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A magnet made of alnico, a ferromagnetic iron alloy, with its keeper

Paramagnetism, ferromagnetism and spin waves

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. Ferromagnetic materials are familiar metals that are noticeably attracted to a magnet, a consequence of their substantial magnetic permeability. Magnetic permeability describes the induced magnetization of a material due to the presence of an external magnetic field. This temporarily induced magnetization, for example, inside a steel plate, accounts for its attraction to the permanent magnet. Whether or not that steel plate acquires a permanent magnetization itself depends not only on the strength of the applied field but on the so-called coercivity of the ferromagnetic material, which can vary greatly.

In physics, several different types of material magnetism have been distinguished. Ferromagnetism (along with the similar effect ferrimagnetism) is the strongest type and is responsible for the common phenomenon of magnetism in magnets encountered in everyday life.^[1] Substances respond weakly to magnetic fields with three other types of magnetism—paramagnetism, diamagnetism, and antiferromagnetism—but the forces are usually so weak that they can be detected only by sensitive instruments in a laboratory. An everyday example of a permanent magnet formed from a ferromagnetic material is a refrigerator magnet, such as those used to hold paper on a refrigerator door. The attraction between a magnet and a ferromagnetic material like iron has been described as "the quality of magnetism first apparent to the ancient world and to us today".^[2]

Permanent magnets (materials that can be magnetized by an external magnetic field and remain magnetized after the external field is removed) are either ferromagnetic or ferrimagnetic, as are the materials that are attracted to them. Relatively few materials are ferromagnetic and are typically pure forms, alloys, or compounds of iron, cobalt, nickel, and certain rare-earth metals. Beyond its chemical composition, a material's ferromagnetic properties (or lack thereof) are affected by its crystal structure. Ferromagnetism is vital in industrial applications and modern technologies, as it is the basis for many electrical and electromechanical devices, such as electromagnets; electric motors; generators; transformers; magnetic storage, including tape recorders and hard disks; and nondestructive testing of ferrous materials.

Ferromagnetic materials can be divided into magnetically soft materials like annealed iron, which can be magnetized but do not tend to stay magnetized, and magnetically hard materials, which do. Permanent magnets are made from hard ferromagnetic materials, such as alnico, and ferrimagnetic materials, such as ferrite, that are subjected to special processing in a strong magnetic field during manufacturing to align their internal microcrystalline structure, making them difficult to demagnetize. To demagnetize a saturated magnet, a certain magnetic field must be applied, and this threshold depends on the coercivity of the respective material. Hard materials have high coercivity, whereas soft materials have low coercivity. The overall strength of a magnet is measured by its magnetic moment or, alternatively, the total magnetic flux it produces. The local strength of magnetism in a material is measured by its magnetization.

History and distinction from ferrimagnetism

Ferromagnetic material: all the molecular magnetic dipoles are pointed in the same direction

Ferrimagnetic material: some of the dipoles point in the opposite direction, but their smaller contribution is overcome by the others

Historically, the term ferromagnetism was used for any material that could exhibit spontaneous magnetization: a net magnetic moment in the absence of an external magnetic field; that is any material that could become a magnet. This general definition is still in common use.^[3]

However, in a landmark paper in 1948, Louis Néel showed that there are two levels of magnetic alignment that result in this behavior. One is ferromagnetism in the strict sense, where all the magnetic moments are aligned. The other is ferrimagnetism, where some magnetic moments point in the opposite direction but have a smaller contribution, so there is still a spontaneous magnetization.^[4]^[5]^: 28–29

In the special case where the opposing moments balance completely, the alignment is known as antiferromagnetism. Therefore antiferromagnets do not have a spontaneous magnetization.

