50th Anniversary Symposium on Nuclear Sizes and Shapes 23/06/08

Elizabeth Cunningham

Does the passage of low energy deuterons through a finite 12C foil lead to

small angle tensor polarisation?

50th Anniversary Symposium on Nuclear Sizes and Shapes 23/06/08

Overview

Experimental motivation

Brief description of tensor polarisation

Nuclear scattering

Atomic scattering

Comparison with experiment

Summary

50th Anniversary Symposium on Nuclear Sizes and Shapes 23/06/08

Experimental Result

At University of Cologne using deuterons up to 16 MeV [1] :

‘First attempt to measure spin dichroism, i.e. creation of tensor polarisation in an unpolarised deuteron beam by unpolarised carbon targets.’ [2]

Observed tensor polarization for small scattering angles

Serious implications for designing polarimeters used in deuteron experiments.

Deviation from randomness of approx. 10%.

[1] A. Rouba et al., Proc. 17th Int. Spin Physics Symp.; SPIN06, 2-7 Oct., Kyoto, Japan, AIP Conf. Proc. 915 (2007).

[2] V. Baryshevsky et al., arXiv:hep-ex/0501045, (2005). IiiiiV. Baryshevsky and A. Rouba, arXiv:nucl-th/0706.3808, (2007).

50th Anniversary Symposium on Nuclear Sizes and Shapes 23/06/08

Tensor Interaction

Deuteron has prolate quadrupole deformation along its spin axis.

Different spin projection iiiiiiiiiiiMI = +1,0, -1 gives different apparent cross sectional area of deuteron ‘seen’ by target.

Tensor Potential:

VTR(r,I)VTR(r)[(I ̂ r )2 2/3]

MI = ±1

MI = 0

b

br

r

50th Anniversary Symposium on Nuclear Sizes and Shapes 23/06/08

T20 Polarisation

T20 polarisation: measure of deviation from randomness.

The only type of tensor polarisation which does not tend to zero for scattering in the forward direction.

NMI = probability deuteron has Iz = MI in transmitted beam.

Unpolarised beam, N+1 = N-1 = N0 =1/3, gives: T20 = 0.

T203Iz

2 2

2

T203 N1N 1 2

2

1 3N0

2

50th Anniversary Symposium on Nuclear Sizes and Shapes 23/06/08

Experiment at Cologne University [1,2], measured tensor polarisation in the transmitted deuterons as large as

for small scattering angles and a carbon target thickness of 132 mg/cm2:

Transmitted deuterons are preferentially aligned with their long axis along incident beam direction.

Experimental T20

T20 = 0.18 ± 0.02

50th Anniversary Symposium on Nuclear Sizes and Shapes 23/06/08

Scattering Theory

z

eikz incident plane wave

f(,I) eikr scattered waver

r

V(r,I,) scattering centre Asymptotic Wavefunction:

Used to calculate cross section and T20 polarization for deuterons elastically scattering from an individual 12C nucleus.

Scattering amplitude connects wavefunction and observables:

(r) eikz f(,I)eikr

r

f(,I)

22 e ik.r V(r,I) (k,r) dr

50th Anniversary Symposium on Nuclear Sizes and Shapes 23/06/08

Optical Potential

Optical potential for d-12C nuclear scattering at 11.9 MeV:

V(r,s)VC(r) Vc(r) iWc(r) VLS(r)L I VTR(r)[I ̂ r ]2 2 3 Coulomb[3] Central[3] Spin-Orbit[3] Tensor[4]

Extrapolation from polarisation data for angles greater than 1 deg. Potential depths:

Used to calculate scattering amplitude:

dd

13

Tr (ff)

T201

2

Tr(f [3Iz2 -2]f )

Tr(ff )

Vc = 119.0 MeV, Wc = 5.8 MeV, VLS = 6.2 MeV, VTR = 3.965 MeV

50th Anniversary Symposium on Nuclear Sizes and Shapes 23/06/08

Results - Nuclear

d-12C at 11.9 MeV

T20 in forward direction is of order ~10-5

50th Anniversary Symposium on Nuclear Sizes and Shapes 23/06/08

Coulomb interaction between deuteron and the atom:

Using Born approximation,

- first order approximation

- assumes effect of scattering potential is small

scattering amplitude for atom A A’ becomes:

Atomic Scattering

re-

riR

VdA Ze2

| R r/2|

e2

| R r/2 ri |i1

Z

fM I A M I 'A '

22 M I ' (r),A ' (ri),k' (R) VdA (R,r,ri) k(R),A (ri),M I(r)

50th Anniversary Symposium on Nuclear Sizes and Shapes 23/06/08

Using q = k - k’ and changing the variables so that R’=R+r/2,

gives a simplified expression for T20 polarisation for single atomic scattering of a deuteron from a carbon atom.

Qd = deuteron quadrupole moment = 0.2860 ± 0.0015 fm2 [5].

Atomic T20

T201

2

Tr([3Iz2 -2]MM)

Tr(M M)

23

Qdq2P2 cos(q)

fM IA M I 'A ' Tatom(q) MM I 'M I(q), MM I 'M I

dr e iqr/2M I '* (r)M I (r)

50th Anniversary Symposium on Nuclear Sizes and Shapes 23/06/08

Results - Atomic

Born approx. factor of 2 higher but both give T20 in forward direction of order ~10-5

d-12C at 11.9 MeV

50th Anniversary Symposium on Nuclear Sizes and Shapes 23/06/08

Estimate of the T20 from multiple atomic scattering events.

Using qj2 = kj

2 (scatt2)j and taking P2(cos(qj

)) = -0.5, most likely

value for small q:

To calculate specific case for comparison with experiment, use 2 [6]

Multiple Scattering

T2023

Qd qj2P2 cos(q j

) j=1

N

T2026

Qdk2 (scatt

2 ) jj=1

N

Multiple scattering calculation gives T20 = 1.2x10-4

50th Anniversary Symposium on Nuclear Sizes and Shapes 23/06/08

Summary- Measurement of T20 = 0.18 ± 0.02 for 5-8 MeV deuterons

passing through a 12C target of thickness 132 mg/cm2.

- Calculation of T20 ~ 10-5 for 11.9 MeV deuterons scattering from a single 12C nucleus.

- Calculation of T20 = 1.2x10-4 for 11.9 MeV deuterons scattering from atomic electrons.

- Theoretical calculation about 3 orders of magnitude smaller than experimental measurement.

- Major discrepacy which could have serious implications for designing polarimeters used in deuteron experiments.

50th Anniversary Symposium on Nuclear Sizes and Shapes 23/06/08

Acknowledgements

[1] A. Rouba et al., Proc. 17th Int. Spin Physics Symp.; SPIN06, 2-7 Oct., Kyoto, Japan, AIP Conf. Proc. 915 (2007).

[2] V. Baryshevsky et al., arXiv:hep-ex/0501045, (2005). iiiiiiV. Baryshevsky and A. Rouba, arXiv:nucl-th/0706.3808, (2007).

[3] H. Wilsch and G. Clausnitzer, Nucl. Phys. A160, 609 (1971).

[4] G. Perrin et al., Nucl. Phys. A282, 221 (1977).

[5] D. M. Bishop and L. M. Cheung, Phys. Rev. A20, 381 (1979).

[6] R. C. Johnson and E. J. Stephenson, in preparation.

Thank you to my supervisors Ron Johnson and Jim Al-Khalili.

Thank you for listening…