Geology Reference
In-Depth Information
materials, the atoms are packed within the crystal
lattice in such a way that highly eccentric orbital
shells of adjacent atoms overlap. In this instance,
the electrons that move across these shells will
be forced to satisfy simultaneously the exclu-
sion principle of
both
atoms. This phenomenon
can be described intuitively by the so-called
ex-
change interaction
, which was proposed in 1928
by Heisenberg to explain the very large electro-
magnetic fields that form in ferromagnetic mate-
rials. Let us consider two atoms with unpaired
electrons and assume that they get close each
other. By Pauli's principle, if the spins of the two
electrons align antiparallel to each other, these
electrons will be able to share a common orbital
shell, and this event would
increase
the electro-
static Coulomb energy. Conversely, if the spins
align parallel to each other, Pauli's exclusion
principle will prevent the formation of a shared
orbit, so that the two electrons would move apart
along separate orbits, thus
reducing
the normal
Coulomb interaction. The latter solution is clearly
favoured
by nature. The order of magnitude of
the Coulomb energy that would be required by a
shared orbit is given by:
m
S
a
3
kT
c
U
m;max
D
n
0
4
(3.74)
where
n
is the number of nearest in-plane neigh-
bors. From (
3.65
), (
3.68
), and (
3.72
) it results:
2
s
2
.s
C
1/
2
m
e
e
2
S
2
m
e
D
e
2
¯
m
S
D
D
B
(3.75)
Therefore,
B
a
3
kT
c
U
m;max
D
n
0
4
(3.76)
Taking
n
D
4and
a
D
0.8393 nm (as appropri-
ate for magnetite crystals), we get
T
c
0.0042 K.
However, actually magnetite becomes ferromag-
netic at
858 K! Clearly, the dipole-dipole inter-
action is too weak to explain the ferromagnetic
behavior of this mineral. It can only be a correc-
tiontotheactual
exchange energy
B
B
kT
c
,
and in theory represents a source of anisotropy.
Ironically, the ultimate source of the strongest
form of magnetism results to be a combination of
Coulomb (i.e., electric,
not
magnetic) interaction
between neighbor electrons and the necessity for
these particles to satisfy the
Pauli exclusion prin-
ciple
of quantum physics. This principle states
that in an atomic system the
quantum state
of
an electron, which is specified by a set of four
quantum numbers
, must be different from any
other electron in the system. The four quantum
numbers are:
The
principal quantum number
,
n
, which de-
fines the
size
of the shell where an electron
moves;
The
orbital quantum number
,
l
, associated
with the angular momentum, which deter-
mines the
shape
of the shell;
The
orbital magnetic quantum number
,
m
,
associated with the orbital magnetic moment;
The
spin magnetic quantum number
,
m
S
,
which is associated with the spin magnetic
moment
Therefore, as m
S
D˙
1=2, given a triplet of
quantum numbers (
n
,
l
,
m
), we can find at most
two electrons in an atomic system having these
quantum numbers and opposite spins. In some
e
2
4 ©
0
r
U
e
D
(3.77)
where
r
is the average distance between the two
electrons and the constant "
0
D
8.8542
10
12
C
2
N
1
m
2
is the
free space permittivity
.Taking
r
D
1Ågives:
e
2
4 ©
0
kr
D
1:67
10
5
K
T
c
(3.78)
This value is 10
5
times larger than the mag-
netic dipole interaction calculated from (
3.76
)us-
ing the same distance. Therefore, if the Coulomb
energy is, on average, reduced by the Pauli ex-
clusion principle to a small fraction of the value
required by (
3.77
), say 0.5 %, the variation of
electrostatic energy after the formation of the new
atomic system will give a Curie temperature of
835 K, which can explain the magnitude of the
molecular field. Thus, the parallel alignment of
electron spins in ferromagnetic materials results
