CONTINUUM STARK SPECTROSCOPY

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CONTINUUM STARK SPECTROSCOPY

L. Dimauro, T. Bergeman, P. Mcnicholl, H. Metcalf

To cite this version:

L. Dimauro, T. Bergeman, P. Mcnicholl, H. Metcalf. CONTINUUM STARK SPECTROSCOPY.

Journal de Physique Colloques, 1982, 43 (C2), pp.C2-167-C2-175. �10.1051/jphyscol:1982213�. �jpa-

00221824�

JOURNAL DE PHYSIQUE

Colloque C2, supplément au n°ll, Tome 43, novembre 1982 page C2-167

CONTINUUM STARK SPECTROSCOPY

L. DiMauro, T. Bergeman, P. McNicholl and H. Metcalf

Physics Department, State Vniveraity of New York, Stony Brook, New York 11794, U.S.A.

Résumé - Nous présentons l'étude expérimentale et théorique de la photoionisation par échelon via l'état 3P du sodium dans des champs électriques jusqu'à 3,90 kV/cm.

L'appareillage utilise un jet atomique et l'excitation est réalisée à l'aide de deux lasers puisés dont les faisceaux sont orthogonaux au jet atomique. Le premier laser à 589 nm permet l'excitation des niveaux P. Le second laser autour de 409 nm permet la photoionisation autour de la limite d'ionisation EQ en champ nul. Nous avons observé des ondulations de grande largeur dans le courant photoélectrique au-delà de E0. Leur largeur décroit lorsque l'énergie décroit. Dans certains cas elles s'affinent en des raies bien définies mais asymétriques correspondant aux états de Rydberg déplacés par effet Stark. Nous avons calculé la densité de force d'oscillateur pour ces transitions en utilisant Ides fonctions d'ondes hydrogénoîdes modifiées. L'accord avec les résultats expérimentaux est excellent. Nous avons considéré les asymétries comme le résultat d'interférences de Fano entre la partie quasi-discrète du spectre et les canaux d'ionisation. En utilisant des potentiels du coeur appropriés, et en tenant compte du couplage entre états discrets-continus et continus-continus induits par l'interaction avec le coeur, nous avons obtenu des valeurs q et T des paramètres de Fano en bon accord avec nos mesures.

Abstract - We have studied experimentally and theoretically two-step photoioniza- tion through the 3P state of sodium in an electric field of up to 3.90 kV/cm. The apparatus uses an atomic beam crossed by two pulsed laser beams, one at 589 nm for the first step,and the other swept through the zero field threshold E0 near 409 nm for the ionization. We observe broad undulations in the photocurrent above EQ that narrow with decreasing energy and eventually sharpen into well-defined but

asymmetric transitions to Stark-shifted Rydberg states. We calculate the oscillator strength density for the appropriate transitions using modified hydrogenic wave functions and find excellent agreement for the position and relative strength of all the spectral features. We model the asymmetries as Fano-type interferences between quasi-discrete and continuum ionization channels. Using appropriate core potentials, and taking into account core-induced coupling of both discrete-to-continuum and continuum-to-continuum states, we obtain estimates of Fano's q and Y that agree reasonably well with our measurements.

1. Introduction. - Studies of the energy levels of atoms in external electric fields have undergone increased vigor and activity during the last several years.

The advent of tunable dye lasers that can easily excite levels of high principal quantum number n having high electric field sensitivity has allowed the experimental investigation of the Stark effect in a previously inaccessible domain. There are two major reasons for the extraordinary sensitivity of these levels to electric fields: 1) the states are characterized primarily by atomic binding fields that may easily be 105 times weaker than those of lower-lying levels, and 2) there are very closely lying levels of opposite parity that are readily mixed by the electric field thereby allowing large induced dipole moments that enhance the strength of the interaction with the external field. The result is observable Stark shifts in

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1982213

C2- 168 JOURNAL DE PHYSIQUE

o r d i n a r y l a b o r a t o r y f i e l d s (few kV/cm) t h a t amount t o hundreds of cm-l i n t h e o p t i - c a l spectrum where t u n a b l e l a s e r s have r e s o l u t i o n much b e t t e r t h a n 1 cm-1. I n f a c t , t h e p r i n c i p l e e x p e r i m e n t a l l i m i t t o p r e c i s i o n S t a r k s p e c t r o s c o p y i s p r e s e n t l y t h e p r o d u c t i o n and measurement of t h e e l e c t r i c f i e l d s themselves.

