{'fraction_non_alphanumeric': 0.026729206155427763, 'fraction_numerical': 0.004337070710315803, 'mean_word_length': 4.746354070518738, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 1, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Magnetic Particles

We present an overview of the subject of Magnetic Particles, starting at a level suitable for advanced high-school students and ending at a level suitable for current practitioners in the field. The sub-topics covered include: Types of Magnetic Materials, Relationships Between the Vector Fields, Magnetic Energies, Dynamics of Magnetic Particles, and Magnetocaloric Effects.


Introduction
Ever since the development of the magnetic compass revolutionized navigation in the second century A.D., applications of magnetic materials have become essential in most branches of engineering. In many of these applications the magnetic materials are utilized in the form of microscopic magnetic particles.
Some modern representative examples from electrical and electronics engineering are nonvolatile storage of information on magnetic tapes and disks, magnetic inks, refrigerators that make use of the magnetocaloric effect, ferrofluid vacuum seals, and the microscopic machines known as micro-electromechanical systems (MEMS). It is likely that these current and emergent engineering applications of microscopic and nanoscopic magnetic particles will be joined by many more in the next few decades.
Even though magnetic particles have been used for a very long time, our understanding of their physical behaviors is relatively new and still quite limited. In addition, the current understanding of the physical properties of magnetic particles, and the engineering applications that are made possible by these properties, cannot easily be described without sophisticated mathematics. Some of the reasons for this are listed below.
1) The magnetism of magnetic particles is fundamentally quantum-mechanical in origin. Hence, understanding the properties of magnetic particles requires the use of quantum mechanics, either directly or by the inclusion of quantum-mechanical effects into phenomenological models. 3) For ferromagnetic particles, which are the ones used in most engineering applications, the relationships between the three vector fields H, B, and M depend on the history of the particle -how the field and magnetization have varied in the past, and possibly how the particle was manufactured. 4) Engineering applications of magnetic particles are determined by physical properties that depend on energetic effects which originate from a number of different physical mechanisms. For example, in spherical particles the coercive field (defined below) is due primarily to the energy associated with crystalline anisotropy, while in elongated needle-like single-domain particles the dipole-dipole interactions (the magnetostatic energy) are the most important in determining the coercive field.
Commonly these complications are what make magnetic particles so useful in engineering applications.
For example, the dependence of M on the detailed history of the particle is what makes magnetic recording possible.
Most engineering applications utilize large numbers of magnetic particles, which may differ in size, orientation, composition, etc. However, a detailed understanding of the behavior of such an assortment of particles requires knowledge of the properties of individual particles. The properties of systems consisting of large numbers of particles can be obtained by an appropriate averaging over the size, orientation, shape, type, and location of individual particles. This article will focus on single magnetic particles. It is only in very recent years that it has become possible to study the behavior of individual particles experimentally. This is due to the development of better methods to produce well-characterized magnetic particles and to the development of measurement techniques with resolutions at the nanometer scale. These ultrahigh resolution techniques include magnetic force microscopy (MFM) (1), micro-SQUID devices (2), Lorentz transmission electron microscopy (3), and giant magnetoresistive (GMR) measurements (4).

Types of Magnetic Materials
At the atomic level, a magnetic material is an arrangement of local magnetic moments. Fundamentally, each such magnetic moment is a quantum-mechanical quantity arising from either the intrinsic spin or the orbital motions of the electrons of an atom or molecule. Each of these localized magnetic moments, often called spins for brevity, may be thought of as a small bar magnet. The microscopic structures of the most widely studied types of magnetic materials are shown schematically in Fig. 1. This figure shows the lowest-energy arrangement (the ground state) of the local magnetic moments or spins in (a) ferromagnetic, (b) antiferromagnetic, and (c) ferrimagnetic materials. The total magnetizations due to the spin arrangements shown in Fig. 1 can be found by adding the vectors together. In a ferromagnet the spins add up to a large vector. For an antiferromagnet they add up to zero. For a ferrimagnet there is some cancellation, but the spins still add up to a nonzero value.
The alignment of the spins in Fig. 1 (a)-(c) correspond to a temperature of absolute zero. At nonzero temperatures, the alignment of the spins is somewhat random due to thermal fluctuations. At temperatures above a critical temperature, T c (which is different for different materials), the thermal fluctuations are so large that the spin arrangement becomes essentially random, the total magnetization is zero, and the material becomes paramagnetic (as shown in Fig. 1d). For ferromagnets in zero applied magnetic field the magnetization vanishes at T c and remains zero above T c . Since most engineering applications of magnetic materials use ferromagnets, the remainder of this article will focus on ferromagnetic particles.

