Editing Ferromagnetism (section)

==Explanation==
The [[Bohr–Van Leeuwen theorem]], discovered in the 1910s, showed that [[classical physics]] theories are unable to account for any form of material magnetism, including ferromagnetism; the explanation rather depends on the [[quantum mechanical]] description of [[atom]]s. Each of an atom's electrons has a [[magnetic moment]] according to its [[Spin (physics)|spin]] state, as described by quantum mechanics. The [[Pauli exclusion principle]], also a consequence of quantum mechanics, restricts the occupancy of electrons' spin states in [[atomic orbitals]], generally causing the magnetic moments from an atom's electrons to largely or completely cancel.<ref>{{cite book|last = Feynman|first = Richard P.|author2=Robert Leighton |author3=Matthew Sands|title = The Feynman Lectures on Physics, Vol. 2|url = https://feynmanlectures.caltech.edu/II_37.html|publisher = Addison-Wesley|year = 1963 |pages = Ch. 37}}</ref> An atom will have a ''net'' magnetic moment when that cancellation is incomplete.

===Origin of atomic magnetism===
One of the fundamental properties of an [[electron]] (besides that it carries charge) is that it has a [[Electron magnetic moment|magnetic dipole moment]], i.e., it behaves like a tiny magnet, producing a [[magnetic field]]. This dipole moment comes from a more fundamental property of the electron: its quantum mechanical spin. Due to its quantum nature, the spin of the electron can be in one of only two states, with the magnetic field either pointing "up" or "down" (for any choice of up and down). Electron spin in atoms is the main source of ferromagnetism, although there is also a contribution from the [[Atomic orbital|orbital]] [[angular momentum]] of the electron about the [[Atomic nucleus|nucleus]]. When these magnetic dipoles in a piece of matter are aligned (point in the same direction), their individually tiny magnetic fields add together to create a much larger macroscopic field.

However, materials made of atoms with filled [[electron shell]]s have a total dipole moment of zero: because the electrons all exist in pairs with opposite spin, every electron's magnetic moment is cancelled by the opposite moment of the second electron in the pair. Only atoms with partially filled shells (i.e., [[Unpaired electron|unpaired spins]]) can have a net magnetic moment, so ferromagnetism occurs only in materials with partially filled shells. Because of [[Hund's rules]], the first few electrons in an otherwise unoccupied shell tend to have the same spin, thereby increasing the total dipole moment.

These [[unpaired electron|unpaired dipoles]] (often called simply "spins", even though they also generally include orbital angular momentum) tend to align in parallel to an external magnetic field{{snd}} leading to a macroscopic effect called [[paramagnetism]]. In ferromagnetism, however, the magnetic interaction between neighboring atoms' magnetic dipoles is strong enough that they align with ''each other'' regardless of any applied field, resulting in the [[spontaneous magnetization]] of so-called [[#Magnetic domains|domains]]. This results in the large observed [[magnetic permeability]] of ferromagnetics, and the ability of magnetically hard materials to form [[permanent magnets]].

===Exchange interaction===
{{Main|Exchange interaction}}
When two nearby atoms have unpaired electrons, whether the electron spins are parallel or antiparallel affects whether the electrons can share the same orbit as a result of the quantum mechanical effect called the [[exchange interaction]]. This in turn affects the electron location and the [[Coulomb force|Coulomb (electrostatic) interaction]] and thus the energy difference between these states.

The exchange interaction is related to the Pauli exclusion principle, which says that two electrons with the same spin cannot also be in the same spatial state (orbital). This is a consequence of the [[spin–statistics theorem]] and that electrons are [[fermions]]. Therefore, under certain conditions, when the [[atomic orbital|orbitals]] of the unpaired outer [[valence electron]]s from adjacent atoms overlap, the distributions of their [[electric charge]] in space are farther apart when the electrons have parallel spins than when they have opposite spins. This reduces the [[electrostatic energy]] of the electrons when their spins are parallel compared to their energy when the spins are antiparallel, so the parallel-spin state is more stable. This difference in energy is called the [[exchange energy]]. In simple terms, the outer electrons of adjacent atoms, which repel each other, can move further apart by aligning their spins in parallel, so the spins of these electrons tend to line up.