The u s u a l t h e o r e t i c a l d e s c r i p t i o n of t h e S t a r k e f f e c t , by p e r t u r b a t i o n t h e o r y a p p l i e d t o t h e f a m i l i a r s p h e r i c a l s o l u t i o n s of t h e Schroedinger e q u a t i o n , i s r a p i d l y pushed beyond t h e l i m i t of v a l i d i t y by e x p e r i m e n t s on a l k a l i atoms n e a r t h r e s h o l d because of t h e v a s t number of high-lying s t a t e s t h a t a r e coupled by t h e f i e l d . How- e v e r , f o r a p u r e l y l / r c e n t r a l p o t e n t i a l and a uniform e l e c t r i c f i e l d ( t h e hydrogen atom S t a r k e f f e c t ) , t h e Schroedinger e q u a t i o n i s s e p a r a b l e i n p a r a b o l i c

c o o r d i n a t e s [I]. The r e s u l t i n g one-dimensional e q u a t i o n s a r e s o l v a b l e by numeric methods. There a r e no t r u e bound s t a t e s because a t a s u f f i c i e n t l y l a r g e d i s t a n c e i n t h e d i r e c t i o n of t h e f i e l d t h e e l e c t r o n h a s p e n e t r a t e d a p o t e n t i a l b a r r i e r and i s no l o n g e r bound t o t h e atom. T h e r e f o r e energy does n o t have d i s c r e t e e i g e n v a l u e s and wave f u n c t i o n s e x i s t f o r a l l v a l u e s of energy (though w i t h widely v a r y i n g a m p l i t u d e a t t h e o r i g i n . ) N e v e r t h e l e s s , t w o of t h e t h r e e o r d i n a r y d i f f e r e n t i a l e q u a t i o n s t h a t s e p a r a t e o u t i n p a r a b o l i c c o o r d i n a t e s have d i s c r e t e e i g e n v a l u e s . The f i r s t s u c h e q u a t i o n i s t h e same a s i n s p h e r i c a l c o o r d i n a t e s and g i v e s t h e f a m i l i a r magnetic e i g e n v a l u e m and t h e s o l u t i o n s 8 ( $ ) = eim9. The e i g e n v a l u e of t h e second e q u a t i o n i s n o t r e l a t e d t o f a m i l i a r p h y s i c a l q u a n t i t i e s such a s m o r II ( a n g u l a r momentum), b u t t o t h e number of n o d e s , n l , on t h e u p f i e l d s i d e of t h e n u c l e u s . The e i g e n v a l u e i s a c t u a l l y t h e s e p a r a t i o n c o n s t a n t Z1, which v a r i e s from 0 t o 1 f o r s t a t e s of i n t e r e s t ; Z 1 (nl

+

^!g

+ ^&

Iml ) / n . From t h e n u m e r i c a l l y computed s o l u t i o n s one can c a l c u l a t e p h y s i c a l o b s e r v a b l e s i n t h e u s u a l way. For example, o p t i c a l t r a n s i t i o n s can be d e s c r i b e d i n terms of t h e w e l l known e l e c t r i c d i p o l e and rotating-wave approx- i m a t i o n . From n e a r t h e z e r o - f i e l d t h r e s h o l d and above, t h e s e hydrogen S t a r k wave- f u n c t i o n s p r o v i d e a good f i r s t approximation t o a l k a l i atom S t a r k e f f e c t s . Some s i g n i f i c a n t e x c e p t i o n s a r i s i n g from t h e p r e s e n c e of t h e c o r e a r e d i s c u s s e d below.

F i g u r e 1 shows t h a t t h e sum of t h e Coulomb b i n d i n g p o t e n t i a l and t h e e x t e r n a l

continuum

4

F i g u r e 1. The combination of Coulomb and uniform f i e l d s produces a p o t e n t i a l w e l l having bound s t a t e s d e g e n e r a t e w i t h unbound ones.

P a r a b o l i c s t a t e s a r e s u b j e c t t o a n a d d i t i o n a l e f f e c t i v e p o t e n t i a l (dashed l i n e ) t h a t may b i n d s t a t e s w i t h energy g r e a t e r t h a n Ec.

S t a r k p o t e n t i a l produces t h r e e energy r e g i o n s , s e p a r a t e d by t h e z e r o - f i e l d i o n i z a t i o n p o t e n t i a l Eo and t h e c l a s s i c a l f i e l d i o n i z a t i o n p o i n t Ec = - 2 q , F = 1 / ( 1 6 n 4 ) [ 2 , 3 ] . A t f i r s t thought i t might seem t h a t t h e r e a r e quasi-bound s t a t e s befow Ec and con- tinuum s t a t e s above i t . Eowever, t h e s e p a r a t i o n i n p a r a b o l i c c o o r d i n a t e s produces an a d d i t i o n a l p o t e n t i a l V t h a t i s analogous t o t h e c e n t r i f u g a l p o t e n t i a l i n

