The magnitude of proton polarization can be measured by NMR technique with a reference. Because the NMR gives the free-induction decay signal, which is a voltage or current. For Boltzmann polarization using strong magnetic field and low temperature, the polarization can be calculated. However, when a reference point is not available, the absolute magnitude of proton polarization can be measured using proton-proton elastic scattering. The principle is the nuclear spin-orbital coupling. That creates left-right asymmetry on the scattering cross section.

Because of spin-orbital interaction:

V_{ls}(r) = f(r) \vec{l} \cdot \vec{s} ,

where f(r) is the distance function, \vec{l} is the relative angular momentum, \vec{s} is the spin of the incident proton. In the following picture, the spin of the incident proton can be either out of the plane (\uparrow ) or into the plan (\downarrow). When the proton coming above, the angular momentum is into the plane (\downarrow ). The 4 possible sign of the spin-orbital interaction is shown. We can see, when the spin is up, the spin-orbital force repulses the proton above and attracts the proton below. That creates an asymmetry in the scattering cross section.

LS.PNG

 

The cross section is distorted and characterized using analysing power A_y. Analyzing power is proportional to the difference between left-right cross-section. By symmetry (parity, time-reversal) consideration, A_y = 1 + P sin(2\theta) (why?), in center of mass frame. In past post, the transformation between difference Lorentz frame. The angle in the A_y has to be expressed in lab angle. The cross section and A_y can be obtained from http://gwdac.phys.gwu.edu/ .


In scattering experiment, the number of proton (yield) is counted in left and right detectors. The yield should be difference when either proton is polarized. The yield is

Y(\theta, \phi) = L \epsilon \sigma_0 (1 + cos(\phi)A_y(\theta) P) ,

where L is the luminosity, \epsilon is the detector efficiency, \sigma_0 is the integrated cross-section of un-polarized beam and target of the detector, P is the polarization of either the target or beam. When both target and the beam are polarized, the cross section is

\sigma = \sigma_0 (1 + (P + P_T)A_y + P P_T C_yy),

where C_yy is spin-spin correlation due to spin-spin interaction of nuclear force.


Using the left-right yield difference, the absolute polarization of the target or the beam can be found using,

\displaystyle A_y P = \frac{Y_L - Y_R}{Y_L + Y_R} ,

where Y_L = Y(\phi =0) and Y_R = Y(\phi=\pi) .