STARK EFFECT IN QUASI-HYDROGENIC SPECIES

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STARK EFFECT IN QUASI-HYDROGENIC SPECIES

E. Luc-Koenig, S. Feneuille, J. Lecomte, S. Liberman, J. Pinard, Ahmed Taleb

To cite this version:

E. Luc-Koenig, S. Feneuille, J. Lecomte, S. Liberman, J. Pinard, et al.. STARK EFFECT IN QUASI-HYDROGENIC SPECIES. Journal de Physique Colloques, 1982, 43 (C2), pp.C2-153-C2-166.

�10.1051/jphyscol:1982212�. �jpa-00221823�

JOURNAL DE PHYSIQUE

Colloque C2, supplément au n°ll, Tome 4S, novembre 1982 page C2-153

STARK EFFECT IN QUASI-HYDROGENIC SPECIES

E. Luc-Koenig, S. Feneuille, J.M. Lecomte, S. Liberman, J. Pinard and A. Taleb Laboratoire Aimé Cotton, Centre National de la Recherche Scientifique, Bâtiment 505, 91405 Orsay Cedex, France

Résumé - L'analyse des spectres de photoionisation d'atomes à un électron optique en présence d'un champ électrique peut être faite en partant du modèle hydrogénoïde.

Le problème de l'effet Stark de l'hydrogène peut être résolu de façon exacte, l'ha- miltonien correspondant étant séparable en coordonnées paraboliques. On définit des densités partielles d'états et on montre qu'au-dessus de la limite classique d'ioni- sation par champ le spectre de l'hydrogène comporte des états quasi-discrets super- posés à des états continus, les différents états n'étant pas couplés. Deux expé- riences concernant les spectres de photoionisation Stark du rubidium montrent l'im- portance des perturbations liées à l'interaction spin-orbite, comme par exemple la stabilisation d'un état Stark par le champ électrique.

Abstract - Photoionization spectra of one-electron atoms in the presence of an ex- ternal electric field are analyzed starting from the knowledge of the hydrogen Stark spectrum. The Coulomb-Stark hamiltonian is separable in parabolic coordinates and thus the problem can be exactly solved. Partial densities of continuum states are defined, and it is shown that above the classical field ionization limit the Stark spectrum of hydrogen consists of quasi-stable states embedded in ionization continua.

The different states are not coupled to each other. Two experiments on the photoioni- zation Stark spectra of rubidium atoms are reported. These spectra are strongly perturbed by the spin-orbit interaction which is responsible for striking anomalies such as the field induced stabilization of a state.

1 . Introduction.

Recently a lot of attention has been paid to photoionization spectra of alkali- atoms in the presence of a static electric field 9" . Various experiments using atomic beams and tunable lasers have been carried out and most of the characteris- tics of such spectra have been explained, at least qualitatively, starting from the theoretical knowledge of the hydrogen Stark spectrum. However the Stark spectrum of hydrogen strongly differs from those of alkali-atoms. The Coulomb potential pos- sesses a dynamical symmetry which allows states with the same value of m^ to cross [l]. In alkali-atoms this symmetry is broken and the two concerned states interact and therefore they cannot cross.

The spectrum corresponding to combined Coulombic and Stark potential is a con- tinuum for any value of the energy E . However this continuum character does not play a significant role in the subcritical regime, that is for energies smaller than the classical field-ionization threshold Ec = ~2vJ (atomic units are used and the ionization potential of the unperturbed atom is chosen as the origin for the energy).

In the subcritical range the most important features are the shifts and broadenings of the discrete states due to the electric field. The main properties of the Stark spectrum (structure of Eydberg states, anticrossings, oscillator strengths,...) can be obtained very accurately by diagonalizing the energy matrix including only those states which have approximately the same energy, and by taking into account explici- tely the non hydrogenic character of the spectrum [2].

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1982212

C2-154 JOURNAL DE PHYSIQUE

In the over critical regime E > E c

,

the Stark spectrum of hydrogen consists of quasi-stable states embedded in ionization continua, but, because of the symmetry of the Coulomb potential there is no interaction between the different states. Any perturbation, breaking this symmetry, couples states with different field ionization properties and modifies the properties of the spectrum. This paper is essentially devoted to the study of the over critical region where new and rather unexpected phenomena have been observed.

2. Stark structure and field-ionization properties of hydrogen.

A

-

Exact wavefunctions for the hydrogen Stark effect.

The Stark hamiltonian of hydrogen is se-prable in parabolic coordinates and consequently the problem can be exactly solved [ 3 ]

,

inasmuch as the relativistic corrections are neglected.

The hamiltonian for a hydrogen-like atom of nuclear charge Z in an external electric field $ directed along the +z-axis is

It commutes with both z-projections of the kinetic moment operator L and of the generalized Runge-Lenz vector [I ]

The mutually commuting operators RO

,

LZ and A, form a complete set of operators [5], the eigenvalues of which ( E

,

mj and (22' -2) ) can be used to determine or- thogonal eigenstates of Ho

.

By using the parabolic coordinates [3]

5

= r+ z

,

r) = r

-

^z

,

^{@ =} tan-' y/x

,

the wavefunction is written as imj@

Y(E,~,@) = (~rl)-'/~

~ ( 5 )

~ ( g ) e (2n)-1/~ ( 3 ) and the resulting one-dimensional equations for

5

and r) are

The separation constants Z1 and Z2 represent the "effective charges" res- pectively associated to

5

and g motions; they are coupled through the relation

z1

+ z 2 = z . (5)

Solutions of these equations can be obtained either numerically [5] or semi- analytically from a modified WKB treatment [ 6 ] . These methods do not involve expan- sions in powers of

9

and are valid for any values of E and

5 .

The effective kinetic energies TE and T ( E ~ . 4) differ in the sign of the electric potential and this determines the diffgrence in the behaviour of

F(S)

^and G(~)

.

Indeed classically the electron is located in the regions of space where and Tg are simultaneously positive. Owing to the boundary condition for the va-

T5

riable

5 ,

Z1 is the discrete eigenvalue of the bound-state problem. For fixed values of E

, 9

and

1% 1

the different eigenvalues Z1

( 9 ,

^{E ; n l}

,

^/mi

1)

^{can be}

characterized by the integer n, which is equal to the number of nodes of the function

F(S)

associated with the bound motion. The motion in r) is a free motion, consequently all continuum properties of the wavefunction Y are included

in the G(~) function. Let us remark that the motion in r) is entirely determined by the values of the quantum numbers E

,

mj and n l

,

or by the equivalent set E , mi

,

^Z1

.

Indeed once Z1 is found, the value of Z2 is obtained from (E~. 5).

For any field stength

9 ,

the quantum numbers mj and Z 1 takes only discrete values, but the SchriSdinger equation has a continuous spectrum with respect to the energy E

.

B

-

Parabolic ionization thresholds at negative energy.

