In section 2.3, th re e different sc a tte rin g processes are m en tio n ed for ionisation collisions, th ey are: th e direct scatterin g process (2.45); the exchange sc atterin g process, w here th e em itted and scattered continuum electrons are exchanged (2.46); th e cap tu re sc atterin g process (only for atom s w ith m ore th a n one valence electron), w here two ta rg e t electrons are em itted and the incident electron is captured (2.47). T hree respective s c a tte rin g a m p litu d e s, f = \f\e1Yl, g = |g|e1/2, an d h, a re to re p re s e n t th e se th re e collision processes. The exchange am p litu d e te n d s to be im p o rta n t a t low im p act energy. G e n era lly sp eak in g , th e exchange process could h a p p e n in an y collision ev en t, how ever, only w ith th e p o larise d beam tec h n iq u e s can th e exchange process be th o ro u g h ly studied.
To illu s tr a te th e exchange effect, a sim plified e lastic s c a tte rin g model is applied. The collision atom is supposed to be a light, one valence electro n atom (e.g. , hydrogen, alk alis), an d th e im p act energy is low. H ence th e sp in -o rb it in te ra c tio n in th e sc a tte rin g is negligible. Two electrons, th e in cid en t electron and th e valence electron, are involved in th e collision so th a t only th e direct an d exchange (betw een th e in cid en t a n d v alen ce e le ctro n s) p ro cesses a re co n sid ered h e re , th e c a p tu re am plitude h is neglected.
A ssum ing th a t th e in itia l directions of th e polarised electron an d atom are p a ra lle l or a n tip a ra lle l to each other, th e possible d irect and exchange tran s itio n s a re illu stra te d .
P rocess A m p litu d e Cross section
e
T
+A 4 - -> eT
+ A 4- f \ f t (2.70) e t + A l ^ e i + A t ~ g \ g f , (2.71) e t +AT —> e T +A T f ~ g \ f - g \ 2 > (2.72) e l + A t -» e l + A t f i f f,
(2.73) e i + A t e t +A l ~ gK
(2.74) 64- + a 4- —^ 64* + A 4' f ~ g \ f - g \ 2 - (2.75)In th e se rea ctio n s eT an d e l re p re s e n t th e sp in -u p a n d spin-dow n electrons, respectively, A t and A l are for th e respective sp in -u p an d spin-down collision targ et. E quations (2.70), (2.71), and (2.72) are reactions for spin-up in c id e n t electrons, w hile th e o th er th re e are for spin-dow n in cid en t electrons. T h ere are couple of m ethods for in v e s tig a tin g th e ex ch an g e in te r a c tio n in th e collision, d ep en d in g on e x p e rim e n ta l conditions. O ne can e ith e r p re p a re spin p o larised in c id e n t electro n s, in itially polarised atom s, or both, for th e collision. In in v estig atin g th e final s ta te , one can e ith e r m ea su re th e spin-dependence of th e cross section, an aly se th e sp in p o larisatio n of th e outgoing electrons, or th e p o larisatio n of th e final ta rg e t. Choosing th e g en eral case: th e in cid en t electro n h a s th e p o la risa tio n of P e a n d th e in itia l ta r g e t h a s th e polarisation of P a, th e spin-dependent elastic sc atterin g cross section and the final electron polarisation are given by (Kessler 1991),
c r ( Q )
=
<j 01
+ P e P af i
1
G °J J
►' _ v p a + 2 A p e _ i fe-.- f g P e x P a y i + p e p a\
\flM'
(2.77)w here <j0 is th e cross section for unpolarised electron sc a tte rin g from an
° 0 =
2I f f
+ 2\st +W
-s f
=W + gf
+ \ \ f -gf
■ (2.78) 2.7.1 Spin-dependence of the differential cross sectionIf only th e in cid en t electrons are polarised, th e cross section is given from th e tran sitio n s (2.70) to (2.72) for spin-up incident electrons or from th e tran sitio n s (2.73) to (2.75) for spin-down incident electrons o^. It is easy to see th a t th is spin-dependent cross section, <7^ or cr^, is th e sam e as th e cross section for u n p o larised electrons <J0 in eq. (2.78). Eq. (2.76) fu rth e r illu stra te s th a t, unlike the case of spin-orbit in teractio n , th e cross section h ere no longer depends on the o rien tatio n of th e sc atterin g plane. In stead , it depends on th e direction of atomic polarisation P a. Only th e electron p o larisatio n com ponent along th e direction of P a contributes to th e sp in -d ep en d en t cross section. The spin-dependence of th e cross section is only subject to th e fact th a t both the collision atom an d electron are in itially polarised. The relativ e cross section difference for sc a tte rin g w ith a n tip ara lle l
T
i (P e ■ P a < 0) and parallelT T
(P e • P a > 0) spins is~ ° t t + °TT
Pe pas_.
(2.79)w here SA is th e norm alised spin asym m etry for exchange in teractio n . It is
fg +fg*
_ |/’llglcos(y1- y 2) _
\f + g f - \ f - g f2cro
|f
(2.80)W hen th e exchange am p litu d e is zero, i.e. ^ = 0, it is obvious th a t
SA = 0, no spin-up-dow n asym m etry will be observed.
2.7.2 Behaviour of the polarisation in scattering
Eq. (2.77) i llu s tr a te s how th e ex ch an g e p ro cess a ffects th e polarisation. A gain th e sc a tte rin g plane does not play a role in th e final electron p o larisatio n , an d it is replaced by th e direction of th e atom ic p o larisatio n .
If th e ta rg e t atom is unpolarised, P a = 0, one has, from eq. (2.77),
V y
The final p o larisatio n keeps th e sam e direction of th e in cid en t electron p o la risa tio n . B u t since 0 < l - | ^ | 2/cr0 < 1, th e m ag n itu d e of th e fin al p o la risa tio n is red u ced because of th e exchange process b etw een th e incident electrons and valence electrons.
If th e in cid en t electron beam is unpolarised, th e outgoing electrons could be p olarised from th e electron-electron exchange w ith a polarised targ et. From eq. (2.77) it is