Electrons and Matter Waves — Question 10

Question 10

Energy spread limits path coherence through the spread of matter-wave phase

A narrow nonrelativistic electron beam has mean kinetic energy $K_0=100\,\mathrm{eV}$ and small rms spread $\sigma_K=1.00\,\mathrm{eV}$. For a path difference $\Delta\ell$, define the linearized rms phase spread $\sigma_\phi=\Delta\ell\,\sigma_k$, with $k=\sqrt{2m_eK}/\hbar$. Define the coherence scale by $\sigma_\phi=1$. Use $hc=1240\,\mathrm{eV\,nm}$ and $m_ec^2=511000\,\mathrm{eV}$.

1. Derive $\sigma_k/k_0$ and calculate the mean wavelength.

2. Find the coherence scale in nanometers and in wavelengths.

Question 10 - Solution

Energy spread limits path coherence through the spread of matter-wave phase

(a) Method and calculation

$$\frac{dk}{dK}=\frac{k}{2K}\quad\Rightarrow\quad\boxed{\sigma_k/k_0\simeq\sigma_K/(2K_0)=.00500}.$$$$\boxed{\lambda_0=\frac{1240}{\sqrt{1022000(100)}}\,\mathrm{nm}=.122658\,\mathrm{nm}}.$$

(b) Interpretation and further calculation

$$\ell_c=1/\sigma_k=\frac{2K_0}{\sigma_K}\frac{\lambda_0}{2\pi}.$$$$\boxed{\ell_c/\lambda_0=100/\pi=31.8310},\qquad\boxed{\ell_c=3.90433\,\mathrm{nm}}.$$

Physical check

This coherence scale uses a specified rms phase criterion and a narrow energy distribution. It is not a universal sharp boundary beyond which all interference disappears.