From the study of

Tq it is possible to define parabolic ionization thres-

h o l d s which depend n o t only on t h e f i e l d s t r e n g t h

9

o r on t h e t o t a l energy E

,

b u t a l s o on jmQl and on t h e s e p a r a t i o n c o n s t a n t Z? ( i . e . on n l ). Typical drawings of Tn a r e p r e s e n t e d i n Figure 1. For a f l x e d value of

1

^mR

1

⁽

I

^mR

I ^>

^{1 )}

and not t o o h i g h a v a l u e of nl

,

two very d i f f e r e n t t y p s of drawings can be obtai- ned. For an energy s m a l l e r t h a n (:E

9;;

n l

,

^{mi), T} i s p o s i t i v e i n two d i s t i n c t r a n g e s f o r q : t h e f i r s t one corresponds t o smaly q v a l u e s , t h e second one t o

l a r g e q v a l u e s . These two r e g i o n s a r e s e p a r a t e d by a p o t e n t i a l b a r r i e r where

Tn

i s n e g a t i v e . For such s t a t e s spontaneous i o n i z a t i o n is p o s s i b l e i n a quantum mecha- n i c a l t r e a t m e n t owing t o t u n n e l l i n g through t h e b a r r i e r . A s t h e energy i n c r e a s e s t h e b a r r i e r becomes narrower and lower u n t i l i t f i n a l l y disappears a t

E = E,P(S; n l

,

ImQJ)

.

For g r e a t e r energy v a l u e s t h e r e is no p o t e n t i a l b a r r i e r and i o n i z a t i o n i s possible even according t o c l a s s i c a l mechanics. The q u a n t i t y

E:($; n l

,

^{ImRI )} r e p r e s e n t s t h e p a r a b o l i c i o n i z a t i o n t h r e s h o l d f o r s t a t e s n l

,

l m R I

.

For s t a t e s l m Q l = 1 t h e p a r a b o l i c i o n i z a t i o n t h r e s h o l d i s g i v e n by [3]

where Z2 depends i m p l i c i t l y on

9 .

For t h e d i f f e r e n t s t a t e s of a h i g h l y e x c i t e d manifold w i t h n

>>

[mi

1 ,

i t can be shown by u s i n g t h e r e s u l t of t h e p e r t u r b a t i o n t h e o r y t h a t f o r t h e lowest s t a t e n l = 0 Z 2 does not d i f f e r t o much from 1 : f o r such a s t a t e t h e p a r a b o l i c i o n i z a t i o n t h r e s h o l d is approximately e q u a l t o t h e c l a s - s i c a l i o n i z a t i o n energy Ec = - 2 f l

.

On t h e c o n t r a r y f o r t h e most e x c i t e d s t a t e n l - n and n 2 = 0

,

Z2 i s very s m a l l and of t h e o r d e r of l/n

.

Then t h e para- b o l i c i o n i z a t i o n t h r e s h o l d i s given by [7] : $

-

^{E: n/4}

-

0.21 ( E , P ) ~ / ~

.

For f i x e d

9

and mQ

,

t h e p a r a b o l i c i o n i z a t i o n t k r e s h o l d i n c r e a s e s w i t h n l and becomes g r e a t e r t h a n 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 f o r s u f f i c i e n t l y high v a l u e s of n ,

.

F i w e 7

-

T y p i c a l drawing of t h e e f f e c t i v e k i n e t i c energy f o r s t a t e s ] m R l >, 2

.

^E$ i s t h e p a r a b o l i c c r i t i c a l energy.

C

-

Density of s t a t e s .

The n o r m a l i z a t i o n of the wavefunction I(%, E ; n,

,

mQ) p e r u n i t energy range i s determined by a n asymptotic c o n d i t i o n on q

,

which corresponds t o t h e va- l u e 1/(2n) f o r t h e outgoing f l u x IOut through a s u r f ace of c o n s t a n t r) (r) ++-) [4,6]. Near t h e o r i g i n , t h e amplitude of Y is

Y ( F , E ; n l ,mi) = [ 6 ( $ , E ; n l

, lmj111/2

⁽⁽¹¹⁾

I?l, n

1 - 1 "

For chosen v a l u e s of

5 ,

^{l m R l and n l}

^{, %}

is a continuous f u n c t i o n of t h e energy, which i s c a l l e d t h e p a r t i a l d e n s i t y of s t a t e s w i t h quantum numbers _{l m L l} and n l

.

This f u n c t i o n does n o t vary monotonically a s it i s shown i n F i g u r e 2

.

For e n e r g i e s s m a l l e r t h a n t h e p a r a b o l i c c r i t i c a l energy

~ $ 9 ;

n ,

,

ImR

1

⁾ and f o r a s m a l l n l - v a l u e ,

6

behaves a s a q u a s i d i s c r e t e spectrum : l t i s vanishingly s m a l l except i n v e r y narrow energy r a n g e s En where i t p r e s e n t s very s h a r p peaks w i t h a Lorentzian p r o f i l e of width

rn .

The2 d i f f e r e n t resonances o c c u r r i n g i n t h e curve

% ( 9 , E ; n l

,

ImR

I

⁾ can be l a b e a l e d w i t h t h e quantum number n2 which i s e q u a l t o t h e number of nodes of t h e ~ ( q ) f u n c t i o n i n s i d e t h e c l a s s i c a l bound r e g i o n ( q 2

<

r)

<

q 1 Figure 1 ) . The e n e r g i e s E!, of t h e q u a s i d i s c r e t e s t a t e s c a n be determined v e r y a c c u r a t e l y from p e r t u r b a t & t h e o r y [ 3 ] . The width of t h e resonance, which i s r e l a t e d t o t h e p r o b a b i l i t y of i o n i z a t i o n of t h e q u a s i - s t a b l e s t a t e by

t u n n e l l i n g through t h e p o t e n t i a l b a r r i e r

,

i n c r e a s e s w i t h t h e energy but remains

JOURNAL DE PHYSIQUE

s m a l l e r t h a n lom2 cm-I. For e n e r g i e s g r e a t e r t h a n E g ( 9 ; n l

,

^]mil)

,

t h e p a r t i a l d e n s i t y of s t a t e s looks r e a l l y l i k e a continuous spectrum s i n c e no p o t e n t i a l b a r r i e r remains i n t h e q-motion. A s m a l l number ( 1 o r 2) of broad unresolved s t r u c t u r e s c a n be observed above E!

,

b u t t h e s e s t r u c t u r e s d i s a p p e a r r a p i d l y w i t h i n c r e a s i n g energy. The p a r t i a l d e n s i t y of continuum s t a t e s becomes v e r y s m a l l a t t h e energy Ed(Y; n l

,

/mi 1) which i s g r e a t e r t h a n t h e z e r o f i e l d i o n i z a t i o n energy. The d i s a p - pearance of t h e s t a t e s nl

,

lmR

I ,

^{f o r E}

^>

Ed can be e x p l a i n e d by r e c a l l i n g t h a t

e

i s d i r e c t l y r e l a t e d t o t h e p r o b a b i l i t y of f i n d i n g t h e e l e c t r o n n e a r t h e nucleus ( E ~ . 7) ; consequently t h e p a r t i a l d e n s i t y of s t a t e s nl

,

lmQI i s n o t n e g l i g i b l e only a t s u c h e n e r g i e s t h a t Tg and TV a r e simultaneously p o s i t i v e f o r s m a l l

5

and q

.

These c o n d i t i o n s a r e f u l f i l l e d f o r O , < Z 1 $ Z because the " e f f e c t i v e charges" a s s o c i a t e d w i t h t h e

5

and q motions correspond t h e n t o a t t r a c t i v e

F i m e 2

-

Typical energy-dependence of t h e p a r t i a l d e n s i t y of s t a t e s f o r d i f f e r e n t n l - v a l u e s and f o r a g i v e n f i e l d s t r e n g t h .

p o t e n t i a l s . For f i x e d n l and {mil

,

Z i s a d e c r e a s i n g f u n c t i o n of E

,

which v a n i s h e s f o r E = Ed

(9;

n ,

,

]mi

I .

