Theoretical predictions:

The sta n d a rd assum ption in PWIA (or DWIA) calculations of TDCS for EM S stu d ies is th a t th e in itial sta te , including th e incident electrons an d th e in itial targ e t, is random ly oriented in space. The spin direction of th e in c id e n t electrons and th e s u b s ta te s M L of th e in itia l ta rg e t a re therefore averaged in th is kind of consideration (see eq. 2.3). This is a very successful tre a tm e n t for electron im pact ionisation w hen th e electron and ta rg e t a re u n p o larised . It is, how ever, not a good ap p ro x im atio n for a collision system w here th e in itial s ta te is oriented in a c ertain direction, such as th e p resen t experim ental arra n g em e n t.

V ery recently the theoretical tre a tm e n t of electron im pact ionisation w ith oriented atom s h as been fu rth e r developed. It is predicted th a t u n d er c e rta in k in em atic conditions th e a n g u la r m o m en tu m from th e in itia l oriented ta rg e t can tra n sfe r to th e two outgoing electrons in th e final state. If th e in itia l ta rg e t is oriented by circularly polarised light, th e electron p a ir in th e final sta te contains inform ation w hich can d istin g u ish a rig h t h an d ed coordinate fram e from a left h an d ed one. T his pro p erty is called o rien tatio n al dichroism , which is described by F eh r, B erak d ar, and K lar

(1994). To obtain a concrete concept of th is dichroism, a brief description is p resen te d here.

F o r th e electron im p act io n isatio n w ith th e in itia l ta r g e t b ein g p rep ared in a certain orientation O, the TDCS is

TD C S(O ) = (2 * )4 W ( r-f ( k a , k b )\V\ko&LML )

k0 mlkx l /

Pm lm l (0)(<PLMLko\V\ P J(Äa ,Ä6))).

(5.13)

w here k 0OLML) is an u n d isto rte d in itia l s ta te consisting of a n incident electron w ith m om entum k Q and a ta rg e t s ta te V sta n d s for th e in teractio n betw een the incident electron and targ et, and th e final s ta te is

'¥J(ka , k b). Pm l m l ( 0 ) describes th e population of m agnetic sublevel M L

of th e ta rg e t in a n g u la r m om entum basis, depending on th e direction of th e o rien tatio n O. In th e p resen t ex p erim en t, for in stan ce, th e sodium ta rg e t is oriented in th e z-direction by circu larly polarised lig h t ( cr+ or G~) . R eplacing th e density m a trix elem ent by s ta te m ultipoles p ^ o (O ) a n d in tro d u c in g th e irred u c ib le sp h e rica l te n s o rs P # 0 , ecl- (5.13) is re w ritte n as TD C S(O ) = (2 k)a^ - ' L ( - )k-lPk0 ( O) ko K (5.14) ■(V~f (.ka , k b )\Vk0PK0V0V'\ , k b w here p b M (O) = X ( - )K~L~M (L - M L M \ K 0 )pK0 (O), L L K a n d Pr o(r,r') = S ( - ) M{ L M L - M\ KO) M 0 LMl (5.15) (5.16) The a d v an tag e of u sin g th e s ta te m ultipoles is th e ir reflection pro p erty

p ^ 0 ( - O ) = (~)K Pi^o(O). By reversing th e direction of targ e t orientation O, for in stan ce, rev ersin g th e p o larisatio n of atom s from +z-direction to -z- direction in th e p resen t experim ent, one finds an o rien tatio n al dichroism given by

A = TD C S(O ) - T D C S (-O )

= -2 (2 ;r)4 bsth. Z ( - ) L p K0(O) (5.17)

kg K=odd

Eq. (5.17) shows th a t only odd values K of the sta te m ultipoles contribute to A. A random ly o rien ted in itial ta rg e t is rep resen ted by th e scalar K = 0 only an d of course shows no asym m etry a t all.

W ith in th e f ir s t B orn a p p ro x im a tio n th e e x p re ss io n for th e dichroism reduces to th e vector operator

ABorn= 2 ^ ( 2 K ) i ^ l m ( . a ß ' , ) K x k b, (5.18)

kQ

w here Im (aß*) is th e am plitude of the dichroism obtained from th e tensor o p e ra to r in eq. (5.14). An im plied condition for a non-zero ABorn is

O - ( K x k b) * 0. Two im p o rta n t re m a rk s can be e x tra c te d from th is dichroism expression:

1) In c o n tra st to a non-oriented sta te th e TDCS is no longer cylindrically sym m etric aro u n d th e m om entum tra n sfe r K,

2) To observe a finite value of the dichroism A , the in itial sta te o rien tatio n

O m u st have a com ponent perpendicular to the ( K , k b) plane.

