Q.(a)

(a)(i) Characteristic X-rays are produced when an inner-shell vacancy created by a bombarding electron is filled by an outer electron, emitting a photon of element-specific energy; (ii) the cut-off wavelength/frequency for a 20,000 V tube follow directly from $\lambda_{min}=hc/eV$. (b) A spectrum is dispersed light, occurring in continuous, line, or band form. Both alternatives answered below.

(a)(i) Origin of characteristic X-ray spectra

In an X-ray tube, electrons are accelerated through a high potential and strike a metal target (anode). If an incident electron has enough energy, it can knock out an electron from an inner shell (e.g. the K-shell, closest to the nucleus) of a target atom, creating a vacancy there.

This vacancy is unstable: an electron from a higher shell (L, M, ...) immediately falls into it to fill the gap. In doing so, it loses energy equal to the difference between the two energy levels, which is emitted as an X-ray photon.

Since the energy levels of an atom are unique to each element, the energies (and so wavelengths) of these photons are characteristic of the target material — appearing as sharp, discrete lines (e.g. $K_\alpha$, $K_\beta$ when the vacancy is in the K-shell) superimposed on the smooth, continuous (bremsstrahlung) background spectrum. This discrete line pattern is called the characteristic X-ray spectrum.

(a)(ii) Cut-off wavelength and frequency for V = 20,000 V

The minimum possible wavelength (cut-off/threshold wavelength) is emitted when an incident electron loses its entire kinetic energy $eV$ in a single collision, converting it fully into one X-ray photon:

$eV=\dfrac{hc}{\lambda_{min}}\ \Rightarrow\ \lambda_{min}=\dfrac{hc}{eV}$

Substituting $h=6.626\times10^{-34}$ Js, $c=3\times10^8$ m/s, $e=1.6\times10^{-19}$ C, $V=2\times10^4$ V:

$\lambda_{min}=\dfrac{(6.626\times10^{-34})(3\times10^8)}{(1.6\times10^{-19})(2\times10^4)}=\dfrac{1.9878\times10^{-25}}{3.2\times10^{-15}}\approx6.21\times10^{-11}\ \text{m}\approx0.621\ \text{Å}$

Corresponding cut-off frequency:

$u_{max}=\dfrac{eV}{h}=\dfrac{(1.6\times10^{-19})(2\times10^4)}{6.626\times10^{-34}}=\dfrac{3.2\times10^{-15}}{6.626\times10^{-34}}\approx4.83\times10^{18}\ \text{Hz}$

(b) What is a spectrum? Types of emission spectrum

A spectrum is the pattern obtained when electromagnetic radiation (light) from a source is resolved (dispersed) into its constituent wavelengths or frequencies, typically using a prism or diffraction grating.

Types of emission spectrum:

1. Continuous emission spectrum: contains all wavelengths without any break, produced by incandescent solids, liquids, or highly compressed (dense) gases — e.g. the spectrum of sunlight or a glowing filament (essentially blackbody radiation). …