 Exchange interaction

In physics, the exchange interaction is a quantum mechanical effect without classical analog which increases or decreases the expectation value of the energy or distance between two or more identical particles when their wave functions overlap. For example, the exchange interaction results in identical particles with spatially symmetric wave functions (bosons) appearing "closer together" than would be expected of distinguishable particles, and in identical particles with spatially antisymmetric wave functions (fermions) appearing "farther apart". It is the mechanism that is responsible for some types of ferromagnetism, among other consequences.
Although sometimes erroneously described as a force, the exchange interaction is a purely quantum mechanical effect without any analog in classical mechanics. It is due to the wave function of indistinguishable particles being subject to exchange symmetry, that is, the wave function describing two particles that cannot be distinguished must be either unchanged (symmetric) or inverted in sign (antisymmetric) if the labels of the two particles are changed.
For example, if the expectation value of the distance between two particles in a spatially symmetric or antisymmetric state is calculated, the exchange interaction may be seen.^{[1]}
Both bosons and fermions can experience the exchange interaction provided that the particles in question are indistinguishable.
Exchange interaction effects were discovered independently by physicists Werner Heisenberg^{[2]} and P. A. M. Dirac^{[3]} in 1926.
The exchange interaction is sometimes called the exchange force, but it is not a true force and should not be confused with the exchange forces produced by the exchange of force carriers, such as the electromagnetic force produced between two electrons by the exchange of a photon, or the strong force between two quarks produced by the exchange of a gluon.^{[4]}
Contents
Exchange Interactions between localized electron magnetic moments
Quantum mechanical particles are classified as bosons or fermions. The spinstatistics theorem of quantum field theory demands that all particles with halfinteger 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 electrons 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.
Exchange of spatial coordinates
Taking a hydrogen moleculelike 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 Φ_{a}(r_{1}) for the first electron and Φ_{b}(r_{2}) for the second electron. We assume that Φ_{a} and Φ_{b} 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:

(
Alternatively, we may also construct the overall positionspace wave function by using a symmetric combination of the product wave functions in position space:

(
Treating the exchange interaction in the hydrogen molecule by the perturbation method, the overall Hamiltonian is:
= +
where = ( + ) and =
Two eigenvalues for the system energy are found:

(
where the E_{+} is the spatially symmetric solution and E_{} is the spatially antisymmetric solution. A variational calculation yields similar results. can be diagonalized by using the positionspace functions given by Eqs. (1) and (2). In Eq. (3), C is the Coulomb integral, B is the overlap integral, and J_{ex} is the exchange integral. These integrals are given by:

(

(

(
The terms in parentheses in Eqs. (4) and (6) correspond to: protonproton repulsion (R_{ab}), electronelectron repulsion (r_{12}), and electronproton attraction (r_{a1/a2/b1/b2}).
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.^{[5]}^{[6]}^{[7]}
Inclusion of spin
The symmetric and antisymmetric combinations in Eqs. (1) and (2) did not include the spin variables (α = spinup; β = spin down); there are also antisymmetric and symmetric combinations of the spin variables:

α(1) β(2) ± α(2) β(1)
(
To obtain the overall wave function, these spin combinations have to be coupled with Eqs. (1) and (2). The resulting overall wave functions, called spinorbitals, are written as Slater determinants. When the orbital wave function is symmetrical the spin one must be antisymmetrical and vice versa. Accordingly, E_{+} above corresponds to the spatially symmetric/spinsinglet solution and E_{} to the spatially antisymmetric/spintriplet solution.
J. H. Van Vleck presented the following analysis:^{[8]}
 The potential energy of the interaction between the two electrons in orthogonal orbitals can be represented by a matrix, say E_{ex}. From Eq. (3), the characteristic values of this matrix are C ± J_{ex}. 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 is S(S + 1). The characteristic values of the matrices and are each and . The characteristic values of the scalar product are and , corresponding to the spinsinglet (S = 0)
 and spintriplet (S = 1) states. From Eq. (3) and the aforementioned relations, the matrix E_{ex} is seen to have the characteristic value C + J_{ex} when has the characteristic value 3/4 (i.e. when S = 0; the spatially symmetric/spinsinglet state). Alternatively, it has the characteristic value C  J_{ex} when has the characteristic value +1/4 (i.e. when S = 1; the spatially antisymmetric/spintriplet state). Therefore,

(
 and, hence,

(
 where the spin momenta are given as and .
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 righthand side of Eq. (9), thereby considering the two electrons as simply having their spins coupled by a potential of the form:

(
It follows that the exchange interaction Hamiltonian between two electrons in orbitals Φ_{a} and Φ_{b} can be written in terms of their spin momenta and . This is named the Heisenberg Exchange Hamiltonian or the HeisenbergDirac Hamiltonian in the older literature:

(
J_{ab} is not the same as the quantity labeled J_{ex} in Eq. (6). Rather, J_{ab}, which is termed the exchange constant, is a function of Eqs. (4), (5), and (6), namely,

