STRONG-FIELD RADIATIVE SCATTERING

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STRONG-FIELD RADIATIVE SCATTERING

F. Faisal

To cite this version:

F. Faisal. STRONG-FIELD RADIATIVE SCATTERING. Journal de Physique Colloques, 1985, 46 (C1), pp.C1-223-C1-239. �10.1051/jphyscol:1985122�. �jpa-00224494�

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JOURNAL DE PHYSIQUE

Colloque C1, suppl6ment au n O l , Tome 46, janvier 1985 page C1-223

STRONG-FIELD R A D I A T I V E SCATTERING

F.H.M. Faisal

FakuZtZt fiir P h y s i k , U n i v e r s i t a t BieZefeZd, F. R. G.

R E S U M E

Nous avons trait4 trois aspects actuels e t importants des collisions radiatives impliquant u n Qlectron dans u n champ laser intense : l e probleme fondamental d e l'interaction Coulombienne, l e s collisions avec effets d1interf4rences et la collision e-H avec excitation stationnaire de l'atome. L'analyse rQvele, entre autre, u n certain nombre de nouveaux phQnomhnes de diffusion. L e s rQsultats sont discutks et illustrQs.

ABSTRACT

Three problems of much interest in radiative scattering are analysed. They include the basic problem of radiative Coulomb scattering, interference scattering and the steady state e-H

scattering, in a strong laser field. The analysis reveals, among other things, a number of novel scattering phenomena. The results are discussed and illustrated.

I-RADIATIVE COULOMB SCATTERING

In this first part, we reexamine the Coulomb scattering in a strong external radiation field and point out new prospects in the study of this basic process.

The prototype of electron scattering in a laser field may be defined by the following one-particle Schrodinger equation (units: -75 = e = m = 1)

where in the dlpole limit the vector potential

E A. cos(wt + 6 ) (linear polarization) ' A I ~ )

-

=

A. I t x cosIwt+61 - E sin(wti6) (circular

Y polarization)

with 5 and 6 , the polarization vector and the phase o f the field, respetlvely.

Other symbols have their usual significance.

The interaction between the electron and the radiation field is essentially of longer range than any of the atomic Besides, the field usually extends over macroscopic distances (say more than 10 Bohr radii). Thus the asymptotic solution for the scattering wave-function should incorporate the motion of the electron within the field. The electron motion becomes actually free only when the field strength F O , ( = A .w/c) + 0 , across the fleld boundary (in space or-time). So far as the atomlc potential is concerned, there arise 0

two qualitatively different situations depending on whether the potantlal is short-range 1e.q. exponential), or long-range le.g.Coulombic1 in nature.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1985122

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C 1-224 JOURNAL DE PHYSIQUE

Below w e first reexamine the peculiarities of the asymptotics in the two cases to bring out this difference and then point out some of its physical conse- quences in the radiative Coulomb scattering. In the process we also discover a new eigen-state of the stationary problem in the potential-free case, which is degenerate with the well-known Volkov-state and which satisfies the same asymptotic condition (outside the field) as the Volkov-state but differs from it considerably in the region inside the field. It 1s suggested that the physical difference between these two degenerate states may lie in the different conditions of preparation ( o r the switching on-off conditions) of the electron-field interaction (at the same total energy).

SHORT-RANGE POTENTIAL

In this case V(r) in ( 1 ) can be neglected at large r and a solution of the resulting equation,

may be given by the well known Volkov-type solutlon Ill

1 ^. ^L 1

$LO'(E,t) = - - - ^expt - 5 1 [k - ; A ( t - ) ~ 2 d t . + ik._r)

-

(2711 312 2

For the sake of brevity w e have introduced the symbol Jn(xl_y) such that

m

2 2

= 1 Jm(x Jn-2m(ka0 cos 8 ) (linear)

i

^m=-m ^k

J n ( k O 1 ~ O ) ( 2 Jn(kao cos O k )

.

^x2< < 1) in4

= J ( k a sin 8 1 e ^k

n 0 k (circular) (5b)

2 2 2

The 'continuum shift' i s x 12 = ~ ~ ' 1 4 ~ or F o /2w2, depeyding on linear or circular polarization, respectively; so = a o , a. = F / 2 w ₀ .

Observe that the solution ( 4 ) is essentially "empty" of the field interaction:

According to (61 there is no change whatever in the probability density (or the associated flux) at the point (_r.t) in the presence of the field (from what it has been in the absence of the field). This is a much stronger limitation of ( 4 ) than the limitation implied by the lack of real emission or absorption of photons b y the free electron, in the presence of the running wave field, eq.

(2). In fact ( 6 ) is not a necessary requirement for all solutions of ( 3 ) which must reduce asymptotically (i.e. outside the field region, where F = 0 ) to the

plane wave state. 0

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( s e e eq. (101 b e l o w f o r a c o u n t e r e x a m p l e w h i c h s a t i s f i e s ( 3 1 and t h e s a m e a s y m p t o t i c condition ( 7 1 , as d o e s ( 4 ) , but i s n o t restricted by t h e " e m p t y "

r e l a t i o n ( 6 ) ) .

