
Table of Contents
Extrinsic Properties: Understanding Magnetic Behavior
The explanation and prediction of coercivities and other extrinsic properties is a key problem in permanent magnetism. A major figure of merit is the maximum energy product, and in practice high-energy products require moderately strong coercivities. More generally, the problem lies in understanding hysteresis loops (figure 1.1) in terms of atomic magnetism and microstructure.
The models investigated in the preceding section are too simplified to provide an appropriate description of real materials. For example, from (3.96a) there follows the inequality (1.29)
\(H_{\text{N}} \geq \frac{2K_{1}}{\mu_{0}M_{\text{s}}} - DM_{\text{s}}\). (3.97)
In fact, as shown in figure 1.10, this equation greatly overestimates coercivity: even in highly sophisticated permanent magnets the coercivities are only 10–40% of the theoretical nucleation field. For example, according to (3.96) the coercivity of Nd2Fe14B should be of the order of 8 T, whereas as-cast Nd2Fe14B magnets are quite soft and sophisticated processing are necessary to achieve coercivities as high as 2.5 T. This disagreement is Brown's coercivity paradox. The ultimate reason for the low coercivities encountered in practice is imperfections such as soft-magnetic inclusions. These imperfections reduce the coercivity by giving rise to localized nucleation modes. By comparison, the curling and coherent-rotation modes investigated in section 3.2.4 are delocalized (extended). Furthermore, imperfections are responsible for the commonly observed non-rectangular shapes of the hysteresis loops.
Hysteresis Loops: The Key to Magnetic Performance
Figure 3.23 summarizes the differences between \(M–H\) and \(B–H\) hysteresis loops. The \(B–H\) loop shows \(B = \mu_{0}(H + M)\) rather than \(M\) as a function of \(H\). The physical origin of hysteresis is revealed more clearly in \(M–H\) loops, whereas \(B–H\) loops are useful for determining parameters for technical applications including the maximum energy product. If the magnetization is measured in a closed circuit, it is unnecessary to distinguish between \(H\) and \(H'\), because \(D = 0\). If other sample geometries are used, such as spheres in an open circuit, the field should be corrected by a demagnetizing contribution \(-DM\), and \(M(H')\) and \(B(H')\) plotted. In general, this skewing correction makes the hysteresis loops more rectangular. Authors often use the symbol \(H\) indiscriminately for the internal and external fields, and one must read the text carefully to see whether a demagnetizing-field correction has been made. It should by now be appreciated that the use of an average, homogeneous

Figure 3.23. Hysteresis loops. The dotted rectangle is the energy product. The load line and the working point are important for permanent magnet applications (section 6.1.1).
demagnetizing correction \(-DM\) is limited to macroscopic samples in which the details of the local magnetization \(M(\mathbf{r})\) can be ignored. Inhomogeneous magnetization states, which exist in micro- and nanostructured magnets, cannot be described in terms of macroscopic demagnetizing factors and may yield an overskewing of the loops (figure 3.24(d)).
Figure 3.24 shows some typical \(M–H\) hysteresis loops. Permanent-magnet hysteresis refers to fields applied parallel to the symmetry or \(c\)-axis of the magnet (figure 3.24(a)), although 'in-plane' magnetization measurements are used to determine uniaxial anisotropy constants (section 4.2.2.2). The width of the loop, that is the coercivity, depends on the magnet's processing and tends to be much smaller than the \(\Theta = 0\) Stoner–Wohlfarth prediction of figure 3.1919. On the other hand, grain misalignment and higher-order in-plane anisotropy contributions may give rise to some in-plane coercivity (which is not shown in figure 3.24(a)). Major or limiting hysteresis loops (the dashed curves in figure 3.24(b)) are obtained by starting from a fully aligned magnet where \(M(\mathbf{r}) = M_{\text{s}}\mathbf{e}\). This is achieved by applying a large positive field. The loop is then obtained by monitoring the volume-averaged magnetization \(\langle M \rangle = M_{\text{e}}\) as a function of the external magnetic field \(H\) (the full curve in figure 1.9). Minor loops are obtained if the maximum applied field \(\pm H\) is insufficient for complete saturation. They lie inside the major loop and therefore include a smaller area than the major loop. A particular class of minor loops is recoil loops, where the field varies between \(-H\) and 0 (figure 3.24(b)). The recoil loop shows that hysteresis loops do not necessarily exhibit the inversion symmetry \(M(-H) = -M(H)\). Shifted \(M–H\) loops can be produced, for
19 Anisotropy contributions may give rise to some in-plane coercivity (which is not shown in figure 3.24(a)).

