12.4: Incident and Scattered Flux


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.

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