T H E COULOMB PROBLEM

U n l i k e i n t h e c a s e of s h o r t r a n g e potentials t h e V o l k o v - l i k e - s o l u t i o n ( 4 1 d o e s not g i v e a proper a s y m p t o t i c solution i n the p r e s e n c e of t h e l o n g r a n g e C o u l o m b potential. An a p p r o x i m a t e s o l u t i o n has been suggested and used 1 2 . 3 1 b e f o r e , but t h a t a p p r o x i m a t i o n t o o i s " e m p t y " , i n t h e s e n c e of eq. 16). T h e p r o p e r t y , ( 6 ) , i s unfortunately q u a l i t a t i v e l y unsatisfactory f o r any a p p r o x i m a t e wavefunc- tion i n a Coulomb p o t e n t i a l i n t h e presence o f t h e f i e l d ; s i n c e t h e Coulomb t a i l w i l l certainly c a u s e r e a l ( n o t simply virtual!) e x c h a n g e o f p h o t o n s , i t should modify even t h e a s y m p t o t i c a m p l i t u d e ( n o t just t h e p h a s e ) of t h e w a v e function. In t h e absence of a n e x a c t solution c o n s i d e r t h e following Ansatz:

w h e r e k k unit vector l n t h e i n c i d e n t d i r e c t i o n .

w h e r e q . = - - - " ' 1 ' S t h e S o m m e r f e l d - c o n s t a n t , kn

I t i s easy t o s e e by s u b s t i t u t i o n t h a t ( 8 ) i s a l i m i t i n g s o l u t i o n o f ( 1 ) i n t h e l i m i t s

I i ) F c o n s t a n t , 0 w +

-

^, ^{( l i )} ^w constant. F o -+ 0 , ( i i i ) a. c o n s t a n t , w + 0 , lao = F O l w L ) ,

As a n approximation ( 8 ) p o s e s s e s t h e p h y s i c a l c h a r a c t e r i s t i c s o f a p l a u s i b l e s o l u t i o n i n t h e C o u l o m b field. T h u s , using t h e k n o w n I & / a s y m p t o t i c e x p r e s s i o n o f ( 8 a ) l t i s e a s i l y f o u n d t h a t $ (r t ) g l v e r i s e t o t h e C o u l o m b and field d i s - torted p l a n e w a v e s a l o n g 2 a n d k s Y m i i a r l y d i s t o r t e d s p h e r i c a l w a v e s , for every s u b - c h a n n e l defined by t h e e m i s s i o n o r t h e a b s o r p t i o n o f n-photons. ( I t i s a l s o e a s i l y verified that i n t h e a b s e n c e o f t h e f i e l d . F = 0. ( 8 1 r e d u c e s t o t h e o r d i n a r y C o u l b m b wave.) F i n a l l y , i n t h e a b s e n c e o f 'the C o u l o m b potential.

Z Z ' = 0 , ( 8 ) r e d u c e s t o

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C 1-226 JOURNAL DE PHYSIQUE

1 2

- 2 1(k2 + x )t + ik. ri - n

-

^{( 1 0 )}

The parameter k , rn the wave function ( 8 ) and 110) is readily identified, by comparision with eq. ( 7 ) , as the wave-vector in the asymptotic region [defined to be the reglon out-side the field where Fo = 0).

A NEW EIGEN-STATE DEGENERATE WITH THE VOLKOV STATE

It is worthwhile to note that the particular solution ( 1 0 ) is not simply an arbitrary wave-packet solution of 13) but is the time dependent version of the following exact eigen-state of the stationary Schrodinger equation (with the eigenvalue E _k = K 12 2 + ^{N o w )}

namely.

(Nois the arbitrary $nitial occupation number and may be conveniently normallzed to No = 0. The A -term may be easily included in ( 1 1 ) and the solution 112) expressed in terms o f the 3 lk(a ) functions, ( 5 1 , i f desired.). T h e popular

Volkov-state. n

-

^-0

is agaln an ei en-state of ( 1 1 ) and is degenerate with ( 1 2 ) having the same eigenvalue E 0. ia The difference between these two exact elgensolutions. ( 1 2 ) and ( 1 3 ) , may presumably be interpreted in terms of the possible difference in k the preparation conditions of these states. For example, judging by the appearence of the fourier components k , 1c.f. eqs. 110). 4 112). ( 1 3 ) ) . the state ( 1 2 ) or ( 1 0 ) might be assocyated wlth a "sudden switching", while the state ( 1 3 ) or 14) with an "adiabatic-switching" ( o n or off), of the electron-field interaction. Eq. ( 1 0 ) i s , as required, an exact solution of the potential-free equation (3). Furthermore, ( 1 0 ) goes over to the plane wave solution [see eq. ( 7 ) ) in the absence of the field fFo = 0). N o t e , however that neither ( 8 ) nor ( 1 0 ) are " e m p t y " in the sence of ( 6 ) .

