Magnetic hysteresis

In Chapter 4 we discussed the energies that control the state of magnetization within ferromagnetic particles. Particles will tend to find a configuration of internal magnetization directions that minimizes the energies (although meta-stable states with local energy minima or LEMs are a possibility). The longevity of a particular magnetization state has to do with the depth of the energy well that the magnetization is in and the energy available for hopping over barriers.

The ease with which particles can be coerced into changing their magnetizations in response to external fields can tell us much about the overall stability of the particles and perhaps also something about their ability to carry a magnetic remanence over the long haul. The concepts of long term stability, incorporated into the concept of relaxation time and the response of the magnetic particles to external magnetic fields are therefore linked through the anisotropy energy constant K (see Chapter 4) which dictates the magnetic response of particles to changes in the external field. This chapter will focus on the response of magnetic particles to changing external magnetic fields.

Magnetic remanence is the magnetization in the absence of an external magnetic
field. If we imagine a particle with a single “easy” axis – a so-called “uniaxial”
particle with magnetic anisotropy constant K_{u}, the magnetic energy density
(energy per unit volume) of a particle whose magnetic moment makes
an angle θ to the easy axis direction (Figure 5.1a) can be expressed
as:

As the moment swings around with angle θ to the easy axis, the anisotropy
energy density ϵ_{a} will change as sketched in Figure 5.1b. The energy minima are
when θ is aligned parallel to the easy axis (an axis means either direction along
the axis, so we pick one direction as being 0 and the other as 180^{∘}). In
the absence of a magnetic field, the moment will lie along one of these
two directions. [In reality, thermal energy will perturb this direction
somewhat, depending on the balance of anisotropy to thermal energy, but
for the present discussion, we are assuming that thermal energy can be
neglected.]

When an external field is applied at an angle ϕ to the easy axis (and an angle
ϕ-θ with the magnetic moment; see Figure 5.1a), the magnetostatic interaction
energy density ϵ_{m} given by the dot product of the magnetization and the applied
field (Equation 4.1 in Chapter 4) or:

The two energy densities (ϵ_{a} and ϵ_{m}) are shown as the thin solid and dashed lines
in Figure 5.1c for an applied field of 30 mT aligned with an angle of 45^{∘} to the
easy axis. There is a competition between the anisotropy energy (tending to keep
the magnetization parallel to the easy axis) and the interaction energy (tending
to line the magnetization up with the external magnetic field). Assuming that the
magnetization is at saturation, we get the total energy density of the particle to
be:

| (5.1) |

The total energy density ϵ_{t} is shown as the heavy solid line in Figure 5.1c.

The magnetic moment of a uniaxial single domain grain will find the angle θ that
is associated with the minimum total energy density (ϵ_{min}; see Figure 5.1b,c).
For low external fields, θ will be closer to the easy axis and for higher external
fields (e.g., 30 mT; Figure 5.1c), θ will be closer to the applied field direction
(ϕ).

When a magnetic field that is large enough to overcome the anisotropy energy is
applied in a direction opposite to the magnetization vector, the moment will
jump over the energy barrier and stay in the opposite direction when the field is
switched off. The field necessary to accomplish this feat is called the flipping
field (μ_{o}H_{f}) (also sometimes the “switching field”). [Note the change to the use
of H for internal fields where M cannot be considered zero.) We introduced this
parameter in Chapter 4 (see Equation 4.8) as the microscopic coercivity. Stoner
and Wohlfarth (1948) showed that the flipping field can be found from
the condition that dϵ_{t}∕dθ = 0 and d^{2}ϵ_{t}∕dθ^{2} = 0. We will call this the
“flipping condition”. The necessary equations can be found by differentiating
Equation 5.1:

| (5.2) |

and again

| (5.3) |

Solving these two equations for B and substituting μ_{o}H for B, we get after some
trigonometric trickery:

| (5.4) |

where t = tan^{1
3 }ϕ. In this equation, ϕ is the angle between the applied field and
the easy axis direction opposite to m.

