# Why does a deuteron nucleus have spin 1? Hydrogen spin 1/2?

#### pa5tabear

I'm learning about NMR and it's been stated that deuteron has a spin 1 nucleus and so should exhibit different behavior.

Why? I've heard of electrons having spin, and now protons, but I don't see why the spin of a neutron should matter since it's uncharged.

I'm thinking a spin of 1 means that deuteron shouldn't follow the "n+1" multiplicity rule.

M = 2nI+1

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#### Chopin

Charge and spin are unrelated concepts. The neutron has spin 1/2 just like the proton does. It doesn't interact with the magnetic field in the same way that a proton does because it's uncharged, but for purposes of doing spin sums, like you're doing here, you still have to keep track of the fact that it's spin 1/2.

#### pa5tabear

So do all nucleons have a spin of 1/2? When is positive 1/2 or negative 1/2?

The problem I'm looking at asks about the proton resonance spectrum for deuteroacetaone (D2CHCOCD3). We are concerned with the two deuteron atoms attached to the same carbon as atom as the hydrogen. Because the other methyl group is too far removed to affect the proton(?).

My book says the effect of deuteron splitting on a proton causing splitting into three bands, so with two of them, you end up with splitting into nine bands.

I think the deuterons should cause a splitting of five because n=2(?) because there are two equivalent deuteron nuclei, each with a spin sum of I = 1.

This would result in 25 bands, which apparently is incorrect, according to my book.

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#### Chopin

The terminology is a little confusing. When we say that a particle has "spin 1/2", what that means is that any given particle will either have a value of spin angular momentum of +1/2 or -1/2. A particle of spin 1 will have either -1, 0, or 1; a particle of spin 3/2 will be -3/2, -1/2, 1/2, or 3/2, and so on. Saying a particle is of a given spin basically tells you the maximum spin it can have--any individual particle will have a spin anywhere between positive that value and negative that value, and always going in full integer steps.

To really understand why spin works that way, it's necessary to learn a bit about representations of the rotation group SO(3)--this is covered in most basic quantum mechanics textbooks.

#### Chopin

I know exactly nothing about NMR, but as my previous post mentioned, a particle of spin 1 can take on three different values of spin angular momentum: -1, 0, and 1. This is probably the reason that a deuteron will cause a three-way split.

#### pa5tabear

I know exactly nothing about NMR, but as my previous post mentioned, a particle of spin 1 can take on three different values of spin angular momentum: -1, 0, and 1. This is probably the reason that a deuteron will cause a three-way split.
Is this definitely true? I've heard this about electrons and their spin but this is my first time learning about proton/neutron spins.

#### Chopin

All fermions have spin. In fact, all fermions have half-integer spin--it's mathematically impossible to have the Pauli Exclusion principle for a particle without it (this is called the Spin Statistics Theorem). So yes, the neutron definitely has spin 1/2. You can check http://en.wikipedia.org/wiki/Neutron to confirm.

What I'm less familiar with is the physics of the deuteron, so I can't say with confidence that its spin must always be 1. What I can say is that when you do spin sums, you have the possibility of either the sum or the difference of the two spins, so in the case where you're summing two spin 1/2 particles (like the proton and the neutron), you have the possibility of creating either a spin-0 or a spin-1 particle as a result. If there's some process that prohibits the spin-0 outcome, then the only remaining option is spin-1.

Regardless of that, though, if you can establish that it's a spin-1 particle, then there will definitely be three separate angular momentum eigenstates, corresponding to -1, 0, and 1. This is true of any spin 1 particle, regardless of how it is created.

#### pa5tabear

All fermions have spin. In fact, all fermions have half-integer spin--it's mathematically impossible to have the Pauli Exclusion principle for a particle without it (this is called the Spin Statistics Theorem). So yes, the neutron definitely has spin 1/2. You can check http://en.wikipedia.org/wiki/Neutron to confirm.

What I'm less familiar with is the physics of the deuteron, so I can't say with confidence that its spin must always be 1. What I can say is that when you do spin sums, you have the possibility of either the sum or the difference of the two spins, so in the case where you're summing two spin 1/2 particles (like the proton and the neutron), you have the possibility of creating either a spin-0 or a spin-1 particle as a result. If there's some process that prohibits the spin-0 outcome, then the only remaining option is spin-1.

Regardless of that, though, if you can establish that it's a spin-1 particle, then there will definitely be three separate angular momentum eigenstates, corresponding to -1, 0, and 1. This is true of any spin 1 particle, regardless of how it is created.
Thanks!

So should a spin-0 particle behave differently than a spin-1 particle? I'm reading an overview of NMR and saw this:

"Two or more particles with spins having opposite signs can pair up to eliminate the observable manifestations of spin. An example is helium. In nuclear magnetic resonance, it is unpaired nuclear spins that are of importance.".

It sounds like this is describing the two protons in helium only pairing up when they cancel to form a spin-0 nucleus.

Does this affect helium being inert?

EDIT: I should probably stop posting and just read more. I read just now that a particle with a net spin can absorb a photon at a frequency that depends on its gyromagnetic ratio. I'm not sure what that means about absorbing particles of other frequencies.

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#### Chopin

Right. A spin 0 particle will not have any splitting, since there's only one spin state it can be in. Helium must be in a spin 0 state because the Pauli Exclusion Principle prohibits the two protons from having the same spin. In a deuteron, the proton and neutron are different types of particles, so that doesn't apply.

Also, I looked at this after posting before: the deuteron can form a spin 0 state just like helium, but that's a higher energy state than the spin 1 state, so the spin 1 state is the ground state for the deuteron.

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