I n a i i m i l a r way, one c a n e x p l a i n t h a t a s t a t e w i t h a s u f f i c i e n t l y high value of n l a p p e a r s a t a p o s i t i v e energy E!($; n l

,

I m R / ) corresponding t o Zl = Z

.

For such a h i g h n l - v a l u e , e x h i b i t s a very broad s t r u c t u r e e x t e n d i n g from E: t o E

,

b u t no resonance e x i s t s a t E

<

E: . ( ~ i ~ . 2 ) . Let u s remark t h a t the e n e r g i e s

E ~ P ~ ;

^{n l}

^,

^[mJl) and ~ : ( b ; n l

,

lmil)

>

0

,

which correspond t o well-defined v a l u e s f o r Z 1 r e s p e c t i v e l y 0 and Z )

,

c a n be o b t a i n e d by c o n s i d e r i n g t h e e q u a t i o n ( 4 a ) a s a one-dimensional e i g e n v a l u e problem w i t h r e s - p e c t t o t h e energy E

.

The t o t a l d e n s i t y of s t a t e s w i t h a g i v e n mJ v a l u e ,

'e (9,

E ; mJ) i s t h e sum over a l l p a r t i a l d e n s i t i e s *& ( $, E ; nl

,

^mi!

.

Above t h e zero-f i e l d i o n i z a t i o n l i m i t , t h e t o t a l d e n s i t i e s of s t a t e s c o n s l s t of very broad and numerous s t r u c t u - r e s . Consequently t h e energy dependence of t h e t o t a l d e n s i t y of s t a t e s e x h i b i t s o s c i l l a t i o n s w i t h an almost n e g l i g i b l e modulation depth even a t r e l a t i v e l y high f i e l d s t r e n g t h (-100 k ~ / c m ) . Furthermore no s i g n i f i c a n t d i f f e r e n c e i s observed i n

$

( g ,

E ; \ m i ( ) a s I m J J v a r i e s .

D

-

Density of o s c i l l a t o r s t r e n g t h s .

Recent experiments on t h e S t a r k e f f e c t of one-electron s p e c t r a , such a s Rb [8] and Na [9,10] have shown t h a t t h e p h o t o i o n i z a t i o n c r o s s s e c t i o n p r e s e n t s o s c i l l a t i o n s e x t e n d i n g n e a r and above t h e z e r o - f i e l d i o n i z a t i o n l i m i t , t h e s e s t r u c - t u r e s depending s t r o n g l y on 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 i g h t . These e x p e r i - mental d a t a a r e r e l a t e d t o t h e d e n s i t y of o s c i l l a t o r s t r e n g t h s f o r single-photon e x c i t a t i o n from a weakly e x c i t e d s t a t e of energy Ei

,

c o n s i d e r e d a s a d i s c r e t e s t a t e c p i even i n t h e presence of a n e l e c t r i c f i e l d . The p a r t i a l d e n s i t y of o s c i l - l a t o r s t r e n g t h s f o r a t r a n s i t i o n from c p i towards the continuum s t a t e

Y ( F , E ; n l ,mi) _. .- i s d e f i n e d by

d f / m ( g , E ; n l

.

^mi) = 2 / 3 ( ~ - E i ) ] < c p i ] r q ] ~ ( F , E ; n ,

,

mi)>12

+

^{( 8 )}

where r = z f o r n- p o l a r i z a t i o n and ( x f i y ) m f o r a- p o l a r i z a t i o n e . Experi- mentally! s t a r t i n g from a pure w e l l d e f i n e d i n i t i a l s t a t e cpi and u s i n g p o l a r i z e d l i g h t , one can observe only t h e t o t a l d e n s i t y of o s c i l l a t o r s t r e n g t h s

df/dE(F, E ; mR) which i s t h e sum over t h e p a r t i a l d e n s i t i e s of o s c i l l a t o r s t r e n g t h s f o r a l l nl

-

channels.

The p a r t i a l d e n s i t y of o s c i l l a t o r s t r e n g t h s does n o t depend d i r e c t l y on t h e p a r t i a l d e n s i t y of s t a t e s ; i t i s a nondiagonal q u a n t i t y i n which two d i f f e r e n t or- b i t a l s cpi and Y appear. Thus t h e energy-dependence of % and df/& can be very d i f f e r e n t . Therefore f o r i n t e r p r e t i n g t h e p h o t o i o n i z a t i o n S t a r k s p e c t r a n e a r t h e i o n i z a t i o n l i m i t , t h e important q u a n t i t y t o be c o n s i d e r e d 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 r a t h e r t h a n simply t h e d e n s i t y of s t a t e s . This important r e s u l t was s t r e s s e d from e a r l i e r numerical c a l c u l a t i o n s on t h e p h o t o i o n i z a t i o n s p e c t r a of

t h e ground s t a t e of hydrogen i n t h e presence of a n e x t e r n a l e l e c t r i c f i e l d [ I I ] : only t h e e x p l i c i t c a l c u l a t i o n of t h e energy dependence of t h e t o t a l d e n s i t y of os- c i l l a t o r s t r e n g t h s can g i v e r e l i a b l e informations on t h e s t r u c t u r e of t h e photoioni- z a t i o n s p e c t r a n e a r and above t h e z e r o - f i e l d i o n i z a t i o n l i m i t . Nevertheless some q u a l i t a t i v e p r e d i c t i o n s can be o b t a i n e d from very simple arguments.

F i r s t l y , important c a n c e l l a t i o n e f f e c t s occur i n t h e 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 m a t r i x element < c p i l r q l ~ ( F , E ; n l

,

mi)> when t h e r e e x i s t w e l l d e f i n e d symme- t r y p r o p e r t i e s . The wavefunction Y($,E ; n ,

,

mR) h a s g e n e r a l l y no symmetry pro- p e r t y e x c e p t i n t h e energy r a n g e E

-

~ ~ ( 9 ; n l

,

mR) i n which Z I ^" Z 2

-

^{~ / 2}

^,

^{where Y}

i s symmetrical w i t h r e s p e c t t o t h e z=O plane. Indeed, i n t h e neighbourhood of t h e n u c l e u s , TS and Tq a r e e q u a l , inasmuch a s t h e S t a r k p o t e n t i a l s can be n e g l e c t e d i n Eq. 4. The energy ~ ~ ( 9 ; n l

,

^{mR) i s} l o c a t e d midway between E ~ ( F ; n l

,

^{mR) and}

~ ~ ( 9 ; ; n l

,

n R )

.

The wavefunction cpi f o r t h e weakly e x c i t e d s t a t e R i mRi

,

i s symmetrical o r a n t i s y m m e t r i c a l w i t h r e s p e c t t o t h e z=0 plane according t o whether R i +m is even o r odd. The t r a n s i t i o n o p e r a t o r is a n t i s y m m e t r i c a l f o r

4 .

a n- p o l a r i z a t i b n (mR. =mi) o r symmetrical f o r a a- p o l a r i z a t i o n ( ImR

-

^{m R .}

I

^{= I} )

.

