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In chemistry and
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which relat ...
, the exchange interaction (with an exchange energy and exchange term) is a
quantum mechanic Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, qua ...
al effect that only occurs between
identical particles In quantum mechanics, identical particles (also called indistinguishable or indiscernible particles) are particles that cannot be distinguished from one another, even in principle. Species of identical particles include, but are not limited to, ...
. Despite sometimes being called an exchange force in an analogy to classical force, it is not a true force as it lacks a
force carrier In quantum field theory, a force carrier, also known as messenger particle or intermediate particle, is a type of particle that gives rise to forces between other particles. These particles serve as the quanta of a particular kind of physical fiel ...
. The effect is due to the
wave function A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements m ...
of
indistinguishable particles In quantum mechanics, identical particles (also called indistinguishable or indiscernible particles) are particles that cannot be distinguished from one another, even in principle. Species of identical particles include, but are not limited to, ...
being subject to exchange symmetry, that is, either remaining unchanged (symmetric) or changing sign (antisymmetric) when two particles are exchanged. Both
boson In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0,1,2 ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have odd half-integer s ...
s and
fermion In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks and ...
s can experience the exchange interaction. For fermions, this interaction is sometimes called Pauli repulsion and is related to the
Pauli exclusion principle In quantum mechanics, the Pauli exclusion principle states that two or more identical particles with half-integer spins (i.e. fermions) cannot occupy the same quantum state within a quantum system simultaneously. This principle was formulated ...
. For bosons, the exchange interaction takes the form of an effective attraction that causes identical particles to be found closer together, as in
Bose–Einstein condensation Bose–Einstein may refer to: * Bose–Einstein condensate ** Bose–Einstein condensation (network theory) * Bose–Einstein correlations * Bose–Einstein statistics In quantum statistics, Bose–Einstein statistics (B–E statistics) describ ...
. The exchange interaction alters the
expectation value In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a l ...
of the distance when the wave functions of two or more indistinguishable particles overlap. This interaction increases (for fermions) or decreases (for bosons) the expectation value of the distance between identical particles (compared to distinguishable particles). Among other consequences, the exchange interaction is responsible for
ferromagnetism Ferromagnetism is a property of certain materials (such as iron) which results in a large observed magnetic permeability, and in many cases a large magnetic coercivity allowing the material to form a permanent magnet. Ferromagnetic materials a ...
and the volume of matter. It has no classical analogue. Exchange interaction effects were discovered independently by physicists
Werner Heisenberg Werner Karl Heisenberg () (5 December 1901 – 1 February 1976) was a German theoretical physicist and one of the main pioneers of the theory of quantum mechanics. He published his work in 1925 in a Über quantentheoretische Umdeutung kinematis ...
and
Paul Dirac Paul Adrien Maurice Dirac (; 8 August 1902 – 20 October 1984) was an English theoretical physicist who is regarded as one of the most significant physicists of the 20th century. He was the Lucasian Professor of Mathematics at the Univer ...
in 1926.


"Force" description

The exchange interaction is sometimes called the ''exchange force''. However, it is not a true force and should not be confused with the exchange forces produced by the exchange of
force carrier In quantum field theory, a force carrier, also known as messenger particle or intermediate particle, is a type of particle that gives rise to forces between other particles. These particles serve as the quanta of a particular kind of physical fiel ...
s, such as the
electromagnetic force In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of ...
produced between two electrons by the exchange of a
photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they always m ...
, or the
strong force The strong interaction or strong force is a fundamental interaction that confines quarks into proton, neutron, and other hadron particles. The strong interaction also binds neutrons and protons to create atomic nuclei, where it is called the ...
between two
quark A quark () is a type of elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. All commonly o ...
s produced by the exchange of a
gluon A gluon ( ) is an elementary particle that acts as the exchange particle (or gauge boson) for the strong force between quarks. It is analogous to the exchange of photons in the electromagnetic force between two charged particles. Gluons bind q ...
. Although sometimes erroneously described as a
force In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a ...
, the exchange interaction is a purely quantum mechanical ''effect'' unlike other forces.


