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| In [[physics]], a [[ferromagnetic]] material is said to have '''magnetocrystalline anisotropy''' if it takes more energy to [[magnetization|magnetize]] it in certain directions than in others. These directions are usually related to the [[crystal structure|principal axes]] of its [[crystal lattice]]. It is a special case of [[magnetic anisotropy]].
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| == Causes ==
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| The [[spin-orbit interaction]] is the primary source of magnetocrystalline [[anisotropy]].
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| == Practical relevance ==
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| Magnetocrystalline anisotropy has a great influence on industrial uses of ferromagnetic materials. Materials with high magnetic anisotropy usually have high [[coercivity]]; that is they are hard to demagnetize. These are called "hard" ferromagnetic materials, and are used to make [[permanent magnet]]s. For example, the high anisotropy of [[rare earth element|rare earth]] metals is mainly responsible for the strength of [[rare earth magnet]]s. During manufacture of magnets, a powerful magnetic field aligns the microcrystalline grains of the metal so their "easy" axes of magnetization all point in the same direction, freezing a strong magnetic field into the material.
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| On the other hand, materials with low magnetic anisotropy usually have low coercivity, their magnetization is easy to change. These are called "soft" ferromagnets, and are used to make [[magnetic core]]s for [[transformer]]s and [[inductor]]s. The small energy required to turn the direction of magnetization minimizes [[core losses]], energy dissipated in the transformer core when the alternating current changes direction.
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| ==Microscopic origin==
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| Magnetocrystalline anisotropy arises mostly from [[spin-orbit coupling]].<ref name=Cullity>{{Harvnb|Cullity|Graham|2005}}</ref> This effect is weak compared to the [[exchange interaction]] and is difficult to compute from first principles, although some successful computations have been made.<ref>{{Harvnb|Daalderop|Kelly|Schuurmans|1990}}</ref>
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| ==Thermodynamic theory==
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| 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 {{math|'''<var>M</var>''' {{=}} <var>M<sub>s</sub></var>(<var>α,β,γ</var>)}}, where {{math|<var>M</var><sub>s</sub>}} is the [[saturation magnetization]]. Because of [[time reversal symmetry]], only even powers of the cosines are allowed.<ref name=Landau>{{Harvnb|Landau|Lifshitz|Pitaevski|2004}}</ref> The nonzero terms in the expansion depend on the [[crystal system]] (''e.g.'', [[cubic crystal system|cubic]] or [[hexagonal crystal system|hexagonal]]).<ref name=Landau/> The ''[[Degree of a polynomial|order]]'' of a term in the expansion is the sum of all the exponents of magnetization components, ''i.e.'', {{math| <var>α β</var>}} is second order. | |
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| ===Uniaxial anisotropy===
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| [[File:EasyAxis.png|thumb|A representation of a uniaxial easy axis and easy plane. Arrows represent possible magnetization directions, black for the easy axis and red for the plane.]]
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| 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 {{math|<var>z</var>}} axis is taken to be the main symmetry axis of the crystal, the lowest order term in the energy is<ref>An arbitrary constant term is ignored.</ref>
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| :<math>E/V = K_1 \left(\alpha^2+\beta^2\right) = K_1\left(1-\gamma^2\right). </math><ref>The lowest-order term in the energy can be written in more than one way because, by definition, {{math| <var>α<sup>2</sup>+β<sup>2</sup>+γ<sup>2</sup><var> {{=}} 1}}.</ref>
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| The ratio {{math| <var>E/V</var>}} is an [[energy density]] (energy per unit volume). This can also be represented in [[spherical polar coordinates]] with {{math| <var>α</var> {{=}} cos <var>φ</var> sin <var>θ</var>}}, {{math| <var>β</var> {{=}} sin <var>φ</var> sin <var>θ</var>}}, and {{math| <var>γ</var> {{=}} cos <var>θ</var>}}:
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| :<math>\displaystyle E/V = K_1 \sin^2\theta.</math>
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| The parameter {{math| <var>K</var><sub>1</sub>}}, often represented as {{math| <var>K</var><sub>u</sub>}}, has units of [[energy density]] and depends on composition and temperature.
