TRANSFER OF ORBITAL ANGULAR MOMENTUM POLARIZATION TO NUCLEAR SPIN POLARIZATION IN EXTERNAL FIELDS. - A THEORETICAL STUDY.

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TRANSFER OF ORBITAL ANGULAR MOMENTUM POLARIZATION TO NUCLEAR SPIN

POLARIZATION IN EXTERNAL FIELDS. - A THEORETICAL STUDY.

H. Gabriel, E. Kupfer

To cite this version:

H. Gabriel, E. Kupfer. TRANSFER OF ORBITAL ANGULAR MOMENTUM POLARIZATION TO NUCLEAR SPIN POLARIZATION IN EXTERNAL FIELDS. - A THEORETICAL STUDY..

Journal de Physique Colloques, 1979, 40 (C1), pp.C1-321-C1-323. �10.1051/jphyscol:1979168�. �jpa-

00218448�

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JOURNAL DE PHYSIQUE Colloque C1, suppltfment au no 2, Tome 40, ftfvrier 1979, page C1-321

TRANSFER OF ORBITAL ANGULAR MOMENTUM POLARIZATION TO NUCLEAR SPIN POLARIZATION IN EXTERNAL FIELDS.

-

A THEORETICAL STUDY.

H. Gabriel

-

E. Kupfer

Institut fiir Atom- und Festkdrperphysik, Freie Universitat Berlin Abstract.- The effect of an external magnetic field on the transfer of electron shell to nuclear spin polarization is studied under the condition that the beam traverses the field region adiabatically.

We find that the modification of the nuclear spin orientation'is partially caused by a transfer of electronic alignment. Such an effect would not occur under sudden passage conditions.

R&sum&.- L'effet d'un champ magndtique extdrieur sur le transfert de polarisation d'une couche llectronique 3 u n spin nucldaire est dtudid dans les conditions de tra- versde adiabatique du champ par le faisceau. Nous trouvons qu'une modification de l'orientation du spin nuclgaire est en paftie due 1 un transfert d'alignement dlec- tronique. Un tel effet n'existerait pas dans des conditions de passage brusque.

Introduction.- A means of producing nuclear spin oriented heavy ion beams was reportedj recently [1,2]. The method is based on the transfer of orbital momentum orientation

(generated by ion-surface interactions at grazing incidence) to the nucleus via hyperfine interaction.

It is the purpose of the present note to investigate theoretically, how the transfer of electron shell to nuclear spin

polarization, observed from ion beam tilted target experiments at zero field, is

affected by external fields outside the ion-target interaction regime.

Theoretical model.- The following geom-etry is chosen: The scattered beam is along the x direction, the target normal lies in the x-y plane tilted by a small angle OC from the y direction. The axis of quantization and the direction of observation are parallel to the z axis. Ion beam-tilted target interaction initially generates polarized angular momentum states. The nuclear spins I are distributed isotropic- ally. This is also true for the electronic spins S, as long as unmagnetized targets are used. Let us assume that at time t = o the ion enters a strong magnetic field regime close to the surface. This field

B 11.z decreases adiabatically along the

beam direction, which requires

to be fulfilled ( Y, atomic Larmor frequency).

It is sufficient for our purposes to write the total hamiltonian of the projectile as

H = Ho + A CS +Q.T~

+ H*nq ^{( 2} We assume LS coupling to hold. Ho would give the energy levels of the ion un- modified by Hmagn (which is the sum of the electronic and nuclear magnetic interaction) as well as by the spin-orbit and the

magnetic hyperfine interactions proportioral to A and a, resp.

At time t = o the projectile enters a magnetic field the strength of which guarantees

to be obeyed. Accordingly, I and J are good quantum numbers and the mixed state of the beam ions at t = o Q(t I\ = o) is most conveniently expressed by the direct

product of irreducible tensoroperators T(;)(J~J~) I . (The isotropy of the nuclear spin distribution simplifies

~("(11) to

6:)

(II)! 1,

Q

At a later time t = to the density operator Q(to) is given by n

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979168

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C1-322 JOURNAL DE PHYSIQUE

In the adiabatic approximation the effect of the evolution operator on an energy eigenstate at t = o = follows from the relation [3]

where ~ ~ ( t - 1 is the eigenvalue of H(t) and

Id-> is the corresponding eigenstate.