Ferromagnetic materials

Curie temperatures for some crystalline ferromagnetic materials^[6]^[7]
Material	Curie temp. (K)
Co	1388
Fe	1043
Fe₂O₃^[a]	948
NiOFe₂O₃^[a]	858
CuOFe₂O₃^[a]	728
MgOFe₂O₃^[a]	713
Mn Bi	630
Ni	627
Nd₂Fe₁₄B	593
MnSb	587
MnOFe₂O₃^[a]	573
Y₃Fe₅O₁₂^[a]	560
CrO₂	386
MnAs	318
Gd	292
Tb	219
Dy	88
EuO	69
^ ^a ^b ^c ^d ^e ^f Ferrimagnetic material

Ferromagnetism is an unusual property that occurs in only a few substances. The common ones are the transition metals iron, nickel, cobalt and their alloys, and alloys of rare-earth metals. It is a property not just of the chemical make-up of a material, but of its crystalline structure and microstructure. Their ferromagnetism results from having many unpaired electrons in their d-block in the case of iron and its relatives, or the f-block in the case of the rare-earth metals, a result of Hund's rule of maximum multiplicity. There are ferromagnetic metal alloys whose constituents are not themselves ferromagnetic, called Heusler alloys, named after Fritz Heusler. Conversely, there are non-magnetic alloys, such as types of stainless steel, composed almost exclusively of ferromagnetic metals.

Amorphous (non-crystalline) ferromagnetic metallic alloys can be made by very rapid quenching (cooling) of an alloy. These have the advantage that their properties are nearly isotropic (not aligned along a crystal axis); this results in low coercivity, low hysteresis loss, high permeability, and high electrical resistivity. One such typical material is a transition metal-metalloid alloy, made from about 80% transition metal (usually Fe, Co, or Ni) and a metalloid component (B, C, Si, P, or Al) that lowers the melting point.

A relatively new class of exceptionally strong ferromagnetic materials are the rare-earth magnets. They contain lanthanide elements that are known for their ability to carry large magnetic moments in well-localized f-orbitals.

The table lists a selection of ferromagnetic and ferrimagnetic compounds, along with the temperature above which they cease to exhibit spontaneous magnetization (see Curie temperature).

Unusual materials

Most ferromagnetic materials are metals, since the conducting electrons are often responsible for mediating the ferromagnetic interactions. It is therefore a challenge to develop ferromagnetic insulators, especially multiferroic materials, which are both ferromagnetic and ferroelectric.^[8]

A number of actinide compounds are ferromagnets at room temperature or exhibit ferromagnetism upon cooling. Pu P is a paramagnet with cubic symmetry at room temperature, but which undergoes a structural transition into a tetragonal state with ferromagnetic order when cooled below its T_C = 125 K. In its ferromagnetic state, PuP's easy axis is in the ⟨100⟩ direction.^[9]

In NpFe₂ the easy axis is ⟨111⟩.^[10] Above T_C ≈ 500 K, NpFe₂ is also paramagnetic and cubic. Cooling below the Curie temperature produces a rhombohedral distortion wherein the rhombohedral angle changes from 60° (cubic phase) to 60.53°. An alternate description of this distortion is to consider the length c along the unique trigonal axis (after the distortion has begun) and a as the distance in the plane perpendicular to c. In the cubic phase this reduces to c/a = 1.00. Below the Curie temperature

{\frac {c}{a}}-1=-(120\pm 5)\times 10^{-4},

which is the largest strain in any actinide compound.^[11] NpNi₂ undergoes a similar lattice distortion below T_C = 32 K, with a strain of (43 ± 5) × 10⁻⁴.^[11] NpCo₂ is a ferrimagnet below 15 K.

In 2009, a team of MIT physicists demonstrated that a lithium gas cooled to less than one kelvin can exhibit ferromagnetism.^[12] The team cooled fermionic lithium-6 to less than 150 nK (150 billionths of one kelvin) using infrared laser cooling. This demonstration is the first time that ferromagnetism has been demonstrated in a gas.

