Geology Reference
In-Depth Information
Similarly, at position (
x
,
y
C
dy
,
z
)wehave:
Now we shall prove that this is a consequence of
the Larmor precession discussed in Sect.
3.2
.We
have shown in Sect.
3.2
that the electron orbits are
magnetic dipoles, with a definite magnetic mo-
ment that depends from the angular momentum
(Eq.
3.17
). In normal conditions, the atoms of a
diamagnetic substance have a net magnetic mo-
ment equals zero, because the various magnetic
moments, associated with orbits, electron spins,
etc., balance out. However, when we apply an
external magnetic field, the precession of an elec-
tron orbit about the field direction is equivalent
to an additional microscopic current loop having
radius
a
and an intensity that according to (
3.16
)
is given by:
I.x;y
C
dy;
z
/
D
M
z
.x;y
C
dy;
z
/d
z
M
z
.x;y;
z
/
C
@y
dy
d
z
(3.42)
@M
z
D
Therefore, it is evident from Fig.
3.9
that a
first contribution to the net macroscopic current
flowing in direction
x
is:
@M
z
@y
dyd
z
•I
.x/
1
D
(3.43)
Another contribution to the current flowing
in the
x
direction can be obtained considering a
thin slice of thickness
dy
, parallel to plane
xz
at
elevation
y
. At position (
x
,
y
,
z
), the existence of a
non-zero component of magnetization
M
y
(
x
,
y
,
z
)
is equivalent to the existence of an infinitesimal
rectangular coil having area
dxdz
and magnetic
moment
dm
y
D
M
y
(
x
,
y
,
z
)
dxdydz
.Bythesame
reasoning as before, we see that an additional
contribution to the macroscopic current in the
x
direction will be:
e¨
L
2
I
L
D
(3.46)
This current is associated with an extra angular
momentum, resulting from the twist of the elec-
tron orbit, which is parallel to the external field:
1
2
ea
2
•L
D
m
e
a
v
L
B
ext
D
m
e
a
2
¨
L
B
ext
D
B
ext
(3.47)
@M
y
@
z
dyd
z
wherewehaveused(
3.20
). It should be noted
that •
L
represents an additional vector of angu-
lar momentum and
not
the variation of
L
in a
small time interval, which is indicated as
d
L
(see
Fig.
3.5
). According to (
3.17
), this extra angular
momentum must be associated with an additional
magnetic moment, •
m
, given by:
•I
.x
2
D
(3.44)
Therefore, the total macroscopic current in the
x
direction is given by:
@M
y
@
z
dyd
z
@M
z
@y
D
•I
.x
1
C
•I
.x
2
D
•I
.x/
j
mx
dyd
z
(3.45)
e
2
a
2
4m
e
B
ext
e
2m
e
•L
D
•m
D
(3.48)
This proves that the components of the curl
of
M
coincide with the components of a net
macroscopic current density, resulting from the
microscopic currents that flow within the body.
The magnetization acquired by a body in pres-
ence of an external magnetic field is called
in-
duced magnetization
and is in general a function
of the applied field
B
ext
. The relation between
M
and
B
ext
is different among the three classes of
magnetic behavior of matter. We have mentioned
that all the ordinary substances acquire a weak
magnetization that opposes
B
ext
(diamagnetism).
Thus, the extra magnetic moment •
m
opposes
the external field
B
ext
. Regarding the quantity
a
2
that appears in (
3.48
),itcanbeshownonthe
basis of quantum mechanics considerations that
it represents the averaged square distance of the
electron from the axis of
B
ext
(e.g., Feynman et al.
2006
). If the external field is aligned with the
z
-
axis and
r
(
x
,
y
,
z
) is the instantaneous position
of the electron, then
a
2
Dh
x
2
iCh
y
2
i
. For a spher-
ically symmetric atom,
h
x
2
i
,
where
r
is the distance of the electron from
iDh
y
2
iDh
z
2
iDh
r
2
