# Exchange interaction

The exchange interaction (better to speak of the exchange energy or more generally of the exchange term ) increases or decreases the energy of a physical system of several interacting identical particles compared to the value that would apply in the event that the particles are not identical but are distinguishable. The exchange energy is not caused by a special type of interaction in addition to the fundamental interactions , but rather by the special way in which several particles form a quantum mechanical state when they are identical particles. In the atomic shell z. B. the exchange energy is based mainly on the electrostatic repulsion between the electrons. It results in an additional energy contribution that varies in size depending on the state. B. is of great importance in atomic structure and chemical bonding and also plays a role in the formation of ferromagnetism .

Corresponding exchange terms must also be taken into account when calculating transition probabilities in reactions and cross- sections in impact processes if the particles are identical. Depending on the type of particle, the frequency of a 90 ° deflection can be increased fourfold or, on the contrary, completely suppressed in the event of collisions.

The exchange term is based on the fact that a 2-particle interaction in quantum mechanical formulas for energy or transition probability always produces an additional summand when it comes to two identical particles: While the interaction in question has the same effect in the first summand as it does with distinguishable particles, the new summand - the exchange term - looks as if the interaction had caused the two identical particles to exchange their places, which, because of their indistinguishability, represents the same physical state. The first contribution is also called the direct term (or direct integral ) and represents the direct quantum mechanical analogy to the result obtained according to classical physics for the respective interaction. The second contribution is also called the exchange integral and corresponds to the actual exchange energy, which has no classic counterpart and forms a purely quantum mechanical phenomenon.

## Relationship to Pauli principle, spin and symmetry of the wave function

The exchange energy (or the exchange term) is often associated with the Pauli principle , but is an independent independent phenomenon. The exchange term always occurs when the calculation of a certain interaction is carried out for a system with two (or more) identical particles. Only the sign of the exchange term depends on whether these particles are subject to the Pauli principle or not. (For examples of the exchange term with and without the Pauli principle, see the scattering of two identical particles below.)

The exchange term is also often associated with the spins of the particles concerned, because in some cases its effects appear as if they were caused by an additional interaction of its own between the spins. This applies e.g. B. in LS coupling of electrons in the atomic shell or in ferromagnetism (see below). This connection arises indirectly due to a concatenation of individual circumstances and without additional interaction.

The exchange term is based solely on the indistinguishability of identical particles, which is taken into account in a special way in quantum mechanics . If one calculates the consequence of a certain interaction (e.g. the potential energy due to the electrostatic repulsion) for two distinguishable particles in well-defined single-particle states, the result is the quantum mechanical analogue of the classical result, which is also referred to here as the direct term . However, for a pair of identical particles, it is not allowed to specify at any point which of the particles adopts which of the single-particle states that occur. As a result, only state vectors are possible for the system of several identical particles , which remain the same when two identical bosons are exchanged (symmetric wave function ) or change their sign when two identical fermions are exchanged (antisymmetric wave function). When calculating with such a wave function, the exchange term is automatically obtained in addition to the direct term, whereby the sign is positive or negative depending on the symmetry. The exchange term looks as if one were to calculate the transition amplitude for the process in which two distinguishable particles change simultaneously into the one-particle state of the other due to the interaction just considered. (Such a process is physically pointless for identical particles.)

## Formal considerations

### Two-particle wave function

The starting point is the wave functions for a single particle, with , be referred ..., where for every coordinate is (optionally also of the spins ). If there are several particles, their coordinates are distinguished by a lower index ( ). For a system of two particles, the functions represent those states in which each of the particles occupies a certain state : the particle with coordinate the state , that with coordinate the state . ${\ displaystyle \, \ psi (\ mathbf {x})}$${\ displaystyle \, \ phi (\ mathbf {x})}$${\ displaystyle \, \ mathbf {x}}$${\ displaystyle \ mathbf {x} _ {1}, \, \ mathbf {x} _ {2}, \, \ ldots}$${\ displaystyle \ varphi (\ mathbf {x} _ {1}, \ mathbf {x} _ {2}) = \ psi (\ mathbf {x} _ {1}) \, \ phi (\ mathbf {x} _ {2})}$${\ displaystyle \ mathbf {x} _ {1}}$${\ displaystyle \, \ psi}$${\ displaystyle \ mathbf {x} _ {2}}$${\ displaystyle \, \ phi}$

In the case of identical particles calls but the spin statistics theorem that the Zweiteilchenwellenfunktion over permutation of the particles symmetrically (at bosons ) or antisymmetrical (at fermions is) . Therefore the simple two- particle wave function has to be entangled and reads correctly: ${\ displaystyle \ Psi (\ mathbf {x} _ {1}, \ mathbf {x} _ {2}) = \ pm \ Psi (\ mathbf {x} _ {2}, \ mathbf {x} _ {1 })}$

${\ displaystyle \ Psi (\ mathbf {x} _ {1}, \ mathbf {x} _ {2}) \, = \, \ varphi (\ mathbf {x} _ {1}, \ mathbf {x} _ {2}) \ pm \ varphi (\ mathbf {x} _ {2}, \ mathbf {x} _ {1}) \; = \; \ psi (\ mathbf {x} _ {1}) \, \ phi (\ mathbf {x} _ {2}) \ pm \ phi (\ mathbf {x} _ {1}) \, \ psi (\ mathbf {x} _ {2})}$.

(The omitted here normalization factor is equal if and normalized and orthogonal are.) For the state of two identical particles can be said, that the two-particle states , are occupied by a respective particles, but no more, which of them which the states occupied. ${\ displaystyle 1 / {\ sqrt {2}}}$${\ displaystyle \, \ psi (\ mathbf {x})}$${\ displaystyle \, \ phi (\ mathbf {x})}$${\ displaystyle \ Psi (\ mathbf {x} _ {1}, \ mathbf {x} _ {2})}$${\ displaystyle \, \ psi (\ mathbf {x})}$${\ displaystyle \, \ phi (\ mathbf {x})}$

### Direct term and exchange term when calculating an expected value

When calculating the expected value of an operator , the decomposition of two summands occurs: ${\ displaystyle \ Psi}$

${\ displaystyle \ langle \, \ Psi (\ mathbf {x} _ {1}, \ mathbf {x} _ {2}) \, | {\ hat {O}} | \, \ Psi (\ mathbf {x } _ {1}, \ mathbf {x} _ {2}) \, \ rangle = D \ pm A}$
• the "direct term":    ${\ displaystyle D = \ langle \, \ psi (\ mathbf {x} _ {1}) \, \ phi (\ mathbf {x} _ {2}) \, | {\ hat {O}} | \, \ psi (\ mathbf {x} _ {1}) \, \ phi (\ mathbf {x} _ {2}) \, \ rangle}$
• the "exchange term":  ${\ displaystyle A = \ langle \, \ phi (\ mathbf {x} _ {1}) \, \ psi (\ mathbf {x} _ {2}) \, | {\ hat {O}} | \, \ psi (\ mathbf {x} _ {1}) \, \ phi (\ mathbf {x} _ {2}) \, \ rangle}$

With each of the simple product functions or from above, only the direct term would have resulted. If the operator describes the potential energy of one particle in the field of the other (e.g. the Coulomb repulsion between two electrons), the direct term corresponds exactly to the classically expected result for the potential energy of one charge cloud in the field of the other. The exchange term only comes about through the entanglement and has the form of the transition amplitude for the process in which the two particles exchange their states through their interaction (cf. the matrix element in Fermi's Golden Rule ). The exchange term occurs with a positive sign for bosons and a negative one for fermions. ${\ displaystyle \, \ varphi (\ mathbf {x} _ {1}, \ mathbf {x} _ {2})}$${\ displaystyle \, \ varphi (\ mathbf {x} _ {2}, \ mathbf {x} _ {1})}$${\ displaystyle {\ hat {O}}}$${\ displaystyle {\ hat {O}}}$

### Direct term and exchange term when calculating a transition amplitude

A transition leads from a two-particle state to another, which is also formed of two single-particle states: . An example is the collision of two particles. In the center of gravity system (i.e. viewed from the center of gravity at rest), they fly towards each other from opposite directions in the initial state . In the final state they fly apart, also in the opposite direction, but along a different axis, which is given by the observed deflection angle. ${\ displaystyle \ Psi (\ mathbf {x} _ {1}, \ mathbf {x} _ {2}) \ rightarrow \ Psi '(\ mathbf {x} _ {1}, \ mathbf {x} _ {2 })}$${\ displaystyle \, \ Psi (\ mathbf {x} _ {1}, \ mathbf {x} _ {2})}$${\ displaystyle \ Psi '(\ mathbf {x} _ {1}, \ mathbf {x} _ {2}) \, = \; \ psi' (\ mathbf {x} _ {1}) \, \ phi '(\ mathbf {x} _ {2}) \ pm \ phi' (\ mathbf {x} _ {1}) \, \ psi '(\ mathbf {x} _ {2})}$${\ displaystyle \, \ Psi (\ mathbf {x} _ {1}, \ mathbf {x} _ {2})}$${\ displaystyle \, \ Psi '(\ mathbf {x} _ {1}, \ mathbf {x} _ {2})}$

If stands for the interaction of the two particles with each other, the transition probability is formed from the matrix element (see Fermi's golden rule ). The matrix element (the transition amplitude) consists of two amplitudes, which are added coherently for bosons and subtracted for fermions before the square of the magnitude is formed to determine the transition probability (or the differential cross section ) : ${\ displaystyle {\ hat {O}}}$

${\ displaystyle \ langle \, \ Psi '(\ mathbf {x} _ {1}, \ mathbf {x} _ {2}) \, | {\ hat {O}} | \, \ Psi (\ mathbf { x} _ {1}, \ mathbf {x} _ {2}) \, \ rangle = D \ pm A}$
• "Direct term":    ${\ displaystyle D = \ langle \, \ psi '(\ mathbf {x} _ {1}) \, \ phi' (\ mathbf {x} _ {2}) \, | {\ hat {O}} | \, \ psi (\ mathbf {x} _ {1}) \, \ phi (\ mathbf {x} _ {2}) \, \ rangle}$
• "Exchange term":  ${\ displaystyle A = \ langle \, \ phi '(\ mathbf {x} _ {1}) \, \ psi' (\ mathbf {x} _ {2}) \, | {\ hat {O}} | \, \ psi (\ mathbf {x} _ {1}) \, \ phi (\ mathbf {x} _ {2}) \, \ rangle}$

The index of the coordinates shows that the direct term describes the process where one particle passes from in and the other from in at the same time . The exchange term belongs to the process with exchanged end states: and . In the case of distinguishable particles, these would be alternative, mutually exclusive processes that can be measured individually in a suitable experiment. In the case of identical particles, however, because of their indistinguishability, no measurement can in principle differentiate whether the particles made the direct process or the exchange process in the experiment. For identical particles, these two paths do not even represent real (mutually exclusive) alternatives, because their amplitudes interfere with each other. The summation (or subtraction) of the two amplitudes before the formation of the square of the absolute value is a coherent superposition that requires the simultaneous presence of both values. ${\ displaystyle \, \ psi}$${\ displaystyle \, \ psi '}$${\ displaystyle \, \ phi}$${\ displaystyle \, \ phi '}$${\ displaystyle \ psi \ rightarrow \ phi '}$${\ displaystyle \ phi \ rightarrow \ psi '}$

## Exchange energy of the electrons in the helium atom

### Wave function for two identical particles

For the two electrons, only the Coulomb interactions with the nucleus and with each other are taken into account, while the energies or forces associated with the spin are neglected due to their insignificance. Then the spatial behavior of the electrons is independent of the behavior of their spins, and the two-particle wave functions can be assumed in the product form . It contains the complete set of coordinates of an electron as the position coordinate and the coordinate for the spin alignment . is the spatial wave function, the spin function for the two particles. Since electrons are fermions, the overall condition must be antisymmetric against Teilchenvertauschung: . This is only possible if there is either antisymmetric and symmetric or vice versa, mixed forms cannot exist. ${\ displaystyle \, \ Psi (\ mathbf {x} _ {1}, \, \ mathbf {x} _ {2}) = \ Phi ({\ vec {r}} _ {1}, {\ vec { r}} _ {2}) \ cdot \ chi _ {S} (s_ {z1}, s_ {z2})}$${\ displaystyle \, \ mathbf {x} {\ mathord {=}} ({\ vec {r}}, s_ {z})}$${\ displaystyle {\ vec {r}} = (x, y, z)}$${\ displaystyle s_ {z} = \ pm {\ frac {1} {2}}}$${\ displaystyle \ Phi ({\ vec {r}} _ {1}, {\ vec {r}} _ {2})}$${\ displaystyle \ chi _ {S} (s_ {z1}, s_ {z2})}$${\ displaystyle \, \ Psi (\ mathbf {x} _ {1}, \, \ mathbf {x} _ {2}) = - \ Psi (\ mathbf {x} _ {2}, \, \ mathbf { x} _ {1})}$${\ displaystyle \ Phi ({\ vec {r}} _ {1}, {\ vec {r}} _ {2})}$${\ displaystyle \, \ chi _ {S} (s_ {z1}, s_ {z2})}$

The position function is built up from two orbitals as they are formed in the Coulomb field of the nucleus. For distinguishable particles it would be easy to assume, because it is certain that particle # 1 is in the orbital and particle # 2 in . However, so that either one or the other of the symmetries (required due to the indistinguishability of the particles) arise, the entire position function must have the following form: ${\ displaystyle \ Phi ({\ vec {r}} _ {1}, {\ vec {r}} _ {2})}$ ${\ displaystyle \, \ varphi _ {A} ({\ vec {r}}), \ varphi _ {B} ({\ vec {r}})}$${\ displaystyle \, \ Phi ({\ vec {r}} _ {1}, {\ vec {r}} _ {2}) \, = \, \ varphi _ {A} ({\ vec {r} } _ {1}) \, \ varphi _ {B} ({\ vec {r}} _ {2})}$${\ displaystyle \ varphi _ {A}}$${\ displaystyle \ varphi _ {B}}$

${\ displaystyle \, \ Phi ({\ vec {r}} _ {1}, {\ vec {r}} _ {2}) \, = \, \ varphi _ {A} ({\ vec {r} } _ {1}) \, \ varphi _ {B} ({\ vec {r}} _ {2}) \, \ pm \, \ varphi _ {B} ({\ vec {r}} _ {1 }) \, \ varphi _ {A} ({\ vec {r}} _ {2})}$

(Note: A normalization factor is omitted here and in the following for better clarity. For the antisymmetric case, the choice must be excluded ( Pauli principle )). ${\ displaystyle {\ sqrt {\ tfrac {1} {2}}}}$${\ displaystyle \, \ varphi _ {A} = \ varphi _ {B}}$

### Splitting depending on the symmetry of the spatial wave function

Without interaction between the two electrons, He atoms with symmetrical and antisymmetrical spatial states would have the same energy; they would form a degenerate energy level of the atom. The only interaction between the electrons is the Coulomb potential (which does not affect the spins). This shifts the energy of the state by the expected value . The factor is 1, the factor , as mentioned above, consists of two terms, ${\ displaystyle {\ hat {O}} = e ^ {2} / (4 \ pi \ varepsilon _ {0} | {\ vec {r}} _ {1} - {\ vec {r}} _ {2 } |)}$${\ displaystyle \ langle \ Psi | {\ hat {O}} | \ Psi \ rangle = \ langle \ Phi | {\ hat {O}} | \ Phi \ rangle \ langle \ chi | {\ hat {1}} | \ chi \ rangle}$${\ displaystyle \ langle \ chi | {\ hat {1}} | \ chi \ rangle}$${\ displaystyle \ langle \ Phi | {\ hat {O}} | \ Phi \ rangle = D \ pm A}$

• first the "direct term" , as well as${\ displaystyle D = \ langle \, \ varphi _ {A} ({\ vec {r}} _ {1}) \, \ varphi _ {B} ({\ vec {r}} _ {2}) \ , | {\ hat {O}} | \, \ varphi _ {A} ({\ vec {r}} _ {1}) \, \ varphi _ {B} ({\ vec {r}} _ {2 }) \, \ rangle = \ iint d ^ {3} r_ {1} \, d ^ {3} r_ {2} \, {\ frac {e ^ {2}} {4 \ pi \ varepsilon _ {0 }}} \, {\ frac {| \ varphi _ {A} ({\ vec {r}} _ {1}) | ^ {2} \, | \ varphi _ {B} ({\ vec {r} } _ {2}) | ^ {2}} {| {\ vec {r}} _ {1} - {\ vec {r}} _ {2} |}}}$
• second, the "exchange term" ${\ displaystyle A = \ langle \, \ varphi _ {B} ({\ vec {r}} _ {1}) \, \ varphi _ {A} ({\ vec {r}} _ {2}) \ , | {\ hat {O}} | \, \ varphi _ {A} ({\ vec {r}} _ {1}) \, \ varphi _ {B} ({\ vec {r}} _ {2 }) \, \ rangle = \ iint d ^ {3} r_ {1} \, d ^ {3} r_ {2} \, {\ frac {e ^ {2}} {4 \ pi \ varepsilon _ {0 }}} \, {\ frac {\ varphi _ {A} ({\ vec {r}} _ {1}) \, \ varphi _ {B} ^ {*} ({\ vec {r}} _ { 1}) \, \ varphi _ {B} ({\ vec {r}} _ {2}) \, \ varphi _ {A} ^ {*} ({\ vec {r}} _ {2})} {| {\ vec {r}} _ {1} - {\ vec {r}} _ {2} |}}}$

The direct term gives the result, which corresponds to the classic idea of ​​two repelling charge clouds with spatial densities . That would be the end result for two distinguishable particles. Because of the indistinguishability of the two electrons, however, the exchange term is added, depending on the symmetry of the position function with a positive or negative sign. It is true for the exchange term if the orbitals and do not overlap (which is why the exchange interaction resulting from the Coulomb repulsion does not generally need to be considered when electrons are spatially widely separated). For orbitals in the same atom, however, always applies . ( When applied to the 3d orbitals of certain compounds, for example, this results in what is known as Hund's rule , a kind of “intra-atomic ferromagnetism”, more precisely: paramagnetism .) As a result, the degenerate energy level of the atom is split into two levels, although that of the orbitals occupied by the particles have remained the same in both levels. The cheaper, i.e. lower, energy belongs to the antisymmetric spatial state, the higher to the symmetric. The qualitative cause is that in the antisymmetric spatial state the two electrons are not to be found in the same place ( ) (because it follows from ). The Coulomb repulsion is therefore reduced, which is energetically favorable. Specifically, the rule stated says that it is energetically more favorable by an amount of size 2A , called " Hund's Rule exchange energy ", to place two electrons in different d orbitals with parallel spin instead of them in one and the same d -Orbital, whereby its spin function would then no longer be symmetrical (“parallel spins”), but must be antisymmetrical. Since there are a total of five different (pairwise orthogonal) d orbitals, the maximum intra-atomic magnetism of the 3d ions for Mn ++ is reached with a magnetic moment of five Bohr units, while Cr ++ and Fe ++ four units have. This makes use of the fact that manganese ++ has to accommodate five, chromium ++ four, iron ++ but six 3d electrons. ${\ displaystyle \, | \ varphi _ {A} | ^ {2}, \, | \ varphi _ {B} | ^ {2}}$${\ displaystyle \, \ Phi}$${\ displaystyle \, A = 0}$${\ displaystyle \, \ varphi _ {A} ({\ vec {r}})}$${\ displaystyle \, \ varphi _ {B} ({\ vec {r}})}$${\ displaystyle \, A> 0}$${\ displaystyle D {\ mathord {-}} A,}$${\ displaystyle D {\ mathord {+}} A,}$${\ displaystyle {\ vec {r}} _ {2} = {\ vec {r}} _ {1} = {\ vec {r}}}$${\ displaystyle \, \ Phi ({\ vec {r}} _ {2}, {\ vec {r}} _ {1}) {\ mathord {=}} - \ Phi ({\ vec {r}} _ {1}, {\ vec {r}} _ {2})}$${\ displaystyle \, \ Phi ({\ vec {r}}, {\ vec {r}}) {\ mathord {=}} 0}$

### Relationship to spin

In the Hamilton operator of the atom, as far as considered here, the spins do not appear. Nevertheless, the two levels that have formed through the splitting due to the electron-electron interaction are characterized by different quantum numbers for the total spin. The reason is that a symmetric or antisymmetric position function always has an oppositely symmetric spin function , and that in the case of two particles with spin, depending on the symmetry, this always has a defined total spin, either or (see two identical particles with spin 1 / 2 ). According to their degree of degeneracy with regard to the spin orientation, states with singlet states are called those with triplet states. For the energy levels of the He atom, it follows that the electrons, if they are in two (different) one-particle orbitals, each form a singlet level and a triplet level, whereby the triplet level (i.e. symmetrical when the spins are swapped, antisymmetrical in position) is lower than the corresponding singlet state. ${\ displaystyle \, S}$${\ displaystyle \ Phi}$${\ displaystyle \ chi}$${\ displaystyle {\ frac {1} {2}}}$${\ displaystyle \, S = 0}$${\ displaystyle \, S = 1}$${\ displaystyle \, S = 0}$${\ displaystyle \, S = 1}$

A special case arises when both electrons occupy the same orbital , because only the symmetrical spatial wave function exists for this. A standard example is the configuration 1s 2 of the ground state in the helium atom. The total spin is too fixed, a split does not occur. ${\ displaystyle \, \ varphi _ {A} {\ mathord {=}} \ varphi _ {B}}$${\ displaystyle \, S {\ mathord {=}} 0}$

Overall, the electrostatic repulsion by means of the quantum mechanical exchange interaction leads to the paradoxical consequence that the spin quantum number has a determining influence on the level scheme without the (magnetic) interaction associated with the electron spin being considered at all. Historically, these connections were discovered by Werner Heisenberg in this specific example in 1926 .

### Generalization for diatomic molecules

The sign with which the exchange term is used in energy calculations in diatomic molecules depends on the circumstances, e.g. For example, with electrons - these are fermions - a symmetrical position function of a twofold set of vectors results in a kind of singlet ground state (antisymmetrical spin function, symmetrical position function, homeopolar σ-bond). This state is generally particularly energetically favored (diatomic molecules are mostly diamagnetic ), while the alternative "triplet state" (parallel position of the atomic spins; π bond) energetically prefers an antisymmetric position function, which occurs with paramagnetic molecules, e.g. B. with O 2 . (In the latter case, for example, it is a matter of exchanging two electrons, one of which is in the p x orbital, the other in the p y orbital.)

In the case of bosons , on the other hand, a symmetric or antisymmetric position function also includes a symmetric (or antisymmetric) spin function. There, too, both signs compete in energy calculations. a. the so-called overlap integral of the wave functions is important.

## Exchange term for the scattering of identical particles

In the case of a collision ( see above ) of two identical particles, the direct and the exchange term only differ in that if one describes the deflection by the angle , then the other describes the deflection by the angle (always in the center of gravity system). In particular, both terms are the same when deflected by 90 °. In the event that they are subtracted from one another (i.e. with an antisymmetric spatial wave function), the transition amplitude is therefore zero, i.e. H. Deflection by 90 ° is completely impossible in this case. In the other case (symmetrical spatial wave function) the transition amplitude is exactly doubled, i.e. the frequency of the deflections by 90 ° (due to the square of the magnitude) quadrupled compared to the case of non-identical particles. These surprising consequences of the quantum mechanical formulas were first predicted in 1930 by Nevill Mott and confirmed experimentally shortly afterwards. It is noteworthy that these deviations from the classically expected behavior result solely from the indistinguishability of the two collision partners and are completely independent of all further details of the process under investigation (such as particle type, force law, energy, ...). ${\ displaystyle \ theta}$${\ displaystyle \ pi - \ theta}$

## Exchange energy and magnetic order

The exchange energy resulting from the electrostatic repulsion of the electrons only acts between two electrons with an overlapping probability of their presence. In addition to electrons from the same atom, these can also be electrons from neighboring atoms or, in metals, even delocalized electrons of the conduction band . The consequences belongs in the chemical , the steric hindrance , in the solid-state physics , the long-range magnetic order in the magnetic materials (see, for example Heisenberg model , Ising model ).

Since the split levels of two electrons each - as described above for the He atom - have different total spins, the same split can also be generated using a suitable factor in front of the operator . This factor is commonly abbreviated to and referred to as the exchange interaction or energy. This term also has a second meaning here: one writes the operator , with , and applies it to a state in which one electron has oriented the spin upwards and the other downwards. This state is converted into the state with reversed spin orientations by the ascending and descending operators. The process thus describes a spin exchange. ${\ displaystyle \, {\ hat {\ vec {S}}} ^ {2} = ({\ hat {\ vec {s}}} _ {1} + {\ hat {\ vec {s}}} _ {2}) ^ {2}}$${\ displaystyle {\ hat {\ vec {s}}} _ {1} \ cdot {\ hat {\ vec {s}}} _ {2} = {\ frac {1} {2}} {\ hat { \ vec {S}}} ^ {2} - {\ hat {\ vec {s}}} _ {1} ^ {2} - {\ hat {\ vec {s}}} _ {2} ^ {2 }}$${\ displaystyle J_ {1,2}}$${\ displaystyle {\ hat {\ vec {s}}} _ {1} \ cdot {\ hat {\ vec {s}}} _ {2} = {\ frac {1} {2}} ({\ hat {s}} _ {1 +} \ cdot {\ hat {s}} _ {2 -} + {\ hat {s}} _ {1 -} \ cdot {\ hat {s}} _ {2+} ) + {\ hat {s}} _ {1z} \ cdot {\ hat {s}} _ {2z}}$${\ displaystyle s _ {\ pm} = s_ {x} \ pm is_ {y}}$${\ displaystyle {\ hat {s}} _ {\ pm}}$

In the case of atoms with a magnetic dipole moment , which sit on the lattice positions of a solid, the effective exchange interaction of the electrons of different atoms depends on the atomic distance and can also change the sign. If the parallel position is energetically favorable against the magnetic forces between the neighboring dipoles, a ferromagnet is formed . If the anti-parallel position is favorable, an anti-ferromagnet forms . In metals, the electrons of the conduction band play a key role (e.g. iron , cobalt and nickel : ferromagnetic, manganese : antiferromagnetic). With alternating signs, magnetic spiral structures can arise (e.g. chrome , terbium ). With “competing signs” of the (effective) exchange energy and a disordered arrangement of the atoms (e.g. Eu x Sr 1-x S alloys, with 0.13 <x <0.65), one obtains spin glasses .