It f o l l o w s t h a t t h e p + t i a l d e n s i t y of o s c i l l a t o r s t r e n g t h s v a n i s h e s a t t h e 'energy Es i f one c o n s i d e r s a b s o r p t i o n e i t h e r from a symmetrical s t a t e w i t h n - p o l a r i z e d

F i g u r e 3

-

Energy-dependences of t h e p a r t i a l d e n s i t y of ex- c i t e d s t a t e s (---) and t h e p a r t i a l d e n s i t y of o s c i l l a t o r s t r e n g t h s ( ) i n t h e photo- a b s o r p t i o n spectrum from t h e ground s t a t e of hydrogen to- wards e x c i t e d s t a t e s w i t h a g i v e n nl-value

.

^The t o t a l d e n s i t y is a l s o p r e s e n t e d . O-. 0 and O-. 1 correspond r e s p e

1 8 % total - X I %

c t i v e l y t o n and a p o l a r i z a t i o n s .

9

= 77kW/cm l i g h t , o r from an antisymmetrical s t a t e w i t h a - p o l a r i z e d l i g h t . In t h e o t h e r two c a s e s no c a n c e l l a t i o n e f f e c t s appear i n t h e 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 m a t r i x element a t E =Es

.

F i g u r e 3 p r e s e n t s t h e energy dependences of b o t h t h e p a r t i a l d e n s i t y of continuum s t a t e s and t h e p a r t i a l d e n s i t y of o s c i l l a t o r s t r e n g t h s i n t h e photoabsorgtion spectrum from t h e symmetrical ground s t a t e .Ei = mi. = 0 of hydrogen f o r 9 = 77 k ~ / c m

,

and w i t h a n- and a a - p o l a r i z a t i o n of t h e l$ght. For t h e n- p o l a r i z a t i o n , df/dE v a n i s h e s midway between E: and Ed

,

and t h e r e f o r e e x h i b i t s a r e l a t i v e l y narrow resonance at Eg

.

On t h e c o n t r a r y f o r t h e a - p o l a r i z a t i o n no c a n c e l l a t i o n e f f e c t s occur and and df/dE a r e almost p r o p o r t i o n a l . Consequent- l y t h e t o t a l d e n s i t y of o s c i l l a t o r s t r e n g t h s p r e s e n t s a s i g n i f i c a n t l y modulated s t r u c t u r e i n t h e rc-spectrum i n c o n t r a s t t o t h e almost s t r u c t u r e l e s s a - spectrwn.

In t h e p h o t o i o n i z a t i o n of a n e x c i t e d s t a t e Ri

,

mRi one g e n e r a l l y r e a c h e s two d i f f e r e n t v a l u e s of R

,

R = j i 2 1

,

i n t h e f i n a l s t a t e . Then i n t e r f e r e n c e e f f e c t s between t h e two channels Ri+l and Ri- I a p p e a r , and they a r e a s s o c i a t e d w i t h ad- d i t i o n a l c a n c e l l a t i o n e f f e c t s i n t h e p a r t i a l d e n s i t y of o s c i l l a t o r s t r e n g t h s d f / d ~ ( F , E ; n ~ ,mx)

,

f o r example a t E = E a such a s Z 1 = z ?

.

^By s t u d y i n g t h e

JOURNAL DE PHYSIQUE

e f f e c t i v e k i n e t i c e n e r g i e s ( E ~ . 4 ) , i t c a n be shown t h a t df/dE h a s a second mini- mum a t E = E b corresponding t o Z1 = Z - 2 7

.

^{When df/dE:} h a s s e v e r a l minima sepa- r a t e d by more o r l e s s s h a r p peaks, t h e n t h e t o t a l d e n s i t y of o s c i l l a t o r s t r e n g t h s e x h i b i t s o s c i l l a t i o n s with a s i g n i f i c a n t modulation depth. D e s t r u c t i v e i n t e r f e r e n c e e f f e c t s between t h e two channels Ri+l and Ri-1 appear i n t h e s p e c t r a o b t a i n e d w i t h n- o r a - p o l a r i z e d l i g h t , if t h e magnetic quantum number of t h e lower s t a t e v e r i f i e s r e s p e & i v e l y I m R .

I ^\<

^{R i - I o r I m R .}

I \<

Ri-2

.

The presence of c a n c e l l a - t i o n s i n t h e p a r t i a l d e n s i b of s t a t e s , which do n o t a r i s e from symmetry p r o p e r t i e s , h a s been p o i n t e d o u t i n t h e e x a c t c a l c u l a t i o n of t h e two s t e p p h o t o i o n i z a t i o n spec- t r a from t h e 3 p m R . s t a t e of hydrogen i n t h e presence of a n e l e c t r i c f i e l d [12].

The r e s u l t s a r e r e p a r t e d i n F i g u r e 4 f o r d i f f e r e n t 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 and p h o t o i o n i z i n g l i g h t . Fos t h e n n and t h e a+< s p e c t r a , i . e . f o r t h e t r a n s i t i o n s 3p mR , = 0 -. mR = 0 and 3p m R . = 1 -. mR = 0

,

t h e p a r t i a l c r o s s - s e c t i o n s have two minim&, because t h e c o n t i n u a 's and d a r e reached. On t h e c o n t r a r y f o r t h e u+u+

spectrum ( t r a n s i t i o n 3 p mi. = 1 + m j = 2)

,

df/dE does n o t v a n i s h s i n c e only t h e d continuum i s populated and t h e r e i s no c a n c e l l a t i o n e f f e c t due t o symmetry proper- t i e s a t E =Es

.

For the spectrum a+% ( 3 ~ mi. = I + mR

:

^{1 )}

^,

^{df/m h a s a} a i n g l e minimum a t E =Es due t o symmetry p r o p e r t i e s . 'For t h e f lrst two c a s e s , important modulations a r e observed i n t h e t o t a l d e n s i t y of o s c i l l a t o r s t r e n g t h s , while i n t h e t h i r d one t h e t o t a l spectrum i s very smooth and i n t h e l a s t one only a s l i g h t l y mo- d u l a t e d s t r u c t u r e i s observed.

d -..

---

_{: n, fixed}

1-0

F i g u r e 4

-

The same a s i n f i g u r e 3 but f o r photoabsorp- t i o n from t h e 3 p s t a t e of hydrogen. mj. ^-f mR indica- t e s t h e magnetic o r b i t a l quantum numbers of t h e 3p and of t h e e x c i t e d - s t a t e .

Each modulation above t h e z e r o - f i e l d i o n i z a t i o n l i m i t corresponds t o t h e apgea- r a n c e of a new channel a s s o c i a t e d w i t h a h i g h e r nl-value

.

Consequently each peak is l o c a t e d a t t h e p a r a b o l i c i o n i z a t i o n t h r e s h o l d of t h e corresponding nl -channel.

As s a i d p r e v i o u s l y t h e p o s i t i o n of t h e s t r u c t u r e s c a n be o b t a i n e d by determining t h e e i g e n v a l u e s of t h e one-dimensional potential. g i v e n f o r ImR

1

^{= l by} (-Z/Z + 9 z )

.