Figure 3.24. \(M–H\) loop shapes frequently encountered in practice: (a) parallel and perpendicular loops, (b) minor (full curves) and major (dashed curves) loops, (c) hard (full curve) and comparatively soft (dotted curves) magnet loops, (d) loop skewing. In (d), the dashed and full loops are measured in open and closed circuits, respectively. The full curve can also be obtained by the demagnetizing-field correction \(H \to H - DM\) (skewing), although an overestimation of \(D\) may lead to overskewing (dotted curves). The graphs (e) and (f) are two-phase superpositions of hard and soft loops.
example, by exchange biasing or a constant field created by currents or other permanent magnets. Virgin curves (the dashed curve in figure 1.9 and the dotted curve in figure 3.23) are obtained on increasing \(H\) from zero after thermal demagnetization, that is after heating beyond \(T_{\text{c}}\).
The main difference between permanent magnets (hard magnets, the solid curve in figure 3.24(c)) and soft magnets (the dotted curve in figure 3.24(c)) is the magnitude of the coercivity. In practice, coercivities of rare-earth permanent magnets tend to exceed 1 T [10 kOe], whereas those of advanced
soft magnets are smaller than 10 μT [0.1 Oe]. Figures 3.24(e) and 3.24(f) illustrate the superposition of hard- and soft-magnetic hysteresis loops. Small volume fractions of a soft phase give rise to characteristic low-field loop inflections (shoulders, constrictions), whereas high volume fractions yield quasi-soft-magnetic loops with modifications close to saturation. In nanostructured magnets, intergranular interactions make the loops more single-phase-like (section 3.3.5).
Extrinsic properties
Magnetic properties derived from the hysteresis loop are called extrinsic properties, since they depend on the real structure (morphology) of the magnet as opposed to the atomic structure. As discussed by Kersten (1943), seemingly small micro-structural changes can lead to drastic modifications of extrinsic properties. For example, the coercivity of technical iron doubles by adding 0.01 wt% nitrogen and in some cases mechanical deformation causes the energy product to increase by a factor of more than 50. The following extrinsic properties are most relevant to permanent magnetism (figure 3.2.3).
(i) The average remanent magnetization or remanence \(M_{\text{r}}\) which remains in a magnet after switching off a large magnetic field. Remanence is smaller than the spontaneous magnetization \(M_{\text{s}}\), but in the case of nearly rectangular hysteresis loops \(M_{\text{r}} \approx M_{\text{s}}\).
(ii) The reverse field or coercivity at which the average magnetization vanishes is called the coercive force. This 'intrinsic coercivity' refers to the \(M–H\) loop and is sometimes denoted \(_{M}H_{\text{c}}\) or \(_{M}H_{\text{c}}\). The switching-field coercivity is defined by the maximum slope of the descending branch of the \(M–H\) hysteresis loop. Unlike the reverse field \(_{B}H_{\text{c}}'\) where the flux density vanishes, the 'intrinsic coercivity' \(_{M}H_{\text{c}}\) is the same whether or not the applied field is corrected for the demagnetizing field \(H_{\text{dm}} = -DM\). In other words, since \(M = 0\), both \(_{M}H_{\text{c}}\) and \(_{M}H_{\text{c}}'\) are independent of the sample shape. By comparison, the flux coercivity \(_{B}H_{\text{c}}\) (figure 1.1) depends on sample shape and has to be corrected for the demagnetizing field.
Coercivity describes the stability of the remanent state and gives rise to the classification of magnets into hard-magnetic materials (permanent magnets), semihard materials (storage media) and soft-magnetic materials. Compared to permanent magnets, which exhibit broad hysteresis loops with coercivities of the order of 1 T, semihard materials used in storage media exhibit narrow rectangular hysteresis loops with coercivities of the order of 0.05 T (40 kA m-1 [500 Oe]). The coercivity of storage media is sufficient to assure the remanence of the stored information without requiring powerful and bulky writing facilities.
(iii) The energy product, defined by \((BH) = \mu_{0} \int_{a} H^{2} \text{d}r\), where \(H\) is the field created by the magnet and the index \(a\) refers to the space outside the magnet.
Physically, it is twice the maximum magnetostatic energy stored by a unit volume of the magnet. Good permanent magnets have energy products of at least about 100 kJ m-3 [12.6 MGOe]. The energy product \((BH)\) depends on the sample shape, so that it is useful to define a maximum energy product \((BH)_{max}\) for a magnet of optimum shape. Since the field outside the magnet is often unknown, it is common to determine the energy product from the \(B–H\) loop: it equals the area of the largest second-quadrant rectangle which fits under the \(B–H\) loop (figure 3.23). Note, however, the limited validity of this hysteresis-loop construction (section 3.3.1.2).
For rectangular \(M–H\) hysteresis loops, the working point is given by the magnitude \(\mu_{0}(1 - D)M_{\text{s}}/D\) of the slope of the load line, which is known as the permeance coefficient. In the high-coercivity limit, the theoretical energy product \(\mu_{0}M_{\text{s}}^{2}/4\) equals that of an ideally rectangular hysteresis loop having \(M_{\text{r}} = M_{\text{s}}\) and \(H_{\text{c}} = M_{\text{s}}/2\). By comparison, in the case of low-coercivity magnets such as carbon steel and alnico the order of magnitude of the energy product is given by the upper bound \(\mu_{0}M_{\text{s}}H_{\text{c}}\). The energy product of some rare-earth magnets is approaching their theoretical limit (figure 1.15).
Energy Product of Ellipsoids of Revolution
A transparent interpretation of extrinsic properties exists for a uniformly magnetized ellipsoid of revolution having the magnetization direction and the easy axis parallel to the axis of revolution. In this case, the energy stored outside the magnet is \(D(1 - D)\mu_{0}M_{\text{s}}^{2}V/2\), so that \((BH) = D(1 - D)\mu_{0}M_{\text{s}}^{2}V\) (section 2.1.7.2). This expression is minimized by \(D = 1/2\), corresponding to the slightly oblate shape of figure 1.4.
This mechanism depends on the size of the ellipsoid, because the coercivity must be sufficiently high to ensure easy-axis alignment. To realize the energy product corresponding to \(D = 1/2\) the magnetization must be stable in the absence of an extended magnetic field. This means \(H_{\text{N}} = 0\) and for macroscopic ellipsoids we obtain from (3.96a) the requirement \(H_{0} = M_{\text{s}}/2\). The corresponding coercivity is \(M_{\text{s}}/2\). However, in the limit of coherent rotation equation (3.88a) yields the condition \(H_{0} \geq M_{\text{s}}/4\). This means that the magnetocrystalline anisotropy necessary to realize the energy product \(\mu_{0}M_{\text{s}}^{2}/4\) decreases with the particle size. Moreover, slightly elongated small particles should exhibit an energy product of nearly \(2\mu_{0}M_{\text{s}}^{2}/9\) in the soft-magnetic limit \(K_{1} = 0\).
These large energy products are of no practical importance, because they cannot be realized in the bulk20, but they throw some doubt on the energy-product interpretation of hysteresis loops. For example, a small, slightly elongated soft particle has a high energy product but a very small coercivity, so that the working
20 Compaction of small particles leads to interaction effects and re-introduces curling-type cooperative modes.
point, as defined by the rectangle in figure 2.23, cannot lie on the \(B–H\) loop. Similar arguments apply to nanostructured materials consisting of imperfectly coupled magnetic grains (section 3.3.5).
Magnetization Processes
The explanation of hysteresis loops of magnetic materials amounts to the description of magnetization processes. This task is complicated by the fact that one set of magnetic data may be reproduced by different micromagnetic theories. There are several aspects of magnetic hysteresis. First, one has to distinguish between reversible and irreversible magnetization processes. The term reversible denotes small magnetization changes around a particular local or global energy minimum. For example, low-field magnetization curves and the approach to saturation are largely reversible. Hysteresis is caused by irreversible processes, which involve jumps between local energy minima in the intermediate field range. There are truly irreversible zero-temperature and finite-temperature time-dependent magnetization processes in permanent magnets. Thermally activated transitions between local minima yield a slight time dependence of the extrinsic properties (magnetic viscosity, section 3.4).
As discussed in section 3.2.1, practically all magnetization processes relevant to permanent magnetism are caused by spatially inhomogeneous magnetization rotations, where the magnitude of the magnetization remains constant. The problem is therefore to explain coercivity and other extrinsic properties in terms of coherent and incoherent magnetization rotations.
The Origin of Coercivity: Factors Controlling Magnetic Resistance
From a basic point of view, coercivity means that a magnetization configuration \(M(\mathbf{r})\) is captured in a local energy minimum. For example, figure 3.1 shows how a local energy minimum vanishes at the nucleation field. However, Brown's paradox and equation (3.97) indicate that there exist incoherent magnetization processes in real materials which are much more harmful to coercivity than curling and coherent rotation. An empirical way of describing this undesired coercivity reduction is through the Kronmüller equation
\(H_{\text{c}} = \alpha_{K} \frac{2K_{1}}{\mu_{0}M_{\text{s}}} - D_{\text{eff}}M_{\text{s}}\) (3.98)
where \(\alpha_{K} < 1\) and \(D_{\text{eff}}\) is an effective magnetostatic interaction parameter (compare with (3.97) section 3.3.3). The parameter \(\alpha_{K}\) is of the order of 1% in as-cast magnets but can be as high as 40% in magnets whose production involves highly sophisticated processing routes. The parameters \(\alpha\) and \(D_{\text{eff}}\) are usually determined by plotting \(H_{\text{c}}M_{\text{s}}\) versus \(M_{\text{s}}^{2}\) for different temperatures.
In the context of permanent magnetism, the following basic coercivity mechanisms are of particular importance:

Figure 3.25. Nucleation and magnetic reversal.

