The book ** Simple Models of Magnetism** by Ralph Skomski, first published in 2008, is now available in paperback format. The book has been welcomed by the scientific community as a "highly readable and thorough account" and of "great use to graduate students and experts in the field alike". The original set of printed copies has been sold out, although hardback orders are still being accepted by Oxford University Press and produced from a laser printer. The 2012 paperback edition differs from the hardcover edition by various minor additions and corrections, most notably two added special-topic sections on the "new" quantum mechanics involving the Berry phase and on orbital-free density-functional theory.

The Berry phase is a phase factor in the quantum-mechanical wave function that was overlooked until 1984 and now penetrates many areas of physics, including magnetism. Orbital-free density functional theory (OF-DFT) or "true" density-functional theory is a quantum-mechanical approach without wave functions.

### Contents

*1. Introduction: The Simplest Models of Magnetism*

2.1. Atomic Origin of Exchange. - 2.2. Magnetic Ions. - 2.3. Exchange between Local Moments. - 2.4. Itinerant Magnetism.

*3. Models of Magnetic Anisotropy*

3.1. Phenomenological Models. - 3.2. Models of Pair Anisotropy. - 3.3. Spin-Orbit Coupling and Crystal-Field Interaction. - 3.4. The Single-Ion Model of Magnetic Anisotropy. - 3.5 Other Anisotropies.

4.1. Stoner-Wohlfarth Model. - 4.2. Hysteresis. - 4.3. Coercivity. - 4.4. Grain-Boundary Models.

*5. Finite-Temperature Magnetism*

5.1. Basic Statistical Mechanics. - 5.2. Spin-Space Modeling. - 5.3. Mean-Field Models. - 5.4. Critical Behavior. - 5.5. Temperature Dependence of Anisotropy.

6.1. Quantum Dynamics and Resonance. - 6.2. Relaxation. - 6.3. Coarse-Grained Models. - 6.4. Slow Magnetization Dynamics.

*7. Special Topics and Interdisciplinary Models*

7.1. Disordered Magnets and Spin Glasses. - 7.2. Soft Matter, Transport, and Magnetism. - 7.3. Bruggeman Model. - 7.4. Nanostructures, Thin Films, and Surfaces. - 7.5. Beyond Magnetism.

*Appendices*

A.1. Units and Constants. - A.2. Mathematics - A.3. Basic Quantum Mechanics. - A.4. Electromagnetism. - A.5. Magnetic Materials. - A.6. Forgotten and Reinvented.

The total length is slightly more than 350 pages. All chapters have exercise sections, the references (about 320) contain full paper titles, and there are more than 750 index terms.

#### 1. Introduction: The Simplest Models of Magnetism

###### 1.1. Field and magnetization

1.2. The circular-current model

1.3. Paramagnetic spins

1.4. Ising model and exchange

1.5. The viscoelastic model of magnetization dynamics

By the second half of the 19th century, Maxwell's equations had established the relation between different electromagnetic fields, and scientists and engineers were aware of the dipolar character of magnetostatic forces and interactions (fig. 1). What is the atomic origin of the magnetization, and how does it involve quantum mechanics and relativistic physics? What determines the hard or soft character of a steel magnet? How to explain the Curie temperature, and why can't we ascribe it to magnetostatic interaction between atomic dipole moments? How can magnetic properties by tuned by systematically varying crystal structure, chemical composition, and nanostructure? Which ways are there to exploit magnetism in computer science and in other areas of advanced technology? Myriads of questions like these have arisen every decade and turned magnetism into a field of intense research. The modeling of magnetic phenomena and materials is a crucial aspect of this research.

###### 19th century magnetism: (a) field created by a horseshoe magnet, (b) mechanical force on a ferromagnetic body, (c) flux lines in a magnetic medium, (d) dipole character of magnetism, and (e) currents as one source of magnetic fields.

To provide an introduction to magnetism and to magnetic modeling, we start with some well-known and extremely simple models. In fact, some aspects of the models are simplistic rather than simple, we may ask the question:

*What is wrong with the simplest models of magnetism?*

However, in spite of their very limited applicability, these models are not useless. First, they hint at typical problems encountered in magnetism and provide a basis and motivation for the models in the main chapters of the book. Second, even the simplest models describe a piece of reality if used in an adequate context.

