Multiple nearest-neighbor exchange model for the frustrated magnetic molecules Mo72Fe30 and Mo72Cr30
Christian Schröder, Ruslan Prozorov, Paul Kögerler, Matthew D. Vannette, Xikui Fang, Marshall Luban, Akira Matsuo, Koichi Kindo, Achim Müller, Ana Maria Todea
aa r X i v : . [ phy s i c s . a t m - c l u s ] M a y Multiple nearest-neighbor exchange model for the frustrated magnetic molecules { Mo Fe } and { Mo Cr } Christian Schr¨oder ∗ Department of Electrical Engineering and Computer Science,University of Applied Sciences Bielefeld, D-33602 Bielefeld,Germany & Ames Laboratory, Ames, Iowa 50011, USA
Ruslan Prozorov, Paul K¨ogerler, Matthew D. Vannette, Xikui Fang, and Marshall Luban
Ames Laboratory & Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA
Akira Matsuo and Koichi Kindo
Institute for Solid State Physics, University of Tokyo,Kashiwanoha 5-1-5, Kashiwa, Chiba 277-8581, Japan
Achim M¨uller and Ana Maria Todea
Fakult¨at f¨ur Chemie, Universit¨at Bielefeld, D-33501 Bielefeld, Germany (Dated: October 22, 2018)Our measurements of the differential susceptibility ∂M/∂H of the frustrated magnetic molecules { Mo Fe } and { Mo Cr } reveal a pronounced dependence on magnetic field ( H ) and temper-ature ( T ) in the low H - low T regime, contrary to the predictions of existing models. Excellentagreement with experiment is achieved upon formulating a nearest-neighbor classical Heisenbergmodel where the 60 nearest-neighbor exchange interactions in each molecule, rather than beingidentical as has been assumed heretofore, are described by a two-parameter rectangular probabil-ity distribution of values of the exchange constant. We suggest that the probability distributionprovides a convenient phenomenological platform for summarizing the combined effects of multiplemicroscopic mechanisms that disrupt the idealized picture of a Heisenberg model based on a singlevalue of the nearest-neighbor exchange constant. PACS numbers: 75.10.Jm, 75.10.Hk, 75.40.Cx,75.50.Xx,75.50.EeKeywords: Quantum Spin Systems, Classical Spin Models, Magnetic Molecules, Heisenberg model, Frustra-tion
I. INTRODUCTION
In recent years there has been extensive research onmagnetic molecules as these are novel, realizable systemsfor exploring magnetic phenomena in low-dimensionalmagnetic materials.
Among these diverse systemsthe pair of Keplerate structural type magnetic moleculesabbreviated as { Mo Fe } and { Mo Cr } , each host-ing a highly symmetric array of 30 exchange-coupledmagnetic ions (“spin centers”), serve as highly attrac-tive targets for the investigation of frustrated magneticsystems. In these molecules the magnetic ions Fe III (spin s = 5 /
2) and Cr
III (spin s = 3/2) occupy the30 symmetric sites of an icosidodecahedron, a closedspherical structure consisting of 20 corner-sharing trian-gles arranged around 12 pentagons (diamagnetic polyox-omolybdate fragments). This is a zero-dimensional ana-logue of the planar kagome lattice that is composed ofcorner-sharing triangles arranged around hexagons. Auseful theoretical framework that has been employed in studying these magnetic molecules is based on anisotropic Heisenberg model, where each magnetic ion iscoupled via intra-molecular isotropic antiferromagneticexchange to its four nearest neighbor magnetic ions, andall of the 60 intra-molecular exchange interactions are of equal strength (henceforth, “single- J model”). Un-fortunately the quantum Heisenberg model of the twomagnetic molecules is intractable using either analyti-cal or matrix diagonalization methods. Nevertheless, thenearest-neighbor exchange constant for each molecule hasbeen established by comparing experimental data above30 K for the temperature-dependent zero-field suscepti-bility with data obtained by simulational methods us-ing the quantum and classical Monte Carlo methods.For temperatures below about 30 K, where the