I Direction of motion of particles with total spin under magnetic field

[sal1854]

TL;DR Summary

The magnetization force imposes paramagnetic materials to move in one direction. What about the fact that their unpaired electrons can have a spin up or down? Shouldn't they be able to move in both directions depending on the spin?

According to Chapter 8 of Griffiths' book Introduction to Electrodynamics, the magnetization force that acts on a magnetic dipole is

$$F_M=\nabla (m \cdot B)$$,

where $m$ is the magnetic moment and $B$ is the magnetic field.

For a paramagnetic or diamagnetic particle

$$m=\dfrac{\chi}{(1+\chi)\mu_0}B$$

where $\chi$ is the magnetic susceptibility (also shown in this wiki page [1]).

Therefore, if a particle is paramagnetic ($\chi>0$) the $F_M$ acting on it will be in the direction of the $\nabla B^2$ (and the opposite direction for a diamagnetic one)! So, all paramagnetic materials will move in the same direction?

What about the fact that the unpaired electrons of said paramagnetic particle can have a "spin-up" or "spin-down" (Stern–Gerlach experiment [2])? Shouldn't then the paramagnetic particles move randomly both "up" or "down" under the magnetization force??

Links:
[1]: https://en.wikipedia.org/wiki/Magnetization
[2]: https://en.wikipedia.org/wiki/Stern–Gerlach_experiment

 

[DrClaude]

For a paramagnetic or diamagnetic particle

$$m=\dfrac{\chi}{(1+\chi)\mu_0}B$$

where $\chi$ is the magnetic susceptibility (also shown in this wiki page [1]).

That's only true for particles without a permanent magnetic moment. What you get then is a field-induced magnetic moment.

If a particle has a permanent magnetic moment, then usually only this moment need be considered (unless the external field is very strong).