RADIATIVE COULOMB SCATTERING AMPLITUDES Using ( 8 ) and ( 1 0 ) In the amplitude integral

one finds for the T-matrix element for the absorption ( N < O ) or emission lN>O) of N-photons

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w h e r e k' = k:' polnts t o t h e scattered d l r e c t l o n *k' ⁼ ( @ , + I . ' _{m n}^{= k}_{- m}- ; _-n

1 s t h e Plnterm!dlate m o m e n t u m transfer. It should be noted t h a t ( 1 4 ) contalns both t h e o f f - s h e l l C o u l o m b t-matrlx 1 5 1 .

w h e r e

6 ( k ,A1 = a r g r(l + i n m ) - qm I n ( 2 km/Al

0 m

and t h e u s u a l o n - s h e l l t - m a t r i x 1 6 1

4 *

w h e r e c o s 0 = k - k ' .

T h e p r e s e n c e of t h e o f f - s h e l l t i n ( 1 4 ) h a s s o m e i n t e r e s t l n g c o n s e - q u e n c e s t o be discussed below. n o t e e n passant t h a t if o n e m a k e s t h e f u r t h e r a p p r o x i m a t i o n o f neglecting t h e o f f - d i a g o n a l i n t e r m e d i a t e a m p l i t u d e s ( s e t t i n g n = m e v e r y w h e r e i n ( 1 4 1 ) t h e n o n e o b t a i n s a KrGger-Jung t y p e o f formula 1 7 1 , albeit modified by t h e " c o n t i n u u m s h i f t " and by t h e o n - s h e l l r e n o r m a l i z a - t i o n f a c t o r 3 (q' /_a ) (In v i e w o f w h a t i s j u s t said and t h e c o n d i t i o n s (il and

0 'mm

(ivil a b o v e , ~t would a p p e a r that t h e r e s t r i c t i o n t o l o w f r e q u e n c i e s i n t h e d e r i v a t i o n s o f t h e u s u a l K r u g e r - J u n g p o t e n t i a l scattering f o r m u l a . m a y n o t be a necessary condition.) Naturally t h e Kroll-Watson formula 181 and t h e Born-approximation 191 a r e a l s o recovered f r o m ( 1 4 ) under f u r t h e r approxima- t i o n s .

R A D I A T I V E - R Y D E E R G RESONANCES

Note t h e s l m p l e R y d b e r g - l l k e poles at t h e negatlve lnteger a r g u m e n t s o f t h e g a m m a functlon ril + l Q m ) , w h l c h a p p e a r I n t h e w a v e f u n c t l o n ( 8 ) o r I n t h e amplitude, eqs. (141 and ( 1 5 1 , f o r t h e e m l s s l o n s u b - c h a n n e l s . T h l s g l v e s r l s e t o a pronounced s e q u e n c e of resonances ( c a l l t h e m r a d l a t l v e - R y d b e r g or R-R r e s o - nancesi at the e l e c t r o n scattering e n e r g y , k 2 1 2 , satlsfylng:

Eq. (171 s a y s t h a t w h e n e v e r t h e e l e c t r o n energy and t h e energy of an underlying Rydberg s t a t e o f principal q u a n t u m n u m b e r , p + 1 , i s bridged by t h e energy o f m ( s t i m u l a t e d , e m i t t e d ) p h o t o n s , t h e r e w o u l d a r i s e a r e s o n a n c e I n t h e scattering c r o s s section d u e t o t h e temporary c a p t u r e o f t h e scattering e l e c t r o n . T h i s i s s c h e m a t i c a l l y s h o w n i n Fig. 1 below. F r o m ( 1 6 ) i t i s c l e a r t h a t t h e R-R r e s o - n a n c e s

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J O U R N A L DE PHYSIQUE

Fig. 1 - ^A sch matlc showing a radiative-Rydberg resonance between the incident electron with k 12 and a Rydberg state, p=l. h

can be explored by varying either the electron energy at a fixed frequency or tuning the laser frequency at a fixed electron energy. It is also clear that these resonances must Play an essential role in the ordinary Bremsstrahlung /lo/

and i n the stimulated electron-ion recombination problems, which also give rise to the interesting phenomenon of photon-amplification 1111. Actually the R-R resonances will not be ideally sharp, as i n ( A 7 1 , but will be broadened essentially by the electron reemission width (say y wlth combinations of p and m ,

as dictated by eq. (171). P

OKUBO-FELDMAN DISCONTINUITY

Another interesting possibility arising from the radiative coulomb scattering is the following. A well-known but intriguing feature of the off-shell Coulomb t-matrix ( s e e eq. (1511 has been the existence of a discontinuity of its magni- tude above an6 below the on-shell energy. This was first studied by Okubo and Feldman 1121 and by Mappleton 1131. Ford 151 has given a particularly lucid description of the same. Making explicit analysis with a finite cut-off length and going to the infinite limit, Ford concludes that the Okubo-Feldman discontinuity is quite real. T h e transition region itself extends over a small interval around the on-shell energy, and coincides with the latter as the cut-off length goes to infinity. The discontinuous modulus of the t -matrix is I5 1

The scattering amplitude ( 1 4 ) indicates an opportunity to explore the lscontin- uous nature of the off-shell Coulomb t-matrix. For example, for x < < ¹^and incident electron at right angles to the (linear) polarization directlon, ko = , (00 = "12, q O = 0). (11) reduces to

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(where 3 are now ordinary Beisel functions). Therefore, for small but non-zero values zf (k'Na+O) e.g. k' = (8.4) with 0 = n/2 f 6 and any convenient 4 , say

Q" 0 or n ) one flnds, to the first order in ~k','~~). from (19)

T(O) tcl*O,*;) on-shell.

