Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 14: Problem 19

By considering flux densities, explain the appearance of the factor \(k_{\alpha} / k_{\alpha_{\rho}}\) in eqn 14.93 for the differential crosssection for scattering from an initial state \(\alpha_{0}\) to a final state \(\alpha\)

Short Answer

Expert verified
The factor \(k_{\alpha} / k_{\alpha_{\rho}}\) represents the change in flux density from the initial state to the final state in the differential crosssection for scattering. The flux density, associated with the magnitude of the wave vectors, changes according to this factor and affects the outcome of the equation.

Step by step solution

01

Understanding the terms in the equation

Before delving into the explanation, it's important to understand what each symbol in the equation represents. Here, \(k_{\alpha}\) and \(k_{\alpha_{\rho}}\) are wave vectors - these represent the magnitude and direction of a wave. The subscript '\(\alpha\)' denotes the final state, and '\(\alpha_{\rho}\)' denotes the initial state.
02

Understand the concept of flux densities

Flux density is a measure of the amount of something (in this case a wave) passing through a given area within a given time. Flux density, in the context of this equation, could be associated with the wave vectors according to their magnitudes.
03

Showing the factor \(k_{\alpha} / k_{\alpha_{\rho}}\) in terms of flux densities

Now, consider how the factor \(k_{\alpha} / k_{\alpha_{\rho}}\) reflects the change in flux density from the initial state to the final state. If \(k_{\alpha}\) is greater than \(k_{\alpha_{\rho}}\), this means the wave vector (and thus the flux density) has increased from the initial state to the final state. Vice versa, if \(k_{\alpha}\) is less than \(k_{\alpha_{\rho}}\), it means the flux density has decreased. This factor is thus a measure of change in the flux density from the initial state to the final state, and thus finds its appearance in the differential crosssection formula.

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

Consider the scattering of an electron by an atom of atomic number \(Z .\) The interaction potential energy can be approximated by the screened Coulomb potential energy \(V(r)=-\left(Z e^{2} / 4 \pi \varepsilon_{0} r\right) \mathrm{e}^{-r / a},\) where \(a\) is the screening length. (a) Use the Born approximation to calculate the differential cross-section for scattering from the screened Coulomb potential. (b) Proceed to evaluate the integral scattering cross-section. (c) In the limit \(a \rightarrow \infty, V(r)\) becomes exactly the Coulomb potential energy. Evaluate the differential and integral cross-sections obtained in parts (a) and (b) in this limit.

The reactance matrix, \(K\), defined in relation to the scattering matrix through \(K=\mathrm{i}(1-S)(1+S)^{-1},\) also appears in scattering theory. Show for elastic scattering by a central potential with partial wave \(l\) that \(K\) is a \(1 \times 1\) matrix with element \(K_{l}=\tan \delta_{l}\)

Equation 14.3 gives the form of the S matrix for a one-dimensional system in which a particle is scattered from an abrupt blip in the potential energy. Write down the analogous expression for scattering from a comparable dip in the potential energy. Proceed to compute the transmission probability for positive energies given that the particle is incident from the left.

A particle of mass \(m\) is scattered off a central potential \(V(r)\) of the form $$V(r)=\left\\{\begin{array}{lll} \infty & \text { if } & r=0 \\\0 & \text { if } & 0V_{0},\) find an expression for the S-wave scattering phase shift \(\delta_{0} .\) Hint. Require the wavefunction and its first derivative to be continuous at \(r=a\) and at \(r=b\)

The incoming Green's function is given by $$G^{(-)}\left(r, r^{\prime}\right)=\frac{\mathrm{e}^{-i k\left|r-r^{\prime}\right|}}{\left|r-r^{\prime}\right|}$$ Show that \(G^{(-)}\) is a solution of the equation $$\left(\nabla^{2}+k^{2}\right) G\left(r, r^{\prime}\right)=-4 \pi \delta\left(r-r^{\prime}\right)$$ Hint. Use an analysis similar to that given in Further information 14.1. Although the incoming Green's function does not yield the desired asymptotic form of the stationary scattering state \((\mathrm{eqn} 14.14), G^{(-)}\) appears in some of the formal expressions of scattering theory.
See all solutions

Recommended explanations on Chemistry Textbooks

Kinetics

Read Explanation

Nuclear Chemistry

Read Explanation

Inorganic Chemistry

Read Explanation

The Earths Atmosphere

Read Explanation

Chemical Analysis

Read Explanation

Physical Chemistry

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