Since we are primarily interested in the study of ground states lifetime corrections will be.suppressed. They could be easily Yaken into account if necessary.

;In the zero-field regime F is a good quantum number and

1

(IJ T M ~ ) is an appropriate set of basis vectors. The corresponding basis of irreducible tensor operators is denoted by

g(k)

(F1F2 1.

4

Neglecting cascade effects the time evolution of the state multipoles in the zero-field region which is entered by the ion at to, for times t > to is easily written as

with

The nuclear spin polarization at time t follows from (6) by averaging over the irrelevant electronic degrees of freedom, yielding

The coupling coefficient on the right-hand side of ( 8 ) is proportional to a 6j symbol and expliaitely given in121

.

^Further

simplification of ( 8 ) arises by averaging over the time window8 large compared to the-hyperfine period

( 9 )

-L

/

( ~:[rr) 7'333) / T(:(FFJ

$?(FF,~.)

.

3 F

The q = o component 'of the nucleaq spin

orientation now reads

3(:'( F F , ~ ~ ) .

This quantity is proportional to<IZ).

The state multipole

9':)

( F F , to) can be expressed as

<FMF191FM~>to .

The elements of the density matrix

< F M ~ ( Q

I

^{F M ~ )} can easily be traced back to the initiaPly prepared states with the help of eqs. (4) and ( 5 ) .

Example.- We study the generation of nuclear spin orientation for an ion beam with I = 1/2, L = 1, S = 1/2, e.g. a 1 3 c II- ground state. The energy eigenvalue

problem can be solved analytically in this case. The expressions are not given ex- plicitely. Instead, the term scheme of the 2p State as a function of B is displayed in Fig. 1

MI MI

,

112 312

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In the present example (10) reduces to

The pair of triples specifies the coupling of I and J to F in the order (1J)F.

Using eqs. (41, ( S ) , (11) and taking into account that the initial nuclear spin distribution is isotropic, (12) can be

If the initial electronic spin distribution is isotropic, as can be assumed after ion scattering from nonmagnetic solid surfaces, eQ. (13) is further simpiified to

This adiabatic passage result should be compared with the corresponding expression for B = o (Leaving any other feature of the experimental arrangment unchanged)

We notice a slight enhancement of the cofactor of the initial electronic orientation in (14) as compared to (15).

Moreover, the adiabatically varying magnetic field causes the nuclear spin orientation to depend also on the initial electronic alignment9 o(LL,t (21 = 0). Thi.s mixing-in would not occur under sudden passage conditions. The contribution of the electronic alignment does not necessarily lead to an overall enhancement of

9

^:⁽^' ¹¹⁾

,

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sinceSo(LL) and yo(LL) may have opposite signs. The following table relates the nuclear orientation to different sets of initial states9 (LL,t = (k) 01.

9

The numbers for the electronic state multipoles in the first and third column are taken from [4l and [5], resp. (The state multipoles given there had to be

transformed to our geometry). The numbers in the second and fourth column refer to a foil perpendicular to the beam. They are taken because 13c I1 ground state multipoles are unknown and for illustrative purposes, only., The corresponding nuclear spin orientations are given for the adiabatic and the zero-field case in the two bottom lines as indicated. It is obvious that, in our example, the nuclear spin orientation amounts to only a few percent for the initial electronic alignment to be generated by passage through a perpendicular foil. This is a small effect in view of other mechanisms that may destroy the nuclear spin

orientation.

It is difficult to predict without detailed calculation whether larger magnetic field effects would occur for states with different sets of quantum numbers.

This work is supported by the Deutsche Forschungsgemeinschaft References

[l] H.J. Andrs, H.J. Pldhn, A. Gaupp, R. Frdhling, Z . Physik A

281,

15 (1977) 121 H.J. Andra, H. Winter: to be

published

[ 3 ] A. Messiah, Quantum Mechanics, North- Holland Publ. Comp., Amsterdam 1976, p. 752

[ 4 ] H. J. AndrZ, R. Frdhling, H. J. Pldhn,

J.D. Silver, Phys. Rev. Lett.

37,

1212 (1976)

1 5 1 H. Schrdder, Z . Physik _A 284, 125 (1978)