A negative muon is captured into the orbit of ( )

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Nuclear Physics · Nuclear constituents, size and binding

The Nucleus Resolved, Ch How big a nucleus is, and what the mass surface knows

Problem

A negative muon is captured into the

1s

orbit of

^{27}\mathrm{Al}

(

Z=13

), whose rms charge radius is

\langle r^2\rangle^{1/2} = 3.06\,\mathrm{fm}

. Take

m_\mu c^2 = 105.658\,\mathrm{MeV}

m_ec^2 = 0.510999\,\mathrm{MeV}

, the atomic mass of

^{27}\mathrm{Al}

27\times 931.494\,\mathrm{MeV}/c^2

, and the electron Bohr radius

a_0 = 52917.7\,\mathrm{fm}

. ((a)) Compute the muon's reduced-mass Bohr radius

a_\mu = a_0(m_e/\mu_{\rm red})/Z

for this atom. ((b)) Use the first-order finite-size shift

\Delta E = \dfrac{2\pi}{3}\dfrac{Ze^2}{4\pi\epsilon_0}|\psi(0)|^2\langle r^2\rangle

, with

|\psi(0)|^2 = 1/(\pi a_\mu^3)

for a

1s

state, to find

\Delta E

. ((c)) Compare

\Delta E

with the point-nucleus Bohr binding energy

E_1 = 13.6\,\mathrm{eV}\times Z^2\times(\mu_{\rm red}/m_e)

, and say whether first-order perturbation theory is trustworthy here. Judge also the approximation

\psi(r) \approx \psi(0)

across the nucleus, by comparing the equivalent sharp radius

R = \sqrt{5/3}\,\langle r^2\rangle^{1/2}

with

a_\mu