M = -1

M * +1

F ig u r e 5.15 The calculation of the TDCSs for electron-im pact ionisation of hydrogen ( 2 p ; M L - ±1) in coplanar asym m etric geom etry w ith (a) Born ap p ro x im atio n , (b) BBK approxim ation. k0 an d ka a re th e resp ectiv e m om enta of th e incident and th e scattered electrons, K is th e m om entum tra n s fe r. The in cid en t energy is E 0 = 250 eV, th e energy of th e ejected e le ctro n is E b = 5 eV, th e s c a tte rin g an g le is 6a = 3°. T h e a n g u la r m o m en tu m q u a n tis a tio n axis is p e rp e n d icu lar to th e s c a tte rin g p lan e (taken from F eh r et al. 1994).

T he c a lcu latio n s of th e TDCS for th e o rie n te d h y d ro g en ( 2p; Ml = ±1) s ta te are show n in figure 5.15 an d figure 5.16, w ith coplanar

asy m m e tric an d n o n -co p lan ar sym m etric g eo m etries, resp ectiv ely . In figure 5.15 (from F e h r et al. 1994), the cr1-excitation lig h t is p e rp en d icu lar

to th e scatterin g plane. The TDCS calculated w ithin the FBA is shown in figure 5.15(a). It is clearly seen th a t th ere is a large difference betw een the two cross sections for ML = +1 and M L = -1 w ith an in cid en t energy of E 0 = 250eV. The dichroism is eq u al to zero w hen K\\kb. In s te a d of c y lin d ric a l sy m m etry th e TDCS h a s a refle ctio n sy m m e try in th e sc a tte rin g plane, T D C S (0 ,£ 6) = T D C S (-0,Ä £), w here k'b is th e vector kh

reflected about th e direction K . Figure 5.15(b) is th e sam e as in figure 5.15(a) except th a t th e correlated final sta te h a s been used (i.e., th e BBK ap proxim ation for th e final s ta te w avefunction). The dichroism is even la rg e r here, and th e reflection sym m etry about th e m om entum tra n s fe r direction is broken.

Q o

F ig u re 5.16 The calculation of th e TDCSs for electron-im pact ionisation of hydrogen (2 p \ M L - ±1) in non-coplanar sym m etric geom etry w ith PWIA and BBK approxim ations. The kinem atics is given by figure 5.8, an d an incident electron energy of E0 = 800eV is used. The solid curve an d the longer dashed curve are th e BBK approxim ation for ML = +1 an d ML = -1 sta te s respectively, and the sh o rter dashed curve is th e PWIA calculation for both Ml = +1 and ML = -1 states (taken from K lar 1994).

F ig u re 5.16 shows th e TDCS calcu latio n for th e n o n -co p la n ar sym m etric geom etry (K lar 1994). The k in em atics a re th e sam e as in figure 5.8, and an incident electron energy of E0 = 800eV is used. In th is geom etry, th e la s e r b ein g p e rp e n d ic u la r to th e x-y. p lan e , only th e

com ponents of th e K , kb vectors w ith in th e x-y plane contribute to the dich ro ism in th e firs t B orn a p p ro x im atio n (FBA). As 0 = 0 th e two outgoing electrons are in th e sc a tte rin g p lan e w hich is p a rallel to th e la se r beam direction. Obviously no dichroism is given a t th is point w hen th e FBA holds. The com parison of BBK an d PW IA approxim ations is given in th is figure. W ith th e BBK approxim ation th e dichroism is very large, even for such a high in cid en t energy. The changing of th e out-of­ plane angle 0 —> - 0 is the reflection of th e kb vector about K . T hus th ere a re two m ethods of in v estig atin g th e dichroism : e ith e r changing th e p

s ta te o rien tatio n direction from M L = +1 to M L = -1 on one side of 0 a n g u la r range, or probing th e cross sections a t both sides of 0 angle (0 —> -0 ). In th e first Born approxim ation th ese two m ethods are equivalent, however, th e BBK calculation gives a slig h t difference betw een th ese two m ethods because th e reflection sym m etry about 0 = 0 is lost. The PWIA calculation, on th e o th er hand, gives absolutely th e sam e cross section for

Ml = +1 an d Ml = -1 s ta te s an d sy m m etry ab o u t 0 = 0 because of th e s p h e ric a lly sy m m etric tr e a tm e n t of th e ta r g e t from th e o rig in a l c o n sid eratio n . U n fo rtu n a te ly , th e p re s e n t ex p erim e n t w as designed before th e above theoretical developm ents. N eith er th e polarisation flip of th e laser lig h t from cr+ to G~, nor th e fu ll-an g u lar-ran g e scan from 0 to -0 w ere perform ed w hen we did th e m ea su re m e n ts, hence inform ation about th e dichroism is not available from th e c u rre n t data.