Work Check: Neutron Scattering

[WWCY]

Homework Statement

Hi all, could someone assist me in checking through my work? Many thanks in advance!

An image of the problem is attached below (problem 1b)

Homework Equations

Far field approximation of a scattered wavefunction:
$$\psi_s (\vec{r}) \approx \Psi_i \ r^{\frac{1-d}{2} } \ e^{ikr} \ f(\Omega)$$
where $\Psi_i$ is the amplitude of the incident wavefunction, $d$ is the dimensionality of the problem and $f$ is the scattering amplitude.

Probability current density of the scattered wave
$$J_s = \frac{\hbar}{m} \Im \Big( \psi ^*_s \nabla \psi _s \Big)$$

The $r$ component:

$$J_{s,r} = \frac{\hbar k}{m} |\Psi _i|^2 |f(\Omega)|^2 r^{1-d}$$
$$J_i = \frac{\hbar k}{m} |\Psi _i|^2$$
where $J_i$ is the incident current

The Attempt at a Solution

[/B]
I wanted to write $J_{s,r}$ in terms of $J_{i}$ since (I think) the number of particles scattered is proportional to the probability "scattered".

Total scattering cross section is $\sigma _t = 10^{23} \sigma _{nuclei} = 10$. Since the scattering is isotropic and $d\sigma _t / d\Omega = |f|^2$,
$$|f|^2 = \sigma _t / 4 \pi$$

Thus
$$J_{s,r} = \frac{J_i \sigma _t}{4\pi r^2}$$
for $d = 3$. And if i denote neutron flux density as $n_s$ and $n_i$ for scattered and incident flux,
$$n_{s} = \frac{n_i \sigma _t}{4\pi r^2}$$

is this right?

 

[mark726]

I am happy to assist you in checking your work. Your equations for the far field approximation of a scattered wavefunction and the probability current density of the scattered wave seem correct. However, I am not sure about your attempt to write $J_{s,r}$ in terms of $J_i$ and your calculation for the scattered neutron flux density.

Firstly, the incident current density, $J_i$, is not equal to the incident flux, $n_i$. The incident current density is a vector quantity that represents the flow of particles per unit area per unit time, while the incident flux is a scalar quantity that represents the number of particles per unit area per unit time. Therefore, the two quantities cannot be equated.

Additionally, your equation for $J_{s,r}$ only holds for the case of isotropic scattering in three dimensions. If the scattering is anisotropic or occurs in a different number of dimensions, the equation will be different. Furthermore, your use of the total scattering cross section, $\sigma_t$, is not entirely correct. The total scattering cross section is the sum of all possible scattering cross sections, which may include contributions from different types of scattering (e.g. elastic, inelastic, etc.). It is not necessarily equal to 10 times the scattering cross section of a single nucleus.

In conclusion, your attempt at a solution is not entirely correct. I suggest reviewing your equations and considering the dimensions and definitions of the quantities involved. It may also be helpful to provide more context or information about the problem, as well as any given values or assumptions. I hope this helps. Best of luck with your work!