Now we can derive the so-called “microscopic coercivity” (H_{k}) introduced in
Section 4.1.6 in Chapter 4. Microscopic coercivity is the maximum flipping field
for a particle. When magnetic anisotropy of a particle is dominated by uniaxial
anisotropy constant K_{u} and ϕ is zero (antiparallel to the easy direction
nearest the moment), μ_{o}H_{k} = 2K_{u}
M_{s}. Using the values appropriate for
magnetite (K_{u} = 1.4 x 10^{4} Jm^{-3} and M_{s} = 480 mAm^{-1} we get μ_{o}H_{k} =
58 mT. To see why this would indeed result in a flipped moment, we
plot the behavior of Equations 5.1 - 5.3 in Figure 5.2. The minimum in
total energy ϵ_{t} occurs at an angle of θ = 180^{∘} (Figure 5.2a) and the
first and second derivatives satisfy the flipping condition by having a
common zero crossing (θ = 0 in Figure 5.2b). There is no other applied
field value for which this is true (see, e.g., the case of a 30 mT field in
Figure 5.2c,d).

The flipping condition depends not only on the applied field magnitude but also
on the direction that it makes with the easy axis (see μ_{o}H_{f} versus ϕ in
Figure 5.3). When ϕ is parallel to the easy axis (zero) (and anti-parallel to m,
μ_{o}H_{f} is 58 mT as we found before. μ_{o}H_{f} drops steadily as the angle between the
field and the easy axis increases until an angle of 45^{∘} when μ_{o}H_{f} starts to
increase again. According to Equation 5.4, μ_{o}H_{f} is undefined when ϕ = 90^{∘}, so
when the field is applied at right angles to the easy axis, there is no field
sufficient to flip the moment.

In this section we will develop the theory for predicting the response of substances to the application of external fields, in experiments that generate hysteresis loops. We will define a number of parameters which are useful in rock and paleomagnetism. For computational details in estimating these parameters from hysteresis data, see Appendix C.1.

Let us begin by considering what happens to single particles when subjected to applied fields in the cycle known as the hysteresis loop. From the last section, we know that when a single domain, uniaxial particle is subjected to an increasing magnetic field the magnetization is gradually drawn into the direction of the applied field. If the flipping condition is not met, then the magnetization will return to the original direction when the magnetic field is removed. If the flipping condition is met, then the magnetization undergoes an irreversible change and will be in the opposite direction when the magnetic field is removed.

Imagine a single domain particle with uniaxial anisotropy. Because the particle is
single domain, the magnetization is at saturation and, in the absence of
an applied field is constrained to lie along the easy axis. Now suppose
we apply a magnetic field in the opposite direction (see track # 1 in
Figure 5.4a). When B reaches μ_{o}H_{f} in magnitude, the magnetization flips to the
opposite direction (track #2 in Figure 5.4) and will not change further
regardless of how high the field goes. The field then is decreased to zero and
then increased along track #3 in Figure 5.4 until μ_{o}H_{f} is reached again.
The magnetization then flips back to the original direction (track #4 in
Figure 5.4a).

Applying fields at arbitrary angles to the easy axis results in loops of
various shapes (see Figure 5.4b). As ϕ approaches 90^{∘}, the loops become
thinner. Remember that the flipping fields for ϕ = 22^{∘} and ϕ = 70^{∘} are
similar (see Figure 5.3) and are lower than that when ϕ = 0^{∘}, but the
flipping field for ϕ = 90^{∘} is infinite, so that “loop” is closed and completely
reversible.

