N has 7 protons and 7 neutrons per atom. So the spin is S=1 for one N-atom. Why is this? I know that they do pairs in the shell model. So I only need to look at the spin of a single proton and a single neutron in N. If both spins are parallel it is 1 or -1. But why is 0 not an option for this (antiparallel spin ex. S = $\frac{1}{2}$ for proton and S=$-\frac{1}{2}$ for neutron)?

Therefore the spin of $N_2$ must be 0 or 2, right? 1 isnt possible because 0 isn't an option for a single N-atom as far as I understand.

I just do not understand the detail. I need to tell if the angular momentum quantum number of $N_2$ can be even, uneven or both if the wave function of the nucleus is antisymmetric. So I need to tell first if the wavefunction is symmetric or not.

  • $\begingroup$ From Blatt & Weiskopf: "Since nuclei are built up of neutrons and protons, each possesses an angular momentum I which is the combined effect of the intrinsic spin of the constituents and of the angular momentum of the orbital motion within the nucleus" $\endgroup$
    – Jon Custer
    Nov 20 '20 at 16:23

In the nuclear shell model, $^{14}$N is seen as an inert $^{12}$C core, with 2 extra nucleons. The ground state for the strong nuclear force is going to put these nucleons in the same spatial state (a symmetric S state) and the same spin state (which for spin 1/2 particles is the spin-1 triplet state). Anti-symmetrization of the wave function is accomplished in the isospin sector via the singlet isospin wave function:

$$ |I=0, I_3=0\rangle = \frac 1 {\sqrt 2}[|p\rangle|n\rangle - |n\rangle|p\rangle]$$

It's important to note that individual nucleons do not have a definite "proton" or "neutron" identity with isospin. It is exactly the same as a spin 1/2 particle not having a definite spin.

So we can treat each $^{14}$N as indistinguishable vector bosons.

You can work out the Clebsch-Gordan coefficients for combining two vectors, but the general structure is:

$$ {\bf 3} \otimes {\bf 3} = {\bf 5}_S \oplus {\bf 3}_A \oplus {\bf 1}_S $$

which means the spin-2 and spin-0 multiplets are symmetric under interchange and the spin-1 combination is antisymmetric. For example:

$$|J=1,J_z=0\rangle = \frac 1 {\sqrt 2}[|1,+1\rangle_1|1,-1\rangle_2 - |1,-1\rangle_1|1,+1\rangle_2]$$


$$|J=0,J_z=0\rangle = \frac 1 {\sqrt 3}[|1,+1\rangle_1|1,-1\rangle_2 + |1,-1\rangle_1|1,+1\rangle_2 - |1,0\rangle_1|1,0\rangle_2 ]$$

The molecular wave function then needs to have the same symmetry as the spin wave function to make the overall wave function symmetric.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.