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I am now playing with a permanent magnet made of Neodymium.

It is impressively strong.

A question is, is the magnetization direction of the magnet fixed relative to its crystal structure?

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The neodymium magnet (Nd$_2$Fe$_{14}$B) has a tetragonal crystal structure with the $c$ axis much longer than the $a$ and $b$ axes. I did google for an image of the unit cell but couldn't find a nice one - you may have more luck. Anyhow, the crystal can be magnetised in any direction, but it is much more easily magnetised in the direction of the $c$ axis and a much higher magnetic field can be produced.

The magnets are polycrystalline, and they're made by pressing powdered Nd$_2$Fe$_{14}$B into the required shape and sintering it. However when the magnet is being pressed an external magnet field is used to make the $c$ axes of the individual crystals all line up the same way. So even though the magnet is polycrystalline the crystal domains within it have their $c$ axes lined up (mostly) in the same direction, and the magnet behaves like a single large crystal. So while the magnet can be magnetised in any direction it is much more easily magnetised, and to a much higher field strength, along the axis that the $c$ axes are aligned with.

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what is the interaction making the c-axis easier axis? – Jiang-min Zhang Jun 25 '14 at 11:35
The technology of pressing the powder under a magnetic field is the reason why do not try to throw the magnet hard. It will "explode". – HolgerFiedler Jun 25 '14 at 17:53

Practically yes, but the true answer is a little bit complexer.

Below the level of the crytals, there are the magnetic domains. In a domain have the atoms the same orientation (and thus, the same magnetic dipole moment).

enter image description here

In the case of a magnetized ferromagnetic material, these magnetic pole of these domains are directing to the same (or to the nearly same) direction.

To change the direction of the magnetic field of the ferromagnetic material, you need to apply so strong magnetic field, which is enough to "redirect" the orientation of the atoms. Of course, it is much easies, if there is an elevated temperature as well, because it also softens the metallic bonds between the atoms.

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Okay, let us focus on a single domain. Is the magnetization direction fixed relative to the lattice structure of the domain? I suspect yes. Because the property of a crystal is anisotropy. – Jiang-min Zhang Jun 25 '14 at 11:15
@Jiang-minZhang Yes, it is. – peterh Jun 25 '14 at 11:21
@ Peter Horvath. If it is so, then another question arises. What is the source of the magnetization? Spins or the orbital motion of the electrons? If it is the spin, then there must be some spin-orbital coupling. Otherwise, the magnetization direction can be arbitrary. – Jiang-min Zhang Jun 25 '14 at 11:27
@Jiang-minZhang Afaik, there need to be an unpaired electron spin with some spin-orbital coupling thing. There is a lot of pshysicists here, I suggest to you to make a new question - it will probably get a lot of upvotes. – peterh Jun 25 '14 at 11:37

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