a and c o n v e r t i n g t h i s d i r e c t l y v i a e q u a t i o n

(5) t o d i s t r i b u t i o n s i n n and L. S i n c e e a c h o r b i t h a s a d e f i n i t e n and a d e f i n i t e L, b o t h

JOURNAL DE PHYSIQUE

determined by t h e i n i t i a l v a l u e of r , t h e two CONCLUSION

v a r i a b l e s a r e c o m p l e t e l y c o r r e l a t e d . I f Pn I n e x p e r i m e n t s where t h e c l a s s i c a l d e n o t e s t h e d i s t r i b u t i o n i n n when P. i s n o t c a p t u r e r a d i u s is l a r g e enough t o i n c l u d e measured, and PP. t h e d i s t r i b u t i o n i n P. when a number o f s u r f a c e e l e c t r o n s , t h e r e may be n i s n o t measured, then we have: a tendency toward p r e f e r e n t i a l l y p o p u l a t i n g

2 (8) s t a t e s o f c e r t a i n v a l u e s o f II = Z/v. I f s o .

pP. da. = ( k / v )P. dn. , e<imax

2 then f o r t i m e s l o n g compared t o t h e mean

P dn = k t (v n + ~ ' / n ) - ~ dn.

n (9) l i v e s of t h e l o w e s t - l y i n g such s t a t e s , t h e

I n g e n e r a l terms t h e p r e d i c t i o n i s f o r t h e p o p u l a t i o n of low-n c i r c u l a r o r b i t s w i t h roughly c o n s t a n t p r o b a b i l i t y almost up t o ESP. and then t h e Rydberg s e r i e s w i t h

max'

L = E p o p u l a t e d w i t h t h e u s u a l n-3 law f o r max

l a r g e n. The d i v i s i o n of p o p u l a t i o n between t h e s e two groups of l e v e l s i s roughly

power-law exponent may r e f l e c t t h e p r o p e r t i e s of t h e c o r r e s p o n d i n g Rydberg s e r i e s ( n , L ) , even though no d i r e c t c a s c a d e s from t h i s s e r i e s t o t h e l e v e l of i n t e r e s t a r e allowed.

ACKNOWLEDGEMENT

Thanks t o L. J . C u r t i s f o r c a l l i n g a t t e n t i o n t o t h i s problem and f o r many s t i m u l a t i n g d i s c u s s i o n s .

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W. Brandt, R.H. R i t c h i e , Nucl. I n s t r . Meth. 132, 43 (1976). A. C r i s p i n , G.N. Fowler, Rev. Mod. Phys. 41, 290 (1970). R.A. McCorkle, G . J . I a f r a t e , Phys. Rev. L e t t . 2, 1263 (1977).

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