Changes

The lengthening of the wave-length happens when a photon interacts with a free electron in the material. The effect can be calculated along the following lines.

The lengthening of the wave-length happens when a photon interacts with a free electron in the material. The effect can be calculated along the following lines.

1) '''Conservation of energy''': We assume that before the collision, the electron is nearly at rest, so its energy is entirely its rest-energy <math>E_e=m_e c^2</math>. Here, <math>m_e</math> is the rest mass of the electron. The photon has an energy of <math>E_{\gamma}= h f</math>, where <math>h</math> is Planck's constant, and <math>f</math> is its initial frequency. After the interaction, the photon's frequency changed to <math>f'</math>, so its energy is now <math>E'_{\gamma} = h f'</math>. The energy of the electron after the collision is <math>E'_e = \sqrt{p^2_e c^2 + m^2_e c^4}</math>, and we get the equation;

1) '''Conservation of energy''': We assume that before the collision, the electron is nearly at rest, so its kinetic energy is zero: <math>E_e=0</math>. The photon has an energy of <math>E_{\gamma}= h f</math>, where <math>h</math> is Planck's constant, and <math>f</math> is its initial frequency. After the interaction, the photon's frequency changed to <math>f'</math>, so its energy is now <math>E'_{\gamma} = h f'</math>. The kinetic energy of the electron after the collision is <math>E'_e = \sqrt{p^2_e c^2 + m^2_e c^4} - m^2_e c^2</math>. (Here, <math>m_e</math> is the mass of the electron.)  We get the equation;

<center><math> h f + m_e c^2 = h f' +\sqrt{p^2_e c^2 + m^2_e c^4} </math></center>

<center><math> h f = h f' +\sqrt{p^2_e c^2 + m^2_e c^4} - m_e c^2</math></center>

(here, <math>\vec{p_e}</math> is the momentum of the electron after the event)

(here, <math>\vec{p_e}</math> is the momentum of the electron after the event)

2) '''Conversation of momentum''': As we assume that the electron is nearly at rest at first, its initial momentum is zero. The momentum of the photon is at first <math>\vec{p_p}</math> and then <math>\vec{p'_p}</math>. The seize of the momentum of a photon can be calculated via <math>p = E/c = {h f}/c</math>.

2) '''Conversation of momentum''': As we assume that the electron is nearly at rest at first, its initial momentum is zero. The momentum of the photon is at first <math>\vec{p_p}</math> and then <math>\vec{p'_p}</math>. The momentum of a photon can be calculated via <math>p = E/c = {h f}/c</math>.

Looking at the picture, we see that  

Looking at the picture, we see that  

<math> \frac{1}{h f'} - \frac{1}{h f'} = \frac{1}{m_e c^2} (1 - \cos \theta)</math>

<math> \frac{1}{h f'} - \frac{1}{h f'} = \frac{1}{m_e c^2} (1 - \cos \theta)</math>

As for a photon <math>f \lambda = c</math>, we can bring this is Compton's form:

As for a photon <math>f \lambda = c</math>, we can bring this into Compton's form:

<math>\lambda' - \lambda = \Delta \lambda = \frac{h}{m_e c} (1 - \cos \theta)</math>

<math>\lambda' - \lambda = \Delta \lambda = \frac{h}{m_e c} (1 - \cos \theta)</math>

This describes exactly the effect observed by Compton! Compton himself used a similar derivation, also including the relativistic energy expression.  

This describes exactly the effect observed by Compton! Compton himself used a similar derivation, also including the relativistic energy expression.

== Probability of Compton Scattering ==

== Probability of Compton Scattering ==