Relationships Between the Vector Fields
The fundamental relationship between the vector fields is given by where µ 0 = 4π × 10 −7 H/m is the permeability of free space. Equation (1) is true for all materials, even for nonlinear ones.
For a linear homogeneous and isotropic material the relationship between M and H is linear and given by where χ m is a material dependent quantity called the magnetic susceptibility. In this case it is possible to write where µ is the permeability of the medium and the parameter µ r is its relative permeability.
In ferromagnetic materials, however, the relationship between the three vector fields is generally nonlinear and history dependent. Thus simple relationships such as those of Eq. (2) and Eq. (3) are not justified. In this case it is necessary to talk about a hysteresis loop. Figure 2 shows a hysteresis loop, with identification of the coercive field H c , the remanent (spontaneous) magnetization M r , and the saturation magnetization M s (the maximum magnetization of a magnetic particle in a strong field). The area of the hysteresis loop corresponds to the work which must be done in taking the magnetic material through one cycle of the applied field. This work is converted to heat and represents a major source of energy loss in devices such as transformers and motors. It was first studied systematically by C. P. Steinmetz (6).
One useful classification of magnetic materials for engineering purposes is into soft and hard magnets.
Soft magnetic materials are usually used for cores of transformers, generators, and motors, and the heads in magnetic tape and disk devices; applications which require a low coercive field, H c , small hysteresis loop area to minimize heat generation, or high permeability. Hard magnetic materials are used for electric sensors, loudspeakers, electric meters, magnetic recording media, and other uses that require a high coercive field, a high remanence, or a large hysteresis loss.
Not all hysteresis loops have the shape shown in Fig. 2. In fact, for single-crystal ferromagnets the shape of the hysteresis loop is usually dependent on the orientation of the applied field with respect to the crystalline axes. For materials composed of many different magnetic particles or grains, the nonlinear effects of each particle must be added to obtain a composite hysteresis loop. For such composite materials, or for bulk materials with impurities, the hysteresis loop is not smooth but contains small jumps, called Barkhausen jumps, that correspond to successive switching of small regions of the material (7). These loops can sometimes be parameterized using a model called the Preisach model (8).

Magnetic Energies
The energies that are relevant to the properties of magnetic materials arise from a variety of physical effects. Which ones are most important for a particular engineering application depend on the composition of the particular piece of material, its mesoscopic structure (grain size, local stress, etc.), its surface properties, and its size and shape.

Exchange Energy
The exchange energy comes from the quantum-mechanical overlap and hybridization of the exchange integrals between atoms and molecules. Since the interaction constant J is due to the overlap of orbitals, it is a short-range interaction which often does not extend beyond nearest-neighbor pairs of lattice sites.
For most materials J is somewhat temperature and stress dependent, due to changes in interatomic distances. It is also dependent on the local environment of a particular magnetic atom. Consequently exchange interactions for atoms at a surface, near a grain boundary, or near an impurity atom may be different from the exchange interaction of the same kind of atom in the bulk of a crystal. Often the Hamiltonian for a magnetic material is written as a sum of Heisenberg terms, such as H = J S i · S j , where the three-dimensional spins S i and S j are located at nearest-neighbor sites i and j of the crystalline lattice.

Crystalline Anisotropy Energy
When magnetic atoms are arranged on a crystal lattice, their spins have a lower energy if they are aligned along certain directions. These are called the easy axes. The directions which require the highest energy for the orientation of the magnetic spins are the hard directions or hard axes. Consequently, certain magnetization directions are preferred. One consequence of crystalline anisotropy is that hysteresis loops, such as the one shown in Fig. 2, depend on the angles between the applied magnetic field and the crystal axes. In magnetic materials which are not perfect crystals the orientation of the easy axes in different grains will most likely be different.
For example, iron is a cubic crystal, and the three cube edges are the easy axes. The anisotropy energy of Fe is expressed as where the α's are the direction cosines for the angles between the cube edges and the direction of the magnetization. Here K 1 and K 2 depend on temperature and are zero above T c .

