​A spin-one-half projectile of mass m and energy E = h^2k^2/2m scatters off an infinitely heavy spin-one-half target. ​

Question

A spin-one-half projectile of mass m and energy E = \(\frac{h^2k^2}{2m}\) scatters off an infinitely heavy spin-one-half target. The interaction Hamiltonian is

\(H_{int} = A\sigma_1.\sigma _2 \frac{e^{-\mu r}}r (\mu > 0),\)

where σ₁ and σ₂ are the Pauli spin operators of the projectile and target respectively. Compute the differential scattering cross section \(\frac{d\sigma}{d\Omega}\) in lowest order Born approximation, averaging over initial and summing over final states of spin polarization. Express \(\frac{d\sigma}{d\Omega}\) as a function of k and the scattering angle e.

Answer 1

Suppose that the projectile is incident on the target along the z-axis, i.e., k₀ = ke_z,. 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