P

s p h e r i c a l c o o r d i n a t e s . For t h e downf i e l d s i d e of t h e n u c l e u s , V (z)=Zl/z+(m -1) 2 /8z2 ( f o r x=y=O). From t h e above e x p r e s s i o n f o r Z 1 , i t i s s e e n t h a t i n c r e a s e s f o r nl>O s o t h a t s t a t e s of h i g h n l , up t o n-m-1, may s t i l l be bound even though t h e y a r e f a r above Ec ( s e e Fig. 1 ) . For a g i v e n n = nl

+

^ng

+

^Iml

+

^1, t h o s e s t a t e s h i g h e s t i n energy (maximum nl) l i e f a r t h e s t t o t h e cathode s i d e and i o n i z e most s l o w l y . Thus between Ec and Eo, t h e r e c o e x i s t many n e a r l y uniform c o n t i n u a of s t a t e s , a l o n g w i t h broad and s h a r p quasi-bound s t a t e s , ( i n s t a n t a n e o u s , f a s t , and slow i o n i z a t i o n

r a t e s ) depending on t h e e f f e c t i v e " p a r a b o l i c " p o t e n t i a l s Vp. Above Eo t h e r e a r e no r e s o n a n t s t a t e s b u t under c e r t a i n c o n d i t i o n s o u t l i n e d below,one o b s e r v e s broad u n d u l a t o r y s p e c t r a l s t r u c t u r e . All o f t h e s e s t a t e s a r i s e from t h e same Hamiltonian and may be computed by t h e same methods. They d i f f e r only i n t h e i r i o n i z a t i o n r a t e , depending on t h e i r p o s i t i o n r e l a t i v e t o t h e e f f e c t i v e p o t e n t i a l b a r r i e r i n t h e t h i r d p a r a b o l i c v a r i a b l e .

2. Experimental D e s c r i p t i o n . - I n o u r e x p e r i m e n t s ( s e e f i g u r e 2) sodium atoms i n a t h e r m a l beam a r e s t e p w i s e i o n i z e d by two l a s e r p u l s e s (one tuned t o D-line e x c i t a - t i o n l i n a c o n s t a n t e l e c t r i c f i e l d . The f i r s t l a s e r p r e p a r e s a 3 P s t a t e and t h e

continuum

- - - _{t o -}

ELECTI*)U DETECTOR

1'"

F i g u r e 2. The energy l e v e l s used i n t h e experiment a r e shown a t t h e l e f t . The c o r e of t h e a p p a r a t u s i s shown a t t h e r i g h t . The l a s e r beams and atomic beam i n t e r s e c t n e a r t h e bottom p l a t e where t h e f i e l d homogeneity i s b e s t . E l e c t r o n s a r e a c c e l e r a t e d upward.

second, swept l a s e r i o n i z e s t h e atom. We u s e v a r i o u s combinations of l i n e a r l y p o l a r i z e d l i g h t (e.g;, a?r means p o l a r i z a t i o n of f i r s t l a s e r i s l t o E f i e l d and second one i s //to E f i e l d ) . The dye l a s e r p u l s e s were of .L 5 n s e c d u r a t i o n , e s s e n t i a l l y s i m u l t a n e o u s , and r e p e a t e d a t 12 Ez. The wavelength was measured t o i 0.5 cm-I w i t h a p r e c i s i o n e t a l o n c a l i b r a t e d w i t h t h e H atomic l i n e a t 4101.7

2.

The l a s e r beams i n t e r s e c t e d t h e atomic beam between two f i e l d p l a t e s spaced about 0.7 cm a p a r t , one of which was a 75% t r a n s p a r e n t mesh. The S t a r k f i e l d a c c e l e r a t e d p h o t o e l e c t r o n s through t h i s g r i d i n t o a m u l t i c h a n n e l p l a t e where t h e y were d e t e c t e d w i t h v e r y h i g h e f f i c i e n c y . The a p p l i e d v o l t a g e measurements were a c c u r a t e t o 0.2%, t h e p l a t e s p a c i n g was measured t o 1%, t h e f i e l d i s t h e r e f o r e known t o about 1%, and i t s inhomogeneity i n t h e i n t e r a c t i o n r e g i o n i s e s t i m a t e d t o b e l e s s t h a n 1%. There was no i o n i z a t i o n p u l s e s o t h a t we o n l y observed " s t a t e s " t h a t f i e l d i o n i z e a t r a t e s g r e a t e r t h a n about

lo5

Hz.

