The superexchange interaction was developed by Kramers,

Anderson, Goodenough and Kanamori. Kramers in 1934 realized the fact that there

was a magnetic interaction between two magnetic ions i.e Mn even though separated by a non magnetic ion i.e O. This model was further developed by Anderson in 1950. In the same year, Goodenough and Kanamori proposed some semi-empirical rules for superexchange interaction between these ions. These cumulative rules for a superexchange interaction are called Goodenough-Kanamori-Anderson (GKA) rules. These rules predict a strong antiferromagnetic structure when the magnetic ion-ligand-magnetic ion angle is 180º with partially filled d shells, and a weak ferromagnetism when the angle is 90º.

I am also writing basic GKA rules (those I collected from literature), these are based on the symmetry relations and electron occupancy of the overlapping atomic orbitals

a) When the two ions have orbital lobes pointing toward each other to have a

reasonably overlap, the exchange is antiferromagnetic

b) When the lobes are d (3z2-r2) type orbitals in the octahedral case, particularly in

the 180 º position in which these lobes point directly toward a ligand and each

other, one obtain particularly large superexchange.

c) When d (xy) orbitals are in the 180º position to each other, they can interact via pπ

orbitals on the ligand and there is an antiferromagnetism.

d) When the orbitals are arranged in such a way that they are expected to be in

contact but to have no overlap integral- most notably a d(3z2-r2) and d (xy) in

180º position, where the overlap is zero by symmetry, the rule gives

ferromagnetism

I hope this can help to understand magnetism even in different systems..

Try to check the following links:

http://boulder.research.yale.edu/Boulder-2003/reading/pwa_sex1.pdf

http://www2.binghamton.edu/physics/docs/super-exchange.pdf

http://www.znaturforsch.com/ab/v57b/s57b0323.pdf

Superexchange has something to do with virtual movement of electrons between atom sites. It is this hopping that opens a gap between rotationally equivalent degenerate spin configurations, and this gap is about the size of superexchange.

I recommend P. W. Anderson' s 1959 paper about superexchange.

The link to Anderson´s paper about superexchange interactions theory:

http://prola.aps.org/pdf/PR/v115/i1/p2_1

Thanks you vert much fir your answers.Yours

The superexchange phenomenon is a kind of indirect magnetic interaction. It takes place between two cations with unpaired spins bound by an anion without unpaired spins. The coupling can be either ferro- or antiferromagnetic. The so called Anderson-Goodenough-Kanamori rules determine which of the two types of coupling is present. Superexchange is short ranged in comparison to other types of magnetic interection based on direct exchange of electrons.

The exchange interaction between two magnetic ions in magnetic compounds arises through the anions situated between the ions and is called superexchange interaction. It was first proposed by Kramers in 1034. The details were studied later by Anderson, Goodenough, Kanamori, Wollan et al., Yamashita, Kondo and Slater. Goodenough, Kanamori and Wollan introduced the symmetry of d-orbitals into the theory and Anderson discussed the superexchange interaction on the basis of Mott insulator. He simplified and unified various mechanisms giving rise to superexchange into two types: (1) the direct exchange interaction and (2) kinetic exchange due to electron transfer. If you wish to go to the details from the horse's mouth then read the review article of Anderson "Exchange in insulators" in Magnetism vol. 1, ed. T. Rado and H. Suhl, Academic press, p. 25-83. Otherwise you can read a nice pedagogic exposition by W. Nolting and A. Ramakanth in Quantum Theory of Magnetism, Springer Verlag (2009) p. 209-216.

The superexchange interaction was developed by Kramers,

Anderson, Goodenough and Kanamori. Kramers in 1934 realized the fact that there

was a magnetic interaction between two magnetic ions i.e Mn even though separated by a non magnetic ion i.e O. This model was further developed by Anderson in 1950. In the same year, Goodenough and Kanamori proposed some semi-empirical rules for superexchange interaction between these ions. These cumulative rules for a superexchange interaction are called Goodenough-Kanamori-Anderson (GKA) rules. These rules predict a strong antiferromagnetic structure when the magnetic ion-ligand-magnetic ion angle is 180º with partially filled d shells, and a weak ferromagnetism when the angle is 90º.

