Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 14: Problem 4

The differential cross-section for the Yukawa potential using the Born approximation is given in Example 14.3 Plot it as a function of the angle \(\theta\) for (a) zero energy, (b) moderate energy \((k \approx \alpha),\) and (c) high energy \((k \gg \alpha)\) For the plots, choose the range of the \(y\) -axis to be 0 to \(\left\\{\left(2 \mu V_{0}\right) /\left(\hbar^{2} \alpha^{2}\right)\right\\}^{2} .\) For moderate energy, take \(k=\alpha / 2 ;\) for high energy, take \(k=10 \alpha\)

Short Answer

Expert verified
Without values for the specific constants, it's not possible to provide a definitive answer to the problem. However, the general approach has been detailed out in the steps. The differential cross section for Yukawa potential will be calculated using above steps and plotted against the angle for 0, moderate and high energy. The higher the energy, the lower and wider the peak of the cross section showing a larger angle scattering at higher energies.

Step by step solution

01

Compute the scattering amplitude

The scattering amplitude \(f(θ)\) is given by \[f(θ) = -\frac{2\mu}{\hbar^2} \int_0^\infty{ V(r) sin(kr) dr}\]. Now, substitute the Yukawa Potential into the above expression to calculate the amplitude.
02

Apply the Born approximation

The Born approximation approximates the scattering amplitude as \(f(θ) ≈ -\frac{2\mu}{\hbar^2} \int_0^\infty{ V_0 e^{-r/a}sin(kr)dr}\), where \(k\) is the wave number related to energy. This integral will yield the scattering amplitude as a function of \(k\) and \(θ\).
03

Calculate the differential cross-section

The differential cross-section given by the Born approximation is \(d\sigma /dΩ = |f(θ)|^2\). Compute this using the amplitude obtained in the previous step.
04

Plot the differential cross-section

Now we want to plot \(d\sigma /dΩ\) as a function of \(θ\) for different \(k\). Remember, \(k\) depends on energy. Plot it for (a) zero energy (b) moderate energy (where \(k=α/2\)) (c) high energy (where \(k = 10α\)). Set the range of the y-axis to be between 0 and \(\left\{\left(2 \mu V_{0}\right) /\left(\hbar^{2}\alpha^{2}\right)\right\}^{2}.\)
05

Analyze the plots

The plots should show how the differential cross-section varies with angle for the different energy levels (based on the value of \(k\)). The higher the energy, hence higher \(k\), the peak of the cross-section lowers and widens, showing larger angle scattering at higher energies.

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

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 \(r=b\)

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 \(\alpha\)

Consider the differential cross-section for elastic scattering given in eqn \(14.46 .\) At a given energy, sketch its dependence on the scattering angle \(\theta\) when the \(l=1\) partial wave dominates the scattering. Do the same for the \(l=0\) and \(l=2\) partial waves.

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.
See all solutions

Recommended explanations on Chemistry Textbooks

Making Measurements

Read Explanation

The Earths Atmosphere

Read Explanation

Nuclear Chemistry

Read Explanation

Inorganic Chemistry

Read Explanation

Physical Chemistry

Read Explanation

Chemistry Branches

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