T y p i c a l s c a n s ( f i g u r e s 3 and 4 ) show s p e c t r a l s t r u c t u r e i n t h e p r o d u c t i o n of p h o t o e l e c t r o n s a s t h e f r e q u e n c y of t h e p h o t o i o n i z i n g l a s e r was swept through about

JOURNAL DE PHYSIQUE

E (optical)

-

2 4 0 0 0 cm"

F i g u r e 3. P a r t of t h e spectrum a t 2.15 kV/cm showing broad u n d u l a t i o n s above Eo and narrow f e a t u r e s below

Eo.

cdculoted position of (n,

,

n2.m)

U

420

4 1 0

400

390

380

370

F i g u r e 4. P a r t of t h e spectrum a t 3.90 kV/cm showing t h e c a l c u l a t e d p o s i t i o n s of t h e peaks. Note t h e asymmetric l i n e s h a p e s and t h e r e s u l t i n g s h i f t of t h e peaks. Also n o t e t h a t t h e m=l peaks a r e v e r y n e a r l y symmetrical.

100 cm-I ( A

2

4085A, l i n e w i d t h % = 0 . 4 cm-l)

.

F i e l d changes produced s h i f t s of t h e s p e c t r a i n agreement w i t h o u r c a l c u l a t i o n s . There a r e broad f e a t u r e s above Eo which narrow w i t h d e c r e a s i n g energy and e v e n t u a l l y s h a r p e n i n t o w e l l - d e f i n e d t r a n s i t i o n s

t o S t a r k - s h i f t e d Rydberg s t a t e s . The s h a r p e s t r e s o n a n c e s have lowest v a l u e s of n2 f o r f i x e d nl because t h e y a r e h i g h e s t i n energy f o r g i v e n n = nl+n2+lml+l; t h e e l e c t r o n d i s t r i b u t i o n i s f u r t h e s t t o t h e c a t h o d e s i d e of t h e n u c l e u s .

The p e r c e n t modulation a m p l i t u d e we o b s e r v e i n t h e u n d u l a t i o n s above Eo i s a b o u t t h e same ( 6 ? 2%) f o r nn and on l a s e r p o l a r i z a t i o n s , and f o r 3 2 ~ 1 / 2 and 3 2 ~ 3 / 2 i n t e r - m e d i a t e s t a t e s . No modulations a r e o b s e r v a b l e (amplitude <2%) f o r T O and oa

p o l a r i z a t i o n s . T h i s i s c o n s i s t e n t w i t h o u r c a l c u l a t i o n s t h a t show t h a t modulations appear o n l y f o r mL = 0 + 0 t r a n s i t i o n s and maxima o c c u r a t low v a l u e s of n2 ( b u t n o t always p r e c i s e l y w i t h n2 = 0 ) .

3. T h e o r e t i c a l D i s c u s s i o n , E > E

.-

Some p h y s i c a l i n s i g h t about t h e b r o a d f e a t u r e s above Eo may be g a i n e d by conside%ng t h e extreme c a s e where t h e e f f e c t of t h e Coulomb f i e l d on t h e f i n a l s t a t e i s n e g l e c t e d . The wave f u n c t i o n f o r t h e f i n a l s t a t e

i s t h e n t h a t of an e l e c t r o n i n a uniform f i e l d : a one-dimensional Airy f u n c t i o n c h a r a c t e r i z e d by a c l a s s i c a l t u r n i n g p o i n t and t h e u n i f o r m p o t e n t i a l g r a d i e n t . E l e c t r i c d i p o l e t r a n s i t i o n p r o b a b i l i t i e s depend on t h e i n t e g r a l of t h e s e P i r y f u n c t i o n s m u l t i p l i e d by t h e o r i g i n a l , low-.lying s t a t e s and t h e o p e r a t o r r. A s t h e energy of t h e f i n a l s t a t e i s r a i s e d by s m a l l m-ounts, i t s c l a s s i c a l t u r n i n g p o i n t s h i f t s and t h e nodes and a n t i n o d e s of t h e Airy f u n c t i o n s h i f t p a s t t h e o r i g i n of c o o r d i n a t e s where t h e t r a n s i t i o n moment i n t e g r a l h a s i t s major c o n t r i b u t i o n from t h e tightly-bound o r i g i n a l s t a t e . Thus t h e t r a n s i t i o n p r o b a b i l i t y o s c i l l a t e s w i t h energy i n a simple and c a l c u l a b l e way, showing s t r u c t u r e i n t h e continuum a t e n e r g i e s v e r y much h i g h e r than t h e z e r o f i e l d i o n i z a t i o n t h r e s h o l d , Eo.

