Hi,
just a follow-up on the definition of asymmetries:
Arenhoevel defines polarizations and asymmetries for deuterium such that
the polarized cross section is written as
sigma = sigma_0 (1 + P1d AedV + P2d AdT
= sigma_0 (1 + sqrt(3/2) hPz AedV + sqrt(1/2) Pzz AdT (1)
where -1<Pz<1 and -2<Pzz<1. The quantities P1d,P2d are "deuteron
polarization parameters" that determine the "deuteron density matrix".
They are related to the usual vector Pz and tensor polarization Pzz by
P1d=sqrt(3/2)*Pz and P2d=sqrt(1/2)*Pzz.
Using spin-dependent yields n(h,Pz,Pzz), and defining
n0=n(1,1,1)+n(-1,1,1)+n(1,-1,1)+n(-1,-1,1)+n(1,0,-2)+n(-1,0,-2),
the raw asymmetries are
sqrt(3/2)*hPz*AedV =
(3/2)*[n(1,1,1)-n(-1,1,1)-n(1,-1,1)+n(-1,-1,1)]/n0 (the factor 3/2
compensates for 4 weights in the numerator versus 6 weights in the
denominator), hence
hPz*AedV = sqrt(3/2)*[n(1,1,1)-n(-1,1,1)-n(1,-1,1)+n(-1,-1,1)]/n0,
and
sqrt(1/2)*Pzz*AdT =
(1/2)*[n(1,1,1)+n(-1,1,1)+n(1,-1,1)+n(-1,-1,1)-2*n(1,0,-2)-2*n(-1,-0,-2)]/n0,
(the factor 1/2 compensates for 12 weights in the numerator versus 6
weights in the denominator, hence
Pzz*AdT = sqrt(1/2)*
[n(1,1,1)+n(-1,1,1)+n(1,-1,1)+n(-1,-1,1)-2*n(1,0,-2)-2*n(-1,-0,-2)]/n0.
Note that formula (1) is a convention for the asymmetries, other authors
may define the asymmetries through
sigma = sigma_0 (1 + hPz AedV + Pzz AdT), (2)
in which case the beam-vector asymmetry is sqrt(3/2) larger and the tensor
asymmetry sqrt(1/2) smaller (which I find more "natural"; I would like to
call formula (1) "unfortunate").
Tim, on your slide 5 from yesterday your Eq.(2) is inconsistent with
Eq.(3) in the above sense. You may want to double-check the conventions in
the plot on slide 11 and the other ones, too.
Best regards
Michael
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