Nuclear Shell Model, spin and parity

[UniPhysics90]

Homework Statement

What is the parity and possible spins predicted by the single particle shell model for the ground state of the 19K40 nucleus

(I'm most unsure about the spins)

Homework Equations

The Attempt at a Solution

This has 19 protons and 21 neutrons.

Odd-odd nucleus so parity is product of the parity of the shells.

Protons: 'Hole' in 1d3/2, l=2 so parity =+
Neutrons: 'Hole' in 1f7/2, l=3 so parity =-

So the parity is +*-=-

Spin of odd-odd nucleus can take values between |j1-j2| and |j1+j2| where j1 is total angular momentum of proton, and j2 is of neutron.

For proton: l=2 so j=5/2 or 3/2
Neutron: l=3 so j=7/2 or 5/2

Spin can take values of 2,3,4,5,6

Is my method for spins correct? I don't really understand what I'm doing.

Thanks

 

[mark726]

for your question!

Your method for determining the parity of the 19K40 nucleus is correct. As for the possible spins, you are on the right track but there are a few things to consider.

Firstly, the total angular momentum of the nucleus is the sum of the angular momenta of the individual particles, so for the proton, the possible values of j are 5/2 and 3/2, and for the neutron, the possible values are 7/2 and 5/2. This means that the total angular momentum of the nucleus can take values of 3/2, 5/2, 7/2, and 9/2.

Additionally, the ground state of a nucleus is typically the state with the lowest energy, which in this case would be the state with the lowest possible spin. This means that the most likely spin for the ground state of the 19K40 nucleus is 3/2. However, it is possible for the nucleus to be in a higher spin state due to other factors such as nuclear interactions.

In summary, the possible spins predicted by the single particle shell model for the ground state of the 19K40 nucleus are 3/2, 5/2, 7/2, and 9/2. I hope this helps clarify things for you. Keep up the good work!