Looking at the spin of 7N7 and 9F10

[bluestar]

I have worked out the total spin for the elements up to Neon however I hit a snag with Nitrogen and Fluorine.

The shell sequence of nitrogen 7N7 is 1s2, 2s2, 2p3 for both the proton and neutron and the total angular spin is 1

Likewise, Fluorine 9F10 is 1s2, 2s2, 2p5 for the proton and 1s2, 2s2, 2p6 for the neutron and the total angular spin is ½

In the nitrogen isotope it would appear that the 3 protons and neutrons are not paired and occupy separate states in the 2p shell. Thus the total orbital angular momentum would be 3 and the total spin would be 3/2 for the protons and additional for the neutrons suggesting the total angular momentum of 9 which is wrong.

So do 2 of the protons in different states in the 2p shell pair up their orbital and spin angular momentum? Likewise do 2 of the neutrons in the 2p shell pair up their orbital and spin angular momentum? If so, this would leave yield an orbital momentum of 1 for the proton and 1 for the neutron; likewise, the spin of the two separate nucleons sum up to 1. This would then yield a total angular momentum of 2 + 1 = 3 with is also wrong.

The only way I can see to achieve a total angular momentum of 1 is if the spin of the proton and neutron were both negative, which I considered wrong because unpaired particle spin were always positive. So can an unpaired particle have a negative spin?

With respect to 9F10 there are 2 sets of paired protons plus 1 separate in the 2p shell and all neutrons are paired. This would produce an orbital angular momentum of 1 and a spin of ½ for a total of 3/2; however JANICE indicates this isotope has only ½ spin suggesting the proton spin was -1/2.

Can anyone shed any light on the total spin of these two isotopes?

 

[PeterDonis]

Are you looking at the nuclear shells or the electron shells? They're not the same. What you're giving look like electron shell configurations.

Check out the Wikipedia page: http://en.wikipedia.org/wiki/Nuclear_shell_model" . According to the nuclear shell model as described there (see the graph on the right in the "predicted magic numbers" section), 7N7 would have 1s(1/2) - 2, 1p(3/2) - 4, 1p(1/2) - 1 for protons, and the same for neutrons. The total spin would be 1 (1/2 each for the unpaired proton and neutron).

9F10 would have 1s(1/2) - 2, 1p(3/2) - 4, 1p(1/2) - 2, 1d(5/2) - 1 for protons, and the same except 1d(5/2) - 2 for neutrons. The total spin would be 1/2 for the unpaired proton.

(Pedantic note: these are, of course, the ground states of these nuclei. This actually does make a difference for the 7N7 case, because there is also a spin-0 state possible for that nucleus, with the same shell configuration, where the unpaired proton and neutron have opposite spins; however, this state has higher energy than the spin-1 state where the unpaired proton and neutron spins are aligned. I've been unable to find a reference online with a good simple explanation of why that is, but I suspect it's because the magnetic moments of the proton and neutron are of opposite signs--the proton is about +2.8 and the neutron is about -2, in the standard units for magnetic moments of elementary particles--which means that when their spins are aligned, their magnetic moments attract each other--opposite magnetic poles attract just as opposite charges do--so that the nucleus as a whole is a little bit more tightly bound--lower total energy--whereas when the spins are opposite, the magnetic moments repel each other and the nucleus as a whole is a little bit less tightly bound--higher total energy.)

 

[sir-pinski]

Firstly, it is important to note that the total spin of an atom is not simply the sum of the individual spins of its constituent particles. The total spin of an atom is determined by a combination of the spins and orbital angular momenta of its electrons and nucleons.

In the case of nitrogen and fluorine isotopes, the total spin can be affected by the arrangement of the protons and neutrons in the 2p shell. As you have correctly pointed out, in the case of 7N7, the 3 protons and neutrons in the 2p shell are not paired, leading to a total orbital angular momentum of 3 and a total spin of 3/2. This is incorrect as the total spin of 7N7 is actually 1, as you have mentioned.

In order to achieve a total spin of 1, it is possible that two of the protons in the 2p shell pair up their spins and orbital angular momenta, as you have suggested. This would result in a total orbital angular momentum of 1 for the proton, and a total spin of 1 for the neutron. This would then lead to a total angular momentum of 2 + 1 = 3, which is correct.

It is also possible for unpaired particles to have negative spins, as spin is a quantum mechanical property and can take on both positive and negative values. However, the convention is to assign positive values to unpaired particles, which is why it may seem unusual to have a negative spin for an unpaired particle.

For 9F10, the situation is slightly different as there are 2 sets of paired protons and all neutrons are paired. This would result in a total orbital angular momentum of 1 and a total spin of 1/2, as you have mentioned. However, the spin of the proton is actually -1/2, which is why the total spin of 9F10 is only 1/2. This is because the proton and neutron in the 2p shell are in a spin-flip state, where the proton's spin is opposite to the neutron's spin. This results in a cancellation of spin, leading to a total spin of 1/2.

In summary, the total spin of an atom is determined by a combination of the spins and orbital angular momenta of its electrons and nucleons. The arrangement of these particles in the 2p shell can