T = - l o t c o + t c o , l l off-shell (below)

Now the difference signal defined by

where

depends essentially on the off-shell t -matrix. Using ( 2 0 ) in ( 2 1 ) and remembering the dependence 0 t . t on A (seeceq. 115)) one finds

We observe that for w < < k 12 this difference should not vanish, due to the 2 presence of the Okubo-Feldman discontinuity accross the shell-radius (c.f.eq.

( 1 8 ) ) . We note parenthetically that an apparent interference term which at the

first sight should be added to the r.h.s. of ( 2 2 ) does not contribute to the signal In the limit A + 0 (infinite cut-off limit), since it oscillates infin- itely rapidly about a mean value o f zero. The right hand side of (22) in fact turns out to be independent of A (see eq. (18)). If this (eq. 1211, or simi- l a r ) rather delicate signal may be measured then we would have some verification of the important (e.g. in the Fadeyev-Lovelace formulation o f the atomlc three-body problems /I&/) and yet rather adhoc construct (within the formal scattering theory designed, essentially, fpr short range forces) of the off-shell Coulomb t-matrix.

11-COHERENT LASER EXCITATION AND THE ELASTIC SCATTERING

We now consider a second topic of much current,interest. The problem discussed concerns the nature of the elastic signals for the scattering of atomic projectiles against a target which is coherently excited by a laser pulse. Before analysing aspects o f this problem i t i s worthwile t o consider the time-dependent behaviour of the target atom i n a strong field.

EVOLUTION OF A TARGET ATOM IN A LASER FIELD

In a near ( o r on)- resonant field the target atoms may no longer be in the Boltzmannian ground state; the field can drive the atoms far from the equili- brium. Furthermore, the atoms may actually begin to decay due primarily to mul- tlpt~oton i o n i z a t ~ o n .

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JOURNAL DE PHYSIQUE

Fig.2 - Time dependent ionization of H-atom. a ) Ionization probability of 1s-state, b) survival probability of 1s-state and c ) ionization spectrum.

- 1

y = 0.09 nsec, Fo = .001 (a.u.) and w = .375 1a.u.)

Fig. 2 shows the result of theoretical calculations I151 fo$ the evolution of an H-atom in a square laser pulse of intensity I = ^{35 GWIcm}, resonant with the 1s-2p transition. One observes (in the middle pannel of this figure) that the H-atom, which has been initially lt=Ol I n the ground state, quickly leaves that state (essentially for the 2p-state but also to elsewhere) and returns to it repeatedly. But the chance that it will return to the ground state decreases progressively as the interaction with the field continues. In fact the ground state H-atom decays steadily in the ionization continuum; this is evidenced by the steady growth of the probability of ionization, P . ( t ) , as seen i n the left pannel. In the present case it takes the whole atoT1?g break apart completely, in a time interval a little more t h a n , ( t = y = ) 0.09 ns. It i s also interesting to observe (in the right pannel) how the ionized electrons energy is distributed and how this spectrum evolves with the interaction time. We observe two peaks i n the spectrum, separated by =20 meV. Note that at least two photons must be absorbed, at the 1s-2p resonant frequency, for the H-atom to decay. In fact the spectrum shows that the peaks occur on either side of the unperturbed energy A € = 0 l2hw above the 1s-ground state). The doublet is a reflection of the optical analog o f the Autler Townes ( A - T ) splitting o f the 1s-2p resonance transition: the ionizing electron simply monitors the split-states; the peaks separation in the spectrum corresponds to the distance between the splitted A-T components. Note that with increasing time the width of the peaks diminish and the two rather sharp lines tend to evolve.

A very similar behaviour is expected for the evolution of alkali atoms (e.g. Na atoms with a near-resonant 3s- 3 p transition) in a laser field. What would be the elastic scattering signal like for a beam of inert projectiles (e.g. a noble gas) from such tarsets in a laser field?

THE SPLITTING OF THE ELASTIC SCATTERING SIGNAL

From the above consideration it is apparent that the scattering slgnal from a superposition 'of target states created by a near (or on)-resonant laser pulse depends on the duration (and, as w e shall see, also on the condition of the application) of the field. Thus if the pulse i s strong it must be short enough.

tlpulse) < tldecay), for the states to be prepared i n a coherent

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superposition-state (during its natural lifetime further, the pulse rise time tlriie) < linlllabi) [where QlRabi) = (A' + p';ll'f;s the two-level general- ized Rabi-frequency with the detuning A , and the coupling strength p ) , then the pulse will effectively switch on "suddenly" ( a t a time s a y , t = O ) .

