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I am actually calculating the nuclear spin of Sodium 23. Here we have 11 protons and 12 neutrons. Now both the nuclei are short of the magic numbers. When I use the shell model for protons and neutrons separately, I found 3 protons in the $1d_{5/2}$ sub-shell and 4 neutrons in the same $1d_{5/2}$ sub-shell. So because of two pairings, neutrons give spin as 0 and because of a pairing in protons, one proton is left out which should give spin as ${1/2}$. But in the book its, $I={3/2}$. Please can anyone explain the fact how the spin of Na nucleus is ${3/2}$. Thank you in advance.

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up vote 2 down vote accepted

The number of neutrons is even, so it indeed means that they contribute spin zero and positive parity.

The spin and parity comes from the "last proton" because the number of protons is odd.

The dependence of the energy on the angular momentum is such that the pairs at a high value of $J$ are preferred (lower in energy) due to the special, spin-dependent features of the strong nuclear force (features invisible in the single-nucleon model). That's true despite the fact that the single-particle shells with a lower $J$ could be preferred.

It follows that among the 3 protons in $1d_{5/2}$, the pair really chooses $j_z=\pm 5/2$, the maximum value (in the absolute value). The remaining slots $j_z=\pm 1/2$ and $\pm 3/2$ are available for the last proton. The last proton also prefers the higher value of $J$ so it will sit in the $J=3/2$ state. It's a $d$-shell, i.e. $l=2$, so the parity is $(-1)^l=+1$.

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Hi LM thanks for the clear explanation. Now suppose if we had two more protons in the $d_{5/2}$, then the spin would have been 1/2. Am i doing it right or not? please provide some info. (PS: I was actually reading your "Why are there spinors?" in your reference frame blog yesterday. So i am doubly happy after seeing your reply,,,Many many thanks) –  bluesquare Dec 19 '13 at 12:04
    
It's aluminum-25 whose spin is actually 5/2 and parity plus, see en.wikipedia.org/wiki/Isotopes_of_aluminum - thanks for your kindness. ... The spin 5/2 isn't that easy to see and the superreliable rules to get the spin in all cases probably doesn't work but in analogy with Hund's rules in chemistry, it's probably favorable to have a larger total J of the protons. You may imagine that the 2 pairs in this case fill the 1/2 and 3/2 pairs of states while the 5/2 is unpaired. If someone knows how to be sure how they pick the states, it is not me. ;-) –  Luboš Motl Dec 20 '13 at 14:59
    
You must understand that it's a very complicated system with nonlinear Yang-mills couplings etc. etc. and one calculates the energy level. It's unreasonable to expect that there exists a kindergarten-comprehensible algorithm to reliably answer a nontrivial aspect of the answer. –  Luboš Motl Dec 20 '13 at 14:59
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