In 2018, a team of University of Minnesota physicists demonstrated that body-centered tetragonal ruthenium exhibits ferromagnetism at room temperature.^[13]

Electrically induced ferromagnetism

Recent research has shown evidence that ferromagnetism can be induced in some materials by an electric current or voltage. Antiferromagnetic LaMnO₃ and SrCoO have been switched to ferromagnetic by a current. In July 2020 scientists reported inducing ferromagnetism in the abundant diamagnetic material iron pyrite ("fool's gold") by an applied voltage.^[14]^[15] In these experiments the ferromagnetism was limited to a thin surface layer.

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 atoms. Each of an atom's electrons has a magnetic moment according to its 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.^[16] 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 magnetic dipole moment, i.e., it behaves like a tiny magnet, producing a magnetic field. This dipole moment comes from the more fundamental property of the electron that it has 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). The spin of the electrons in atoms is the main source of ferromagnetism, although there is also a contribution from the orbital angular momentum of the electron about the 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 shells 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 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 a shell tend to have the same spin^{[dubious – discuss]}, thereby increasing the total dipole moment.

These 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 – 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 domains. This results in the large observed magnetic permeability of ferromagnetics, and the ability of "hard" magnetic materials to form permanent magnets.

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 (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 orbitals of the unpaired outer valence electrons 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,^[17] which tends to align the dipoles antiparallel. In certain doped semiconductor oxides RKKY interactions have been shown to bring about periodic longer-range magnetic interactions, a phenomenon of significance in the study of spintronic materials.^[18]

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 1000 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, ferrimagnetic, and antiferromagnetic substances. These mechanisms include direct exchange, RKKY exchange, double exchange, and superexchange.

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 strains. 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 blocking temperature at which a transition to superparamagnetism occurs.^[19]

Magnetic domains

The above 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. The reason for this is that a bulk piece of ferromagnetic material is divided into tiny regions called magnetic domains^[20] (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 object 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 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 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

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 by the process of the spins of the electrons in atoms near the wall in one domain turning 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, creating 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 magnetization changes in thousands of tiny discontinuous jumps as the domain walls suddenly "snap" past defects.

This magnetization as a function of the external field is described by a 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) a magnetized material, subjecting it to vibration by hammering it, or applying a rapidly oscillating magnetic field from a 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 demagnetizing the material.

Commercial magnets are made of "hard" ferromagnetic or ferrimagnetic materials with very large magnetic anisotropy such as alnico and 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

As the temperature 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 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, 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 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]}