Figure 3.26. Virgin curves for pinning-controlled and nucleation-controlled permanent magnets.
(i) coherent rotation in aligned fine particles;
(ii) magnetization curling in homogeneous single-domain particles;
(iii) incoherent nucleation in slightly inhomogeneous magnets;
(iv) domain-wall pinning in strongly inhomogeneous magnets;
(v) localized rotation in isotropic nanostructures.
The first three mechanisms produce nucleation-controlled coercivities, that is the formation of reversed domains is inhibited by high nucleation fields. By comparison, pinning coercivity means that reverse domains, once nucleated, do not immediately grow because the domain walls are unable to move (section 3.3.4).
The difference between nucleation-controlled and pinning-controlled coercivities is illustrated in figures 3.25 and 3.26. The second quadrant of the major loop is shown in figure 3.25. Starting from the completely aligned
magnetization (A), a reverse field gives rise to nucleation at (B). The nucleation field establishes a lower bound to the coercivity (C), but due to domain-wall pinning, the coercivity is often higher than \(H_{\text{N}}\) (C'). On the other hand, figure 3.19 shows that grain misalignment reduces the magnetization at which nucleation occurs, so that nucleation and pinning are more difficult to separate in isotropic magnets (section 3.3.5). There are two main experimental methods used to distinguish between pinning and nucleation. First, microscopy can be used to investigate the microstructure and to monitor domain-wall propagation. Second, virgin curves (initial curves) of microstructured magnets are very different for pinning-controlled and nucleation-controlled magnets (figure 3.26). After thermal demagnetization, domain walls in nucleation-controlled particles are very mobile, so that saturation is achieved in very low fields. By contrast, pinning centres impede the domain wall motion in both the virgin-curve and major-loop regimes.
Coercivity and Microstructure
The microstructure of a magnetic material determines the coercivity mechanism. In the following sections we describe the magnetic material by the energy functional
\(E = \int \left(A\left(\frac{\nabla M}{M_{\text{s}}}\right)^{2} - K_{1}\frac{(\mathbf{n} \cdot M)^{2}}{M_{\text{s}}^{2}} - \mu_{0}M \cdot H\right) \text{d}V\) (3.99)
where \(A(\mathbf{r})\), \(M_{\text{s}}(\mathbf{r})\), \(K_{1}(\mathbf{r})\) and \(\mathbf{n}(\mathbf{r})\) describe the magnet's real structure. Compared to the more general expression (3.52), this energy functional implies a closed-circuit configuration where \(H' = H\) and ignores higher-order anisotropy constants. Note that macroscopic demagnetizing fields amount to a curling-type contribution which, according to equation (3.94), can be incorporated as a demagnetizing field. On the other hand, atomic demagnetizing fields are negligible due to their weakness. This means that (3.99) amounts to the neglect of stray fields at grain edges etc (Kronmüller 1987, Schrefl and Fidler 1998) which is a fair approximation in many cases.
Defect-free microstructures give rise to high coercivities of the order of \(H_{0} = 2K_{1}/\mu_{0}M_{\text{s}}\) (section 3.2), but we will see that crystal inhomogeneities larger than a few nanometres tend to have a detrimental effect. In particular, sintered magnets consist of crystallites whose radii are often larger than 1 μm. Figure 3.27 shows the schematic microstructure of a crystallite. Localized nucleation processes are caused by inhomogeneities and occur in the bulk (A) or at the surface (B). To obtain nucleation-controlled magnets it is necessary to reduce the number of morphological inhomogeneities, for example by annealing or liquid-phase sintering. Otherwise it is helpful to try to pin the domain walls (C). The physics of inhomogeneous nucleation and pinning will be investigated in sections 3.3.3 and section 3.3.4, respectively.

