Physics Bootcamp

Photoelectric effect demonstrated that energy in light came in bunches or quanta called photons. If photons really do exist as particles, each photon would have energy as well as momentum. We have seen that the energy of a photon is equal to \(hf\text{.}\) Since a photon must travel at the speed of light, its rest mass must be zero as shown in the chapter on relativity. This gives the following expression for the momentum of a photon.

where I have replaced \(f/c = \lambda\text{,}\) where \(\lambda\) is the wavelength of the electromagnetic wave. We will often refer to \(f\) and \(\lambda\) the frequency and wavelength of the photon, even though they refer to the frequency and wavelength of the electromagnetic wave that the photon belongs to.

Arthur Compton of Washington University in St. Louis, Missouri analyzed the energy and momentum conservation in the process of collision of a photon with an electron and found that, the photon coming at an angle to the original direction after collision will have a different wavelength than the wavelength of the incident photon. The predicted shift in wavelength was small, of the order of picometers, and was independent of the wavelength of the incoming photon. Since X-rays have wavelengths in the picometer range the experiment is usually done with X-rays as Compton had done. Also, since the work function of a surface is in the electron-volt (eV) range while the energy of a photon in X-ray has energy in the keV range, we ignore the work function completely.

Collision of a photon with a free electron:

Figure 53.5.1 shows schematically the situation before collision and after collision. The electron initially is taken to be an almost free electron and at rest. This assumption for the electron is justified since the speeds of the photon and the recoiled electron will be in the relativistic regime while the speeds of a loosely bound electron in the material will be non-relativistic.

Let \(\vec p_0\) be the momentum of the incoming photon, \(\vec p_\theta\) the momentum of the outgoing photon at angle \(\theta\text{,}\) and \(\vec p_e\) be the momentum of the recoiled electron. The vector equation of conservation of momentum will be

These vectors form a triangle as shown in Figure 53.5.1. The law of cosine applied to the triangle gives the following relation between the magnitudes of the momenta and angle \(\theta\text{.}\)

with the momenta given by the following:

where \(v\) is the speed of the recoiled electron, \(\lambda_0\) is the wavelength of the incoming photon, and \(\lambda_\theta\) the wavelength of the outgoing photon at angle \(\theta\text{.}\) Therefore, we get the following equation from the conservation of momentum

The conservation of energy will give

Since the velocity of the recoiled electron is not measured in the experiment we will eliminate \(v\) from Eqs. (53.5.5) and (53.5.6). A long and tedious calculation gives the following result.

The quantity \(\Delta\lambda = \lambda_{\theta} - \lambda_0\) is called the Compton shift and the quantity \(\lambda_c = \dfrac{h}{mc}\) is called the Compton wavelength of the particle of mass \(m\text{,}\) which is the mass of electron here. Putting the value of \(h\text{,}\) \(c\text{,}\) and \(m\) in \(\dfrac{h}{mc}\) we find the Compton wavelength of an electron to be

According to Equation (53.5.7) no shift will be seen in the forward direction, \(\theta = 0\) and a maximum shift will occur at the back scattered direction, \(\theta = 180^{\circ}\text{.}\) At \(90^{\circ}\) the shift will equal the Compton wavelength of the object that the photon was scattered from.