[Physics] Form Factor in Rutherford Scattering

Question

My question relates to Rutherford Scattering of particles. When we calculate the "differential cross-section" expression for a nucleus with finite size, it is said that the expression is almost the same as that if the nucleus was a point only a "Form Factor" multiplies,which is nothing but the fourier transform of the charge density, I want to know the derivation of this expression with the form factor. I cant find it anywhere.

Answer 1

My discussion essentially follows this article.

Form Factors are an intuitive and simple tool used to describe the scattering particles from extended targets. Here I'm going to show how the Form Factor comes about in the context of the scattering of spinless electrons

As with many scattering experiments, the quantity we are interested in is the differential cross section $\frac{d\sigma}{d\Omega}$ of our scattered electrons off our target. The differential cross section is related to the scattering amplitudes through the relation: $$\frac{d\sigma}{d\Omega}=\frac{k}{k_i}|f(\theta,\phi)|^2$$ where $\theta$ is the scattered angle.

The scattering amplitudes $f(\theta,\phi)$ can be obtained in approximate form using the Born Approximation. To first order (and up to a normalization) the Born Approximation can be written as: $$f_{B1}=\int\phi_{k_f}^*(\vec{r})V(\vec{r})\phi_{k_i}(\vec{r})d^3\vec{r}$$

In the first Born Approximation the initial incoming wave and the outgoing waves are assumed to be plane waves of the form: \begin{align} \phi_{k_i}(\vec{r})=e^{ik_i\cdot r}\\ \phi_{k_f}(\vec{r})=e^{ik_f\cdot r} \end{align}

We can describe an extended charge distribution by $Ze\rho(\vec{r})$ with $$\int\rho(\vec{r})d^3\vec{r}=1$$

In this case, the potential experienced by an electron located at $\vec{r}$ is given by the Coloumb potential: $$V(\vec{r})=-\frac{Ze^2}{4\pi\epsilon_0}\int\frac{\rho(\vec{r}')}{|r-r'|}d^3\vec{r}'$$

And substituting this potential into the general expression for the first Born Approximation to the scattering amplitudes $f(\theta,\phi)$ gives $$f_{B1}=-\frac{Ze^2}{4\pi\epsilon_0}\int e^{\frac{iq\cdot r}{h}}\int\frac{\rho(\vec{r}')}{|r-r'|}d^3\vec{r}d^3\vec{r}'$$

Making the substitution $\vec{R}=\vec{r}-\vec{r}'$ and noticing that $d^3\vec{R}=d^3\vec{r}$ $$f_{B1}=-\frac{Ze^2}{4\pi\epsilon_0}\int \frac{e^{\frac{iq\cdot \vec{R}}{h}}}{\vec{R}}d^3\vec{R} \left[\int e^{\frac{iq\cdot \vec{r}'}{h}}\rho(\vec{r}')d^3\vec{r}'\right]$$

This bracked factor is known as the Form Factor, $F(q)$.

$$F(q)=\int e^{\frac{iq\cdot \vec{r}'}{h}}\rho(\vec{r}')d^3\vec{r}' $$

It can be shown that when the expression for $f_{B1}$ is used to determine $\frac{d\sigma}{d\Omega}$, that:

$$\frac{d\sigma}{d\Omega}=\left(\frac{Ze}{4E}\right)^2\frac{1}{sin^4(\theta/2)}|F(q)|^2$$

This expression can be interpreted intuitively as Rutherford scattering modulated by the square of the Form Factor. In other words, electron scattering off an extended source is equal to scattering off a point source modulated by the form factor.