 #1
Diracobama2181
 74
 2
 Homework Statement:

Suppose I am given the scattering cross section $$\sigma(\theta)=\alpha+\beta cos(\theta)+\gamma cos^2(theta)$$
a) Find the scattering amplitude.
b) Express α, β and γ in terms of the phase shifts δl
(c) Are there any constraints on the magnitudes of α, β and γ if the
scattering amplitude is not allowed to grow any faster than ln E as the
energy E becomes very large?
(d) Deduce the total scattering crosssection and show that it is consistent with the optical theorem.
 Relevant Equations:

$$\frac{d\sigma}{d \Omega}=f(\theta)^2$$
$$\sigma=\frac{4 \pi}{k}\sum_{l=0}^{\infty}(2l+1)sin^2(\delta_{l})$$
a) I have $$d\sigma=\beta sin(\theta)d(\theta)+2\gamma sin(\theta)cos(\theta) d\theta$$
and $$d \Omega=2\pi sin(\theta) d \theta$$
so $$\frac{d\sigma}{d \Omega}=\frac{\beta}{2\pi}+2\gamma cos(\theta)=f(\theta)^2$$
b) $$\sigma(\theta)=\alpha+\beta cos(\theta)+\gamma cos^2(theta)=\sigma=\frac{4 \pi}{k}\sum_{l=0}^{\infty}(2l+1)sin^2(\delta_{l})$$
Stuck here. Not sure if this is sufficient.
c) Also having issues with this one and deciding how to tackle it.
d) Waiting on doing this one until I can finish the previous two parts.
Has my setup so far been fine and are there any tips or suggestions on how I should tackle these problems?
Thanks in advance.
and $$d \Omega=2\pi sin(\theta) d \theta$$
so $$\frac{d\sigma}{d \Omega}=\frac{\beta}{2\pi}+2\gamma cos(\theta)=f(\theta)^2$$
b) $$\sigma(\theta)=\alpha+\beta cos(\theta)+\gamma cos^2(theta)=\sigma=\frac{4 \pi}{k}\sum_{l=0}^{\infty}(2l+1)sin^2(\delta_{l})$$
Stuck here. Not sure if this is sufficient.
c) Also having issues with this one and deciding how to tackle it.
d) Waiting on doing this one until I can finish the previous two parts.
Has my setup so far been fine and are there any tips or suggestions on how I should tackle these problems?
Thanks in advance.