The suddenly-swltched target states will evolve (in the resonant two-level approximation) as

where

IA 1 > = cos 0 Il.no> - sin Q 12.n 0 - I > ; ( 2 4 I

and

IA > = sin 0 Il,n > ⁺ cos 0 12,n - I > ;

2 0 0 ( 2 5 )

are the two well known perturbed eigenstates; -mi4 5 8 6 1 ~ 1 4 , tan 2 0 = @/A and I l , n o > , 12.n#> are the product states, latom> [field). The perturbed eigen-frequencies are

and

(and they are reserved for A<O).

where w l and w are the Bohr-frequencies of the levels Il> and 12,;

A z w 2 - w 1 - ^{W ;}2 ~ z - h 1 = n .

Note that i f initially l$(O))= I l , n o > then

<hll$(0)> = Cos Q and <A21$(0)> = sin Q

REDUCTION OF THE PURE-STATE INTO A MIXTURE (IN THE [A>-REPRESENTAITON)

Consider now that the on-set of the collision, between a pair of target and p r o -

jectile atoms, occurs at t = t O The density matrix of the target at this time i s (from e q . (23))

Clearly, the target is i n a pure-state at this time ( [ ~ ( t I 1 2 = ~ ( t ~ ) ) . But, the instant t o , for all pairs of colliding atoms, is essen!ially a random quan- tity; so the observable signal depends on the average o f ( 2 8 ) over the distribution of to. Assuming an uniform distribution w e get

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JOURNAL DE PHYSIQUE

C O S ~ 8 -i,-- ( e ipT - 1 ₎ cos Q sin Q R T

= lim

T-r-

i 2 - ^-iRT-

RT ( e ¹ I cos 8 sin 0 , sin2 Q

One sees from ( 2 9 ) that the off-diagonal correlation between the dressed states ] A l > and IA2 is important but for a few Rabi-periods, T = 0 (l/Q).

In the limit the correlation is negligible and w e get,

THE SCATTERING SIGNAL IN THE VICINITY OF THE INCIDENT ENERGY.

Thus the scattering signal becomes effectively that from a mixture of states ([Dl 2 f [81) in the IA>-representation:

where the arnplltude rnatrlx

the scattered wave vectors are 5 . k 1 2 ^andk2, with

k12 ^; - f!! p )'I2 h

where k i s the incident (relative) wave-vector and p i s the reduced mass.

Hence Prom ( 3 1 ) and (32) one finds that the "elastic signal" splits up into three components, separated by the Rabi-energy hR, from each other:

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where the central peak has the intensity

and the intensities of the side bands are

and

with

(To isolate them clearly the energy resolution would require to be better than 6 R !

INTERFERENCE-SCATTERING

It would appear that the signal ( 3 3 ) is merely a n - incoherent sum of cross-sections. This is indeed so but in the IA>-representation. The signal ( 3 3 ) , when analysed in the representation of the asymptotically free-states, reveals the hidden interference scattering between the scattering amplitudes directly arising from the atomic states 1 1 ) and 12>.

Consider a typical amplitude in the ]A>-representation,

where T is the transition operator and I k > , Ik. .> are plane waves

0 1 3

It is sufficient to show the existence of the interference scattering i n the simplest (FBA) approximation of T 2 V. (For,the heavy-particle beam collisions of interest, with k o l a r g e , this is often a sufficient approximation.) In this case, clearly.

where the interaction potential V=V(_R,x); f! is the relative coordinate and 2 are the target coordinates.

Using eqs. (34)-(36) in (33) one easily finds the following results.

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C1-234 JOURNAL DE PHYSIQUE

Thus, not only the central peak ( 3 7 ) , but also the splitted components 138) and 139), depend on the direct interference scattering between the two elastic amplitudes f lko+b) and f2+2(ko+k), originating from the bare states Il> and 12> o f thel+iarget atoms. We observe that the scattering signal may be more accurately computed, i f necessary, from the T-matrix elements, ( 3 5 ) , in the IA>-representation; this is because the dressed propagator ?'can be derived explicitly and used i n the- well-known equation

L L

The dressed-propagator at E . =

I-!

⁺^*Ai ^,for the present case. is found

1 2lJ

2 ik. . IE-j' I

0 1 3

G ( ) = - - - , I) > ? - - - ^<A,l

1 2f12 j=l

'

^I^R-B'^I ³

THE ADIABATIC CASE

Until now w e have restricted ourselves to the "sudden switching" case. If tlrise) > lIR(Rabi) (which is usually satisfied for 1A1 > line-widths) then the pulse switches on adiabatically and according to the adiabatic theorem the state Ih,> becomes occupied (for A > O ) , while IA > remains empty. It is seen by an analysis completely similar to the previous case (c.f. eqs. 2 (24)-(26)). that the "elastic signal" now splits-up into a doublet:

It is also clear from the preceeding analysis that the adiabatic-signal (41).

like the sudden signal ( 3 3 1 , results from the interference of the usual elastic scattering amplitudes from the target states / I > and 12).

