Page 241 - Electromagnetics

In most paramagnetic materials the alignment of the permanent moment of neigh-
boring atoms is random. However, in the subsets of paramagnetic materials known as
ferromagnetic, anti-ferromagnetic, and ferrimagnetic materials, there is a strong coupling
between the spin moments of neighboring atoms resulting in either parallel or antiparal-
lel alignment of moments. The most familiar case is the parallel alignment of moments
within the domains of ferromagnetic permanent magnets made of iron, nickel, and cobalt.
Anti-ferromagnetic materials, such as chromium and manganese, have strongly coupled
moments that alternate in direction between small domains, resulting in zero net mag-
netic moment. Ferrimagnetic materials also have alternating moments, but these are
unequal and thus do not cancel completely.
Ferrites form a particularly useful subgroup of ferrimagnetic materials. They were first
developed during the 1940s by researchers at the Phillips Laboratories as low-loss mag-
netic media for supporting electromagnetic waves [65]. Typically, ferrites have conduc-
7
0
tivities ranging from 10 −4 to 10 S/m (compared to 10 for iron), relative permeabilities
in the thousands, and dielectric constants in the range 10–15. Their low loss makes them
useful for constructing transformer cores and for a variety of microwave applications.
Their chemical formula is XO · Fe 2 O 3 , where X is a divalent metal or mixture of metals,
such as cadmium, copper, iron, or zinc. When exposed to static magnetic fields, ferrites
exhibit gyrotropic magnetic (or gyromagnetic) properties and have permeability matrices
of the form (2.32). The properties of a wide variety of ferrites are given by von Aulock
[204].
To determine the permeability matrix of a ferrite we will model its electrons as simple
spinning tops and examine the torque exerted on the magnetic moment by the application
of an external field. Each electron has an angular momentum L and a magnetic dipole
moment m, with these two vectors anti-parallel:

m(r, t) =−γ L(r, t)
where
q e 11
γ = = 1.7592 × 10 C/kg
m e
is called the gyromagnetic ratio.
Let us first consider a single spinning electron immersed in an applied static magnetic
field B 0 . Any torque applied to the electron results in a change of angular momentum as
given by Newton’s second law
dL(r, t)
T(r, t) = .
dt
We found in (3.179) that a very small loop of current in a magnetic field experiences
a torque m × B. Thus, when first placed into a static magnetic field B 0 an electron’s
angular momentum obeys the equation
dL(r, t)
=−γ L(r, t) × B 0 (r) = ω 0 (r) × L(r, t) (4.108)
dt
where ω 0 = γ B 0 . This equation of motion describes the precession of the electron spin
axis about the direction of the applied field, which is analogous to the precession of a
gyroscope [129]. The spin axis rotates at the Larmor precessional frequency ω 0 = γ B 0 =
γµ 0 H 0 .
We can use this to understand what happens when we insert a homogeneous ferrite
material into a uniform static magnetic field B 0 = µ 0 H 0 . The internal field H i experienced
by any magnetic dipole is not the same as the external field H 0 , and need not even be in

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