How to Calculate the Form Factor of a Nucleus?

[sunrah]

Homework Statement

Calculate form factor of nucleus (A, Z given). Radius $R = 1.2\cdot10^{-15}A^{\frac{1}{3}}$

Homework Equations

$F(\textbf{q}) = \frac{1}{Ze} \int d^{3}\textbf{r} \rho(r)e^{i\textbf{q}\cdot \textbf{r}}$

The Attempt at a Solution

using polar coords $d^{3}\textbf{r} = r^{2}dr \sin{\theta}d\theta d\phi$

and know that we can write

$e^{i\textbf{q}\cdot \textbf{r}} = e^{iqr\cos{\theta}}$

so then use substitution $x = \cos{\theta}$

but my question is about $\rho$. this is spatial density distribution. This is not just ρ = Atomic number / volume, or is it? How do I find this?

Also is this connected to radial Fermi distribution? if so, how?

 

[mark726]

To calculate the form factor of a nucleus, you will need to know its spatial density distribution, which is not simply the atomic number divided by the volume. The spatial density distribution of a nucleus is determined by its composition and structure, and can vary greatly depending on the specific nucleus being studied.

One way to find the spatial density distribution is through experimental methods, such as scattering experiments or electron microscopy. These experiments can provide information about the distribution of nucleons (protons and neutrons) within the nucleus.

Another approach is to use theoretical models, such as the radial Fermi distribution, which describes the distribution of nucleons within a nucleus based on their energy levels. This can be used to calculate the spatial density distribution and then the form factor.

To connect this to the equation for the form factor, you will need to integrate the spatial density distribution over the volume of the nucleus. This will give you the average spatial density, which can then be used in the form factor equation. Keep in mind that the form factor also depends on the momentum transfer, so you will need to consider the specific values of A and Z given in the problem.