This model p o t e n t i a l was i n t r o d u c e d from s e m i - c l a s s i c a l arguments [ 8 ] ; t h e e x a c t c a l c u l a t i o n a l l o w s us t o j u s t i f y t h e r e s u l t s o b t a i n e d i n t h e c l a s s i c a l approach. The s t r u c t u r e s i n t h e t o t a l d e n s i t y of o s c i l l a t o r s t r e n g t h s a r e v e r y s h a r p when c o n t i - nuum s t a t e s mR=O a r e e x c i t e d , because t h e p a r t i a l d e n s i t y of s t a t e s mR=O r i s e s very a b r u p t l y towards i t s maximum value a t E =EP ( ~ i g u r e s 3 and 4 show t h e d i f f e - r e n c e i n behaviour of t h e p a r t i a l d e n s i t y of s t a x e s w i t h mj=O and mR#O ). A t E=E$ t h e wavefunction of a s t a t e mj=O i s h i g h l y c o n c e n t r a t e d a l o n g t h e p o s i t i v e z-axis w i t h n l - n o d e s , and i t corresponds t o t h e c l a s s i c a l s t a b l e t r a j e c t o r y which undergoes c y c l i c a l motion about t h e p o s i t i v e z-axis [ 8 ] . These c o n s i d e r a t i o n s a l l o w u s t o e x p l a i n why s t r u c t u r e s i n t h e S t a r k p h o t o i o n i z a t i o n s p e c t r a a r e p r e f e r e n t i a l l y

observed whenever upper s t a t e s , m j = O a r e e x c i t e d , s p e c i a l l y f o r m R . = O + mi= 0 t r a n s i t i o n s ( s e e Figures 3 and 4 ) . I n such c a s e s , a l l t h e t h r e e f u n c t i o n s o c c u r r i n g i n t h e t r a n s i t i o n matrix element ((Pi

,

Y

,

^z) t a k e very l a r g e v a l u e s on t h e z - a x i s , and t h e p a r t i a l d e n s i t y of o s c i l l a t o r s t r e n g t h s h a s very h i g h maxima, s p e c i a l l y a t E

= ~ g .

I n t h e o t h e r c a s e s , t h e s e t h r e e f u n c t i o n s have s i g n i f i c a n t v a l u e s i n d i f f e - r e n t s p a t i a l r a n g e s s o d f / d ~ undergoes c o n s i d e r a b l e s p a t i a l smoothing of t h e in- t e r f e r e n c e e f f e c t s .

3. Non-hydrogenic e f f e c t s i n t h e S t a r k p h o t o i o n i z a t i o n s p e c t r a of Rb

.

When a p e r t u r b a t i o n b r e a k s t h e dynamical symmetry s p e c i f i c t o t h e Coulomb-Stark p o t e n t i a l a d i f f e r e n t f i e l d ioniza.tion p r o c e s s occurs [I?] : s t a t e s which a r e s t a b l e w i t h r e s p e c t t o pure t u n n e l l i n g e f f e c t , c a n i o n i z e due t o mixing w i t h an a l r e a d y

i o n i z i n g s t a t e . This process a p p e a r s i n many-electron atoms i n which t h e c e n t r a l p o t e n t i a l a c t i n g on t h e o u t e r e l e c t r o n i s no l o n g e r Coulombic. Thus s t a t e s w i t h t h e same mR v a l u e b u t w i t h d i f f e r e n t n , quantum number, can be s t r o n g l y mixed. The symmetry can a l s o be broken by r e l a t i v i s t i c i n t e r a c t i o n s c o u p l i n g s t a t e s w i t h t h e same m. v a l u e . This k i n d of mixing e x i s t s in a l l atoms even i n Wdrogen. Any

J .

w r t u r b a t i o n which couples a s t a b l e s t a t e t o a continuum, can s t r o n g l y p e r t u r b t h e p h o t o i o n i z a t i o n s p e c t r a . In t h i s s e c t i o n we describe two r e c e n t experiments d e a l i n g w i t h t h e S t a r k p h o t o i o n i z a t i o n s p e c t r a of Rb

.

Rather unexpected phenomena have been observed, which have been a t t r i b u t e d t o symmetry b r e a k i n g due t o v e r y weak per- t u r b a t i o n induced by t h e s p i n - o r b i t i n t e r a c t i o n .

A

-

Field-induced s t a b i l i z a t i o n of a w e l l d e f i n e d S t a r k s t a t e of Rb

In t h i s experiment 1141, ^{R b} atoms i n t h e i r ground s t a t e a r e photoionized by t h e l i g h t of a single-mode pulsed dye l a s e r propagating i n a d i r e c t i o n perpendi- c u l a r t o b o t h t h e atomic beam and t h e e l e c t r i c f i e l d . The produced i o n s a r e counted and numerous h i g h - r e s o l u t i o n p h o t o i o n i z a t i o n s p e c t r a a r e r e c o r d e d f o r d i f f e r e n t f i e l d s t r e n g t h s

9 .

The e x p e r i m e n t a l map of e x c i t e d s t a t e s observed w i t h a n-po-

l a r i z a t i o n of t h e l i g h t i s p r e s e n t e d i n P i g . 5. The s t u d i e d r e g i o n l i e s very c l o s e t o t h e saddle-point l i n e (E = -2@ )

.

Studying t h e time of a r r i v a l of t h e produ- ced i o n s o n t o t h e d e t e c t o r a f t e r t h e s h o r t l i g h t p u l s e e x c i t a t i o n , a n anomalous dis- p e r s i o n i n time i s observed. This anomaly i s l o c a t e d very p r e c i s e l y on curve A

.

Figure 5

-

Experimental map of t h e s t a t e s of rubidium i n a s t a t i c e l e c t r i c f i e l d observed from t h e ground s t a t e w i t h a n - p o l a r i z a t i o n .

The l i f e t i m e of s t a t e A e x h i b i t s an anomalous b e h a v i o w ; t h e l o c a t i o n i s i n d i c a t e d by the star.

SPL Saddle p o i n t l i n e E = -2%

JOURNAL DE PHYSIQUE

F i m e 6

Lifetime of t h e s t a t e A a g a i n s t t h e e l e c t r i c f i e l d

9 .

The measurement of t h e l i f e t i m e of t h e s t a t e A a s a f u n c t i o n of t h e e l e c t r i c f i e l d e x h i b i t s a n important i n c r e a s e (a f a c t o r of 30) i n ^s very narrow range of f i e l d s t r e n g t h ( A $ 8 ~ / c m ) a s i t i s shown i n Fig. 6. The f i e l d - i o n i z a t i o n p r o c e s s of t h e s t a t e A cannot be a s c r i b e d t o t u n n e l l i n g e f f e c t , because t h e i o n i z a t i o n r a t e does not i n c r e a s e r e g u l a r l y w i t h i n c r e a s i n g f i e l d s t r e n g t h . It a r i s e s t h e n necessa- r i l y from s t a t e mixing. We have searched f o r a neighbouring s t a b l e s t a t e by r e c o r - d i n g t h e s p e c t r a o b t a i n e d by adding a s t r o n g pulsed e l e c t r i c f i e l d , but no o t h e r s t a b l e s t a t e has been observed. So t h e anomaly must be e x p l a i n e d a s r e s u l t i n g from t h e mixing of t h e s t a t e A with an u n s t a b l e s t a t e . Jn f a c t t h e mixing of a s t a b l e s t a t e w i t h a r a p i d l y i o n i z i n g s t a t e may sometimes l e a d t o a l o c a l i z e d decrease i n t h e f i e l d i o n i z a t i o n r a t e of t h e s t a b l e s t a t e , a s it is shown by studying a r a t h e r simple model. We c o n s i d e r two n o n - i n t e r a c t i n g S t a r k s t a t e s of hydzogen, which c r o s s a t a g i v e n f i e l d s t r e n g t h

PC

^; t h e d i f f e r e n c e i n t h e e n e r g i e s of t h e s e s t a t e s va- r i e s l i n e a r l y w i t h

5

:

-

^E2(

9 )

⁼

~ ( 9 -

&)

.

