Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 39: Problem 63

Evaluate the form factor and the Coulomb-scattering 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 the thin-shell and solid-sphere charge distributions? Explain.

Short Answer

Expert verified
#Answer# The form factor F(k)^2 is given by the square of the magnitude of the scattering amplitude: \(F(k)^2 = \left|\frac{Ze^2}{4\pi \epsilon_0} \int \frac{e^{ik \cdot r}}{r^2} d^3r \right|^2\) After evaluating the form factor and differential cross-section for both the thin spherical shell and a solid sphere, we can compare the results to determine if the scattering experiment can distinguish between the two charge distributions. If the resulting form factors and differential cross-sections are significantly different, then the scattering experiment can distinguish between the thin shell and solid sphere charge distributions; otherwise, it cannot.

Step by step solution

01

Find the electric field due to a thin shell

Since the charge is distributed uniformly over the surface of the sphere, we can use Gauss's law to find the electric field outside the sphere. Gauss's law states that: \(\oint \vec{E} \cdot d \vec{A} = \frac{Q_{enc}}{\epsilon_0}\) Here, \(Q_{enc}\) is the enclosed charge, and \(\epsilon_0\) is the vacuum permittivity. For a sphere of radius r (r > a), the integral on the left-hand side becomes: \( \oint \vec{E} \cdot d \vec{A} = E(r) \cdot 4\pi r^2\) As the charge enclosed by the sphere of radius r is Ze, we get: \(E(r) \cdot 4\pi r^2 = \frac{Ze}{\epsilon_0}\) Now, we can solve for the electric field E(r), \(E(r) = \frac{Ze}{ 4\pi \epsilon_0 r^2}\)
02

Calculate the scattering amplitude

We can express the scattering amplitude in terms of the Fourier transform of the electric field: \(f(k) = \frac{Ze^2}{4\pi \epsilon_0} \int \frac{e^{ik \cdot r}}{r^2} d^3r\) Here, k is the momentum transfer vector, given by \(k = k_i - k_f\) where \(k_i\) is the initial momentum and \(k_f\) is the final momentum of the scattered electron. The integral is evaluated over all space.
03

Evaluate the form factor

The form factor is given by the square of the magnitude of the scattering amplitude f(k): \(F(k)^2 = \left| \frac{Ze^2}{4\pi \epsilon_0} \int \frac{e^{ik \cdot r}}{r^2} d^3r \right|^2\) We need to evaluate this integral to calculate the form factor.
04

Calculate the differential cross-section

The differential cross-section of the Coulomb scattering is given by: \(\frac{d\sigma}{d\Omega} = \left| \frac{f(k)}{k^2} \right|^2\) Here, we'll need to substitute the expression for the scattering amplitude f(k) and evaluate it.
05

Comparison with a solid sphere

After calculating the form factor and differential cross-section for both the thin spherical shell and a solid sphere, we will compare the results to determine if this scattering experiment can distinguish between the two charge distributions. If the resulting form factors and differential cross-sections are significantly different, then the scattering experiment can distinguish between the thin shell and solid sphere charge distributions. Otherwise, it cannot.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The text describes and sketches the basic Feynman diagram for the fundamental process involved in the decay of the free neutron: One of the neutron's \(d\) -quarks converts to a \(u\) -quark, emitting a virtual \(W^{-}\) boson, which decays into an electron and an electron anti-neutrino (the only decay energetically possible). Similarly describe and sketch the basic (tree-level) Feynman diagram for the fundamental process involved in each of the following decays: a) \(\mu^{-} \rightarrow e^{-}+\nu_{\mu}+\bar{\nu}_{e}\) b) \(\tau^{-} \rightarrow \pi^{-}+\nu_{\tau}\) c) \(\Delta^{++} \rightarrow p+\pi^{+}\) d) \(K^{+} \rightarrow \mu^{+}+\nu_{\mu}\) e) \(\Lambda \rightarrow p+\pi\)

Draw a quark-level Feynman diagram for the decay of a neutral kaon into two charged pions, \(K^{0} \rightarrow \pi^{+}+\pi^{-}\).

\( \mathrm{~A} 6.50-\mathrm{MeV}\) alpha particle is incident on a lead nucleus. Because of the Coulomb force between them, the alpha particle will approach the nucleus only to a minimum distance, \(r_{\text {min }}\) a) Determine \(r_{\text {min }}\). b) If the kinetic energy of the alpha particle is increased, will the particle's distance of approach increase, decrease, or remain the same? Explain.

The de Broglie wavelength, \(\lambda\), of a 5-MeV alpha particle is \(6.4 \mathrm{fm}\), as shown in this chapter, and the closest distance, \(r_{\text {min }}\), to the gold nucleus this alpha particle can get is \(45.5 \mathrm{fm}\) (calculated in Example 39.1). Based on the fact that \(\lambda \ll r_{\text {min }}\), one can conclude that, for this Rutherford scattering experiment, it is adequate to treat the alpha particle as a a) particle. b) wave.

In a positron annihilation experiment, positrons are directed toward a material such as a metal. What are we likely to observe in such an experiment, and how might it provide information about the momentum of electrons in the metal?
See all solutions

Recommended explanations on Physics Textbooks

Linear Momentum

Read Explanation

Circular Motion and Gravitation

Read Explanation

Fundamentals of Physics

Read Explanation

Electric Charge Field and Potential

Read Explanation

Atoms and Radioactivity

Read Explanation

Scientific Method Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free