6.3 The compton effect  (Page 2/12)

Compton shift

As given by Compton, the explanation of the Compton shift is that in the target material, graphite, valence electrons are loosely bound in the atoms and behave like free electrons. Compton assumed that the incident X-ray radiation is a stream of photons. An incoming photon in this stream collides with a valence electron in the graphite target. In the course of this collision, the incoming photon transfers some part of its energy and momentum to the target electron and leaves the scene as a scattered photon. This model explains in qualitative terms why the scattered radiation has a longer wavelength than the incident radiation. Put simply, a photon that has lost some of its energy emerges as a photon with a lower frequency, or equivalently, with a longer wavelength. To show that his model was correct, Compton used it to derive the expression for the Compton shift. In his derivation, he assumed that both photon and electron are relativistic particles and that the collision obeys two commonsense principles: (1) the conservation of linear momentum and (2) the conservation of total relativistic energy.

In the following derivation of the Compton shift, $E_f$ and ${\overrightarrow{p}}_f$ denote the energy and momentum, respectively, of an incident photon with frequency f . The photon collides with a relativistic electron at rest, which means that immediately before the collision, the electron’s energy is entirely its rest mass energy, $m_0{c}^2.$ Immediately after the collision, the electron has energy E and momentum $\overrightarrow{p},$ both of which satisfy [link] . Immediately after the collision, the outgoing photon has energy ${\tilde{E}}_f,$ momentum ${\overrightarrow{\tilde{p}}}_f,$ and frequency $f^{\prime }.$ The direction of the incident photon is horizontal from left to right, and the direction of the outgoing photon is at the angle $\theta ,$ as illustrated in [link] . The scattering angle $\theta$ is the angle between the momentum vectors ${\overrightarrow{p}}_f$ and ${\overrightarrow{\tilde{p}}}_f,$ and we can write their scalar product:

Following Compton’s argument, we assume that the colliding photon and electron form an isolated system. This assumption is valid for weakly bound electrons that, to a good approximation, can be treated as free particles. Our first equation is the conservation of energy for the photon-electron system:

The left side of this equation is the energy of the system at the instant immediately before the collision, and the right side of the equation is the energy of the system at the instant immediately after the collision. Our second equation is the conservation of linear momentum for the photon–electron system where the electron is at rest at the instant immediately before the collision:

The left side of this equation is the momentum of the system right before the collision, and the right side of the equation is the momentum of the system right after collision. The entire physics of Compton scattering is contained in these three preceding equations––the remaining part is algebra. At this point, we could jump to the concluding formula for the Compton shift, but it is beneficial to highlight the main algebraic steps that lead to Compton’s formula, which we give here as follows.