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| The [[minima]] in this energy with respect to {{math| <var>θ</var>}} satisfy
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| :<math>\frac{\partial E}{\partial \theta} = 0 \qquad \text{and} \qquad \frac{\partial^2 E}{\partial \theta^2} > 0.</math>
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| If {{math| <var>K</var><sub>1</sub> > 0}},
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| the directions of lowest energy are the {{math| ± <var>z</var>}} directions. The {{math| <var>z</var>}} axis is called the ''easy axis''. If {{math| <var>K</var><sub>1</sub> < 0}}, there is an ''easy plane'' perpendicular to the symmetry axis (the [[basal plane]] of the crystal). | |
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| Many models of magnetization represent the anisotropy as uniaxial and ignore higher order terms. However, if {{math| <var>K</var><sub>1</sub> < 0}}, 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 crystal system|hexagonal]], [[tetragonal crystal system|tetragonal]] or [[rhombohedral crystal system|rhombohedral]]).<ref name=Landau/>
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| <gallery>
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| image:Hexagonal lattice.svg|The hexagonal lattice cell.
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| image:Tetragonal.png|The tetragonal lattice cell.
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| image:Rhombohedral.svg|The rhombohedral lattice cell.
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| </gallery>
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| ====Hexagonal system====
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| [[File:EasyCone.png|thumb|A representation of an easy cone. All the minimum-energy directions (such as the arrow shown) lie on this cone.]]
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| In a [[hexagonal crystal system|hexagonal system]] the {{math|<var>c</var>}} axis is an axis of sixfold rotation symmetry. The energy density is, to fourth order,
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| :<math>\displaystyle E/V = K_1 \sin^2\theta + K_2 \sin^4\theta + K_3\sin^4\theta\cos 6\phi </math>.
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| The uniaxial anisotropy is mainly determined by the first two terms. Depending on the values {{math| <var>K</var><sub>1</sub> }} and {{math| <var>K</var><sub>2</sub>}}, there are four different kinds of anisotropy (isotropic, easy axis, easy plane and easy cone):<ref name=Cullity7-7>{{harvnb|Cullity|Graham|2005|p=7-7}}</ref>
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| * {{math| <var>K</var><sub>1</sub> {{=}} <var>K</var><sub>2</sub> {{=}} 0}}: the ferromagnet is [[isotropic]].
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| * {{math| <var>K</var><sub>1</sub> > 0}} and {{math| <var>K</var><sub>2</sub> > -<var>K</var><sub>1</sub>}}: the {{math|<var>c</var>}} axis is an easy axis.
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| * {{math| <var>K</var><sub>1</sub> > 0}} and {{math| <var>K</var><sub>2</sub> < -<var>K</var><sub>1</sub>}}: the basal plane is an easy plane.
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| * {{math| <var>K</var><sub>1</sub> < 0}} and {{math| <var>K</var><sub>2</sub> < -<var>K</var><sub>1</sub>/2}}: the basal plane is an easy plane.
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| * {{math| -2<var>K</var><sub>2</sub> < <var>K</var><sub>1</sub> < 0}}: the ferromagnet has an ''easy cone'' (see figure to right).
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| 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.
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| Below are some room-temperature anisotropy constants for hexagonal ferromagnets. Since all the values of {{math| <var>K</var><sub>1</sub>}} and {{math| <var>K</var><sub>2</sub>}} are positive, these materials have an easy axis.
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| {| class="wikitable"
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| |+ Room-temperature anisotropy constants ({{math| × 10<sup>4</sup> J/m<sup>3</sup> }}). <ref name=Cullity/>
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| |-
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| ! Structure !! <math>K_1</math> !! <math>K_2</math>
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| |-
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| | {{Cobalt}} || <math>45</math> || <math>15</math>
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| |-
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| | α{{Iron|2}}{{Oxygen|3}} ([[hematite]]) || <math>120</math><ref name=Dunlop/>
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| |-
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| | {{Barium}}{{Oxygen}}· 6{{Iron|2}}{{Oxygen|3}} || <math>3</math>
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| |-
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| | {{Yttrium}}{{Cobalt|5}} || <math>550</math>
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| |-
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| | {{Manganese}}{{Bismuth}} || <math>89</math> || <math>27</math>
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| |}
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| Higher order constants, in particular conditions, may lead to first order magnetization processes [[FOMP]].
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| ====Tetragonal and Rhombohedral systems====
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| The energy density for a tetragonal crystal is<ref name=Landau/>
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| :<math>\displaystyle E/V = K_1 \sin^2\theta + K_2 \sin^4\theta + K_3\sin^4\theta \sin 2\phi </math>.