References

^ Chikazumi, Sōshin (2009). Physics of ferromagnetism. English edition prepared with the assistance of C. D. Graham, Jr. (2nd ed.). Oxford: Oxford University Press. p. 118. ISBN 9780199564811.
^ Bozorth, Richard M. Ferromagnetism, first published 1951, reprinted 1993 by IEEE Press, New York as a "Classic Reissue". ISBN 0-7803-1032-2.
^ Somasundaran, P., ed. (2006). Encyclopedia of surface and colloid science (2nd ed.). New York: Taylor & Francis. p. 3471. ISBN 9780849396083.
^ Cullity, B. D.; Graham, C. D. (2011). "6. Ferrimagnetism". Introduction to Magnetic Materials. John Wiley & Sons. ISBN 9781118211496.
^ Aharoni, Amikam (2000). Introduction to the theory of ferromagnetism (2nd ed.). Oxford: Oxford University Press. ISBN 9780198508090.
^ Kittel, Charles (1986). Introduction to Solid State Physics (sixth ed.). John Wiley and Sons. ISBN 0-471-87474-4.
^ Jackson, Mike (2000). "Wherefore Gadolinium? Magnetism of the Rare Earths" (PDF). IRM Quarterly. 10 (3). Institute for Rock Magnetism: 6. Archived from the original (PDF) on 2017-07-12. Retrieved 2016-08-08.
^ Hill, Nicola A. (2000-07-01). "Why Are There so Few Magnetic Ferroelectrics?". The Journal of Physical Chemistry B. 104 (29): 6694–6709. doi:10.1021/jp000114x. ISSN 1520-6106.
^ "Neutron diffraction study of PuP: The electronic ground state". Phys. Rev. B. 14 (9): 4064–4067. 1976. Bibcode:1976PhRvB..14.4064L. doi:10.1103/PhysRevB.14.4064. {{cite journal}}: Unknown parameter |authors= ignored (help)
^ "Magnetic properties of neptunium Laves phases: NpMn₂, NpFe₂, NpCo₂, and NpNi₂". Phys. Rev. B. 11 (1): 530–544. 1975. Bibcode:1975PhRvB..11..530A. doi:10.1103/PhysRevB.11.530. {{cite journal}}: Unknown parameter |authors= ignored (help)
^ ^a ^b "Lattice distortions measured in actinide ferromagnets PuP, NpFe₂, and NpNi₂" (PDF). J. Phys. Colloque C4, Supplement. 40 (4): C4-68–C4-69. Apr 1979. Archived (PDF) from the original on 2011-05-09. {{cite journal}}: Unknown parameter |authors= ignored (help)
^ G.-B. Jo; Y.-R. Lee; J.-H. Choi; C. A. Christensen; T. H. Kim; J. H. Thywissen; D. E. Pritchard; W. Ketterle (2009). "Itinerant Ferromagnetism in a Fermi Gas of Ultracold Atoms". Science. 325 (5947): 1521–1524. arXiv:0907.2888. Bibcode:2009Sci...325.1521J. doi:10.1126/science.1177112. PMID 19762638. S2CID 13205213.
^ Quarterman, P.; Sun, Congli; Garcia-Barriocanal, Javier; D. C., Mahendra; Lv, Yang; Manipatruni, Sasikanth; Nikonov, Dmitri E.; Young, Ian A.; Voyles, Paul M.; Wang, Jian-Ping (2018). "Demonstration of Ru as the 4th ferromagnetic element at room temperature". Nature Communications. 9 (1): 2058. Bibcode:2018NatCo...9.2058Q. doi:10.1038/s41467-018-04512-1. PMC 5970227. PMID 29802304.
^ "'Fool's gold' may be valuable after all". phys.org. Retrieved 17 August 2020.
^ Walter, Jeff; Voigt, Bryan; Day-Roberts, Ezra; Heltemes, Kei; Fernandes, Rafael M.; Birol, Turan; Leighton, Chris (1 July 2020). "Voltage-induced ferromagnetism in a diamagnet". Science Advances. 6 (31): eabb7721. Bibcode:2020SciA....6B7721W. doi:10.1126/sciadv.abb7721. ISSN 2375-2548. PMC 7439324. PMID 32832693.
^ Feynman, Richard P.; Robert Leighton; Matthew Sands (1963). The Feynman Lectures on Physics, Vol. 2. Addison-Wesley. pp. Ch. 37.
^ Chikazumi, Sōshin (2009). Physics of ferromagnetism. English edition prepared with the assistance of C. D. Graham, Jr. (2nd ed.). Oxford: Oxford University Press. pp. 129–130. ISBN 9780199564811.
^ Assadi, M. H. N.; Hanaor, D. A. H. (2013). "Theoretical study on copper's energetics and magnetism in TiO₂ polymorphs". Journal of Applied Physics. 113 (23): 233913-1–233913-5. arXiv:1304.1854. Bibcode:2013JAP...113w3913A. doi:10.1063/1.4811539. S2CID 94599250.
^ Aharoni, Amikam (1996). Introduction to the Theory of Ferromagnetism. Clarendon Press. ISBN 0-19-851791-2.
^ Feynman, Richard P.; Robert B. Leighton; Matthew Sands (1963). The Feynman Lectures on Physics. Vol. I. Pasadena: California Inst. of Technology. pp. 37.5–37.6. ISBN 0465024939.

