Scattering of waves in a symmetrical potential (using python)

Question

I'm looking at scattering of waves in a symmetrical potential as part of a research project.

If a plane wave $e^{(ikr)}$ is incident on a spherically symmetric potential $V(r)$ the scattered wave is given by $S(r,\theta) = \frac{f(\theta)e^{(ikr)}}{ r}$

My understanding is that $f(\theta)=\frac{1}{k}\sum_{\ell=0}^{\infty}(2\ell +1)e^{in_{\ell}}\sin(n_{\ell})P_{\ell}\cos(\theta)$

in which $P_{\ell}(\theta)$ is the usual Legendre polynomial of order $\ell$.

The phase-shift $n_\ell$ can be obtained by the limit which I have defined to be:

$n_\ell= \lim_{r\rightarrow \infty}n_{\ell}(r)$,

where:

$\frac{d}{dr} [\tan n_{\ell}(r)] = −kr^2V(r)[j_\ell(Kr) − y_\ell(Kr)\tan(n_L(r))]^2$

in which $j_\ell$ and $y_\ell$ are the spherical Bessel functions of order $\ell$. At $r = R$ we have

$\tan(n_{\ell}(R))=\frac{j_{\ell}(kR)}{y_{\ell}(kR)}$

Use this formalism to investigate the scattering of a $He^3$ atom by an ion via a potential of the form

$V(r)=\frac{A^2}{r^4}$.

I have completed the mathematical requirements for this project however I am really stuck on the coding element of it.

I want to use python 3.7 to solve this in the following way:

Take $A = 35.3$ and $R = 7.0$ and compute $n_\ell(\infty)$ numerically for $k = 0.05$ and $k = 0.5$.

Use the resulting phase shifts to compute the differential cross-section $|f(\theta)|^2$ as a function of theta for the two values of $K$ given above.

I can plot the results and compare myself.

Any help would be great.

Answer 1

I would throw an answer for you but as Wolfgang said in his comment, it's not clear what you are looking for here.

First of all, I'm not familiar with the physics of your system and I would only describe a procedure to solve this problem from a mathematical point of view.

Basically, you are looking for $n_{\ell} = \lim_{r \rightarrow \infty} n_{\ell}(r)$.

You have this ODE for $n_{\ell}(r)$:

$$\frac{d}{dr}(\tan(n_{\ell}(r))) = -kr^{2}V(r)(j_{\ell}(Kr) - y_{\ell}(Kr)\tan(n_{\ell}(r)))^{2}$$

Take $\tan(n_{\ell}(r)) = \mathcal{U}(r)$, so:

$$\frac{d \mathcal{U}(r)}{dr} = -kr^{2}V(r) (j_{\ell}(Kr) - y_{\ell}(Kr) \mathcal{U}(r))^{2}$$

You could solve this ODE numerically by scipy.integrate.odeint, if you know the value of $\mathcal{U}(0)$.

When you found $\mathcal{U}(r)$, you have: $n_{\ell}(r) = \arctan(\mathcal{U}(r))$.

In order to find $n_{\ell}$ as the limit of $r \rightarrow \infty$, I suggest you to continue integration until a big number (you need to decide what this big number is based on a length scale in your system). Then finding $f(\theta)$ is just a simple for loop to calculate it based on your formula and you could easily plot $f(\theta)$ vs. $\theta$.