Electrons and Matter Waves — Question 8

Question 8

A wavelength tolerance sets the upper energy for the nonrelativistic formula

For an electron, $\lambda_{\mathrm{nr}}=hc/\sqrt{2m_ec^2K}$ and $\lambda_{\mathrm{exact}}=hc/\sqrt{K(K+2m_ec^2)}$. Require the relative overestimate $\lambda_{\mathrm{nr}}/\lambda_{\mathrm{exact}}-1$ to be at most $\delta=.0100$. Use $hc=1240\,\mathrm{eV\,nm}$ and $m_ec^2=511000\,\mathrm{eV}$.

1. Derive the maximum permitted K as a function of $\delta$.

2. Evaluate the bound and the error at $K=10.0\,\mathrm{keV}$.

Question 8 - Solution

A wavelength tolerance sets the upper energy for the nonrelativistic formula

(a) Method and calculation

$$\sqrt{1+K/(2m_ec^2)}\le1+\delta.$$$$\boxed{K\le2m_ec^2(2\delta+\delta^2)}.$$

(b) Interpretation and further calculation

$$\boxed{K_{\max}=20542.2\,\mathrm{eV}=20.5422\,\mathrm{keV}}.$$$$K=10^4\,\mathrm{eV}:\quad\boxed{\delta_{10}=\sqrt{1+10^4/1022000}-1=.00488046}.$$

Physical check

A one-percent error criterion on wavelength is not identical to a one-percent criterion on kinetic energy or velocity.