#### 2. Models of Exchange

##### 2.1. Atomic Origin of Exchange

###### 2.1.1. One-electron wave functions

2.1.2. Two-electron wave functions

2.1.3. Hamiltonian and spin structure

2.1.4. Heisenberg model

2.1.5. Independent-electron approximation

2.1.6. Correlations

2.1.7. *Hubbard model

2.1.8. *Kondo model

The competition between the kinetic and Coulomb energies decides whether the spin state is ferromagnetic (??) or antiferromagnetic (??). The kinetic energy is realized by interatomic hopping and accompanied by an energy reduction due to hybridization. The Pauli principle forbids the occupancy of low-lying or 'bonding' state by electrons with parallel spin, so that the corresponding ground-state spin structure is ??. Ferromagnetism is a many-electron effect and means that some electrons occupy excited one-electron states. The necessary energy is supplied by the Coulomb repulsion between electrons, which punishes ?? occupancies of one-electron orbitals. The corresponding exchange constant *J*, defined as half the energy difference between the ?? and ?? states, depends on parameters such as the interatomic distance and the number of electrons per atom. An exactly solvable two-electron model considers two atoms and one atomic orbital per atom. Aside from direct exchange, which is always positive, *J* reflects the relative strength of the Coulomb integral compared to the hopping integral. As a rule, pronounced interatomic hopping destroys the parallel spin alignment. Finally, a two-electron model is used to introduce and discuss various models and approximations, including the Heisenberg, Hubbard model, and Kondo models, and to separate independent-electron or Hartree-Fock contributions from correlation corrections.

**Energy levels of noninteracting electrons (a) paramagnetism and (b) ferromagnetism. Ferromagnetism occurs if the Coulomb interaction, as parameterized by ***I*, is larger than the one-electron level splitting **D***E* ~ 1/*D*. In the Stoner model (right), ferromagnetism this means *D*(*E*_{F}) > 1/I. The dotted line describes the onset of ferromagnetism, *D*(*E*) = 1/*I*.

*I*, is larger than the one-electron level splitting

*E*~ 1/

*D*. In the Stoner model (right), ferromagnetism this means

*D*(

*E*

_{F}) > 1/I. The dotted line describes the onset of ferromagnetism,

*D*(

*E*) = 1/

*I*.

##### 2.2. Magnetic Ions

###### 2.2.1. Atomic orbitals

2.2.2. Angular-momentum algebra

2.2.3. Vector model and Hund's rules

2.2.4. Spin and orbital moment

The magnetism of solids, including transition-metal magnets, retains many features of atomic or ionic magnetism. Ionic magnetic moments are created by the intra-atomic exchange between inner-shell electrons. The electrons are described by the quantum numbers *n*, *l*, and *m*, as obtained from the Schrödinger equation, and obey the corresponding angular-momentum commutation rules. For practical reasons, magnetic ions are divided into iron-series (3d), palladium-series (4d), platinum-series (5d), and actinide (5f) ions, but other electrons, such as 2p electrons, may also be involved. Atomic moments contain both spin and orbital contributions but the survival of ionic moments in a crystalline environment depends on the considered element. As a rule, Hund's-rules 4f moments are conserved, whereas 3d orbital moments are largely quenched by the crystal field. The orbital moment contributes to magnetization and is an important requirement for magnetocrystalline anisotropy.

##### 2.3. Exchange between Local Moments

###### 2.3.1. Exchange in oxides

2.3.2. Ruderman-Kittel exchange

2.3.3. Zero-temperature spin structure

Interatomic exchange between magnetic moments localized on individual atoms is well described by Heisenberg interactions *J*_{ij} between atomic spins of constant magnitude. Depending on the respective positive or negative sign of the *J*_{ij}, the favors parallel or antiparallel alignment of neighboring spins, which often translates into ferromagnetic (FM), ferrimagnetic (FI), or antiferromagnetic (AFM) order. Ferri- and antiferromagnetic spin structures involve the formation of magnetic sublattices, which may be spontaneous (AFM) or indicative of nonequivalent crystallographic sites (FI). In addition, there exist noncollinear or incommensurate spin structures due to competing exchange interactions, for example in many rare-earth elements. Noncollinear spin configurations may also be caused by external magnetic fields, as exemplified by the spin-flop transition in anti-ferromagnets. Oxides are often ferri- or antiferromagnetic, and typical exchange mechanism are super-exchange and double exchange. Interactions between local moments in metals are fairly well approximated by the RKKY model, which yields an oscillatory exchange mediated by conduction electrons.

##### 2.4. Itinerant Magnetism

###### 2.4.1. Free electrons, Pauli susceptibility, and the Bloch model

2.4.2. Band structure

2.4.3. Stoner model and beyond

2.4.4. *Itinerant antiferromagnets

The magnetism of Fe, Co, and Ni, as well as that of typical transition-metal alloys, is delocalized or *itinerant*. In a simple one-electron picture, the electrons fill the available delocalized states until the Fermi level is reached. This explains the widespread occurrence of noninteger magnetic moments in metallic ferromagnets. Nonmagnetic metals have two equally populated ? and ? subbands; and an applied magnetic field transfers a few electrons from the ? band to the ? band. This is known as Pauli paramagnetism, but the corresponding spin polarization is small, typically less than 0.1%. In itinerant ferromagnets, the atomic orbitals hybridize and form bands. The corresponding one-electron energies, as epitomized by the band width, compete against Hund's-rules intra-atomic exchange, and ferromagnetism is realized in *narrow* bands. The simplest model of itinerant ferromagnetism is the Bloch model, where the intra-atomic exchange is evaluated for free electrons. A more sophisticated model is the Stoner model, which relates the onset of ferromagnetism to the density of states (DOS) at the Fermi level. The density of states exhibits a strong dependence on the crystal structure, which makes it difficult to predict the ferromagnetic moment from the atomic composition. Itinerant magnets with approximately half-filled bands exhibit a strong trend towards antiferromagnetism, because the hybridization energy of half-filled ? and ? bands is lower than that of completely-filled ? bands.