Figure 3.27. Microstructural interpretation of nucleation and pinning. Nucleation may occur inside the particle (A) or at the surface (B), whereas pinning coercivity (C) is due to a limited domain-wall mobility. This figure assumes a common c-axis and therefore ignores the intermediate magnetization configurations relevant in random-anisotropy magnets (section 3.3.5).
Fine-particle Magnetism
In very small elongated particles there is some coercivity due to shape anisotropy, even if \(K_{1}\) is very low. This shape anisotropy is different from that associated with the archetypal horseshoe shape of steel magnets: from (3.96a) we see that shape alone cannot stabilize the magnetization of a macroscopic piece of material.
Shape anisotropy was exploited in bonded iron and cobalt particles and is still used in alnicos and various recording media. The former materials, known as lodex permanent magnets, exhibited coercivities of up to about 0.1 T and energy products approaching 30 kJ m-3. The fairly low remanence, about 0.74 T, arises from the volume of the lead matrix. A similar coercivity mechanism is realized in alnico magnets, which consist of elongated Fe–Co particles in an Ni–Al matrix. The Fe–Co needles have a length of up to 1 μm, but their diameter barely exceeds a few tens of nanometres. Using (3.96b) and taking \(D = 0\), \(A = 10\ pJ\ m^{-1}\ (10^{-11}\ J\ m^{-1})\), \(\mu_{0}M_{\text{s}} = 2.43\ T\) and \(R = 20\ nm\) we obtain the coercivity \(\mu_{0}H_{\text{c}} = 0.088\ T\), which is a typical result for alnico magnets.
Note that fine particle magnets are also known as 'elongated single-domain (ESD) particles'. This usage is unfortunate, because it gives the false impression that coherent rotation and single-domain magnetism are just two names for the same phenomenon. In fact, hard-magnetic powder particles having radii slightly smaller than \(R_{\text{sd}}\) are single domain but exhibit incoherent nucleation.
Nucleation in Inhomogeneous Permanent Magnets: Mechanisms and Applications
As emphasized by Aharoni (1962), Brown's paradox is related to the large anisotropy constants \(K_{1}\) in hard magnets. The point is that local reductions
of \(K_{1}\), associated with morphological inhomogeneities, reduce the nucleation field. Illustratively speaking, small reverse fields cause nucleation in soft regions and lead to direct magnetic reversal in particles whose radii are smaller than the critical single-domain radius \(R_{\text{sd}}\) (section 3.2.2). Nucleation in particles having a multidomain ground state may give rise to domain formation, but in practice the reverse field tends to destroy the domains immediately after formation (chapter 5).
Theoretical Description
Let us consider a magnet described by the energy functional (3.99) and subject to the external field \(\mathbf{H} = H\mathbf{e}_{z}\). Ideally aligned (textured) magnets are described by \(\mathbf{n} = \mathbf{e}_{z}\), that is the easy axis is parallel to the applied field. Putting (3.93) into (3.99), and minimizing \(E\) with respect to the small magnetization component \(\mathbf{m}\) we obtain (Skomski and Coey 1993)
\(-2\nabla(A(\mathbf{r})\nabla\mathbf{m}) + (2K_{1}(\mathbf{r}) + \mu_{0}M_{\text{s}}(\mathbf{r})H)\mathbf{m} = 0\). (3.100)
Since \(m_{x}\) and \(m_{y}\) are decoupled in this equation and yield degenerate sets of nucleation fields, it is sufficient to consider the function \(m_{x}(\mathbf{r})^{21}\).
For the moment, we will restrict ourselves to inhomogeneous anisotropy profiles \(K_{1}(\mathbf{r})\), so that (3.100) reduces to
\(-A\nabla^{2}m_{x} + K_{1}(\mathbf{r})m_{x} = \frac{\mu_{0}}{2}M_{\text{s}}H_{\text{N}}m_{x}\). (3.101)
This equation is reminiscent of Schrödinger's equation for a particle moving in a three-dimensional potential \(K_{1}(\mathbf{r})\), so that one can apply ideas familiar from quantum mechanics to discuss micromagnetics. In particular, the nucleation field corresponds to the quantum-mechanical ground-state energy, and the small transverse magnetization or nucleation mode \(m_{x}\) has its analogue in the wavefunction.
Case Studies
Equations (3.100) and (3.101) have been solved for a number of cases. Let us start with the simple example of a spherical soft inclusion of radius \(R\) (figure 3.28(a)) in a hard matrix. Putting \(K_{1} = 0\) inside the sphere and solving (3.101) for \(m_{x}(R) = 0\) yields the nucleation mode (figure 3.29(a))
\(m_{x} = \frac{C_{0}}{r} \sin \frac{\pi r}{R}\) (3.102)
and the nucleation field \(H_{\text{N}} = 2\pi^{2}A / \mu_{0}M_{\text{s}}R^{2}\). It is interesting to rewrite \(H_{\text{N}}\) in terms of the anisotropy field of the hard matrix
\(H_{\text{N}} = H_{0} \frac{\delta_{\text{h}}^{2}}{R^{2}}\) (3.103)

Figure 3.28. Sofi inclusion in a hard matrix: ( a ) spherical inclusion of radius R and < b> layered structures.

Figure 3.29. Radial plots of \(K_{1}\) (full lines) and \(m_{x}\) (dashed curves) in arbitrary units.
where \(\delta_{\text{h}}\) is the domain-wall width of the hard phase and \(H_{0}\) is the anisotropy field \(2K_{1}/\mu_{0}M_{\text{s}}\). Since typical wall widths \(\delta_{\text{h}}\) are of the order of 5 nm, soft inhomogeneities much larger than a few nanometres may be harmful to coercivity. The validity of (3.102) and (3.103) is restricted to \(H_{\text{N}} \ll H_{0}\), because \(m_{x}(R) = 0\) is only approximately true for finite \(H_{0}\) (figure 3.29b). If \(H_{\text{N}}\) is comparable to \(H_{0}\), (3.103) slightly overestimates the nucleation field.
The \(R\) dependence of the nucleation field is shown in figure 3.3022.
In reality, the hard and soft phases have different sets of material parameters: in the hard phase, \(K_{1}(\mathbf{r}) = K_{\text{h}}\), \(M_{\text{s}}(\mathbf{r}) = M_{\text{h}}\) and \(A(\mathbf{r}) = A_{\text{h}}\), whereas in the soft phase \(K_{1}(\mathbf{r}) = 0\), \(M_{\text{s}}(\mathbf{r}) = M'\) and \(A(\mathbf{r}) = A_{\text{s}}\). For a soft spherical inclusion we obtain from (3.100) the eigenvalue equation
\(\frac{A_{\text{s}}}{A_{\text{h}}}\left\{R\sqrt{\frac{\mu_{0}M'H_{\text{N}}}{2A_{\text{s}}}}\cot\left(R\sqrt{\frac{\mu_{0}M'H_{\text{N}}}{2A_{\text{s}}}}\right)-1\right\}+1+R\sqrt{\frac{2K - \mu_{0}M_{\text{h}}H_{\text{N}}}{2A_{\text{h}}}} = 0\) (3.104)
where the indices h and s refer to the hard and soft regions, respectively.
22 The \(1/R^{2}\) dependence arises from the sharp boundary between the hard and soft phases, whereas smooth boundaries yield a \(1/R\)-type law.