quantumMonte Carlo method proves to be ineffective for the twomagnetic molecules due to frustration effects, the clas-sical Heisenberg model is at present the only practicalplatform for establishing the dependence of the magne-tization M ( H, T ) on external magnetic field H and tem-perature T . A rigorous analytical result for the classical,nearest-neighbor, single- J Heisenberg model states that,in the zero temperature limit, M is linear in H until M saturates (saturation fields H s = 17.7 T and 60.0 T for { Mo Fe } and { Mo Cr } , respectively). The practi-cal relevance of the classical Heisenberg model in describ-ing these magnetic molecules even at low temperatureswas strikingly demonstrated in an earlier experiment on { Mo Fe } at 0.4 K, showing an overall linear depen-dence of M on H and its saturation at approximately17.7 T. The ground state envisaged by the classical single- J model is characterized by high-symmetry spin frustra-tion. In particular, for H = 0, in the ground state thespins are coplanar with an angular separation of 120 ◦ between the orientations of nearest-neighbor spins. Onincreasing the external field H the spin vectors gradu-ally tilt towards the field vector, until full alignment isachieved when H = H s , while their projections in theplane perpendicular to the field vector retain the 120 ◦ pattern for nearest-neighbor spins. It would be reason-able to expect that M ( H, T ) is an analytic function ofits variables and thus ∂M/∂H ≈ M s /H s for H < H s and k B T ≪ J , essentially independent of both H and T in these intervals, where J is the exchange constantof the single- J classical Heisenberg model. Indeed ourclassical Monte Carlo calculations confirm this behavior.As shown here, new and crucial features of M ( H, T )become accessible upon examining the differential sus-ceptibility, ∂M/∂H , as a function of H and T . Whatemerge are significant conflicts, spelled out in Secs. II Aand II B between the results of our measurements anda theory based on the classical single- J model as con-cerns both the T and H dependence of ∂M/∂H below5 K. However, and this is the central idea of this pa-per, agreement with experiment is achieved upon adopt-ing a refined classical Heisenberg model of { Mo Fe } and { Mo Cr } , where we drop the assumption ofa single, common value for the nearest-neighbor ex-change constant, using instead a rectangular probabilitydistribution for the 60 nearest-neighbor intra-molecularinteractions.The layout of this paper is as follows. In Sec. II A wepresent our experimental results for ∂M/∂H as a func-tion of T in the low-temperature range for { Mo Fe } and { Mo Cr } for several low field values. In Sec. II Bwe present our experimental results for ∂M/∂H as afunction of H . In Sec. III we show that the modelsystem of a classical isosceles spin triangle exhibits thesame qualitative features as the results of our experi-ments summarized in Sec. II A. In Sec. IV, using theclassical Heisenberg model of the icosidodecahedron andMonte Carlo simulational methods, we show that excel-lent agreement between a model calculation based ona two-parameter distribution of exchange constants andour experimental data of Sec. II can be achieved. Fi-nally, in Sec. V we summarize our findings, discuss thebroader implications of our results, and identify severalopen questions. II. EXPERIMENTA. ∂M/∂H versus T Measurements of ∂M/∂H versus T were performedat Ames Laboratory on polycrystalline samples of { Mo Fe } and { Mo Cr } (see experimental sectionsof Refs. 6 and 7) prepared by employing optimized syn-thesis methods and re-crystallization steps to minimize paramagnetic impurities. Our data was obtained byusing a self-resonating LC circuit driven by a tunneldiode. Briefly, a tank circuit consisting of a small coilof inductance L and a capacitor, C , is kept at constanttemperature of 5 ± .