External links

Media related to Ferromagnetism at Wikimedia Commons
Electromagnetism – ch. 11, from an online textbook
Sandeman, Karl (January 2008). "Ferromagnetic Materials". DoITPoMS. Dept. of Materials Sci. and Metallurgy, Univ. of Cambridge. Retrieved 2019-06-22. Detailed nonmathematical description of ferromagnetic materials with illustrations
Magnetism: Models and Mechanisms in E. Pavarini, E. Koch, and U. Schollwöck: Emergent Phenomena in Correlated Matter, Jülich 2013, ISBN 978-3-89336-884-6

[fi-8] ^ ^a ^b ^c ^d ^e ^f Ferrimagnetic material

[Chikazumi-1] Chikazumi, Sōshin (2009). Physics of ferromagnetism. English edition prepared with the assistance of C. D. Graham, Jr. (2nd ed.). Oxford: Oxford University Press. p. 118. ISBN 9780199564811.

[bozorth-2] Bozorth, Richard M. Ferromagnetism, first published 1951, reprinted 1993 by IEEE Press, New York as a "Classic Reissue". ISBN 0-7803-1032-2.

[3] Somasundaran, P., ed. (2006). Encyclopedia of surface and colloid science (2nd ed.). New York: Taylor & Francis. p. 3471. ISBN 9780849396083.

[4] Cullity, B. D.; Graham, C. D. (2011). "6. Ferrimagnetism". Introduction to Magnetic Materials. John Wiley & Sons. ISBN 9781118211496.

[5] Aharoni, Amikam (2000). Introduction to the theory of ferromagnetism (2nd ed.). Oxford: Oxford University Press. ISBN 9780198508090.

[6] Kittel, Charles (1986). Introduction to Solid State Physics (sixth ed.). John Wiley and Sons. ISBN 0-471-87474-4.

[7] Jackson, Mike (2000). "Wherefore Gadolinium? Magnetism of the Rare Earths" (PDF). IRM Quarterly. 10 (3). Institute for Rock Magnetism: 6. Archived from the original (PDF) on 2017-07-12. Retrieved 2016-08-08.

[9] Hill, Nicola A. (2000-07-01). "Why Are There so Few Magnetic Ferroelectrics?". The Journal of Physical Chemistry B. 104 (29): 6694–6709. doi:10.1021/jp000114x. ISSN 1520-6106.

[Lander-10] "Neutron diffraction study of PuP: The electronic ground state". Phys. Rev. B. 14 (9): 4064–4067. 1976. Bibcode:1976PhRvB..14.4064L. doi:10.1103/PhysRevB.14.4064. {{cite journal}}: Unknown parameter |authors= ignored (help)

[Aldred-11] "Magnetic properties of neptunium Laves phases: NpMn₂, NpFe₂, NpCo₂, and NpNi₂". Phys. Rev. B. 11 (1): 530–544. 1975. Bibcode:1975PhRvB..11..530A. doi:10.1103/PhysRevB.11.530. {{cite journal}}: Unknown parameter |authors= ignored (help)

[Mueller-12] "Lattice distortions measured in actinide ferromagnets PuP, NpFe₂, and NpNi₂" (PDF). J. Phys. Colloque C4, Supplement. 40 (4): C4-68–C4-69. Apr 1979. Archived (PDF) from the original on 2011-05-09. {{cite journal}}: Unknown parameter |authors= ignored (help)

[13] G.-B. Jo; Y.-R. Lee; J.-H. Choi; C. A. Christensen; T. H. Kim; J. H. Thywissen; D. E. Pritchard; W. Ketterle (2009). "Itinerant Ferromagnetism in a Fermi Gas of Ultracold Atoms". Science. 325 (5947): 1521–1524. arXiv:0907.2888. Bibcode:2009Sci...325.1521J. doi:10.1126/science.1177112. PMID 19762638. S2CID 13205213.

