13. Ch. 5, part 3

Lecture 13 - Magnetism in matter

Magnetization in matter varies on a microscopic scale, because of electron currents in atomic and molecular orbitals and because of the magnetization density associated with intrinsic magnetic moment of the electrons (and also of nuclei, but the effective nuclear magneton is smaller than |e|h̄/2M_{e}c by a factor of about 2000). The separation of currents into “microscopic” currents that give rise to M and “macroscopic” currents that are caused by batteries (or by varying magnetic fields) is not always clear-cut. We will assume that such a separation is possible, and consider only simple cases.

13.1. Macroscopic equations

Associated with the magnetization density M there is a vector potential

A(x) = ∫ \frac{M(x′)× (x − x′)}{|x − x′|^{3}} d^{3}x′ = ∫ M(x′) × ∇′ \frac{1}{|x − x′|} d^{3}x′

Integrating by parts, this becomes

A(x) = ∫ \frac{∇′ × M(x′)}{|x − x′|} d^{3}x′

showing that there is an effective magnetization current

J_{M} = c∇ × M

Then the B field obeys the equations

∇ · B = 0
∇ × B = \frac{4π}{c} J + 4π∇ × M

The macroscopic equations are obtained by a suitable average of these, and have the same form. From now on in this lecture, M and B are averaged quantities. It is also convenient to introduce the field

H = B − 4πM

so that the macroscopic equations become

∇ · B = 0
∇ × H = \frac{4π}{c} J

in analogy with

∇ · D = 4πρ
∇ × E = 0

H is called the magnetic field and B is called the magnetic flux density. The trouble with these names is that they suggest that the fundamental field is H, in analogy to electrostatics, where the fundamental field is E, while actually the fundamental field is B.

As in the electrical case, the boundary conditions at the interface of two media are that B_{n} is continuous and H_{∥} is continuous in the absence of a surface current density.

A major difference from the electric case is that many materials exhibit a spontaneous magnetization and do not have a simple local relation between M and H that is analogous to P = χ^{(e)}E. If there is a relation

M = χ^{(m)}H

where χ^{(m)} is the magnetic susceptibility, then one defines the magnetic permeability μ = 1 + 4πχ^{(m)} and obtains

B = μH

in analogy with D = ϵE. Because this relation makes life simple, one tries to use it as much as possible, allowing μ to depend on H and even on the history of the sample.

13.2. Magnetic susceptibility

There are many types of magnetic behavior in matter. Strictly speaking, none of them can be discussed classically, because one can show that classically, there is no magnetism at thermal equilibrium. This theorem comes from the basic fact that the magnetic force on a moving particle of charge q and velocity v, the Lorentz force

F = \frac{q}{c} v × B

is such that F·v = 0. Thus the energy of any particle is not changed when a magnetic field is switched on and the thermal distribution, which depends only on energy, is unchanged.

More formally, the Hamiltonian of a particle of mass M, classically or quantally, is

H = \frac{1}{2M} (p − \frac{q}{c}A(x))^{2} + V(x)

All thermodynamic results are derivable from the partition function Z. Classically

Z_{cl} = ∫ d^{3}p d^{3}x exp(−H/k_{B}T)

The integral over momenta can be performed by changing variable from p to p−qA(x)/c, leaving a Z_{cl} that does not depend on A:

Z_{cl} = (2πMk_{B}T)^{3/2} ∫ d^{3}x exp (−V(x)/k_{B}T)

Quantum theory is essential to describe magnetism in the real world. To see how things work out, we take a constant field in the z direction. The vector potential depends on the gauge. A popular gauge choice in quantum theory is the Landau gauge, where

A_{x} = − By A_{y} = 0 A_{z} = 0

but in our case it is more elegant to use the symmetrical gauge, where

A =\frac{1}{2} B × x

Then

H = \frac{p^{2}}{2M} − \frac{q}{2Mc} B · x × p + \frac{q^{2}B^{2}}{8Mc^{2}} (x^{2} + y^{2}) + V(x)

Let us consider separately the two field-dependent terms in H, first for atoms, then for molecules and solids

13.2.1. Diamagnetism and paramagnetism of atoms

The diamagnetic term

\frac{q^{2}B^{2}}{8Mc^{2}} (x^{2} + y^{2})

always raises the energy. For atoms or ions

⟨x^{2} + y^{2}⟩ = \frac{2}{3} ⟨r^{2}⟩

is almost temperature independent. The energy change for an atom or ion containing Z electrons is then

\frac{Zq^{2}B^{2}}{12Mc^{2}} ⟨r^{2}⟩

if we understand that

⟨r^{2}⟩ = \frac{∫ r^{2}ρ(x) d^{3}x}{∫ ρ(x) d^{3}x} = \frac{∫ r^{2}ρ(x) d^{3}x}{Z}

Equating the energy change to −\frac{1}{2}B· m^{(dia)} we obtain the induced dipole moment

m^{(dia)} = −\frac{Ze^{2} ⟨r^{2}⟩}{6M_{e}c^{2}} B

We should use B_{local} instead of B, but the difference turns out to be insignificant. With N atoms per unit volume, the diamagnetic susceptibility is then

χ^{(dia)} = −\frac{NZe^{2} ⟨r^{2}⟩}{6M_{e}c^{2}}

For atomic hydrogen ⟨r^{2}⟩ = 3a_{0}^{2} and recalling χ^{(el)} = (9/2)a_{0}^{3} we find −χ^{(dia)}/χ^{(el)} = (1/9)e^{2}/M_{e}c^{2}a_{0} = (1/9) (e^{2}/h̄c)^{2} ≃ 5.92·10^{−6}. The ratio e^{2}/M_{e}c^{2}a_{0} is also equal to ⟨v^{2}⟩ /c^{2}, in agreement with the earlier remark that magnetism is a relativistic effect. In general, the magnitude of χ^{(dia)} is Z · 10^{−6} per mole.

