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| '''Spontaneous magnetization''' is the appearance of an ordered [[Spin (physics)|spin]] state ([[magnetization]]) at zero applied magnetic field in a [[ferromagnetic]] or [[ferrimagnetic]] material below a [[critical point (physics)|critical point]] called the [[Curie temperature]] or {{math|<var>T<sub>C</sub></var>}}.
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| ==Overview==
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| Heated to temperatures above {{math|<var>T<sub>C</sub></var>}}, ferromagnetic materials become [[paramagnetic]] and their magnetic behavior is dominated by [[spin wave]]s or [[magnon]]s, which are [[boson]] collective excitations with energies in the meV range. The magnetization that occurs below {{math|<var>T<sub>C</sub></var>}} is a famous example of the [[spontaneous symmetry breaking|"spontaneous" breaking]] of a global symmetry, a phenomenon that is described by [[Goldstone's theorem]]. The term "symmetry breaking" refers to the choice of a magnetization direction by the spins, which have spherical symmetry above {{math|<var>T<sub>C</sub></var>}}, but a preferred axis (the magnetization direction) below {{math|<var>T<sub>C</sub></var>}}.
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| ==Temperature dependence==
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| To first order, the temperature dependence of spontaneous magnetization at low temperatures is given by [[Felix Bloch|Bloch's]] Law: <ref>{{harvnb|Ashcroft|Mermin|1976|p=708}}</ref>
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| :<math>M(T) = M(0)\left(1-(T/T_c\right)^{3/2}),</math>
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| where {{math|M(0)}} is the spontaneous magnetization at absolute zero. The decrease in spontaneous magnetization at higher temperatures is caused by the increasing excitation of spin waves. In a particle description, the spin waves correspond to magnons, which are the massless [[Goldstone boson]]s corresponding to the [[continuous symmetry|broken symmetry]]. This is exactly true for an isotropic magnet.
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| [[Magnetic anisotropy]], that is the existence of an easy direction along which the moments align spontaneously in the crystal, corresponds however to "massive" magnons. This is a way of saying that they cost a minimum amount of energy to excite, hence they are very unlikely to be excited as <math>T\rightarrow 0</math>. Hence the magnetization of an anisotropic magnet is harder to destroy at low temperature and the temperature dependence of the magnetization deviates accordingly from the Bloch's law. All real magnets are anisotropic to some extent.
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| Near the Curie temperature,
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| :<math>M(T) \propto \left(T-T_c\right)^\beta,</math> | |
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| where {{math|<var>β</var>}} is a [[critical exponent]] that depends on composition. The exponent is {{math|0.34}} for {{iron}} and {{math|0.51}} for {{nickel}}.<ref>{{harvnb|Chikazumi|1997|pp=128–129}}</ref>
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| An empirical interpolation of the two regimes is given by
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| :<math> \frac {M(T)}{M(0)} = \left (1-(T/T_c\right)^{\alpha})^{\beta},</math>
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| it is easy to check two limits of this interpolation that follow laws similar to the Bloch law, for <math> T \rightarrow 0</math>, and the critical behavior, for <math> T \rightarrow T_C </math>, respectively.
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| ==Notes and references== | |
| <references/>
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| ==Further reading==
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| *{{cite book
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| | last = Ashcroft
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| | first = Neil W.
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| | last2 = Mermin
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| | first2 = N. David
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| | author2-link = N. David Mermin
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| | title = Solid State Physics
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| | publisher = [[Holt, Rinehart and Winston]]
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| | year = 1976
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| | isbn = 0-03-083993-9
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| | ref = harv
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| }}
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| *{{cite book
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| |last = Chikazumi
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| |first = 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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| }}
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| [[Category:Magnetic ordering]]
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