Suppose that the projectile is incident on the target along the z-axis, i.e., k0 = kez,. In lowest order Born approximation, the scattering amplitude is
where q = k0 - k, q = 2k sin \(\frac \theta 2\). Denote the total spin of the system by S. Then S = \(\frac 12 (\sigma_1 + \sigma_2)\) and
If the initial states of spin of the projectile and target are \(\binom 10_P =\alpha_P, \binom10 _T= \alpha_T\) respectively, then the initial state of spin of the system is \(\Theta_{11} = \alpha _P \alpha_T\), the scattered wave function is \(f_1 (\theta) \frac{e^{ikr}}{r} \Theta_{11}\), and the corresponding differential scattering cross section is given by
we can obtain the remaining differential cross sections:
Averaging over the initial states (i) and summing over the final states (f) of spin polarization, we obtain