Why do compasses always align with the magnetic field, when the spin of individual particles only align half of the time?

Question

If I have an electron orbiting a nucleus and I apply a magnetic field there will be two effects:

• Larmor precession given by the cross product of the magnetic moment and the field of the experiment

$$ \vec{\tau}=\vec{\mu} \times \vec{B} $$

• An attractive or repulsive force given by the dot product of the magnetic moment and the field of the experiment

$$ F = -\nabla(-\vec{\mu} \cdot \vec{B}) $$

Fine, I understand this

But then, when I put a non-magnetic piece of metal in a magnetic field it is always results in an attractive force. Why?

Given that the alignment of the atoms and their electrons is random, the dot product of the magnetic moment of each atom with the field is just as likely to be positive or negative, isn't it? In fact that's exactly what we see in the in things like the Stern-Gerlach experiment, and yet compasses and big pieces of metal are always attracted by magnetic fields

I understand that this is a symmetry break in the ground level, the electrons in the non-magnetic metal can have less energy by being aligned with the field and thus being attracted by it, which makes that configuration more likely and almost guaranteed after a short period of time, but why? When I try to calculate the energy I don't see why being aligned with the field gives them less energy than being anti-aligned. Both configurations seem to be identical in terms of energy

Answer 1

For a paramagnetic metal, the magnetic field polarizes the atoms in the material and, for $\mu>1$, will attract them. The neutron has an intrinsic, permanent magnet moment, which means that, whether it is attracted or repelled depends on its orientation.