The origin of the magnetic interaction between simple paramagnetic centers is well described in many excellent books and reviews to which the interested reader is directed.4,26–28 Here we simply recall that the interaction can be often considered as the formation of a weak covalent bond between two suitably chosen buildingblocks. We may imagine that on each buildingblock the unpaired electrons are in molecular orbitals which are usually denoted as ‘‘magnetic orbitals.’’29,30 If the two magnetic orbitals have a nonzero overlap, the spins will pair (anti- ferromagnetic coupling) and a weak covalent bond will form between the two magnetic centers.
By contrast, if the overlap is zero the spins will tend to stay parallel to each other (ferromag- netic coupling), provided that areas of relatively large overlap density are present. These qualitative rules were proposed in the 1960s by Goodenough and Kanamori.31–33 A simple clear example was worked out by Kahn et al.34 who synthesized a dinuclear compound of formula CuVO(fsa)2enCH3OH (Figure 5). The unpaired electron on the copper(II) center is in thed(x2y2) orbital, while on the oxovanadium(IV) center the unpaired electron lies in thed(xy) orbital. It is apparent that the two orbitals are orthogonal to each other, independent of the angle Cu—O—V at the bridging oxygen ligand. However the magnetic orbital is not exactly localized on the metal ions, but will have a nonzero density on the oxygen ligand as well. In fact, the interaction between the magnetic orbitals is named exchange interactionif they overlap directly or superexchange if they overlap at a formally diamagnetic bridging ligand. It is this area of common overlap that determines the observed ferromagnetic coupling. If the two magnetic centers did not interact with each other a simple paramagnetic behavior would be expected, which corresponds to the sums of the Curie laws, Equation (1), for the two ions. A simple way of determiningwhether a compound follows the Curie law is to plotmTvs. Tin a wide temperature range as shown in Figure 6. It is apparent that the experimental curve lies above the theoretical one for two noninteractingspinsSẳ1/2,
mTẳNA2Bðg2Cu ỵ g2Vị
4kB ð7ị
The deviations from the Curie law are associated with the fact that the interaction between the copper(II) and oxovanadium(IV) centers gives rise to a singlet state and a triplet state, separated by an energy. At low temperature the triplet will be selectively populated and the value ofmT will tend to the value expected for Sẳ1, 1.00 emu K mol1. From the fit of the temperature dependence of mT, which reflects the variation of the thermal population of the singlet and
V Cu
N N
Figure 5 ORTEP view of the structure of CuVO(fsa)2enCH3OH (after Kahn et al.).34 The carbon and oxygen atoms are shown in pale and dark gray, respectively. The other atoms are labeled.
triplet states, it is possible to obtain the energy separation . Therefore magnetic measurement can directly provide information on the interaction between the two metal centers.
Exchange and superexchange interactions are described, in the simplest way, by the spin Hamiltonian:35
HẳJS1S2 ð8ị
Different notations whereJis substituted by –J,2J, þ2Jare often encountered in the literature.
They are all equivalent to each other from a theoretical point of view, but of course comparison between the spin Hamiltonian parameters can be made only if the same convention is used.
According to this spin Hamiltonian, a ferromagnetic coupling corresponds to a negative Jwhile an antiferromagnetic coupling is associated with a positive J.
The eigenstates of Equation (8)can be grouped into multiplets having different values of the total spinSand the susceptibility can be easily calculated by takinginto account the energyE(S) and the Boltzmann population of the different multiplets:
mẳNAg22B 3kBT
P
S
SðS ỵ1ịð2S ỵ1ịeEðSị=kBT P
S
ð2Sỵ 1ịeEðSị=kBT ð9ị
where the summations extend over all permitted Svalues, i.e.,jS1S2j SS1þS2.
The above treatment is valid for pairs but can be easily extended to larger numbers of interactingions by includingin the spin Hamiltonian (Equation (8)) all possible pairwise inter- actions and extendingin a similar way the sums in Equation (9). The number of S states is (2Siþ1) for a pair of spinsSibut increases rapidly as more spins are added. More details on the procedure for the calculation of the thermodynamic properties of high-nuclearity spin clusters will be available in Volume 7.
Perhaps the most complete study of exchange interactions in a class of molecule-based magnetic materials containingpairs of different transition-metal ions has been performed on the series of Prussian-blue analogs.36The compounds have general formula CnAp[B(CN)6]qxH2O, where C is a monovalent cation, usually nonmagnetic, while A and B are magnetic cations, octahedrally coordinated to the N- and C-donor atoms, respectively, of the bridging cyanide ligands. Some typical structures are shown inFigure 7. In octahedral symmetry thed-orbitals are of eithereg- or t2g-symmetry, as depicted inFigure 8. The t2g-orbitals on a B center have nonzero overlap with the t2g-orbitals on the A center, but zero overlap with the eg orbitals. Therefore the t2g–t2g
interaction is expected to be antiferromagnetic, while thet2geginteraction must be ferromagnetic.
0 0.8
0.9 1 1.1
100
50 150
T (K)
χT (cm3 K mol–1)
200 250 300
Figure 6 Temperature dependence of themTproduct of CuVO(fsa)2enCH3OH (reproduced by permis- sion of the American Chemical Society fromJ. Am. Chem. Soc1978,100, 3931–3933).
