## Introduction to Magnetism

All forms of magnetism have a quantum mechanical origin. Although classical effects like Lenz's law are sometimes invoked in a handwaving discussion of magnetism, the Bohr-van Leeuwen theorem shows that magnetism is strickly a quantum mechanical effect. A good historical overview of magnetism can be found in John H. Van Vleck's Nobel lecture, Quantum Mechanics: The Key to Understanding Magnetism (Nobel Prize 1977). Quantum mechanics is difficult and many body quantum mechanics, which is necessary to understand ferromagnetism, is more difficult.

Some materials, like iron, exhibit a spontaneous magnetization in the absence of an applied magnetic field. However, all materials show an induced magnetization when a magnetic field is applied. The magnetic induction field $\vec{B}$ is,

$$\vec{B}=\mu_{0}\left(\vec{H}+\vec{M}\right)$$

where $\vec{H}$ is the magnetic intensity and $\vec{M}$ is the magnetization. The magnetic intensity is the same inside the material and outside the material while the magnetic induction field $\vec{B}$ changes as it enters a material. Typically, the quantity under experimental control is the magnetic intensity $\vec{H}$. For materials that do not exhibit a spontaneous magnetization, the response to small applied $\vec{H}$ fields is generally expressed in terms of the magnetic susceptibility $\chi_M$,

$$\vec{M} = \chi_M\vec{H}.$$

The magnetic induction field be written as,

$$\vec{B}=\mu_{0}\left(\vec{H}+\vec{M}\right)=\mu_{0}\left(1+\chi_M\right)\vec{H}= \mu_{r}\mu_{0}\vec{H},$$

where $\mu_{r} = 1+\chi_M$ is the relative permeability.

For anisotropic materials and higher applied fields, the magnetization is written as a Taylor expansion of the $\vec{H}$ field in Einstein notation as,

$$M_i= \chi_{ij}H_j+\chi^{(2)}_{ijk}H_jH_k+\chi^{(3)}_{ijkl}H_jH_kH_l+\cdots$$

Here $\chi_{ij}$ is the linear magnetic susceptibility, and $\chi^{(2)}_{ijk}$ is a rank-3 tensor that describes the quadratic part of the nonlinear response.

Materials that have no spontaneous magnetization (their magnetization is zero when no external magnetic field is applied) are called diamagnetic if the linear suceptibility is negative and paramagnetic if the linear susceptibility is positive.