(
However, with orthogonal orbitals (in which B = 0), for example with different orbitals in the same atom, J_{ab} = J_{ex}.
Effects of exchange
If J_{ab} 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 HeitlerLondon model of chemical bonding, but this model of ferromagnetism has severe limitations in solids (see below). If J_{ab} is negative, the interaction favors electrons with antiparallel spins, potentially causing antiferromagnetism. The sign of J_{ab} is essentially determined by the relative sizes of J_{ex} and the product of CB^{2}. This can be deduced from the expression for the difference between the energies of the triplet and singlet states, E_{}  E_{+}:

(
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 moments) is negligibly small compared to this electric interaction.
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).
Normally, exchange interactions are very shortranged, confined to electrons in orbitals on the same atom (intraatomic exchange) or nearest neighbor atoms (direct exchange) but longerranged interactions can occur via intermediary atoms and this is termed Superexchange.
Direct exchange interactions in solids
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 manyelectron system gives:.

(
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_{ab} above not the exchange integral J_{ex}. The exchange integral J_{ex} 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 a,

(
For a bodycentered cubic lattice,

(
and for a facecentered cubic lattice,

(
The form of Eq. (14) corresponds identically to the 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_{ij}S_{ji}. The Ising model was invented by Wilhelm Lenz in 1920 and solved for the onedimensional case by his doctoral student Ernst Ising in 1925. The energy of the Ising model is defined to be:

(
Limitations of the Heisenberg Hamiltonian and the localized electron model in solids
Because the Heisenberg Hamiltonian presumes the electrons involved in the exchange coupling are localized in the context of the HeitlerLondon, or valence bond (VB), theory of chemical bonding, it is an adequate model for explaining the magnetic properties of electrically insulating narrowband ionic and covalent nonmolecular solids where this picture of the bonding is reasonable. Nevertheless, theoretical evaluations of the exchange integral for nonmolecular 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_{C} ≈ 2⟨J⟩/3k_{B} where ⟨J⟩ is the exchange interaction averaged over all sites). The Heisenberg model thus cannot explain the observed ferromagnetism in these materials.^{[9]} In these cases, a delocalized, or HundMullikenBloch (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 spinonly 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 nonintegral values for the spinonly magnetic moment per atom. However, with ferromagnets μ_{S} = − gμ_{B}[S(S + 1)]^{1 / 2} (g = 2.0023 ≈ 2) tends to overestimate the total spinonly magnetic moment per atom. For example, a net magnetic moment of 0.54 μ_{B} 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.^{[10]} By contrast, an isolated Ni atom (electron configuration = 3d^{8}4s^{2}) in a cubic crystal field will have two unpaired electrons of the same spin (hence, ) and would thus be expected to have in the localized electron model a total spin magnetic moment of μ_{S} = 2.83μ_{B} (but the measured spinonly magnetic moment along one axis, the physical observable, will be given by ). Generally, valence s and p electrons are best considered delocalized, while 4f electrons are localized and 5f and 3d/4d electrons are intermediate, depending on the particular internuclear distances.^{[11]} In the case of substances where both delocalized and localized electrons contribute to the magnetic properties (e.g. rareearth systems), the RudermanKittelKasuyaYosida (RKKY) model is the currently accepted mechanism.
See also
 Doubleexchange mechanism
 Exchange symmetry
 Pauli exclusion principle
 Slater determinant
 Superexchange
 Holstein–Herring method
References
 ^ David J. Griffifths, "Introduction to Quantum Mechanics", Second Edition, pp. 207210
 ^ Mehrkörperproblem und Resonanz in der Quantenmechanik, W. Heisenberg, Zeitschrift für Physik 38, #6–7 (June 1926), pp. 411–426. DOI 10.1007/BF01397160.
 ^ 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.
 ^ Exchange Forces, HyperPhysics, Georgia State University, accessed June 2, 2007.
 ^ Derivation of the Heisenberg Hamiltonian, Rebecca Hihinashvili, accessed on line October 2, 2007.
 ^ Quantum Theory of Magnetism: Magnetic Properties of Materials, Robert M. White, 3rd rev. ed., Berlin: SpringerVerlag, 2007, section 2.2.7. ISBN 3540651160.
 ^ The Theory of Electric and Magnetic Susceptibilities, J. H. van Vleck, London: Oxford University Press, 1932, chapter XII, section 76.
 ^ Van Vleck, J. H. "Electric and Magnetic Susceptibilities, Oxford, Clarendon Press, p. 318 (1932).
 ^ see, for example, Stuart, R. and Marshall, W. Phys. Rev. 120, 353 (1960).
 ^ Elliot, S. R. "The Physics and Chemistry of Solids", John Wiley & Sons, New York, p. 615 (1998)
 ^ J. B. Goodenough "Magnetism and the Chemical Bond" Interscience Publishers, New York, pp. 517 (1966).
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