Paleomagnetism and Earth History - Quantitative Plate Tectonics - page 175

of Néel ( 1949 ), which represents still today the
fundamental starting point for a description of
single-domain TRM.
Let us consider an assemblage of spheroidal
SD grains aligned with the external field H .
This configuration could be attained by an
ensemble of grains characterized by uniaxial
anisotropy , meaning that there exists a unique
preferred (“easy”) axis of magnetization (in
either direction). In this instance, each grain has
two minimum-energy states, with magnetization
M 1 DC M S (grain moment aligned with H )or
M 2 D - M S . In general, from ( 3.69 ) we see that the
potential energy associated with a grain having
magnetization M and volume V is given by:

and the electric field of the crystal lattice (spin-
orbit interactions) (e.g., Kachkachi et al. 2000 ).
Although weak, these interactions operate over a
long range, so that their contribution is not gener-
ally negligible. Indeed, they introduce anisotropy
in the system, that is, a dependence of the total
energy from the direction of magnetization. Here
we shall consider only the dipole-dipole inter-
actions, which play an important role in small
systems such as SD grains. The potential energy
associated with these forces is given by ( 3.73 ) .
Considering all dipole pairs in a SD grain, the
total magnetostatic energy of the dipole-dipole
interaction can be written as follows:

" 3 S i r ij S j r ij
r ij

#

0 B
4

X

S i S j
r ij

U D D

U D V M B D 0 V M H (6.1)

i ¤ j

(6.3)

Therefore, the minimum-energy states of a
grain in the assemblage will be: U 1 D - 0 VM S H
and U 2 DC 0 VM S H . If the number of grains
is sufficiently large, the equilibrium configura-
tion for this assemblage is a Maxwell-Boltzmann
distribution of the magnetization directions, in
which the average magnetization is determined
by the Boltzmann partition function:

where the sum is extended to all dipole pairs at
sites i and j and r ij is their distance. In the case
of ellipsoidal grains with volume V , it is possi-
ble to show that ( 6.3 ) determines a macroscopic
shape anisotropy , which is a form of magnetic
anisotropy (Kachkachi et al. 2000 ), and the total
magnetostatic energy for the dipole-dipole inter-
actions assumes the following simple expression:

X

2

M i e U i =kT

2 0 V D x M x C D y M y C D z M z
(6.4)

1

e U i =kT D M S tanh 0 VM S H

U D D

i D 1

M eq D

2

kT

X

iD1

where D x , D y ,and D z are positive quantities
called demagnetizing factors for reasons that will
be clear shortly. These quantities depend from the
grain geometry and size. For a prolate spheroid
with axes X , Y D X ,and Z > X , they are given
by:

(6.2)

This formula shows that M eq D 0for H D 0. If
N is the total number of grains in the ensemble, at
any time t we have that n grains are in a state with
parallel alignment with the external field (state
1), and N - n have anti-parallel alignment. To
evaluate the probability of a transition between
the two states, we must first determine the energy
barrier separating them. To this purpose, it is
necessary to take into account that in addition
to the strong exchange interaction described in
Chap. 3 between the atoms of a ferromagnetic
material, there exist dipole-dipole interactions
and interactions between the magnetic moments

lg 1 C e
1 e

2e

1 e 2
2e 3

D z D

1
2 .1 D z /

D x D D y D (6.5)

where 0 < e < 1 is the ellipsoid eccentricity:

q 1 .X=Z/ 2

e D