Exchange interaction
The exchange interaction (better one speaks only of the exchange energy, or more generally 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 distinguishable. The exchange energy corresponds to no additional type of interaction in addition to the fundamental interaction, but it is based, for example, atoms mainly on the electrostatic repulsion between the electrons. It results in a different size depending on the state of additional energy contribution, for example, in atomic structure, and the conclusion of the ferromagnetism is of great importance, and for the chemical bond plays a role.
The exchange energy based on the fact that in the case of two identical particles interaction in quantum mechanical formulas always a second term brings forth: While the relevant interaction in the first summand is just as affecting as distinguishable particles, sees the new summand - the exchange term - look as whether the interaction would have caused the two identical particles to interchange their places, but this is the same physical condition due to their indistinguishability. The first contribution is also called direct term ( or direct integral) and is the direct quantum-mechanical analogy is to the result obtained after classical physics for the respective interaction. The second contribution is also called exchange integral and corresponds to the actual exchange of energy, which has no classical counterpart and is a purely quantum mechanical phenomenon.
Appropriate exchange terms must also be considered in the calculation of transition probabilities in reactions and cross sections in collision processes, if it is identical particles. For example, may be enhanced by 90 ° depending on the particle four-fold or completely inhibited the distraction.
- 3.1 wave function for two identical particles
- 3.2 splitting depending on the symmetry of the spatial wave function
- 3.3 Relationship to the spin
- 3.4 Generalization for diatomic molecules
Relation to Pauli principle, spin and symmetry of the wave function
The energy exchange (or the exchange term ) is often associated with the Pauli principle, but is a more general independent thereof or independent phenomenon. The exchange term occurs as a result of a specific interaction for ever, if the calculation for a system with two (or more) identical particles is to apply. Whether the particles are subject to the Pauli principle or not, has an effect only on the sign of the exchange term. ( For examples of the exchange term with and without Pauli principle see the scattering of two identical particles below).
With the spin of that particles of the exchange term is also often associated. Here spinless particles ( see, eg, below the exchange term in the scattering of alpha particles ) formally assigned to the integer zero spin ( particles with integer spin, including zero, generally form the so-called bosons, which include the Higgs boson is one that also has zero spin ), while fermions have half-integer spin, such as electrons. If this relationship exists in certain areas, such as in the LS- coupling in the atomic shell, then because of a series of individual circumstances. Then, however, one can parameterize the effects sometimes so that the exchange energy appears as a measure of the strength of effective interaction between spins ( see below the application in ferromagnetism ).
The exchange term is based solely on the indistinguishability of identical particles. While you can assign a system of distinguishable particles, at least in its basic states without quantum entanglement, each particle individually a well-defined single-particle means indistinguishability in quantum mechanics that can be set at any point, which occupies the particles which the occurring single-particle states. As a result, only state vectors are possible for the system of several identical particles, which remain the same when interchanging two identical bosons (symmetric wave function ) or when interchanging two identical fermions change sign ( antisymmetric wave function ). If one calculates the result of a specific interaction (eg, the potential energy due to the electrostatic repulsion ), the result for a state of distinguishable particles, the direct term, ie the quantum mechanical analogue of the classical result. But if we take the symmetric or antisymmetric wave function is obtained in addition to the exchange term. He has an appearance as if you consider the transition amplitude for a process to change over to the result of the just considered interaction both particles simultaneously in the single-particle state of the other. ( Such a process would be physically meaningless for identical particles, therefore it exists only for distinguishable particles. )
Formal considerations
Zweiteilchenwellenfunktion
The starting point are the wave functions for a single particle to be with, called ..., where stands for all coordinates ( optionally also of the spins). For multiple particles whose coordinates are distinguished by a subscript (). For a system of two particles, the functions those conditions show in which each of the particles occupying a certain state: the particles with the coordinate state, the coordinate of the condition.
In the case of identical particles but calls the spin- statistics theorem, that the Zweiteilchenwellenfunktion against permutation of the particles is symmetric (with bosons ) or antisymmetric (for fermions ). Therefore must the simple Zweiteilchenwellenfunktion are entangled and is correct:
( The omitted here normalization factor has the value if and normalized and orthogonal. ) For the state of two identical particles one can still say that the two single-particle states are occupied by each a particle, but not which of them which the states occupied.
Direct term and exchange term when calculating an expectation value
In calculating the expectation value of an operator occur because of the decomposition of two summand:
- The "direct term ":
- The " exchange term ":
Each of the simple product functions or from above, only the direct term would have resulted. If the operator is the potential energy of a particle in the field of the other describes (eg, the Coulomb repulsion between two electrons) corresponds to the direct expression just the classically expected result for the potential energy of a charge cloud in the field of the other. The exchange term comes only through the entanglement is concluded and if the shape of the transition amplitude for the process in which the two particles exchange their states by their interaction (see the matrix element in Fermi's Golden Rule). The exchange term occurs for bosons with positive sign in fermions with negative.
