Fundamentals of Electromagnetics - RF/Microwave Interaction with Biological Tissues - page 14

which combines the applied and induced fields, hence the external source field
and the induced magnetization. The permeability of the material is defined by

( ) =

mm = 1 + c mm H m (1.25)

- 1

0 d 0 r

It is the product of the permeability of vacuum m 0 (in henrys per meter) by
the relative permeability m r (dimensionless) of the material. Equation (1.25)
defines at the same time the magnetic susceptibility c d . The use of permeabil-
ity, relative permeability, and magnetic susceptibility is limited to isotropy, loss-
lessness, and linearity, which is far from being always the case. It should be
stressed again, however, that biological tissues are essentially nonmagnetic.
Magnetization is a very complicated phenomenon [3, 4-7]. It may be due to a
variety of mechanisms, which can be summarized here only briefly.
All materials are diamagnetic: Diamagnetism is a general property of matter.
It is due to the fact that electrons placed in a magnetic field have their rotation
speed increased, which results in an induced magnetic moment opposed to the
applied magnetic field, hence in a decrease of magnetization. The diamagnetic
susceptibility is negative and very small, normally of the order of -10 -8 to
-10 -5 . It does not depend explicitly upon the temperature; it depends however
upon the density of the material. A number of metals used in engineering are
only diamagnetic, for instance, copper, zinc, gold, silver, cadmium, mercury, and
lead. As an example, the relative permeability of copper is 0.999991.
When the atoms of a material have a permanent magnetic moment, this
moment tends to align itself with the applied magnetic field to minimize the
magnetic energy. This phenomenon is similar to electric polarization. It is a
rather slow phenomenon, correctly described by a first-order law. It charac-
terizes paramagnetism , in which the magnetization is positive: It reinforces the
effect of the applied field. It should be noted that paramagnetic materials are
also diamagnetic. Diamagnetism, however, is much smaller than paramagnet-
ism and is not observable in this case. Paramagnetic magnetization is modeled
similarly to electric polarization. It depends upon temperature and density.
The paramagnetic susceptibility is positive and, in general, much smaller
than 1. Some metals are paramagnetic, for example, aluminum, platinum,
manganese, magnesium, and chromium. As for electric polarization, when the
material density increases, modeling becomes much more difficult, and classi-
cal physics yields wrong models. Classical physics almost completely fails when
trying to establish quantitative models. It can, however, yield some very illu-
minating insight on the phenomena involved with the dielectric character of
materials, in particular about the influence of frequency.
The relative permeability related to this phenomenon is

mm m
r

=¢- ¢ j (1.26)

r r

It is a complex quantity, with real and imaginary parts. The imaginary part is
a measure of the magnetic losses. These are often expressed also as the tangent
of the loss angle :