Can nuclei spin have any direction? I'm aware that the spin has no classical analogue, and at the same time it is sometimes useful to think of it as a spinning of a small sphere.
When we follow the analogy one would say that the spin momentum will point 'up' or 'down' respect, say, an external field.
My simple question is this one:

*

*Does the total spin indeed have any direction as a vector?

If you can give me some hints and ideas, I could explore this further, but I am trying to understand how the total magnetic moment's direction of a gas would change, in response to an external, static magnetic field.
I am aware the spins would precess and so on, but am just confusing about the "direction" of the spin.
 A: Suppose  you prepare the  spin 1/2 nucleus  in the spin state
$$
e^{i\phi/2} (\cos\theta/2) |+\rangle +  e^{-i\phi/2} \sin(\theta/2) |-\rangle
$$ by ensuring that it is in the  $+1/2$ eigenstate of the spin when  measured in the direction specified by polar angles $\theta$, $\phi$. This is the best you   can do to get the spin pointing in the  $\theta$, $\phi$  direction, and  can arrange  it by using a magnetic field pointing in that direction. If you now decide to measure along the $z$ axis the probability $P_\pm$ of getting $S_z=\pm 1/2$ is
$$
P+= \cos^2(\theta/2), \quad P_- = \sin^2(\theta/2).
$$
A: how the total magnetic moment's direction of a gas would change, in response to an external, static magnetic field.
We have charges and an external magnetic field. Since the charges move, and in the vast majority of cases not parallel to the magnetic field, they move - according to the Lorentz force - on curved paths between the collisions with each other.
In this process, the magnetic dipoles of the charges - i.e. the unbound electrons and the ions as a whole - are aligned in the direction of the external magnetic field.
Does the total spin indeed have any direction as a vector?
The spin is in the same direction as the magnetic moment, so that the summary alignment of the spins is in the direction of the external magnetic field.