I am also writing basic GKA rules (those I collected from literature), these are based on the symmetry relations and electron occupancy of the overlapping atomic orbitals

a) When the two ions have orbital lobes pointing toward each other to have a

reasonably overlap, the exchange is antiferromagnetic

b) When the lobes are d (3z2-r2) type orbitals in the octahedral case, particularly in

the 180 º position in which these lobes point directly toward a ligand and each

other, one obtain particularly large superexchange.

c) When d (xy) orbitals are in the 180º position to each other, they can interact via pπ

orbitals on the ligand and there is an antiferromagnetism.

d) When the orbitals are arranged in such a way that they are expected to be in

contact but to have no overlap integral- most notably a d(3z2-r2) and d (xy) in

180º position, where the overlap is zero by symmetry, the rule gives

ferromagnetism

I hope this can help to understand magnetism even in different systems..

Besides reading the articles nd book (sub-) chapters I can also recommend to make drawings of the coupling situations of two magnetic sites, to recognize all the states involved in the coupling mechanism and to write down a Hamiltonian for this in the Hubbard model terminology. Then you can transfer the model to some software (such as scilab, which is free, or matlab etc) and solve the Eigenvalue problem (numerically). Then you can play with Coulomb and hopping energies and see, how the solutions behave. I did this and presented both the equations and solutions (not too hard, actually) to my students in a magnetism lecture and they appreciated seeing that they could actually put their own hands on the phenomena. I think that fostered their understanding. My starting point was the book by Nolting which Tapan recommended. And in comparison to other books it is probably one of the more correct ones (and not taking too many shortcuts, which quite a few others - sometimes incorrectly - do).

Thanks you a lot for yours answers!

Some can undergo Antiferromagnetic while the the others may undergo Ferromagnetic superexchange. This depends upon the type of interaction between the orbitals of two paramagnetic cations and the intervening diamagnetic anion.

So both Antiferromagnetic and Ferromagnetic Superexchange need separate discussion

[I] Superexchange describes the interaction between moments on ions too far apart to be connected by direct exchange, but coupled over a relatively long distance through a non-magnetic material( generally O^-2).

[II] Consider the interacting cations A&B and the intermediate O. It is generally assumed that the nature and the strength of interaction for a given pair of cations is a function primarily of the distance between AO and OB and the angle AOB. The closer the angle AOB to 180 degrees, the stronger will be the superexchange interaction. { in other cases, sp hybridof anions may be considered}

Both Antiferromagnetic and ferromagnetic superexchange are briefly explained as:

[III] ANTIFERROMAGNETIC SUPEREXCHANGE:{ See attached file ;Figure-I; MnO; Zoom-150-200 }

The idea can be illustrated by two Mn^2 ions[ t2g^3eg^2] and one intervening O^2-. The Mn^2 -- O2- —Mn^2 are arranged along the one axis, forming a 180 degree position[both eg orbitals undergo sigma interaction with pz]. There are two spins in the O^2- site (the up-spin state + and down-spin state − ). Because of the overlap of wave function , one of p electrons (with down-spin state − ) from the O^2- HOPS OVER to one of Mn^2 ions {LHS}. [We assume that a part of one electron hops]. Note that only the down-state electron of p-electrons can move to the Mn^2 site {LHS}. The remaining unpaired p electron on the O^2- site then enters into a DIRECT EXCHANGE with the other Mn^2+ ion {RHS}. Such interaction between Mn^2 ions coupled through O^2- is called antiferromagnetic superexhange.

It is a strong interaction . The magnetic moment decreases and becomes less than even the normal paramagnetic substance.Its experimental magnetic moment is 4.58 BM.

[IV]FERROMAGNETIC SUPEREXCHANGE See 2nd and 3rd Figures:

Consider two Fe^4 ions[ t2g^3eg^1] and one intervening O2-.

One overlapping half filled 3d orbital belongs to eg set and the other to the half filled t^2g set.There can be two types of interactions called 180 degree and 90 degree interactions represented separately as:

[i]180 degree interaction

The eg set of orbital of Fe^4{RHS} can undergo sigma interaction with pz to enter into a DIRECT EXCHANGE while the other from the O^2- HOPS OVER to the t2g set of orbital of the other Fe^4{LHS} to undergo week pi interaction with px orbital of the anion.

[ii] 90 degree interaction

It is the reverse of 180 degree interaction.

Such ferromagnetic superexchange interaction is much weaker .Such substances still have enough ferromagnetic character{ Experimental magnetic moment of this ferromagnetic oxide=3.5BM/ Fe .

Angewandte Chemie International Edition . 50(52):12547-50[2011].