View source for Diamagnetism

{{short description|Magnetic property of ordinary materials}}
[[File:Diamagnetic graphite levitation.jpg|thumb|[[Pyrolytic carbon]] has one of the largest diamagnetic constants{{clarify|date=June 2023}} of any room temperature material.  Here a pyrolytic carbon sheet is levitated by its repulsion from the strong magnetic field of [[neodymium magnet]]s]]
{{Condensed matter physics}}

'''Diamagnetism''' is the property of materials that are repelled by a [[magnetic field]]; an applied magnetic field creates an [[induced magnetic field]] in them in the opposite direction, causing a repulsive force.  In contrast, [[paramagnetic]] and [[ferromagnetic]] materials are attracted by a magnetic field. Diamagnetism is a [[quantum mechanical]] effect that occurs in all materials; when it is the only contribution to the magnetism, the material is called diamagnetic. In paramagnetic and ferromagnetic substances, the weak diamagnetic force is overcome by the attractive force of [[magnetic dipole]]s in the material. The [[Permeability (electromagnetism)|magnetic permeability]] of diamagnetic materials is less than the [[Vacuum_permeability|permeability of vacuum]], ''μ''<sub>0</sub>. In most materials, diamagnetism is a weak effect which can be detected only by sensitive laboratory instruments, but a [[superconductivity|superconductor]] acts as a strong diamagnet because it entirely expels any magnetic field from its interior (the [[Meissner effect]]).

Diamagnetism was first discovered when [[Anton Brugmans]] observed in 1778 that [[bismuth]] was repelled by magnetic fields.<ref>{{cite journal|title=Diamagnetic Levitation&nbsp;– Historical Milestones|author=Gerald Küstler|journal= Rev. Roum. Sci. Techn. Électrotechn. Et Énerg.|volume=52, 3|pages=265–282|year=2007 |url=http://revue.elth.pub.ro/viewpdf.php?id=55}}</ref> In 1845, [[Michael Faraday]] demonstrated that it was a property of matter and concluded that every material responded (in either a diamagnetic or paramagnetic way) to an applied magnetic field. On a suggestion by [[William Whewell]], Faraday first referred to the phenomenon as ''diamagnetic'' (the prefix ''dia-'' meaning ''through'' or ''across''), then later changed it to ''diamagnetism''.<ref>{{cite journal |last=Jackson |first=Roland |date=21 July 2014 |title=John Tyndall and the Early History of Diamagnetism |journal=Annals of Science |volume=72 |issue=4 |pages=435–489 |doi=10.1080/00033790.2014.929743 |pmid=26221835 |pmc=4524391 }}</ref><ref>{{cite encyclopedia |date=June 2017 |title=diamagnetic, adj. and n |encyclopedia=OED Online |publisher=Oxford University Press}}</ref>

A simple [[rule of thumb]] is used in chemistry to determine whether a particle (atom, ion, or molecule) is paramagnetic or diamagnetic:<ref>{{Cite web|url=https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplemental_Modules_(Physical_and_Theoretical_Chemistry)/Physical_Properties_of_Matter/Atomic_and_Molecular_Properties/Magnetic_Properties|title=Magnetic Properties|date=2013-10-02|website=Chemistry LibreTexts|language=en|access-date=2020-01-21}}</ref> If all electrons in the particle are paired, then the substance made of this particle is diamagnetic; If it has unpaired electrons, then the substance is paramagnetic.

== Materials ==
[[File:Diamagnetic material interaction in magnetic field.png|thumb|left|Diamagnetic material interaction in [[magnetic field]]. On keeping diamagnetic materials in a magnetic field, the electron orbital motion changes in such a way that magnetic dipole moments are induced on the atoms / molecules in the direction opposite to the external magnetic field]]

Diamagnetism is a property of all materials, and always makes a weak contribution to the material's response to a magnetic field. However, other forms of magnetism (such as [[ferromagnetism]] or [[paramagnetism]]) are so much stronger such that, when different forms of magnetism are present in a material, the diamagnetic contribution is usually negligible. Substances where the diamagnetic behaviour is the strongest effect are termed diamagnetic materials, or diamagnets. Diamagnetic materials are those that some people generally think of as ''non-magnetic'', and include [[Water (properties)|water]], [[wood]], most organic compounds such as petroleum and some plastics, and many metals including [[copper]], particularly the heavy ones with many [[core electrons]], such as [[mercury (element)|mercury]], [[gold]] and [[bismuth]]. The magnetic susceptibility values of various molecular fragments are called [[Pascal's constants]] (named after {{ill|Paul Pascal (physicist)|fr|Paul Pascal|lt=Paul Pascal}}).

