Two fundamental parameters in rock magnetism are the transition sizes between SP and SD behavior, and between SD and MD behavior. These transitions are of interest not only to paleomagnetists, but also to environmental scientists who use magnetic properties to determine the sources of materials (e.g., natural versus industrial), and to biomagnetists (e.g., SD-size biomagnetic material appears to be used by some animals to sense the external magnetic field; see Moskowitz [this issue].
A recent example is the renewed interest in SP magnetite. Although incapable of carrying a RM, SP magnetites appear to be abundant in many ancient, remagnetized carbonate rocks. Evidence for such SP populations is provided by the distinctive, ``wasp-waisted'' hysteresis loops that remagnetized carbonates exhibit, but which unremagnetized carbonates do not. ``Wasp-waisted'' loops have generated much recent excitement, because they could provide a ``remagnetization fingerprint'' and thus an invaluable diagnostic tool for paleomagnetists [ Jackson, 1990; McCabe and Channel, 1994; Channel and McCabe, 1994]. Furthermore, it appears that SP magnetite might have precipitated from fluids driven tectonically on a regional scale. Thus an improved understanding and detection of SP magnetite might shed light on the nature of some tectonic events.
In the following, we focus on the SD-MD ``transition'' in
equi-dimensional magnetite, the size of which is denoted by d
.
Many calculations of d
were made before modern numerical theory
was applied to domain theory, and all yielded values near 0.05
m at room temperature [e.g., Stacey, 1963] obtained 0.03
m and Butler and Banerjee, [1975] obtained 0.08 
m for
d
). However, several recent two- and three-dimensional
micromagnetic calculations now show there is no sharp transition
between SD and MD structures [e.g., Williams and Dunlop, 1989,
1990; Newell et al. 1993, Thompson et al., 1994, Enkin
and Williams, 1994]. (In a one-dimensional calculation, the magnetization
is allowed to be a function of only one coordinate in a Cartesian
coordinate system; in two dimensions, it is a function of two coordinates,
etc.) Instead, the magnetization in a nearly uniformly magnetized grain
becomes increasingly nonuniform as the grain size increases.
m magnetite grain as calculated by Newell et al. [1993]:
the ``vortex'' state (Figure 1a) and the ``diagonal'' state (Figure 1b).
The vortex state is usually the lower energy state and it is this state
that is found in three-dimensional calculations that assume
uniaxial anisotropy [ Enkin and Williams, 1994]. As the size is
increased further, the diagonal state becomes more and more like a
three-domain grain and the vortex state becomes more like a two-domain
grain. m, above which there is
a rapid decrease in the magnetic moment (for the vortex state); the
moment in the grain is reduced to half its maximum value by
0.08 
m. It appears that a value somewhere between 0.06 and
0.08 
m is the closest analogy to the older definition of the
critical size transition between SD and MD structure. (But note
that the structures above d
do not resemble the conventional
view of domains.). Note that this ``critical size'' is essentially
the same as found previously using simpler models. Because there
is a very narrow range in critical size estimates for d
we
conclude that an estimate near 0.07 
m for equi-dimensional
magnetite is robust. In addition, the larger SD grains reverse
directions via a vortex mode, which has a lower activation energy
than the coherent rotation mode assumed by Neel; consequently,
``blocking'' temperatures, etc., will be significantly lower [ Enkin
and Williams, 1994]. When the grain size is significantly greater
than a micron, quasi-two-dimensional [ Ye and Merrill, 1991] and
two-dimensional [ Xu et al., 1994] calculations indicate the
structures within the grain become more like traditional domains
separated by traditional domain walls. (Traditional domains in
magnetite are uniformly magnetized regions oriented at 180
,
109
, or 71
with respect to each other. In small grains
there is a blurring of domains and walls, such that it is difficult
to discern where a domain ends and a wall begins---as is evidenced
by
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