The spin-orbit constant

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

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

, fitted from the

2g_{9/2}

2g_{7/2}

splitting in

^{209}\mathrm{Pb}

, is applied unchanged (ignoring its known dependence on mass number) to estimate the

1f_{7/2}

1f_{5/2}

splitting near calcium (

A = 40

). ((a)) Using

\Delta E_{so} = C\hbar^2(2\ell+1)/2

with

\ell = 3

, find the predicted

1f

splitting. ((b)) Using

\hbar\omega = 41A^{-1/3}\,\mathrm{MeV}

, find

\hbar\omega

for

A = 40

and express the splitting as a fraction of

\hbar\omega