Looks like you brought em in kicking and screaming. Good job.
p1 = h*f1/c
p2 = h*f2/c
Dp = h*Df/c
Df = Dp*c/h
Or if you prefer
Dl = h/(Dp*c)
The scattering of photons from charged particles is called Compton scattering after Arthur Compton who was the first to measure photon-electron scattering in 1922. When the incoming photon gives part of its energy to the electron, then the scattered photon has lower energy and according to the Planck relationship has lower frequency and longer wavelength. The wavelength change in such scattering depends only upon the angle of scattering for a given target particle. The constant in the Compton formula above can be written