Eikonal Approximation: Find total Scattering Cross Section

In summary, the differential cross section is given by..1.) \frac{d\sigma}{d\Omega}= |f(\vec{k}',\vec{k})|^2Where ##f(\vec{k}',\vec{k})## is the scattering amplitude.2.) The optical theorem is a easy way to find the total scattered cross section from a potential. It is given by..\text{Im}\left(f(\theta=0)\right)=\frac{\sigma_{tot}}{4\pi}3.) The expression for the Eikonal approximation is..f(\vec{k
  • #1
Xyius
508
4

Homework Statement


Using the Eikonal approximation
(1) Determine the expression for the total scattering cross section of a particle in a potential V(r)
(2) Using this result, compute the total scattered cross section for the following potential.

[tex]
V(r)=
\begin{cases}
V_0, \text{for } r < a \\
0 , \text{for } r >a
\end{cases}
[/tex]

Where ##V_0 > 0##

Homework Equations



The differential cross section is given by..

1.) [tex]\frac{d\sigma}{d\Omega}= |f(\vec{k}',\vec{k})|^2[/tex]

Where ##f(\vec{k}',\vec{k})## is the scattering amplitude.

2.) The optical theorem is a easy way to find the total scattered cross section from a potential. It is given by..

[tex]\text{Im}\left(f(\theta=0)\right)=\frac{\sigma_{tot}}{4\pi}[/tex]

3.) The expression for the Eikonal approximation is..

[tex]f(\vec{k}',\vec{k})=-i k \int_0^{\infty}db b J_0(kb\theta)[e^{2 i \Delta(b)}-1][/tex]

Where..

[tex]\Delta(b)=\frac{-m}{2k\hbar^2}\int_{-\infty}^{\infty}V(\sqrt{b^2+z^2})dz[/tex]

Where ##V(\sqrt{b^2+z^2})## means ##V## OF ##\sqrt{b^2+z^2}##, not times.

##b## is the impact parameter, and in the book they say that ##l## can be treated as a continuous variable since we are at high energies, and they say that ##l = bk ##. Not sure if this helps.

The Attempt at a Solution



I used the optical theorem (equation 2) to get..

[tex]\sigma_{tot}=4 \pi \text{Im} \left(f(\vec{k}',\vec{k})\right)=-4 \pi k \int_0^{\infty}db b J_0(kb\theta)[\text{Re}(e^{2i\Delta(b)})-1]_{\theta=0}[/tex]

So it seems that all I would need to do is calculate the integral, but I am having trouble finding ##\Delta(b)## because the limits are infinity. When I plug in the potential I get..

[tex]\Delta(b)=\frac{-m}{2k\hbar^2}\int_{-\infty}^{\infty}V_0dz[/tex]

Which diverges? Clearly the limits simplify to something that's not infinity on both ends. But the variable of integration is z, which has no limit in either direction. I found some examples, but the ones I found use the gaussian potential and do not change the limits of integration.
 
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  • #2
The range of integration should be: b²+z²<a² .
 
  • #3
So I guess the integration is only over the scatterer?

So if I have ##b^2+z^2=a^2 \rightarrow z=\sqrt{a^2-b^2}##. So the limits of integration should be from ##z=-\sqrt{a^2-b^2}## to ##z=\sqrt{a^2-b^2}##?
 
  • #4
The limits are over the whole space.
But your potential V(r) is not.
 
  • #5


I would suggest first checking your integral for any potential errors, as it does seem to diverge. You can also try using numerical methods to evaluate the integral and see if it converges to a reasonable value. Additionally, you can try solving the integral for the specific potential given in the problem, as it may have a closed-form solution.

If you are unable to find a solution using the Eikonal approximation, you can also try using other methods such as the Born approximation or the partial wave analysis. These methods may provide more accurate results for the total scattering cross section.

In general, it is important to carefully consider the limits of integration when using approximations in physics, as they can greatly affect the final result. It may also be helpful to consult with other experts in the field or consult other resources for guidance on solving these types of problems.
 

1. What is the Eikonal Approximation method used for?

The Eikonal Approximation is a method used in theoretical physics to calculate the total scattering cross section of particles. It is based on the scattering amplitude and takes into account the interactions between the particles being scattered.

2. How does the Eikonal Approximation calculate the total scattering cross section?

The Eikonal Approximation calculates the total scattering cross section by integrating the phase shift function over all possible scattering angles. The phase shift function represents the change in phase of the scattered particle compared to the incident particle due to the interactions between them.

3. What are the limitations of the Eikonal Approximation method?

The Eikonal Approximation method is most accurate for small scattering angles and weak interactions between particles. It also assumes that the scattering process is elastic, meaning that the kinetic energy of the particles remains unchanged after scattering. This may not be the case for some particles and interactions.

4. How does the Eikonal Approximation compare to other methods of calculating the total scattering cross section?

The Eikonal Approximation is a simpler and more computationally efficient method compared to other methods, such as the Green's function approach. However, it is less accurate and may not be suitable for certain systems or interactions.

5. Can the Eikonal Approximation be applied to all types of particles and interactions?

The Eikonal Approximation can be applied to a wide range of particles and interactions, as long as the assumptions of the method are met. However, it may not be suitable for describing highly complex or non-elastic scattering processes.

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