Answer to Question #103179 in Electricity and Magnetism for Ajay

Question

Why do magnetic moments in ferromagnetic materials, tend to align parallel to each other?
Establish the relation B = μ0(H + M), for a ferromagnetic material.

Answer 1

The descriptive theory of ferromagnetism was developed by the P. Weiss (1865-1940). Sequential quantitative theory based on quantum mechanics was developed by V. Heisenberg (1901-1976).

According to Weiss, ferromagnets at temperatures below the Curie point have spontaneous magnetization regardless of the presence of an external magnetizing field. Spontaneous magnetization, however, is in apparent contradiction to the fact that many ferromagnetic materials are not magnetized even at temperatures below the Curie point. To eliminate this contradiction, Weiss introduced the hypothesis that a ferromagnet below the Curie point is divided into a large number of small macroscopic regions-domains that are spontaneously magnetized to saturation.

It is now established that the magnetic properties of ferromagnets are determined by the spin magnetic moments of electrons. It is also established that only crystalline substances with incomplete internal electron shells with uncompensated spins can have ferromagnetic properties. In such crystals (Fe, Co, Ni - d elements), forces can occur that force the spin magnetic moments of the electrons to Orient themselves parallel to each other, which leads to the appearance of spontaneous magnetization regions. These forces, called exchange forces [1], are of a quantum nature-they are caused by the wave properties of electrons.

Ferromagnetic materials [2] spontaneously divide into magnetic domains because the exchange interaction is a short-range force, so over long distances of many atoms the tendency of the magnetic dipoles to reduce their energy by orienting in opposite directions wins out. If all the dipoles in a piece of ferromagnetic material are aligned parallel, it creates a large magnetic field extending into the space around it. This contains a lot of magnetostatic energy. The material can reduce this energy by splitting into many domains pointing in different directions, so the magnetic field is confined to small local fields in the material, reducing the volume of the field.

The vector $\vec B$ the induction of the resulting magnetic field in a ferromagnet is equal to the vector sum of the magnetic inductions of the external (magnetizing) $\vec B_0$ and internal fields $\vec B_{int}$.

$\vec B=\vec B_0+\vec B_{int}$

The external magnetic inductions is related to the magnetic field in a vacuum by a simple ratio (SI base unit)

$\vec B_0=\mu_0\cdot \vec H$

The internal fields magnetic inductions is related to the the vector of magnetization $\vec M$ of material

$\vec B_{int}=\mu_0 \cdot \vec M$.

Thus we have $\vec B=\mu_0\cdot(\vec H+\vec M)$

[1] Terunobu Miyazaki, Hanmin Jin: "The Physics of Ferromagnetism" Springer-Verlag, Berlin 2012, p.61

[2] https://en.wikipedia.org/wiki/Ferromagnetism