This energy difference can be orders of magnitude larger than the energy differences associated with the [[magnetic dipole–dipole interaction]] due to dipole orientation,<ref name=Chikazumi2>{{cite book |last=Chikazumi |first=Sōshin |title=Physics of ferromagnetism |year=2009 |publisher=Oxford University Press |location=Oxford |isbn=978-0-19-956481-1 |edition=2nd |others=English edition prepared with the assistance of C.&nbsp;D. Graham, Jr. |pages=129–130}}</ref> which tends to align the dipoles antiparallel. In certain doped semiconductor oxides, [[RKKY interaction]]s have been shown to bring about periodic longer-range magnetic interactions, a phenomenon of significance in the study of [[Spintronics|spintronic materials]].<ref>{{cite journal |last1=Assadi |first1=M.&nbsp;H.&nbsp;N. |last2=Hanaor |first2=D.&nbsp;A.&nbsp;H. |title=Theoretical study on copper's energetics and magnetism in TiO<sub>2</sub> polymorphs |journal= Journal of Applied Physics |year=2013 |volume=113 |issue=23 |pages=233913-1–233913-5 |doi=10.1063/1.4811539 |arxiv=1304.1854 |bibcode=2013JAP...113w3913A |s2cid=94599250}}</ref>

The materials in which the exchange interaction is much stronger than the competing dipole–dipole interaction are frequently called ''magnetic materials''. For instance, in iron (Fe) the exchange force is about 1,000 times stronger than the dipole interaction. Therefore, below the Curie temperature, virtually all of the dipoles in a ferromagnetic material will be aligned. In addition to ferromagnetism, the exchange interaction is also responsible for the other types of spontaneous ordering of atomic magnetic moments occurring in magnetic solids: antiferromagnetism and ferrimagnetism. There are different exchange interaction mechanisms which create the magnetism in different ferromagnetic,<ref>{{Cite journal |last1=García |first1=R. Martínez |last2=Bilovol |first2=V. |last3=Ferrari |first3=S. |last4=de la Presa |first4=P. |last5=Marín |first5=P. |last6=Pagnola |first6=M. |date=2022-04-01 |title=Structural and magnetic properties of a BaFe12O19/NiFe2O4 nanostructured composite depending on different particle size ratios |url=https://www.sciencedirect.com/science/article/pii/S030488532101132X |journal=Journal of Magnetism and Magnetic Materials |volume=547 |pages=168934 |doi=10.1016/j.jmmm.2021.168934 |s2cid=245150523 |issn=0304-8853|url-access=subscription }}</ref> ferrimagnetic, and antiferromagnetic substances—these mechanisms include [[Exchange interaction#Direct exchange interactions in solids|direct exchange]], [[RKKY interaction|RKKY exchange]], [[double exchange]], and [[superexchange]].

===Magnetic anisotropy===
{{Main|Magnetic anisotropy}}

Although the exchange interaction keeps spins aligned, it does not align them in a particular direction. Without [[magnetic anisotropy]], the spins in a magnet randomly change direction in response to [[thermal fluctuations]], and the magnet is [[superparamagnetic]]. There are several kinds of magnetic anisotropy, the most common of which is [[magnetocrystalline anisotropy]]. This is a dependence of the energy on the direction of magnetization relative to the [[crystallographic lattice]]. Another common source of [[anisotropy]], [[inverse magnetostriction]], is induced by internal [[deformation (mechanics)|strains]]. [[Single-domain (magnetic)|Single-domain magnets]] also can have a ''shape anisotropy'' due to the magnetostatic effects of the particle shape. As the temperature of a magnet increases, the anisotropy tends to decrease, and there is often a [[superparamagnetism#Blocking temperature|blocking temperature]] at which a transition to superparamagnetism occurs.<ref name=Aharoni>{{cite book|last=Aharoni|first=Amikam|author-link=Amikam Aharoni|title=Introduction to the Theory of Ferromagnetism|publisher=[[Clarendon Press]]|year=1996|isbn=0-19-851791-2|url=https://archive.org/details/introductiontoth00ahar|url-access=registration}}</ref>

===Magnetic domains===
[[File:Electromagnetic dynamic magnetic domain motion of grain oriented electrical silicon steel.gif|thumb|Electromagnetic dynamic magnetic domain motion of grain-oriented electrical silicon steel]]
[[File:Weiss-Bezirke1.png|thumb|[[Kerr micrograph]] of a metal surface showing magnetic domains, with red and green stripes denoting opposite magnetization directions]]
{{Main|Magnetic domain}}

The spontaneous alignment of magnetic dipoles in ferromagnetic materials would seem to suggest that every piece of ferromagnetic material should have a strong magnetic field, since all the spins are aligned; yet iron and other ferromagnets are often found in an "unmagnetized" state. This is because a bulk piece of ferromagnetic material is divided into tiny regions called ''[[magnetic domain]]s''<ref name="Feynman">
 {{cite book
  | last = Feynman | first = Richard P.
  | author2 = Robert B. Leighton>
  | author3 = Matthew Sands
  | title = The Feynman Lectures on Physics
  | volume = I
  | publisher = California Inst. of Technology
  | year = 1963
  | location = Pasadena
  | pages = 37.5–37.6
  | url = {{google books|plainurl=y|id=bDF-uoUmttUC|page=4}}
  | isbn = 0-465-02493-9
  }}</ref> (also known as ''Weiss domains''). Within each domain, the spins are aligned, but if the bulk material is in its lowest energy configuration (i.e. "unmagnetized"), the spins of separate domains point in different directions and their magnetic fields cancel out, so the bulk material has no net large-scale magnetic field.