⁽⁹⁾

The i o n i z a t i o n r a t e f o r the two s t a t e s i s determined by t u n n e l l i n g and we suppose t h a t t h e f i r s t s t a t e i s s l i g h t l y unstable : i t s width TI i n c r e a s e s r a p i d l y with

9

C151 :

1 / 2 I ' , ( F ) = a e x p ( ~ ~ )

.

⁽¹⁰⁾

This resonance i s a s s o c i a t e d w i t h a narrow continuum w i t h a d e n s i t y of s t a t e s [5]

given by a normalized Lorentzian f u n c t i o n

e , ( E , @ ) =

f

r 1 ( 9 ) / [ ( E - E ~ ( B ) ) ~

+

( P , ( F ) / ~ ) ~ ]

.

^{( 1 1 )}

A s

9

i n c r e a s e s t h e width of t h e Lorentzian i n c r e a s e s and i t s h e i g h t d e c r e a s e s . In t h e neighbourhood of the n u c l e u s t h e wavefunction of t h i s s t a t e , normalized p e r u n i t energy, i s given by ( c f . Eq. 7) :

Y ~ E , , = , E , I

,

~

l a ] - o

~ ~

.

^{( 1 2 )}

We suppose t h e second s t a t e t o be v e r y s t a b l e : i t would correspond t o a quasi-dis- c r e t e s t a t e w i t h a n e g l i g i b l e width. This second s t a t e i s d e s c r i b e d by t h e wave- f u n c t i o n fp2(r) which does n o t desend on $

.

In t h e r e a l atom t h e s e two s t a t e s are coupled through a p e r t u r b a t i o n

~ ( 2 )

which v a n i s h e s f o r l a r g e

I f 1 .

Due t o t h i s c o u p l i n g t h e q u a s i - d i s c r e t e s t a t e i o n i - z e s and i t s k i d t h i s e q u a l t o [16] :

A t a g i v e n f i e l d

9 ,

t h e width of t h e s t a b l e s t a t e i s p r o p o r t i o n a l t o t h e d e n s i t y of s t a t e s d e s c r i b i n g t h e broad s t a t e measured a t t h e energy of t h e d i s c r e t e s t a t e . It i s obvious t h a t

r ^{( 3 )}

does n o t i n c r e a s e r e g u l a r l y w i t h i n c r e a s i n g

9 .

Generally t h e l i f e t i m e r 2 ( 9 f cc [P?( P ) ] - l d e c r e a s e s i n a g i v e n range of

9 ,

a phenomenon which c a n be e x p l a i n e d s ~ m p l y by d i a g o n a l i z i n g t h e complex-energy m a t r i x formed w i t h , t h e unperturbed e n e r g i e s of t h e two s t a t e s , t h e i r decay r a t e s and t h e i r mutual coupling [IT]. For p a r t i c u l a r v a l u e s f o r t h e s e t of parameters A ,

PC ,

a and $ a more i r r e g u l a r v a r i a t i o n i s observed a s i t is shown i n F i g u r e 7 : i n a d d i t i o n t o t h e p r e v i o u s l y d e s c r i b e d minimum, t h e r e i s a narrow maximum f o l l o w e d by a second minimum. I n a g i v e n range of f i e l d s t r e n g t h s , r2 p r e s e n t s a maximum corresponding t o a s t a b i l i z a t i o n of t h e s t a t e . In c o n c l u s i o n , by t a k i n g i n t o ac- count e x p l i c i t e l y both t h e f i e l d and energy dependences of t h e d e n s i t y of s t a t e s a s s o c i a t e d w i t h t h e u n s t a b l e s t a t e i t i s p o s s i b l e t o show t h a t t h e a n t i c r o s s i n g of t h i s s t a t e w i t h a s t a b l e s t a t e l e a d s sometimes t o a very w e l l l o c a t e d i n c r e a s e of t h e l i f e t i m e of t h e s t a b l e s t a t e . To i n t e r p r e t t h e anomaly i n t h e f i e l d - i o n i z a t i o n

F i m e

'Z -

C a l c u l a t e d f i e l d dependence f o r t h e l i f e t i m e 22 of t h e s t a b l e s t a t e ( 2 1 , 4 , 1 )

,

coupled t o t h e u n s t a b l e s t a t e ( 2 3 , 0 , 0 ) through t h e s p i n - o r b i t e i n t e r a c t i o n .

9

= 2132 v/cm

-

1 A = -0.0241 V-I

a = 9.134 cm-I $ = 0 . 0 2 3 8 7 ' cm r a t e of t h e s t a t e A

,

a s due t o t h e p o c e s s d e s c r i b e d above

,

i t i s necessary t o i d e n t i f y , i n t h e corresponding range ( E ,

5 )

of t h e hydrogen spectrum, two s t a t e s w i t h d i f f e r e n t f i e l d - i o n i z a t i o n r a t e s . I f t h e s p i n - o r b i t i n t e r a c t i o n is n e g l i g i b l e only m -0 s t a t e s can be e x c i t e d from t h e ground s t a t e Ri= mRi = 0 by u s i n g a a-

B-

Figure 8

-

S t a r k s t r u c t u r e of mR=O s t a t e s of hydrogen.

The s t a t e s a r e l a b e l l e d by (n,nl,mR)

----

Experimentally observed s t a t e A

... (21,4,1) s t a t e of hydrogen

I

^\

Anomalous behaviour i n t h e ^-281 2050 21W 2150 22W

l i f e t i m e of t h e s t a t e A

.

^{F I} v ~ ~ - 9

JOURNAL DE PHYSIQUE

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 i g h t . We have i d e n t i f i e d a l l m j = 0 s t a t e s of hydro- gen p r e s e n t i n t h e range of energy and f i e l d s t r e n g t h s t u d i e d . They a r e very s t a b l e s t a t e s ^{( T >} 10-3 s ) except f o r t h e s t a t e r k 2 3 nl=O which i s r e l a t i v e l y u n s t a b l e : i t s l i f e t i m e decreases from 3.1 t o 2.10-3 s when

5

i n c r e a s e s from 2050 t o 2250 ~ / c m . This s t a t e , a s s o c i a t e d with a very low d e n s i t y of s t a t e s , cannot be e f - f i c i e n t l y populated by a b s o r p t i o n from the ground s t a t e , and i s not observed i n t h e experiment. There is no mj=O s t a t e corresponding t o t h e observed s t a t e A

,

^{b u t}

f o r r e l a t i v e l y heavy atoms such a s Rb t h e s p i n - o r b i t i n t e r a c t i o n cannot be neglec- t e d and m . r a t h e r t h a n m j i s a n exact quantum number. Therefore w i t h n - p o l a r i -

4 .