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| Note that the {{math| <var>K</var><sub>3</sub>}} term, the one that determines the basal plane anisotropy, is fourth order (same as the {{math| <var>K</var><sub>2</sub>}} term). The definition of {{math| <var>K</var><sub>3</sub>}} may vary by a constant multiple between publications.
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| The energy density for a rhombohedral crystal is<ref name=Landau/>
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| :<math>\displaystyle E/V = K_1 \sin^2\theta + K_2 \sin^4\theta + K_3\cos\theta\sin^3\theta \cos 3\phi </math>.
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| ===Cubic anisotropy===
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| [[File:MagneticAnisotropyCubicAnisotropyPositive.png|thumb|right|Energy surface for cubic anisotropy with {{math| <var>K</var><sub>1</sub> > 0}}. Both color saturation and distance from the origin increase with energy. The lowest energy (lightest blue) is arbitrarily set to zero.]]
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| [[File:MagneticAnisotropyCubicAnisotropyNegative.png|thumb|right|Energy surface for cubic anisotropy with {{math| <var>K</var><sub>1</sub> < 0}}. Same conventions as for {{math| <var>K</var><sub>1</sub> > 0}}.]]
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| In a [[cubic crystal system|cubic crystal]] the lowest order terms in the energy are<ref name=Landau/><ref name=Cullity/>
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| :<math> E/V = K_1 \left(\alpha^2\beta^2+\beta^2\gamma^2+\gamma^2\alpha^2\right) + K_2\alpha^2\beta^2\gamma^2.</math>
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| If the second term can be neglected, the easy axes are the {{math| < 100 >}} axes (''i.e.'', the {{math| ± <var>x</var>}}, {{math| ± <var>y</var>}}, and {{math| ± <var>z</var>}}, directions) for {{math| <var>K</var><sub>1</sub> > 0}} and the {{math| < 111 >}} directions for {{math| <var>K</var><sub>1</sub> < 0}} (see images on right).
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| If {{math| <var>K</var><sub>2</sub>}} is not assumed to be zero, the easy axes depend on both {{math| <var>K</var><sub>1</sub>}} and {{math| <var>K</var><sub>2</sub>}}. 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.
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| {| class="wikitable"
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| |+ Easy axes for {{math| <var>K</var><sub>1</sub> > 0}}. <ref name=Cullity/>
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| |-
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| ! Type of axis !! <math>K_2 = +\infty</math> to <math>-9K_1/4</math> !! <math>K_2 = -9K_1/4</math> to <math>-9K_1</math> !! <math>K_2 = -9K_1</math> to <math>-\infty</math>
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| |-
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| ! scope="row" | Easy
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| | <math>\langle 100 \rangle</math> || <math>\langle 100 \rangle</math> || <math>\langle 111 \rangle</math>
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| |-
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| ! scope="row" | Medium
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| | <math>\langle 110 \rangle</math> || <math>\langle 111 \rangle</math> || <math>\langle 100 \rangle</math>
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| |-
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| ! scope="row" | Hard
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| | <math>\langle 111 \rangle</math> || <math>\langle 110 \rangle</math> || <math>\langle 110 \rangle</math>
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| |}
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| {| class="wikitable"
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| |+ Easy axes for {{math| <var>K</var><sub>1</sub> < 0}}. <ref name=Cullity/>
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| |-
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| ! Type of axis !! <math>K_2 = +\infty</math> to <math>-9K_1/4</math> !! <math>K_2 = -9K_1/4</math> to <math>-9K_1</math> !! <math>K_2 = -9K_1</math> to <math>-\infty</math>
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| |-
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| ! scope="row" | Easy
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| | <math>\langle 111 \rangle</math> || <math>\langle 110 \rangle</math> || <math>\langle 110 \rangle</math>
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| |-
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| ! scope="row" | Medium
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| | <math>\langle 110 \rangle</math> || <math>\langle 111 \rangle</math> || <math>\langle 100 \rangle</math>
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| |-
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| ! scope="row" | Hard
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| | <math>\langle 100 \rangle</math> || <math>\langle 100 \rangle</math> || <math>\langle 111 \rangle</math>
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| |}
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| Below are some room-temperature anisotropy constants for cubic ferromagnets. The compounds involving {{Iron|2}}{{Oxygen|3}} are [[Ferrite (magnet)|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.