Direct term and exchange term in calculation of a transition amplitude
A transition leads from a two-particle state to another, which is also formed of two single-particle states. One example is the collision of two particles. In the center of mass system (ie viewed from the stationary center of gravity off) they fly in the initial state from the opposite direction to each other. In the final state they fly apart, also in the opposite direction, but along a different axis, which is determined by the observed deflection angle.
Stands for the interaction of the two particles together, the transition probability is formed from the matrix element (see Fermi's Golden Rule ). The matrix element ( the transfer amplitude) consists of two amplitude, which adds coherently at bosons are subtracted at fermions before for determining the transition probability (or the differential cross -section), the square sum is formed:
- " Direct term ":
- " Exchange term ":
On the index of the coordinates can be read that the direct term describes the process where the same passes a particle of in and the other from to. The exchange term is part of the process with reversed end states: and. For distinguishable particles, the alternative, mutually exclusive processes that can be measured individually in a suitable experiment would be. For identical particles but can be distinguished by any measurement due to their indistinguishability principle, whether the particles have made the direct process or the exchange process in the experiment. For identical particles, these two ways are not even real ( mutually exclusive ) alternatives is because their amplitudes interfere with each other. Summing (or subtracting ) the two amplitudes prior to formation of the square value is a coherent superposition of which requires the simultaneous presence of both values .
Exchange energy of electrons in the helium atom
Wave function for two identical particles
For the two electrons only the Coulomb interaction with the nucleus and with each other are taken into account, while related to the spin energies or forces are neglected due to their insignificance. Then the spatial behavior of the electrons is independent of the behavior of their spins, and the Zweiteilchenwellenfunktionen can be recognized in the product form. This is the full set of coordinates of an electron with a spatial coordinate and a coordinate for the spin orientation. is the local wave function, the spin function for the two particles. Since electrons are fermions, the overall state must be antisymmetric against Teilchenvertauschung. This is only possible if either is anti-symmetric and symmetric or vice versa, mixed forms can not exist.
The local feature is composed of two orbitals, how they form in the nuclear Coulomb field. For distinguishable particles would be easy to use, because it is clear that particle # 1 is located in the orbital and particle # 2 in. This, however, either the one or the other of the ( required because of the indistinguishability of the particles ) symmetries arises, the entire local function must have the following form:
(Note: A normalization factor is here and hereinafter omitted for better clarity for the antisymmetric case, the election must be excluded ( Pauli principle). ).
Splitting depending on the symmetry of the spatial wave function
Without interaction of the two electrons with each other would He atoms with symmetric and antisymmetrischem location state the same energy; they would form a degenerate energy level of the atom. As interaction between the electrons alone, the Coulomb potential is now recognized (that does not act on the spins ). Thus, the energy of the state shifts to the expected value. The factor is 1, the factor is, as mentioned above, consists of two terms,
- First, the " direct term ", and
- Secondly, the " exchange term "
The direct term is the result back, which corresponds to the classical notion of two repulsive charge clouds with the spatial densities. That would already be the final result for two distinguishable particles. Because of the indistinguishability of the two electrons, however, is in addition to the exchange term, depending on the symmetry of the spatial wave function with positive or negative sign. It is true for the exchange term, if the orbitals and do not overlap (which is why you do not need to look at the object created by the Coulomb repulsion exchange interaction in spatially widely separated electrons ia). But is global for orbitals in the same atom. ( This is, for example, when applied to the 3d orbitals of certain compounds, the so-called Hund's rule, a type of "inner nuclear Ferromagnetism ", or more precisely. Paramagnetism ) As a result of the degenerate energy levels of the atom is split into two levels, although from the particles occupied orbitals have remained the same in both levels. The more favorable, ie lower energy part of the antisymmetric spatial state, the higher, the symmetrical. Qualitative cause is that the antisymmetric spatial state, the two electrons are not to be found in the same place () ( as follows ). The Coulomb repulsion is therefore reduced, which is energetically favorable. Specifically, stating the specified rule that it called by an amount the size 2A, in English " Dog 's Rule exchange energy", is energetically more favorable to place two electrons with parallel spin in different d- orbitals rather than in one-and- same d orbital to accommodate, in which case their spin function would no longer symmetric ( " parallel spins " ), but must be antisymmetric. Since there are a total of five different (pairwise orthogonal ) d orbitals are, the maximum intra- atomic magnetism is achieved in the 3d - ions for Mn with a magnetic moment of five Bohr units, while having Cr and Fe four units. This exploits that manganese five, four chrome , iron but six 3d electrons must accommodate.