#### 3. Models of Magnetic Anisotropy

##### 3.1. Phenomenological Models

###### 3.1.1. Uniaxial anisotropy

3.1.2. Second-order anisotropy of general symmetry

3.1.3. Higher-order anisotropies of nonuniaxial symmetry

3.1.4. Cubic anisotropy

3.1.5. Anisotropy coefficients

3.1.6. Anisotropy fields

The dependence of the magnetic energy on the magnetization angle is usually parameterized in terms of anisotropy constants. The simplest phenomenological anisotropy model is lowest-order or second-order uniaxial anisotropy, *E*_{a}/*V* = *K*_{1}sin^{2}*q*, where *K*_{1} is the first uniaxial anisotropy constant. Most magnetic materials exhibit additional higher-order and, depending on their crystal symmetry, nonuniaxial contributions. These are often, but not always, small corrections to *K*_{1}. An important exception is cubic anisotropy, where the second-order terms are zero by symmetry, and the leading term is fourth-order. Other parameterization tools are anisotropy coefficients, such as *k*_{2}, and anisotropy fields, such as 2*K*_{1}/µ_{o}*M*_{s}.

##### 3.2. Models of Pair Anisotropy

###### 3.2.1. Dipolar interactions and shape anisotropy

3.2.2. Demagnetizing factors

3.2.3. Applicability of the shape-anisotropy model

3.2.4. The Néel model

Magnetostatic dipole interactions are one source of magnetic anisotropy, although typical magnetostatic anisotropies are much smaller than the leading magnetocrystalline anisotropy contribution of electronic origin. Aside from a lattice contribution to the magneto-crystalline anisotropy, magnetostatic anisotropy manifests itself as macroscopic shape anisotropy. Due to magnetization inhomogenities, shape anisotropy is limited to very small length scales. An atomic pair-anisotropy model is the Néel model, where the anisotropy is parameterized in terms of bond directions. The Néel model yields the correct symmetry of the anisotropy contributions but fails to reproduce the single-ion origin of the anisotropy of most bulk materials and surfaces.

##### 3.3. Spin-Orbit Coupling and Crystal-Field Interaction

###### 3.3.1. Relativistic origin of magnetism

3.3.2. Hydrogen-like atomic wave functions

3.3.3. Crystal-field interaction

3.3.4. Quenching

3.3.5. Spin-orbit coupling

Spin-orbit coupling and crystal-field interaction are key requirements for magnetocrystalline anisotropy. Spin-orbit coupling is a higher-order terms in the relativistic Pauli expansion in terms of the small parameter *v*/*c*, where *v* is the velocity of the electrons. Magnetic interactions, such as spin-orbit coupling, tend to be much smaller than the leading electrostatic and exchange interactions, but the high effective nuclear charge of inner electrons in rare-earth atoms enhances the spin-orbit coupling. In solids, the spin-orbit coupling competes against the crystal-field splitting, which favors the suppression (quenching) of the orbital moment. Quenched orbitals have a standing-wave character and adapt more easily to the crystal field than unquenched or running-wave orbitals. The outcome of the competition between spin-orbit coupling and crystal-field interaction determines the degree of quenching and the magnitude of the magnetic anisotropy. 3d electrons tend to undergo strong quenching. For example, iron has a magnetization of about 2.2 µ_{B}, but only about 5% of this moment are of orbital origin. The opposite is true for the 4f electrons in rare-earth ions, which are close to the nucleus and therefore combine a weak crystal-field interaction with a strong spin-orbit coupling.

##### 3.4. The Single-Ion Model of Magnetic Anisotropy

###### 3.4.1. Rare-earth anisotropy

3.4.2. Point-charge model

3.4.3. The superposition model

3.4.4. Transition-metal anisotropy

Single-ion anisotropy is usually much stronger than pair anisotropy and combines electrostatic crystal-field and relativistic spin-orbit interactions. The spin-orbit coupling creates current loops that interact with the anisotropic crystal field. Rare-earth anisotropy involves unquenched wave functions, and the magnetic anisotropy energy is approximately equal to the electrostatic energy of rigid 4f ions in the crystal field. This is exploited in models such as the point-charge model and the superposition model. In 3d magnets, the orbital moments are largely quenched, and spin-orbit coupling is a small perturbation to the leading crystal-field and hopping terms. Heavy transition-metal atoms combine strong crystal-field interaction with strong spin-orbit coupling and are intermediate between 3d and 4f atoms.

