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.



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 .

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) .