Before we go on, it is useful to consider for a moment how hysteresis
measurements are made in practice. Measurements of magnetic moment m as a
function of applied field B are made on a variety of instruments, such as a
vibrating sample magnetometer (VSM) or alternating gradient force
magnetometer (AGFM). In the latter, a specimen is placed on a thin stalk
between pole pieces of a large magnet. There is a probe mounted behind the
specimen that measures the applied magnetic field. There are small coils on the
pole pieces that modulate the gradient of the applied magnetic field (hence
alternating gradient force). The specimen vibrates in response to changing
magnetic fields and the amplitude of the vibration is proportional to the moment
in the axis of the applied field direction. The vibration of the specimen stalk is
measured and calibrated in terms of magnetic moment. The magnetometer is
only sensitive to the induced component of m parallel to the applied field B_{o},
which is m_{||} = mcosϕ (because the off axis terms are squared and very small,
hence can be neglected.) In the hysteresis experiment, therefore, the
moment parallel to the field m_{||} is measured as a function of applied field
B.

In rocks with an assemblage of randomly oriented particles with uniaxial
anisotropy, we would measure the sum of all the millions of tiny individual loops.
A specimen from such a rock would yield a loop similar to that shown in
Figure 5.5a. If the field is first applied to a demagnetized specimen, the initial
slope is the (low field) magnetic susceptibility (χ_{lf}) first introduced in Chapter 1.
From the treatment in Section 5.1 it is possible to derive the equation
χ_{lf} = μ_{o}M_{s}^{2}∕3K_{u} for this initial (ferromagnetic) susceptibility (for more, see
O’Reilly 1984).

If the field is increased beyond the flipping field of some of the magnetic grains
and returned to zero, the net remanence is called an isothermal remanent
magnetization (IRM). If the field is increased to +B_{max}, all the magnetizations
are drawn into the field direction and the net magnetization is equal to the sum
of all the individual magnetizations and is the saturation magnetization M_{s}.
When the field is reduced to zero, the moments relax back to their individual
easy axes, many of which are at a high angle to the direction of the saturating
field and cancel each other out. A loop that does not achieve a saturating field
(red in Figure 5.5a is called a minor hysteresis loop, while one that does is called
the outer loop.

The net remanence after saturation is termed the saturation remanent
magnetization M_{r} (and sometimes the saturation isothermal remanence sIRM).
For a random assemblage of single domain uniaxial particles, M_{r}∕M_{s} = 0.5. The
field necessary to reduce the net moment to zero is defined as the coercive field
(μ_{o}H_{c}) (or coercivity).

The coercivity of remanence μ_{o}H_{cr} is defined as the magnetic field required
to irreversibly flip half the magnetic moments (so the net remanence
after application of a field equal to -μ_{o}H_{cr} to a saturation remanence is
0). The coercivity of remanence is always greater than or equal to the
coercivity and the ratio H_{cr}∕H_{c} for our random assemblage of uniaxial
SD particles is 1.09 (Wohlfarth, 1958). Here we introduce two ways of
estimating coercivity of remanence, illustrated in Figure 5.5. If, after
taking the field up to some saturating field +B_{max}, one first turned the
field off (the descending curve), then increased the field in the opposite
direction to the point labeled μ_{o}H′_{cr}, and one were to then switch the field
off again, the magnetization would follow the dashed curve up to the
origin. For single domain grains, the dashed curve would be parallel to the
lower curve (the ascending curve). So, if one only measured the outer
loop, one could estimate the coercivity of remanence by simply tracing
the curve parallel to the lower curve (dashed line) from the origin to
the point of intersection with the upper curve (circled in Figure 5.5a).
This estimate is only valid for single domain grains, hence the prime in
μ_{o}H_{cr}′.

An alternative means of estimating coercivity of remanence is to use a so-called
ΔM curve (Jackson et al., 1990) which is obtained by subtracting the ascending
loop from the descending loop (see Figure 5.5b). When all the moments are
flipped into the new field, the ascending and descending loops join together and
ΔM is 0. ΔM is at 50% of its initial value at the field at which half the moments
are flipped (the definition of coercivity of remanence); this field is here termed
μ_{o}H_{cr}.