We note that for A < 0 , the roles of I A l > and IA2> are reversed (c.f. eqs. 126)) and the adiabatic signal turns out to be.

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We conclude that the elastic signal splits up into a triplet separated by hR.

when (the coherent superposition of the two states o f ) the target is prepared by a "sudden-pulse", and into a doublet if the target is prepared by an "adiabatic pulse". In the latter case the sidepeak changes from the red to the blue side of the central peak, as the detuning changes sign.

111-LOW ENERGY RADIATIVE e-H SCATTERING

We now turn our attention from the coherent excitation to the steady state excitation of the target atoms and consider the important problem of electron scattering from such targets. Perhaps the most interesting aspects of radiative electron-atom scattering phenomena occur at l o w electron energies. E. where E . 4 4 w . (in this situation neither the Born-approximation nor the so-called 1'

l i w frequency approximation may be applicable). We devote this last section to a brief discussion of a number of interesting phenomena which occur in the steady-state conditions at high field strengths and at l o w scattering energies with specific reference to the radiative e-H collision.

To keep the calculations managable and yet physically correct w e have adopted a pseudo- potential approach 1161 in which the potential parameters are empirical- ly determined with respect to the known data on the low-energy e-H scattering (in the absence of the fleld).

0.4

I N C I D E N T E L E C T R O N E N E R G Y

Fig.3. - Low energy e-H elastic scattering cross section showing the resonance at a 9.558 eV (width a 0.04 e V ) , obtained from the pseudo-potential model, Scales are i n a.u..

Fig.3 shows the kind of result one obtains in this way when the field is off.

The elastic 15-1s cross section at l o w scattering energies is shown. It also reproduce the famous e-H scattering resonance 1171 at 9.6 eV (width 0.04 eV see magnification in the inset). The same pseudo-potentials yield the H- (negative-ion) bound state at -0.75 eV (not shown in the figure). The cross-section at zero-energy is seen to compare with the s andard two-state close-coupling

f

calculation 1181 which gives o(E=O) 2 254.75 a. (although the actual value could be somewhat less/18/). It appears that the 2 - s t a t e . p s e u d o - p o t e n t i a l model used here fits the known l o w energy scattering data physically correctly. The model, therefore, has been extended 1161 to include the effect o f the external field. We discuss some results of the influence of the field on the electron scattering processes.

A BOUND NEGATIVE ION-STATE RESONANTLY ENHANCES THE ELASTIC SCATTERING

Consider the e-H scattering with the field o n , at an electron energy far below the inelastic threshold (Eth = 10.2 eV) and at a photon frequency w = .05 au.

The result i s shown in figure 4.

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Fig.4 - The field modified e-H elastic cross section at F o = . 0 0 5 a.u.

w = . 0 5 a.u.. Sharp structure at ².6 eV i s an one photon "capture-escape"

resonance. Small structure at s 1 . 9 5 eV i s due to a two-photon resonance. The scales are in a.u..

We see a dramatic change i n the field modified elastic cross section -0 1 0 ' 1 0

(with no net emission o r absorption of photons); it exhibits a sharp strGctuPe which "reflects" the bound H-negative ion-state into the elastic channel. At

E. = . 0 2 2 4 tau1 the photon energy f i w = . 0 5 (au) matches approximately with the

binding energy of H - , E = - . 0 2 7 6 ( a u ) and we have a resonant capture o f the elastic electron i n &he H-state (as it loses the energy of one-photon by the stimulated emission). But the captured electron can also reabsorb a laser photon and return to its positive energy elastic channel; the delay 1 1 9 1 caused by the temporary "capture and escape" episodes shows up as a new resonance in the elastic channel. We note that a smaller two-photon "capture-escape" resonance i s also discernable at an electron energy which is .fiw = . 0 5 laul above the primary one-photon resonance. Clearly the main resonance may be used to deter- mine the electron-affinity by elastic scattering in the field.

SUB-THRESHOLD ELECTRONS CAN EXCITE THE TARGET

Another interesting phenomenon, at electron energies below the first inelastic excitation threshold of the H-atom, is the resonant excitation of the n = 2 level, by the sub-threshold electrons. This process may permit one to probe experimen- tally a portion of the off-shell electron scattering amplitude. "Off-shell"

calculations / 2 0 - 2 2 1 , up till now, have been confined to the above threshold electron energies where the effect occurs along with the direct collisional excitation but can be separated. With the sub-threshold electrons, on the other hand, the dominant process itself i s the off-shell one.

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Fig.5 - Resonant sub-threshold exci- tation o f H(n=2) b y l o w energy electrons.

F o = 0.01 a . u . , w = 0.094 a.u..

The scales are i n a.u..

