Centrifuge-based uranium enrichment: Natural uranium is composed of two isotopes: \({ }^{238} \mathrm{U}\) and \({ }^{235} \mathrm{U}\), with percentages of \(99.27 \%\) and \(0.72 \%\), respectively. If uranium hexafluoride gas \(\mathrm{UF}_{6}\) is injected into a rapidly spinning hollow metal cylinder with inner radius \(R\), the equilibrium pressure of the gas is largest at the inner radius and isotopic concentration differences between the axis and the inner radius allow enrichment of the concentration of \({ }^{235} \mathrm{U}\).

(a) Write down the Lagrangian \(\mathcal{L}\left(\left\{q_{k}, \dot{q}_{k}ight\}ight)\) for particles of mass \(m\) moving in a cylindrical coordinate system rotating at angular velocity \(\omega\) and use a Legendre transformation

\[
\mathscr{H}\left(\left\{q_{k}, p_{k}ight\}ight)=\sum_{k} p_{k} \dot{q}_{k}-\mathcal{L}
\]

to show that the one-particle Hamiltonian \(\mathscr{H}\) in that cylindrical coordinate system is

\[
\mathscr{H}\left(r, \theta, z, p_{r}, p_{\theta}, p_{z}ight)=\frac{p_{r}^{2}}{2 m}+\frac{\left(p_{\theta}^{2}-m r^{2} \omegaight)^{2}}{2 m r^{2}}+\frac{p_{z}^{2}}{2 m}
\]

Ignore the internal degrees of freedom of the molecules since they will not affect the density as a function of position. Show that the one-particle partition function shown here can be written as

\[
Q_{1}(V, T)=\frac{1}{h^{3}} \int_{-\infty}^{\infty} d p_{r} \int_{-\infty}^{\infty} d p_{\theta} \int_{-\infty}^{\infty} d p_{z} \int_{0}^{R} d r \int_{0}^{2 \pi} d \theta \int_{0}^{H} d z \exp (-\beta \mathscr{H})
\]

by constructing the Jacobian of transformation between the cartesian and the cylindrical coordinates for the phase space integral. Evaluate the partition function \(Q_{1}\) in a closed form and determine the Helmholtz free energy of this system.

(b) Determine the number density \(n(r)\) as a function of the distance \(r\) from the axis for the \(N\) molecules of gas in the rotating cylinder. Show that, in the limit \(\omega ightarrow 0\), the density becomes uniform with the value \(n=N / \pi R^{2} H\). Find an expression for the ratio of the pressure at the inner radius of the cylinder \(R\) to the pressure at the axis of the cylinder as a function of \(\omega\) and \(R\).

(c) Evaluate the pressure ratios for the two isotopically different \(\mathrm{UF}_{6}\) gases at room temperature for the case \(\omega R=500 \mathrm{~m} / \mathrm{s}\). Show that the pressure ratio for \({ }^{238} \mathrm{U}\) is approximately \(20 \%\) larger than the pressure ratio for \({ }^{235} \mathrm{U}\) so that extracting gas near the axis results in an enriched concentration of \({ }^{235} \mathrm{U}\). A series of centrifuges can be used to raise the concentration of \({ }^{235} \mathrm{U}\) to create a fissionable grade of uranium for use in power-generating reactors or in nuclear weapons. Not surprisingly, this technology is a major concern for possible nuclear proliferation.