Figure 3.30. Nucleation fields of soft-magnetic spherical inclusions in a hard matrix (schematic).
Note that the ratio \(A_{\text{s}}/A_{\text{h}}\) yields only minor corrections to \(H_{\text{N}}\), because both quantities are of the order of \(10\ pJ\ m^{-1}\).
Another geometry of interest is that of layered inclusions (figure 3.28(b)) and multilayers. For multilayers one obtains from (3.101)
\(\frac{A_{\text{s}}}{A_{\text{h}}}\sqrt{\frac{\mu_{0}H_{\text{N}}M'}{2A_{\text{s}}}}\tan\left(\frac{L_{\text{s}}}{2}\sqrt{\frac{\mu_{0}H_{\text{N}}M'}{2A_{\text{s}}}}\right)=\sqrt{\frac{2K - \mu_{0}H_{\text{N}}M_{\text{h}}}{2A_{\text{h}}}}\tanh\left(\frac{L_{\text{h}}}{2}\sqrt{\frac{2K - \mu_{0}H_{\text{N}}M_{\text{h}}}{2A_{\text{h}}}}\right)\) (3.105)
where \(L_{\text{h}}\) and \(L_{\text{s}}\) denote the thicknesses of the hard and soft layers, respectively. The thickness dependence of \(H_{\text{N}}\) established by this equation is similar to figure 3.30.
Perturbation Theory
In lowest-order perturbation theory, the solution of the eigenvalue problem (3.101) is
\(H_{\text{N}}=\frac{2\langle K_{1}(\mathbf{r})\rangle}{\mu_{0}M_{\text{s}}}\) (3.106)
where \(\langle K_{1}(\mathbf{r})\rangle\) is the volume average of the anisotropy constant. Since \(K(\mathbf{r}) \approx 0\) in the soft phase, \(\langle K_{1}(\mathbf{r})\rangle = f_{\text{m}}K\), where \(f_{\text{m}}\) is the volume fraction of the hard phase. This approximation means that the lowest-order nucleation field is given by the volume-averaged anisotropy constant. This is reasonable if the spatial extension of the inhomogeneities is small, so that the exchange interaction assures a homogeneous magnetization. In terms of figure 3.30, the validity of (3.106) is restricted to the plateau region.
Equation (3.106), whose quantum-mechanical analogue is the 'virtual-crystal' approximation used to describe disordered alloys, can be used to calculate the energy products of ideally structured two-phase permanent magnets (Skomski and Coey 1993). The idea is to exploit the surplus anisotropy in modern rare-earth intermetallics to improve the permanent magnetic performance of soft phases such as Fe, which have a low anisotropy but a spontaneous magnetization \(M'\) which is larger than the magnetization \(M_{\text{h}}\) of hard phases. The \(\Theta = 0\) curve in figure 3.19 and the line ABC in figure 3.25 indicate that (3.101) yields rectangular hysteresis loops. The energy product is maximized by adjusting the volume fraction \(f\) of the hard phase so that \(H_{\text{c}} = H_{\text{N}} = (M_{\text{s}})/2\). This yields the energy product
\((BH)_{\text{max}} = \frac{\mu_{0}M^{2}}{4}\left(1 - \frac{\mu_{0}(M' - M_{\text{h}})M'}{2K}\right)\) (3.107)
realized by
\(f_{\text{m}} = \frac{\mu_{0}M^{2}}{4K_{\text{h}}}\). (3.108)
Since \(M' > M_{\text{h}}\), the energy product (3.107) is larger than the theoretical maximum energy product \(\mu_{0}M_{\text{h}}^{2}/4\) of the hard phase. Physically, the hard regions act as a skeleton to stiffen the magnetization direction of the soft regions.
To give an example, (3.107) and (3.108) predict that Sm2Fe17N3/Fe nanoscale hybrids could exhibit a theoretical energy product of 870 kJ m-3 with a volume fraction of the hard phase of only about 10%. Note that (3.107) and (3.108) are independent of the shape of the soft regions, so long as their size lies in the plateau region in figure 3.30. Even higher energy products, up to about 1 MJ m-3, are predicted if iron is replaced by Fe65Co35, where \(\mu_{0}M_{\text{s}} = 2.45\) T. The practical problem, however, is to realize a structure where the soft regions are sufficiently small to avoid nucleation at small fields while having the hard regions crystallographically aligned23. A conceivable solution is the use of multilayers consisting of hard and soft magnetic layers, but the advantages of these hypothetical'megajoule' magnets—with very high energy product and low raw material costs—may be largely outweighed by the demanding processing requirements for all but thin-film applications.
Second-order perturbation theory leads to the consideration of the correlation expression
\(\langle(K_{1}(\mathbf{r}) - \langle K_{1}\rangle)(K_{1}(\mathbf{r}') - \langle K_{1}\rangle)\rangle \approx K_{0}^{2}\text{e}^{-|\mathbf{r}-\mathbf{r}'|/R_{0}}\) (3.109)
and yields
\(H_{\text{N}} = \frac{2\langle K_{1}\rangle}{\mu_{0}M_{\text{s}}} - \frac{2K_{0}^{2}R_{0}^{2}}{\mu_{0}M_{\text{s}}A}\). (3.110)
Equation (3.106), whose quantum-mechanical analogue is the 'virtual-crystal' approximation used to describe disordered alloys, can be used to calculate the energy products of ideally structured two-phase permanent magnets (Skomski and Coey 1993). The idea is to exploit the surplus anisotropy in modern rare-earth intermetallics to improve the permanent magnetic performance of soft phases such as Fe, which have a low anisotropy but a spontaneous magnetization \(M'\) which is larger than the magnetization \(M_{\text{h}}\) of hard phases. The \(\Theta = 0\) curve in figure 3.19 and the line ABC in figure 3.25 indicate that (3.101) yields rectangular hysteresis loops. The energy product is maximized by adjusting the volume fraction \(f\) of the hard phase so that \(H_{\text{c}} = H_{\text{N}} = (M_{\text{s}})/2\). This yields the energy product
\((BH)_{\text{max}} = \frac{\mu_{0}M^{2}}{4}\left(1 - \frac{\mu_{0}(M' - M_{\text{h}})M'}{2K}\right)\) (3.107)
realized by
\(f_{\text{m}} = \frac{\mu_{0}M^{2}}{4K_{\text{h}}}\). (3.108)
Since \(M' > M_{\text{h}}\), the energy product (3.107) is larger than the theoretical maximum energy product \(\mu_{0}M_{\text{h}}^{2}/4\) of the hard phase. Physically, the hard regions act as a skeleton to stiffen the magnetization direction of the soft regions.
To give an example, (3.107) and (3.108) predict that Sm2Fe17N3/Fe nanoscale hybrids could exhibit a theoretical energy product of 870 kJ m-3 with a volume fraction of the hard phase of only about 10%. Note that (3.107) and (3.108) are independent of the shape of the soft regions, so long as their size lies in the plateau region in figure 3.30. Even higher energy products, up to about 1 MJ m-3, are predicted if iron is replaced by Fe65Co35, where \(\mu_{0}M_{\text{s}} = 2.45\) T. The practical problem, however, is to realize a structure where the soft regions are sufficiently small to avoid nucleation at small fields while having the hard regions crystallographically aligned23. A conceivable solution is the use of multilayers consisting of hard and soft magnetic layers, but the advantages of these hypothetical'megajoule' magnets—with very high energy product and low raw material costs—may be largely outweighed by the demanding processing requirements for all but thin-film applications.
Second-order perturbation theory leads to the consideration of the correlation expression
\(\langle(K_{1}(\mathbf{r}) - \langle K_{1}\rangle)(K_{1}(\mathbf{r}') - \langle K_{1}\rangle)\rangle \approx K_{0}^{2}\text{e}^{-|\mathbf{r}-\mathbf{r}'|/R_{0}}\) (3.109)
and yields
\(H_{\text{N}} = \frac{2\langle K_{1}\rangle}{\mu_{0}M_{\text{s}}} - \frac{2K_{0}^{2}R_{0}^{2}}{\mu_{0}M_{\text{s}}A}\). (3.110)
Here \(r_{0}\) is the average size of the hard and soft regions. This equation shows how anisotropic inhomogeneities affect the nucleation field: in the case of extended inhomogeneities the exchange stiffness \(A\) is unable to suppress the switching of soft regions.
Nucleation-Controlled Permanent Magnets
Equations (3.103) and (3.110) indicate that inhomogeneities much larger than the Bloch-wall width of the main phase are harmful to coercivity. In other words, nucleation-controlled permanent magnetism requires the removal of all inhomogeneities larger than a small multiple of the domain-wall width. This is a rather difficult task, since a single inhomogeneity is able to initiate the magnetic reversal of a whole particle or crystallite. Traditional ways of obtaining high nucleation fields are high-temperature annealing of the bulk material and use of powder or polycrystalline material with small crystallite sizes. The latter requirement is helpful in restricting the effect of scattered inhomogeneities to a small volume fraction of the whole magnet.
There are several ways of isolating magnetic crystallites. A very simple method is bonding of powder particles no larger than a few micrometres in a metal or polymer matrix. Matrix materials used in practice are polymers such as PVC and rubber, although metals such as zinc can be used as well. Nucleation-controlled coercivity in sintered magnets pre-supposes the formation of suitable particle interfaces on heat treatment. Roughly speaking, the heat treatment smooths the particle interfaces and eliminates crystal imperfections. In practice, it is helpful to support this process by off-stoichiometric additives. Examples of nucleation-controlled permanent magnets are sintered Nd2Fe14B and BaFe12O19 and bonded BaFe12O19, SmCo5, Nd2Fe14B and Sm2Fe17N3 magnets (chapter 5).
Pinning: Its Role in Magnetic Hysteresis and Coercivity
Nucleation leads to the formation of reversed magnetic domains which grow under the action of the reverse field. Domain growth involves domain-wall motion. If the nucleation leads to complete reversal then the coercivity \(H_{\text{c}}\) equals the nucleation field \(H_{\text{N}}\) and the coercivity is said to be nucleation controlled. Often, however, the domain-wall motion is impeded by lattice inhomogeneities (figure 3.31). This mechanism, called pinning, yields coercivities higher than \(H_{\text{N}}\). As a rule, nucleation dominates in more or less ideal crystals, whereas pinning is typical of materials with pronounced nanoscale inhomogeneities24. Thus, if one fails to create coercivity by reducing the number of nucleation inhomogeneities one may still get coercivity by creating pinning centres.

