Exchange double

Mixed-valence trinuclear systems involve two groups  

The localisation of the extra electron (hole) on a definite centre is denoted by , i.e. A BC), AB C) and ABC ). Each ket is constructed by means a coupling scheme, for example 

In accordance with the addition of angular momenta the intermediate spin numbers are 

Since only one possibility exists for the intermediate states Sas = SA+s and SCs = Sc + s, a single term arises in the recoupling formula. For electron delocalisation 5 = +1/2 whereas for the case of hole delocalisation s = —1/2 can be applied. 

For a homonuclear system, SA = SB = Sc = S, (which is the most frequent case of mixed-valence systems), the 9/-symbol can be expressed through 6/-symbols and the recoupling coefficients become [9] 

This simple electron hopping explanation has later been revised to incorporate the strong electron-phonon coupling caused by the Jahn-TeUer splitting of the Mn ions. The conduction is due to the hopping of a magnetic polaron rather than a bare electron [3]. 

A rigorous description of electron hopping in the presence of other unpaired electrons can be written down for the simple model system defined in Fig. 6.4. The two electrons in the ai and b orbitals provide a static background of unpaired electrons for the mobile electron in the 02 b2 channel. Choosing Ms = there are 24 ways to distribute the three electrons over the four orbitals, but only six of them 

2 Find the four determinants (or linear combinations of determinants) that represent a triplet spin coupling of the electrons on center a or b. What is the spin coupling of the other two (linear combinations) of determinants What do you expect for the relative energies of the two groups  

The interaction matrix elements are readily written down using the Slater-Condon rules given in Chap. 1. We will work out three examples and leave the others as 

K = KaxU2 = Kb- b2 is ih on-site exchange interaction and K = Ka bi is the intersite exchange. Two approximations have been made to obtain this matrix. In the 

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Ammonium Perchlorate. The commercial AP product is manufactured by the double-exchange reaction of sodium perchlorate and ammonium chloride (102,103). 

Schematically, the double exchange interaction can be written in the form of a spin Hamiltonian operator (working on the system spin AB —for short, S) as...

And the combination of Heisenberg superexchange plus Zener double exchange results in zero-field energy levels in terms of the system spin S... 

FIGURE 11.7 Spin ladders for the dinuclear FeIII-Fe11 cluster. The two metal ions are subject to superexchange (7) and double exchange (B) with J B,J B, or J = B/4. The three ladders are not normalized to the same energy scale. 

The one-electron reduced cluster, [3Fe-4S]° has a system spin 5 = 2, which is envisioned to be the result of parallel coupling (i.e., through double exchange) a ferric 5 = 5/2 and a ferrous 5 = 2 ion into a delocalized pair with 5 = 9/2, and subsequently coupling this structure antiparallel to the remaining 5 =5/2 iron into an overall system spin 5 = 2 (Papaefthymiou et al. 1987). 

Bastardis, R., Guihery, N., and Suaud, N. 2007. Relation between double exchange and Heisenberg model spectra application to the half-doped manganites. Physical Review B 75 132403 (1 1). 

De Gennes, P.-G. 1960. Effects of double exchange in magnetic crystals. Physical Review 118 141-154. 

Ding, X.-Q., Bominaar, E.L., Bill, E., Winkler, H., Traitwein, A.X., Driieke, S., Chaudhuri, P., and Wieghardt, K. 1990. Mossbauer and electron paramagnetic resonance study of the double exchange and Heisenberg-exchange interactions in a novel binuclear Fe(II/III) delocalized-valence compound. Journal of Chemical Physics 92 178-186. 

Belinsky MI. 1996. Hyperfine evidence of strong double exchange in multimetallic [Fe4S4]-Fe) active center of Escherichia coli sulfite reductase. J Biol Inorg Chem 1 186-8. 

Achim C, Golinelli M-P, Bomiaaar EL, et al. 1996. Mossbauer study of Cys56Ser mutant 2Fe ferredoxin from Clostridium pasteuriemum evidence for double exchange in an [Fc2S2] cluster. J Am Chem Soc 118 8168-9. 

Double Exchange in Reduced Clusters and Novel Clusters with MFe S Cores... 

The literature on double exchange is not very extensive. The main papers dealing with the problem discussed here are the original work by Anderson and Hasegawa (10), and papers by Karpenko (33), Borshch et al. (30), Belinskii et al. (34) and Girend (31). Particularly noteworthy is the analysis by Noodleman and Baerends (35) in their theoretical study of the electronic structure of ferredoxins with Fe2S2 cores. Quite recently, Belinskii (32) has published an extensive theoretical study for mixed-valence trimers with D3h symmetry. 

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