Born Approximation Scattering Formula
The differential scattering cross-section, $ \frac{d\sigma}{d\Omega} $, in the first Born approximation is given by:
$ \frac{d\sigma}{d\Omega} = \left(\frac{m}{2\pi\hbar^2}\right)^2 |V(\vec{q})|^2 $
where $ m $ is the particle mass, $ \hbar $ is the reduced Planck constant, and $ V(\vec{q}) $ is the Fourier transform of the scattering potential $ V(\vec{r}) $. The vector $ \vec{q} = \vec{k}_f - \vec{k}_i $ is the momentum transfer vector, with $ \vec{k}_i $ and $ \vec{k}_f $ being the initial and final wave vectors, respectively.
Fourier Transform of Delta Potential
The potential is given as $ V(\vec{r}) = g\delta^3(\vec{r}) $. The Fourier transform $ V(\vec{q}) $ is calculated as:
$ V(\vec{q}) = \int V(\vec{r}) e^{-i\vec{q}\cdot\vec{r}} d^3r $ $ V(\vec{q}) = \int g\delta^3(\vec{r}) e^{-i\vec{q}\cdot\vec{r}} d^3r $
Using the property of the Dirac delta function ($ \int f(\vec{r})\delta^3(\vec{r}) d^3r = f(\vec{0}) $), we get:
$ V(\vec{q}) = g e^{-i\vec{q}\cdot\vec{0}} = g e^0 = g $
The Fourier transform of the delta potential is simply the constant $ g $.
Differential Scattering Cross-Section Calculation
Substitute the calculated $ V(\vec{q}) = g $ into the Born approximation formula:
$ \frac{d\sigma}{d\Omega} = \left(\frac{m}{2\pi\hbar^2}\right)^2 |g|^2 $
Since $ g $ is a positive constant, $ |g|^2 = g^2 $. Thus, the differential scattering cross-section is:
$ \frac{d\sigma}{d\Omega} = \left(\frac{mg}{2\pi\hbar^2}\right)^2 $
Angular Dependence Analysis
The resulting expression for $ \frac{d\sigma}{d\Omega} $ is a constant value: $ \left(\frac{mg}{2\pi\hbar^2}\right)^2 $. This value depends only on constants ($ m $, $ g $, $ \hbar $) and does not contain the scattering angles $ \theta $ or $ \phi $.
Therefore, the differential scattering cross-section is independent of both $ \theta $ and $ \phi $.
Important Questions from Scattering Cross Section Phase Shift Method
Consider the potential $U(r)$ defined as $$U(r) = -U_0 \frac{e^{- \alpha r}}{r}$$ where $ \alpha$ and $U_0$ are real constants of appropriate dimensions. According to the first Born approximation, the elastic scattering amplitude calculated with $U(r)$ for a (wave-vector) momentum transfer $q$ and $ \alpha \to 0$, is proportional to
(Useful integral: $ \int_0^{ \infty} \sin(qr)e^{- \alpha r} dr = \frac{q}{ \alpha^2+q^2}$)
- The scattering of particles by a potential can be analyzed by Born approximation. In particular, if the scattered wave is replaced by an appropriate plane wave, the corresponding Born approximation is known as the first Born approximation. Such an approximation is valid for
- Consider an elastic scattering of particles in $l = 0$ states. If the corresponding phase shift $\delta_0$ is $90^\circ$ and the magnitude of the incident wave vector is equal to $\sqrt{2}\pi$ fm$^{-1}$ then the total scattering cross section in units of fm$^2$ is ________.
- Protons and $\alpha$-particles of equal initial momenta are scattered off a gold foil in a Rutherford scattering experiment. The scattering cross sections for proton on gold and $\alpha$-particle on gold are $\sigma_p$ and $\sigma_\alpha$ respectively. The ratio $\sigma_\alpha/\sigma_p$ is _________
- Consider the scattering of neutrons by protons at very low energy due to a nuclear potential of range $r_0$. Given that,
$cot(kr_0 + \delta) \approx -\frac{\gamma}{k}$
where $\delta$ is the phase shift, $k$ the wave number and $(-\gamma)$ the logarithmic derivative of the deuteron ground state wave function, the phase shift is
