Electrons and Matter Waves — Question 5

Question 5

Electron interference can be used to infer accelerating voltage

A coherent nonrelativistic electron beam passes through slits separated by $d=1.00\,\mu\mathrm m$. The distant screen is $L=1.00\,\mathrm m$ away and the fringe spacing is $\Delta y=.100\,\mathrm{mm}$. Use the small-angle relation $\Delta y=\lambda L/d$; neglect initial kinetic energy. Use $hc=1240\,\mathrm{eV\,nm}$ and $m_ec^2=511000\,\mathrm{eV}$.

1. Infer the wavelength and accelerating potential U.

2. Predict the new fringe spacing if U is increased by a factor of four.

Question 5 - Solution

Electron interference can be used to infer accelerating voltage

(a) Method and calculation

$$\boxed{\lambda=d\Delta y/L=1.00\times10^{-10}\,\mathrm m=.100\,\mathrm{nm}}.$$$$\boxed{eU=\frac{(hc)^2}{2m_ec^2\lambda^2}=150.450\,\mathrm{eV}},\quad U=150.450\,\mathrm V.$$

(b) Interpretation and further calculation

$$\lambda\propto K^{-1/2}\propto U^{-1/2}.$$$$\boxed{\Delta y_{\mathrm{new}}=\Delta y/2=.0500\,\mathrm{mm}}.$$

Physical check

The voltage determines longitudinal momentum; transverse coherence and a stable apparatus are also needed for observable fringes.