Figure 5.5a is the loop created in the idealized case in which only uniaxial
ferromagnetic particles participated in the hysteresis measurements; in fact the
curve is entirely theoretical. In “real” specimens there can be paramagnetic,
diamagnetic AND ferromagnetic particles and the loop may well look like
that shown in Figure 5.6. The initial slope of a hysteresis experiment
starting from a demagnetized state in which the field is ramped from zero
up to higher values is the low field magnetic susceptibility or χ_{lf} (see
Figure 5.6). If the field is then turned off, the magnetization will return again
to zero. But as the field increases passed the lowest flipping field, the
remanence will no longer be zero but some isothermal remanence. Once all
particle moments have flipped and saturation magnetization has been
achieved, the slope relating magnetization and applied field reflects only the
non-ferromagnetic (paramagnetic and/or diamagnetic) susceptibility, here
called high field susceptibility, χ_{hf}. In order to estimate the saturation
magnetization and the saturation remanence, we must first subtract the
high field slope. So doing gives us the blue dashed line in Figure 5.6
from which we may read the various hysteresis parameters illustrated in
Figure 5.5b.

In the case of equant grains of magnetite for which magnetocrystalline anisotropy
dominates, there are four easy axes, instead of two as in the uniaxial case
(see Chapter 4). The maximum angle ϕ between an easy axis and an
applied field direction is 55^{∘}. Hence there is no individual loop that goes
through the origin (see Figure 5.7). A random assemblage of particles with
cubic anisotropy will therefore have a much higher saturation remanence.
In fact, the theoretical ratio of M_{r}∕M_{s} for such an assemblage is 0.87,
as opposed to 0.5 for the uniaxial case (Joffe and Heuberger, 1974).

In superparamagnetic (SP) particles, the total magnetic energy E_{t} = ϵ_{t}v (where
v is volume) is balanced by thermal energy kT. This behavior can be modeled
using statistical mechanics in a manner similar to that derived for paramagnetic
grains in Section 3.2.2 in Chapter 3 and summarized in Appendix A.2.2. In
fact,

| (5.5) |

where γ = M_{s}Bv
kT and N is the number of particles of volume v, is a reasonable
approximation. The end result, Equation 5.5, is the familiar Langevin function
from our discussion of paramagnetic behavior (see Chapter 3); hence the term
“superparamagnetic” for such particles.

The contribution of SP particles for which the Langevin function is valid with
given M_{s} and d is shown in Figure 5.8a. The field at which the population
reaches 90% saturation B_{90} occurs at γ ~ 10. Assuming particles of magnetite
(M_{s} = 480 mAm^{-1}) and room temperature (T = 300∘K), B_{90} can be evaluated
as a function of d (see Figure 5.8b). Because of its inverse cubic dependence on
d, B_{90} rises sharply with decreasing d and is hundreds of tesla for particles a few
nanometers in size, approaching paramagnetic values. B_{90} is a quick guide to
the SP slope (the SP susceptibility χ_{sp}) contributing to the hysteresis
response and was used by Tauxe et al. (1996) as a means of explaining
distorted loops sometimes observed for populations of SD/SP mixtures.
B_{90} (and χsp) is very sensitive to particle size with very steep slopes
for the particles at the SP/SD threshold. The exact threshold size is
still rather controversial, but Tauxe et al. (1996) argue that it is ~ 20
nm.

For low magnetic fields, the Langevin function can be approximated as ~1 3γ . So we have:

| (5.6) |

We can rearrange Equation 4.11 in Chapter 4 to solve for the volume at which a uniaxial grain passes through the superparamagnetic threshold we find:

| (5.7) |

Comparing this expression with that derived for ferromagnetic susceptibility in
Section 5.2.1, we find that χ_{sp} is a factor of ln(Cτ) ≃ 27 larger than the
equivalent single domain particle.

