Spontaneous magnetization is the appearance of an ordered

spin Spin or spinning most often refers to: * Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning * Spin, the rotation of an object around a central axis * Spin (propaganda), an intentionally b ...

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. Movement within this field is described by direction and is either Axial or Di ...

) at zero applied magnetic field in a

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 ...

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 when ...

material below a critical point called the

Curie temperature In physics and materials science, the Curie temperature (''T''C), or Curie point, is the temperature above which certain materials lose their permanent magnetic properties, which can (in most cases) be replaced by induced magnetism. The Cur ...

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, d ...

and their magnetic behavior is dominated by

spin wave A spin wave is a propagating disturbance in the ordering of a magnetic material. These low-lying collective excitations occur in magnetic lattices with continuous symmetry. From the equivalent quasiparticle point of view, spin waves are known as ...

s or

magnon A magnon is a quasiparticle, a collective excitation of the electrons' spin structure in a crystal lattice. In the equivalent wave picture of quantum mechanics, a magnon can be viewed as a quantized spin wave. Magnons carry a fixed amount of e ...

s, 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 odd half-integer s ...

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

global symmetry In physics, a 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 ''continuo ...

, a phenomenon that is described by

Goldstone's theorem In particle and condensed matter 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 in par ...

. 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 Bloch is a surname of German origin. Notable people with this surname include: A–F * (1859-1914), French rabbi *Adele Bloch-Bauer (1881-1925), Austrian entrepreneur *Albert Bloch (1882–1961), American painter * (born 1972), German motor journal ...

T^3/2 law: :

M(T) = M(0)\left(1-(T/T_c\right)^),

where is the spontaneous magnetization at

absolute zero Absolute zero is the lowest limit of the thermodynamic temperature scale, a state at which the enthalpy and entropy of a cooled ideal gas reach their minimum value, taken as zero kelvin. The fundamental particles of nature have minimum vibration ...

. The decrease in spontaneous magnetization at higher temperatures is caused by the increasing excitation of

spin waves A spin wave is a propagating disturbance in the ordering of a magnetic material. These low-lying collective excitations occur in magnetic lattices with continuous symmetry. From the equivalent quasiparticle point of view, spin waves are known as ...

. In a particle description, the spin waves correspond to magnons, which are the massless

Goldstone boson In particle and condensed matter 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 in parti ...

s corresponding to the

broken symmetry In physics, symmetry breaking is a phenomenon in which (infinitesimally) small fluctuations acting on a system crossing a critical point decide the system's fate, by determining which branch of a bifurcation is taken. To an outside observ ...

. This is exactly true for an isotropic magnet.

Magnetic anisotropy In condensed matter physics, magnetic anisotropy describes how an object's magnetic properties can be different depending on direction. In the simplest case, there is no preferential direction for an object's magnetic moment. It will respond to ...

, 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_c-T\right)^\beta,

where is a

critical exponent Critical or Critically may refer to: *Critical, or critical but stable, medical states **Critical, or intensive care medicine *Critical juncture, a discontinuous change studied in the social sciences. *Critical Software, a company specializing in ...

that depends on the

universality class In statistical mechanics, a universality class is a collection of mathematical models which share a single scale invariant limit under the process of renormalization group flow. While the models within a class may differ dramatically at finite s ...

of the magnetic interaction. Experimentally the exponent is 0.34 for

iron Iron () is a chemical element with symbol Fe (from la, ferrum) 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, right in f ...

and 0.51 for

nickel Nickel is a chemical element with 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 slow to ...

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

\frac  = \left (1-(T/T_c\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_C

, respectively.

Overview

Temperature dependence

See also

Notes and references

Further reading