Two neutrons approach each other in a relative -wave ( )

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

The Nucleus Resolved, Ch The deuteron and what two nucleons do to each other

Problem

Two neutrons approach each other in a relative

s

-wave (

L=0

). Square-well fits at a common range

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

give depths

V_{pp}=21.12\,\mathrm{MeV}

V_{nn}=21.28\,\mathrm{MeV}

and

V_{np}^{\mathrm{singlet}}=21.65\,\mathrm{MeV}

for the three

^1S_0

channels, while the just-bound depth at this range is

23.2\,\mathrm{MeV}

(the depth at which a well first holds a state at zero energy). ((a)) Using antisymmetry of the two-neutron state,

(-1)^{L+S+T}=-1

, and the fact that

T_z=(N-Z)/2=+1

for two neutrons, determine the allowed total spin

S

and isospin

T

for this

L=0

pair. ((b)) Predict, using the well depths above, whether a bound dineutron exists. ((c)) Explain why the fact that the bound deuteron channel (

^3S_1

T=0

) has

S=1

, while every

T=1

realisation of the

L=0

channel (pp, nn, and np singlet) is forced to

S=0

and stays unbound, is itself evidence that the nucleon-nucleon force depends on spin or, equivalently in the

s

wave, on isospin.