Moving domain walls around is much easier than flipping the magnetization of an entire particle coherently. The reason for this is the same as the reason that it is easier to move a rug by lifting up a small wrinkle and pushing that through the rug, than to drag the whole rug by the same amount. Because of the greater ease of changing magnetic moments in multidomain (MD) grains, they have lower coercive fields and saturation remanence is also much lower than for uniformly magnetized particles (see typical MD hysteresis loop in Figure 5.9a.)

The key to understanding multi-domain hysteresis is the reduction in
multi-domain magnetic susceptibility χ_{md} from “true” magnetic susceptibility
(χ_{i}) because of self-demagnetization. The true susceptibility would be that
obtained by measuring the magnetic response of a particle to the internal field H_{i}
(applied field minus the demagnetizing field -NM – see Section 4.1.5; see
Dunlop 2002a). Recalling that the demagnetizing factor is N, the so-called
screening factor f_{s} is (1 + Nχ_{i})^{-1} and χ_{md} = f_{s}χ_{i}. If we assume that χ_{md} is
linear for fields less than the coercivity, then by definition χ_{md} = M_{r}
H_{c} (see
Figure 5.9b). From this, we get:

By a similar argument, coercivity of remanence (H_{cr}) is suppressed by the
screening factor which gives coercivity so:

Putting all this together leads us to the remarkable relationship noted by Day et al. (1977; see also Dunlop 2002a):

| (5.8) |

When χ_{i} H_{c}
M_{s} is constant, Equation 5.8 is a hyperbola. For a single mineralogy, we
can expect M_{s} to be constant, but H_{c} depends on grain size and the state
of stress which are unlikely to be constant for any natural population
of magnetic grains. Dunlop (2002a) argues that if the main control on
susceptibility and coercivity is domain wall motion through a terrain of variable
wall energies, then χ_{i} and H_{c} would be inversely related and gives a
tentative theoretical value for χ_{i}H_{c} in magnetite of about 45 kAm^{-1}.
This, combined with the value of M_{s} for magnetite of 480 kAm^{-1} gives a
value for χ_{i} H_{c}
M_{s} ~ 0.1. When anchored by the theoretical maximum for
uniaxial single domain ratio of M_{r}∕M_{s} = 0.5, we get the curve shown in
Figure 5.9c. The major control on coercivity is grain size, so the trend
from the SD limit down toward low M_{r}∕M_{s} ratios is increasing grain
size.

There are several possible causes of variability in wall energy within a magnetic
grain, for example, voids, lattice dislocations, stress, etc. The effect of voids is
perhaps the easiest to visualize, so we will consider voids as an example of why
wall energy varies as a function of position within the grain. We show a
particle with lamellar domain structure and several voids in Figure 5.10.
When the void occurs within a uniformly magnetized domain (left of
figure), the void sets up a demagnetizing field as a result of the free poles
on the surface of the void. There is therefore, a self-energy associated
with the void. When the void is traversed by a wall, the free pole area is
reduced, reducing the demagnetizing field and the associated self-energy.
Therefore, the energy of the void is reduced by having a wall bisect it.
Furthermore, the energy of the wall is also reduced, because the area of
the wall in which magnetization vectors are tormented by exchange and
magnetocrystalline energies is reduced. The wall gets a “free” spot if it
bisects a void. The wall energy E_{w} therefore is lower as a result of the
void.

In Figure 5.11, we show a sketch of a hypothetical transect of E_{w} across a
particle. There are four LEMs labelled a-d. Domain walls will distribute
themselves through out the grain in order to minimize the net magnetization of
the grain and also to try to take advantage of LEMs in wall energy.