Diamagnetic materials, like water, or water-based materials, have a relative magnetic permeability that is less than or equal to 1, and therefore a [[magnetic susceptibility]] less than or equal to 0, since susceptibility is defined as {{nowrap|''χ''<sub>v</sub> {{=}} ''μ''<sub>v</sub> − 1}}. This means that diamagnetic materials are repelled by magnetic fields. However, since diamagnetism is such a weak property, its effects are not observable in everyday life. For example, the magnetic susceptibility of diamagnets such as water is {{nowrap|''χ''<sub>v</sub> {{=}} {{val|-9.05|e=-6}}}}. The most strongly diamagnetic material is [[bismuth]], {{nowrap|''χ''<sub>v</sub> {{=}} {{val|-1.66|e=-4}}}}, although [[pyrolytic carbon]] may have a susceptibility of  {{nowrap|''χ''<sub>v</sub> {{=}} {{val|-4.00|e=-4}}}} in one plane. Nevertheless, these values are orders of magnitude smaller than the magnetism exhibited by paramagnets and ferromagnets. Because ''χ''<sub>v</sub> is derived from the ratio of the internal magnetic field to the applied field, it is a dimensionless value.

In rare cases, the diamagnetic contribution can be stronger than paramagnetic contribution. This is the case for [[gold]], which has a magnetic susceptibility less than 0 (and is thus by definition a diamagnetic material), but when measured carefully with [[X-ray magnetic circular dichroism]], has an extremely weak paramagnetic contribution that is overcome by a stronger diamagnetic contribution.<ref>{{cite journal|author=Motohiro Suzuki, Naomi Kawamura, Hayato Miyagawa, Jose S. Garitaonandia, Yoshiyuki Yamamoto, and Hidenobu Hori|title=Measurement of a Pauli and Orbital Paramagnetic State in Bulk Gold Using X-Ray Magnetic Circular Dichroism Spectroscopy|journal = Physical Review Letters|volume = 108|issue = 4|pages = 047201|date=24 Jan 2012|doi=10.1103/PhysRevLett.108.047201|pmid = 22400883|bibcode=2012PhRvL.108d7201S}}</ref>


{|class="wikitable sortable"
|+Notable diamagnetic materials<ref name=hyper>{{cite web |last=Nave |first=Carl L. |title=Magnetic Properties of Solids |website=Hyper Physics |url=http://hyperphysics.phy-astr.gsu.edu/Hbase/tables/magprop.html |access-date=2008-11-09}}</ref>
!Material!! ''χ''<sub>v</sub> [× 10<sup>−5</sup> (SI units)]
|-
|Superconductor
| −10<sup>5</sup>
|-
|[[Pyrolytic carbon]]
| −40.9
|-
|[[Bismuth]]
| −16.6
|-
|[[Neon]]
| −6.74
|-
|[[Mercury (element)|Mercury]]
| −2.9
|-
|[[Silver]]
| −2.6
|-
|[[Diamond|Carbon (diamond)]]
| −2.1
|-
|[[Lead]]
| −1.8
|-
|[[Graphite|Carbon (graphite)]]
| −1.6
|-
|[[Copper]]
| −1.0
|-
|[[Water]]
| −0.91
|}

=== Superconductors ===
[[File:EXPULSION.png|thumb|Transition from ordinary [[electrical conductivity|conductivity]] (left) to [[superconductivity]] (right). At the transition, the [[superconductor]] expels the magnetic field and then acts as a perfect diamagnet.]]
[[superconductivity|Superconductors]] may be considered [[Superdiamagnetism|perfect diamagnets]] ({{nowrap|''χ''<sub>v</sub> {{=}} −1}}), because they expel all magnetic fields (except in a thin surface layer) due to the [[Meissner effect]].<ref>{{cite book | last=Poole | first=Charles P. Jr. |author1-link=Charles P. Poole |year=2007 |title=Superconductivity |edition=2nd |publisher=Academic Press |location=Amsterdam |isbn=9780080550480 |page=23}}</ref>