Exchange interactions between localized electron magnetic moments

Quantum mechanical particles are classified as bosons or fermions. The
spin–statistics theorem In quantum mechanics, the spin–statistics theorem relates the intrinsic spin of a particle (angular momentum not due to the orbital motion) to the particle statistics it obeys. In units of the reduced Planck constant ''ħ'', all particles tha ...
of
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
demands that all particles with
half-integer In mathematics, a half-integer is a number of the form :n + \tfrac, where n is an whole number. For example, :, , , 8.5 are all ''half-integers''. The name "half-integer" is perhaps misleading, as the set may be misunderstood to include number ...
spin Spin or spinning most often refers to: * Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning * Spin, the rotation of an object around a central axis * Spin (propaganda), an intentionally b ...
behave as fermions and all particles with
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language o ...
spin behave as bosons. Multiple bosons may occupy the same
quantum state In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in t ...
; however, by the
Pauli exclusion principle In quantum mechanics, the Pauli exclusion principle states that two or more identical particles with half-integer spins (i.e. fermions) cannot occupy the same quantum state within a quantum system simultaneously. This principle was formulated ...
, no two fermions can occupy the same state. Since
electron The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no kno ...
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.


Exchange of spatial coordinates

Taking a hydrogen molecule-like system (i.e. one with two electrons), one may attempt to model the state of each electron by first assuming the electrons behave independently, and taking wave functions in position space of \Phi_a(r_1) for the first electron and \Phi_b(r_2) for the second electron. We assume that \Phi_a and \Phi_b are orthogonal, and that each corresponds to an energy eigenstate of its electron. Now, one 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 position–space 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 Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...
, composed of the Hamiltonian of the unperturbed separate hydrogen atoms \mathcal^ and the pertubation \mathcal^ is: :\mathcal = \mathcal^ + \mathcal^ where \mathcal^ = -\frac\Delta_-\frac\Delta_-\frac-\frac and \mathcal^ = \left(\frac + \frac - \frac - \frac \right) The first two terms denote the kinetic energy, the following terms the potential energy corresponding to: proton–proton repulsion (''R''ab), electron–electron repulsion (''r''12), and electron–proton attraction (''r''a1/a2/b1/b2). All quantities are assumed to be
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
. Two eigenvalues for the system energy are found: where the ''E''+ is the spatially symmetric solution and ''E'' is the spatially antisymmetric solution, corresponding to \Psi_ and \Psi_ respectively. A variational calculation yields similar results. \mathcal can be diagonalized by using the position–space functions given by Eqs. (1) and (2). In Eq. (3), ''C'' is the two-site two-electron Coulomb integral (It may be interpreted as the repulsive potential for electron-one at a particular point \Phi_a(\vec r_1)^2 in an electric field created by electron-two distributed over the space with the probability density \Phi_b(\vec r_2)^2), \mathcal is the overlap integral, and ''J''ex is the exchange integral, which is similar to the two-site Coulomb integral but includes exchange of the two electrons. It has no simple physical interpretation, but it can be shown to arise entirely due to the anti-symmetry requirement. These integrals are given by: 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.


Inclusion of spin

The symmetric and antisymmetric combinations in Equations (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: 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-orbitals, are written as
Slater determinant In quantum mechanics, a Slater determinant is an expression that describes the wave function of a multi-fermionic system. It satisfies anti-symmetry requirements, and consequently the Pauli principle, by changing sign upon exchange of two electr ...
s. When the orbital wave function is symmetrical the spin one must be anti-symmetrical and vice versa. Accordingly, ''E''+ above corresponds to the spatially symmetric/spin-singlet solution and ''E'' to the spatially antisymmetric/spin-triplet solution. J. H. Van Vleck presented the following analysis: :''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, \langle (\vec_a + \vec_b)^2 \rangle is S(S+1). The characteristic values of the matrices'' \langle \vec_a^\rangle ''and'' \langle \vec_b^\rangle ''are each'' \tfrac(\tfrac + 1) = \tfrac ''and'' \langle(\vec_a + \vec_b)^2\rangle = \langle\vec_a^\rangle + \langle\vec_b^\rangle + 2\langle\vec_a \cdot \vec_b\rangle. ''The characteristic values of the scalar product'' \langle\vec_a \cdot \vec_b\rangle ''are'' \tfrac(0 - \tfrac)= -\tfrac ''and'' \tfrac(2 - \tfrac) = \tfrac, ''corresponding to both the spin-singlet'' (''S'' = 0) ''and spin-triplet'' (''S'' = 1) ''states, respectively. :From Eq. (3) and the aforementioned relations, the matrix'' Eex ''is seen to have the characteristic value'' ''C'' + Jex ''when'' \langle\vec_a \cdot \vec_b\rangle ''has the characteristic value −3/4'' (i.e. ''when'' S = 0; ''the spatially symmetric/spin-singlet state). Alternatively, it has the characteristic value'' C − Jex ''when'' \langle \vec_a \cdot \vec_b\rangle'' has the characteristic value +1/4 (i.e. when'' S = 1; ''the spatially antisymmetric/spin-triplet state). Therefore,'' :''and, hence,'' :''where the spin momenta are given as'' \langle\vec_a\rangle ''and'' \langle\vec_b\rangle. 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: It follows that the exchange interaction Hamiltonian between two electrons in orbitals ''Φa'' and ''Φb'' can be written in terms of their spin momenta \vec_a and \vec_b . This interaction is named the Heisenberg exchange Hamiltonian or the Heisenberg–Dirac 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 \mathcal = 0), for example with different orbitals in the ''same'' atom, ''J''ab = ''J''ex.


