The deuteron's measured spectroscopic quadrupole moment is

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

Q_{sp}=+0.286\,\mathrm{fm^2}

, definitely not zero, even though its dominant component is the

\ell=0

^3S_1

state. Write the deuteron's quadrupole moment as

Q_{sp}=(1-P_D)Q_S+P_DQ_D+2\sqrt{P_D(1-P_D)}\,Q_{SD},

where

Q_S

Q_D

and

Q_{SD}

are, respectively, the quadrupole moment of the pure

^3S_1

state, of the pure

^3D_1

state, and the

^3S_1

^3D_1

interference matrix element, and

P_D

is the

D

-state probability. A particular model of the deuteron's radial wavefunctions gives the interference matrix element

Q_{SD}=0.65\,\mathrm{fm^2}

; the pure

D

-state term is small enough to be dropped at the level of this estimate. ((a)) State the value of

Q_S

and justify it. ((b)) Show that, for a small admixture,

Q_{sp}\approx2\sqrt{P_D}\,Q_{SD}

, and explain in one sentence which operator property makes the interference term, not the population term, the leading one. ((c)) Use the measured

Q_{sp}

and the given

Q_{SD}

to estimate

P_D

, and check that the smallness assumed in part (b) is satisfied.