I have been tasked to find the rate at which scattered electrons will be detected on an annular detector in the far-field. The exact question I'm working with is:
Suppose that 1keV electrons, initially incident along the optic axis $\hat{\mathbf{z}}$, scatter from the Coulomb field of a stationary silicon nucleus before being detected by an annular detector centered on the optic axis in the far-field. If the flux of incident electrons is \begin{equation} |\mathbf{j}_0| = 10^8 \mathring{\text{A}}^{-2} \text{sec$^{-1}$} , \end{equation} and if the inner and outer edges of the annular detector correspond to scattering angles of 50 mrad and 150 mrad, respectively, at what rate will the scattered electrons be detected?
For Coulomb scattering, I know that
\begin{equation} \dfrac{d\sigma}{d\Omega} = \dfrac{4m^2}{\hbar^2} \bigg ( \dfrac{Ze^2}{4\pi \epsilon_0 } \bigg )^2 \dfrac{1}{|\mathbf{k}-\mathbf{k}_0|^4} \end{equation}
Here, $\mathbf{k}_0$ is the incident plane wave vector and $\mathbf{k}$ is the scattered plane wave vector. My question is regarding the $\mathbf{k}$-vectors. I'm not sure how to set them up in terms of the spherical-coordinate $\theta$ (the polar angle). Since we are told the detector is in the far-field, can the small angle approximation, $\sin(\theta) \approx \theta$ be used here? In this case, $\mathbf{k}_0 = 0$ and $\mathbf{k} = \theta$.
Any help here would be appreciated. I know how to solve the rest of the problem, but I'm unsure on how to work with the $\mathbf{k}$-vectors here.