###### Point-charge model, as applied to the crystal-field interaction of a prolate ion, such as Sm^{3+}: (a) cubic crystal-field and (b-c) tetragonal crystal fields of opposite crystal-field parameter *A*_{2}^{0}. For the shown prolate ion, (b) and (c) are easy-axis and easy-plane, respectively. For instance, in (c), the negative 4f charges at the ends of the ion are repelled by the negative crystal-field charges, which would force the magnetization (arrow) from the *c*-axis direction into the *a*-*b* plane.

##### 3.5. Other Anisotropies

###### 3.5.1. Magnetoelasticity

3.5.2. Anisotropic exchange

3.5.3. Models of surface anisotropy

Magnetoelastic anisotropy is caused by mechanical strain and tends to yields substantial anisotropy contributions in soft magnets. Physically, it is equivalent to magnetocrystalline anisotropy, because a strained cubic lattice can be considered as an unstrained lattice with reduced symmetry. Surface and interface anisotropies are of magnetocrystalline origin, too. A key feature is that their strengths and symmetries depend on the indexing of the surfaces. Other anisotropies involving spin-orbit coupling are the Dzyaloshinski-Moriya interaction in magnets with very low symmetry and the anisotropic exchange, which must not be confused with anisotropic exchange bonds. The so-called unidirectional anisotropy in exchange-coupled magnets is a biasing effect that does not involve spin-orbit coupling.

#### 4. Micromagnetic Models

##### 4.1. Stoner-Wohlfarth Model

###### 4.1.1. Aligned Stoner-Wohlfarth particles

4.1.2. Angular dependence

4.1.3. Spin reorientations and other first-order transitions

4.1.4. Limitations of the Stoner-Wohlfarth model

A simple but powerful micromagnetic model is the Stoner-Wohlfarth or coherent rotation model. It assumes a rigid exchange coupling between the atomic in a ferromagnetic body and reproduces the exact micromagnetic behavior in the limit of very small particles. This includes structural inhomogeneous particles and features such as a grain boundaries, but the length-scale requirements are quite stringent. The model is a useful starting point for the discussion of the angular dependence of magnetization curves and predicts spin reorientation transitions, for example from easy-axis to easy-cone magnetism. A key prediction of the Stoner-Wohlfarth model is that the coercivity is equal to the anisotropy field. This is rarely observed, due to the size of particles encountered in practice. In large particles, magnetostatic interactions lead to incoherent magnetic reversal, even in the absence of morphological inhomogenities and in single-domain particles.

##### 4.2. Hysteresis

###### 4.2.1. Micromagnetic free energy

4.2.2. *Magnetostatic self-interaction

4.2.3. *Exchange stiffness

4.2.4. Linearized micromagnetic equations

4.2.5. Micromagnetic scaling

4.2.6. Domains and domain walls

The local magnetization **M**(**r**) is determined by the competition between interatomic exchange, anisotropy, Zeeman energy, and magnetostatic self-interaction. The energy contributions establish a complicated nonlinear and nonlocal problem, involving metastable magnetic energy minima and leading to a history dependence known as hysteresis. In some cases, it is possible to linearize the micromagnetic equations. In the Stoner-Wohlfarth model, the hysteresis is independent of the exchange, which is assumed to be sufficiently strong to ensure a rigid coupling between the spins. In reality, this is rarely the case, although exchange favors relatively smooth magnetization variations. Magnetostatic contributions dominate in macroscopic magnets, where they lead to the formation of magnetic domains, separated by domain walls. The wall width *d*_{B} = p(*A*/*K*_{1})^{1/2} reflects competition between exchange and anisotropy, but there are no comparable simple relations for the domain size.

##### 4.3. Coercivity

###### 4.3.1. Nucleation

4.3.2. Pinning

4.3.3. Phenomenological coercivity modeling

As other extrinsic or hysteretic properties, coercivity is strongly real-structure dependent. Aside from coherent rotation, important coercivity mechanisms are, curling, localized nucleation, and domain-wall pinning. Nucleation refers to the onset of magnetization reversal and determines the coercivity in nearly defect-free magnets. With increasing size, the nucleation mechanism in perfect ellipsoids of revolution changes from coherent rotation to curling. The curling mode costs some exchange energy but is magnetostatically favorable due to vortex-like flux closure. However, both coherent rotation and curling greatly overestimate the coercivity of most magnetic materials. This disagreement, known as Brown's paradox, is solved by considering localized nucleation due to imperfections. Micromagnetic localization costs exchange energy, too, but is favorable from the point of view of anisotropy, because it exploits local anisotropy minima. The transition from coherent rotation to curling or localized nucleation is unrelated to the single-domain character of the magnet, and magnetization reversal in single-domain particles is not necessarily coherent. *Pinning* means that the motion of domain wall is impeded by imperfections. It determines the coercivity in strongly inhomogeneous magnets. Some pinning mechanisms are Kersten pinning, Gaunt-Friedel pinning and weak pinning. The above considered micromagnetic models must be distinguished from phenomenological models and methods, such as Preisach models and remanence plots.

