Question

Evaluate the form factor and the differential cross section, \(d \sigma / d \Omega\) for a beam of electrons scattering off a thin spherical shell of total charge \(Z e\) and radius \(a\). Could this scattering experiment distinguish between thinshell and solid-sphere charge distributions? Explain.

Short answer

Question: Evaluate the form factor and differential cross-section for a beam of electrons scattering off a thin spherical shell of total charge \(Ze\) and radius \(a\), and determine if this scattering experiment can distinguish between thin shell and solid-sphere charge distributions. Answer: The form factor for a thin spherical shell is given by: \(F(q) = \frac{3Ze}{(qa)^{3}}( \sin(qa) - qa \cos(qa))\), and the differential cross-section is given by \(\frac{d\sigma}{d\Omega} = \frac{d\sigma _{Mott}}{d\Omega} |F(q)|^2\), with \(\frac{d\sigma _{Mott}}{d\Omega} = (\frac{Ze^2}{4 \pi \epsilon_0})^2 \frac{1}{(2E \sin^2(\frac{\theta}{2}))^2}\). The scattering experiment can distinguish between thin shell and solid sphere charge distributions if their form factors or differential cross-sections show significant differences. This can be achieved by evaluating and comparing these quantities for both thin shell and solid sphere charge distributions.

Step-by-step solution

Understanding the Form Factor and the Differential Cross-Section

The form factor, denoted by \(F(q)\), is a measure of how the target's spatial charge distribution affects scattered wave amplitudes. In the case of electron scattering, \(q\) is the magnitude of the momentum transfer, given by \(q = 2 k \sin(\frac{θ}{2})\), where \(k\) is the magnitude of the incident beam's momentum, and θ is the scattering angle. The differential cross-section, denoted by \(d\sigma/d\Omega\), represents the probability of scattering in a specific solid angle \(d\Omega\) around the direction of interest.

Evaluating the Form Factor

For a thin spherical shell, the charge distribution is given by: ρ(r) = \(\frac{Ze}{4 \pi a^{2}} \delta(r-a)\) Since the form factor is the Fourier transform of the charge distribution, we can evaluate it as follows: \(F(q) = \int_{V}\rho(r) e^{-iqr} d^{3}r\) For spherical coordinates, with \(r\) ranging from 0 to \(a\), and \(θ\) ranging from 0 to \(2 \pi\), it can be expressed as: \(F(q) = \frac{Ze}{4\pi a^{2}} \int_{0}^{2 \pi}d\phi \int_{0}^{\pi}\sin(\theta)d\theta \int_{0}^{\infty}\delta(r-a)e^{-iqr}r^{2}dr\) Upon calculating the integral, the form factor becomes: \(F(q) = \frac{3Ze}{(qa)^{3}}( \sin(qa) - qa \cos(qa))\)

Evaluating the Differential Cross-Section

The Mott differential cross-section is given by the following formula: \(\frac{d\sigma}{d\Omega} = \frac{d\sigma _{Mott}}{d\Omega} |F(q)|^2\) Where \(\frac{d\sigma _{Mott}}{d\Omega}\) is the Mott differential cross-section for scattering off a point charge, given by: \(\frac{d\sigma _{Mott}}{d\Omega} = (\frac{Ze^2}{4 \pi \epsilon_0})^2 \frac{1}{(2E \sin^2(\frac{\theta}{2}))^2}\) Substituting the obtained form factor and the Mott cross-section into the equation for differential cross-section, we can find the expression for \(\frac{d\sigma}{d\Omega}\).

Distinguishing Thin Shell and Solid Sphere Charge Distributions

The scattering experiment can distinguish between thin shell and solid sphere charge distributions if their form factors or differential cross-sections show significant differences. Evaluating these quantities for both thin shell and solid sphere charge distributions, calculating the difference in the obtained results, and analyzing the discrepancies will allow us to determine if the scattering experiment can distinguish between the two distributions. In conclusion, following the above-mentioned steps, we can evaluate the form factor and differential cross-section, and analyze the differences in the results for thin shell and solid sphere charge distributions to determine whether the scattering experiment can distinguish between the two charge distributions.