# Nuclear Shells and angular momentum?

1. Nov 12, 2009

### jeebs

Hi,
I have a bismuth Bi-209 nucleus, and I have to find the angular momentum of the one nucleon outside the full shells, and hence find the spin of this nucleus as a whole.

What I said was that it has 83 protons and 209-83 = 126 neutrons, and since 82 and 126 are magic numbers, there is a full shell of neutrons. This means the extra proton outside the full shell of protons corresponding to the magic number of 82 is the nucleon this question deals with.

I then said that this extra proton must determine the spin of the nucleus as a whole, since the paired neutrons and paired protons all cancel each others' spins.

Now for the angular momentum, j = l + s.
I can get the spin if I know j and l, the total and orbital angular momenta respectively.

The orbital angular momentum I can get from $$L = \sqrt{l(l+1)}\hbar$$
but then I have to know what the orbital angular momentum quantum number l is.

I looked at an energy level diagram in my notes that appears to show the magic number 82 occurring at the 1h11/2 energy level, meaning that, from
....s p d f g h
l : 0 1 2 3 4 5
I have l=5, so I know the magnitude of this proton's angular momentum.

First off, I was wondering, is there some way I could have deduced that l=5 without looking at the diagram, like say, if I was in an exam?

Secondly, I'm not sure where I go from here. I have the orbital angular momentum since I have l=5, but I do not have the total angular momentum, so how do I find out what the spin is?

Thanks.