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Figure 5 exhibits a typical result of such excitation cross-section, o 0 ' 2 - 1 '

i n which the electron borrows a photon from the field and excites &he upper state of the atom by the collisional transfer of its enhanced kinetic energy to the atom. What is more, the process becomes resonant i n nature due to the presence of the elastic scattering resonance ( a t z 9.6 eVl. We also note the existence of a threshold cusp / 4 , 1 6 , 2 3 / (at s 10.2 e V ) and a secondary resonance at s 0.35 (au). We should emphasise that the resonant sub-threshold excitations should occur in other systems as well. Excitation of vibrational states of diatomic molecules, for example H2 and N2, by very low energy electrons in the presence of a strong infrared laser (e.g. a CO -laser a t 6 w = - 1 1 7 e V ) should also exhibit the resonant sub-threshold excitation phenomena. 2

LOW ENERGY ANGULAR DISTRIBUTION

The field can affect the differential scattering cross sections d u e to both symmetry and dynamical reasons. In the absence o f the field, the scattering process maintains its cylindrical symmetry about the incident beam direction.

which leads to the usual azimuthal invariance of the angular distributions of the scattered electrons. The very presence of the field destroyes this invari- a n c e , since in general it provides an extra quantization direction in the laboratory (except perhaps if the polarization direction lies along the incident electron beam direction). Besides, the distribution with respect to the polar angle, in a given azimuthal plane also can be overwhelmingly modified.

Fig.6 - Low energy e-H differential scattering cross section in the azimuth plane q = 45' modified by a circularly polarized field. Electrons energy E . = 1.36 e V ; F = .005 a.u., w = 0.05 a.u.. See text for description.

0

Figure 6 shows an example of the change in the low-energy ( E . = .05 a.u.) angular. distribution for the e-H scattering, due to the p+esence of the field

( 4 w = .05 a.u., peak field strength Fo = .005a.u.) when a circularly polarized

photon-beam is directized a) along the incident electron-beam direction (left-pannel) b) along 45' [middle-pannel) , and c ) perpendicular [right-pannel) , to the electron-beam direction. The outer circles in these pannels correspond to the usual spherical distribution of the scattered low-energy electrons (when the field is off).

CONVERSION OF ELECTRON ENERGY INTO PHOTON ENERGY BY SCATTERING

Just as the electron can borrow energy from the field, the field may also gain energy from the electron during the collision. At specific electron energies it may happen that the field gains more energy from the electron than the other way around, which could lead to an amplification of the field intensity.

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- 1 5

I N C I D E N T E L E C T R O N E N E R G Y r a u . )

Fig. 7 - Total photon absorption coefficient showing resonant "negative absorption" or gain due to "capture-escape" scattering of electrons against H at field strength F = .001. .OD25 and .005 (a.u.). The scales are in (a.u.1.

In figure 7 w e show the coefficient, a ( L ) , of the net absorpt.ion of photons by the scattering system (as a function 07 the incident electron energy Ei, for a fixed field frequency, w = .05 au) where both the elastic scattering and the atomic excitation are allowed for /16/.

Fig. 7 clearly exhibits how at preferred electron energies the absorption coefficient becomes negative and hence changes into a "gain-coefficient". Observe also that the "gain-resonance" deepens, broadens, as well as shifts towards the

lower energies. the field strength increases, from

F O = O D 1 (a.u. ) = 5.1 1;' VIcm, through .OOPS (a.u. 1 to .005 (I.u. 1 . Here w e have a case of photon-amplification by the radiative scattering of electrons.

IV-SUMMARY

Three aspects of scattering in strong radiation fields are analysed. They include I) a reexamination o f the basic process of radiative Coulombic scatter- i n g , 11) interference scattering o f neutrals from a superposition o f target states i n a strong laser pulse, and 111) the steady state e-H radiative scattering. They revea1,among other things,existence of sequences of radiative Rydberg resonances and the possibility of testing the Okubo-Feldrnan discontinuity in Coulomb scattering. It is predicted that the elastic signal for scattering of inert projectiles + active targets i n a strong resonant field will split up into a triplet (sudden-pulse) or a doublet (adiabatic-pulse). Radiative e-H scattering i s used to exhibit a number of novel scattering phenomena such as elastic resonances due to the bound negative ion states, resonant excitations by sub-threshold electrons and light amplification by the "capture-escape" scatter- Ing.

A C K N O W L E D G E M E N T S

I would like to thank Mr.L.Dimou for calculating and preparing the graphs of the results presented in section 111. This research has been partially supported by Deutsche Forschungsgerneinschaft under project number DFGIFa-16011-1 Oz:2825.

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REFERENCES

1 . D.M. V o l k o w ( 1 9 3 5 ) Z . P h y s . 1 4 , 2 5 0

2 . M. J a i n a n d N. T z o a r . P h y s . R e v . A 1 8 , ( 1 9 7 8 ) 5 3 8 .

3 . P . C a v a l i e r e . G. F e r r a n t e a n d C. L e o n e . J . P h y s . B 1 3 , ( 1 9 8 0 1 4 4 9 5 .