###### Pinning in a submillimeter piece of iron containing a small inhomogenity. The external magnetic field points in the ? direction and increases from left to right.

##### 4.4. Grain-Boundary Models

###### 4.4.1. Boundary conditions

4.4.2. Spin structure at grain boundaries

4.4.3. Models with atomic resolution

4.4.4. Nanojunctions

The spin structure of magnets is modified by imperfections such as grain boundaries and nanojunctions. On a continuum level, grain boundaries are modeled by taking into account the appropriate boundary conditions. Even for well-localized and weak imperfections, the magnetization perturbation extends several nanometers into the adjacent ferromagnetic regions. Micromagnetic problems with atomic resolution can, in principle, be calculated from first principles, but the large number of affected atoms and the involved small energies make these calculations very difficult. In fact, models with atomic resolution tend to yield rather small corrections to the continuum results. At granular interfaces, both the reduced grain-boundary exchange and grain misalignment contribute to the perturbation of the spin structure. Changes in the interatomic exchange yield large magnetization gradients, whereas anisotropy changes at grain boundaries of hard-magnetic materials are much less effective in perturbing the spin structure.

#### 5. Finite-Temperature Magnetism

##### 5.1. Basic Statistical Mechanics

###### 5.1.1. Probability and partition function

5.1.2. *Fluctuations and response

5.1.3. Phase transitions

5.1.4. Landau theory

Finite-temperature equilibrium amounts to the minimization of the free energy *F* = *E* - *TS*, where the entropy *S* describes thermal disorder. Zero-temperature equilibrium means that only the lowest-lying state is occupied, but for nonzero temperatures T > 0, the interaction with the heat bath leads to the population of excited states. The probability of finding an equilibrium spin configuration (index µ) is given by the Boltzmann distribution exp(-*E*_{µ}/*k*_{B}*T*). Thermal averages are conveniently obtained as derivatives of the partition function *Z* = S_{µ} exp(-*E*_{µ}/*k*_{B}*T*). The partition function leads to general relationships, such as *F* = *k*_{B}*T* ln*Z* and the fluctuation-response theorem relating real-space correlations to the susceptibility, and to model-specific predictions. The main challenge is the large number spin configurations, which increases exponentially with the size of the magnet. A topic of particular interest is phase transitions, especially the continuous (or second-order) phase transition at the critical or Curie temperature *T*_{c}. A simple phenomenological free-energy model is the Landau model, which treats the critical behavior on a mean-field level.

##### 5.2. Spin-Space Modeling

###### 5.2.1. Heisenberg models

5.2.2. Ising, XY, and other n-vector models

5.2.3. *Other discrete and continuum spin models

5.2.4. Ionic excitations

5.2.5. Spin fluctuations in itinerant magnets

The modeling of atomic spins is a key aspect of finite-temperature magnetism. The simplest model is the Ising model, where each atom has two spin states *s*_{i} = ± 1. The Ising model captures some essential features of magnetism but ignores the quantum-mechanical effects and amounts to the unphysical prediction of infinite magnetic anisotropy. An isotropic model is the Heisenberg model, which exists in form of classical and quantum-mechanical realizations. A generalization of Ising and Heisenberg models is the *n*-vector model, which also includes the classical limits of models such as the XY model. The quantum-mechanical Heisenberg model provides an adequate description of typical magnetic ions. In magnetic or exchange fields, the energy levels of the magnetic ions split into multiplets whose finite-temperature occupancy is described by Brillouin functions. A very complicated situation is encountered in itinerant magnets, such as Fe, Co, and Ni. The Stoner model greatly overestimates the Curie temperature, because thermal excitations create spin disorder and break the assumed Bloch symmetry of the wave functions. As a consequence, itinerant moments are fairly well-conserved at the Curie temperature and the spin structure resembles that of localized magnets. An exception are very weak itinerant ferromagnets, such as ZrZn_{2}, whose finite-temperature behaviour is determined by long-wavelength spin-fluctuations close to the Stoner limit.

###### Spin structure in itinerant magnets: (a) zero temperature, (b) Stoner picture and (c) localized or Heisenberg picture. In (b) and (c), the temperature is just below *T*_{c}. In iron-series transition metals, the spin structure at *T*_{c} is close to (c), with little reduction in the magnitude of the moment.