005 K. The sample is mounted ona sapphire holder, which is inserted into the coil withoutmaking contact. In the absence of the sample, the cir-cuit resonates at the resonant frequency 2 πf = 1 / √ LC .When a sample with susceptibility χ is inserted into thecoil, the resonant frequency changes from f to f ( χ ) dueto the change of the coil inductance, L = d Φ /dI , whereΦ is the total magnetic flux through the coil and I is thecurrent in the coil. This current generates a magneticfield of about 20 mOe. If the magnetic perturbation dueto the sample is small, the shift of the resonant frequency FIG. 1: Temperature dependence of the measured differen-tial susceptibility ∂M/∂H for { Mo Fe } (upper panel) andfor { Mo Cr } (lower panel). Values of H are listed in thelegends. is given by ∆ ff ≈ ∆ L ( χ )2 L = 4 πχ (1 − N ) V V . Here N is the demagnetization factor, ∆ L ( χ ) = | L ( χ ) − L | ≪ L is the change of the coil inductance, V is the sample volume and V is volume of the coil,and χ is the dynamic magnetic susceptibility at our typ-ical resonant frequency of 10 MHz. This frequency isstill much lower than characteristic frequencies and theresponse can be considered as static, i.e., χ can be iden-tified with ∂M/∂H . Measured frequency shifts for thesamples described in this work are of the order of 1 to10 Hz, whereas the experimental resolution of the setup isabout 0.01 Hz which corresponds to a smallest detectablemagnetic moment of about 5 picoemu. For easy compari-son with other experiments, the measured frequency shiftis proportional to the change in the total magnetic mo-ment of the sample at a given frequency, ∆ M = C ∆ f ,where C is a calibration constant.In Fig. 1 we show our experimental data for ∂M/∂H versus T for { Mo Fe } and { Mo Cr } , respectively.The calibration of the experimental data was achieved bymatching the effective magnetic moment inferred fromthe measured frequency shift to low-field (3 - 50000Gs) ac (10 to 1000 Hz) and dc susceptibility measure-ments performed on a Quantum Design MPMS, whichshow that at high temperatures dynamic effects can beneglected. The assumption of static behavior is sup-ported by the fact that all curves collapse onto eachother for T > ∂M/∂H on H and T . As remarked in Sec. I, this behavior is con-trary to that predicted by the single- J model, specifically M ≈ Hχ ( k B T /J ) ≈ Hχ (0), that is, ∂M/∂H is inde-pendent of H and T in the regime H ≪ H s , k B T ≪ J . B. ∂M/∂H versus H In Fig. 3 we show our experimental data for ∂M/∂H versus H for both { Mo Fe } (upper panel) and { Mo Cr } (lower panel). These data sets were obtainedby measuring the magnetization using pulsed magneticfields (typical sweep rate 15000 T /s ) with a standard in-ductive method at facilities of Okayama University, To-hoku University, and University of Tokyo. Also shownare the corresponding classical Monte Carlo simulationalresults for the single- J model (dashed curve) for the in-dicated temperatures.The inadequacy of the single- J model is striking inthat the simulational data differ from the experimentaldata in four important ways. First , the experimentaldata exhibit a steep rise for decreasing low fields, and thelower the temperature the steeper the rise. This is con-sistent with our experimental findings in Sec. II A for the T - dependence of ∂M/∂H . These features are entirelyabsent for the single- J model; in particular, ∂M/∂H is essentially independent of field in the low-field regime. Second , the local minimum in the experimental data issignificantly broader than that predicted by the single- J model. Third , according to the single- J model a localminimum in ∂M/∂H versus H emerges at T = 0 K at afield H = H s /3 (approx. 6 T and 20 T for { Mo Fe } and { Cr Fe } , respectively) and it is enhanced withincreasing T . By contrast, our experimental data forfields in the vicinity of H s /3 differ insignificantly withtemperature. Fourth , for { Mo Fe } the experimentaldata for 0.42 K shows a decrease with increasing fieldabove 10 T, quite distinct from the pattern of the single- J model. III. CLASSICAL ISOSCELES SPIN TRIANGLES
