Geology Reference
In-Depth Information
absolute temperature of the body, and
M
s
is the
saturation value
of magnetization. The function
L
in (
3.52
) is called
Langevin function
and tends
asymptotically to unity, so that
M
!
M
s
as
Ÿ
!1
. The existence of a saturation magnetiza-
tion indicates that the induced magnetization can-
not increase arbitrarily as we raise the magnitude
of the external field or decrease the temperature.
Given an atomic magnetic moment
m
,thereisal-
ways a maximum value of magnetization that can
be attained by the substance, which corresponds
to a perfect alignment of the elementary atomic
moments.
The shape of the Langevin function is illus-
trated in Fig.
3.10
.If
n
indicates the density of
magnetic dipoles, then:
the nucleous. Therefore,
a
2
D
(2/3)
h
r
2
i
and (
3.48
)
can be rewritten as follows:
e
2
ǝ
r
2
Ǜ
6m
e
B
ext
•m
D
(3.49)
The total magnetization can be calculated us-
ing the definition (
3.38
). If
N
is the total number
of atoms and
Z
is the average number of electrons
per atom, we have:
V
X
i
NZe
2
6m
e
V
ǝ
r
2
Ǜ
B
ext
1
M
D
m
i
D
6m
e
ǝ
r
2
Ǜ
B
ext
ne
2
D
(3.50)
where
n
D
NZ/V
is the density of electrons.
Therefore,
M
has opposite direction with respect
to
B
ext
. This relation can be rewritten as follows:
M
s
D
mn
(3.53)
For
M
D
M
s
(that is, for
L
(Ÿ)
D
1) all the
atomic dipoles are aligned with the external
field
B
ext
. Equation (
3.52
) implies that for
mB
ext
100
kT
the induced magnetization
practically coincides with the saturation
magnetization. Conversely, for
mB
ext
<<
kT
the
function
L
is approximately a linear function with
slope
1/3:
¦
0
B
ext
M
D
(3.51)
The dimensionless quantity ¦, which is neg-
ative in the case of diamagnetic materials, is
called
magnetic susceptibility
and its value de-
pends from the substance. Common diamagnetic
substances are water, wood, many organic mate-
rials and most metals.
Let us consider now the case of paramagnetic
materials. In this instance, there are permanent
magnetic moments at atomic scale, which are
independent each other and in normal conditions
are randomly oriented due to thermal agitation.
Therefore, the total magnetization (
3.38
) is zero
in absence of external field. However, if we apply
a magnetic field, the magnetic moments tend to
align to the external field, determining a non-zero
net magnetization. The magnitude of the induced
magnetization is determined by the
equation of
Langevin
:
M
M
s
Š
mB
ext
3kT
(3.54)
Substituting (
3.53
)gives:
nm
2
B
ext
3kT
M
Š
(3.55)
This equation is
Curie's law
. It predicts that
the magnetization is inversely proportional to the
temperature. A comparison of this result with
(
3.51
) shows that the magnetic susceptibility of
a paramagnetic material is positive and assumes
the value:
M
M
s
D
L.Ÿ/
coth .Ÿ/
1
Ÿ
I
Ÿ
mB
ext
kT
(3.52)
0
nm
2
3kT
¦
D
(3.56)
We have seen that the magnetization of dia-
magnetic and paramagnetic materials is a vector
having the same direction of the external field
B
ext
where
m
is the permanent magnetic moment of
the atoms,
k
is the Boltzmann constant,
T
is the
