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 = g_j j μ_N
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, ¹⁸O¹⁶O
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 shell¹⁸O
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} g_j μ_N (I won't work out g_j or put the value of μ_N in)

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

 

[Flucky]

I think I may be wrong about the electric quadrupole moments, it's possible that only protons contribute to Q. So because the only thing that changes with oxygen isotopes is neutron number Q will be 0 for all of them?

EDIT: scratch that neutrons do actually contribute as they attract the protons slightly.