Editing Diamagnetism (section)

== 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|access-date=26 November 2021|archive-date=16 December 2021|archive-url=https://web.archive.org/web/20211216155913/https://zenodo.org/record/2057083|url-status=live}}</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 |archive-date=27 June 2013 |archive-url=https://web.archive.org/web/20130627030334/http://physics.ucsc.edu/~peter/231/magnetic_field/node5.html |url-status=dead }}</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-1/2 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. |year=1993 |title=Aharonov-Bohm ballistic billiards |journal=Physica B: Condensed Matter |volume=189 |issue=1–4 |pages=204–209 |bibcode=1993PhyB..189..204L |doi=10.1016/0921-4526(93)90161-x}}</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.