Of c o u r s e , i n t h e r e g i o n n e a r t h e o r i g i n where t h e t r a n s i t i o n mom-lt i n t e g r a l i s determined, one cannot n e g l e c t t h e Coulomb p o t e n t i a l . The wave f u n c t i o n i s n o t t h e simple Airy f u n c t i o n suggested above, b u t h a s a cusp a s a r e s u l t of t h e Coulomb w e l l ( s e e Fig. 5 ) . A s t h e e x c i t a t i o n energy i s r a i s e d , t r a n s i t i o n s t o s t a t e s of

STARK WAVE FUNCTIONS

S ^In,,

^{E S ~ )}

Figure 5. The o s c i l l a t o r s t r e n g t h d e n s i t y depends on t h e i n t e g r a l of ( i n i t i a l s t a t e ) x ( t r a n s i t i o n o p e r a t o r ) x ( f i n a l s t a t e ) . The product of t h e

f i r s t two i s p l o t t e d on t h e lowest t r a c e . Other p l o t s show wave f u n c t i o m o f a p a r t i c u l a r nl f o r v a r i o u s e n e r g i e s along z a x i s ( u n i t s depend on e l e c t r i c f i e l d ) , a s w e l l a s wave f u n c t i o n a t one energy f o r a new nl.

h i g h e r nl quantum number a r e allowed, and new c o n t r i b u t i o n s t o t h e wave f u n c t i o n a p p e a r , each having one more node between t h e o r i g i n and t h e c l a s s i c a l t u r n i n g p o i n t . There s t i l l remains some e x c i t a t i o n p r o b a b i l i t y t o t h e former s t a t e , b u t i t s energy dependence i s r a t h e r weak compared w i t h t h a t of t h e t r a n s i t i o n t o t h e newly a c c e s s i b l e s t a t e . Thus an o s c i l l a t i o n i n t h e e x c i t a t i o n r a t e appears with t h e a d d i t i o n of each new node of t h e wave f u n c t i o n between t h e o r i g i n and t h e c l a s s i c a l t u r n i n g p o i n t .

Undulations appear only f o r mi = 0 + 0 t r a n s i t i o n s because t h e s e wavefunctions a r e most c o n c e n t r a t e d a l o n g t h e z a x i s , where they have a s u c c e s s i o n of nodes. The nodes move a s a f u n c t i o n of energy s o t h a t t h e t r a n s i t i o n s t r e n g t h s from lower mQ = 0 s t a t e s e x h i b i t s h a r p maxima. For mQ

#

0 s t a t e s , $ i s z e r o everywhere a l o n g t h e z axis. The t r a n s i t i o n i n t e g r a l s e n t a i l c o n s i d e r a b l e s p a t i a l smoothing of t h e i n t e r f e r e n c e e f f e c t s i f e i t h e r i n i t i a l o r f i n a l s t a t e h a s m i f 0. The r e s u l t a n t

C2-172 JOURNAL DE PHYSIQUE

p e a k s a r e s o broad t h a t t h e r e i s o n l y a n e g l i g i b l e m o d u l a t i o n i n t h e t o t a l t r a n s i - t i o n p r o b a b i l i t y .

These o s c i l l a t o r y f e a t u r e s have been observed i n a t o m i c s p e c t r a and d i s c u s s e d i n s e v e r a l r e c e n t p a p e r s 14-71. There h a s even been some d i s a g r e e m e n t about t h e a p p r o p r i a t e d e s c r i p t i o n f o r t h i s s p e c t r a l s t r u c t u r e i n t h e continuum. S i m i l a r f e a t u r e s i n t r a n s i t i o n s from bound t o continuum s t a t e s had been p r e d i c t e d [8! a s

" i n t e r n a l d i f f r a c t i o n " e f f e c t s o v e r 50 y e a r s ago, and unambiguously observed [9] i n m o l e c u l a r s p e c t r a . These u n d u l a t i o n s a r e c h a r a c t e r i s t i c o f excimer e m i s s i o n

s p e c t r a

[lo]

a s w e l l as o t h e r m o l e c u l a r t r a n s i t i o n s

f113.

I n b o t h t h e a t o m i c S t a r k - i o n i z a t i o n s p e c t r a d i s c u s s e d h e r e and t h e m o l e c u l a r Franck-Condon o s c i l l a t i o n s , t h e s p e c t r a l s t r u c t u r e a r i s e s from u n d u l a t i o n s i n t h e t r a n s i t i o n p r o b a b i l i t y and n o t from t h e p r e s e n c e of r e s o n a n c e s i n t h e continuum of f i n a l energy s t a t e s . I t i s t h e t a s k of continuum S t a r k s p e c t r o s c o p y t o s t u d y t h e s h a p e s , w i d t h s , and p o s i t i o n s of t h e s e s p e c t r a l f e a t u r e s b o t h above and below Eo, and t o p r o v i d e a c o n s i s t e n t , s i m p l e , g e n e r a l d e s c r i p t i o n of t h e observed s p e c t r a .