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| {| class="wikitable"
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| |+ Room-temperature anisotropy constants ({{math| × 10<sup>4</sup> J/m<sup>3</sup> }}). <ref name=Cullity/>
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| |-
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| ! Structure !! <math>K_1</math> !! <math>K_2</math>
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| |-
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| | {{Iron}} || <math>4.8</math> || <math>\pm 0.5</math>
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| |-
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| | {{Nickel}} || <math> -0.5</math> || <math> -0.2</math>
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| |-
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| | {{Iron}}{{Oxygen}}· {{Iron|2}}{{Oxygen|3}} ([[magnetite]]) || <math>-1.1</math>
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| |-
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| | {{Manganese}}{{Oxygen}}· {{Iron|2}}{{Oxygen|3}} || <math>-0.3</math>
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| |-
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| | {{Nickel}}{{Oxygen}}· {{Iron|2}}{{Oxygen|3}} || <math>-0.62</math>
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| |-
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| | {{Magnesium}}{{Oxygen}}· {{Iron|2}}{{Oxygen|3}} || <math>-0.25</math>
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| |-
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| | {{Cobalt}}{{Oxygen}}· {{Iron|2}}{{Oxygen|3}} || <math>20</math>
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| |-
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| |}
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| ==Temperature dependence of anisotropy==
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| 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.<ref name=Cullity/> Some materials also have an ''isotropic point'' at which {{math| <var>K</var><sub>1</sub> {{=}} 0}}. [[Magnetite]] ({{Iron|3}}{{Oxygen|4}}), a mineral of great importance to [[rock magnetism]] and [[paleomagnetism]], has an isotropic point at 130 [[kelvin]].<ref name=Dunlop>{{harvnb|Dunlop|Özdemir|1997}}</ref>
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| 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.<ref name=Dunlop/>
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| ==Magnetostriction==
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| 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 [[magnetostriction]]. To keep the lattice from deforming, a [[Stress (mechanics)|stress]] must be applied. If the crystal is not under stress, magnetostriction alters the effective magnetocrystalline anisotropy. If a ferromagnet is [[single domain (magnetic)|single domain]] (uniformly magnetized), the effect is to change the magnetocrystalline anisotropy parameters.<ref name=Chikazumi>{{harvnb|Chikazumi|1997|loc=chapter 12}}</ref>
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| In practice, the correction is generally not large. In hexagonal crystals, there is no change in {{math| <var>K</var><sub>1</sub>}}.<ref name=Ye>{{harvnb|Ye|Newell|Merrill|1994}}</ref> In cubic crystals, there is a small change, as in the table below.
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| {| class="wikitable"
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| |+ Room-temperature anisotropy constants {{math| <var>K</var><sub>1</sub>}} (zero-strain) and {{math| <var>K</var><sub>1</sub>′}} (zero-stress) ({{math| × 10<sup>4</sup> J/m<sup>3</sup> }}). <ref name=Ye/>
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| |-
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| ! Structure !! <math>K_1</math> !! <math>K'_1</math>
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| |-
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| | {{Iron}} || <math>4.7</math> || <math>4.7</math>
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| |-
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| | {{Nickel}} || <math>-6.0</math> || <math>-5.9</math>
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| |-
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| | {{Iron}}{{Oxygen}}· {{Iron|2}}{{Oxygen|3}} ([[magnetite]]) || <math>-1.1</math> || <math>-1.4</math>
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| |}
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| ==See also==
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| *[[Anisotropy energy]]
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| ==Notes and references==
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| {{Reflist|3}}
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| ==Further reading==
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| {{Refbegin|2}}
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| *{{cite book
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| |last1 = Chikazumi
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| |first1 = Sōshin
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| |title = Physics of Ferromagnetism
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| |publisher = [[Clarendon Press]]
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| |year = 1997
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| |isbn = 0-19-851776-9
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| |ref = harv
| |
| }}
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| *{{Cite book
| |
| |first1 = B.D.
| |
| |last1 = Cullity
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| |first2 = C.D.
