Calculating the Wavelength of Monochromatic Radiation in Compton Scattering

[Pouyan]

Homework Statement

Compton scattering can be used both to measure the direction and energy of photons in nuclear physics experiments. For a particular preparation a spectrum of Compton scattered electrons was measured which clearly corresponded to a generally monochromatic gamma radiation. The maximum electron energy was measured to 150 keV. Calculate the wavelength of the incoming monochromatic radiation.

Homework Equations

What do I know :

Compton formula : λ'-λ = (h/mc) * (1-cos(θ))
Energy conservation: hc/λ + m * c^2 = hc/λ' +γ*m *c^2
E_befor = hc/λ , E_after = hc/λ'

The Attempt at a Solution

The correct solution is :
E_max = 150 keV = 2.4 * 10^-14

λ'-λ = (h/mc) * (1-cos(θ)) and maximum occurs when cosθ = -1 then :
λ'-λ = 2*(h/mc)
Further we have :
E_befor + m * c² = E_after + γ*m *c²
E_befor - E_after = (γ-1)mc² = E_kinetic
E_kinetic = hc((1/λ)-(1/λ')) †
We take E_kinetic = 150 keV = 2.4 * 10^-14
and solve λ'= 2*(h/mc) + λ
and just put everything in † and solve λ which is = 0.044nm

My question is why can't we solve this by thinking :

hc/λ + m * c^2 = hc/λ' +γ*m *c^2 = Constant = 2.4 * 10^-14 J

and just take hc/λ + m * c^2 = 2.4 * 10^-14 J ? I don't get the same λ as the solution and I know it's wrong but why is this wrong?!

 

[vela]

Why would the maximum electron energy be equal to the total energy of the photon-electron system? That's what you're claiming in your method.