[14] Quarterman, P.; Sun, Congli; Garcia-Barriocanal, Javier; D. C., Mahendra; Lv, Yang; Manipatruni, Sasikanth; Nikonov, Dmitri E.; Young, Ian A.; Voyles, Paul M.; Wang, Jian-Ping (2018). "Demonstration of Ru as the 4th ferromagnetic element at room temperature". Nature Communications. 9 (1): 2058. Bibcode:2018NatCo...9.2058Q. doi:10.1038/s41467-018-04512-1. PMC 5970227. PMID 29802304.

[Phys-15] "'Fool's gold' may be valuable after all". phys.org. Retrieved 17 August 2020.

[Voigt-16] Walter, Jeff; Voigt, Bryan; Day-Roberts, Ezra; Heltemes, Kei; Fernandes, Rafael M.; Birol, Turan; Leighton, Chris (1 July 2020). "Voltage-induced ferromagnetism in a diamagnet". Science Advances. 6 (31): eabb7721. Bibcode:2020SciA....6B7721W. doi:10.1126/sciadv.abb7721. ISSN 2375-2548. PMC 7439324. PMID 32832693.

[17] Feynman, Richard P.; Robert Leighton; Matthew Sands (1963). The Feynman Lectures on Physics, Vol. 2. Addison-Wesley. pp. Ch. 37.

[Chikazumi2-18] Chikazumi, Sōshin (2009). Physics of ferromagnetism. English edition prepared with the assistance of C. D. Graham, Jr. (2nd ed.). Oxford: Oxford University Press. pp. 129–130. ISBN 9780199564811.

[19] Assadi, M. H. N.; Hanaor, D. A. H. (2013). "Theoretical study on copper's energetics and magnetism in TiO₂ polymorphs". Journal of Applied Physics. 113 (23): 233913-1–233913-5. arXiv:1304.1854. Bibcode:2013JAP...113w3913A. doi:10.1063/1.4811539. S2CID 94599250.

[Aharoni-20] Aharoni, Amikam (1996). Introduction to the Theory of Ferromagnetism. Clarendon Press. ISBN 0-19-851791-2.

[Feynman-21] Feynman, Richard P.; Robert B. Leighton; Matthew Sands (1963). The Feynman Lectures on Physics. Vol. I. Pasadena: California Inst. of Technology. pp. 37.5–37.6. ISBN 0465024939.