zed l i g h t ~t 1s a l s o p o s s i b l e t o e x c i t e s t a t e s mj=1/2 mj=l from t h e ground s t a t e . We have shown t h a t t h e l o c a t i o n of t h e very s t a b l e s t a t e n = 2 1 , n l = 4 mj=l does n o t d i f f e r s i g n i f i c a n t l y from t h e r e c o r d e d p o s i t i o n of t h e s t a b l e s t a t e A (Fig.8). Further- more assuming t h a t t h e anomaly observed i n t h e l i f e t i m e of t h e s t a t e A a r i s e s from t h e l e v e l a n t i c r o s s i n g e f f e c t due t o t h e s p i n - o r b i t i n t e r a c t i o n i n v o l v i n g t h e s t a b l e s t a t e ( 2 1 , 4 , 1 ) and t h e s l i g h t l y u n s t a b l e s t a t e (23,0,0)

,

we have determined t h e v a l u e s of t h e parameters a

,

p

, Sc

and A o c c u r r i n g i n t h e p r e v i o u s l y d e s c r i b e d model and we have v e r i f i e d t h a t t h i s s e t of parameters i s a s s o c i a t e d with a n impor- t a n t i n c r e a s e of t h e l i f e t i m e of t h e s t a b l e s t a t e i n a very narrow range of f i e l d s t r e n g t h s . Such a r e s u l t s u p p o r t s t h e v a l i d i t y of t h e t h e o r e t i c a l i n t e r p r e t a t i o n of t h e anomalous behaviour i n t h e f i e l d - i o n i z a t i o n p r o p e r t i e s of the s t a t e A

.

B

-

P h o t o i o n i z a t i o n S t a r k s p e c t r a from t h e e x c i t e d s t a t e 5 2 ~ 3 / 2 F = MF=4 of 8 5 ~ b

.

This experiment i s performed i n an atomic beam of 8 5 ~ b and 8 7 ~ b atoms i n t h e i r ground s t a t e . The atoms a r e s u b j e c t e d t o two l a s e r e x c i t a t i o n s i n t h e pre- sence of an e l e c t r i c f i e l d . A cw single-mode high r e s o l u t i o n dye l a s e r right-hand- p o l a r i z e d i s tuned and servo-locked on t h e t r a n s i t i o n 5 2 ~ 1 / 2 F=3 ⁺ 5 2 ~ 3 / 2 F=4 of t h e i s o t o p e 8 5 ~ b

.

Consequently an optical-pumping between-magnetic s u b l e v e l s occurs and t h e use of a cw l a s e r permits t h e e x c i t a t i o n of 8 5 ~ b atoms i n t h e pure s t a t e 5 2 ~ 7 / 2 F = MF= 4

,

which corresponds t o t h e well-defined magnetic o r b i t a l quantum number mR=l

.

The i o n i z i n g l i g h t i s o b t a i n e d from a t u n a b l e pulsed dye l a s e r which i s a+

,

x or o- p o l a r i z e d . The produced i o n s a r e counted by a n e l e c t r o n m u l t i p l i e r . S p e c t r a corresponding t o t h e t h r e e d i f f e r e n t p o l a r i z a t i o n s of t h e pulsed l a s e r have been r e c o r d e d ( ~ i g . 9 ) . I n t h e energy range E

-

-200 cm-I t h e s t r u c t u r e s of each of t h e t h r e e s p e c t r a c a n be c l a s s i f i e d i n two d i f f e r e n t groups, t h e widths and t h e p r o f i l e s b e i n g very s i m i l a r i n a g i v e n group. For exam- p l e i n t h e o+u+ spectrum one observes dissymmetrical s t r u c t u r e s looking l i k e Fano p r o f i l e s [I61 w i t h a parameter q approximately e q u a l t o 1 ; between two such reso- nances t h e r e appears a very narrow symmetrical s t r u c t u r e . Some resonances a r e loca- t e d a t t h e same energy i n t h e t h r e e s p e c t r a b u t they correspond t o very d i f f e r e n t p r o f i l e s ; f o r example t h e narrow s t r u c t u r e s of t h e a+a- spectrum a r e a s s o c i a t e d w i t h t h e broad ones of t h e a+n spectrum and r e c i p r o c a l l y . By u s i n g t h e hydrogenic model we t r y t o l a b e l t h e observed resonances by t h e p a r a b o l i c quantum numbers ( n

,

n l

,

m4)

.

I f one assumes t h a t m j i s a n e x a c t quantum number t h e m j v a l u e s of t h e final s t a t e a r e r e s p e c t i v e l y 0 , +I

,

and +2 f o r t h e

a+u-' ,

a+n and u+a+

s p e c t r a . With t h i s assumption, only one resonance out of two can be i d e n t i f i e d i n each s p e c t r a . More p r e c i s e l y i n t h e t h r e e s p e c t r a only t h e broad s t r u c t u r e s can be l a b e l l e d . However r e c a l l i n g t h a t t h e r e i s an a p p a r e n t degeneracy i n t h e t h r e e spec- tra and t h a t rubidium i s a r e l a t i v e l y heavy atom, i t is p o s s i b l e t o a s s e r t t h a t t h e symmetry b r e a k i n g w i t h r e s p e c t t o mj a r i s e s from the spin-orbit i n t e r a c t i o n which c o u p l e s e x c i t e d s t a t e s w i t h t h e same m . v a l u e b u t w i t h d i f f e r e n t m j v a l u e s . The s t r u c t u r e s of t h e u+u-

,

o+n and a+& s p e c t r a a r e a s s o c i a t e d r e s p e c t i v e l y w i t h f i n a l s t a t e s m j = 1/2

,

3/2 and 5/2 and correspond t o t h e two s e r i e s (mg = 1/2

,

m j =mj-1/2) and (ms= -1/2

,

ma= m.+l/2)

.

Then a l l resonances observed i n t h e t h r e e s p e c t r a c a n be i d e n t i f i e d , an$ t h e two s e r i e s correspond t o d i f f e r e n t mS- v a l u e s : t h e broad s t r u c t u r e s a r e a s s o c i a t e d w i t h ms = 1/2 meanwhile t h e narrow resonances correspond t o m,=-1/2

,

independently of t h e s t u d i e d spectrum. However i n t h i s a n a l y s i s , t h e same l d e n t i f i c a t i o n ( n n l mj) i s a s c r i b e d t o two resonances

Figure 9

Recordings of the two-step photoionization spectra in the rubidium atom.

9

= 9950 ~ / c m

.

Various light polarizations are used : A : a+a-

,

B : ~ + n

,

^{C : a+a+}

(n

,

^n,

,

^m ) : parabolic quantum numbers labelling the resonances.

C2-164 JOURNAL DE PHYSIQUE

w i t h very d i f f e r e n t p r o f i l e s according t o whether t h e s e s t r u c t u r e s a r e observed in t h e spectrum m.

-

mR+1/2 o r mi-1/2

.