Figure 3.31. Pinning geometries: (a) a plane wall, (b) two rod-like pinning centres and (c) a three-dimensional model. The magnetic field is applied in the \(y\)-\(z\)-plane and moves the wall in the \(x\)-direction.
Pinning and Domain Wall energy
Depending on the geometry of the pinning centres and their spatial distribution there are various pinning models. Figure 3.31 shows some model geometries and the corresponding wall configurations in a small reverse field. The simplest pinning model (figure 3.31(a)) considers a plane domain wall of area \(L^{2}\). Local variations in \(K_{1}\) and \(A\) then lead to a dependence of the wall energy \(\gamma = (4AK_{1})^{1/2}\) on the wall position \(x\) (figure 3.32), and the magnetic energy is then to a fair approximation
\(E(x)=\gamma(x)L^{2}-2\mu_{0}M_{\text{s}}HxL^{2}\). (3.111)
Here the last term on the right-hand side describes the Zeeman interaction of the wall. Minimizing this function with respect to \(x\) yields the propagation field
\(H_{\text{p}}=\frac{(\text{d}\gamma(x)/\text{d}x)_{\text{max}}}{2\mu_{0}M_{\text{s}}}\). (3.112)
When the reverse field exceeds the propagation field then the magnetization jumps to the next minimum, which is called a Barkhausen jump. In practice, \(H_{\text{c}}\approx H_{\text{p}}\) and appropriately determined propagation fields are used to estimate the coercivity of pinning-controlled magnets.
Pinning by Small Defects
When the size of the inhomogeneities is smaller than the wall thickness \(\delta_{\text{B}}\), then (3.112) overestimates the pinning force. In fact, when the thickness \(b\) of the inhomogeneity goes to zero then \(H_{\text{p}}\) vanishes because the wall can move freely. Figure 3.33 shows \(H_{\text{p}}\) as a function of the thickness of the pinning centres. For small defects the propagation field is linear with \(b\), and we see that pinning is most pronounced when the thickness is comparable to the domain-wall width.

Figure 3.32. Pinning of u plane wall (a ) in a soft region and ( b) by a hard inclusion

Figure 3.33. Dependence of the pinning coercivity on the size of the pinning centre. The dashed curve is obtained from (3.112).
Note that the position and the height of the propagation-field maximum exhibit a minor dependence on the shape of the inhomogeneity (Becker and Döring 1939).
Equation (3.112) shows that the origin of the pinning force is the gradient of the wall energy rather than the size of the inhomogeneity. This means that sharp boundaries may be very effective pinning centres. In the case of step-like wall energy profiles \(\gamma(x)\), the propagation field approaches a plateau value for \(b > \delta_{\text{B}}\) and there is no descending branch in the \(H_{\text{p}}(b)\) curve.
The trapping of walls by a small number of powerful pinning centres, as in figure 3.32, is called strong pinning. By contrast, pinning caused by a large number of very small pinning centres, such as atomic defects, is called weak pinning. In the case of weak pinning, the wall energy is averaged over a distance of the order of \(\delta_{\text{B}}\), so that the density of pinning centres determines the pinning strength.
Domain-Wall Curvature
The configurations of figures 3.31(b) and 3.31(c) show that pinning may lead to the deformation of the wall. In general, the deformation of domain walls reduces \(H_{\text{p}}\), because it may provide a possibility to circumvent the energy barriers associated with pinning sites. The microstructure figure 3.31(c) leads to curved domain walls where the position \(x\) of a wall element depends on \(y\) and \(z\). The wall energy is
\(E = -2\mu_{0}M_{\text{s}}H\int x(y,z)\text{d}y\text{d}z + \gamma\int\sqrt{1 + \left(\frac{\partial x}{\partial y}\right)^{2} + \left(\frac{\partial x}{\partial z}\right)^{2}}\text{d}y\text{d}z\). (3.113)
To calculate the curvature of the wall we approximate the curved part of the wall by a segment of a sphere of radius \(R\) and minimize the energy \(E\) with respect to \(R\). The result is surprisingly simple: \(R = \gamma/\mu_{0}M_{\text{s}}H\). Since \(R\) cannot be smaller than the radius of \(R_{0}\), the pinning ring in figure 3.31(c), the pinning mechanism breaks down when the field exceeds \(\gamma/\mu_{0}M_{\text{s}}R_{0}\). Taking \(\mu_{0}M_{\text{s}} = 1.5\ T\), \(\gamma = 30\ mJ\ m^{-2}\) and \(R_{0} = 100\ nm\) we obtain the breakthrough field \(\mu_{0}H = 0.25\ T\). Similar results are obtained for other configurations, such as figure 3.31(b). This shows that very coarse pinning skeletons destroy coercivity, independently of the strength of the pinning.
Pinning Mechanisms
In steel magnets, magneto-elastic inhomogeneities (section 3.1.2.3) and martensitic lattice distortions (section 5.2.1) act as pinning centres. More common, however, is pinning at minority-phase inclusions where the intrinsic properties are different from the main phase. In practice, pinning-controlled permanent magnetism presupposes a sufficiently high density of pinning centres.
Pinning coercivity in Sm2Co17-based magnets is achieved by an off-stoichiometric net composition. SmCo5 is characterized by a very high anisotropy, of the order of 17 MJ m-3, which makes it possible to produce nucleation-controlled SmCo5 magnets. The saturation magnetization of Sm2Co17 is somewhat higher than that of SmCo5, but its comparatively low magnetocrystalline anisotropy, about 3.3 MJ m-3, makes it difficult to create coercivity in pure Sm2Co17 material. The outcome of the technological development of Sm-Co 2:17 magnets has been the production of pinning-type hybrids where Sm2Co17 crystallites are surrounded by a SmCo5 boundary phase (figures 3.34 and 5.22)25. The Sm2Co17 cells, whose size is of the order of 100 nm, are responsible for the saturation magnetization and consist of hexagonal Th2Ni17-type platelets in a rhombohedral Th2Zn17-type matrix. The 1:5 boundary phase enhances the coercivity by pinning the domain walls.

