A nucleon in an orbital of orbital angular momentum and spin couples to total angular…

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Nuclear Physics · The nuclear force and nuclear models

The Nucleus Resolved, Ch The shell model: a mean field, a spin-orbit term and the magic numbers

Problem

A nucleon in an orbital of orbital angular momentum

\ell

and spin

s=\tfrac12

couples to total angular momentum

\vec{\jmath}=\vec\ell+\vec s

, so that

\vec\ell\cdot\vec s = \frac12\left(\hat{\jmath}^{\,2}-\hat\ell^{\,2}-\hat s^{\,2}\right).

((a)) Using the eigenvalues of

\hat{\jmath}^{\,2}

\hat\ell^{\,2}

and

\hat s^{\,2}

, find

\langle\vec\ell\cdot\vec s\rangle

for

j=\ell+\tfrac12

and for

j=\ell-\tfrac12

, and hence show that the energy splitting between the two members of the doublet, produced by a spin-orbit term with

j=\ell+\tfrac12

lying lower, can be written

\Delta E_{so} = C\hbar^2\,\frac{2\ell+1}{2},

where

C\hbar^2

is a positive constant fixing the sign and radial matrix element of the spin-orbit potential. ((b)) The constant fitted to the

2g_{9/2}

2g_{7/2}

pair in

^{209}\mathrm{Pb}

C\hbar^2 = 0.55\,\mathrm{MeV}

. Assuming the same constant applies in the same mass region, predict the splitting of a

1i_{13/2}

1i_{11/2}

doublet (

\ell=6