400 Magnetism: General Introduction
The observed exchange interaction J can then be expressed as a sum of pairwise interactions between magnetic orbitals on the two sites:
Jẳ 1 4S1S2
X
i;j
Jij ð10ị
where i and j run over all magnetic orbitals on the centers 1 and 2, respectively. For instance, chromium(III) has a d3electron configuration, with three unpaired electrons in the t2g-orbitals, while copper(II) and nickel(II) have d9 and d8 configurations with one and two unpaired electrons in the eg orbitals, respectively. In the CrIIICuII derivative the interactingmagnetic orbitals are d(x2y2) on copper(II) and d(xz), d(yz), d(xy) on chromium(III), so that overall ferromagnetic coupling is expected. Indeed the coupling was found to be ferromagnetic, as shown in Figure 9. Similar considerations hold for the CrIIINiII derivative. On the other hand, vanadium(II) has a d3 configuration with three unpaired electrons in the t2g-orbitals and the CrIIIVIIinteraction is antiferromagnetic. The coupling constants observed throughout the series are given inTable 2 along with the number of ferromagnetic and antiferromagnetic pathways.
The above treatment considers that the unpaired electrons are localized on the metal centers.
While this is certainly a good approximation in the majority of cases, another mechanism based on spin polarization may sometimes be operative.37Spin polarization is of fundamental import- ance for organic radicals,38where in general the unpaired spin density is delocalized on a number of atoms.27,28,39 It has been invoked to explain magnetic coupling in [Fe(Cp*)2]þ(TCNE),40,41 the first organometallic compound with magnetic orbitals of s and p character to show a
(a) (b) (c)
Figure 7 Typical structures of cubic Prussian-blue analogs: (a) A(III)[B(III)(CN)6], A1B1; (b) Cs(I)A(II) [B(III)(CN)6], Cs1A1B1; (c) {A(II)}3[B(III)(CN)6]2xH2O, A3B2. [B(CN)6] are the dark solid octahedra surrounded by CN (very small spheres), A are the light small spheres, C are the gray medium size spheres
in (b); in (c) H2O are shown by the small light-gray spheres.
(N≡C)5B–C≡N–A(N≡C)5 y x y y
y
z
z z z
B C N C N A C N A
(a) (b) (c)
Figure 8 Local magnetic orbitals in an isolated (NC)5_B_CN_A(NC)5 binuclear unit: (a) t2g magnetic orbitals at the B site; (b)t2gmagnetic orbitals at the A site; (c)egmagnetic orbitals at the A site.
TC (K) 300
200
100
Ti V Cr
CrII
Mn MnII
Fe FeII
Co CoII
Ni NiII
Cu CuII
Zn d6 Z
d4
d3 d3
d5
d7 d8 d9 (eg)1 3 F
3 F
6 F 6 F 6 F 6 F
6 AF 3 AF 9 AF 9 AF
9 AF (t2g)3 (t2g)3
[Cr(CN)6]3–
AII3CrIII2 stoichiometry VII
Figure 9 Experimental variation of TC with Z, the atomic number of the A(II) cation in the series {A(II)}3[Cr(III)(CN)6]2, electronic structures and expected exchange pathways (black dots ferrimagnets;
gray dots ferromagnets) (after Verdagueret al.).36
Table 2 Electronic structure, exchange pathways, and interactions in AII-NC-BIIImagnetic pairs.a CrIII(t2g3)
VII (t2g3)
CrII (t2g3)(eg)
MnII (t2g3)(eg2)
FeII (t2g4)(eg2)
CoII (t2g5)(eg2)
NiII (t2g6)(eg2)
CuII (t2g6)(eg3)
AF pathways 9 9 9 6 3 0 0
F pathways 0 3 6 6 6 6 3
Interactiona AF AF AF af af F F
a Upper case notation indicates a strong coupling, either ferromagnetic (F) or antiferromagnetic (AF) in nature, while lower-case notation indicates a weak coupling.
N
N
N
N
Fe
Figure 10 Schematic view of the [Fe(Cp*)2]þand the TCNEions. The noncarbon atoms have been labeled.
402 Magnetism: General Introduction
transition to ferromagnetic order at4 K. A sketch of the interactingcation and anion is shown inFigure 10. At the lowest level of approximation the unpaired electron of the cation is localized on the iron ion. However, for a more correct description one must take into account spin correlation effects, by consideringexcited states which mix into the ground state. Formally this corresponds to transferringto the Cp* ligand a fraction of unpaired electron, with a spin opposed to that on the metal center. The negative spin density on the Cp* ring interacts antiferromagnet- ically with the spin density on the TCNEmoiety, thus giving rise to an effective ferromagnetic couplingbetween the iron(III) and TCNEcenters.42It must be stressed that other justifications for the observed ferromagnetic coupling are also possible.
Spin polarization has also been invoked in order to justify the observed alternance of ferro- and antiferromagnetic coupling in dinuclear molybdenum(V) complexes bridged by pyridine type ligands.43–46 The sketch of Figure 11 is self explanatory: when the bridge is 4,40-bipyridine the couplingis antiferromagnetic, while for 4,50-bipyridine the couplingis ferromagnetic. It must be stressed that the procedure outlined in Figure 11 for the spin polarization mechanism is over- simplified, and one should take into account other factors, such as the angle between the aromatic rings. However, the model is simple and has certainly some predictive power for experimentalists.