The deuteron's binding energy is held fixed while the assumed square-well range is varied

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 deuteron and what two nucleons do to each other

Problem

The deuteron's binding energy

B = 2.2246\,\mathrm{MeV}

is held fixed while the assumed square-well range

R_{\rm w}

is varied. Take

\mu_{\rm red}c^2 = 469.46\,\mathrm{MeV}

and

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

, and use the matching condition

k_{\rm in}\cot(k_{\rm in}R_{\rm w}) = -\kappa

with

V_0 - B = \hbar^2k_{\rm in}^2/2\mu_{\rm red}

. ((a)) Find the well depth

V_0

needed for

R_{\rm w} = 1.7\,\mathrm{fm}

. ((b)) Find the well depth

V_0

needed for

R_{\rm w} = 2.5\,\mathrm{fm}

. ((c)) Someone claims the depth must scale as

V_0 \propto 1/R_{\rm w}^2

, since that is how the just-bound threshold depth scales. Using the fitted value

V_0 = 33.7\,\mathrm{MeV}

R_{\rm w} = 2.1\,\mathrm{fm}

as a reference, check whether this simple scaling reproduces your answers to (a) and (b).