I want to derive the basic equations for the rate of electron capture, taking into account the relativistic kinematics explicitly

$$p^{+}+e^{-}\rightarrow n^{0}+\nu_{e}$$

The idea is to treat electronic wave function like a particle-in-the-box, rather than use full blown atomic orbitals. But I want to do the full calculation over the kinematics, and compute the matrix elements, form factors, etc.

I am assuming one can derive this to using something like the differential cross section for 2+2 body scattering

$$1+2\rightarrow 3+4$$


(in the C.M. frame, $\mathbf{p_{e}}=-\mathbf{p_{p}}$)

and the rate is obtained by the integrating the differential cross section, to get something like


where $\sigma$ is the unpolarized cross section, and where $$\big\Vert\psi_{e}(0)\big\Vert$$ is the electronic density at the origin (the nuclear charge)

The matrix elements $M_{i,f}$ can be computed from the Lagrangian for the Weak interaction.

I am trying to get to an expression for the rate of capture which is something like


where $\mathbf{k}$ is the momentum 3-vector for the outbound neutrino $\nu_{e}$

Note that the rate calculations integrate over the solid angle of the neutrino


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