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| In [[physics]], the '''exchange interaction''' is a [[quantum mechanic]]al effect between [[identical particles]]. It is due to the [[wave function]] of [[identical particles|indistinguishable particles]] being subject to [[exchange symmetry]], that is, either remaining unchanged (symmetric) or changing its sign (antisymmetric) when two particles are exchanged. Both [[boson]]s and [[fermion]]s can experience the exchange interaction. For fermions, it is sometimes called '''Pauli repulsion''' and related to the [[Pauli exclusion principle]]. For bosons, the exchange interaction takes the form of an attraction that causes identical particles to be found closer together, as in [[Bose–Einstein condensation]].
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| The exchange interaction alters the [[Expectation value (quantum mechanics)|expectation value]] of the [[energy]] when the wave functions of two or more indistinguishable particles overlap. It increases (for fermions) or decreases (for bosons) the expectation value of the distance between identical particles (as compared to distinguishable particles).<ref>David J. Griffiths, "Introduction to Quantum Mechanics", Second Edition, pp. 207–210</ref> Among other consequences, the exchange interaction is responsible for [[ferromagnetism]] and for the volume of matter. It has no [[classical mechanics|classical]] analogue.
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| Exchange interaction effects were discovered independently by physicists [[Werner Heisenberg]]<ref>Mehrkörperproblem und Resonanz in der Quantenmechanik, W. Heisenberg, ''Zeitschrift für Physik'' '''38''', #6–7 (June 1926), pp. 411–426. DOI [http://dx.doi.org/10.1007/BF01397160 10.1007/BF01397160].</ref> and [[Paul Dirac]]<ref>[http://links.jstor.org/sici?sici=0950-1207%2819261001%29112%3A762%3C661%3AOTTOQM%3E2.0.CO%3B2-X On the Theory of Quantum Mechanics], P. A. M. Dirac, ''Proceedings of the Royal Society of London, Series A'' '''112''', #762 (October 1, 1926), pp. 661—677.</ref> in 1926.
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| =="Force" description==
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| {{for|interaction mediation by exchange of particles|force carrier}}
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| The exchange interaction is sometimes called the ''exchange force''. However, it is not a true force and should not be confused with the [[exchange force]]s produced by the exchange of [[force carrier]]s, such as the [[electromagnetic force]] produced between two electrons by the exchange of a [[photon]], or the [[strong force]] between two [[quark]]s produced by the exchange of a [[gluon]].<ref>[http://hyperphysics.phy-astr.gsu.edu/hbase/forces/exchg.html Exchange Forces], HyperPhysics, [[Georgia State University]], accessed June 2, 2007.</ref>
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| Although sometimes erroneously described as a [[force]], the exchange interaction is a purely quantum mechanical effect unlike other forces.
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| ==Exchange Interactions between localized electron magnetic moments==
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| Quantum mechanical particles are classified as bosons or fermions. The [[spin-statistics theorem]] of [[quantum field theory]] demands that all particles with [[half-integer]] [[spin (physics)|spin]] behave as fermions and all particles with [[integer]] spin behave as bosons. Multiple bosons may occupy the same [[quantum state]]; by the [[Pauli exclusion principle]], however, no two fermions can occupy the same state. Since [[electron]]s have spin 1/2, they are fermions. This means that the overall wave function of a system must be antisymmetric when two electrons are exchanged, i.e. interchanged with respect to both spatial and spin coordinates. First, however, exchange will be explained with the neglect of spin.