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| |last2 = Graham
| |
| |title = Introduction to Magnetic Materials
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| |publisher = [[John Wiley & Sons]]
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| |year = 2005
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| |isbn = 0-201-01218-9
| |
| |ref = harv
| |
| }}
| |
| *{{Cite journal
| |
| |last1 = Daalderop
| |
| |first1 = G. H. O.
| |
| |last2 = Kelly
| |
| |first2 = P. J.
| |
| |last3 = Schuurmans
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| |first3 = M. F. H.
| |
| |year = 1990
| |
| |title = First-principles calculation of the magnetocrystalline anisotropy energy of iron, cobalt, and nickel
| |
| |journal = [[Phys. Rev. B]]
| |
| |volume = 41
| |
| |issue = 17
| |
| |pages = 11919–11937
| |
| |doi = 10.1103/PhysRevB.41.11919
| |
| |ref = harv
| |
| |bibcode = 1990PhRvB..4111919D }}
| |
| *{{cite book
| |
| |last1 = Dunlop
| |
| |first1 = David J.
| |
| |last2 = Özdemir
| |
| |first2 = Özden
| |
| |title = Rock Magnetism: Fundamentals and Frontiers
| |
| |publisher = [[Cambridge Univ. Press]]
| |
| |year = 1997
| |
| |isbn = 0-521-32514-5
| |
| |ref = harv
| |
| }}
| |
| *{{Cite book
| |
| |first = L. D.
| |
| |last = Landau
| |
| |author-link=Lev Landau
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| |first2 = E. M.
| |
| |last2 = Lifshitz
| |
| |author2-link = Evgeny Lifshitz
| |
| |first3 = L. P.
| |
| |last3 = Pitaevski
| |
| |title = Electrodynamics of Continuous Media
| |
| |edition = Second
| |
| |volume = 8
| |
| |series = [[Course of Theoretical Physics]]
| |
| |publisher = [[Elsevier]]
| |
| |year = 2004
| |
| |origyear = First published in 1960
| |
| |isbn = 0-7506-2634-8
| |
| |ref = harv
| |
| }}
| |
| *{{Cite journal
| |
| |last1 = Ye
| |
| |first1 = Jun
| |
| |last2 = Newell
| |
| |first2 = Andrew J.
| |
| |last3 = Merrill
| |
| |first3 = Ronald T.
| |
| |year = 1994
| |
| |title = A re-evaluation of magnetocrystalline anisotropy and magnetostriction constants
| |
| |journal = [[Geophysical Research Letters]]
| |
| |volume = 21
| |
| |issue = 1
| |
| |pages = 25–28
| |
| |doi = 10.1029/93GL03263
| |
| |ref = harv
| |
| |bibcode = 1994GeoRL..21...25Y }}
| |
| {{Refend}}
| |
| | |
| {{DEFAULTSORT:Magnetocrystalline Anisotropy}}
| |
| [[Category:Magnetic ordering]]
| |
| [[Category:Orientation]]
| |
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Many engagement rings become a set with a matching wedding engagement ring. Buying the set is a choice because merchants also save bankroll. Also, buying her a wedding band along the woman's diamond ring shows your dedication and commitment to explain with the wedding. It might not seem important to you in the time, but to her it indicates everything!
Women wear necklace on the neck. Women are wearing Pendants and necklaces since years. However, since the inception, necklaces have changed a bunch. The pattern, kinds of have re-structured. Now Pendants are prepared of many materials like gold, silver and yellow metal. In today's economy, the prices of gold are soaring and thus, many women are purchasing silver bracelet. One will obtain the best designs in many different styles on the inside market. Moreover, the regarding pendant made from silver is pretty new. A silver pendant is a very small piece of ornament, which an individual hangs at the neck in the chain. A single solitaire is incredibly popular the particular market and girls love put on it.
It is vital to give peace the opportunity and publicize it in a silent route. You cannot continue shouting peace at work or onto the streets. Down the road . however say it is often a quite non offensive way, without sounding political. By putting on the symbol around your neck, ears or finger, you can not only promote the concept of world peace but also promote peace within private personal neighborhood.
Round Brilliant - The round brilliant is present day day version of the round by using a lot more sparkle and shine. It's not the most well liked choice in diamond ring.
Currently Deva is designing gold plated lapel pins for the campaign of Ceasar Mitchell, city council president. Deva is positioning Custom Identity as a brandname that can transition easily into constantly working out for charitable events. Custom Identity LLC., was among the list of sponsors for this Communities going to school Back to Camp Weekend, an event that helps supports underpriveleged children.
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