Figure 3.34. The schematic microstructure of a pinning-type Sm2Co17 magnet.
Nanostructured Permanent Magnets: Advancements and Opportunities
An interesting class of permanent magnetic materials are isotropic nanostructures such as melt-spun Nd2Fe14B and mechanically alloyed Sm2Fe17N3/Fe (section 5.3.4). Essentially, the materials consist of exchange-coupled grains whose radii are typically of the order of 50 nm or less. From a fundamental point of view, these nanostructures are closely related to amorphous magnets and random-anisotropy spin glasses, because their behaviour reflects competing exchange and anisotropy contributions. Yet, room-temperature coercivity of amorphous alloys is very low (10-6T for Fe80B20 and less than 1 μT for more sophisticated soft magnets), whereas that of properly processed rare-earth permanent magnets may exceed 4 T.
On a continuum level, the materials are described by (3.99), but in contrast to the \(K_{1}(\mathbf{r})\) disorder (section 3.3.3) there is also an easy-axis randomness \(n(\mathbf{r})\). Essentially, the exchange interaction favours parallel spin alignment, whereas the random easy axes tend to misalign the spins. Intergranular exchange in permanent magnets enhances the remanence while reducing the coercivity (section 3.3.5.2), but in magnetic recording media intergranular exchange is undesirable, because it reduces the achievable storage density by forming extended exchange-coupled magnetic regions (interaction domains).
Interaction-free Grains
A crude approximation is to treat the nanocrystalline magnet as an ensemble of misaligned but interaction-free Stoner–Wohlfarth particles (section 3.2.3) where the easy axis \(n\) is different in each grain but the local magnetization \(M(\mathbf{r})\) is parallel to \(n\). In practice, this approach can be used to describe

Figure 3.35. Zero-field spin configurations of interaction-free crystallites. Each arrow represents a grain or particle.
the weak-coupling regime, where \(K_{1}\) dominates or the intergranular exchange coupling is very weak. The net hysteresis loop is then a superposition of the hysteresis loops of the individual grains as shown in figure 3.19.
Due to the Stoner–Wohlfarth character of the loops it is comparatively easy to create coercivity. However, the ensemble average of the magnetization
\(\langle M_{z}\rangle = M_{\text{s}}\frac{\int_{0}^{\pi}P(\Theta)\cos\Theta\sin\Theta\text{d}\Theta}{\int_{0}^{\pi}P(\Theta)\sin\Theta\text{d}\Theta}\) (3.114)
is rather low for isotropic grain ensembles. In this equation, \(P(\Theta)\) is the probability that the easy axis of the crystallites makes an angle \(\Theta\) with the \(z\)-axis (field axis). After thermal demagnetization, all magnetization directions have equal probability, so that \(P(\Theta) = 1\) for all \(\Theta\) and \(\langle M_{z}\rangle = 0\). In the remanent state, \(M_{z} > 0\) for all grains, so that \(P(\Theta) = 1\) for \(\Theta < \pi/2\) and \(P(\Theta) = 0\) for \(\Theta > \pi/2\) (figure 3.35). The remanence calculated from (3.114) is then \(M_{\text{r}} = \langle M_{z}\rangle = M_{\text{s}}/2\). This yields a rather disappointing theoretical product \(\mu_{0}M_{\text{r}}^{2}/M_{\text{s}}^{2} = 1/16\) for isotropic uniaxial magnets (table 5.7). In random assemblies of cubic crystallites or easy-plane crystallites \(M_{\text{r}}/M_{\text{s}} > 1/2\), but it is very difficult to create usable coercivity in these materials, because the multiple easy axes are associated with higher-order anisotropy contributions.
The width of the hysteresis loop is determined by the coercivities of the individual grains (figure 3.19). In the case of misalignment angles \(\Theta < \pi/4\), the individual coercivities equal the Stoner–Wohlfarth nucleation or switching field
\(H_{\text{N}}=\frac{2K_{1}}{\mu_{0}M_{\text{s}}}\frac{1}{(\cos^{2/3}\Theta + \sin^{2/3}\Theta)^{3/2}}\). (3.115)
For \(\Theta > \pi/4\), the coercivity is realized by coherent rotation:
\(H_{\text{c}}=\frac{K_{1}}{\mu_{0}M_{\text{s}}}\sin2\Theta\). (3.116)