##### 5.3. Mean-Field Models

###### 5.3.1. Mean-field Hamiltonians

5.3.2. Basic mean-field predictions

5.3.3. *Ornstein-Zernike correlations

5.3.4. Magnetization and Curie temperature

5.3.5. *Mean-field Curie temperature of n-vector models

5.3.6. Two-sublattice magnetism

5.3.7. Merits and limitations of mean-field models

The explanation of ferromagnetic order requires the consideration of interatomic exchange interactions. Mean-field models place individual atomic spins in an exchange field created by neighboring atoms. This maps the interaction problem to the problem of noninteracting spins in a magnetic field, except that the field must be calculated selfconsistently from the magnetization. Linearization of the mean-field equation yields the Curie temperature *T*_{c} ~ *zJ*, where *z* is the number of interaction neighbors and *J* is the interatomic exchange. Below *T*_{c}, the mean-field equations have two ferromagnetic solutions ± *M*_{s}(*T*). The mean-field model is easily generalized to two or more sublattices, where it yields complicated spin structures, such as ferrimagnets, antiferromagnets, and complicated spin structures. In the most general case of *N* sublattices (or *N* non-equivalent atomic sites) it yields *N* coupled algebraic equations. The mean-field approximation provides a reasonable description of the finite-temperature magnetization for a broad range of models, including Ising and Heisenberg models. It also describes critical fluctuations, albeit on an dimensionality-independent Ornstein-Zernike level. Mean-field theory breaks down at very low temperatures, where the excitations have the character of spin waves, and near *T*_{c}, where critical fluctuations interfere. In particular, mean-field models tend to overestimate the Curie temperature. This failure is most pronounced in one-dimensional magnets where *T*_{c} = 0 but mean-field theory predicts *T*_{c} > 0.

##### 5.4. Critical Behavior

###### 5.4.1. One-dimensional models

5.4.2. Superparamagnetic clusters

5.4.3. *Ginzburg criterion

5.4.4. Fluctuations and criticality

5.4.5. Renormalization group

The behavior of the magnetization near the critical point is largely determined by long-range thermodynamic fluctuations. Mean-field models provide a poor description of this regime, because they assume that the interatomic exchange translates into a local exchange field. Critical behavior is described by scaling laws such *M*_{s} ~ (*T*_{c} ? *T*)^{b}, *c* ~ |*T* ? *T*_{c}|^{-}^{g}, and *x* ~ |*T* ? *T*_{c}|^{?}^{n}, where the exponents generally differ from the mean-field predictions. In addition, critical fluctuations tend to reduce the Curie temperature. The deviations from mean-field behavior depend on the spatial dimensionality *d* of the magnet and on the considered model (spin dimensionality *n* and long- or short-range character of the exchange). However, critical exponents are independent of structural details such as the number z of nearest neighbors. By gauging the role of fluctuations as a function of *d*, one finds that mean field exponents are essentially exact for *d* ? 4 (Ginzburg criterion). There are very few exact solutions in two or more dimensions, most notably the Onsager solution for the two-dimensional Ising model. A famous approach to treat critical fluctuations is renormalization-group analysis, where one exploits that the correlation length diverges at the critical point. The idea is to iteratively rescale the size of the magnet, *x*' = *x*/*b*, and to exploit that *x*' = *x*/*b* = ? at the critical point.

##### 5.5. Temperature Dependence of Anisotropy

###### 5.5.1. Callen and Callen model

5.5.2. Rare-earth anisotropy

5.5.3. Sublattice modeling

Room-temperature anisotropy energies per atom are much smaller than *k*_{B}*T*, indicating that the finite-temperature magnetic anisotropy relies on the support by interatomic exchange. The exchange suppresses the excitation of atomic spins onto states with reduced anisotropy. Simple ferromagnets, such as Fe and Co, obey the *m* = *n*(*n*+1)/2 power laws predicted by the Callen and Callen model. For example, 2nd- and 4th-order anisotropies are characterized by *m* = 3 and *m* = 10. However, the applicability of the model is an exception rather than the rule. In the single-ion model of rare-earth anisotropy, which describes high-performance permanent magnets such as SmCo_{5} and Nd_{2}Fe_{14}B, the temperature dependence is determined by the intersublattice exchange and qualitatively very different from the Callen and Callen predictions. Pictorially, intramultiplet excitations destroy the net asphercity of the 4f charge clouds by thermally randomizing the directions of the rare-earth moment. Other compounds have exponents that deviate from the Callen and Callen predictions, for example *m* = 2 and *m* = 1 for the 2nd-order anisotropies of *L*1_{0} and actinide magnets, respectively.