== Demonstrations ==
=== Curving water surfaces ===
If a powerful magnet (such as a [[supermagnet]]) is covered with a layer of water (that is thin compared to the diameter of the magnet) then the field of the magnet significantly repels the water. This causes a slight dimple in the water's surface that may be seen by a reflection in its surface.<ref>{{cite web  |last=Beatty |first=Bill |year=2005 |title=Neodymium supermagnets: Some demonstrations—Diamagnetic water |website=Science Hobbyist |url=http://amasci.com/amateur/neodymium.html#water |access-date=26 September 2011}}</ref><ref>{{cite web |author=Quit007 |year=2011 |title=Diamagnetism Gallery |website=DeviantART |url=http://quit007.deviantart.com/gallery/23787987 |access-date=26 September 2011}}</ref>

=== Levitation ===<!-- This section is linked from [[Magnetic levitation]] -->
{{main|Magnetic levitation#Diamagnetic levitation}}
[[File:Frog diamagnetic levitation.jpg|thumb|A live [[frog]] levitates inside a {{convert|32|mm|2|abbr=on}} diameter vertical bore of a [[Bitter solenoid]] in a magnetic field of about 16 [[tesla (unit)|teslas]] at the [[Radboud University Nijmegen|Nijmegen High Field Magnet Laboratory]].<ref>{{cite web |year=2011 |title=Diamagnetic Levitation |website=High Field Laboratory |publisher=[[Radboud University Nijmegen]] |url=https://www.ru.nl/hfml/research/levitation/diamagnetic-levitation/ |access-date=26 September 2020}}</ref> ]]

Diamagnets may be levitated in stable equilibrium in a magnetic field, with no power consumption. [[Earnshaw's theorem]] seems to preclude the possibility of static magnetic levitation. However, Earnshaw's theorem applies only to objects with positive susceptibilities, such as ferromagnets (which have a permanent positive moment) and paramagnets (which induce a positive moment). These are attracted to field maxima, which do not exist in free space. Diamagnets (which induce a negative moment) are attracted to field minima, and there can be a field minimum in free space.

A thin slice of [[pyrolytic graphite]], which is an unusually strongly diamagnetic material, can be stably floated in a magnetic field, such as that from [[Rare-earth magnet|rare earth]] permanent magnets. This can be done with all components at room temperature, making a visually effective and relatively convenient demonstration of diamagnetism.

The [[Radboud University Nijmegen]], the [[Netherlands]], has conducted experiments where water and other substances were successfully levitated. Most spectacularly, a live frog (see figure) was levitated.<ref>{{cite web |year=2011 |title=The Real Levitation |website=High Field Laboratory |publisher=[[Radboud University Nijmegen]] |url=http://www.ru.nl/hfml/research/levitation/diamagnetic/ |access-date=26 September 2011}}</ref>

In September 2009, NASA's [[Jet Propulsion Laboratory]] (JPL) in Pasadena, California announced it had successfully levitated mice using a [[superconducting magnet]],<ref>{{cite journal |last1=Liu |first1=Yuanming |last2=Zhu |first2=Da-Ming |last3=Strayer |first3=Donald M. |last4=Israelsson |first4=Ulf E. |year=2010 |title=Magnetic levitation of large water droplets and mice |journal=[[Advances in Space Research]] |doi=10.1016/j.asr.2009.08.033 |bibcode=2010AdSpR..45..208L |volume=45 |issue=1 |pages=208–213}}</ref> an important step forward since mice are closer biologically to humans than frogs.<ref>{{cite web |last=Choi |first=Charles Q. |date=2009-09-09 |title=Mice levitated in lab |website=Live Science |url=http://www.livescience.com/animals/090909-mouse-levitation.html |access-date=26 September 2011}}</ref> JPL said it hopes to perform experiments regarding the effects of microgravity on bone and muscle mass.

Recent experiments studying the growth of protein crystals have led to a technique using powerful magnets to allow growth in ways that counteract Earth's gravity.<ref>{{cite web |last=Kleiner |first=Kurt |date=10 August 2007 |title=Magnetic gravity trick grows perfect crystals |website=[[New Scientist]] |url=https://www.newscientist.com/article/dn12467-magnetic-gravity-trick-grows-perfect-crystals.html |access-date=26 September 2011}}</ref>

A simple homemade device for demonstration can be constructed out of bismuth plates and a few permanent magnets that levitate a permanent magnet.<ref>{{cite web |date=2 December 2008 |title=Fun with diamagnetic levitation |publisher=ForceField |url=http://www.fieldlines.com/other/diamag1.html |access-date=26 September 2011 |url-status=dead |archive-url=https://web.archive.org/web/20080212011654/http://www.fieldlines.com/other/diamag1.html |archive-date=12 February 2008 }}</ref>

== Theory ==
The electrons in a material generally settle in orbitals, with effectively zero resistance and act like current loops. Thus it might be imagined that diamagnetism effects in general would be common, since any applied magnetic field would generate currents in these loops that would oppose the change, in a similar way to superconductors, which are essentially perfect diamagnets. However, since the electrons are rigidly held in orbitals by the charge of the protons and are further constrained by the [[Pauli exclusion principle]], many materials exhibit diamagnetism, but typically respond very little to the applied field.

