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In
condensed matter physics Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter, especially the solid and liquid phases which arise from electromagnetic forces between atoms. More generally, the ...
, magnetic anisotropy describes how an object's
magnetic Magnetism is the class of physical attributes that are mediated by a magnetic field, which refers to the capacity to induce attractive and repulsive phenomena in other entities. Electric currents and the magnetic moments of elementary particles ...
properties can be different depending on direction. In the simplest case, there is no preferential direction for an object's
magnetic moment In electromagnetism, the magnetic moment is the magnetic strength and orientation of a magnet or other object that produces a magnetic field. Examples of objects that have magnetic moments include loops of electric current (such as electromagnets ...
. It will respond to an applied
magnetic field A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...
in the same way, regardless of which direction the field is applied. This is known as magnetic
isotropy Isotropy is uniformity in all orientations; it is derived . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence ''anisotropy''. ''Anisotropy'' is also used to describe ...
. In contrast, magnetically anisotropic materials will be easier or harder to magnetize depending on which way the object is rotated. For most magnetically anisotropic materials, there are two easiest directions to magnetize the material, which are a 180° rotation apart. The line parallel to these directions is called the easy axis. In other words, the easy axis is an energetically favorable direction of
spontaneous magnetization Spontaneous magnetization is the appearance of an ordered spin state (magnetization) at zero applied magnetic field in a ferromagnetic or ferrimagnetic material below a critical point called the Curie temperature or . Overview Heated to tempera ...
. Because the two opposite directions along an easy axis are usually equivalently easy to magnetize along, the actual direction of magnetization can just as easily settle into either direction, which is an example of
spontaneous symmetry breaking Spontaneous symmetry breaking is a spontaneous process of symmetry breaking, by which a physical system in a symmetric state spontaneously ends up in an asymmetric state. In particular, it can describe systems where the equations of motion or t ...
. Magnetic anisotropy is a prerequisite for hysteresis in ferromagnets: without it, a ferromagnet is
superparamagnetic Superparamagnetism is a form of magnetism which appears in small ferromagnetic or ferrimagnetic nanoparticles. In sufficiently small nanoparticles, magnetization can randomly flip direction under the influence of temperature. The typical time bet ...
.


Sources

The observed magnetic anisotropy in an object can happen for several different reasons. Rather than having a single cause, the overall magnetic anisotropy of a given object is often explained by a combination of these different factors: ;
Magnetocrystalline anisotropy In physics, a ferromagnetic material is said to have magnetocrystalline anisotropy if it takes more energy to magnetize it in certain directions than in others. These directions are usually related to the principal axes of its crystal lattice. I ...
: The atomic structure of a
crystal A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macr ...
introduces preferential directions for the
magnetization In classical electromagnetism, magnetization is the vector field that expresses the density of permanent or induced magnetic dipole moments in a magnetic material. Movement within this field is described by direction and is either Axial or Dia ...
. ; Shape anisotropy: When a particle is not perfectly spherical, the
demagnetizing field The demagnetizing field, also called the stray field (outside the magnet), is the magnetic field (H-field) generated by the magnetization in a magnet. The total magnetic field in a region containing magnets is the sum of the demagnetizing fields ...
will not be equal for all directions, creating one or more easy axes. ; Magnetoelastic anisotropy: Tension may alter magnetic behaviour, leading to magnetic anisotropy. ; Exchange anisotropy: Occurs when
antiferromagnetic In materials that exhibit antiferromagnetism, the magnetic moments of atoms or molecules, usually related to the spins of electrons, align in a regular pattern with neighboring spins (on different sublattices) pointing in opposite directions. ...
and
ferromagnetic Ferromagnetism is a property of certain materials (such as iron) which results in a large observed magnetic permeability, and in many cases a large magnetic coercivity allowing the material to form a permanent magnet. Ferromagnetic materials ...
materials interact.


