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 socalled 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 nonferromagnetic (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 multidomain hysteresis is the reduction in multidomain magnetic susceptibility χ_{md} from “true” magnetic susceptibility (χ_{i}) because of selfdemagnetization. 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 socalled 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 selfenergy 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 selfenergy. 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 ad. 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 5.11bg). 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 stairstep 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.