Dipole-Dipole Energy
The classical interaction between two magnetic dipoles m 1 and m 2 (which are vector quantities) is This gives a long-range interaction between the dipoles. Since each spin in Fig. 1 represents a magnetic dipole, this dipole-dipole interaction must be considered in dealing with magnetic particles. In many engineering applications where there are no time varying external fields, the microscopic motion of each individual spin (due to precession and random thermal fluctuations) need not be taken into account, and the familiar equations of magnetostatics are recovered. Consequently, the dipole-dipole interaction is sometimes called the magnetostatic energy. In particular, from magnetostatics the energy stored in the fields is given by where the integral is over all ferromagnetic bodies. This may be understood as the integral of the interaction of each dipole with the field H created by all the other magnetic dipoles.
In general the field inside a uniformly magnetized ferromagnetic body is not uniform, except in very special cases, such as general ellipsoids. For a general ellipsoid which is uniformly magnetized, the demagnetizing field inside the ellipsoid can be written as The demagnetizing field is due to the magnetic field caused by all the other dipoles in the particle. When M is parallel to one of the principal axes of the ellipsoid, the tensor is diagonal, and its three diagonal elements are called the demagnetizing factors. Then the magnetostatic self-energy of a uniformly magnetized ellipsoid of volume V is given by since the tensor is diagonal. The demagnetizing factors may not be equal, so some directions will be preferred. This gives a shape anisotropy term in the energy of a ferromagnet. Note that the preferred directions from shape anisotropy can be different from the preferred directions for crystalline anisotropy. For small enough particles the energy cost of a domain wall is always higher than the gain resulting from the reduction in the magnetostatic energy. As a result, sufficiently small particles consist of a single magnetic domain. Nanometer-sized Co particles created by electron beam lithography may have only one or a few domains. Figure 3 shows pictures of an array of these particles.

Substrate and Surface Energies
The surface of a magnetic particle can influence its physical behavior as much or even more than the bulk. This is because the energies associated with the exchange interactions and the crystalline anisotropy are extremely sensitive to the local environment. The environment of a magnetic atom is different at the surface and in the bulk. Surface effects become increasingly important as the particle size is decreased, due to the increased surface-to-volume ratio.
Surface and interface effects can have important engineering applications. For example, giant magnetoresistive (GMR) materials can be grown by coupling a ferromagnetic film or particle to an antiferromagnetic bulk, leading to an exchange interaction between the spins of the two types of materials. This gives an exchange bias which leads to an asymmetric hysteresis loop. This is highly desirable because it enhances the signal-to-noise ratio of the read heads used in magnetic recording applications.
Surfaces can also affect the response of a magnetic particle to changes in its environment. For example, if the direction of the external field is reversed, reversal of the magnetization of the particle may be initiated at the surface. This can make the coercive field of small particles significantly smaller than it would be in the absence of surfaces.

Strain
The application of stress to a magnetic material results in a strain response. The sensitivity of the local magnetization to the local environment means that strain in the lattice will change the local energies (the exchange energy, the crystalline anisotropy energy, and the dipole-dipole energy), which can alter the behavior of the magnetization locally.

Temperature
The energy scales associated with the effects discussed above must be compared with the thermal energy k B T , which causes the randomness of the spins seen in Fig. 1(d). Here k B is Boltzmann's constant, and the temperature T is given in Kelvin. If the thermal energy is comparable to the other energies, it can affect engineering applications. For instance, at nonzero temperature there is some probability that the orientation of the magnetization of a particle will change spontaneously. This can lead to loss of data integrity in magnetic recording media (9).

Dynamics of Magnetic Particles
The magnetization dynamics of magnetic particles are important for engineering applications. For example, consider a bit of information to be stored on one of the single-domain particles shown in Fig. 3. The coding may be that a North-South orientation corresponds to Boolean 0 and a South-North orientation to Boolean 1. Engineering questions involving the magnetization dynamics include: 1) How strong a field must be applied to change a 0 to a 1 or vice versa? 2) How long can a particle encoded with a bit be exposed to a stray field without losing data integrity, and how strong a stray field can be tolerated? 3) If a bit is written today, how long can it be trusted to remain as it was written?