At the molecular level

The magnetic anisotropy of a
benzene Benzene is an organic chemical compound with the molecular formula C6H6. The benzene molecule is composed of six carbon atoms joined in a planar ring with one hydrogen atom attached to each. Because it contains only carbon and hydrogen a ...
ring (A),
alkene In organic chemistry, an alkene is a hydrocarbon containing a carbon–carbon double bond. Alkene is often used as synonym of olefin, that is, any hydrocarbon containing one or more double bonds.H. Stephen Stoker (2015): General, Organic, an ...
(B),
carbonyl In organic chemistry, a carbonyl group is a functional group composed of a carbon atom double-bonded to an oxygen atom: C=O. It is common to several classes of organic compounds, as part of many larger functional groups. A compound containing a ...
(C), alkyne (D), and a more complex molecule (E) are shown in the figure. Each of these unsaturated functional groups (A-D) create a tiny magnetic field and hence some local anisotropic regions (shown as cones) in which the shielding effects and the chemical shifts are unusual. The bisazo compound (E) shows that the designated proton can appear at different chemical shifts depending on the photoisomerization state of the azo groups. The ''trans'' isomer holds proton far from the cone of the benzene ring thus the magnetic anisotropy is not present. While the ''cis'' form holds proton in the vicinity of the cone, shields it and decreases its chemical shift. This phenomenon enables a new set of
nuclear Overhauser effect The nuclear Overhauser effect (NOE) is the transfer of nuclear spin polarization from one population of spin-active nuclei (e.g. 1H, 13C, 15N etc.) to another via cross-relaxation. A phenomenological definition of the NOE in nuclear magnetic res ...
(NOE) interactions (shown in red) that come to existence in addition to the previously existing ones (shown in blue).


Single-domain magnet

Suppose that a ferromagnet is single-domain in the strictest sense: the magnetization is uniform and rotates in unison. If the
magnetic moment In electromagnetism, the magnetic moment is the magnetic strength and orientation of a magnet or other object that produces a magnetic field. Examples of objects that have magnetic moments include loops of electric current (such as electromagnets ...
is \boldsymbol and the volume of the particle is V, the magnetization is \mathbf = \boldsymbol/V = M_s \left(\alpha,\beta,\gamma\right), where M_s is the
saturation magnetization Seen in some magnetic materials, saturation is the state reached when an increase in applied external magnetic field ''H'' cannot increase the magnetization of the material further, so the total magnetic flux density ''B'' more or less levels off ...
and \alpha, \beta, \gamma are
direction cosines In analytic geometry, the direction cosines (or directional cosines) of a vector are the cosines of the angles between the vector and the three positive coordinate axes. Equivalently, they are the contributions of each component of the basis to ...
(components of a
unit vector In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat"). The term ''direction vec ...
) so \alpha^2 + \beta^2 + \gamma^2 = 1. The energy associated with magnetic anisotropy can depend on the direction cosines in various ways, the most common of which are discussed below.


Uniaxial

A magnetic particle with uniaxial anisotropy has one easy axis. If the easy axis is in the z direction, the
anisotropy energy Anisotropic energy is energy that is directionally specific. The word anisotropy means "directionally dependent", hence the definition. The most common form of anisotropic energy is magnetocrystalline anisotropy, which is commonly studied in ferrom ...
can be expressed as one of the forms: :E = KV \left(1 - \gamma^2 \right) = KV \sin^2\theta, where V is the volume, K the anisotropy constant, and \theta the angle between the easy axis and the particle's magnetization. When shape anisotropy is explicitly considered, the symbol \mathcal is often used to indicate the anisotropy constant, instead of K. In the widely used
Stoner–Wohlfarth model The Stoner–Wohlfarth model is a widely used model for the magnetization of single-domain ferromagnets. It is a simple example of magnetic hysteresis and is useful for modeling small magnetic particles in magnetic storage, biomagnetism, rock magn ...
, the anisotropy is uniaxial.


Triaxial

A magnetic particle with triaxial anisotropy still has a single easy axis, but it also has a hard axis (direction of maximum energy) and an intermediate axis (direction associated with a
saddle point In mathematics, a saddle point or minimax point is a point on the surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero (a critical point), but which is not a local extremum of the functio ...
in the energy). The coordinates can be chosen so the energy has the form :E = K_aV\alpha^2 + K_bV\beta^2. If K_a > K_b > 0, the easy axis is the z direction, the intermediate axis is the y direction and the hard axis is the x direction.


Cubic

A magnetic particle with cubic anisotropy has three or four easy axes, depending on the anisotropy parameters. The energy has the form :E = KV \left(\alpha^2\beta^2 + \beta^2\gamma^2 + \gamma^2\alpha^2\right). If K > 0, the easy axes are the x, y, and z axes. If K < 0, there are four easy axes characterized by x = \pm y = \pm z.


See also

*
Fluorescence anisotropy Fluorescence anisotropy or fluorescence polarization is the phenomenon where the light emitted by a fluorophore has unequal intensities along different axes of polarization. Early pioneers in the field include Aleksander Jablonski, Gregorio Weber ...


References


Further reading

* {{DEFAULTSORT:Magnetic Anisotropy Magnetic ordering Orientation (geometry)