How Do I Solve These Scattering Cross Section Problems?

In summary, the author is having problems with solving for the scattering amplitude and is looking for help. He has found that using the equation for spherical harmonic Y(θ), solving for sin(θ), and then taking the inverse cosine gives him the amplitude. However, he is still having trouble with getting rid of the beta term.
  • #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.
 
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  • #2
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:
  • #3
So, I talked with my professor, and apparently, there was a typo. It should be that $$\frac{d\sigma}{d\Omega}=\alpha+\beta cos(\theta)+\gamma cos^2(\theta)$$.
 
  • #4
Charles Link said:
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.
 
  • #5
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).
 
  • #6
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??
 

1. What is a scattering cross section problem?

A scattering cross section problem involves calculating the probability of a particle scattering off a target, based on its properties and the properties of the target. It is commonly used in fields such as nuclear physics and astrophysics.

2. How do I approach solving a scattering cross section problem?

The first step is to gather all the relevant information, such as the properties of the particle and the target, as well as any known equations or formulas. Then, use these to calculate the cross section using the appropriate method, such as the Born approximation or the partial wave method.

3. What are some common challenges when solving scattering cross section problems?

One of the main challenges is dealing with the complexity of the equations involved. Additionally, it can be difficult to accurately determine the properties of the particle and target, which can greatly affect the results.

4. Are there any tips for solving scattering cross section problems more efficiently?

It can be helpful to break down the problem into smaller, more manageable parts. Also, double-checking all calculations and equations can help avoid mistakes. Additionally, utilizing computer programs or simulations can make the process faster and more accurate.

5. How are scattering cross section problems used in scientific research?

Scattering cross section problems are used to study the behavior of particles and their interactions with different targets. This information can then be applied to various fields, such as nuclear energy, medical imaging, and astrophysics. They are also used to test and validate theoretical models and predictions.

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