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Wave Properties of particles. Schrodinger Equation.

Problem 6.60

A narrow stream of monoenergetic electrons falls at an angle of incidence \(\theta=30^{\circ}\) on the natural facet of an aluminium single crystal. The distance between the neighbouring crystal planes parallel to that facet is equal to \(d=0.20 \mathrm{nm}\). The maximum mirror reflection is observed at a certain accelerating voltage \(V_{0} .\) Find \(V_{0}\) if the next maximum mirror reflection is known to be observed when the accelerating voltage is increased \(\eta=2.25\) times.

Reveal Answer
V0=π2ℏ2/2me(η−1)2d2sin⁡2θ=0.15keVV_{0}=\pi^{2} \hbar^{2} / 2 m e(\sqrt{\eta}-1)^{2} d^{2} \sin ^{2} \theta=0.15 \mathrm{keV}
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