# Relation between nuclear shape and it spin number

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• Renato Murback
Renato Murback
TL;DR Summary
What is the relation between the nuclear shape (or charge distribution) and the nuclear spin?
I'm beginner in NMR studies and now I'm studying quadrupole interaction. In this case, spins larger than 1/2 have a non-spherical charge distribution and this asymmetry interacts with external electric fields.
But the question is: What is the relation between the nuclear shape (or charge distribution) and the nuclear spin? I know spin 1/2 has a spherical shape, and larger spins have differents shapes.... But why? How nuclear spins implies the nuclear shape?
Thank you very much for your attention

Welcome to PF, Renato.

Paging @Dale

Renato Murback
spin 1/2 = magnetic dipole.
spin 3/2 = magnetic octupole
etc.

See the Wigner-Eckhart Theorem.

Renato Murback
how is these spin numbers related with the nuclear shape? what is the relation between them?

If you go to Google, select Images, and type in Nuclear Quadrupole (e.g.) you will get plenty of pictures showing you the shape.

I don't want to know the shape. I want to know the physical relation between the shape and the spin number. How does spin implies the shape? Or how does nuclear shape implies the spin number?

I told you the "physical reason". It's the Wigner-Eckhart Theorem.

Renato Murback
I'll learn about that. Thank you

berkeman
While not a proof, here is an example that shows how Wigner-Eckhart works. Suppose I want the quadrupole moment of a spin-½ system. That's <½| ℚ | ½> where ℚ is the quadrupole operator. ℚ | ½> = a | 3/2 > + b | 5/2 > where a and b are numbers. Messy to calculate, but just numbers. But since <1/2 | 3/2 > and <1/2 | 5/2 >are both zero, we don't need to: <½| ℚ | ½> = 0. A spin-½ system cannot have a quadrupole moment.

Renato Murback

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