The story of an electron kicked by a wave.

The Compton Effect

Abstract
The photoelectric effect used the fact that energy is conserved with a collision between a photon and an electron at rest in a metal. the involved energy of the incident photon is on the same order of magnitude as the binding energy of an electron to a nucleus , that is few eV. However, if the energy of the photon is large compared to the binding energy of the electron, that is several KeV, therefore, both conservation of momentum and energy could be considered. Compton used this fact in an experiment of scattered x-ray radiation off of a graphite block to measure the inrease of the wavelength of the x-rays.

Introduction
The Compton effect was observed by Arthur Compton in 1923. This Compton Scattering is about an interaction between an incident gamma photon and an electron at rest. The incident particle-wave loses enough energy to an orbital electron to cause its ejection, that is to ionize the related atom. After the collision, the incident photon becomes lower in energy, then infrequency, and then large in wavelength with an emission direction different from that of before the collision. Compton scattering is considered to be the principal absorption mechanism for gamma rays in the intermediate energy range 100 keV to 10 MeV.

- P₀ Incident photon's momentum ,
- P_e0 Stationary electron's momentum,
- P₁ Scattered photon's momentum,
- P_e1 Recoil electron's momentum,
-  a , b  Scattering and recoil 
angles respectively.

Theory

The conservation of momentum is written by:

P₀ + 0 = P₁ + P_e1
Solving for P_e1, we have:
P_e1² =( P₀ - P₁) ² = P₀² + P₁² -2 P₀P₁ = P₀² + P₁² -2 P₀P₁cos (b)
We know that:
P₀ = hv₀/c
P₁ = hv₁/c
Then:
P_e1² = (hv₀/c)² + (hv₁/c)² -2 (hv₀/c)( hv₁/c) cos (b) (Eq.1)

The conservation of enegy is written by:

E₀+ E_e0=E₁+E_e1
We know that :
E₀=hv₀and
E_e0=m_ec²and
E₁=hv₁ and
E_e1=sqrt (p_e1²c²+m_e²c⁴)
Solving for p_e1², we have:
p_e1²c²= E_e1² - m_e²c⁴
Using :
E_e1= E₀+ E_e0-E₁
we get:
p_e1²c² = (hv₀+m_ec²-hv₁)² - m_e²c⁴ (Eq.2)
Equating (Eq.1) and (Eq.2), we have:
(hv₀)² + (hv₁)² -2 (hv₀)( hv₁) cos (b) = (hv₀)²+ (hv₁)² - 2hv₀hv₁ + 2 (hv₀ -hv₁) m _ec²
(hv₀)( v₁) cos (b) = hv₀ v₁ - (v₀ -v₁) m _ec²
hv₀ v₁(1-cos (b)) = (v₀ -v₁) m _ec²
hc/w₀ c/w₁(1-cos (b)) = (c/w₀ -c/w₁) m _ec²
hc (1-cos (b) = (w₁ - w₀) m _ec²
Where w =c/v is the incident photon's wavelength.
w₁ - w₀ =h (1-cos (b))/ m _ec

That is :
w₁ - w₀ =W_C(1-cos (b))
Where W_C= h/ m _ec is known as the Compton wavelength.

Results and discussion

Conclusion

Contact :

ajaja@sympatico.ca