= Solution
A <muonic atom> binds a negative <muon> in the nuclear <Coulomb potential>. The <muon> is a <spin one-half> <lepton> with the <electron>'s charge but about 207 times its <mass>. For a point <nucleus> of <mass> $M_N$ and charge $Ze$, the <reduced mass> and nonrelativistic hydrogenic scales, with <fine-structure constant> $\alpha$, are
$$
\mu_r=\frac{m_\mu M_N}{m_\mu+M_N},\qquad
a_\mu=\frac{\hbar}{Z\alpha\mu_r c},\qquad
E_n=-\frac{\mu_r c^2(Z\alpha)^2}{2n^2}.
$$
For a heavy <nucleus> $\mu_r\simeq m_\mu$, giving a radius roughly $1/207$ of the corresponding electronic radius and transition <energies> roughly 207 times larger. For hydrogen, <nuclear recoil> matters: $m_\mu\simeq106\,\mathrm{MeV}/c^2$ gives $\mu_r\simeq95\,\mathrm{MeV}/c^2$ and $a_\mu\simeq284\,\mathrm{fm}$ rather than the infinite-nuclear-mass value about $255\,\mathrm{fm}$.
A slowed negative <muon> is captured into an excited orbit and undergoes a <radiative cascade>, ejecting atomic <electrons> and emitting muonic X-rays. Inner orbits penetrate the <nuclear charge distribution>, particularly at large $Z$. Replacing a point charge by an extended charge makes the central attraction less singular and shifts states with substantial nuclear overlap. Measured transition <energies> consequently constrain the <nuclear charge radius>. Sensitivity is particularly strong for $S$ states, whose contact density scales as $(Z\alpha\mu_r)^3$.
An instructive experiment is <Randolf Pohl>'s team's muonic-hydrogen spectroscopy. A stopped-muon target prepared a metastable $2S$ population; a tunable pulsed laser drove a $2S\to2P$ transition, followed by a detectable $2P\to1S$ X-ray. Scanning laser frequency located a resonance whose interpretation included the <proton> finite-size contribution to the <Lamb shift>. Delayed detection selected the long-lived population against prompt cascade backgrounds. Laser-off or detuned measurements provide a background control; <energy> and time selection alone would not establish a laser-induced resonance.<The size of the proton by Randolf Pohl and others>[Nature 466, pp. 213–216, Fig. 1 experimental principle and Figs. 4–5 time spectra and resonance]{ref}
\Image[/past-exam-of-the-particle-and-nuclear-physics-course-of-the-university-of-cambridge/2016/paper-1-muonic-scales.png]
{title=<reduced mass> changes the orbital scale and nuclear-size sensitivity}
{description=Original calculated hydrogenic radial probabilities, plotted against physical radius with probability measured per logarithmic interval. The Bohr radii are stated alongside; these are point-nucleus model curves, not measured nuclear densities.}
Extracting a radius requires <quantum electrodynamics>, including <vacuum polarization>, recoil and nuclear-structure corrections. A free <muon> decays on a microsecond timescale; a bound negative <muon> can instead undergo <muon capture>, for example $\mu^-p\to n\nu_\mu$. These processes limit observation time. In heavy atoms, relativistic motion and nuclear overlap make the simple Bohr formulas only qualitative. \b[Muonic spectroscopy converts a smaller orbital scale into enhanced sensitivity to nuclear charge structure], provided atomic and nuclear corrections are controlled.
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