Metastability
Arguably the most important concept needed to understand the dynamics of magnetic particles is metastability. One familiar example of this ubiquitous natural phenomenon is the supercooling of water.
That metastability should be relevant to magnetism is suggested by the fact that the magnetization M of a magnetic particle is history dependent. Figure 4 (a) shows the standard picture of metastability in a bistable system, such as a uniaxial single-domain magnetic particle. There are two free-energy minima, one of which is only a local minimum (the metastable state), and one of which is the global minimum (the equilibrium state). If the system is in the metastable state, a free-energy barrier, ∆F , must be overcome before the system can relax to the equilibrium state. The average lifetime of the metastable state, τ , is given by where τ 0 is an inverse attempt frequency. In magnetic particles τ 0 is typically taken to be on the order of an inverse phonon frequency, usually τ 0 ≈ 10 −10 sec. More detailed analysis shows that τ 0 depends on the curvature of the free energy at both the metastable minimum and the saddle point (the maximum that separates the metastable and equilibrium states).
The system can get into the metastable state due to its history. How can it get out? There are two possible answers to this question. Think in analogy to a pitcher half-filled with water, which is held above a sink. The water is the system, the pitcher is the metastable state, and the sink is the equilibrium state. One way to get the water into the sink is to shake the pitcher to make the water splash out.
The more vigorously it is shaken, the faster the water will splash out. This method of escape from a metastable state is analogous to a magnetic material at a nonzero temperature. As seen from Eq. (7), the lifetime increases as the height of the barrier increases, whereas it decreases as the temperature increases.
Another way of getting the water into the sink is to tip the pitcher. As the pitcher continues to tip, at some point the water starts to flow out. This corresponds to an escape from the metastable state of a magnetic particle, which is valid even at zero temperature. In particular, the barrier ∆F depends on the applied field H appl , and changing H appl corresponds to tipping the pitcher. At a particular value of the field ∆F equals zero, and the metastable state disappears. This situation is illustrated in Fig. 4(b). The field that must be applied for ∆F to vanish is often called the nucleation field, H nucl .
This pitcher analogy also illustrates an important consideration concerning the escape from a multidimensional metastable well at zero temperature. In particular, how far the pitcher must be tipped before the water starts to spill out depends on the direction in which it is tipped. The smallest angle is needed if the pitcher is tipped to make the water spill from the spout -the lowest part of the rim.
Similarly, the nucleation field, H nucl , often depends on the direction of the applied field. However, for thermally driven escape the decay always proceeds across the saddle point (analogous to the spout) as the temperature (analogous to the amplitude of the shaking) is increased, or as the waiting time is increased at fixed temperature.
The waiting time is related to another physical quantity of interest, the probability that a magnetic particle starting in the metastable state at t=0 has never left it at time t. This probability is often called P not (t). It is only in the last decade that it has become possible to measure P not (t) for individual single-domain magnetic particles (1,2). For thermal escape over a single barrier one has where τ is given by Eq. (7). Equation (7) is the fundamental equation needed to understand the dynamics of a magnetic particle.
All one requires is a knowledge of ∆F . This, however, requires a knowledge of the energy of the spin configuration at the saddle point. Since ∆F enters in an exponential, the value of τ is extremely sensitive to small changes in ∆F .

Coherent Rotation
Consider a spherical single-domain uniaxial magnetic particle with crystalline anisotropy constant K and a uniform magnetization M in an applied magnetic field along the easy axis. If all the spins in the particle point in the same direction, the total energy is E = KV sin 2 (θ) − M s V H cos(θ). Here the saturation magnetization M s makes an angle θ with the easy axis, and the volume of the particle is V . In this case it is possible to obtain the zero-temperature free-energy barrier exactly, namely In this model all the spins rotate coherently and act like one large spin.
Consequently, particles that behave this way are called superparamagnetic particles. Note that the zero-temperature energy barrier has been assumed to be valid at finite temperatures in this model of a superparamagnet. Since superparamagnetism implies escape over a single barrier, P not (t) is given by Eq. (8).
The particle volume V enters the metastable lifetime in an exponential, so a small change in particle size can lead to extremely large changes in τ . For example, for an iron sphere with a radius of 115Å the lifetime τ ≈ 0.07 s. If the radius is 140Å, then τ ≈ 1.5 × 10 5 s (10).
The coherent rotation mode described above is often called the Néel-Brown reversal mode (10) since Néel derived Eq. (7) with ∆F =KV /k B T and Brown wrote a differential equation for a random walk in the metastable well to obtain a non-constant prefactor for Eq. (7). It is also possible to obtain a zerotemperature hysteresis curve using the same assumptions as above, namely a uniform magnetization and that only the external field and a uniaxial anisotropy are important. This is called the Stoner-Wohlfarth model (10). This model gives an upper bound for the coercive field as H c ≤ 2K/M s .
An equivalent analysis for the case in which shape anisotropy is important can also be performed.
Again, with the assumption that all the spins always point in the same direction, the analysis is the same except that K now arises from shape anisotropy. The assumption that the spin configuration at the saddle point has all spins pointing in the same direction is only sometimes valid. There are other zerotemperature reversal modes where the zero-temperature saddle point has spins in other configurations.
Examples include modes descriptively named buckling, curling, and fanning. The dominant mode depends on the geometry and size of the magnetic particle.

Nucleation and Growth
As the volume V increases, the average rate for magnetization reversal via coherent rotation quickly becomes too small to be practically important. This was illustrated by the example with iron particles discussed above. Other reversal modes with lower free-energy barriers can then come into play, especially for highly anisotropic materials, in which domain walls are relatively thin.
At nonzero temperatures, thermal fluctuations continually create and destroy small "droplets" of spins aligned with the applied field. The free energy of such a droplet consists of two competing parts: a positive part due to the interface between the droplet and the metastable background, and a negative part due to the alignment of the spins in the droplet with the field. For a droplet of radius R, these terms are proportional to R d−1 and −|H|R d , respectively. Here d is the spatial dimension of the particle, and these relations hold both for three-dimensional particles (d=3) and for particles made of a thin film (d=2). Small droplets most likely shrink. However, if a droplet manages to become larger than a critical radius, R c ∝ 1/|H|, it will most likely continue to grow and eventually bring the whole particle into the stable magnetization state. The free-energy barrier associated with such a critical droplet is proportional to 1/|H| d−1 , which is independent of the particle size! Since the droplets can only grow at a finite speed, in sufficiently large particles or for sufficiently strong fields, new critical droplets may nucleate at different positions while the first droplets are still growing.
The droplet switching mechanism sketched above gives rise to three regimes of field strengths and particle sizes: 1: For sufficiently weak fields and/or small particles, a critical droplet would be larger than the entire particle. As a result, the saddle point configuration consists of an interface which cuts across the particle, so that ∆F ∝ V (d−1)/d , independent of |H| to lowest order. The behavior in this regime is essentially superparamagnetic, even though the dependence of ∆F on the particle size is somewhat weaker than predicted for uniform rotation. In this regime both Eq. (7) and Eq. (8) are valid.
2: For stronger fields and/or larger particles, the magnetization reverses by the action of a single droplet of the stable phase. The free-energy barrier is independent of the particle volume, but because the droplet can nucleate anywhere in the particle, the average lifetime is inversely proportional to V . We call this decay regime the Single-Droplet regime. A series of snapshots of a computer simulation of Single-Droplet decay is shown in Fig. 5. In this regime Eq. (8) is still valid.
3: For yet stronger fields and/or larger particles, the decay occurs via a large number of nucleating and growing droplets. In this regime, which we call the Multi-Droplet regime, the average lifetime is independent of V . A series of snapshots of a computer simulation of Multi-Droplet decay is shown in Fig. 6. In this regime Eq. (8) is no longer valid, and P not (t) takes the form of an error function (11,12).

Switching Fields
Two rather similar quantities that are often measured for magnetic particles are the switching field, H sw , and the coercive field, H c . The former is defined as the magnitude of the field for which a particle switches with probability 1/2 within a given waiting time after field reversal. The latter is the value of the sinusoidal field at which the hysteresis loop crosses the field axis, as shown in Fig. 2. Both depend weakly on the time scale of the experiment (waiting time or inverse field frequency), but they are qualitatively similar over a wide range of time scales. A collection of experimentally measured coercive fields for various materials are shown as functions of the particle size in Fig. 7(a). The increase in H c for small sizes is due to the superparamagnetic behavior of the particles, while the decrease for larger sizes is due to the dipole-dipole interaction, which causes large particles to break up into multiple domains. For particles in the nanometer range, which are single-domain in equilibrium, the crossovers between the three nucleation driven magnetization reversal mechanisms described above give rise to very similar size dependences (11,12), as seen in Fig. 7(b).

Domain Boundary Movement
If the magnetic particle is multidomain in zero field, the application of a magnetic field will cause the domain wall(s) to move. In this case the dynamics of the magnetization is dominated by the domain-wall movement. Typically there will be pinning sites due to impurities, grains, and surfaces, that the domain wall must overcome before it can move. These obstacles can either be overcome by applying a sufficiently large field, or by waiting for the random thermal fluctuations to move the domain wall past the pinning centers. This is analogous to the zero-temperature and finite-temperature reversal mechanisms in singledomain particles. If the domain wall moves due to random thermal fluctuations the magnetization of the particle will change slowly with time, a phenomenon called magnetization creep (12).

Magnetic Viscosity
Consider a large number of identical non-interacting particles. If a strong field is applied and then removed, the remanent magnetization will decay with time as M r (t) = M r (0) exp (−t/τ ) as particles cross the barrier separating the two equilibrium states. However, in the typical case the particles are not identical, and there is a distribution of lifetimes, P(τ ). In this case the time decay is Under some circumstances and for certain specific distributions for P(τ ) Eq. (9) can be approximated by M r (t) ≈ C − S ln(t/t 0 ) where C, S, andt 0 are constants. This logarithmic decay of the magnetization is called magnetic viscosity. It must be emphasized that this logarithmic equation is a valid approximation only under certain specific circumstances and even then only for a limited range of t (10).

Magnetocaloric Effects
Magnetocaloric effects of magnetic particles have engineering applications principally in refrigeration.
Current applications are mostly to ultra-low temperature refrigeration, but near-term applications to higher temperatures seem promising. Given that at constant volume the P dV work term is zero, the first law of thermodynamics becomes where U is the internal energy of the magnetic particle and the symbol d − denotes an infinitesimal change rather than a true differential. This must be considered because of the history dependence of M . For an adiabatic change, d − Q = T dS = 0, where S is the entropy. Then the relation can be obtained (14), where C H is the heat capacity at constant field. Thus an increase in H produces a rise in temperature, and vice versa, which is the magnetocaloric effect. This can be utilized as a cooling mechanism by placing the magnetic material in a strong applied field, and then turning off the applied field. This will lead to a decrease in temperature due to adiabatic demagnetization according to Eq. (11).  Shown on the loop is the location of the average coercive field, H c . This is the field which must be applied to make the magnetization equal to zero. Also shown is the remanent, or spontaneous, magnetization, M r . This is the magnetization when the applied field is equal to zero. In this particular model the value of M r is known exactly. The saturation magnetization, M s , corresponds to all the spins aligned, as in Fig. 1(a). The fluctuations on the hysteresis loop are due to random thermal noise, which is typically seen in hysteresis loops for small particles at nonzero temperatures. This loop is for a temperature of 0.97T c . The loop would become smoother at lower temperatures and for larger particles than the 64×64 lattice used here. The loop area can be a complicated function of H 0 , ω, and temperature (5). The data used to generate this figure are from a Monte Carlo simulation, courtesy of Dr. Scott W. Sides.   4. A metastable well is illustrated in (a). Even though the lowest-energy state is on the right, an energy barrier ∆F must be overcome before the metastable state on the left can decay to the equilibrium state. As the external magnetic field is increased, the barrier decreases, and it becomes zero at a particular field called the nucleation field, H nucl , as illustrated by the heavy curve in (b). There is no metastable state for higher fields, as illustrated by the light curve in (b).