Relationship to the spin
The Hamiltonian of the atom, to the extent considered here, the spins do not occur. Nevertheless, the two levels, which have been formed by the splitting due to the electron -electron interaction, characterized by different quantum numbers for the total spin. The reason is that for a symmetric or antisymmetric spatial wave function always heard an opposite symmetric spin function, and that this spin, depending on the symmetry always has a defined total spin in the case of two particles, either or (see Two identical particles with spin 1 / 2). After her degree of degeneracy with respect to the spin orientation states with hot singlet, with the triplet states. For the energy levels of the He atom follows that the electrons when they Einteilchenorbitalen sit in two ( mutually different ), each form a singlet and a triplet level, the triplet level (that is symmetric under interchange of the spins, antisymmetric in the place) is lower than the corresponding singlet.
A special case arises when both electrons occupy the same orbital, because this only exists the symmetric spatial wave function. A standard example is the configuration 1s2 ground state in the helium atom. The total spin is to set a splitting does not occur.
Overall, the results by means of electrostatic repulsion of the quantum mechanical exchange interaction to the paradoxical result is that the spin quantum number, is replaced by a decisive influence on the level diagram, without the (magnetic) interaction associated with the electron spin is considered at all. Historically, in 1926 these relationships by Werner Heisenberg were discovered in this concrete example.
Generalization for diatomic molecules
Which sign of the exchange term in energy calculations in diatomic molecules is received depends on the circumstances, for example, results in electron - these are fermions - a symmetric spatial wave function of a dual vector set some kind singlet ground state ( antisymmetric spin function, symmetric spatial wave function homopolar, σ bond). This condition is generally energetically more favored ( diatomic molecules are generally diamagnetic ), while the alternate " triplet state " ( the parallel position of the nuclear spins; π bond), an anti-symmetric position function energetically favored, which occurs with paramagnetic molecules, such as O2. ( In the latter case, it is z, as the exchange of two electrons, one of which is the other is in px orbital, the py orbital. )
For bosons, however, belongs to a symmetric or antisymmetric spatial wave function and a symmetric (or antisymmetric ) spin function. But even there it compete with energy calculations both signs, which, inter alia, arrives at the so-called overlap of the wave functions.
Exchange term in the scattering of identical particles
For an impact ( so ) of two identical particles, the direct and the exchange term only in that if one describes the deflection by the angle, then the other, the deflection by the angle (always in the center of mass system ). In particular, both terms are the same for deflection by 90 °. In the case that they are subtracted from each other ( ie when antisymmetric spatial wave function ), the transition amplitude is therefore zero, ie deflection by 90 ° is completely impossible in this case. In the other case (symmetric spatial wave function ) is the transition amplitude exactly double the frequency of the deflections of 90 ° thus quadrupled (because of the magnitude squared ) compared to the case of non-identical particles. These surprising consequences of quantum mechanical formulas were first predicted in 1930 by Nevill Mott and confirmed shortly afterwards experimentally. It is remarkable that these differences stemming only from classically expected behavior of the indistinguishability of the two collision partners and all other details of the process under investigation (such as particle type, energy law, energy, ... ) are completely independent. It is also maintained during the transition to the classical limit.
Exchange energy and magnetic order
The energy exchange which results from the electrostatic repulsion of the electrons is only between two electrons with overlapping probability. This may be in addition to electrons of the same atom but also electrons of neighboring atoms or, in metals, even delocalized electrons of the conduction band. One of the consequences is one in chemistry, steric hindrance, in solid state physics, the long-range magnetic order in magnetic materials ( see, eg, Heisenberg model, Ising model ).
Since the split levels of two electrons each - as described above for the He atom - have different total spin, one can generate the same splitting and using a suitable factor in front of the operator. This factor is usually with abbreviated and referred to as the exchange interaction or energy. This name has also a second sense: to write the operator with, and applies it to a state in which an electron has aligned the spin up and and the other down. This condition is converted by the up and lowering operators in the state with reversed spin orientations. The process thus describes a spin exchange.
When the atoms with magnetic dipole moment, sitting on the lattice sites of a solid, the effective exchange interaction of the electrons of different atoms from the atomic distance is dependent and can also switch the sign.
If, contrary to the magnetic forces between the adjacent dipoles parallel position is energetically favorable, a ferromagnet forms.
If the antiparallel favorable, an anti- ferromagnet forms.
In metals, the electrons of the conduction band is significantly involved ( for example, iron, cobalt and nickel: ferromagnetic, manganese: anti- ferromagnetic).
In alternating sign magnetic spiral structures can arise (eg chromium, terbium ).
In " competing sign " of the (effective) exchange energy and disordered arrangement of atoms (eg EuxSr1 - xS alloys with 0.13