Compton Scattering in Newtonian Physics

[rpardo]

Homework Statement

Derive the equation non-relativistic equation for Compton scattering

(mc^2) [(1/E2)-(1/E1)]+cos(theta)-[((E1-E2)^2)/(2E1E2)]=1

E1 = Incident Photon's energy
E2 = Scattered Photon's energy
theta= scattering angle
m = mass of electron
c = s

Here is the lab
In other words derive equation 2

http://www.physics.uoguelph.ca/~reception/2440/RelativMech-Compton-Jan10-08.pdf

Homework Equations

Conservation of Energy,conservation of mass, cosine law

The Attempt at a Solution

I've derived the first equation on the lab (relativistic approach)
I am really stumped on how to derive this...its been 4 hours and counting...
Any help at all would be appreciated. I feel like I'm really close but the algebra is the problem

Thanks in advance for your help guys and gals,

Truly appreciated

 

[vela]

How did you set up the problem?

Here are some suggestions to make the algebra easier:

1. Write the kinetic energy of the electron as $p^2/(2m)$.
2. Don't break the conservation of momentum equation into components.
3. Isolate the electron's momentum in the conservation of momentum equation on one side and then square the equation.

 

[kerimek]

.


I would first like to commend you for taking on this challenging problem and for seeking help when you are stuck. It shows great determination and dedication to your studies.

Now, to derive the non-relativistic equation for Compton scattering, we need to make some assumptions and simplifications. First, we will assume that the energies of the incident and scattered photons are much smaller than the rest energy of the electron, which is mc^2. This means that the energies can be approximated as E1 ≈ mc^2 and E2 ≈ mc^2.

Next, we will use the conservation of energy and momentum to derive the equation. The initial energy of the system is E1, and the final energy is E2. The initial momentum is zero since the photon is at rest, and the final momentum is p2 = E2/c (using the relativistic expression for momentum).

Conservation of energy: E1 = E2 + K2 (where K2 is the kinetic energy of the scattered electron)

Conservation of momentum: p2 = p1 + p2cosθ (where p1 is the momentum of the incident photon and p2cosθ is the momentum of the scattered photon in the x-direction)

Solving for p2 and substituting into the energy equation, we get:

E1 = (E2^2)/(2mc^2) + E2 + K2

Using the approximation for E1 and E2, and expanding the kinetic energy term using the non-relativistic expression (K2 = p2^2/2m), we get:

mc^2 = (m^2c^4)/(2mc^2) + mc^2 + (p2^2)/(2m)

Simplifying and rearranging, we get:

(p2^2)/(2m) = (m^2c^4)/(2mc^2)

Substituting p2 = E2/c, we get:

(E2^2)/(2m) = (m^2c^2)/2

Multiplying both sides by 2 and rearranging, we get the desired equation:

mc^2[(1/E2) - (1/E1)] + cosθ = 1

This equation is valid for non-relativistic energies, and it reduces to the relativistic equation when E1 and E2 are large compared to the rest energy of the electron.

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