A SEMICLASSICAL MODEL FOR THE ANGULAR MOMENTUM TRANSFER BY BEAM-TARGET INTERACTION BY BEAM-TARGET INTERACTION

HAL Id: jpa-00218446

https://hal.archives-ouvertes.fr/jpa-00218446

Submitted on 1 Jan 1979

HAL

is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire

HAL, est

destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

A SEMICLASSICAL MODEL FOR THE ANGULAR MOMENTUM TRANSFER BY BEAM-TARGET INTERACTION BY BEAM-TARGET INTERACTION

J. Burgdörfer, H. Gabriel

To cite this version:

J. Burgdörfer, H. Gabriel. A SEMICLASSICAL MODEL FOR THE ANGULAR MOMENTUM

TRANSFER BY BEAM-TARGET INTERACTION BY BEAM-TARGET INTERACTION. Journal

de Physique Colloques, 1979, 40 (C1), pp.C1-315-C1-317. �10.1051/jphyscol:1979166�. �jpa-00218446�

JOURNAL DE PHYSIQUE Colloque C1, suppldment au no 2, Tome 40, fgvrier 1979, page C1-315

A SEMICLASSICAL MODEL FOR THE ANGULAR MOMENTUM TRANSFER BY BEAM-TARGET INTERACTION

J, Burgdarfer

-

^{H. Gabriel}

Institut fiir Atom- und FestkBrperphysik der Freien Universitat Berlin Abstract.- It is shown within a semiclassical model that the production of atomic orientation by electron capture can be explaie-dby symmetry breaking. In a beam-foil passage experiment symmetry breaking of the effective projectile-electron Coulomb interaction results fro'm the

variation of shielding at the solid-vacuum boundary. In grazing incidence scattering experiments or for ion collisions in gaseous targets the full symmetry is lowered by selection of the impact parameter regime.

Resume.- O n montre dans un modsle semi-classique que la production d'orientation atomique par capture df81ectrons peut sfexpliquer par une rupture de symetrie. Dans une experience de passage d'un ion 2 travess une feuille la rupture de la symgtrie de l'intgraction Coulombienne effective entre projectile et Qlectron resuite de la variation de 1'Qcran 2 la limite solide vide. Dans les experiences de diffusion en incidence rasante ou pour les collisions ioniques dans des cibles gazeuses la sym6trie compl2te est diminuse par selection du regime du paramltre dtimpact.

It is the aim of the present note to formulate within a semiclassical treatment general conditions for the occurence of atomic orientation by electron capture from localized target states. The dependence of the Stokes parameters on the angle

between the s u r f a c e n m a l go and the beam axis in the case of tilted foil experiments will also be discussed.

It is known [I] that the Stokes parameters may be written as

(1)

I -I.(.?, + R ( " < ~ L ; - I-">>)

in terms of expectation values of the components of the angular momentum

operator

L.

(We restrict to spin S = o ions mocing along the z axis. Light is observed in the y direction). Within a classical treatment we need to replace. these quantum mechanical averages by appropriate

classical statistical averages over the corresponding quantities. In order to achieve this the physical model must be

The semiclassical model.- Consider a projectile at sufficiently high velocity

( v - 3 1 a.u.1 allowing the initial and P

final state velocities of the active

electron with respect to the binding ion to be neglected. (The Born approximation,could be used if we were interested in a quantum mechanical treatment of electron capture.) The projectile is assumed to be a structure- less ion of effective charge Z which sees an environmental electron passing by with a velocity

ve

^{= -v}

.

We interprete the

-P

quantum mechanical target-electron density as a classical continuum charge density n(g +

r).

The ion is located at Ij at a given time. The position of an electron with respect to the projectile is denoted by

r.

Assume, furthermore, that the electmn trajectories remain straight lines un- distorted by Rutherford scattering.

In our classical treatment capture of an electron from the continuum just described into an orbit of the projectile cannot be calculated from first principles. Let us assume that this event occurs wi%h the statistical weight

( 5 )

'F ( ~ ~ r , ~ )

⁼

f 4 ( ! 3 ) ? ( T , ~ ? ,

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979166

C1-316 JOURNAL DE PHYSIQUE

In the factorized form (an approximation The term proportional to n(R) drops out due suggested by the quantum mechanical treat-

ment) P ( 2 ,

ve)

gives the probability that an electron at 2 be captured into a bound state of the projectile at a given

5.

^{It is}

physically reasonable to assume that P(2,

ve)

has axial symmetry with respect to the beam ( 2 ) axis. Different weight is given to the various ion positions by the screening factor E-I(~) which allows us to account for noticeable differences in the electron capture processes at different ion positions

R

due to changes e.g. in the effective Coulomb interaction or due to spatial restrictions in the collision parameters. Examples will be given later.

Transfer of an electron from the continuum to a bound state is accompanied by a transfer of orbital momentum. Due to the short interaction time (%10-~~s) this will be the dominating effect even for S

4

0.

