X Rays and the Ordering of the Elements — Question 1

Question 1

The X-ray continuum cutoff depends on accelerating voltage

Electrons initially at rest are accelerated through a potential difference V and strike a massive target. A single photon can take at most the incident electron kinetic energy; neglect target recoil. Use $hc=1240\,\mathrm{eV\,nm}$.

1. Derive the minimum emitted wavelength and evaluate it at $V=40.0\,\mathrm{kV}$.

2. Find the voltage required to reduce that minimum wavelength by 25 percent.

Question 1 - Solution

The X-ray continuum cutoff depends on accelerating voltage

(a) Method and calculation

$$K=eV,\quad E_{\gamma,\max}=eV\Rightarrow\boxed{\lambda_{\min}=hc/(eV)}.$$$$\boxed{\lambda_{\min}=1240/40000=.0310\,\mathrm{nm}}.$$

(b) Interpretation and further calculation

$$\lambda_{\min}\prime=.75\lambda_{\min},\quad V\prime/V=\lambda_{\min}/\lambda_{\min}\prime.$$$$\boxed{V\prime=40.0/.75=53.33\,\mathrm{kV}}.$$

Physical check

The cutoff describes the bremsstrahlung continuum and follows total energy conservation. Characteristic line wavelengths depend primarily on atomic structure and need not shift when the voltage changes.