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Semiclassical Mechanics with Molecular Applications

M. S. Child

Chapter 9

Atom–atom scattering - all with Video Answers

Educators

Chapter Questions

Problem 1

(i) Use eqn $(9.7)$ and (9.25) to derive expressions for the classical deflection function, $\chi(\lambda)$, and the semiclassical phase shift, $\eta(\lambda)$, for scattering by a hard sphere of radius $d$ at angular momentum $l+1 / 2=\lambda<k d$.
(ii) Show that the classical differential and total cross-sections are given by $(\mathrm{d} \sigma / \mathrm{d} \Omega)_{\mathrm{cl}}=d^2 / 4$ and $\sigma_{\mathrm{cl}}=\pi d^2$ respectively. Secondly, use the random phase approximation $\sin ^2 \eta(\lambda) \simeq 0.5$ to show that $\sigma_{\text {semi }}=2 \pi d^2$ for $k d \gg 1$.
(iii) Compute the partial-wave sum for $\mathrm{d} \sigma / \mathrm{d} \Omega$ over the range $0<\theta<\pi$ at $k d=10.0$, using the above form for $\eta(\lambda)$. Truncate the sum at $\lambda=k d$ and estimate the ratio $\sigma / \pi d^2$.

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Problem 2

Combine the random phase approximation to $f(\theta)$ with the approximation $P_l(\cos \theta) \simeq \sqrt{\theta / \sin \theta} J_0(\lambda \theta)$ from problem 2.7 to show that the small-angle hard sphere scattering amplitude may be estimated as
$$
\begin{aligned}
f(\theta) & \simeq \frac{1}{\mathrm{i} k} \sqrt{\frac{\theta}{\sin \theta}} \int_0^{k d} \lambda J_0(\lambda \theta) \mathrm{d} \lambda=\frac{d}{\mathrm{i} \sqrt{\theta \sin \theta}} J_1(k d \theta) \\
& \simeq \frac{k d^2}{2 \mathrm{i}} \sqrt{\frac{\theta}{\sin \theta}} \exp \left[-\frac{(k d \theta)^2}{8}\right]
\end{aligned}
$$
over the angular region $0<\theta<\pi / k d$, over which $P_l(\cos \theta)>0$ for $\lambda \leqslant k d$. Estimate the small angle 'shadow' contribution to the integrated cross-section, and compare it with the discrepancy between the random phase value $2 \pi d^2$ and the value $\pi d^2$ implied by equivalence between the classical and semiclassical differential cross-sections over the full angular region.
[Hints: $\int_0^a x J_0(x) \mathrm{d} x=a J_1(a)=\left(a^2 / 2\right)\left(1-\left(a^2 / 8\right) \ldots\right) \simeq\left(a^2 / 2\right) \mathrm{e}^{-a^2 / 8}$ for $a<3$.

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04:04

Problem 3

Show by means of the substitution in eqn (9.19) from problem 2.7 , that
$$
P_l(\cos \theta) \simeq(\theta / \sin \theta)^{1 / 2} J_0(\lambda \theta)=\left(\frac{\theta}{\sin \theta}\right)^{1 / 2} \frac{1}{\pi} \int_0^\pi \exp (-\mathrm{i} \lambda \theta \cos \varphi) \mathrm{d} \varphi .
$$
Deduce, by stationary phase integration with respect to $\lambda$, that $f(\theta) \simeq f_a(\theta)+$ $f_g(\theta)$, where $f_a(\theta)$ is the $a$ branch contribution to $(9.51)$ and $f_g(\theta)$ is the glory scattering amplitude
$$
\begin{aligned}
f_g(\theta)= & \frac{\exp (-3 \mathrm{i} \pi / 4)}{k}\left(\frac{2 \pi \theta}{\sin \theta}\right)^{1 / 2} \frac{1}{\pi} \\
& \times \int_0^\pi \frac{\lambda(\theta, \varphi)}{\sqrt{\left|\chi^{\prime}[\lambda(\theta, \varphi)]\right|}} \exp \{2 \mathrm{i} \eta[\lambda(\theta, \varphi)]-\mathrm{i} \lambda(\theta, \varphi) \theta \cos \varphi\} \mathrm{d} \varphi,
\end{aligned}
$$
where $\chi^{\prime}(\lambda)=\mathrm{d} \chi / \mathrm{d} \lambda$ and $\lambda(\theta, \varphi)$ is defined such that
$$
\chi[\lambda(\theta, \varphi)]=2(\partial \eta / \partial \lambda)=\theta \cos \varphi
$$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
04:22