The [[Bohr–Van Leeuwen theorem]] proves that there cannot be any diamagnetism or paramagnetism in a purely classical system. However, the classical theory of Langevin for diamagnetism gives the same prediction as the quantum theory.<ref name=Kittel>{{cite book  |last = Kittel |first = Charles |author-link=Charles Kittel |title = [[Introduction to Solid State Physics]] |publisher = [[John Wiley & Sons]] |edition = 6th |year = 1986  |pages= 299–302|isbn = 978-0-471-87474-4}}</ref> The classical theory is given below.

=== Langevin diamagnetism ===

[[Paul Langevin]]'s theory of diamagnetism (1905)<ref>{{Cite journal|last=Langevin|first=Paul|author-link=Paul Langevin|date=1905|title=Sur la théorie du magnétisme|journal=Journal de Physique Théorique et Appliquée|language=fr|volume=4|issue=1|pages=678–693|doi=10.1051/jphystap:019050040067800|issn=0368-3893|url=https://zenodo.org/record/2057083}}</ref> applies to materials containing atoms with closed shells (see [[dielectrics]]). A field with intensity {{math|<var>B</var>}}, applied to an [[electron]] with charge {{math|<var>e</var>}} and mass {{math|<var>m</var>}}, gives rise to [[Larmor precession]] with frequency {{math|<var>ω</var> {{=}} <var>eB</var> / 2<var>m</var>}}. The number of revolutions per unit time is {{math|<var> ω</var> / 2{{pi}}}}, so the current for an atom with {{math|<var>Z</var>}} electrons is (in [[SI units]])<ref name=Kittel/>

:<math> I = -\frac{Ze^2B}{4 \pi m}.</math>

The [[magnetic moment]] of a current loop is equal to the current times the area of the loop. Suppose the field is aligned with the {{math|<var>z</var>}} axis. The average loop area can be given as <math>\scriptstyle  \pi\left\langle\rho^2\right\rangle</math>, where <math>\scriptstyle \left\langle\rho^2\right\rangle</math> is the mean square distance of the [[electrons]] perpendicular to the {{math|<var>z</var>}} axis. The magnetic moment is therefore

:<math> \mu = -\frac{Ze^2B}{4 m}\langle\rho^2\rangle.</math>

If the distribution of charge is spherically symmetric, we can suppose that the distribution of {{math|<var>x,y,z</var>}} coordinates are [[independent and identically distributed]]. Then <math>\scriptstyle \left\langle x^2 \right\rangle \;=\; \left\langle y^2 \right\rangle \;=\; \left\langle z^2 \right\rangle \;=\; \frac{1}{3}\left\langle r^2 \right\rangle</math>, where <math>\scriptstyle \left\langle r^2 \right\rangle</math> is the mean square distance of the electrons from the nucleus. Therefore, <math>\scriptstyle \left\langle \rho^2 \right\rangle \;=\; \left\langle x^2\right\rangle \;+\; \left\langle y^2 \right\rangle \;=\; \frac{2}{3}\left\langle r^2 \right\rangle</math>.  If <math>n</math> is the number of atoms per unit volume, the volume [[magnetic susceptibility|diamagnetic susceptibility]] in SI units is<ref>{{Cite book|title=[[Introduction to Solid State Physics]]|last=Kittel|first=Charles|publisher=John Wiley & Sons|year=2005|isbn=978-0471415268|edition=8|chapter=Chapter 14: Diamagnetism and Paramagnetism|author-link=Charles Kittel}}</ref>

:<math>\chi = \frac{\mu_0 n \mu}{B} = -\frac{\mu_0e^2  Zn }{6 m}\langle r^2\rangle.</math>
In atoms, Langevin susceptibility is of the same order of magnitude as [[Van Vleck paramagnetism|Van Vleck paramagnetic susceptibility]].