According to eqs (1) to 14) statistical averages of well defined functions f

(z(r, ^R

⁾ of the angular momentum density

(6)

L - t ~ , R ) = ( r ^- x m ~ ) n ( B + r )

must be performed with the help of (5) yielding

The integration includes all possible ion trajectories as well as the total target- electron continuum.

Atomic orientation.- We illustrate the analysis with S defined in (4) which is a measure of circular polarization. It is now given by

( 8 )

s=

^caxs:

^{*Jd3R -} R - " B ~ J ~ ~ ~ P(T,

&)

Expanding n(R

+

2) into a Taylor series around

-g

we arrive at

to axial symmetry of P(2, 1,).

The importance and physical meaning will now be demonstrated in two cases.

Case a): E(R) = const. According to this assumption all ion positions at every possible trajectory are equivalent con- cerning capture of an electron. In the example of electron transfer from the surface to the projectile after foil passage, E = const corresponds to ignoring any screening of the ionic charge by the sea of target electrons. 1% the case of electron capture from atomic collisions in gases E = const would apply for isotropic- ally distributed impact parameters. In boi5 cases we arrive at

(10) S . 0

by transforming (9) into surface integrals which vanish, since kim n(R) = 0 for localized states. IR/+ 09

Case b): Discontinuous E(R). It is known

[ 2 , 3 ] that at ion velocities v p 5 2 a.u.

the projectile is screened in the bulk of carbon foils. A drastic change of the screening effects occurs at the solid- vacuum boundary. We idealize the real situation by jumping from = (perfect screening) to = 1 (no screening) and arrive at the following first-order result

Circular polarisation clearly appears as a result of surface effects in our model. It was shown in[4] that S is mainly determined by the mean electronic density g r a d i e n t o e side the foil. Notice that eq. (11) yields SoCsinB due to the vector product of the surface normal n and

ve

= -v

-0

.

^This

-P

angular dependence is in good agreement with the experimental findings of S/I [5,6]. We have also studied under which conditions proauction of atomic orientation via

gaseous collisions is theoretically predicted by our semiclassical model.

The effect was recently found experiment- ally [ ⁷³

.

It forms out from (9) that

S 0 is expected because of symmetry breaking of the effective Coulomb interaction as produced by a selective impactparameter regime. (The same argument holds for ion-surface scattering at grazing incidence). Additional effects like a change in sign of S/I at v il a.u. fall off the range of validity of the present P model.

Angular dependence of the Stokes parameters.- The Stokes parameters M and C can be

analyzed in the same manner as S. Their angular dependence in tilted-foil

experiments can most easily be calculated if we choose

a form we were led to by the fact that in high-energy regime electrons are captured into mL = 0 states predominantly. Further- more we assume circul-ar orbits. The expres- sion (12) is, of course, an idealization

of the real situation, which leads to

C = 0 . By performing the integration over

r one finds that

-

(13)

s

⁼

^a- k B

M

⁼

b c c d h z Z p ( a , b , c > O )

C - 0

Comparison with quantum mechanical calculations.- Electron capture into a hydrogenic 2p state was calculated by us*

the Brinkman-Kramers approximation. The initial state was chosen to be a hydrogenic 1s state together with condition that the ion-electron intekaction be restricted to a semiinfinite half space (as in case b) above). The Stokes parameters can be ex- pressed in terms of irreducible tensor operators which must be calculated numerically. Fig. 1 shows S,M and C as a function of as calculated within.the Brinkman-Kramers approximation (full curves). The broken curves display S and M taken from (13) by adjusting S,M at 6 = 0'

and go0. Despite the oversimplifications of our model the agreement between the corresponding results is surprisingly good in the entire 6 range. The deviations are of l h e same order of magnitude as the

difference between Cclassical = 0 and

C q. m. .(: 0 as apparent from Fig. 1.

Valuable discussions with Dr. Schrbder, Dr. Kupfer and Prof. Andr3 are gratefully acknowledged.

This work is supported by the Deutsche.

Forschungsgemeinschaft

.

References

[I]

U. Fano, J. Macek, .Rev. Mod. Phys. 4 5 , 553 (1973)

[2J W. Brandt, in "Atomic Collisions in Solids", eds.: S. Datz, B.R. Appleton, C.D. Moak, Plenum Press, New York

(1975), p. 261

[ 3 1 M.C. Cross, in "inelastic ion-surface

collisions", eds.: N.H. Tolk, J.C. Tully, W. Heiland, C.W. White, Academic Press, New York (19771, p.253

[ 4 ] H. Schrcder, E. Kupfer, Z. Physik

A 249, 13 (1976)

[5] H.G. Berry, Rep. Prog. Phys.

9,

¹⁵⁵

(1977)

[61

G . J. Pedrazzini

,

R. B. Gardiner, C.H. Liu: Phys. Lett.

s,

23 (1977) 171 W. Wittmann, H. ^J. Andrd, to be

published