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 Most ferromagnetic materials are metals, since the conducting electrons are often responsible for mediating the ferromagnetic interactions. It is therefore a challenge to develop ferromagnetic insulators, especially [[Multiferroics|multiferroic]] materials, which are both ferromagnetic and [[ferroelectric]].<ref>{{Cite journal |last=Hill |first=Nicola A. |date=2000-07-01 |title=Why Are There so Few Magnetic Ferroelectrics? |journal=The Journal of Physical Chemistry B |volume=104 |issue=29 |pages=6694–6709 |doi=10.1021/jp000114x |issn=1520-6106}}</ref>
-A number of [[actinide]] compounds are ferromagnets at room temperature or exhibit ferromagnetism upon cooling. [[Plutonium|Pu]] [[Phosphorus|P]] is a paramagnet with [[Cubic crystal system|cubic symmetry]] at [[room temperature]], but which undergoes a structural transition into a [[Tetragonal crystal system|tetragonal]] state with ferromagnetic order when cooled below its ''T''<sub>C</sub>&nbsp;=&nbsp;125&nbsp;K. In its ferromagnetic state, PuP's [[easy axis]] is in the ⟨100⟩ direction.<ref name=Lander>{{cite journal |authors=Lander G. H., Lam D. J. |title=Neutron diffraction study of PuP: The electronic ground state |journal=Phys. Rev. B |year=1976 |volume=14 |issue=9 |pages=4064–4067 |doi=10.1103/PhysRevB.14.4064 |bibcode=1976PhRvB..14.4064L }}</ref>
+A number of [[actinide]] compounds are ferromagnets at room temperature or exhibit ferromagnetism upon cooling. [[Plutonium|Pu]][[Phosphorus|P]] is a paramagnet with [[Cubic crystal system|cubic symmetry]] at [[room temperature]], but which undergoes a structural transition into a [[Tetragonal crystal system|tetragonal]] state with ferromagnetic order when cooled below its ''T''<sub>C</sub>&nbsp;=&nbsp;125&nbsp;K. In its ferromagnetic state, PuP's [[easy axis]] is in the ⟨100⟩ direction.<ref name=Lander>{{cite journal |authors=Lander G. H., Lam D. J. |title=Neutron diffraction study of PuP: The electronic ground state |journal=Phys. Rev. B |year=1976 |volume=14 |issue=9 |pages=4064–4067 |doi=10.1103/PhysRevB.14.4064 |bibcode=1976PhRvB..14.4064L }}</ref>
-In [[Neptunium|Np]]Fe<sub>2</sub> the easy axis is ⟨111⟩.<ref name=Aldred>{{cite journal |authors=Aldred A. T., Dunlap B. D., Lam D. J., Lander G. H., Mueller M. H., Nowik I. |title=Magnetic properties of neptunium Laves phases: NpMn<sub>2</sub>, NpFe<sub>2</sub>, NpCo<sub>2</sub>, and NpNi<sub>2</sub> |journal=Phys. Rev. B |year=1975 |volume=11 |issue=1 |pages=530–544 |doi=10.1103/PhysRevB.11.530 |bibcode=1975PhRvB..11..530A }}</ref> Above {{nowrap|''T''<sub>C</sub> ≈ 500&nbsp;K}}, NpFe<sub>2</sub> is also paramagnetic and cubic. Cooling below the Curie temperature produces a [[rhombohedral]] distortion wherein the rhombohedral angle changes from 60° (cubic phase) to 60.53°. An alternate description of this distortion is to consider the length ''c'' along the unique trigonal axis (after the distortion has begun) and ''a'' as the distance in the plane perpendicular to ''c''. In the cubic phase this reduces to {{nowrap|''c''/''a'' {{=}} 1.00}}. Below the [[Curie temperature]]
+In [[Neptunium|Np]]Fe<sub>2</sub> the easy axis is ⟨111⟩.<ref name=Aldred>{{cite journal |authors=Aldred A. T., Dunlap B. D., Lam D. J., Lander G. H., Mueller M. H., Nowik I. |title=Magnetic properties of neptunium Laves phases: NpMn<sub>2</sub>, NpFe<sub>2</sub>, NpCo<sub>2</sub>, and NpNi<sub>2</sub> |journal=Phys. Rev. B |year=1975 |volume=11 |issue=1 |pages=530–544 |doi=10.1103/PhysRevB.11.530 |bibcode=1975PhRvB..11..530A }}</ref> Above {{nowrap|''T''<sub>C</sub> ≈ 500&nbsp;K}}, NpFe<sub>2</sub> is also paramagnetic and cubic. Cooling below the Curie temperature produces a [[rhombohedral]] distortion wherein the rhombohedral angle changes from 60° (cubic phase) to 60.53°. An alternate description of this distortion is to consider the length ''c'' along the unique trigonal axis (after the distortion has begun) and ''a'' as the distance in the plane perpendicular to ''c''. In the cubic phase this reduces to {{nowrap|''c''/''a'' {{=}} 1.00}}. Below the Curie temperature
 : <math>\frac{c}{a} - 1 = -(120 \pm 5) \times 10^{-4},</math>
@@ Line 123: / Line 123: @@
 ===Electrically induced ferromagnetism===