Figure 3.36. The angular dependence of the coercivity for coherent rotation.
In this regime, the nucleation field \(H_{\text{N}}\) is of academic interest only, describing singularities in the third quadrant of the \(M - H\) loop. Figure 3.36 shows \(H_{\text{N}}\) and \(H_{\text{c}}\) as a function of \(\Theta\).
The angular dependence of the coercivities (3.115) and (3.116) is different to the angular dependence \(H_{\text{c}}(\Theta) = H_{0} / \cos \Theta\), which was proposed by Kondorski (1937) to describe domain-wall pinning. The latter equation means that only the field projection parallel to the magnetization contributes to domain-wall motion.
The hysteresis loop of an interaction-free ensemble of powder particles or crystallites is obtained by integrating over all individual loops. To calculate the magnetization one can use (3.114), but now the angular distribution function \(P(\Theta)\) is field dependent. Typical results are shown in figure 3.37. As mentioned above, for uniaxial magnets the remanence \(M_{\text{r}} = M_{\text{s}} / 2\). For iron-type (\(K_{1} > 0\)) and nickel-type (\(K_{1} < 0\)) cubic magnets, \(M_{\text{r}} / M_{\text{s}}\) equals 0.832 and 0.866, respectively. Putting \(K_{2} = 0\) yields the coercivity \(H_{\text{c}} = 4970 H_{0}\). This means that properly designed, randomly oriented nanostructured magnets can have very high coercivities. A good example has been mechanically alloyed Sm2Fe17N3, where a coercivity in excess of 4 T has been achieved (Lui et al. 1992).
Intergranular Exchange and Remanence Enhancement
It is possible to improve the remanence of uniaxial grain assemblies (figure 3.35) by intergranular exchange, because exchange favours parallel spin alignment throughout the magnet. This is known as remanence enhancement. On the other hand, very strong intergranular interactions lead to the breakdown of the picture of individual grains and the magnetic reversal becomes a cooperative effect involving many grains. This is accompanied by a strong coercivity reduction. Consider the limit where the exchange ensures a homogeneous magnetization on a macroscopic scale. Putting \(M(\mathbf{r}) = eM_{\text{s}}\) in (3.99), where the unit vector \(e\)

Figure 3.37. Hysteresis loops for isotropic powder samples: (full curve) \(K_{2} = 0\) and (dashed curve) \(K_{2} = 0.4K_{1}\).
denotes the magnetization direction, we see that the average \(\langle(\mathbf{n} \cdot \mathbf{e})^{2}\rangle = 1/3\) is independent of \(\mathbf{e}\) and the anisotropy energy does not depend on the magnetization direction.
Figure 3.38 shows \(M_{\text{r}}\) and \(H_{\text{c}}\) as a function of the grain size. For dimensional reasons the strength of the intergranular exchange can be expressed in terms of the dimensionless parameter \(A/K_{1}R^{2}\), where \(R\) is an average grain radius. This parameter shows that intergranular exchange is most effective in the limit of small grain sizes. For atomic disorder, where \(R \approx a\), the magnetic energy (3.99) can be rewritten as a sum involving Heisenberg spin operators and is then referred to as the HPZ model (Harris, Plischke and Zuckermann 1973).
It is convenient to introduce a magnetic interaction length \(R_{0}\) which scales as \(\delta_{0} = (A/K_{1})^{1/2}\) but depends on structural details and is roughly equal to the Bloch-wall width. In the limit of weak exchange, perturbation theory can be used to calculate the remanence enhancement (Callen et al 1977), and from (3.99) one obtains in lowest order
\(M_{\text{r}} = M_{\text{s}}\left(\frac{1}{2} + \frac{R_{0}^{2}}{R^{2}}\right)\). (3.117)
The strong-exchange limit \(A/K_{1}R^{2} \ll 1\), realized for example in soft-magnetic amorphous alloys, cannot be described in terms of perturbation expressions such as (3.117). In fact, the cooperative character of the magnetization gives rise to dimensionality-dependent power laws describing \(H_{\text{c}}\) and \(M_{\text{r}}\) as a function of \(R\). Consider, for example, the coercivity \(H_{\text{c}} \approx 2K_{\text{eff}}/\mu_{0}M_{\text{s}}\), where \(K_{\text{eff}}\) is some effective, volume-averaged anisotropy constant. Due to the random easy-axis alignment the volume average of the anisotropy vanishes, but local anisotropy fluctuations yield some coercivity as a function of the size \(L_{\text{f}} \gg R\) of the exchange-coupled regions. The number

Figure 3.38. Remanence and coercivity of isotropic nanomagnets as a function of the average grain size \(R\) (schematic). This curve, where \(R_{0}\) is a nanoscale interaction length, ignores localized nucleation processes which reduce \(H_{\text{c}}\) and \(M_{\text{r}}\) in the limit of very large grains.
of grains in an exchange-correlated region is of the order of \((L_{\text{f}}/R)^{d}\), where \(d\) is the spatial dimensionality of the magnet, so that square-root fluctuations associated with the finite number of grains yield \(K_{\text{eff}} \approx K_{1}(R/L_{\text{f}})^{d/2}\). On the other hand, dimensional analysis shows that the magnetic correlation length \(L_{\text{f}} \approx (A/K_{\text{eff}})^{1/2}\). Hence, \(K_{\text{eff}}\) and \(H_{\text{c}}\) scale as \(R^{2d/(4 - d)}\) in the soft-magnetic limit of small grain sizes, that is as \(R^{6}\) in three dimensions. Since the coercivity is associated with square-root fluctuations of a large number of pinning centres (grains) it may be interpreted as a kind of weak-pinning coercivity (section 3.3.4.2). Note that \(d = 4\) is a marginal dimension and, as in the case of thermal fluctuations (section 2.3.3), mean-field theory is qualitatively exact in more than four dimensions27.
A consequence of intergranular exchange is that two-phase hysteresis-loop inflections such as those shown in figure 3.24(e) vanish in fine-grained magnetic nanostructures. By contrast, ensembles of microcrystallites can often be described as interaction-free particles, although exchange and magnetostatic interactions tend to modify the individual loops. A simple model is the Preisach model, where the interactions appear as random magnetic fields acting on the individual crystallites (Becker and Döring 1939). However, figure 3.11 and section 3.2.4 show that internal interaction fields are unable to give an appropriate description of cooperative magnetization processes. In fact, the validity of the internal-field approach is restricted to the non-cooperative ensembles, where the width of the switching-field distribution \(P(H_{\text{c}})\) of the (non-interacting) crystallites is larger than the magnitude of the interaction fields.
Real-Structure Modelling
Equation (3.117) and related mean-field treatments of the random-anisotropy problem cannot be used when more than two structural or magnetic length scales are involved. This refers in particular to two-phase nanostructures consisting of hard and soft regions, where the soft phase is used to enhance the remanence beyond figure 3.38. In practice, one encounters sharp grain boundaries and at interfaces between misaligned hard grains the spin alignment is limited to the vicinity of the grain boundary. The width of the perturbed region, about \((A/K_{1})^{1/2}\), is smaller than the grain size in typical nanostructured permanent magnets, so that the remanence enhancement is restricted to the grain-boundary region.
In isotropic hard-soft composites, the low anisotropy of the soft phase means that the spin perturbation responsible for the remanence enhancement extends well into the soft phase. As a rule, the low cubic anisotropy of the soft phase is a small perturbation. However, the behaviour of soft magnetic regions larger than the coherence length (3.91) is strongly affected by magnetostatic interactions. In this quite complicated case the magnetic properties can be predicted from numerical calculations (Schrefl and Fidler 1998).