#### 6. Magnetization Dynamics

##### 6.1. Quantum Dynamics and Resonance

###### 6.1.1. Spin precession

6.1.2. Uniform magnetic resonance

6.1.3. Spin waves

6.1.4. Spin dynamics in inhomogeneous magnets

The dynamics of paramagnetic ions has the character of spin precession, and essential features of this picture carry over to ferromagnets. This is exploited in the modeling of magnetic resonance and spin waves. Spin waves are wave-vector dependent excited states that contribute to both finite-temperature equilibrium and nonequilibrium magnetic properties. For example, the low-temperature spontaneous magnetization of three-dimensional Heisenberg ferromagnets is significantly smaller than the mean-field prediction, and Bloch's spin-wave arguments indicate that there is no long-range isotropic ferromagnetism in two or less dimensions. The magnetization of spin waves is generally delocalized but always corresponds to an integer number of switched spins. In perfect ferromagnets, the spin waves form a continuum of delocalized states, but real-structure imperfections and nanoscale structural features lead to spin-wave localization and discrete spin-wave levels.

###### Magnetization processes: (a) thermally activated and (b) field-induced. In most systems, thermal activation is a small correction to the leading field-dependent or 'static' mechanism.

##### 6.2. Relaxation

###### 6.2.1. Damped precession

6.2.2. *Physical origin of relaxation

6.2.3. *A mechanical model

The precession of magnetization vectors is damped by the interaction between relevant macroscopic (magnetic) and irrelevant or heat-bath degrees of freedom. The latter include, for example, lattice vibrations (phonons). The damping or relaxation time depends on the interactions between different subsystems, as described by Fermi's golden rule. However, the deterministic character of the Schrödinger equation forbids irreversible processes, and the same is true for the closely related Liouville-von Neumann equation. An example is Zermelo's recurrence objection, which states that any system eventually returns to its original state. The reason for the reversibility is the consideration of both relevant and irrelevant degrees of freedom. In reality, there is a separation of microscopic (reversible) and macroscopic (irreversible) time scales, and relaxation is obtained by integration over all microsopic or irrelevenat degrees of freedom. A simple mechanical analog is a system of masses coupled by harmonic springs. Relaxation proceeds towards local energy minima, as opposed to thermally activated magnetization processes, although the two phenomena have similar physical origins.

##### 6.3. Coarse-Grained Models

###### 6.3.1. Master equation

6.3.2. Fokker-Planck equations

6.3.3. Langevin models

The time-dependent many-body Schrödinger equation can, in principle, be used to predict the evolution of any magnetic system. However, this is neither practical nor necessary, because individual heat-bath degrees of freedom do not contain any specific information about the magnetic behavior. Coarse-grained models abstract from quantum-mechanical features operative on atomic length and time scales. Simplifying somewhat, there are descriptions based on three different types of equations: (i) master or rate equations, (ii) Fokker-Planck equations, and (iii) Langevin equations. These approaches are physically largely equivalent, although the master equation is able to describe macroscopic magnetization jumps, whereas the Fokker-Planck and Langevin equations interpret macroscopic magnetization changes as a chain of microscopic events. Like the master equation, the Fokker-Planck equation deals with the probability of magnetization configurations, whereas the Langevin equation governs the local magnetization vector as a function of random thermal forces.

##### 6.4. Slow Magnetization Dynamics

###### 6.4.1. Magnetic viscosity and sweep-rate dependence

6.4.2. Superposition model of magnetic viscosity

6.4.3. Asymptotic behavior*

6.4.4. Energy-barrier models

6.4.5. *Linear and other laws

6.4.6. Superparamagnetism

6.4.7. *Fluctuations

The nonequilibrium character of magnetic hysteresis leads to a time dependence of the extrinsic properties known as magnetic viscosity. Their corresponding activated magnetization reversal reflects the cooperative thermal excitation of nanoscale activation volumes and is a small correction to the leading static magnetization processes. For example, freshly magnetized permanent magnets loose a few mT of their magnetization within the first few hours, and the coercivity decreases with decreasing sweep rate d*H*/d*t*. As a crude rule, magnetic viscosity is described by a logarithmic time dependence. A simple derivation of the logarithmic law assumes an ensemble of independent relaxation processes, but it can also be considered as an inverted Arrhenius law. The involved energy barriers are smaller than or comparable to 25 *k*_{B}*T*, so that the reversal usually requires the support by a magnetic field. The field-dependence of the energy barriers is usually described by a power-law exponent 3/2, although highly symmetric energy landscapes yield an exponent 2. These two exponents cover a wide range of coherent and incoherent magnetization processes, including various types of pinning and nucleation. The slow dynamics of nanoparticles is a consideration in several areas of magnetism, such as magnetic recording and ferrofluids. On length scales of a very few nanometers, the behavior blends into equilibrium thermodynamics and acquires the character of giant thermodynamic fluctuations.

