12.4: Incident and Scattered Flux

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The box normalisation corresponds to one particle per volume \(L^3\), so that the number of particles crossing unit area perpendicular to the beam per unit time is just given by the magnitude of the incident velocity divided by \(L^3\):

\[\text{incident flux } = \frac{|{\bf p}|/m}{L^3} = \frac{\hbar k}{mL^3} \nonumber\]

Using the Golden Rule, we have that the rate of transitions between the initial state of wave-vector \({\bf k}\) and final states whose wave-vectors \({\bf k}'\) lie in the element of solid angle d\(\Omega\) about the direction \((\theta, \phi)\) of the wave-vector \( {\bf k}'\), is given by

\[R = \frac{2\pi}{\hbar} |V_{ {\bf k'k}}|^2 \frac{L^3}{8\pi^3} \frac{mk}{\hbar^2} d\Omega\nonumber\]

but this is just the number of particles scattered into d\(\Omega\) per unit time. To get the scattered flux we simply divide by d\(\Omega\) to get the number per unit time per unit solid angle.