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| ===Exchange of spatial coordinates===
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| Taking a hydrogen molecule-like system (i.e. one with two electrons), we may attempt to model the state of each electron by first assuming the electrons behave independently, and taking wave functions in position space of <math>\Phi_a(r_1)</math> for the first electron and <math>\Phi_b(r_2)</math> for the second electron. We assume that <math>\Phi_a</math> and <math>\Phi_b</math> are orthogonal, and that each corresponds to an energy eigenstate of its electron. Now, we may construct a wave function for the overall system in position space by using an antisymmetric combination of the product wave functions in position space:
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| {{NumBlk|:|<math>\Psi_A(r_1,r_2)= \frac{1}{\sqrt{2}}[\Phi_a(r_1) \Phi_b(r_2) - \Phi_b(r_1) \Phi_a(r_2)]</math>|{{EquationRef|1}}}}
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| Alternatively, we may also construct the overall position–space wave function by using a symmetric combination of the product wave functions in position space:
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| {{NumBlk|:|<math>\Psi_S(r_1,r_2)= \frac{1}{\sqrt{2}}[\Phi_a(r_1) \Phi_b(r_2) + \Phi_b(r_1) \Phi_a(r_2)]</math>|{{EquationRef|2}}}}
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| Treating the exchange interaction in the hydrogen molecule by the perturbation method, the overall [[Hamiltonian (quantum mechanics)|Hamiltonian]] is:
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| <math>\mathcal{H}</math> = <math>\mathcal{H}^{(0)}</math> + <math>\mathcal{H}^{(1)}</math>
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| where <math>\mathcal{H}^{(0)} = -\frac{\hbar^2}{2m}\left(\nabla^2_{1} + \nabla^2_{2}\right)-\frac{e^2}{r_{1}}-\frac{e^2}{r_{2}}</math> and <math>\mathcal{H}^{(1)} = \left(\frac {e^2}{R_{ab}} + \frac {e^2}{r_{12}} - \frac {e^2}{r_{a1}} - \frac {e^2}{r_{b2}}\right)</math>
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| Two eigenvalues for the system energy are found:
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| {{NumBlk|:|<math>\ E_{+/-} = E_{(0)} + \frac{C \pm J_{ex}}{1 \pm B^2}</math>|{{EquationRef|3}}}}
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| where the ''E''<sub>+</sub> is the spatially symmetric solution and ''E''<sub>−</sub> is the spatially antisymmetric solution. A variational calculation yields similar results. <math>\mathcal{H}</math> can be diagonalized by using the position–space functions given by Eqs. (1) and (2). In Eq. (3), ''C'' is the '''Coulomb integral''', ''B'' is the '''overlap integral''', and ''J''<sub>ex</sub> is the '''exchange integral'''. These integrals are given by: | |
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| {{NumBlk|:|<math> C = \int \Phi_a(r_1)^2 \left(\frac{1}{R_{ab}} + \frac{1}{r_{12}} - \frac{1}{r_{a1}} - \frac{1}{r_{b2}}\right) \Phi_b(r_2)^2 \, dr_1\, dr_2</math>|{{EquationRef|4}}}}
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| {{NumBlk|:|<math> B = \int \Phi_b(r_2) \Phi_a(r_2) \, dr_2</math>|{{EquationRef|5}}}}
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| {{NumBlk|:|<math> J_{ex} = \int \Phi_a^{*}(r_1) \Phi_b^{*}(r_2) \left(\frac{1}{R_{ab}} + \frac{1}{r_{12}} - \frac{1}{r_{a1}} - \frac{1}{r_{b2}}\right) \Phi_b(r_1) \Phi_a(r_2) \, dr_1\, dr_2</math>|{{EquationRef|6}}}}
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| The terms in parentheses in Eqs. (4) and (6) correspond to: proton–proton repulsion (''R''<sub>ab</sub>), electron–electron repulsion (''r''<sub>12</sub>), and electron–proton attraction (''r''<sub>a1/a2/b1/b2</sub>).
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| Although in the hydrogen molecule the exchange integral, Eq. (6), is negative, Heisenberg first suggested that it changes sign at some critical ratio of internuclear distance to mean radial extension of the atomic orbital.<ref>[http://phycomp.technion.ac.il/~riki/Heisenberg.html Derivation of the Heisenberg Hamiltonian], Rebecca Hihinashvili, accessed on line October 2, 2007.</ref><ref>''Quantum Theory of Magnetism: Magnetic Properties of Materials'', Robert M. White, 3rd rev. ed., Berlin: Springer-Verlag, 2007, section 2.2.7. ISBN 3-540-65116-0.</ref><ref>''The Theory of Electric and Magnetic Susceptibilities'', J. H. van Vleck, London: Oxford University Press, 1932, chapter XII, section 76.</ref>
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| ===Inclusion of spin===
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| The symmetric and antisymmetric combinations in Eqs. (1) and (2) did not include the spin variables (α = spin-up; β = spin down); there are also antisymmetric and symmetric combinations of the spin variables:
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| {{NumBlk|:|<math>\alpha(1)</math> <math>\beta(2)</math> ± <math>\alpha(2)</math> <math>\beta(1)</math>|{{EquationRef|7}}}}
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| To obtain the overall wave function, these spin combinations have to be coupled with Eqs. (1) and (2). The resulting overall wave functions, called [[spin-orbital]]s, are written as [[Slater determinant]]s. When the orbital wave function is symmetrical the spin one must be anti-symmetrical and vice versa. Accordingly, ''E''<sub>+</sub> above corresponds to the spatially symmetric/spin-singlet solution and ''E''<sub>−</sub> to the spatially antisymmetric/spin-triplet solution.