Ferromagnetic materials spontaneously divide into magnetic domains because the [[exchange interaction]] is a short-range force, so over long distances of many atoms, the tendency of the magnetic dipoles to reduce their energy by orienting in opposite directions wins out. If all the dipoles in a piece of ferromagnetic material are aligned parallel, it creates a large magnetic field extending into the space around it. This contains a lot of [[magnetostatics|magnetostatic]] energy. The material can reduce this energy by splitting into many domains pointing in different directions, so the magnetic field is confined to small local fields in the material, reducing the volume of the field. The domains are separated by thin [[Domain wall (magnetism)|domain walls]] a number of molecules thick, in which the direction of magnetization of the dipoles rotates smoothly from one domain's direction to the other.

===Magnetized materials===
[[File:Moving magnetic domains by Zureks.gif|thumb|Moving domain walls in a grain of [[silicon steel]] caused by an increasing external magnetic field in the "downward" direction, observed in a Kerr microscope. White areas are domains with magnetization directed up, dark areas are domains with magnetization directed down.]]

Thus, a piece of iron in its lowest energy state ("unmagnetized") generally has little or no net magnetic field. However, the magnetic domains in a material are not fixed in place; they are simply regions where the spins of the electrons have aligned spontaneously due to their magnetic fields, and thus can be altered by an external magnetic field. If a strong-enough external magnetic field is applied to the material, the domain walls will move via a process in which the spins of the electrons in atoms near the wall in one domain turn under the influence of the external field to face in the same direction as the electrons in the other domain, thus reorienting the domains so more of the dipoles are aligned with the external field. The domains will remain aligned when the external field is removed, and sum to create a magnetic field of their own extending into the space around the material, thus creating a "permanent" magnet. The domains do not go back to their original minimum energy configuration when the field is removed because the domain walls tend to become 'pinned' or 'snagged' on defects in the crystal lattice, preserving their parallel orientation. This is shown by the [[Barkhausen effect]]: as the magnetizing field is changed, the material's magnetization changes in thousands of tiny discontinuous jumps as domain walls suddenly "snap" past defects.

This magnetization as a function of an external field is described by a [[Hysteresis loop|hysteresis curve]]. Although this state of aligned domains found in a piece of magnetized ferromagnetic material is not a minimal-energy configuration, it is [[metastable]], and can persist for long periods, as shown by samples of [[magnetite]] from the sea floor which have maintained their magnetization for millions of years.

Heating and then cooling ([[Annealing (metallurgy)|annealing]]) a magnetized material, subjecting it to vibration by hammering it, or applying a rapidly oscillating magnetic field from a [[degaussing|degaussing coil]] tends to release the domain walls from their pinned state, and the domain boundaries tend to move back to a lower energy configuration with less external magnetic field, thus [[demagnetization|demagnetizing]] the material.

Commercial [[magnet]]s are made of "hard" ferromagnetic or ferrimagnetic materials with very large magnetic anisotropy such as [[alnico]] and [[ferrite (magnet)|ferrites]], which have a very strong tendency for the magnetization to be pointed along one axis of the crystal, the "easy axis". During manufacture the materials are subjected to various metallurgical processes in a powerful magnetic field, which aligns the crystal grains so their "easy" axes of magnetization all point in the same direction. Thus, the magnetization, and the resulting magnetic field, is "built in" to the crystal structure of the material, making it very difficult to demagnetize.

===Curie temperature===
{{Main|Curie temperature}}
As the temperature of a material increases, thermal motion, or [[entropy]], competes with the ferromagnetic tendency for dipoles to align. When the temperature rises beyond a certain point, called the [[Curie temperature]], there is a second-order [[phase transition]] and the system can no longer maintain a spontaneous magnetization, so its ability to be magnetized or attracted to a magnet disappears, although it still responds [[Paramagnetism|paramagnetically]] to an external field. Below that temperature, there is a [[spontaneous symmetry breaking]] and magnetic moments become aligned with their neighbors. The Curie temperature itself is a [[critical point (thermodynamics)|critical point]], where the [[magnetic susceptibility]] is theoretically infinite and, although there is no net magnetization, domain-like spin correlations fluctuate at all length scales.

The study of ferromagnetic phase transitions, especially via the simplified [[Ising model|Ising]] spin model, had an important impact on the development of [[statistical physics]]. There, it was first clearly shown that [[mean field theory]] approaches failed to predict the correct behavior at the critical point (which was found to fall under a ''universality class'' that includes many other systems, such as liquid-gas transitions), and had to be replaced by [[renormalization group]] theory.{{citation needed|date=December 2012}}