Geology Reference
In-Depth Information
very efficient carriers of remnant magnetization.
In fact, while the magnetization of MD grains
tends to decay with time, that of SD particles is
stable and can carry paleomagnetic information
over a long time interval (up to billion years).
When the size of magnetite grains is between
1 and 10 m, the particles are called pseudo -
single - domain (PSD) grains. They form a class of
grains exhibiting intermediate values of the ratio
M r / M s and the coercive field H c . These grains
contain a small number of domains and may carry
a substantial and stable remnant magnetization.
Grain size distributions of many igneous and
sedimentary rocks have a peak within the mag-
netite PSD field, although they have only a small
percentage of particles within the true SD field.
Therefore, PSD grains can be important carriers
of paleomagnetic information.
So far, we have been mainly concerned with
the phenomenological aspect of ferromagnetism,
after having mentioned that it arises from the cou-
pling of electron spins between neighbor atoms.
However, we have not yet explained the nature
of this interaction, and the different ways through
which the electron spins are coupled within a
crystal lattice. Understanding this subject will
require some basic quantum mechanics concepts,
because ultimately the magnetism of matter can-
not be explained by classical electrodynamics
(see e.g., Feynman et al. 2006 ). At first glance,
we could think that the alignment of magnetic
moments in ferromagnetic materials results from
their magnetic interaction. However, we are go-
ing to prove that this is not the case. Let us
consider the angular momentum of an electron
in an atom, which includes a component of or-
bital motion about the nucleus and one associ-
ated with the spinning about its own axis. These
components also originate magnetic moments,
one arising from the orbital motion, m ,andthe
other associated with spinning, which will be
indicated as m S . A famous theorem, due to Bohr
and van Leeuwen, shows that the orbital moment
m cannot produce a net magnetization, even in
presence of external field B ext . Thus, let us focus
on the spin moment m S .Let S be the intrinsic
angular momentum associated with the spinning
of the electron. Quantum mechanics shows that in
this case the relation between magnetic moment
and angular momentum is slightly different from
( 3.17 ), so that the ratio of m S to S is twice the ratio
of m to L :
e
m e S
m S D
(3.65)
Another difference with respect to classical
mechanics is that in the case of atomic systems
it is not possible to determine unambiguously the
absolute direction of S (or L ) at any given time
t . However, it is possible to show that at time t
the projection of S onto any arbitrary axis n can
assume only a finite number of values:
S n D .s k/ ¯I k D 0;1;:::;2s (3.66)
where the quantity s is called spin of the electron
(or, in general, of the particle) and the constant
- h D h /2  D 1.054571726(47) 10 34 [Js] is the
reduced Planck's constant . A similar relation can
be written for the orbital momentum L .There-
fore, at any given time there are only 2 s C 1
possible values for the component of S along an
arbitrary axis, for example the z -axis. Such dis-
cretization of the physical variables is one of the
consequences of quantum mechanics. Equation
( 3.66 ) indicates that the maximum magnitude of
the projection of S onto n is s - h , whereas classical
mechanics would give S . Therefore, we would
expect that s D S / - h . However, this conclusion
would be wrong. The average squared magnitude
of S along an arbitrary axis n is given by:
D .S n/ 2 E
2s
X
1
.s k/ 2
2
D
¯
2s
C
1
" s 2 .2s C 1/ C
k
D
0
2s 2 .2s C 1/ #
X
2s
2
D ¯
k 2
2s
C
1
kD0
X
2s
2
2sC1
C ¯
k 2
2 s 2
kD0
2sC1 6 2s.2s C 1/.4s C 1/
2
C ¯
2 s 2
1
2 s.s C 1/
D
3 ¯
(3.67)
 
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