Domain walls move in response to external magnetic fields (see Figure 5.11b-g). Starting in the demagnetized state (Figure 5.11b), we apply a magnetic field that increases to saturation (Figure 5.11c). As the field increases, the domain walls move in sudden jerks as each successive local wall energy high is overcome. This process, known as Barkhausen jumps, leads to the stair-step like increases in magnetization (shown in the inset of Figure 5.11g). At saturation, all the walls have been flushed out of the crystal and it is uniformly magnetized. When the field decreases again, to say +3 mT (Figure 5.11d), domain walls begin to nucleate, but because the energy of nucleation is larger than the energy of denucleation, the grain is not as effective in cancelling out the net magnetization, hence there is a net saturation remanence (Figure 5.11e). The walls migrate around as a magnetic field is applied in the opposite direction (Figure 5.11f) until there is no net magnetization. The difference in nucleation and denucleation energies was called on by Halgedahl and Fuller (1983) to explain the high stability observed in some large magnetic grains.

Day et al. (1977) popularized the use of diagrams like that shown in Figure 5.9c
which are known as Day diagrams. They placed quasi-theoretical bounds on the
plot whereby points with M_{r}∕M_{s} ratios above 0.5 were labelled single domain
(SD), and points falling in the box bounded by 0.5 > M_{r}∕M_{s} > 0.05 and
1.5 < H_{cr}∕H_{c} < 5 were labelled pseudo-single domain (PSD). Points with
M_{r}∕Ms below 0.05 were labelled multi-domain (MD). This paper has been cited
over 800 times in the literature and the Day plot still serves as the principle
way that rock and paleomagnetists determined domain state and grain
size.

The problem with the Day diagram is that virtually all paleomagnetically useful
specimens yield hysteresis ratios that fall within the PSD box. In the early 90s,
paleomagnetists began to realize that many things besides the trend from SD to
MD behavior that control where points fall on the Day diagram. Pick and
Tauxe (1994) pointed out that mixtures of SP and SD grains would have
reduced M_{r}∕M_{s} ratios and enhanced H_{cr}∕H_{c} ratios. Tauxe et al. (1996)
modelled distributions of SP/SD particles and showed that the SP-SD
trends always fall above those observed from MD particles (modelled in
Figure 5.9c).

Dunlop (2002a) argued that because M_{r} for SP grains is zero, the suppression of
the ratio M_{r}∕M_{s} is directly proportional to the volume fraction of the SP
particles. Moreover, coercivity of remanence remains unchanged, as it is entirely
due to the non-SP fraction. Deriving the relationship of coercivity, however, is
not so simple. It depends on the superparamagnetic susceptibility (χ_{sp}), which in
turn depends on the size of the particle and also the applied field (see
Section 5.2.4). In his simplified approach, Dunlop could only use a single (small)
grain size, whereas in natural samples, there will always be a distribution of grain
sizes. It is also important to remember that volume goes as the cube of the radius
and for a mixture to display any SP suppression of M_{r}∕M_{s} almost all of
the particles must be SP. It is impossible that these would all be of a
single radius (say 10 or 15 nm); there must be a distribution of sizes.
Moreover, Dunlop (2002a) neglected the complication in SP behavior as the
particles reach the SD threshold size, whereas it is expected that many (if
not most) natural samples containing both SP and SD grain sizes will
have a large volume fraction of the larges SP sizes, making their neglect
problematic.

Hysteresis ratios of mixtures of SD and MD particles will also plot in the “PSD”
box. Dunlop (2002a) derived the theoretical behavior of such mixtures on the
Day diagram. The key equations are 1) Equation 9 from Dunlop (2002)
which governs the behavior of the ratio M_{r}∕M_{s} as a function of the
volume fraction of single domain material (f_{SD}) and multi-domain material
(f_{MD}):

2) Equation 10 from Dunlop (2002a) which governs the behavior of coercivity:

and 3) Equation 11 from Dunlop (2002a) which governs the behavior of coercivity of remanence in SD/MD mixtures:

where χ_{SD} and χ_{MD} are the susceptibilities of the SD and MD fractions
respectively and (χ_{r})_{SD} and (χ_{r})_{MD} are the M_{r} vs H_{cr} slopes of the SD and MD
remanences respectively. What we need to calculate the SD/MD mixing curve are
values for the various parameters for single domain and multi domain
end-members. These were measured empirically for the MV1H bacterial
magnetosomes (see Chapter 6) and commercial magnetite (041183 of Wright
Company) by Dunlop and Carter-Stiglitz (2006) and shown in Table ??. Using
the linear mixing model of Dunlop (2002a), we plot the theoretical mixing curve
predicted for these empirically constrained end-members as the heavy red line in
Figure 5.9c.

