Geology Reference
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t
t
⎛
⎜
⎞
⎟
θθ
()
t
=
exp
−
0
0
with
t
0
/
mB
. Tauxe (2010) prefers not to use the
small angle approximation and arrives at the more rig-
orous solution found in Nagata (1961) after neglecting
the inertial term:
=
π
d
3
η
θ
θ
t
t
⎛
⎝
⎞
⎠
=
⎛
⎝
⎞
⎠
⎛
⎜
⎞
⎟
0
tan
tan
exp
−
2
2
0
For either solution, the end result is essentially the
same. Because of magnetite's strong spontaneous
magnetization, the magnetic grain will be more easily
magnetized along its long axis (should it have a shape
anisotropy) or along its 111 crystallographic axis
(should the grain be equidimensional). Once the grain
settles through the water column, is deposited and
then incorporated into the sediment column, it should
keep its alignment and provide a record of the direction
of the geomagnetic fi eld during deposition. However,
as many workers have pointed out, when reasonable
values for a grain's magnetic moment (5 × 1 0
− 17
A m
2
),
the strength of the geomagnetic fi eld (50 μT) and the
viscosity of water (10
Fig. 2.1
Magnetic torque on a magnetic particle settling in
the water column or settled into a large pore space after
deposition. The geomagnetic fi eld exerts a torque on the
magnetic moment of the particle, both shown by arrows.
2
I
d
dt
θ
d
dt
θ
−
3
Pa s) are used in the traditional
DRM theory equations, the time of alignment is of the
order just one second.
+
λ
+
mB
sin
θ
=
0
2
where
I
is the moment of inertia of the grain (in this
case assumed to be spherical, I =
π
d
5
ρ
/60);
λ
is the
coeffi cient of viscous drag between the particle and
water (
λ
=
π d
3
η
where
η
is viscosity of the water);
d
is
diameter of the grain; and
ρ
is the grain density. The
third term is the torque (restoring force) of the mag-
netic fi eld
B
acting on the magnetic moment of the
particle
m
. In traditional treatments, the small angle
approximation is used and the equation becomes linear
with sin
θ ~
θ
. With the small angle approximation, the
equation becomes that of a damped, simple harmonic
oscillator meaning that the grain swings back and
forth about the fi eld direction with decaying amplitude
until it becomes aligned. The second approximation
made is that, for very small particles,
d
5
becomes
vanishingly small and the fi rst term can be made to
disappear. This gives a fi rst - order linear differential
equation:
πη
d
mB
3
3 14159
.
××
10
−
6
10
−
3
t
=
=
=
13
.
0
−
17
510
×
×
50
This result would predict that all the grains in a sedi-
mentary rock should be perfectly aligned in fi elds of the
same strength as the Earth's fi eld and that DRM inten-
sity should be independent of geomagnetic fi eld inten-
sity. However, early re-deposition experiments with
glacial clays (Johnson
et al
. 1948) showed a strong
linear dependence between laboratory magnetic fi eld
strength and DRM intensity. This result shows that the
simple traditional notion of how a rock acquires a
DRM needs modifi cation.
Collinson (1965) tackled this problem by introduc-
ing a misaligning mechanism, Brownian motion, to
mitigate the aligning torque of the geomagnetic fi eld.
He modeled the Brownian motion with Langevin
theory (see detailed treatment in Butler's or Tauxe's
paleomagnetism textbooks) that is also used to describe
the origin of paramagnetism in materials:
d
dt
θ
λ
+
mB
θ
= 0
pDRM
pDRM
sat
=
mB
kT
kT
mB
The solution of this equation shows that the alignment
improves as an exponential function of time:
⎛
⎝
⎞
⎠
−
⎛
⎞
⎠
coth
⎝
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