- #1

Diracobama2181

- 75

- 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 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?

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.