Consistency of atomic magnetic dipole spatial orientation wi

[draconicsheep]

I've been recently trying to understand the concept of paramagnetism, but I feel like I'm running into 2 conflicting models.

Stern–Gerlach seems to suggest that electron spins always point up or down to an incident magnetic field, regardless of their spatial orientation. Similarly, when thinking about topological insulators or electrons in the same orbital, spin is thought of as up or down without really pointing in a particular direction in x, y, z space.

However, in the usual picture of paramagnetism, the dipoles of atoms and their larger magnetic domains point in 3d space, and can have any angle relative to one another (without considering ordering), not just 180 degrees.

I guess I'm not seeing how the up-down-only picture of electron spin can be reconciled with the solid state picture of atomic magnetic dipoles, wherein atoms have magnetic dipoles pointing in 3d space.

 

[Simon Bridge]

Atomic magnetic moments are, in general, randomly oriented... is a solid, the atoms are usually too far apart to affect each other... like you can sit two bar magnets on a table at any orientation if they are far enough apart.
Close together, though, they resist all but two orientations. Give it a go.
There will be an in-between situation where the static friction of the table is just enough to prevent spontaneous rotation into a stable state... then you thump the table and everything snaps into place.
That should help you think about it. The quantum situation is a bit more complicated... in most materials, atomic magnetic moments prefer to be counter-aligned... but there are other forces at play.

Some other things to condider: in an atom by itself, electron magnetic moments are influenced by each other, by the orbital moment, and the nuclear magnetic moment... these fields impose a z direction. The up-down picture is a result of this. The SG experiment involves individual atoms where the applied field is non-uniform... imposing a more glibal z direction... the atomic z axis then orients either up or down wrt this.

 

[draconicsheep]

Given the explanation provided, then, wherein an incident field imposes a spin value in the z-direction, how can we know the domain orientation in x, y, z space of dipoles in a ferromagnetic material? Reading physics.stackexchange.com/questions/166566/… , uncertainty seems to suggest that we can't know the direction of electron spin in x, y, z space but only in 1 dimension at a time. But the existence of ferromagnetic domains with measurable directions relative to one another seems to violate that.

 

[Simon Bridge]

In magnetic materials, there are stable states where the individual magnetic moments line up ... this gives rise to the magnetic domains.
This does not violate the uncertainty in the x-y plane ... we are only measuring the z direction.

But recall - any attempt to measure the x-y components is the same as imposing a new z-direction and just calling it "x" or "y" or whatever. All that happens is the measurement makes the orthogonal components uncertain.
These are just labels: "z" points in any direction you decide.

When you read the theory - take care not to confuse several different uses for "z axis".