4. T h e o r e t i c a l D i s c u s s i o n , E < E,.

-

More demanding comparisons w i t h d a t a c a n o n l y b e made w i t h c a r e f u l c a l c u l a t i o n s of a p p r o p r i a t e p h y s i c a l o b s e r v a b l e s . Because t h e s p e c t r u m of h i g h l y e x c i t e d atoms i n e l e c t r i c f i e l d s shows broad a s w e l l a s s h a r p t r a n s i t i o n s , t h e most n a t u r a l s p e c t r a l q u a n t i t y i s t h e o s c i l l a t o r s t r e n g t h d e n s i t y (OSD) d e s c r i b e d by Luc-Koenig and B a c h e l i e r [12]. I n o r d e r t o c a l c u l a t e t h i s , we f i r s t compute n u m e r i c a l f i n a l - s t a t e wave f u n c t i o n s i n p a r a b o l i c c o o r d i n a t e s f o r a Coulomb-plus-Stark f i e l d a s d e s c r i b e d above (one c o o r d i n a t e shown i n f i g u r e 5 ) . We u s e t h e s e w i t h hydrogen-like wave f u n c t i o n s f o r t h e lower s t a t e , a p p r o p r i a t e l y m o d i f i e d f o r t h e quantum d e f e c t . I n o u r e x p e r i m e n t s on sodium t h i s is t h e 3 2 ~ e x c i t e d s t a t e b e c a u s e we u s e a two s t e p e x c i t a t i o n p r o c e s s . To a c c o u n t f o r t h e f i n e s t r u c t u r e of t h i s i n t e r m e d i a t e s t a t e we compute t h e 3 2 ~ d e n s i t y m a t r i c e s i n a n mQmsmI b a s i s b e c a u s e t h e f i r s t (yellow) quantum produces a m i x t u r e of mQ=O and 1 s u b l e v e l s . T h i s computation a c c o u n t s f o r t h e p o l a r i z a t i o n of t h e e x c i t i n g l a s e r , and assumes s t a t i s t i c a l p o p u l a t i o n of ground s t a t e ( 3 2 ~ ) s u b l e v e l s . The continuum d e n s i t y m a t r i x i s d i a g o n a l i n mkmsml b e c a u s e of t h e v e r y weak c o u p l i n g between Q , S and I i n t h e continuum s t a t e s and b e c a u s e a l l e l e c t r o n s a r e d e t e c t e d .

We c a l c u l a t e t h e t r a n s i t i o n moment i n t e g r a l , c a l c u l a t e t h e OSD f o r v a r i o u s f i n a l s t a t e s a t any g i v e n e n e r g y , and t h e n p l o t i t f o r comparison w i t h t h e data[5].

T y p i c a l r e s u l t s , shown i n f i g u r e 6 , d e m o n s t r a t e t h a t t h e p o s i t i o n and approximate i n t e n s i t i e s of v e r y many of t h e s p e c t r a l f e a t u r e s c a l c u l a t e d t h i s way a g r e e v e r y w e l l w i t h t h e d a t a shown i n f i g u r e s 3 and 4 . Comparably good agreement i s found f o r v a r i o u s p o l a r i z a t i o n s of t h e e x c i t i n g l i g h t beams. D e t a i l e d d e s c r i p t i o n of t h e l i n e s h a p e s , h o w e v e r , r e q u i r e s a more c a r e f u l s p e c i f i c a t i o n o f t h e e x p e r i m e n t a l c o n d i t i o n s a s w e l l a s i n c l u s i o n of t h e e f f e c t s o f t h e atomic c o r e e l e c t r o n s on t h e f i n a l s t a t e . More c a r e f u l e x a m i n a t i o n of t h e s h a r p r e s o n a n c e s i n t h e sodium p h o t o i o n i z a t i o n s p e c t r u m shown i n f i g u r e 4 r e v e a l s d e f i n i t e asymmetries; t h e r e i s a c l e a r s h a r p edge and d i p on one s i d e of some of t h e peaks. The resemblance t o t h e c l a s s i c Fano [13] p r o f i l e s found i n many i o n i z a t i o n phenomena h a s l e d u s t o develop a t h e o r e t i c a l model based on i n t e r f e r e n c e between t r a n s i t i o n s t o s h a r p s t a t e s of s p e c i f i c nl and continuum s t a t e s o f t h e same e n e r g y , b u t o f lower nl. T h i s i n t e r - f e r e n c e i s brought a b o u t by c o u p l i n g induced by t h e non-hydrogenic p a r t of t h e c e n t r a l p o t e n t i a l from t h e c o r e e l e c t r o n s . Thus, t h e v a r i o u s components of t h e OSD a t a p a r t i c u l a r e n e r g y shown i n f i g u r e 6 cannot b e s i m p l y added i n c o h e r e n t l y . It i s n e c e s s a r y t o a c c o u n t f o r t h e s t a t e mixing produced by t h e c o r e f i r s t , and t h e n t o s q u a r e t h e sum t o c a l c u l a t e t h e OSD. Such mixing i s v e r y much weaker f o r m#O s t a t e s t h a n f o r m=O b e c a u s e t h e i r wave f u n c t i o n s a r e r e l a t i v e l y s m a l l i n t h e c o r e

( n o n - p e n e t r a t i n g o r b i t s ) . These a r e r e s p o n s i b l e f o r t h e s h a r p , s y m m e t r i c a l p e a k s i n F i g . 4. T h i s mixing a s p e c t of t h e t h e o r e t i c a l d e s c r i p t i o n c o n s t i t u t e s t h e major d i f f e r e n c e between t h e s p e c t r u m of hydrogen [12,14] and t h a t of t h e a l k a l i s .

