Nuclear Shells and angular momentum?

[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 1h_11/2 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.

 

[NateD]

The spin of the nucleus will be determined by the total angular momentum of the extra proton, which is equal to the orbital angular momentum plus the spin angular momentum. Since you know the orbital angular momentum, you can calculate the total angular momentum by adding the spin angular momentum. The spin angular momentum is usually given in multiples of the reduced Planck's constant, h-bar. For example, if your total angular momentum is 6h-bar, the spin angular momentum is 1h-bar. Once you have the total angular momentum, you can calculate the spin of the nucleus by dividing the total angular momentum by the reduced Planck's constant (h-bar). For example, if you have a total angular momentum of 6h-bar, the spin of the nucleus is 6/h-bar, or 6. Hope this helps!