Effects of exchange

If ''Jab'' is positive the exchange energy favors electrons with parallel spins; this is a primary cause of
ferromagnetism Ferromagnetism is a property of certain materials (such as iron) which results in a large observed magnetic permeability, and in many cases a large magnetic coercivity allowing the material to form a permanent magnet. Ferromagnetic materials a ...
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 below). If ''Jab'' 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 C \mathcal. This sign 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. In general, the direct magnetic interaction between a pair of electrons (due to their
electron magnetic moment In atomic physics, the electron magnetic moment, or more specifically the electron magnetic dipole moment, is the magnetic moment of an electron resulting from its intrinsic properties of spin and electric charge. The value of the electron magne ...
s) 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 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 Superexchange, or Kramers–Anderson superexchange, is the strong (usually) antiferromagnetic coupling between two next-to-nearest neighbour cations through a non-magnetic anion. In this way, it differs from direct exchange, in which there is coupl ...
.


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 many-electron 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 body-centered cubic lattice, and for a face-centered cubic lattice, The form of Eq. (14) corresponds identically to the
Ising model The Ising model () (or Lenz-Ising model or Ising-Lenz model), named after the physicists Ernst Ising and Wilhelm Lenz, is a mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables that represent m ...
of ferromagnetism except that in the Ising model, the dot product of the two spin angular momenta is replaced by the scalar product ''SijSji''. 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:


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 Heitler–London, or
valence bond In chemistry, valence bond (VB) theory is one of the two basic theories, along with molecular orbital (MO) theory, that were developed to use the methods of quantum mechanics to explain chemical bonding. It focuses on how the atomic orbitals of ...
(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''C ≈ 2⟨''J''⟩/3''k''B where ⟨''J''⟩ is the exchange interaction averaged over all sites). The Heisenberg model thus cannot explain the observed ferromagnetism in these materials. 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 \mu_S = - g \mu_ (S+1) (''g'' = 2.0023 ≈ 2) tends to overestimate the total spin-only magnetic moment per atom. For example, a net magnetic moment of 0.54 μB 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. By contrast, an isolated Ni atom (electron configuration = 3''d''84''s''2) in a cubic crystal field will have two unpaired electrons of the same spin (hence, \vec = 1) and would thus be expected to have in the localized electron model a total spin magnetic moment of \mu_S = 2.83 \mu_ (but the measured spin-only magnetic moment along one axis, the physical observable, will be given by \vec_S = g \mu_ \vec = 2 \mu_). 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.J. B. Goodenough: ''Magnetism and the Chemical Bond'', Interscience Publishers, New York, pp. 5–17 (1966). In the case of substances where both delocalized and localized electrons contribute to the magnetic properties (e.g. rare-earth systems), the Ruderman–Kittel–Kasuya–Yosida (RKKY) model is the currently accepted mechanism.


See also

* Double-exchange mechanism * Exchange symmetry *
Pauli exclusion principle In quantum mechanics, the Pauli exclusion principle states that two or more identical particles with half-integer spins (i.e. fermions) cannot occupy the same quantum state within a quantum system simultaneously. This principle was formulated ...
*
Slater determinant In quantum mechanics, a Slater determinant is an expression that describes the wave function of a multi-fermionic system. It satisfies anti-symmetry requirements, and consequently the Pauli principle, by changing sign upon exchange of two electr ...
*
Superexchange Superexchange, or Kramers–Anderson superexchange, is the strong (usually) antiferromagnetic coupling between two next-to-nearest neighbour cations through a non-magnetic anion. In this way, it differs from direct exchange, in which there is coupl ...
* Holstein–Herring method * Spin-exchange interaction * Multipolar exchange interaction * Antisymmetric exchange


Notes


References


External links


Exchange Mechanisms
in E. Pavarini, E. Koch, F. Anders, and M. Jarrell: Correlated Electrons: From Models to Materials, Jülich 2012,



{{DEFAULTSORT:Exchange Interaction Pauli exclusion principle Quantum chemistry