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| [[John Hasbrouck Van Vleck|J. H. Van Vleck]] presented the following analysis:<ref>Van Vleck, J. H. "Electric and Magnetic Susceptibilities, Oxford, Clarendon Press, p. 318 (1932).</ref>
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| :''The potential energy of the interaction between the two electrons in orthogonal orbitals can be represented by a matrix,'' ''say'' ''E''<sub>ex</sub>. ''From Eq. (3), the characteristic values of this matrix are'' ''C'' ± ''J''<sub>ex</sub>. ''The characteristic values of a matrix are its diagonal elements after it is converted to a diagonal matrix. Now, the characteristic values of the square of the magnitude of the resultant spin <math>(\vec{s}_a + \vec{s}_b)^2 </math> is <math>S(S+1)</math>. The characteristic values of the matrices'' <math>\vec{s}_a^{\;2}</math> ''and'' <math>\vec{s}_b^{\;2}</math> ''are each'' <math>\tfrac{1}{2}(\tfrac{1}{2} + 1) = \tfrac{3}{4}</math> ''and'' <math>(\vec{s}_a + \vec{s}_b)^2 = \vec{s}_a^{\;2} + \vec{s}_b^{\;2} + 2\vec{s}_a \cdot \vec{s}_b </math>. ''The characteristic values of the scalar product'' <math>\vec{s}_a \cdot \vec{s}_b</math> ''are'' <math>\tfrac{1}{2}(0 - \tfrac{6}{4})= -\tfrac{3}{4}</math> ''and'' <math>\tfrac{1}{2}(2 - \tfrac{6}{4}) = \tfrac{1}{4}</math>, ''corresponding to the spin-singlet'' (''S'' = 0)
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| :''and spin-triplet'' (''S'' = 1) ''states. From Eq. (3) and the aforementioned relations, the matrix'' ''E''<sub>ex</sub> ''is seen to have the characteristic value'' ''C'' + ''J''<sub>ex</sub> ''when'' <math> \vec{s}_a \cdot \vec{s}_b </math> ''has the characteristic value −3/4'' (i.e. ''when'' ''S'' = 0; ''the spatially symmetric/spin-singlet state). Alternatively, it has the characteristic value'' ''C'' − ''J''<sub>ex</sub> ''when'' <math> \vec{s}_a \cdot \vec{s}_b </math> ''has the characteristic value +1/4 (i.e. when'' ''S'' = 1; ''the spatially antisymmetric/spin-triplet state). Therefore,''
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| {{NumBlk|:|<math>E_{ex} - C + \frac{1}{2}J_{ex} + 2J_{ab} \vec{s}_a \cdot \vec{s}_b = 0 </math>|{{EquationRef|8}}}}
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| :''and, hence,'' | |
| {{NumBlk|:|<math>E_{ex} = C - \frac{1}{2}J_{ex} - 2J_{ab} \vec{s}_a \cdot \vec{s}_b </math>|{{EquationRef|9}}}}
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| :''where the spin momenta are given as'' <math> \vec{s}_a </math> ''and'' <math> \vec{s}_b </math>.
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| Dirac pointed out that the critical features of the exchange interaction could be obtained in an elementary way by neglecting the first two terms on the right-hand side of Eq. (9), thereby considering the two electrons as simply having their spins coupled by a potential of the form:
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| {{NumBlk|:|<math>\ -2J_{ab} \vec{s}_a \cdot \vec{s}_b </math>|{{EquationRef|10}}}}
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| It follows that the exchange interaction Hamiltonian between two electrons in orbitals Φ<sub>a</sub> and Φ<sub>b</sub> can be written in terms of their spin momenta <math> \vec{s}_a </math> and <math> \vec{s}_b </math>. This is named the [[Heisenberg model (classical)|Heisenberg Exchange Hamiltonian]] or the Heisenberg–Dirac Hamiltonian in the older literature:
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| {{NumBlk|:|<math>\mathcal{H}_{Heis} = -2J_{ab} \vec{s}_a \cdot \vec{s}_b </math>|{{EquationRef|11}}}}
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| ''J''<sub>ab</sub> is not the same as the quantity labeled ''J''<sub>ex</sub> in Eq. (6). Rather, ''J''<sub>ab</sub>, which is termed the '''exchange constant''', is a function of Eqs. (4), (5), and (6), namely,
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| {{NumBlk|:|<math>\ J_{ab} = \frac{1}{2} (E_+ - E_-) = \frac{J_{ex}- CB^2}{1-B^4}</math>|{{EquationRef|12}}}}
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| However, with orthogonal orbitals (in which ''B'' = 0), for example with different orbitals in the ''same'' atom, ''J''<sub>ab</sub> = ''J<sub>ex</sub>''.
