Elong-15-05-06

Relating t20 and Azz

The Boden '91 paper, the only existing t20 measurement taken with a solid ND3 target, does a nice job of describing the relationship between the measured ratio and t20. Eq. 1 describes their ratio,

<math>R_{exp}=\frac{d\sigma_p}{d\sigma_u}=1+P_{zz} \left( \frac{d\sigma_T-d\sigma_u}{d\sigma_u} \right) = 1+P_{zz} \left( R_T-1 \right) </math>

<math>R_T-1 = \frac{1}{P_{zz}} \left( \frac{d\sigma_p}{d\sigma_u} -1 \right).</math>

This matches closely to our Azz,

<math>A_{zz}=\frac{2}{fP_{zz}}\left( \frac{d\sigma_p}{d\sigma_u}-1\right), </math>

where the <math>f=1</math> in the Boden result, since they are tagging deuterons and have negligible contamination.

The two are related by

<math>R_T-1=\frac{1}{2}A_{zz}.</math>

Later on in the paper, <math>(R_T-1)</math> is related to t20 by neglecting the (small) magnetic contributions to get

<math>t_{20}=-\sqrt{8}(R_T-1),</math>

which we can relate to our Azz above by

<math>t_{20}=-\sqrt{2}A_{zz}.</math>

Calculating the elastic Azz at x=2 and smearing it using the spectrometers' momentum resolution, we can relate the uncertainty in t20 to Azz by

<math>\delta t_{20} = \sqrt{2}\delta A_{zz},</math>

which yields the plot below of statistical uncertainty we would expect for longitudinally-tensor-polarized electron-deuteron scattering.