The odd-proton nuclide ( , , ) has ground state spin-parity ; filling the Nilsson…

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

The Nucleus Resolved, Ch Collective motion: the deformed nucleus, vibrations, rotations and giant resonances

Problem

The odd-proton nuclide

^{165}\mathrm{Ho}

(

Z=67

N=98

A=165

) has ground state spin-parity

\tfrac72^-

; filling the Nilsson levels of this chapter assigns this state

K=J=\tfrac72

in the deformed field. Its measured static (spectroscopic) electric quadrupole moment (Chapter 2) is

Q_{\rm sp}=+3.58\,e\,\mathrm b

. ((a)) Using the mass formula's odd-even pairing term, restated from Chapter 1 as

\Delta_{\rm pair}\approx12A^{-1/2}\,\mathrm{MeV}

, compute

\Delta_{\rm pair}

for

A=165

. ((b)) Using the rotor's projection factor

Q_{\rm sp}=Q_0\dfrac{3K^2-J(J+1)}{(J+1)(2J+3)}

with

K=J=\tfrac72

, find the intrinsic quadrupole moment

Q_0

. ((c)) Taking

R_0=1.2A^{1/3}\,\mathrm{fm}

, compute

\beta_2

from

Q_0=\dfrac{3}{\sqrt{5\pi}}ZR_0^2\beta_2

. ((d)) Ground-band moments of inertia in this region are typically about half the rigid-sphere value. Using the size of

\Delta_{\rm pair}

found in part (a), explain briefly why pairing correlations of this strength can still keep

\mathcal I

well below

\mathcal I_{\rm rigid}

at the low spins of the

^{165}\mathrm{Ho}

ground band.