Some Nuclear Properties — Question 10

Question 10

A nuclear quadrupole moment distinguishes shape from volume

For a uniform charge ellipsoid with semiaxes a,a,c and total charge Ze, use $\langle x^2\rangle=\langle y^2\rangle=a^2/5$ and $\langle z^2\rangle=c^2/5$. Define intrinsic quadrupole moment $Q_0=\int\rho_q(3z^2-r^2)dV$.

1. Derive $Q_0$ and determine its sign for prolate and oblate shapes.

2. For $Z=40$, $a=4.00\,\mathrm{fm}$ and $c=5.00\,\mathrm{fm}$, calculate $Q_0/e$ and the radius of an equal-volume sphere.

Question 10 - Solution

A nuclear quadrupole moment distinguishes shape from volume

(a) Method and calculation

$$Q_0=Ze[2\langle z^2\rangle-\langle x^2\rangle-\langle y^2\rangle]=\boxed{\frac{2Ze}{5}(c^2-a^2)}.$$$$\boxed{c>a:\ Q_0>0;\qquad c<a:\ Q_0<0;\qquad c=a:\ Q_0=0}.$$

(b) Interpretation and further calculation

$$\boxed{Q_0/e=(2\times40/5)(25-16)=144\,\mathrm{fm^2}}.$$$$\frac{4\pi R^3}{3}=\frac{4\pi a^2c}{3}\Rightarrow\boxed{R=(a^2c)^{1/3}=4.309\,\mathrm{fm}}.$$

Physical check

This is an intrinsic, body-fixed shape moment. The observed laboratory quadrupole moment also depends on the nuclear angular-momentum state.