4 . L . D . L a n d a u a n d E.M. L l f s h i t z , Q u a n t u m M e c h a n i c s 2 n d E d . P e r g a m o n P r e s s , O x f o r d ( 1 9 6 5 1 p p . 5 1 8 , 5 6 5 .

5 . W.F. F o r d , P h y s . R e v . 1 3 3 . ( 1 9 6 4 ) 8 1 6 1 6

6 . L . I . S c h l f f , Q u a n t u m M e c h a n i c s 3 r d E d . M c G r a w - H i l l . K o g a k u s h a L t d . , T o k y o ( 1 9 6 8 ) p . 1 4 0

7 . H . K r u g e r a n d C. J u n g , P h y s . R e v . A 1 7 , ( 1 9 7 8 ) 1 7 0 6 . 8 . F . V . B u n k i n a n d M.V. F e d o r o v , JETP 2 2 , ( 1 9 6 6 1 8 4 4 ,

N . K . Rahman, P h y s . R e v . A 8 , ( 1 9 7 4 ) 8 0 4 ,

I . V . H e r t e l a n d L . H a h n , J . p h y s . 8 5 , ( 1 9 7 2 ) 1 9 9 5 . 9 . N.M. K r o l l a n d K.M. W a t s o n . P h y s . R e v . A 8 , ( 1 9 7 3 ) 8 0 4 . 1 0 . V . P . Z h d a n o v . S o v . P h y s . JETP 5 2 . 1 ( 1 9 8 0 1

L . G . O y a c h k o v , G.A. K o b z e r a n d G . E . N o r m a n , I b i d 3 8 , ( 1 9 7 3 ) 6 9 7 1 1 . F . H . M . F a l s a l , A . L a m i a n d N . K . Rahman, J . P h y s . 8 1 4 , ( 1 9 8 1 ) 1 5 6 9 . 1 2 . 5 . O k u b o a n d D . F e l d m a n , P h y s . R e v . 1 1 7 , ( 1 9 6 0 ) 2 9 2 .

1 3 . R . A . M a p p l e t o n . J . M a t h . P h y s . 2 , 4 8 2 ( 1 9 6 1 ) ; I b i d 3 , ( 1 9 6 2 ) 2 9 7 1 4 . J . C . Y . C h e n a n d T . I s h l h a r a , P h y s . R e v . 1 8 6 , ( 1 9 6 9 ) 2 5

1 5 . F.H.M. F a l s a l a n d J.M. M o l o n e y , J . P h y s . 8 . 1 4 . ( 1 9 8 1 1 3 6 0 3 . 1 6 . F . H . M . F a i s a l , i n L a s e r A s s i s t e d C o l l i s i o n a n d R e l a t e d T o p i c s ,

e d . N . K . Rahman a n d C. G u i d o t t i . H a r w o o d A c a d . P u b l i s h e r s , New Y o r k ( 1 9 8 2 1 , p . 2 8 7 ;

L . D i m o u a n d F.H.M. F a i s a l , i n C o l l i s i o n s a n d H a l f - C o l l i s i o n s W i t h L a s e r s , e d . N . K . Rahman a n d C. G u i d o t t i , H a r w o o d A c a d . P u b l i s h e r s ,

-

N e w E r k ( 1 9 8 4 1 , p . 1 2 1 :

F.H.M. F a i s a l , Comments I n A t . M o l . P h y s . ( 1 9 8 4 1 ( i n p r e s s ) 1 7 . 6 . 3 . S c h u l z , R e v . Mod. P h y s . 4 5 , ( 1 9 7 3 ) 3 7 8 .

1 8 . N . F . M o t t a n d H.S.W. M a s s e y , T h e T h e o r y o f A t o m i c C o l l i s i o n s , 3 r d . E d . O x f o r d ( 1 9 6 5 ) p . 5 3 0 .

1 9 . E . P . W i g n e r . P h y s . R e v . 9 8 . ( 1 9 6 5 1 1 4 5 .

2 0 . N . K . Rahman a n d F . H . M . F a i s a l , J . P h y s . B 9 , ( 1 9 7 6 ) L 2 7 5 ; P h y s . L e t t . 5 7 A , 4 2 6 ( 1 9 7 6 ) ; J . P h y s . 8 1 1 , ( 1 9 7 8 ) 2 0 0 3 .

2 1 . F.H.M. F a i s a l i n C o h e r e n c e a n d C o r r e l a t i o n i n A t o m i c C o l l i s i o n s e d . H. K l e i n p o p p e n a n d J . F . W i l l i a m s , P l e n u m P r e s s , n . Y . ( 1 9 8 0 1 , p . 4 7 9 . 2 2 . S . J e t z k e , F . H . M . F a i s a l , R. H i p p l e r a n d H . O . L u t z , Z . P h y s i k ( 1 9 8 4 ) ,

( i n p r e s s ) .

2 3 . C. J u n g a n d H . S . T a y l o r , P h y s . R e v . A 2 3 , ( 1 9 8 1 ) 1 1 1 5 .

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