Geology Reference
In-Depth Information
e
m
e
S
B
ext
This quantity does not depend from the se-
lected axis of projection
n
. In particular, we have:
D
.S
i/
2
E
U
m
D
(3.70)
D
.S
j/
2
E
D
.S
k/
2
E
Equation (
3.66
) puts a constrain on the possi-
ble values of
D
D
-
h
/2. There-
fore, the magnetic energy, just like the projec-
tion of
S
along an axis, can assume only two
values:
S
B
ext
, which are
˙
Therefore,
D
S
x
C
S
y
C
S
z
E
D
.S
i/
2
E
ǝ
S
2
Ǜ
D
D
e
¯
2m
e
B
˙
B
B
D
.S
j/
2
E
D
.S
k/
2
E
D
3
D
.S
k/
2
E
U
m
D˙
(3.71)
C
C
where the quantity:
Finally, using (
3.67
) and taking into account
that <
S
2
> does not depend from the direction
of any axis, so that <
S
2
>
D
S
2
, we obtain the
following general result:
2m
e
D
9:27400968.20/
10
24
JT
1
(3.72)
e
¯
B
S
2
D¯
s
.s
C
1/
(3.68)
is called the
Bohr magneton
. Let us consider now
a neighbor magnetic dipole
m
0
S
at position
r
.In
this instance,
B
ext
has the form (
3.21
)and
U
m
represents the magnetic potential of dipole-dipole
interaction. We have:
This important relation shows that for an
atomic system the maximum projection of
S
along an arbitrary axis is never
S
, but it is smaller
than this quantity, because
s
-
h
is always less
than the square root of
-
hs
(
s
C
1). Therefore,
in the quantum mechanical world the
observed
projection of angular momentum can never be
exactly aligned with any particular axis! An
electron has spin
s
D
½, so that there are only two
possible values for the projection of
S
along an
arbitrary axis: (
S
n
)
D˙
U
m
D
m
0
S
B
ext
.r/
"
3
.
m
S
r
/
m
0
S
r
r
5
#
m
S
m
0
S
r
3
0
4
Š
(3.73)
-
h
/2, which correspond
to the two possible (“up” and “down”) states
associated with its spinning. Now let us consider
the interaction between the magnetic moment
of an electron and a magnetic field. When the
electron is placed in an external magnetic field,
it acquires an additional potential energy that
depends from the component of
m
S
along
B
ext
:
As we may have expected, this expression
is symmetric with respect to
m
S
and
m
0
S
.
If this were the interaction responsible for
the alignment of spins in ferromagnetic
materials, then the thermal energy correspond-
ing to the Curie temperature
T
c
,whichis
kT
c
(
k
D
1.3806488(13)
10
23
[JK
1
]beingthe
Boltzmann constant), would have the same order
of magnitude of the strongest field energy of
dipolar interaction. In fact, the former represents
the threshold thermal energy separating ordered
states from paramagnetic behavior, while the
latter would be a measure of the aligning energy.
A rough estimate can be obtained considering
that the nearest neighbor separation coincides
with the lattice constant
a
. Therefore, setting
m
S
D
m
0
S
in (
3.73
), we would have, for in-plane
interactions:
U
m
D
m
S
B
ext
(3.69)
This equation shows that the magnetic energy
depends from the orientation of the magnetic
moment with respect to the external field: if
m
S
?
B
ext
, then the potential energy is maxi-
mum (
U
m
D
0), whereas
U
m
assumes its mini-
mum value (
U
m
D
m
S
B
ext
) when the magnetic
moment is aligned with the field. Substituting
(
3.65
)gives:
