In
chemistry
Chemistry is the science, scientific study of the properties and behavior of matter. It is a natural science that covers the Chemical element, elements that make up matter to the chemical compound, compounds made of atoms, molecules and ions ...
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 r ...
, 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 fi ...
.
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 ...
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
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, ...
, 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 an ...
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) describe ...
.
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 ...
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 breakthrough paper. In the subsequent series ...
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 force In physics the term exchange force has been used to describe two distinct concepts which should not be confused.
Exchange of force carriers in particle physics
The preferred meaning of exchange force is in particle physics, where it denotes a forc ...
s 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 fi ...
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 ...
, 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 n ...
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.
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 p ...
, 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 spin 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 ...
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 i ...
; 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 kn ...
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
for the first electron and
for the second electron. We assume that
and
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
and the pertubation
is:
:
where
and
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
and
respectively. A variational calculation yields similar results.
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
in an electric field created by electron-two distributed over the space with the probability density
,
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-orbital
In atomic theory and quantum mechanics, an atomic orbital is a function describing the location and wave-like behavior of an electron in an atom. This function can be used to calculate the probability of finding any electron of an atom in any spe ...
s, 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 elect ...
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,
is
. The characteristic values of the matrices''
''and''
''are each''
''and''
. ''The characteristic values of the scalar product''
''are''
''and''
, ''corresponding to both the spin-singlet'' (''S'' = 0) ''and spin-triplet'' (''S'' = 1) ''states, respectively.
: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/spin-singlet 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/spin-triplet 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 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 ''Φ
a'' and ''Φ
b'' can be written in terms of their spin momenta
and
. 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
= 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
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 ...
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 ''J
ab'' is negative, the interaction favors electrons with antiparallel spins, potentially causing
antiferromagnetism
In materials that exhibit antiferromagnetism, the magnetic moments of atoms or molecules, usually related to the spins of electrons, align in a regular pattern with neighboring spins (on different sublattices) pointing in opposite directions. ...
. The sign of ''J''
ab is essentially determined by the relative sizes of ''J''
ex and the product of
. 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 magnet ...
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
The dihydrogen cation or hydrogen molecular ion is a cation (positive ion) with formula . It consists of two hydrogen nuclei (protons) sharing a single electron. It is the simplest molecular ion.
The ion can be formed from the ionization of a n ...
(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.
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
,
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 ...
of ferromagnetism except that in the Ising model, the dot product of the two spin angular momenta is replaced by the scalar product ''S
ijS
ji''. 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
The Stoner criterion is a condition to be fulfilled for the ferromagnetic order to arise in a simplified model of a solid. It is named after Edmund Clifton Stoner.
Stoner model of ferromagnetism
Ferromagnetism ultimately stems from Pauli exc ...
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
(''g'' = 2.0023 ≈ 2) tends to overestimate the total
spin-only 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. 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,
) and would thus be expected to have in the localized electron model a total spin magnetic moment of
(but the measured spin-only magnetic moment along one axis, the physical observable, will be given by
). 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 The double-exchange mechanism is a type of a magnetic exchange that may arise between ions in different oxidation states. First proposed by Clarence Zener, this theory predicts the relative ease with which an electron may be exchanged between two s ...
*
Exchange symmetry
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, ...
*
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 elect ...
*
Superexchange
*
Holstein–Herring method The Holstein– Herring method, also called the surface Integral method, or Smirnov's method is an effective means of getting the exchange energy splittings of asymptotically degenerate energy states in molecular systems. Although the exchange energ ...
*
Spin-exchange interaction
*
Multipolar exchange interaction
*
Antisymmetric exchange
In Physics, antisymmetric exchange, also known as the Dzyaloshinskii–Moriya interaction (DMI), is a contribution to the total magnetic exchange interaction between two neighboring magnetic spins, \mathbf_i and \mathbf_j . Quantitatively, it i ...
Notes
References
External links
Exchange Mechanismsin 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