What is a Nuclear Quadrupole Moment?

[anchal2147]

Homework Statement

The Question is as follow:

For an infinite three-dimensional harmonic oscillator potential well with oscillator frequency $\omega$, the radial wave functions for the lowest s-state and the lowest d-state are, respectively,

$$
R_{1 s}(r)=2 v^{3 / 4} \pi^{-1 / 4} e^{-\frac{1}{2} v r^{2}} \quad R_{1 d}(r)=\frac{4}{\sqrt{15}} v^{7 / 4} \pi^{-1 / 4} e^{-\frac{1}{2} v r^{2}} r^{2}
$$

Where, the oscillator length parameter $ v=M \omega / \hbar $, and $M$ is the mass of a nucleon. Find the root-mean-square radii in each of these states for $\hbar \omega=15 \mathrm{MeV}$. Compare the values obtained with the measured deuteron radius. For the radial wave function given above, what is the value of the off-diagonal matrix element $\left\langle R_{1 s}\left|r^{2}\right| R_{1 d}\right\rangle$ ? Use this model to calculate the quadrupole moment of the deuteron assuming that the wave function is predominantly made of the ${ }^{3} S_{1}$ - state with a $4 \%$ admixture of the ${ }^{3} D_{1}$ - state.

**In the last step of the problem it is asking to use the model to find quadrupole moment having states with admixture of ${ }^{3} S_{1}$ - state with a $4 \%$ admixture of the ${ }^{3} D_{1}$ - state.**

I cant get any approach how to start with this??

Relevant Equations

Spherical harmonics

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[berkeman]

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