SD/MD | M_{r}∕M_{s} | χ (A m^{-1}T^{-1}) | χ_{r} (MA m^{-1}T^{-1}) | μ_{o}H_{c} (mT) | μ_{o}H_{cr} (mT) |

SD | 0.5 | 5.2 | 4.55 | 46 | 52.5 |

MD | 0.02 | 4.14 | 0.88 | 5.56 | 26.1 |

If a population of SD particles are so closely packed as to influence one another,
there will be an effect of particle interaction. This will also tend to suppress the
M_{r}∕M_{s} ratio, drawing the hysteresis ratios down into the PSD box. Finally, the
PSD box could be populated by pseudo-single domain grains themselves. Here we
will dwell for a moment on the meaning of the term “pseudo-single domain”,
which has evolved from the original posed by Stacey (1961; see discussion in
Tauxe et al. 2002). In an attempt to explain trends in TRM acquisition Stacey
envisioned that irregular shapes caused unequal domain sizes, which would give
rise to a net moment that was less than the single domain value, but considerably
higher than the very low efficiency expected for large MD grains. The modern
interpretation of PSD behavior is complicated micromagnetic structures
that form between classic SD (uniformly magnetized grains) and MD
(domain walls) such as the flower or vortex remanent states (see, e.g.,
Figure 4.5 in Chapter 4). Taking all these factors into account means that
interpretation of the Day diagram is far from unique. The simple calculations
of Dunlop (2002a) are likely to be inappropriate for almost all natural
samples.

Hysteresis loops can yield a tremendous amount of information yet much of this
is lost by simply estimating the set of parameters M_{r},M_{s},H_{cr},H_{c},χ_{i},χ_{hf}, etc.
Mayergoyz (1986) developed a method using what are known as First Order
Reversal Curves or FORCs to represent hysteresis data. The most recent way of
dealing with FORCs is that of Harrison and Feinberg (2008) which is illustrated
in Figure 5.12. In the FORC experiment, a specimen is subjected to a saturating
field, as in most hysteresis experiments. The field is lowered to some field
μ_{o}H_{a}, then increased again through some value μ_{o}H_{b} to saturation (see
Figure 5.12a). The magnetization curve between μ_{o}H_{a} and μ_{o}H_{b} is a “FORC”.
A series of FORCs (see Figure 5.12b) can be generated to the desired
resolution.

To transform FORC data into some useful form, Harrison and Feinberg (2008) use a locally-weighted regression smoothing technique (LOESS). For a given measurement point P LOESS fits a second-order polynomial function of the form

| (5.9) |

to the measured magnetization surface in a specified region (for example the
circle shown in Figure 5.12b) where the a_{i} are fitted coefficients. The
LOESS technique takes a user defined number of the nearest neighbors (see
inset to Figure 5.12b) for an arbitrary shaped region over which the
data are smoothed. The coefficient -a_{6}(H_{a},H_{b}) is the FORC density at
the point. A FORC diagram is the contour plot of the FORC densities,
rotated such that μ_{o}H_{c} = μ_{o}(H_{b} - H_{a})∕2 and μ_{o}H_{u} = μ_{o}(H_{a} + H_{b})∕2.
Please note that because H_{a} < H_{b}, data are only possible for positive
H_{c}.