=== In metals ===
The Langevin theory is not the full picture for [[metals]] because there are also non-localized electrons. The theory that describes diamagnetism in a [[Fermi gas|free electron gas]] is called '''Landau diamagnetism''', named after [[Lev Landau]],<ref>Landau, L. D. "Diamagnetismus der metalle." Zeitschrift für Physik A Hadrons and Nuclei 64.9 (1930): 629-637.</ref> and instead considers the weak counteracting field that forms when the electrons' trajectories are curved due to the [[Lorentz force]]. Landau diamagnetism, however, should be contrasted with [[Paramagnetism#Pauli paramagnetism|Pauli paramagnetism]], an effect associated with the polarization of delocalized electrons' spins.<ref name="ntnu">{{cite web|url=http://phy.ntnu.edu.tw/~changmc/Teach/SS/SS_note/chap11.pdf |archive-url=https://web.archive.org/web/20060504001229/http://www.phy.ntnu.edu.tw/~changmc/Teach/SS/SS_note/chap11.pdf |archive-date=2006-05-04 |url-status=live|title=Diamagnetism and paramagnetism|website=NTNU lecture notes|last=Chang|first=M. C.|access-date=2011-02-24}}</ref><ref>{{cite web |first1=Nikos |last1=Drakos |first2=Ross |last2=Moore |first3=Peter |last3=Young |title=Landau diamagnetism |url=http://physics.ucsc.edu/~peter/231/magnetic_field/node5.html |website=Electrons in a magnetic field |date=2002 |access-date=27 November 2012}}</ref> For the bulk case of a 3D system and low magnetic fields, the (volume) diamagnetic susceptibility can be calculated using [[Landau quantization]], which in SI units is

:<math>\chi = -\mu_0\frac{e^2}{12\pi^2 m\hbar}\sqrt{2mE_{\rm F}},</math>

where <math>E_{\rm F}</math> is the [[Fermi energy]]. This is equivalent to <math>-\mu_0\mu_{\rm B}^2 g(E_{\rm F})/3</math>, exactly <math display="inline">-1/3</math> times Pauli paramagnetic susceptibility, where <math>\mu_{\rm B}=e\hbar/2m</math> is the [[Bohr magneton]] and <math>g(E)</math> is the [[density of states]] (number of states per energy per volume). This formula takes into account the spin degeneracy of the carriers (spin ½ electrons).

In [[doping (semiconductor)|doped semiconductors]] the ratio between Landau and Pauli susceptibilities may change due to the [[Effective mass (solid-state physics)|effective mass]] of the charge carriers differing from the electron mass in vacuum, increasing the diamagnetic contribution. The formula presented here only applies for the bulk; in confined systems like [[quantum dot]]s, the description is altered due to [[Potential well#Quantum confinement|quantum confinement]].<ref>{{Cite journal|last1=Lévy|first1=L.P.|last2=Reich|first2=D.H.|last3=Pfeiffer|first3=L.|last4=West|first4=K.|title=Aharonov-Bohm ballistic billiards|journal=Physica B: Condensed Matter|volume=189|issue=1–4|pages=204–209|doi=10.1016/0921-4526(93)90161-x|bibcode=1993PhyB..189..204L|year=1993}}</ref><ref>{{Cite journal|last1=Richter|first1=Klaus|last2=Ullmo|first2=Denis|last3=Jalabert|first3=Rodolfo A.|title=Orbital magnetism in the ballistic regime: geometrical effects|journal=Physics Reports|volume=276|issue=1|pages=1–83|doi=10.1016/0370-1573(96)00010-5|arxiv=cond-mat/9609201|bibcode=1996PhR...276....1R|year=1996|s2cid=119330207}}</ref> Additionally, for strong magnetic fields, the susceptibility of delocalized electrons oscillates as a function of the field strength, a phenomenon known as the [[De Haas–Van Alphen effect]], also first described theoretically by Landau.

== See also ==
*[[Antiferromagnetism]]
*[[Magnetochemistry]]
*[[Moses effect]]
*{{annotated link|Diamagnetic inequality}}

== References ==
{{reflist}}

== External links ==
*{{Commons category inline}}
*[https://feynmanlectures.caltech.edu/II_34.html The Feynman Lectures on Physics Vol. II Ch. 34: The Magnetism of Matter]

{{magnetic states}}

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[[Category:Electric and magnetic fields in matter]]
[[Category:Magnetic levitation]]
[[Category:Magnetism]]