Magnetic anisotropy - easy directions of magnetization

[klopjob]

Homework Statement

We have a very thin plate of single crystalline Ni (fcc). The plane of the plate corresponds to the (001)-plane of the crystal. The crystal has a quadratic shape and is cut so that the edges corresponds to the crystallographic [110]-directions. The shape anisotropy makes the domain magnetization lie in the (001)-plane. The first shape anisotropy constant K₁ is negative. Which are the easy directions of magnetization in the crystal?

The Attempt at a Solution

The key sheet says: Since K₁ is negative, it implies the easy directions are of type [110]. The literature and http://en.wikipedia.org/wiki/Magnetocrystalline_anisotropy" on the other hand clearly says the easy directions are of type [100] for K₁>0 and [111] for K₁<0 if the second anisotropy constant K₂ is assumed to be zero. Then why isn't [111] the easy direction in this case? Is it simply because it doesn't lie in the (001)-plane? How do we go from here to say that the [110] types are the easy direction, is it a coincidence this is the same as the crystal edges? Furthermore, can we really say these are the easy and not 'medium' directions?

 

[mark726]

Thank you for your post. I would like to provide some clarification on the concept of magnetocrystalline anisotropy and its relation to the crystal structure and shape anisotropy.

Magnetocrystalline anisotropy refers to the preferred direction of magnetization in a crystal due to its atomic structure. This anisotropy is dependent on the crystallographic direction and can be described by the first and second shape anisotropy constants, K1 and K2, respectively.

In the case of a thin plate of single crystalline Ni with a quadratic shape, the plane of the plate corresponds to the (001)-plane of the crystal. This means that the crystallographic [001]-direction is perpendicular to the plane of the plate. The shape anisotropy, which is dependent on the shape of the crystal, causes the domain magnetization to lie in the (001)-plane.

Now, to determine the easy directions of magnetization in the crystal, we need to consider the sign of the first shape anisotropy constant, K1. As you correctly pointed out, for K1>0, the easy directions are of type [100], and for K1<0, the easy directions are of type [111]. However, in this case, K1 is negative, which means that the easy directions are of type [110]. This is because the [110]-directions lie in the (001)-plane and are also perpendicular to the crystal edges, as stated in the problem.

To answer your question, it is not a coincidence that the [110]-directions are the easy directions in this case. This is because the shape anisotropy, which is dependent on the shape of the crystal, causes the domain magnetization to preferentially align along these directions. And yes, we can say that these are the easy directions and not the medium directions, as the easy directions have the lowest energy state for the magnetization.

I hope this helps clarify the concept of magnetocrystalline anisotropy and its relation to the crystal structure and shape anisotropy. If you have any further questions, please don't hesitate to ask.