I n Ref. 1 3 Fano u s e s t h e c o u p l i n g between d i s c r e t e s t a t e s and one o r more c o n t i n u a , a l o n g w i t h r e l a t i v e t r a n s i t i o n i n t e n s i t i e s , t o d e r i v e asymmetric i o n i z a - t i o n s p e c t r a l s h a p e s c h a r a c t e r i z e d by a w i d t h p a r a m e t e r

r

and an asymmetry

F i g u r e 6. C a l c u l a t e d v a l u e s of o s c i l l a t o r s t r e n g t h d e n s i t y a t 3.59 kV/cm f o r t h e t r a n s i t i o n s s t u d i e d h e r e . Note t h e n a r r o w i n g of t h e r e s o n a n c e s t o s h a r p t r a n s i t i o n s a t low e n e r g i e s . The r e l a t i v e s t r e n g t h s of t h e l+l and 0+0 p a r t s depend on t h e e x c i t a t i o n p r o c e s s . The s m a l l numbers r e f e r t o t h e n2 r e g i o n w i t h i n e a c h n l m a n i f o l d . These a r e s h a r p l y d e f i n e d b e c a u s e n 2 i s e q u a l t o t h e number of n o d e s i n TI f u n c t i o n between t h e o r i g i n and t h e p o t e n t i a l maximum.

p a r a m e t e r q . We f o l l o w t h i s a p p r o a c h , t a k i n g t h e u n p e r t u r b e d s t a t e s t o b e t h e hydrogen S t a r k wave f u n c t i o n s d e s c r i b e d above, and t h e l o c a l i z e d sodium i o n

p o t e n t i a l a s t h e i n t e r a c t i o n t e r m , t o c a l c u l a t e

r

and q e x p l i c i t l y . We compute t h e i o n c o r e p o t e n t i a l from Hartree-Fock sodium i o n wave f u n c t i o n s 1151 and p o l a r i z a - b i l i t y t e r m s 1161; t h e r e s u l t p r o v i d e s a c c u r a t e v a l u e s of t h e sodium S , P , and D quantum d e f e c t s . I n s p i t e of t h e a p p a r e n t a p p e a l o f t h i s d i r e c t u s e of t h e i o n p o t e n t i a l a s t h e c o u p l i n g between d i s c r e t e and h y d r o g e n i c continuum c h a n n e l s , t h e r e s u l t g i v e s p a r a m e t e r s

r

and q a b o u t 1 0 t i m e s l a r g e r t h a n t h o s e d e t e r m i n e d by l e a s t - - s q u a r e s f i t t i n g t o o u r e x p e r i m e n t a l l i n e s h a p e s . Although o t h e r k i n d s o f c a l c u l a t i o n s c a n b e done [173, we have chosen t o p u r s u e t h i s one f u r t h e r .

T h i s approach f a i l s b a d l y b e c a u s e i t n e g l e c t s t h e c o u p l i n g between d i f f e r e n t c o n t i n u a and between d i f f e r e n t e n e r g y e i g e n f u n c t i o n s w i t h i n e a c h continuum. To improve t h e a p p r o x i m a t i o n , we " p r e d i a g o n a l i z e " t h e continuum c h a n n e l s by means of t h e Lippmann-Schwinger i n t e g r a l e q u a t i o n ,

w i t h a G r e e n ' s f u n c t i o n t h a t i n c l u d e s a l l continuum c h a n n e l s b u t o m i t s t h e q u a s i - bound s t a t e . The G r e e n ' s f u n c t i o n i s

C2-174 JOURNAL DE PHYSIQUE

Here t h e sum r u n s o v e r j from n =O t o n .=n -1. F ( 5 ) and G(v) a r e s o l u t i o n s of t h e e q u a t i o n s o b t a i n e d by s e p a r a t i n i ' t h e Sch!ded%nger e q u a t i o n i n p a r a b o l i c c o o r d i n a t e s 5 and TI. The wave f u n c t i o n i s $(nl,n2,m) = F ( E ) G ( ~ ) ~ ~ ~ @ / =

