- Compton Scattering - Electron Momentum

[phil ess]

URGENT - Compton Scattering - Electron Momentum

Homework Statement

An x-ray photon of initial energy 1x10^5 eV traveling in the +x direction is incident on a free
electron at rest. The photon is scattered at right angles into the +y direction. Find the components of momentum of the recoiling electron.

Homework Equations

Lots

The Attempt at a Solution

Since the photon recoils at 90 degrees, I'm assuming that the electron recoils at 45 degrees, so the x and y components of its momentum are equal. Then I just need to find the momentum of the electron after collision:

Ephoton = 1.602x10^-14 J = hc/lambda => lambda1 = 1.2398x10^-11 m

Then using the compton equation:

delta lambda = (h/melectron*c)(1-cos 90) = 3.5135x10^-12 m

Which gives the final energy of the photon via:

lambda2 = lambda1 + delta lambda = 1.59115x10^-11 m
Ephtoton' = hc/lambda2 = 1.2484x10^-14

Then the energy lost by the photon is gained by the electron, whose total energy becomes:

Eelectron' = rest energy + photon energy = melectron*c^2 + (1.602-1.2484)x1-^-14 = 8.5407x10^-14

The energy gained by the electron is in the form of kinetic energy, so we can find its speed:

KE' = (1.602-1.2484)x10^-14 = 1/2 melectron*v^2 => v = 8.811x10^7 m/s

Finally the relativistic momentum of the electron is given by:

p=gamma mv
E=gamma mc^2

=> v/c=pc/E => pelectron' = Ev/c^2 = 8.3729x10^-23

But the momentum of the initial photon is:

pphoton = h/lambda1 = 5.3437x10^-23

So momentum is not conserved? I have tried this problem so many times my head hurts! Can anyone see where I've gone wrong? Any help is greatly aprreaciated!

 

[alphysicist]

Hi phil ess,

Homework Statement

An x-ray photon of initial energy 1x10^5 eV traveling in the +x direction is incident on a free
electron at rest. The photon is scattered at right angles into the +y direction. Find the components of momentum of the recoiling electron.

Homework Equations

Lots

The Attempt at a Solution

Since the photon recoils at 90 degrees, I'm assuming that the electron recoils at 45 degrees,

I don't think you can assume this. The recoil angle for the electron is something you'll need to find while doing the problem.

so the x and y components of its momentum are equal. Then I just need to find the momentum of the electron after collision:

Ephoton = 1.602x10^-14 J = hc/lambda => lambda1 = 1.2398x10^-11 m

Then using the compton equation:

delta lambda = (h/melectron*c)(1-cos 90) = 3.5135x10^-12 m

Which gives the final energy of the photon via:

lambda2 = lambda1 + delta lambda = 1.59115x10^-11 m
Ephtoton' = hc/lambda2 = 1.2484x10^-14

Then the energy lost by the photon is gained by the electron, whose total energy becomes:

Eelectron' = rest energy + photon energy = melectron*c^2 + (1.602-1.2484)x1-^-14 = 8.5407x10^-14

The energy gained by the electron is in the form of kinetic energy, so we can find its speed:

KE' = (1.602-1.2484)x10^-14 = 1/2 melectron*v^2 => v = 8.811x10^7 m/s

Finally the relativistic momentum of the electron is given by:

p=gamma mv
E=gamma mc^2

=> v/c=pc/E => pelectron' = Ev/c^2 = 8.3729x10^-23

But the momentum of the initial photon is:

pphoton = h/lambda1 = 5.3437x10^-23

So momentum is not conserved? I have tried this problem so many times my head hurts! Can anyone see where I've gone wrong? Any help is greatly aprreaciated!