-Recent research has shown evidence that ferromagnetism can be induced in some materials by an electric current or voltage.   Antiferromagnetic LaMnO<sub>3</sub> and SrCoO have been switched to ferromagnetic by a current. In July 2020 scientists reported inducing ferromagnetism in the abundant [[diamagnetic]] material [[iron pyrite]] ("fool's gold") by an applied voltage.<ref name="Phys">{{cite news |title='Fool's gold' may be valuable after all |url=https://phys.org/news/2020-07-gold-valuable.html |access-date=17 August 2020 |work=phys.org |language=en}}</ref><ref name="Voigt">{{cite journal |last1=Walter |first1=Jeff |last2=Voigt |first2=Bryan |last3=Day-Roberts |first3=Ezra |last4=Heltemes |first4=Kei |last5=Fernandes |first5=Rafael M. |last6=Birol |first6=Turan |last7=Leighton |first7=Chris |title=Voltage-induced ferromagnetism in a diamagnet |journal=Science Advances |date=1 July 2020 |volume=6 |issue=31 |pages=eabb7721 |doi=10.1126/sciadv.abb7721 |pmid=32832693 |pmc=7439324 |bibcode=2020SciA....6B7721W |language=en |issn=2375-2548 |doi-access=free }}</ref> In these experiments the ferromagnetism was limited to a thin surface layer.
+Recent research has shown evidence that ferromagnetism can be induced in some materials by an electric current or voltage. Antiferromagnetic LaMnO<sub>3</sub> and SrCoO have been switched to ferromagnetic by a current. In July 2020 scientists reported inducing ferromagnetism in the abundant [[diamagnetic]] material [[iron pyrite]] ("fool's gold") by an applied voltage.<ref name="Phys">{{cite news |title='Fool's gold' may be valuable after all |url=https://phys.org/news/2020-07-gold-valuable.html |access-date=17 August 2020 |work=phys.org |language=en}}</ref><ref name="Voigt">{{cite journal |last1=Walter |first1=Jeff |last2=Voigt |first2=Bryan |last3=Day-Roberts |first3=Ezra |last4=Heltemes |first4=Kei |last5=Fernandes |first5=Rafael M. |last6=Birol |first6=Turan |last7=Leighton |first7=Chris |title=Voltage-induced ferromagnetism in a diamagnet |journal=Science Advances |date=1 July 2020 |volume=6 |issue=31 |pages=eabb7721 |doi=10.1126/sciadv.abb7721 |pmid=32832693 |pmc=7439324 |bibcode=2020SciA....6B7721W |language=en |issn=2375-2548 |doi-access=free }}</ref> In these experiments the ferromagnetism was limited to a thin surface layer.
 ==Explanation==
@@ Line 129: / Line 129: @@
 ===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 the more fundamental property of the electron that it has quantum mechanical [[Spin (physics)|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). The spin of the electrons 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.
+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 the more fundamental property of the electron that it has 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). The spin of the electrons 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 a shell tend to have the same spin{{dubious|date=August 2022}}, thereby increasing the total dipole moment.
@@ Line 137: / Line 137: @@
 ===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 mechanics|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.
+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.
+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=9780199564811 |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 1000 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]].
+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 1000 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, ferrimagnetic, and antiferromagnetic substances. These mechanisms include [[Exchange interaction#Direct exchange interactions in solids|direct exchange]], [[RKKY interaction|RKKY exchange]], [[double exchange]], and [[superexchange]].
-There are different exchange interaction mechanisms which create the magnetism in different ferromagnetic, 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>
+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===
@@ Line 182: / Line 181: @@
 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.
+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===

v t e Magnetism
Magnetic response	diamagnetism superdiamagnetism paramagnetism superparamagnetism Van Vleck paramagnetism
Magnetic states	altermagnetism antiferromagnetism ferrimagnetism ferromagnetism superferromagnetism ferromagnetic superconductor helimagnetism metamagnetism mictomagnetism spin glass amorphous magnetism spin ice