^{As i t} was discussed previously, i n a photo- a b s o r p t i o n spec&& one does not observe t h e d e n s i t y of e x c i t e d s t a t e s but only t h e d e n s i t y of o s c i l l a t o r s t r e n g t h s , which depends s t r o n g l y on t h e coupling between dis.

c r e t e and continuwn s t a t e s . I n hydrogenic approximation, e x c i t e d s t a t e s a s s o c i a t e d w i t h a given m j value c o n s i s t both of continua (E

,

n l

,

mi

,

^ms) a s s o c i a t e d w i t h t h e wavef unctions Y+(E) and T(E) r e s p e c t i v e l y f o r ms = 1/2 and -1/2

,

and of d i s c r e t e s t a t e s ( n

,

n l

,

m j

.,

^{ms) noted by cp+ and cp-} i n the same way. I n t h e rubidium atom t h e s e s t a t e s I n t e r a c t through AV

,

t h e non-coulombic p a r t of t h e e l e c t r o s t a t i c p o t e n t i a l , and through A t h e spin-orbit i n t e r a c t i o n ( ~ i g . l o ) , and t h e e i g e n f u n c t i o n @(E) can be expanded i n terms of cp+

,

^rp'

,

^{Y+(E' )} and Y!-(E' )

.

I n t h e study of e l e c t r i c d i p o l e t r a n s i t i o n s connecting t h e lower s t a t e 5 2 ~ 1 / 2

,

mA = 1/2

,

m j = 1 t o t h e upper s t a t e Q(E)

,

only t h e components Y+(E* ) and cp of t h e wavefunction c o n t r i b u t e t o t h e value of the t r a n s i t i o n matrix element. So t h e width of t h e p r o f i l e a s s o c i a t e d with t h e d i s c r e t e s t a t e cpf ^{( o r} cp- ) i s deter- mined by t h e s t r e n g t h of t h e i n t e r a c t i o n A V + A ( o r A ) connecting t h e s t u d i e d s t a t e t o t h e continuum Y+(E')

.

The parameter q [16] depends on t h e same i n t e r - a c t i o n and on d and D t h e matrix elements of the t r a n s i t i o n operator connecting t h e lower s t a t e t o cp+ and y + ( E t ) r e s p e c t i v e l y . For t h e p r o f i l e a s s o c i a t e d w i t h cp-

,

q vanishes. We have s t u d i e d a simple model c o n s i s t i n g i n t h r e e d i s c r e t e

F i w e 10

Schematic r e p r e s e n t a t i o n of t h e d i s c r e t e and continuous e x c i t e d

U 5 ' p 3 mi.1

s t a t e s w i t h a given mj value.

/2

The couplings a r e

A t h e s p i n - o r b i t i n t e r a c t i o n . AV t h e non-coulombic part of

t h e c e n t r a l p o t e n t i a l .

s t a t e s of e n e r g i e s El

,

E2 and Eg coupled t o one continuum. The s t a t e s 1 and J correspond t o s t a t e s cpf and a r e associated w i t h t h e same parameters

r

and q ; t h e s t a t e 2 corresponds t o s t a t e cp- (q2=0)

.

The r e s u l t i n g a b s o r p t i o n p r o f i l e s [16] a r e c a l c u l a t e d and some r e s u l t s a r e r e p o r t e d on Fig. 11. To i n t e r p r e t q u a l i t a - t i v e l y t h e a+< spectrum, we suppose t h a t

r2

(due t o t h e spin-orbit i n t e r a c t i o n of p s t a t e s ) i s s m a l l e r t h a n

rl

(due t o t h e non-hydrogenic c h a r a c t e r of s s t a - t e s ) . For a r e l a t i v e l y small value f o r q l

,

t h e p r o f i l e c o n s i s t s i n r e l a t i v e l y symmetrical s t r u c t u r e s , a narrow one l y i n g between two broad resonances. If one assumes t h a t

r l = r 2

and t h a t ql

-

¹

^,

t h e n t h e r e s u l t i n g p r o f i l e remoduces t h e structures observed i n t h e a+a+ spectrum. Indeed f o r s t a t e s m.-5/2

,

t h e i n t e r a c -

J -

t i o n s A and AV a r e small and of the same o r d e r of magnitude. This simple analy- sis c l e a r l y demonstrates t h a t t h e S t a r k photoionization s p e c t r a of Rb a r e s t r i k - i n g l y perturbed by t h e s p i n - o r b i t i n t e r a c t i o n , even i n the v i c i n i t y of t h e zero- f i e l d i o n i z a t i o n l i m i t . Compared t o the S t a r k photoionization s p e c t r a from t h e 3 2 PYl2 mj=l s t a t e of Na [ l o ] t h e s t r u c t u r e s a r e twice a s numerous i n t h e Rb

F i g w e 1 1

F r o f i l e s c a l c u l a t e d i n t h e t h r e e - s t a t e model a ) a+a+ spectrum b ) a+a- spectrum

s p e c t r a , a r e s u l t which can only be understood by t a k i n g i n t o account t h e spin-orbit i n t e r a c t i o n .

However t h e p e r t u r b a t i o n s AV and A a r e so l a r g e t h a t i t is not p o s s i b l e t o analyze t h e d i f f e r e n t s t r u c t u r e s independently from one another, by studying t h e coupling of a s i n g l e d i s c r e t e s t a t e w i t h one continuum. It i s necessary t o consider a s a whole t h e e f f e c t s of t h e c o r e i n t e r a c t i o n and those of the spin-orbit i n t e r a c - t i o n . A s u i t a b l e method has been developed w i t h i n t h e frame-work of the non r e l a t i - . v i s t i c multichannel quantum d e f e c t theory [18] and s u c c e s s f u l l y a p p l i e d t o t h e ana- l y s i s of t h e S t a r k photoionization spectrum of Na 3 2~ [ l g ] . I n t h i s work only t h e e f f e c t s due t o t h e s p h e r i c a l l y symmetric i o n i c core a r e introduced through 3/2 quantum d e f e c t s . This coupling merely d i l u t e s t h e resonances l o c a t e d near the zero- f i e l d i o n i z a t i o n t h r e s h o l d without s h i f t i n g t h e i r p o s i t i o n s o r c r e a t i n g new peaks.

I n conclusion t h e experimental study of t h e S t a r k photoionization s p e c t r a of Rb has c l e a r l y shown t h e important r o l e played by t h e s p i n - o r b i t i n t e r a c t i o n even near t h e z e r o - f i e l d i o n i z a t i o n l i m i t . Indeed, although weak, t h i s coupling l e a d s t o t h e observation of new resonances i n t h e photoionization of a s t a t e with a well- defined mi value. The observed p r o f i l e s depend s t r o n g l y on t h e p o l a r i z a t i o n of t h e l i g h t because t h e c r u c i a l q u a n t i t y i s not simply t h e d e n s i t y of e x c i t e d s t a t e s but t h e d e n s i t y of o s c i l l a t o r s t r e n g t h s . Furthermore a t t h e a n t i c r o s s i n g of s t a t e s with d i f f e r e n t mi v a l u e s , t h e spin-orbit i n t e r a c t i o n s t r o n g l y perturbs t h e f i e l d -

i o n i z a t i o n p r o p e r t i e s of t h e s t a t e s . This property can be explained by i n t r o d u c i n g e x p l i c i t l y both t h e energy and f i e l d dependences of t h e d e n s i t y of continuum s t a t e s . Iastly non-hydrogenic i n t e r a c t i o n s s t r o n g l y modify the photoionization S t a r k s p e c t r a : indeed, i n t h e over c r i t i c a l r e g i o n , t h e S t a r k spectrum of hydrogen c o n s i s t s si- multaneously i n d i s c r e t e and continuum s t a t e s , and a l l arguments deduced from t h e

study of unperturbed bound s t a t e s must be generalized w i t h g r e a t caution.

C2- 166 JOURNAL DE PHYSIQUE

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