# Scattering Cross Section

## 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 cross-section 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?

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I googled the topic, (my expertise here is limited), and I think you are needing the equation $f(\theta)=\frac{1}{2 i k} \sum (2l+1) (e^{2 i \delta_l}-1)P_l(\cos{\theta})$. You can then set like powers of $\cos{\theta}$ equal.

Last edited:
So, I talked with my professor, and apparently, there was a typo. It should be that $$\frac{d\sigma}{d\Omega}=\alpha+\beta cos(\theta)+\gamma cos^2(\theta)$$.

I googled the topic, (my expertise here is limited), and I think you are needing the equation $f(\theta)=\frac{1}{2 i k} \sum (2l+1) (e^{2 i \delta_l}-1)P_l(\cos{\theta})$. You can then set like powers of $\cos{\theta}$ equal.
I considered that, but I don't think that method would quite work since l is a summation to infinity.

Using this new info, I get for a) that $f(\theta)=\frac{\sqrt{4\pi}}{k} \sum_{l=0}^{\infty} \sqrt{2l+1}Y_{l0} (e^{i\delta _l})sin^2{\delta_l}$, where $$Y_{l0}$$ is a spherical harmonic.
For B, I can use $$\sigma=\int |f(\theta)|^2d\Omega=2\pi \int_{0}^{\pi}(\alpha+\beta cos(\theta)+\gamma cos^2(\theta))sin(\theta)d\Omega=4\pi\alpha+\frac{4\pi}{3}\gamma=\frac{4 \pi}{k}\sum_{l=0}^{\infty}(2l+1)sin^2(\delta_{l})$$.
However, this gets rid of $$\beta$$. Also, still not sure where to go for C).

I don't understand how to do this. I believe you need to be given the scattering amplitude f(θ) explicitly to enable the rest of the problem. Perhaps that is the intent??