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| ===Effects of exchange===
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| If ''J<sub>ab</sub>'' is positive the exchange energy favors electrons with parallel spins; this is a primary cause of [[ferromagnetism]] in materials in which the electrons are considered localized in the Heitler–London model of chemical bonding, but this model of ferromagnetism has severe limitations in solids (see [[Exchange_interaction#Limitations_of_the_Heisenberg_Hamiltonian|below]]). If ''J<sub>ab</sub>'' is negative, the interaction favors electrons with antiparallel spins, potentially causing [[antiferromagnetism]]. The sign of ''J''<sub>ab</sub> is essentially determined by the relative sizes of ''J''<sub>ex</sub> and the product of ''CB''<sup>2</sup>. This can be deduced from the expression for the difference between the energies of the triplet and singlet states, ''E''<sub>−</sub> − ''E''<sub>+</sub>:
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| {{NumBlk|:|<math>\ E_{-} - E_{+} = \frac{2(CB^2 - J_{ex})}{1-B^4} </math>|{{EquationRef|13}}}}
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| Although these ''consequences'' of the exchange interaction are magnetic in nature, the ''cause'' is not; it is due primarily to electric repulsion and the Pauli exclusion principle. Indeed, in general, the direct magnetic interaction between a pair of electrons (due to their [[electron magnetic moment]]s) is negligibly small compared to this electric interaction.
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| Exchange energy splittings are very elusive to calculate for molecular systems at large internuclear distances. However, analytical formulae have been worked out for the [[hydrogen molecular ion]] (see references herein).
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| Normally, exchange interactions are very short-ranged, confined to electrons in orbitals on the same atom (intra-atomic exchange) or nearest neighbor atoms ('''direct exchange''') but longer-ranged interactions can occur via intermediary atoms and this is termed [[Superexchange]].
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| ==Direct exchange interactions in solids==
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| In a crystal, generalization of the Heisenberg Hamiltonian in which the sum is taken over the exchange Hamiltonians for all the (''i'',''j'') pairs of atoms of the many-electron system gives:.
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| {{NumBlk|:|<math>\mathcal{H}_{Heis} = \frac{1}{2}(-2J\sum_{i,j} \vec{S}_i \cdot \vec{S}_j\quad) = -\sum_{i,j}J \vec{S}_i \cdot \vec{S}_j </math>|{{EquationRef|14}}}}
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| The 1/2 factor is introduced because the interaction between the same two atoms is counted twice in performing the sums. Note that the ''J'' in Eq.(14) is the exchange constant ''J''<sub>ab</sub> above not the exchange integral ''J''<sub>ex</sub>. The exchange integral ''J''<sub>ex</sub> is related to yet another quantity, called the '''exchange stiffness constant''' (''A'') which serves as a characteristic of a ferromagnetic material. The relationship is dependent on the crystal structure. For a simple cubic lattice with lattice parameter <math>a</math>,
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| {{NumBlk|:|<math> A_{sc} = \frac{J_{ex}S^2}{a}</math>|{{EquationRef|15}}}}
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| For a body-centered cubic lattice,
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| {{NumBlk|:|<math> A_{bcc} = \frac{2J_{ex}S^2}{a}</math>|{{EquationRef|16}}}}
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| and for a face-centered cubic lattice,
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| {{NumBlk|:|<math> A_{fcc} = \frac{4J_{ex}S^2}{a}</math>|{{EquationRef|17}}}}
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| The form of Eq. (14) corresponds identically to the [[Ising model|Ising [statistical mechanical] model]] of ferromagnetism except that in the Ising model, the dot product of the two spin angular momenta is replaced by the scalar product ''S<sub>ij</sub>S<sub>ji</sub>''. The Ising model was invented by Wilhelm Lenz in 1920 and solved for the one-dimensional case by his doctoral student Ernst Ising in 1925. The energy of the Ising model is defined to be:
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| {{NumBlk|:|<math> E = - \sum_{i\neq j} J_{ij} S_{i}^z S_{j}^z \,</math>|{{EquationRef|18}}}}
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| ===Limitations of the Heisenberg Hamiltonian and the localized electron model in solids===
