For a neutron bound by a small separation energy in an orbital outside a compact core

PHYSICS · Multi-part · Fermi University Physics · Nuclear Physics · Nuclear reactions · Nuclear reactions

The Nucleus Resolved, Ch Beams and the edges of the chart: emittance, heavy ions, high spin and the drip lines

Problem

For a neutron bound by a small separation energy

S_n

in an

\ell=0

orbital outside a compact core, the radial wavefunction outside the core is well approximated by the pure exponential tail

u(r)=\sqrt{2\kappa}\,e^{-\kappa r}

, normalised so

\int_0^\infty |u(r)|^2\,\mathrm{d}r=1

, where

\kappa=\sqrt{2\mu_{\rm red}S_n}/\hbar c

and

\mu_{\rm red}

is the neutron-core reduced mass. Treat the core as pointlike compared with

1/\kappa

, so that all of

\langle r^2\rangle

comes from the tail. ((a)) Derive

\langle r^2\rangle=\int_0^\infty r^2|u(r)|^2\,\mathrm{d}r=1/(2\kappa^2)

, and hence the tail's rms radius

1/(\sqrt2\,\kappa)

. ((b)) The ground state of

^{15}\mathrm{C}

1/2^+

, a single neutron in a

2s_{1/2}

orbital outside a

^{14}\mathrm{C}

core, with one-neutron separation energy

S_n=1.22\,\mathrm{MeV}

. Using

\mu_{\rm red}c^2=\tfrac{14}{15}m_nc^2

with

m_nc^2=939.57\,\mathrm{MeV}

and

\hbar c = 197.327\,\mathrm{MeV\,fm}

, compute

\kappa

1/\kappa

and the tail's rms distance from the core. ((c)) A neutron with the same

S_n=1.22\,\mathrm{MeV}

sitting instead in a

1d_{5/2}

orbital (

\ell=2

) does not produce a comparably extended halo. Explain briefly, in terms of the centrifugal barrier, why the

\ell=0

case reaches so much further than an

\ell=2

case of the same binding energy.