# Working out magnetic moment and electric quadrupole moment

Flucky
Could somebody check if I have done this correctly please?

1. Homework Statement

Draw the shell filling for oxygen isotopes and make predictions of their nuclear spin, parity, magnetic moment and electric quadrupole moment.

## Homework Equations

Magnetic moment μ = gj j μN
Electric quadrupole moment Q ≈ -<r2>$\frac{2j - 1}{2(j + 1)}$

## The Attempt at a Solution

So starting off with the isotopes of oxygen: 16O, 17O, 18O

16O
This fills the 1p$\frac{1}{2}$ shell.

spin = 0 because it is even-even
parity = 1 because even-even
μ = 0 because even-even
Q = 0 because it is a closed shell

18O
This corresponds to the 1d$\frac{5}{2}$ shell.

Same values as above except for the electric quadrupole moment:

Q ≈ -<r2>$\frac{2(0) - 1}{2((0) + 1)}$ = -<r2>$\frac{-1}{2}$ = $\frac{1}{2}$<r2>

17O
This corresponds to a valence neutron in the 1d$\frac{5}{2}$ shell.

spin = j = $\frac{5}{2}$
parity = (-1)l = (-1)2 = 1

μ = $\frac{5}{2}$ gj μN (I won't work out gj or put the value of μN in)

Q ≈ -<r2>$\frac{2(\frac{5}{2}) - 1}{2((\frac{5}{2}) + 1)}$ = -$\frac{4}{7}$<r2>