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| Because the Heisenberg Hamiltonian presumes the electrons involved in the exchange coupling are localized in the context of the Heitler–London, or [[Valence bond theory|valence bond]] (VB), theory of chemical bonding, it is an adequate model for explaining the magnetic properties of electrically insulating narrow-band ionic and covalent non-molecular solids where this picture of the bonding is reasonable. Nevertheless, theoretical evaluations of the exchange integral for non-molecular solids that display metallic conductivity in which the electrons responsible for the ferromagnetism are itinerant (e.g. iron, nickel, and cobalt) have historically been either of the wrong sign or much too small in magnitude to account for the experimentally determined exchange constant (e.g. as estimated from the Curie temperatures via ''T''<sub>C</sub> ≈ 2⟨''J''⟩/3''k''<sub>B</sub> where ⟨''J''⟩ is the exchange interaction averaged over all sites). The Heisenberg model thus cannot explain the observed ferromagnetism in these materials.<ref>see, for example, Stuart, R. and Marshall, W. ''Phys. Rev.'' '''120''', 353 (1960).</ref> In these cases, a delocalized, or Hund–Mulliken–Bloch (molecular orbital/band) description, for the electron wave functions is more realistic. Accordingly, the [[Stoner model]] of ferromagnetism is more applicable. In the Stoner model, the spin-only magnetic moment (in Bohr magnetons) per atom in a ferromagnet is given by the difference between the number of electrons per atom in the majority spin and minority spin states. The Stoner model thus permits non-integral values for the spin-only magnetic moment per atom. However, with ferromagnets <math>\mu_S = - g \mu_B [S(S+1)]^{1/2} </math> (''g'' = 2.0023 ≈ 2) tends to overestimate the total [[Spin magnetic moment|spin-only magnetic moment]] per atom. For example, a net magnetic moment of 0.54 μ<sub>B</sub> per atom for Nickel metal is predicted by the Stoner model, which is very close to the 0.61 Bohr magnetons calculated based on the metal's observed saturation magnetic induction, its density, and its atomic weight.<ref>Elliot, S. R. "The Physics and Chemistry of Solids", John Wiley & Sons, New York, p. 615 (1998)</ref> By contrast, an isolated Ni atom (electron configuration = 3''d''<sup>8</sup>4''s''<sup>2</sup>) in a cubic crystal field will have two unpaired electrons of the same spin (hence, <math>\vec{S} = 1</math>) and would thus be expected to have in the localized electron model a total spin magnetic moment of <math>\mu_S = 2.83 \mu_B</math> (but the measured spin-only magnetic moment along one axis, the physical observable, will be given by <math>\vec{\mu}_S = g \mu_B \vec{S} = 2 \mu_B</math>). Generally, valence ''s'' and ''p'' electrons are best considered delocalized, while 4''f'' electrons are localized and 5''f'' and 3''d''/4''d'' electrons are intermediate, depending on the particular internuclear distances.<ref>J. B. Goodenough "Magnetism and the Chemical Bond" Interscience Publishers, New York, pp. 5–17 (1966).</ref> In the case of substances where both delocalized and localized electrons contribute to the magnetic properties (e.g. rare-earth systems), the [[RKKY interaction|Ruderman–Kittel–Kasuya–Yosida (RKKY)]] model is the currently accepted mechanism.
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| ==See also==
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| * [[Double-exchange mechanism]]
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| * [[Exchange symmetry]]
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| * [[Pauli exclusion principle]]
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| * [[Slater determinant]]
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| * [[Superexchange]]
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| * [[Holstein–Herring method]]
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| * [[Spin-exchange interaction]]
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| ==References==
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| {{reflist|2}}
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| ==External links==
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| * [http://www.cond-mat.de/events/correl12/manuscripts/koch.pdf Exchange Mechanisms] in E. Pavarini, E. Koch, F. Anders, and M. Jarrell: Correlated Electrons: From Models to Materials, Jülich 2012, ISBN 978-3-89336-796-2
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| * [http://wpage.unina.it/mdaquino/PhD_thesis/main/node7.html Exchange Interaction and Energy]
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| * [http://www.cmp.liv.ac.uk/frink/thesis/thesis/node68.html Exchange Interaction and Exchange Anisotropy]
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| {{DEFAULTSORT:Exchange Interaction}}
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| [[Category:Pauli exclusion principle]]
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| [[Category:Quantum chemistry]]
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| {{Link GA|es}}
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