PHY294H - Lecture 27

From atoms to magnetism

Magnetic materials are composed of magnetic moments that reside on the atoms of the magnetic materials. Each isolated atom has a magnetic moment that depends primarily on the number of electrons in its outer unfilled energy shells. These electrons have angular momentum and spin both of which contribute to the atomic magnetic moment. Inert atoms have very small magnetic moment while atoms like Iron have a large atomic magnetic moment. When atoms combine to form materials, their electrons are shared with other atoms and consequently the magnetic moment of each atom is changed. Calculation of the magnetic properties thus depends on the details of the atomic bonding in materials which is a quantum physics problem. Nevertheless we can learn a lot about the magnetism of materials from classical EM analysis, as we shall carry out below.

We shall concentrate on three types of magnetic materials: Ferromagnets(e.g. Iron, Permalloy), Paramagnets (e.g. Aluminum) and Diamagnets(e.g. copper, superconductors). Ferromagnets loose their ferromagnetism at the Curie temperature T_c, after which they become paramagnets. Paramagnets (and ferromagnets) are materials where the magnetic moments prefer to align with an applied magnetic field. Diamagnets are materials where the magnetic moments prefer to align opposite the applied magnetic field.

We want to quantitatively describe ferromagnetism, diamagnetism and paramagnetism of materials. To do this we need to introduce some new notation and some new variables. Once we have done this we will relate the new variables that we have introduced to the atomic scale magnetic moments.

Magnetization

We define ${\vec{M}}$ to be the magnetization, which is the magnetic moment per unit volume. (Remember that we already defined the magnetic moment due to a current loop to be $\mu = NIA$. However we are now going to use $\mu$ to be the permeability, so we will use m for the atomic magnetic moment.) From this we see that ${\vec{M}}$ has units of amps/meter. The magnetic field inside the material is given by $\vec{B} = \mu_0 \vec{M}$. If place this material in a magnetic field $\vec{B}_0$, then the field inside the material is given by,

$$\vec{B} = \vec{B}_0 + \mu_0 \vec{M}$$ (1)

We also define the magnetic intensity by,

$$\vec{H} = {1\over \mu_0} \vec{B} - \vec{M}$$ (2)

Or equivalently,

$$\vec{B}_0 = \mu_0 \vec{H}\ \ \ ;\ \ \ \vec{B} = \mu_0 \vec{H} + \mu_0 \vec{M}$$ (3)

The reason that we introduce the magnetic intensity $\vec{H}$ is that when we apply a magnetic field to a material, in most cases the magnetic moment changes linearly, that is,

$$\vec{M} = \chi_m \vec{H}$$ (4)

Note that $\chi_m$ is dimensionless. $\chi_m$ is the magnetic susceptibility and is the most basic magnetic properties of materials. Diamagnets like superconductors have negative $\chi_m$, while ferromagnets have large positive values of $\chi_m$. Using the relations (3) and (4) it is evident that,

$$\vec{B} = \vec{B}_0 + \mu_0 \vec{M} = \mu_0 \vec{H} + \mu_0 \chi_m \vec{H}$$ (5)

We thus have,

$$\vec{B} = \mu_0(1+\chi_m) \vec{H} = \mu \vec{H}$$ (6)

where $\mu$ is the permeability. This should not be confused with the magnetic moment for a current ring, which is also called $\vec{\mu}$. This is horrible notation, but it is entrenched in the area. We shall have to live with it.

Example: Magnetic field enhancement in a solenoid

For a solenoid with n turns per unit length and carrying current I, we found,

$$B_0 = \mu_0nI$$ (7)

This is the result for a solenoid with an If we place a material inside the solenoid, we have,

$$B = \mu n I = \mu_0(1+\chi_m) n I$$ (8)

From this expression it is evident that the magnetic field inside the solenoid is greatly enhanced if the center (the core) of the solenoid is composed of a magnetic material which has large magnetic susceptibility $\chi_m$, for example permalloy.

Atoms as magnetic dipoles

The magnetic properties of atoms comes from a combination of the orbital motion of the electrons about the nucleus and the intrinsic spin of electrons, protons and neutrons. We can understand at a qualititative level the orbital part using Bohr's model of the atom, however understanding magnetism and in particular spin requires a knowledge of quantum mechanics, so we shall just state the results in that case.

Orbital magnetic moment of the Bohr atom

If a single electron moves in a circular orbit at speed v and with radius r, then the current is

$$I = {e\over 2\pi r} v$$ (9)

The magnetic moment, m_orbital (note the change in notation) is then,

$$m_{orbital} = I \pi r^2 = - {1\over 2} e v r = -{e\over 2m_e} L$$ (10)

where L = m_evr is the angular momentum of the electron. This relation is usually written as,

$$\vec{m}_{orbital} = g_L \vec{L}$$ (11)

where $g_L = -{e\over 2m_e}$ is called the gyromagnetic ratio.