Problem 4

(i) The following mapping equation in the above integral for $f_g(\theta)$,
$$
F(\theta, \psi)=2 \eta[\lambda(\theta, \varphi)]-\lambda(\theta, \varphi) \theta \cos \varphi=A(\theta)-\zeta(\theta) \cos \psi
$$
defines a function $\varphi(\psi)$. Use the approximation
$$
G(\theta, \psi)=\frac{\lambda(\theta, \varphi)}{\sqrt{\left|\chi^{\prime}[\lambda(\theta, \varphi)]\right|}}\left(\frac{\mathrm{d} \varphi}{\mathrm{d} \psi}\right) \simeq p(\theta)+q(\theta) \cos \psi
$$
to obtain the uniform Bessel approximation (Berry 1975; Connor 2004)
$$
f_g(\theta) \simeq \frac{\exp [\mathrm{i} A(\theta)-3 \mathrm{i} \pi / 4]}{k}\left(\frac{2 \pi \theta}{\sin \theta}\right)^{1 / 2}\left\{p(\theta) J_0[\zeta(\theta)]+\mathrm{i} q(\theta) J_0^{\prime}[\zeta(\theta)]\right\},
$$
where $J_0^{\prime}(z)=\mathrm{d} J_0 / \mathrm{d} z$.
(ii) Show that correspondence between the stationary phase points on the two sides of the mapping equation yields $\lambda_b(\theta)=\lambda(\theta, \pi)$ and $\lambda_c(\theta)=\lambda(\theta, 0)$, such the $\chi_b=-\theta$ and $\chi_c=\theta$ respectively, in terms of which
$$
\begin{aligned}
A(\theta) & =\frac{1}{2}\left[2 \eta_b+2 \eta_c+\left(\lambda_b-\lambda_c\right) \theta\right], \\
\zeta(\theta) & =\frac{1}{2}\left[2 \eta_b-2 \eta_c+\left(\lambda_b+\lambda_c\right) \theta\right], \\
p(\theta) & =\frac{1}{2}\left(\frac{\zeta(\theta)}{\theta}\right)^{1 / 2}\left[\left(\lambda_c /\left|\chi_c^{\prime}\right|\right)^{1 / 2}+\left(\lambda_b /\left|\chi_b^{\prime}\right|\right)^{1 / 2}\right], \\
q(\theta) & =\frac{1}{2}\left(\frac{\zeta(\theta)}{\theta}\right)^{1 / 2}\left[\left(\lambda_c /\left|\chi_c^{\prime}\right|\right)^{1 / 2}-\left(\lambda_b /\left|\chi_b^{\prime}\right|\right)^{1 / 2}\right] .
\end{aligned}
$$
[Hint: $\mathrm{d} \varphi / \mathrm{d} \psi$ may be deduced from the second derivative of the transformation identity (compare eqn (B.12)).

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
03:28

Problem 5

Deduce that the results of problem 9.4 reduce under the approximation
$$
\eta(\lambda) \simeq \eta_{\mathrm{g}}-\frac{1}{2} \eta_{\mathrm{g}}^{\prime \prime}\left(\lambda-\lambda_{\mathrm{g}}\right)^2
$$
to the limiting form
$$
f_g(\theta) \simeq \frac{\exp \left(2 \mathrm{i} \eta_{\mathrm{g}}-3 \mathrm{i} \pi / 4\right)}{k}\left(\frac{2 \pi \lambda_g^2 \theta}{\sin \theta\left|\chi_{\mathrm{g}}^{\prime}\right|}\right)^{1 / 2} J_0\left(\lambda_{\mathrm{g}} \theta\right) \quad \text { as } \quad \theta \rightarrow 0 .
$$

Noah Musser
Noah Musser
Numerade Educator