- #1
b2386
- 34
- 0
Hi everyone,
Right now, I am working on a homework problem asking me to derive the Compton effect, which is given by \lambda\prime-\lambda=\frac{h}{m_ec}(1-cos\theta)
A diagram of the situation can be found here: http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/compton.html#c1 (However, in my diagram, the recoil electron and scattered photon are switched with the electron and \phi below the x-axis and the photon and \theta above the x axis.)
To derive the equation, I use the momentum conservation equations and energy conservation equation.
There are two equations for the momentum, one for the x component and one for the y component:
x-component: \frac{h}{\lambda}=\frac{h}{\lambda\prime}cos\theta+\gamma m u cos\phi
y-component: 0=\frac{h}{\lambda\prime}sin\theta-\gamma mu sin\phi
where \gamma=\frac{1}{\sqrt{1-\frac{u^2}{c^2}}}
m=electron mass
u=electron velocity after the collision
My equation for energy conservation is:
h\frac{c}{\lambda}+mc^2=h\frac{c}{\lambda\prime}+\gamma m c^2
To start off, I solved the x component of the momentum for cos(phi) and the y component for sin(phi). I then added the equations together and got (since sin^2 + cos^2 = 1):
\frac{h^2}{\lambda\prime^2\gamma^2m^2u^2}+\frac{h^2}{\lambda^2\gamma^2m^2u^2}-\frac{2h^2cos\theta}{\lambda\prime\lambda\gamma^2m^2u^2}=1
At this point, I am to now eliminate u using the resulting equation and the equation for the energy conservation as given above. However, I am stuck on this next step because it seems as though any algabraic manipulation I try makes the equation extremely complicated. Could anyone please give me some hints as to what step should be next?
Right now, I am working on a homework problem asking me to derive the Compton effect, which is given by \lambda\prime-\lambda=\frac{h}{m_ec}(1-cos\theta)
A diagram of the situation can be found here: http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/compton.html#c1 (However, in my diagram, the recoil electron and scattered photon are switched with the electron and \phi below the x-axis and the photon and \theta above the x axis.)
To derive the equation, I use the momentum conservation equations and energy conservation equation.
There are two equations for the momentum, one for the x component and one for the y component:
x-component: \frac{h}{\lambda}=\frac{h}{\lambda\prime}cos\theta+\gamma m u cos\phi
y-component: 0=\frac{h}{\lambda\prime}sin\theta-\gamma mu sin\phi
where \gamma=\frac{1}{\sqrt{1-\frac{u^2}{c^2}}}
m=electron mass
u=electron velocity after the collision
My equation for energy conservation is:
h\frac{c}{\lambda}+mc^2=h\frac{c}{\lambda\prime}+\gamma m c^2
To start off, I solved the x component of the momentum for cos(phi) and the y component for sin(phi). I then added the equations together and got (since sin^2 + cos^2 = 1):
\frac{h^2}{\lambda\prime^2\gamma^2m^2u^2}+\frac{h^2}{\lambda^2\gamma^2m^2u^2}-\frac{2h^2cos\theta}{\lambda\prime\lambda\gamma^2m^2u^2}=1
At this point, I am to now eliminate u using the resulting equation and the equation for the energy conservation as given above. However, I am stuck on this next step because it seems as though any algabraic manipulation I try makes the equation extremely complicated. Could anyone please give me some hints as to what step should be next?
Last edited:
