Magnetocrystalline anisotropy
Encyclopedia
Magnetocrystalline anisotropy is the dependence of the internal energy of a ferromagnet on the direction of its magnetization
. As a result, certain crystallographic
directions are preferred directions, or easy axes, for the magnetization. It is a special case of magnetic anisotropy
. The spin-orbit interaction
is the primary source of the magnetocrystalline anisotropy
.
Magnetocrystalline anisotropy determines whether a magnetic material can be made into a good hard magnet, a good soft magnet or neither. Hard magnets are an essential component of electromagnetic motors and soft magnets are an essential component of transformers.
and is difficult to compute from first principles, although some successful computations have been made.
(e.g., cubic
or hexagonal
). The order
of a term in the expansion is the sum of all the exponents of magnetization components, i.e., is second order.
The ratio is an energy density
(energy per unit volume). This can also be represented in spherical polar coordinates with , , and :
The parameter , often represented as , has units of energy density
and depends on composition and temperature.
The minima in this energy with respect to satisfy
If ,
the directions of lowest energy are the directions. The axis is called the easy axis. If , there is an easy plane perpendicular to the symmetry axis (the basal plane of the crystal).
Many models of magnetization represent the anisotropy as uniaxial and ignore higher order terms. However, if , the lowest energy term does not determine the direction of the easy axes within the basal plane. For this, higher-order terms are needed, and these depend on the crystal system (hexagonal
, tetragonal
or rhombohedral
).
the axis is an axis of sixfold rotation symmetry. The energy density is, to fourth order,
.
The uniaxial anisotropy is mainly determined by the first two terms. Depending on the values and , there are four different cases:
The basal plane anisotropy is determined by the third term, which is sixth-order. The easy directions are projected onto three axes in the basal plane.
Below are some room-temperature anisotropy constants for hexagonal ferromagnets. Since all the values of and are positive, these materials have an easy axis.
.
Note that the term, the one that determines the basal plane anisotropy, is fourth order (same as the term). The definition of may vary by a constant multiple between publications.
The energy density for a rhombohedral crystal is
.
the lowest order terms in the energy are
If the second term can be neglected, the easy axes are the axes (i.e., the , , and , directions) for and the directions for (see images on right).
If is not assumed to be zero, the easy axes depend on both and . These are given in the table below, along with hard axes (directions of greatest energy) and intermediate axes (saddle point
s) in the energy). In energy surfaces like those on the right, the easy axes are analogous to valleys, the hard axes to peaks and the intermediate axes to mountain passes.
Below are some room-temperature anisotropy constants for cubic ferromagnets. The compounds involving are ferrites
, an important class of ferromagnets. In general the anisotropy parameters for cubic ferromagnets are higher than those for uniaxial ferromagnets. This is consistent with the fact that the lowest order term in the expression for cubic anisotropy is fourth order, while that for uniaxial anisotropy is second order.
, a mineral of great importance to rock magnetism
and paleomagnetism
, has an isotropic point at 130 kelvin
.
Magnetite also has a phase transition
at which the crystal symmetry changes from cubic (above) to monoclinic or possibly triclinic below. The temperature at which this occurs, called the Verwey temperature, is 120 Kelvin.
. To keep the lattice from deforming, a stress must be applied. If the crystal is not under stress, magnetostriction alters the effective magnetocrystalline anisotropy. If a ferromagnet is single domain
(uniformly magnetized), the effect is to change the magnetocrystalline anisotropy parameters.
In practice, the correction is generally not large. In hexagonal crystals, there is no change in . In cubic crystals, there is a small change, as in the table below.
Magnetization
In classical electromagnetism, magnetization or magnetic polarization is the vector field that expresses the density of permanent or induced magnetic dipole moments in a magnetic material...
. As a result, certain crystallographic
Crystal structure
In mineralogy and crystallography, crystal structure is a unique arrangement of atoms or molecules in a crystalline liquid or solid. A crystal structure is composed of a pattern, a set of atoms arranged in a particular way, and a lattice exhibiting long-range order and symmetry...
directions are preferred directions, or easy axes, for the magnetization. It is a special case of magnetic anisotropy
Magnetic anisotropy
Magnetic anisotropy is the direction dependence of a material's magnetic properties. In the absence of an applied magnetic field, a magnetically isotropic material has no preferential direction for its magnetic moment while a magnetically anisotropic material will align its moment with one of the...
. The spin-orbit interaction
Spin-orbit interaction
In quantum physics, the spin-orbit interaction is any interaction of a particle's spin with its motion. The first and best known example of this is that spin-orbit interaction causes shifts in an electron's atomic energy levels due to electromagnetic interaction between the electron's spin and...
is the primary source of the magnetocrystalline anisotropy
Anisotropy
Anisotropy is the property of being directionally dependent, as opposed to isotropy, which implies identical properties in all directions. It can be defined as a difference, when measured along different axes, in a material's physical or mechanical properties An example of anisotropy is the light...
.
Magnetocrystalline anisotropy determines whether a magnetic material can be made into a good hard magnet, a good soft magnet or neither. Hard magnets are an essential component of electromagnetic motors and soft magnets are an essential component of transformers.
Microscopic origin
Magnetocrystalline anisotropy arises mostly from spin-orbit coupling. This effect is weak compared to the exchange interactionExchange interaction
In physics, the exchange interaction is a quantum mechanical effect without classical analog which increases or decreases the expectation value of the energy or distance between two or more identical particles when their wave functions overlap...
and is difficult to compute from first principles, although some successful computations have been made.
Thermodynamic theory
The magnetocrystalline anisotropy energy is generally represented as an expansion in powers of the direction cosines of the magnetization. The magnetization vector can be written , where is the saturation magnetization. Because of time reversal symmetry, only even powers of the cosines are allowed. The nonzero terms in the expansion depend on the crystal systemCrystal system
In crystallography, the terms crystal system, crystal family, and lattice system each refer to one of several classes of space groups, lattices, point groups, or crystals...
(e.g., cubic
Cubic crystal system
In crystallography, the cubic crystal system is a crystal system where the unit cell is in the shape of a cube. This is one of the most common and simplest shapes found in crystals and minerals....
or hexagonal
Hexagonal crystal system
In crystallography, the hexagonal crystal system is one of the 7 crystal systems, the hexagonal lattice system is one of the 7 lattice systems, and the hexagonal crystal family is one of the 6 crystal families...
). The order
Degree of a polynomial
The degree of a polynomial represents the highest degree of a polynominal's terms , should the polynomial be expressed in canonical form . The degree of an individual term is the sum of the exponents acting on the term's variables...
of a term in the expansion is the sum of all the exponents of magnetization components, i.e., is second order.
Uniaxial anisotropy
More than one kind of crystal system has a single axis of high symmetry (threefold, fourfold or sixfold). The anisotropy of such crystals is called uniaxial anisotropy. If the axis is taken to be the main symmetry axis of the crystal, the lowest order term in the energy isThe ratio is an energy density
Energy density
Energy density is a term used for the amount of energy stored in a given system or region of space per unit volume. Often only the useful or extractable energy is quantified, which is to say that chemically inaccessible energy such as rest mass energy is ignored...
(energy per unit volume). This can also be represented in spherical polar coordinates with , , and :
The parameter , often represented as , has units of energy density
Energy density
Energy density is a term used for the amount of energy stored in a given system or region of space per unit volume. Often only the useful or extractable energy is quantified, which is to say that chemically inaccessible energy such as rest mass energy is ignored...
and depends on composition and temperature.
The minima in this energy with respect to satisfy
If ,
the directions of lowest energy are the directions. The axis is called the easy axis. If , there is an easy plane perpendicular to the symmetry axis (the basal plane of the crystal).
Many models of magnetization represent the anisotropy as uniaxial and ignore higher order terms. However, if , the lowest energy term does not determine the direction of the easy axes within the basal plane. For this, higher-order terms are needed, and these depend on the crystal system (hexagonal
Hexagonal crystal system
In crystallography, the hexagonal crystal system is one of the 7 crystal systems, the hexagonal lattice system is one of the 7 lattice systems, and the hexagonal crystal family is one of the 6 crystal families...
, tetragonal
Tetragonal crystal system
In crystallography, the tetragonal crystal system is one of the 7 lattice point groups. Tetragonal crystal lattices result from stretching a cubic lattice along one of its lattice vectors, so that the cube becomes a rectangular prism with a square base and height .There are two tetragonal Bravais...
or rhombohedral
Rhombohedral crystal system
In crystallography, the trigonal crystal system is one of the seven crystal systems, and the rhombohedral lattice system is one of the seven lattice systems...
).
Hexagonal system
In a hexagonal systemHexagonal crystal system
In crystallography, the hexagonal crystal system is one of the 7 crystal systems, the hexagonal lattice system is one of the 7 lattice systems, and the hexagonal crystal family is one of the 6 crystal families...
the axis is an axis of sixfold rotation symmetry. The energy density is, to fourth order,
.The uniaxial anisotropy is mainly determined by the first two terms. Depending on the values and , there are four different cases:
- When , the ferromagnet is isotropic.
- When and , the axis is an easy axis.
- When and , the basal plane is an easy plane.
- When and have different signs, the ferromagnet has an easy cone (see figure to right).
The basal plane anisotropy is determined by the third term, which is sixth-order. The easy directions are projected onto three axes in the basal plane.
Below are some room-temperature anisotropy constants for hexagonal ferromagnets. Since all the values of and are positive, these materials have an easy axis.
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Tetragonal and Rhombohedral systems
The energy density for a tetragonal crystal is.Note that the term, the one that determines the basal plane anisotropy, is fourth order (same as the term). The definition of may vary by a constant multiple between publications.
The energy density for a rhombohedral crystal is
.Cubic anisotropy
In a cubic crystalCubic crystal system
In crystallography, the cubic crystal system is a crystal system where the unit cell is in the shape of a cube. This is one of the most common and simplest shapes found in crystals and minerals....
the lowest order terms in the energy are
If the second term can be neglected, the easy axes are the axes (i.e., the , , and , directions) for and the directions for (see images on right).
If is not assumed to be zero, the easy axes depend on both and . These are given in the table below, along with hard axes (directions of greatest energy) and intermediate axes (saddle point
Saddle point
In mathematics, a saddle point is a point in the domain of a function that is a stationary point but not a local extremum. The name derives from the fact that in two dimensions the surface resembles a saddle that curves up in one direction, and curves down in a different direction...
s) in the energy). In energy surfaces like those on the right, the easy axes are analogous to valleys, the hard axes to peaks and the intermediate axes to mountain passes.
| Type of axis |  to  |  to  |  to  |
|---|---|---|---|
| Easy |  |
 |
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| Medium |  |
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| Hard |  |
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| Type of axis |  to  |  to  |  to  |
|---|---|---|---|
| Easy |  |
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| Medium |  |
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| Hard |  |
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Below are some room-temperature anisotropy constants for cubic ferromagnets. The compounds involving are ferrites
Ferrite (magnet)
Ferrites are chemical compounds consisting of ceramic materials with iron oxide as their principal component. Many of them are magnetic materials and they are used to make permanent magnets, ferrite cores for transformers, and in various other applications.Many ferrites are spinels with the...
, an important class of ferromagnets. In general the anisotropy parameters for cubic ferromagnets are higher than those for uniaxial ferromagnets. This is consistent with the fact that the lowest order term in the expression for cubic anisotropy is fourth order, while that for uniaxial anisotropy is second order.
| Structure |  |  |
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| · (magnetite Magnetite Magnetite is a ferrimagnetic mineral with chemical formula Fe3O4, one of several iron oxides and a member of the spinel group. The chemical IUPAC name is iron oxide and the common chemical name is ferrous-ferric oxide. The formula for magnetite may also be written as FeO·Fe2O3, which is one part... ) |
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Temperature dependence of anisotropy
The magnetocrystalline anisotropy parameters have a strong dependence on temperature. They generally decrease rapidly as the temperature approaches the Curie temperature, so the crystal becomes effectively isotropic. Some materials also have an isotropic point at which . MagnetiteMagnetite
Magnetite is a ferrimagnetic mineral with chemical formula Fe3O4, one of several iron oxides and a member of the spinel group. The chemical IUPAC name is iron oxide and the common chemical name is ferrous-ferric oxide. The formula for magnetite may also be written as FeO·Fe2O3, which is one part...
, a mineral of great importance to rock magnetism
Rock magnetism
Rock magnetism is the study of the magnetic properties of rocks, sediments and soils. The field arose out of the need in paleomagnetism to understand how rocks record the Earth's magnetic field. This remanence is carried by minerals, particularly certain strongly magnetic minerals like magnetite...
and paleomagnetism
Paleomagnetism
Paleomagnetism is the study of the record of the Earth's magnetic field in rocks. Certain minerals in rocks lock-in a record of the direction and intensity of the magnetic field when they form. This record provides information on the past behavior of Earth's magnetic field and the past location of...
, has an isotropic point at 130 kelvin
Kelvin
The kelvin is a unit of measurement for temperature. It is one of the seven base units in the International System of Units and is assigned the unit symbol K. The Kelvin scale is an absolute, thermodynamic temperature scale using as its null point absolute zero, the temperature at which all...
.
Magnetite also has a phase transition
Phase transition
A phase transition is the transformation of a thermodynamic system from one phase or state of matter to another.A phase of a thermodynamic system and the states of matter have uniform physical properties....
at which the crystal symmetry changes from cubic (above) to monoclinic or possibly triclinic below. The temperature at which this occurs, called the Verwey temperature, is 120 Kelvin.
Magnetostriction
The magnetocrystalline anisotropy parameters are generally defined for ferromagnets that are constrained to remain undeformed as the direction of magnetization changes. However, coupling between the magnetization and the lattice does result in deformation, an effect called magnetostrictionMagnetostriction
Magnetostriction is a property of ferromagnetic materials that causes them to change their shape or dimensions during the process of magnetization. The variation of material's magnetization due to the applied magnetic field changes the magnetostrictive strain until reaching its saturation value, λ...
. To keep the lattice from deforming, a stress must be applied. If the crystal is not under stress, magnetostriction alters the effective magnetocrystalline anisotropy. If a ferromagnet is single domain
Single domain (magnetic)
Single domain, in magnetism, refers to the state of a ferromagnet in which the magnetization does not vary across the magnet. A magnetic particle that stays in a single domain state for all magnetic fields is called a single domain particle . Such particles are very small...
(uniformly magnetized), the effect is to change the magnetocrystalline anisotropy parameters.
In practice, the correction is generally not large. In hexagonal crystals, there is no change in . In cubic crystals, there is a small change, as in the table below.
| Structure |  |  |
|---|---|---|
 |
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|
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 |
|
| · (magnetite Magnetite Magnetite is a ferrimagnetic mineral with chemical formula Fe3O4, one of several iron oxides and a member of the spinel group. The chemical IUPAC name is iron oxide and the common chemical name is ferrous-ferric oxide. The formula for magnetite may also be written as FeO·Fe2O3, which is one part... ) |
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