The paramagnetic term can be written in various ways:

−\frac{q}{2Mc} B · x × p = − \frac{q}{2Mc} B · L = −\frac{q}{2Mc} B(xp_{y} − yp_{x})

Classically, we compute Z_{cl} by integrating over the direction and the magnitude of L. This is equivalent to integrating over all values of p and gives a result that does not depend on B, as one can double-check. In the real world, both the direction and the magnitude of L are quantized, and there is also the spin. The full paramagnetic term is

−\frac{e}{2M_{e}c} B· (L + g_{0} S)

for each electron, where g_{0} = 2 · 1.00116. The individual electron’s L and S combine to form a total L and S, and these in turn add up to a total angular momentum J = L + S (not to be confused with the current). In an atom or ion only one value of J contributes significantly to the quantum partition function Z, and to first order in B the paramagnetic interaction is (not obviously) equivalent to

−\frac{eg}{2M_{e}c}B · J

where, taking g_{0} = 2, the Landé g factor is given by

g = \frac{3}{2} + \frac{1}{2} [\frac{S(S + 1) − L(L + 1)}{J(J + 1)}]

In effect, we are back to a magnetic moment

μ = \frac{eg}{2M_{e}c} J

interacting with the field. For small values of μH/k_{B}T, we can treat the problem classically and we find, as in the electric case,

⟨μ_{z}⟩ = \frac{μ^{2}}{3k_{B}T} H

This leads to Curie’s law for the paramagnetic susceptibility

χ^{(para)} = \frac{NJ(J + 1)}{3k_{B}T} (\frac{egh̄}{2M_{e}c})^{2} = \frac{C}{T}  (13.1)

For atomic hydrogen g = 2, J = S = \frac{1}{2}h̄ and

\frac{χ^{(para)}}{−χ^{(dia)}} = \frac{\frac{N}{4k_{B}T} (\frac{eh̄}{M_{e}c})2}{\frac{Ne^{2}a_{0}^{2}}{2M_{e}c^{2}}} = \frac{e^{2}/2a_{0}}{k_{B}T}

where of course e^{2}/2a_{0} = 13.6 eV is the ground state energy, and k_{B}T ≃ 1/40 at room temperature. In general χ^{(para)} is larger than χ^{(dia)} by roughly the ratio of the ground state energy to k_{B}T, or about 500 at room temperature. Thus when J ≠ 0 gases and insulators are paramagnetic, but paramagnetism is a weak effect. At low T or high field the magnetization is a non-linear function of H and “saturates” at M_{sat} = Nμ, as discussed more fully below.

13.2.2. Molecules and solids

If the electron orbitals are localized on atoms or ions, the above treatment applies also to molecules, liquids and solids considered as assemblies of independent atoms. When J ≠ 0 the substance is paramagnetic, but paramagnetism is weak. Even for a solid with N = 10^{23}cm^{−3} the order of magnitude of χ^{(para)} is 1/(40T).

For extended orbitals, things get complicated. Small molecular orbitals have properties similar to those of atoms. In ordinary insulators and semiconductors magnetic effects are weak because the presence of an energy gap “freezes” out the motions of the valence electrons and the total spin is zero.

At the opposite limit are metals, where many electrons are free to move in cyclotron orbits, more or less. These orbits set up a large circulating current, which however is largely canceled by “skipping orbits” around the boundary. The cancellation is exact (by the general theorem) if the electrons can be treated classically, but there are quantum effects at temperatures below the Fermi temperature T_{F}, which is usually thousands of degrees. The net result is a weak Landau diamagnetism that generally increases with H but shows characteristic oscillations. In addition there is the Pauli spin paramagnetism which is given essentially by Curie’s law ( 13.1) with T_{F} in place of T. Thus it is temperatureindependent and weak, χ^{P auli} ≃ 10^{−6}.

13.2.3. Ferromagnetism

It would appear from the above that magnetism in matter is always weak, but this is not true. Electrostatic forces between electrons, coupled with the Pauli exclusion principle, can favor a parallel alignment of the spins of nearby atoms. This exchange interaction is described by −J μ_{i} · μ_{j} for two nearby spins. When all the spins are aligned, it causes an effective molecular field N_{n}J μ to act on each spin. Here N_{n} is the number of nearest neighbors. When the spins are only partially aligned, the mean molecular field is N_{n}J ⟨μ_{z}⟩ , where ⟨μ_{z}⟩ is computed from the self-consistent equation

⟨μ_{z}⟩ = \frac{μ^{2}}{3k_{B}T} (H + N_{n}J ⟨μ_{z}⟩)

This equation leads to the Curie-Weiss law

χ^{(m)} = \frac{C}{T − T_{c}}

showing that the susceptibility blows up at a critical temperature T_{c}. For T < T_{c} spontaneous magnetization occurs.

A spontaneously magnetized piece of material may or may not have a net overall magnetic moment. Generally the direction of magnetization varies in the different magnetic domains within the material. In a soft ferromagnet the domains reorient easily and there is still effectively a linear relation between B and H with a very large effective χ. In a hard ferromagnet the domains do not reorient easily and the phenomenon of hysterisis occurs.