# Deriving the Compton Effect

Tags:
1. Sep 28, 2015

### eje5758

1. The problem statement, all variables and given/known data
I need to derive the equation related to the Compton Effect from the equations for momentum and energy conservation.

2. Relevant equations
(1) Compton Effect: λ' -λ = h/me(1-cosθ)
(2) Conservation of Momentum (x-direction): h/λ= (h/λ')cosθ+γumeucosΦ
(3) Conservation of Momentum (y-direction): 0= (h/λ')sinθ+γumeusinΦ
(4) Conservation of Energy: h(c/λ)+mec2 = h(c/λ')+γumec2

Where,
λ=initial wavelength of a photon
λ'= final wavelength of a photon
θ= the angle in which an electron scatters
Φ= the angle in which the photon scatters
u= speed at which electron scatters
c= speed of light in a vacuum

3. The attempt at a solution
This is a system of equations problem. So I approached the problem by eliminating squaring equations (2) and (3) and eventually eliminating Φ. This gave me [(h/λ-hcosθ/λ')2+ (h2sin2θ)/(λ')2]/γum2u2=1

Things started to get realllly messy from here. I successfully eliminated u, but by that time, the algebra was too far gone to get back to the Compton Effect equation. Any help would be appreciated.

2. Sep 30, 2015

### gleem

The derivation should not be too messy if you kept track of all the constant factors correctly. you might help yourself by for example combining quantities as h/m0cλ into one symbol say α. Just use care in writing down each expression and persist. Leave the cosines alone and rely on identity relationships to simplify the expressions.