Electrons and Matter Waves — Question 4

Question 4

Crystal diffraction provides an experimental determination of the action scale

Nonrelativistic electrons are accelerated from rest through $U=150\,\mathrm V$. First-order Bragg reflection occurs at glancing angle $\theta=15.0^\circ$ from planes separated by $d=.194\,\mathrm{nm}$. Use $2d\sin\theta=\lambda$, $m_e=9.11\times10^{-31}\,\mathrm{kg}$ and $e=1.60\times10^{-19}\,\mathrm C$.

1. Find the measured wavelength.

2. Infer h from the wavelength and accelerating voltage.

Question 4 - Solution

Crystal diffraction provides an experimental determination of the action scale

(a) Method and calculation

$$\boxed{\lambda=2d\sin\theta=1.00422\times10^{-10}\,\mathrm m}.$$$$p=\sqrt{2m_e eU}=6.61272\times10^{-24}\,\mathrm{kg\,m/s}.$$

(b) Interpretation and further calculation

$$h_{\mathrm{inferred}}=p\lambda=2d\sin\theta\sqrt{2m_e eU}.$$$$\boxed{h_{\mathrm{inferred}}=6.64061\times10^{-34}\,\mathrm{J\,s}}.$$

Physical check

The angle is measured from the planes, not their normal. The inferred action scale is limited by the supplied rounded measurements and constants.