# Characteristic and Continuous X-rays

The x-rays are produced in a Coolidge tube when a high energy electron interacts with a heavy metal target.

The high energy electrons get decelerated by Coulomb’s interaction with the target atom. These decelerating electrons emit continuous X-rays (or bremsstrahlung). When an electron loses its entire energy in a single Coulomb's interaction then X-rays of minimum wavelength (cut-off wavelength) are emitted.

The cut-off wavelength of the continuous X-ray spectrum is related to the accelerating potential $V$ of the Coolidge tube by \begin{align} \lambda_\mathrm{cut\ off} & =\frac{hc}{eV} \approx \frac{12400}{V} \;\;\; \mathrm{(Angstrom/V)}. \nonumber \end{align} Typically, accelerating voltage is a few thousand Volts and cut off wavelength is of the order of an Angstrom. The cut-off frequency of continuous x-rays is given by $\nu_\mathrm{cut\ off}=c/\lambda_\mathrm{cut\ off}$. The cutoff wavelength decreases when the accelerating potential of Coolidge tube is increased. The intensity of X-rays increases with an increase in accelerating potential as shown in the figure. The intensity of X-rays depends on multiple factors.

The high energy electrons also knock out an inner shell electron of the target atom. The characteristic X-rays are emitted when an electron (in atom) makes a transition from a high energy state to a low energy state to fill the vacancy. For example, in $K_\alpha$ X-ray series, the electrons makes transition from the high energy state to the K shell of the atom. The frequency of characteristic X-rays is related to the atomic number of the target element by Moseley’s Law $\sqrt{\nu}=a(Z-b)$. The frequency of characteristic X-rays depends on the target material (it is independent of the accelerating potential).

## Solved Problems on X-rays

### Problem from IIT JEE 2008

Which of the following statements is wrong in the context of X-rays generated from a X-ray tube?

1. Wavelength of characteristic X-rays decreases when the atomic number of the target increases.
2. Cut-off wavelength of the continuous X-rays depends on the atomic number of the target.
3. Intensity of the characteristic X-rays depends on the electrical power given to the X-ray tube.
4. Cut-off wavelength of the continuous X-rays depends on the energy of the electrons in the X-ray tube.

Solution: The frequency $\nu$ of characteristic X-rays is related to atomic number $Z$ by Moseley's law, \begin{align} \sqrt{\nu}=a(Z-b). \end{align} which gives, \begin{align} \lambda=\frac{c}{\nu}=\frac{c}{a^2(Z-b)^2}. \end{align} Thus, wavelength decreases with an increase in $Z$. The cut-off wavelength of continuous X-rays corresponds to the maximum energy of an electron in X-ray tube. It is given by, \begin{align} hc/\lambda=eV, \end{align} where $V$ is accelerating potential. The intensity of X-rays depends on the number of electrons striking the target per second, which in turn depends on electrical power given to the X-ray tube as energy of each electron is $eV$.

### Problem from IIT JEE 2000

Electrons with energy 80 keV are incident on the tungsten target of an X-ray tube. K-shell electrons of tungsten have 72.5 keV energy. X-rays emitted by the tube contains only,

1. a continuous X-ray spectrum (Bremsstrahlung) with a minimum wavelength of $\approx 0.155$ Angstrom.
2. a continuous X-ray spectrum with all wavelengths.
3. the characteristic X-ray spectrum of tungsten.
4. a continuous X-ray spectrum with a minimum wavelength of $\approx 0.155$ Angstrom and the characteristic X-ray spectrum of tungsten.

Solution: The energy of incident electron ($E_\text{in}=80$ keV) is sufficient to knock out K-shell electrons (72.5 keV) thereby emitting characteristic X-rays. The minimum wavelength of continuous spectrum corresponds to $E_\text{in}$ and is given by \begin{align} \lambda&=\frac{hc}{E_\text{in}}=\frac{(6.63\times{10}^{-34})\,(3\times{10}^{8})}{(80\times{10}^{3})\,(1.602\times{10}^{-19})}\nonumber\\ &=0.155\;\mathrm{Angstrom}.\nonumber \end{align}

### Problem from IIT JEE 1988

The potential difference applied to an X-ray tube is increased. As a result, in the emitted radiation,

1. the intensity increases.
2. the minimum wavelength increases.
3. the intensity remains unchanged.
4. the minimum wavelength decreases.

Solution: The minimum wavelength of $X$-ray is related to the applied potential by \begin{align} \lambda_\text{min}=\frac{hc}{eV}=\frac{12420}{V}\;\;\mathrm{Angstrom.}\nonumber \end{align} Thus, the minimum wavelength decreases when $V$ is increased. The intensity is the rate of energy flow per unit area. The energy of photons increases when $V$ is increased. Thus, intensity also increases with an increase in the applied potential.

However, intensity of $X$-rays is generally controlled by the filament current. An increase in the filament current increases its temperature which in turn increases the rate of electron emissions (thermionic emissions). Thus, more number of electrons strikes the target leading to increase in number of $X$-ray photons.

## Questions on X-rays

Question 1 (IIT JEE 1998): X-rays are produced in an X-ray tube operating at a given accelerating voltage. The wavelength of the continuous X-rays has values from

A. $0$ to $\infty$.
B. $\lambda_\text{min}$ to $\infty$, where $\lambda_\text{min}>0$.
C. $0$ to $\lambda_\text{max}$, where $\lambda_\text{max}<\infty$.
D. $\lambda_\text{min}$ to $\lambda_\text{max}$ where $0<\lambda_\text{min}<\lambda_\text{max}<\infty$.

Question 2 (IIT JEE 1988): The shortest wavelength of X-ray emitted from an X-ray tube depends on

A. the current in the tube.
B. the voltage applied to the tube.
C. the nature of the gas in tube.
D. the atomic number of the target material.

Question 3 (IIT JEE 1986): When the number of electrons striking the anode of an X-ray tube is increased the____________ of the emitted X-rays increases while when the speeds of the electrons striking the anode are increased the cut-off wavelength of the emitted X-rays___________.

A. cut off wavelength, decreases
B. cut-off wavelength, decreases
C. intensity, increases
D. intensity, decreases

Question 4 (Concepts of Physics): Consider a photon of a continuous X-ray coming from a Coolidge tube. Its energy comes from

A. the kinetic energy of the striking electron
B. the kinetic energy of the free electrons of the target
C. the kinetic energy of the ions of the target
D. an atomic transition in the target