Imagine we travel down the descending magnetization curve (dashed line in
Figure 5.12a) to a particular field μ_{o}H_{a} less than the smallest flipping field in
the assemblage. If the particles are single domain, the behavior is reversible
and the first FORC will travel back up the descending curve. It is only
when |μ_{o}H_{a}| exceeds the flipping field of some of the particles that the
FORC will trace a new curve on the inside of the hysteresis loop. In the
simple single domain, non-interacting, uniaxial magnetite case, the FORC
density in the quadrants where H_{a} and H_{b} are of the same sign must
be zero. Indeed, FORC densities will only be non-zero for the range of
flipping fields because these are the bounds of the flipping field distribution.
So the diagram in Figure 5.12c is nearly that of an ideal uniaxial SD
distribution.

Consider now the case in which a specimen has magnetic grains with
non-uniform magnetizations such as vortex structures or domain walls. Walls
and vortices can move much more easily than flipping the moment of
an entire grain coherently. In fact, they begin to move in small jumps
(from LEM to LEM) as soon as the applied field changes. If a structure
nucleates while the field is decreasing and the field is then ramped back up,
the magnetization curve will not be reversible, even though the field
never changed sign or approached the flipping field for coherent rotation.
The resulting FORC for such behavior would have much of the “action”
in the region where H_{a} is positive. When transformed to H_{u} and H_{c},
the diagram will have the high densities for small H_{c} but over a range
of �H_{u}. The example shown in Figure 5.13 is of a specimen that has
been characterized as “pseudo-single domain”. The FORC diagram in
Figure 5.13b has some of the FORC densities concentrated along the H_{c} axis
characteristic of single domain specimens (e.g., Figure 5.12c), but there
is also concentration along the H_{u} axis characteristic of PSD and MD
specimens.

In many cases the the most interesting thing one learns from FORC diagrams is the degree to which there is irreversible behavior when the field is reduced to zero then ramped back up to saturation (see Figure 5.14). Such irreversible behavior in what Yu and Tauxe (2005) call the “Zero FORC” or ZFORC can arise from particle interactions, domain wall jumps or from the formation and destruction of vortex structures in the magnetic grains.

Fabian (2003) defined a parameter called “transient hysteresis” which is the area between the ascending and descending loops of a ZFORC (shaded area in Figure 5.14). This is defined as:

SUPPLEMENTAL READING: Dunlop and �zdemir (1997), chapters 5 and 11; O’Reilly (1984), pp 69-87; Dunlop (2002a,b)

Problem 1

For a grain with uniaxial anisotropy in an external field, the direction of
magnetization in this grain will be controlled entirely by the uniaxial anisotropy
energy density ϵ_{a} and the magnetic interaction energy ϵ_{m}. The total energy can
be written:

Problem 2 [From Jeff Gee]

In this problem, we will begin to use some real data. The data files used with this book are part of the PmagPy distribution, which you should have already downloaded and installed. [See Preface for instructions.]

The file hysteresis.txt) in the Chapter_5 directory contains data for a single hysteresis loop. Note that the units are as measured: H (Oe), moment (emu) and it is fine to leave them in these units.

a) Read the data into a Pandas DataFrame. Determine the high field slope at |H| > 4000 Oe. Typically one calculates separate slopes for the +H data and -H data and averages these. A general least squares polynomial fit (numpy.polyfit) should do the trick.

b) Use the slope you determined to plot both the original hysteresis loop and the slope-corrected loop (i.e. removing the high field paramagnetic slope).

c) What is the ratio M_{r}∕M_{s} (saturation remanence/saturation magnetization)
for this sample? The coercivity of remanence (H_{cr}) for this sample was estimated
at 264 Oe. Based on the M_{r}∕M_{s} and H_{cr}∕H_{c} ratios, is this sample more likely to
contain single domain or multidomain grains?

d) This small sample has a mass of 10.6 mg. Assuming the magnetic material is
magnetite, estimate the mass fraction of magnetite (92 Am^{2}/kg; note 1 emu/gm
is equivalent to 1 Am^{2}/kg).