# Spin $1$ vs. spin $1/2$

A crystal contains $N$ atoms which possess spin $1$ and magnetic moment $\mu$. Placed in a uniform magnetic field $B$ the atoms can orient themselves in three directions: parallel, perpendicular, and antiparallel to the field. If the crystal is in thermal equilibrium at temperature $T$ find an expression for its mean magnetic moment $M$, assuming that only the interactions of the dipoles with the field $B$ need be considered.

What does it mean for an atom to possess spin $1$ instead of $\pm \frac{1}{2}$? I thought particles can only have one of these 2 values for spin value.

How does a perpendicular orientation affect the probabilities of the other 2 orientations?In the book I am reading through, it gives an example with only a parallel and antiparallel possibility for spin. However, I know that intuitively these can't be the only 2 options. When the dipole flips from parallel to antiparallel, it must swing through a perpendicular state (assuming only $x,y$ directions apply here).

• Fermions that are also fundamental particles have spin 1/2. An atom is a composite object, made up of electrons, protons, and neutrons, each of which have their own spins that add together. Furthermore, there are other angular momenta like the orbital angular momentum of electrons, and they add to make the total angular momentum of the atom. Sometimes this total angular momentum is just called spin, because the algebra is the same. – march Feb 17 '16 at 4:13
• The main consequence of spin 1 is that it makes the particles Bosons (ie excluded from the Pauli Exclusion Principle) and the other is that the magnetic field can orient itself in three directions: parallel, perpendicular, and antiparallel to the field. – Aron Feb 17 '16 at 4:14

A atom is constituted of fermions(proton, neutron and electron). But whenever the atom an atom has even no of constituents it behaves like a boson. This is very easy to understand. All fermions have spin 1/2. So even no of fermions will have integer spin and therefor behave like bosons. Thus atom with spin 1 is possible.

Now given any value of spin S it will have (2S+1) eigenvalues s starting from -S to +S by steps of one. Remember these eigenvalues are discrete. And they give you the orientations. For your case the atom has spin 1. So it will have 3 eigenstates of spin. You named them to be parallel, antiparallel and perpendicular. But remember the spin is never completely parallel nor completely anti-parallel to the Magnetic field. There will always be some uncertainty.

You said:

I know that intuitively these can't be the only 2 options.

But that's wrong. Because if the atom is spin-1/2, then there will only be two options namely the parallel and the anti-parallel one. There will not be any steps in between.