Probably the most important result of observations and theory
of magnetic domains of the past decade is the demonstration that
LEM states exist. Moreover, as discussed in the introduction, this
will have important consequences for magnitudes and stabilities of
virtually all remanent magnetizations. However, the precise number
of LEM states allowed in a given grain depends on the theoretical
model used and, moreover, the range of LEM states calculated does
not always agree with those inferred from observations. Moon and
Merrill [1985] used a one-dimensional calculation to obtain six LEM
states for a 1 
m magnetite cube. In general, as the size of
the grain increases, so does the mean and range in the number of
domains (i.e., LEM states) about the mean. Moon and Merrill [1985]
concluded that while LEM states should exist, the activation energy
between the calculated LEM states were too high. It was expected
that when increasingly accurate two- and three-dimensional
calculations became available, the predicted activation
energies between LEM states would decrease and, consequently, so
would the mean number and range of acceptable LEM states. This
expectation has been realized. For example, Ye and Merrill [1991]
used a quasi-two-dimensional calculation to find four LEM states for
a 
m magnetite cube, while more recently, Xu et al. [1994]
use a two-dimensional calculation to obtain only two LEM states for
the same cube. It is important to recognize that these are not
three ``competing'' models with different LEM state predictions!
Instead the models have steadily improved and the Xu et al. [1994]
model should be presently regarded as providing the best
theoretical estimates for the number of LEM states in magnetite
grains exceeding a micron in size.
The ability to compare theoretical to observed domain
structure has varied with the technology available for imaging
domains and domain walls in grains of different sizes. Only
recently has it been possible to examine directly domain structures
in small grains in the 0.05 
m. size range. Moreover, most
results from these techniques emphasize their potential, rather
than producing definitive tests of theory. Most of these
techniques involve magnetic force microscopy [e.g., Hartmann, 1989
1990, Williams et al ., 1992, Oti and Price, 1993, Gibson and
Schultz, 1993, Hagg et al., 1993, and Lederman et al, 1993], but
other methods also have been applied (e.g., high resolution scanning
Lorentz electron microscopy by Yajima et al., [1993], and magnetic
electron emission holography by Timmermans et al., [1993]. The
effects of interactions between grains can have important
consequences for domain structures, activation energies, etc., and
these are being investigated by such techniques as magnetic force
microscopy [e.g. Gibson and Schultz, 1993] and others
(e.g., by the Wohlfarth-Henkle technique;
[ Proksch and Moskowitz, 1994]. Although
most of these experimental results are preliminary and illustrate
their potential, the results appear consistent with the theories
for magnetic structures in submicron grains; for example
vortex-like structures are observed in needle-shaped SD permalloy
grains [ Gibson and Schultz, 1993]. However, conventional indirect
measures of magnetic structures indicate that the theory contains
some serious omissions. For example, numerous experiments indicate
that a substantially larger proportion of magnetization originates
from magnetite grains that are bigger than d
than permitted by
the theory leading to such a curve as shown in Figure 3, which shows a broad range of domains after
identical thermal treatments. The range in the number of domains
in Figure 3 is significantly larger than that obtained after
demagnetization in an alternating field. (This demagnetization was
repeated many times to obtain a distribution.)
Thus both domain theory and observation appear to have
confirmed the existence of LEM states. But in the case of
titanomagnetites, including the end member magnetite, there is not
good agreement between theory and observation. In general, Bitter
pattern studies reveal far fewer domains (as much as an order of
magnitude less) than what are predicted [e.g., Halgedahl, 1987, Worm
et al., 1991, Heider et al., 1992]. Several explanations have been
advanced to resolve this problem. Williams et al., [1992]
calculated that as the depth of a domain wall decreases with
respect to the surface, the wall becomes increasingly difficult to
image with magnetic colloid. Newell et al. [1993] calculated that
whereas 90
Neel walls would not be well-imaged with the
Bitter technique, 180
Bloch and Neel walls would be
observable. (The magnetization in Neel walls lies in the same
plane as that in the adjacent domains---in contrast to that of
Bloch walls.) In short, these authors argue that the discrepancy
between theory and observation is due to incomplete imaging of all
the domain walls when the Bitter technique is employed.
However, the above possible explanations do not appear to be telling the entire story, since other observational data show similar results to those obtained from the Bitter technique. For example, using the MOKE technique, both Worm et al. [1991] and Heider and Hoffmann [1992] observed only a few domains in moderately large (several tens of microns on a side) synthetic magnetite grains. It is important to point out that the MOKE technique images domains rather than walls, as done in the Bitter technique. Heider and Hoffman [1992] suggest that both techniques might underestimate the number of domains simply because the domain walls often will not be perpendicular to the crystal face on which the domains are observed. Although this almost certainly is a factor, it cannot account for the very large gap between theory and observation. Independently, Worm et al. [1991] and Moon [1991] argue that a high residual stress in the sample may explain the discrepancy. Moon [1991] also suggests there may be a large error in the exchange constant used in the theories. Ye and Merrill [1995a] present arguments against this last possibility and they carry out one- dimensional domain calculations to illustrate that the high residual stress suggestion does appear to be an important factor.
Much better agreement between observed and predicted number of
domains has been recently obtained by Ozdemir and Dunlop [1993] on
a large (several mm) single crystal of magnetite. Bitter patterns
were observed on a [100] surface, revealing the 180
,
71
, and 109
walls with appropriate spacing, as expected
from theory for magnetite. There are good reasons to suspect this
sample has low residual stress relative to many magnetite samples
used in other domain observations [e.g., Worm et al., 1991], adding
support for the argument that the discrepancy between early theory
and observation is primary a result that many of the magnetite
samples used in the experiments contained high residual stress.
Of special interest is the possibility that a SD LEM state
can exist in a large grain (e.g.,
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