Abstract
This is an overview of what the Landau-Lifshitz-Gilbert equations are doing in soft-magnetic materials with dimensions large compared to the exchange length. The surface magnetic charges try to cancel applied magnetic fields inside the soft magnetic material. The exchange energy tries to reach a minimum while meeting the boundary conditions set by the magnetic charges by using magnetization patterns that have a curl but no divergence. It can almost do this, but it still pays to add some divergence to further lower the exchange energy. There are then both positively and negatively charged regions in the bulk. The unlike charges attract one another, but do not annihilate because they are paid for by the reduction in exchange energy. The micromagnetics of soft magnetic materials is about how those charges rearrange themselves. The topology of magnetic charge distributions presents challenges for mathematicians. No one guessed that they like to form helical patterns of extended multiples of charge density.
Highlights
There are two types of curls: bends and twists. This is about the physics of unit vector fields
It is that m is a unit vector that allows one to express the exchange energy, which is the sum of the squares of first derivatives, by second derivatives using the vector Laplacian
Magnetic charges aid in the understanding of the solutions to the Landau–Lifshitz–Gilbert (LLG) equations of micromagnetism
Summary
There are two types of curls: bends and twists (think of vortices and Bloch domain walls). This is about the physics of unit vector fields. It is that m is a unit vector that allows one to express the exchange energy, which is the sum of the squares of first derivatives, by second derivatives using the vector Laplacian. Thinking in terms of curls and divergences has been helpful in visualizing three-dimensional magnetism.
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