,

n o r m a l i z e d t o an o u t g o i n g wave of a m p l i t u d e 1/2.rr. % 1 ( ~ ) i s s h i f t e d 90° from G (q) i n t h e

a s y m p t o t i c r e g i o n , beyond t h e b a r r i e r m t h e

n

p o t e n t i a l . The ? & t e g r a l e q u a t i o n ( 1 ) i s s o l v e d f o r e a c h continuum c h a n n e l by t h e Fredholm m a t r i x i n v e r s i o n p r o c e d u r e [ l 8 ] b e c a u s e t h e Born s e r i e s does n o t converge. A v a r i a b l e mesh g r i d i n t h e 5-rl p l a n e w i t h % 100 p o i n t s w i t h i n a n e f f e c t i v e c o r e r a d i u s of 3 a t o m i c u n i t s y i e l d s n e a r l y

convergent v a l u e s f o r t h e c o u p l i n g e l e m e n t s between t h e q u a s i - d i s c r e t e and continuum c h a n n e l s . The t h e o r e t i c a l v a l u e f o r

r

i s found by summing t h e s q u a r e s of t h e coupl- i n g e l e m e n t s , and r e s u l t s a r e shown i n T a b l e 1 f o r t h r e e r e s o n a n c e s . D i f f e r e n c e s between t h e c a l c u l a t e d and e x p e r i m e n t a l

r

v a l u e s i n Table 1 a r e a t t r i b u t e d t o t h e

l a s e r l i n e w i d t h ( 0 . 5 cm-l) and s c a n n i n g t i m e c o n s t a n t , which were n o t c o n s i d e r e d i n t h e f i t t i n g p r o c e s s . These w i d t h s can n o t be simply combined b e c a u s e of t h e asymmetric l i n e s h a p e s .

The computation of q e n t a i l s a c a l c u l a t i o n of t h e t r a n s i t i o n moments from a lower s t a t e f o r e a c h " p r e d i a g o n a l i z e d " continuum channel; t h e i n t e g r a t i o n e x t e n d s o u t t o 30 a . u . Then q i s found from q-l =

C

qi-l. Table 1 i n d i c a t e s t h a t c a l c u l a t e d and e x p e r i m e n t a l v a l u e s of t h e asymmetry = p a r a m e t e r q a g r e e w e l l -

T a b l e 1

Resonance E x p e r i m e n t a l C a l c u l a t e d

5. Summary

-

I n summary, we b e l i e v e t h a t we have e s t a b l i s h e d a good t h e o r e t i c a l d e s c r i p t i o n of t h e s p e c t r u m of t r a n s i t i o n s t o h i g h l y e x c i t e d s t a t e s i n e x t e r n a l e l e c t r i c f i e l d s t h a t a r e comparable t o o r s t r o n g e r t h a n t h e atomic b i n d i n g f i e l d . The b a s i s f o r t h i s t h e o r y was o u t l i n e d i n Ref. 1121, b u t we have m o d i f i e d i t i n two s u b s t a n t i a l ways. F i r s t , we have c a l c u l a t e d t h e OSD f o r t r a n s i t i o n s from a n e x c i t e d s t a t e t o t h e continuum i n o r d e r t o a p p l y i t t o o u r e x p e r i m e n t a l s t e p w i s e e x c i t a t i o n . Second we have i n c l u d e d some of t h e e f f e c t s o f t h e a t o m i c c o r e , a b s e n t from t h e hydrogen atom, i n o r d e r t o a c c o u n t f o r o u r asymmetric s i g n a l s h a p e s . We have made c a r e f u l measurements of t h e s p e c t r u m of t r a n s i t i o n s t o i o n i z i n g s t a t e s i n sodium i n e x t e r n a l e l e c t r i c f i e l d s of a p p r o p r i a t e s t r e n g t h . The c a l c u l a t e d s p e c t r a l

p a r a m e t e r s a r e i n good agreement w i t h o u r measurements.

We a r e g r a t e f u l t o t h e N a t i o n a l S c i e n c e Foundation (USA) f o r t h e s u p p o r t of t h i s work.

R e f e r e n c e s

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As a n a l t e r n a t i v e , we c o u l d s i m p l y c a l c u l a t e t h e p h o t o i o n i z a t i o n p r o b a b i l i t y vs.

w a v e l e n g t h a n d f i t t h e r e s u l t a n t l i n e s h a p e t o f i n d a n a p p r o p r i a t e q a n d

r.

Such a c a l c u l a t i o n h a s b e e n p e r f o r m e d by D. Harmin a n d s u b m i t t e d t o P h y s i c a l Review L e t t e r s .

L.S. Rodberg a n d R.M. T h a l e r , I n t r o d u c t i o n t o t h e Quantum T h e o r y o f S c a t t e r i n g (Academic P r e s s , New York, 1967) p . 1 5 0 .