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Chapter 14: Problem 6

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.

Short Answer

Expert verified
In this problem, an electron is scattered by an atom. Using Born approximation and the screened Coulomb potential, we managed to calculate the differential scattering cross-section, the total scattering cross-section, and their limits as the screening length goes to infinity.

Step by step solution

01

Use Born Approximation

For the Born approximation, the differential cross-section for scattering is given by: \[\frac{d \sigma}{d \Omega}=\frac{1}{(2 \pi)^{2}} \frac{|f(k, k')|^{2}}{v^{2}}\] Here, the scattering amplitude \(f(k, k')\) is given by the spatial Fourier transform of the potential \(V(r)\). We first calculate the Fourier transform of \(V(r)\), denoted \(V(q)\), where \(q=|\boldsymbol{q}|=|\boldsymbol{k}-\boldsymbol{k}'|\).
02

Derive the differential cross-section

We insert the expression of \(V(r)\) into the spatial Fourier transform, perform the integration, and obtain \(V(q)\). Then we substitute \(V(q)\) into the formula of the differential cross-section in the Born approximation. The resulting expression is the desired differential cross-section in terms of the screening length \(a\), momentum transfer \(q\), atomic number \(Z\) and electron velocity \(v\).
03

Evaluate the Total Scattering Cross-section

To calculate the total scattering cross-section, we need to integrate over the whole solid angle. As the integrand depends only on the magnitude, not the direction, of the momentum transfer, we can use polar coordinates in momentum space. After integration, we can get the total cross-section.
04

Evaluate the Limit of Infinite Screening Length

If \(a\rightarrow\infty\), the screened Coulomb potential \(V(r)\) becomes the ordinary Coulomb potential, \(V(r)=-Z e^{2} / (4 \pi \varepsilon_{0} r)\). Consequently, our previous analysis simplifies dramatically: the Fourier transform of \(V(r)\) becomes simpler, and the differential and total cross-sections greatly simplify as well. After this simplification, we get the results for pure Coulomb scattering.

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Most popular questions from this chapter

Find an expression for the WKB phase shift for S-wave scattering at an energy \(E\) by the potential \(V=A / r^{2}.\)

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.

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 \(\alpha\)

Show for the elastic scattering of a particle by a central potential \(V(r)\) that approaches zero more rapidly than \(1 / r\) as \(r \rightarrow \infty\) that the integral cross-section can be written as $$\sigma_{\mathrm{tot}}=\frac{4 \pi}{k} \mathrm{im} f_{k}(0)$$ where im \(f_{k}(0)\) is the imaginary part of the forward scattering amplitude \((\theta=0) .\) This is the so-called optical theorem. Hint. The Legendre polynomials are required to satisfy \(P_{l}(1)=1\) for all values of \(l\)

Use the Born approximation to calculate the differential cross-section for scattering from the spherical square-well potential (Section 14.5 ). Hint. Use integration by parts to determine the scattering amplitude.
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