Spontaneous magnetization is the appearance of an ordered spin state (

magnetization In classical electromagnetism, magnetization is the vector field that expresses the density of permanent or induced magnetic dipole moments in a magnetic material. Accordingly, physicists and engineers usually define magnetization as the quanti ...

) at zero applied magnetic field in a

ferromagnetic Ferromagnetism is a property of certain materials (such as iron) that results in a significant, observable magnetic permeability, and in many cases, a significant magnetic coercivity, allowing the material to form a permanent magnet. Ferromagne ...

ferrimagnetic A ferrimagnetic material is a material that has populations of atoms with opposing magnetic moments, as in antiferromagnetism, but these moments are unequal in magnitude, so a spontaneous magnetization remains. This can for example occur wh ...

material below a critical point called the Curie temperature or .

Overview

Heated to temperatures above , ferromagnetic materials become

paramagnetic Paramagnetism is a form of magnetism whereby some materials are weakly attracted by an externally applied magnetic field, and form internal, induced magnetic fields in the direction of the applied magnetic field. In contrast with this behavior, ...

and their magnetic behavior is dominated by spin waves or magnons, which are

boson In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0, 1, 2, ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have half odd-intege ...

collective excitations with energies in the meV range. The magnetization that occurs below is an example of the "spontaneous" breaking of a

global symmetry The symmetry of a physical system is a physical or mathematical feature of the system (observed or intrinsic) that is preserved or remains unchanged under some transformation. A family of particular transformations may be ''continuous'' (such ...

, a phenomenon that is described by

Goldstone's theorem In physics, Goldstone bosons or Nambu–Goldstone bosons (NGBs) are bosons that appear necessarily in models exhibiting spontaneous breakdown of continuous symmetries. They were discovered by Yoichiro Nambu within the context of the BCS superc ...

. The term "symmetry breaking" refers to the choice of a magnetization direction by the spins, which have

spherical symmetry In geometry, circular symmetry is a type of continuous symmetry for a planar object that can be rotated by any arbitrary angle and map onto itself. Rotational circular symmetry is isomorphic with the circle group in the complex plane, or the ...

above , but a preferred axis (the magnetization direction) below .

Temperature dependence

To a first order approximation, the temperature dependence of spontaneous magnetization at low temperatures is given by the Bloch law (by

Felix Bloch Felix Bloch (; ; 23 October 1905 – 10 September 1983) was a Swiss-American physicist who shared the 1952 Nobel Prize in Physics with Edward Mills Purcell "for their development of new methods for nuclear magnetic precision measurements and di ...

): :

M(T) = M(0) \left -\left(\frac \right)^ \right

where is the spontaneous magnetization at

absolute zero Absolute zero is the lowest possible temperature, a state at which a system's internal energy, and in ideal cases entropy, reach their minimum values. The absolute zero is defined as 0 K on the Kelvin scale, equivalent to −273.15 ° ...

. 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 bosons corresponding to the broken symmetry. This is exactly true for an isotropic magnet. 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

T\rightarrow 0

. 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 ''T''^3/2 law. All real magnets are anisotropic to some extent. Near the Curie temperature, :

M(T) \propto \left(T_-T\right)^\beta,

where is a critical exponent that depends on the universality class of the magnetic interaction. Experimentally the exponent is 0.34 for

iron Iron is a chemical element; it has symbol Fe () and atomic number 26. It is a metal that belongs to the first transition series and group 8 of the periodic table. It is, by mass, the most common element on Earth, forming much of Earth's o ...

and 0.51 for

nickel Nickel is a chemical element; it has symbol Ni and atomic number 28. It is a silvery-white lustrous metal with a slight golden tinge. Nickel is a hard and ductile transition metal. Pure nickel is chemically reactive, but large pieces are slo ...

. An empirical interpolation of the two regimes is given by :

\frac  = \left (1-(T/T_\right)^)^,

it is easy to check two limits of this interpolation that follow laws similar to the Bloch law, for

T \rightarrow 0

, and the critical behavior, for

T \rightarrow T_

, respectively.

Notes and references

* *{{cite book , last = Chikazumi , first = Sōshin , title = Physics of Ferromagnetism , publisher =

Clarendon Press Oxford University Press (OUP) is the publishing house of the University of Oxford. It is the largest university press in the world. Its first book was printed in Oxford in 1478, with the Press officially granted the legal right to print books ...

, year = 1997 , isbn = 0-19-851776-9

Overview

Temperature dependence

See also

Notes and references

Further reading