#### 7. Special Topics and Interdisciplinary Models

##### 7.1. Disordered Magnets and Spin Glasses

###### 7.1.1. Atomic disorder and electronic structure

7.1.2. *Green Functions

7.1.3. Ferromagnetic order in inhomogeneous magnets

7.1.4. Spin glasses

Atomic disorder has far-reaching consequences for the behavior of magnetic materials. It modifies the electronic structure but does not necessarily destroy ferromagnetism, as exemplified by amorphous ferromagnets. Spin glasses combine disorder with competing exchange, and their ground state is neither ferromagnetic nor antiferromagnetic. The equilibrium and nonequilibrium properties of spin glasses have remained a complex problem, and several models have been developed, such as the Edwards-Anderson (EA) and Sherrington-Kirkpatrick (SK) models. On a mean-field level, the determination of ordering and spin-glass temperatures involves the diagonalization of large random matrices.

##### 7.2. Soft Matter, Transport, and Magnetism

###### 7.2.1. Random walks, polymers, and diffusion

7.2.2. *The n = 0 vector-spin model

7.2.3. Polymers and critical dimensionality

7.2.4. Percolation

7.2.5. Diffusive transport

7.2.6. Gases in magnetic metals

7.2.7. Magnetoresistance

7.2.8. Other transport phenomena involving magnetism

There are many links between magnetism and transport properties. Finite-temperature magnetism exhibits a randomness reminiscent of diffusion processes and polymer chains, and there has been cross-fertilization in both directions. For example, self-avoiding polymer chains can be mapped onto an *n*-vector model with *n* = 0, and the self-interaction behavior of polymer chains is a real-space analog to the onset of mean-field behavior in four-dimensional ferromagnets. Other phase-transition analogies are percolation and gases in metals, where the links are both mathematical (lattice-gas description) and physical (interstitial permanent magnets). Magnetoresistance based on various mechanisms (AMR, GMR, CMR, TMR) is of practical importance, as is superconductivity.

###### How can we understand that mean-field critical exponents are exact in *d* 4 dimensions? In the polymer analogy, mean-field behavior and critical fluctuations correspond to random and self-avoiding walks, respectively, and critical fluctuations have the character of self-intersections between distant parts of a chain. The fractal dimensionality of random walks is two, and in four dimensions, their (self-)intersections become zero-dimensional, or essentially unimportant.

##### 7.3. Bruggeman Model

###### 7.3.1. Static and dynamic properties

7.3.2. *Parameterization

7.3.3. *Selfconsistent materials equations

7.3.4. The response parameter g

7.3.5. Percolation in the Bruggeman model

The Bruggeman model describes linear mechanical, magnetic, electrical, and transport properties for a broad variety of composites. The idea is to start from exact solutions for small volume fractions of a second phase in a main or matrix phase. Arbitrary volume fractions are then treated by selfconsistently embedding the phases in an effective medium. The theory yields materials parameters as functions of the volume fractions and geometries of the phases. Each system is described by a single interaction parameter *g*, which is equal to the percolation threshold of the composite. The predictions of the Bruggeman model, especially the behavior near the percolation threshold, are mean-field like.

##### 7.4. Nanostructures, Thin Films, and Surfaces

###### 7.4.1. Length scales in nanomagnetism

7.4.2. Nanomagnetic effects of atomic origin

7.4.3. Random anisotropy

7.4.4. *Cooperative magnetization processes

7.4.5. Two-phase nanostructures

The magnetism of nano- and thin-film structures is intermediate between atomic and macroscopic magnetism but cannot be reduced to a superposition of the two limits. Characteristic length scales are of order *a*/*a*_{o} = 7.52 nm, that is, comparable to magnetic domain-wall widths. Phenomena of atomic origin are often important on somewhat smaller length scales of 1 to 2 nm. For example, magnetic nanoparticles embedded in a nonmagnetic metallic matrix experience a significant RKKY coupling, in spite of the rapidly oscillating character of the RKKY interaction. Competing nanoscale exchange and anisotropy are described by random-anisotropy models, whereas other micro- and nanomagnetic models describe composite nanostructures and thin-film nanostructures, such as exchange-coupled multilayers. One example is hard-soft two-phase permanent magnets, whose performance goes beyond what is expected from the volume fractions of the involved hard and soft phases.

##### 7.5. Beyond Magnetism

###### 7.5.1. Metallurgy

7.5.2. Biology and medicine

7.5.3. Social sciences

The transparent phase space of magnetic models has lead to various applications of magnetic modeling in other areas of human knowledge. Examples are metallurgy, population dynamics, neurology, and sociology. Order-disorder transitions, gases in metals, and spinodal de–composition in alloys can essentially be described by the Ising model, whereas models of population dynamics are often of the diffusion or Fokker-Planck type. There is a close link between neurology and spin glass models, as epitomized by the Hebb rule. As in magnetism, the quality of a model depends on the parameterization and ranges from a crude qualitative understanding to quantitative predictions. In turn, magnetic modeling has been influenced and reinvigorated by developments in other areas of science and technology.