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I'm currently in the process of computing a differential cross-section for the scattering of a 420 MeV electron by an O-16 nucleus (with a Wood-Saxon charge distribution). The problem is that the resulting graphs don't look anything like they should. I first calculated the form factor for the wood-saxon charge distribution (which is spherically symmetric) using the quad function of scipy.integrate then calculated the Rutherford cross-section, the recoil factor which are quite straightforward. The differential cross-section is then given by

$$\frac{d\sigma}{d\Omega} = \left(\frac{d\sigma}{d\Omega}\right)_\text{Ruth}\left(\frac{1}{1 + \frac{q^2}{2ME}}\right)|F(q)|^2$$

With $M$ the mass of the nucleus, $E$ the energy of the incoming electron, $q$ the transfered momentum and

$$F(q) = \frac{4\pi}{q}\int_0^\infty dr \frac{r\sin(qr/\hbar)}{1+e^{(r-R)/a}}$$

I used natural units for the different quantities. I suspect the problem to come from the form factor but how exactly I don't know. I tried to improve the integration accuracy as much as I could but it didn't help. Also, somehow, having a smaller step for the angle array seems to make it worse. Here is the Python code I made :

import matplotlib.pyplot as plt
import scipy.integrate as integrate
from scipy.special import expit
from scipy.signal import savgol_filter

def wood_saxon_model(R, a, r):
    density = expit((R - r) / a)
    return density

def form_factor(q):
    R = 1.07 * 16**(1/3)
    a = 0.5
    #hbar = 6.582119e-22
    hbar = 1
    integral_result, _ = integrate.quad(lambda r: r * np.sin(q * r / hbar) * wood_saxon_model(R, a, r), 0, np.inf, limit = 1000)
    return 4 * np.pi * hbar * integral_result / q


def rutherford_cs(Z1, Z2, q):
    alpha = 7.29735256e-3
    #hbarc = 1.97327e2
    hbarc = 1
    cs = (Z1 * Z2 * alpha * hbarc)**2 / (q**4 + 1e-10)
    return cs

def recoil(E, M, q):
    return 1 / (1 + q**2 / (2 * M * E))

def cross_section(Z1, Z2, E, M, q):
    return rutherford_cs(Z1, Z2, q) * recoil(E, M, q) * form_factor(q)**2
 
theta = np.linspace(np.pi / 18, 6 * np.pi / 18, 1000)
Z1 = 1
Z2 = 8
E = 420
M = 1.489916863e4
q = 2 * E * np.sin(theta / 2)


wood_saxon_cross_section = [cross_section(Z1, Z2, E, M, x) for x in q]
squared_form_factor = [form_factor(x)**2 for x in q]

#smoothed_cross_section = savgol_filter(wood_saxon_cross_section, window_length=11, polyorder=3)

plt.plot(np.degrees(theta), squared_form_factor)
plt.show()
plt.plot(np.degrees(theta), wood_saxon_cross_section)
plt.yscale('log')
plt.xlabel('Scattering Angle (degrees)')
plt.ylabel('Differential Cross Section')
plt.title('Differential Cross Section')
plt.show()

I'm clueless as to what might be the trouble here. Here is a comparison of what I get vs what kind of thing I should get: What I have What I should have

What I have vs what I should have (image source)

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    $\begingroup$ It seems like you are off by a factor of $10^{30}?$ have you double checked all of your natural units and scaling? That sort of error magnitude makes it seem like some constants are in the wrong place $\endgroup$
    – whpowell96
    Feb 21 at 20:36

1 Answer 1

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Seems that your electron energy of $E=420~\text{MeV}$ messes angle range upon which you calculate cross-section. If you make it smaller by 2 orders of magnitude, like E = 420/10**2, then you will get expected chart (in addition integration will be very fast) :

enter image description here

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