In this Section we give a qualitative explanation forour experimental findings in Sec. II A. We suggest that
FIG. 2: Magnetic field dependence of the measured differen-tial susceptibility ∂M/∂H , shown in red, for { Mo Fe } for T = 0.42 K and 60 mK (upper panel, inset) and for { Mo Cr } for T = 1.3 K and 0.5 K (lower panel, inset). The dashedcurves are the results of the single- J model for these temper-atures. the strong sensitivity of the differential susceptibility on H and T reflects a non-analytic dependence of the mag-netization on these variables, a characteristic already ex-hibited by independent classical isosceles spin triangles.The Hamiltonian of a spin triangle is given in footnote 16in terms of dimensionless quantities. In particular, thethree pair-wise interactions are described by two differentantiferromagnetic exchange constants (positive values of J and J ′ ) and we consider both cases, J ′ < J and J ′ > J .For the equilateral spin triangle ( J ′ = J = J ) at T =0 K the magnetic moment per triangle, M ( H, H and, in particular, itvanishes for H →
0. Specifically, M ( H,
0) = 3
H/H s for − H s < H < H s , where H s = 3 J is the saturation field.Analytical calculation of M ( H,
0) for the correspond-ing isosceles spin triangle is a non-trivial task for generalvalues of
J/J ′ as it is necessary to carefully identify theconfiguration of least energy for arbitrary values of H .The final results are as follows: For the case J ′ > J : M ( H > ,
0) = 1 − J/J ′ + (2 + J/J ′ ) H/H s for 0 < H
0) =
J/J ′ − H/J ′ for0 < H < J ′ − J ; M ( H,
0) = 1 for 2 J ′ − J < H < J ;and M ( H,
0) =
H/J for
J < H < J = H s . M ( H, M ( − H,
0) = − M ( H, H → M ( H, T ) in the low H - low T regime. FIG. 3: Temperature dependence of the differential suscep-tibility ∂M/∂H for a classical isosceles spin triangle (blackcurves) with
J/k B = 1 K and J ′ /k B = 1 . H = 0, 0.1, . . . , 1. Inset: M/M s versus H for thevalues k B T /J = 0 , . , . , . , . , . , . , .
5. Re-sults for the corresponding classical equilateral spin triangle( J /k B = 1 . The form of M ( T, H ) for finite T can, in principle, be derived for this model system by analytical methods,however these calculations are substantially more intri-cate than for the classical equilateral spin triangle (seeSec. II B of O. Ciftja et al. listed in Ref. 11). For practi-cal purposes, the simplest procedure is to use the classi-cal Monte Carlo method for convenient numerical choicesof J ′ /J . In Fig. 3 we display our results for the choices J/k B = 1 K, J ′ /k B = 1 . M is indeed a continuous function of H for any nonzero T , the quantity ∂M/∂H , provided inthe main portion of Fig. 3, exhibits strong temperaturedependence for weak magnetic fields. Indeed, the curvesfan with increasing H in a manner that is strikingly sim-ilar to that shown in Fig. 1. Similar behavior occurs forcases where J ′ < J . By contrast, for the correspond-ing classical equilateral spin triangle ∂M/∂H is virtuallyindependent of T , as is seen in Fig. 3. IV. MULTIPLE NEAREST-NEIGHBORCOUPLINGS
The T and H dependence of ∂M/∂H for the classicalisosceles spin triangle considered in the previous sectionis remarkably similar to that of our experimental datain Sec. II. However, to achieve a more realistic model,in the following we assume that the 60 nearest-neighborexchange interactions between magnetic ions in a givenmolecule are characterized by a probability distributionwith two adjustable width parameters. In the followingSection we rationalize the use of a probability distribu-tion as a convenient way for summarizing the combinedeffects of multiple microscopic mechanisms that disruptthe use of an idealized, single- J model.We simulate each of { Mo Fe } and { Mo Cr } by considering an ensemble of up to 100 indepen-dent systems, for a total of 60 couplings per system.We assign values of the 6000 exchange constants us-ing a random number generator according to the fol-lowing rules: 1) The average value, J n , of the clas-sical exchange constant (in units of Boltzmann’s con-stant) for the n th system is allowed to assume any valuein the interval ((1 − τ ) J , (1 + τ ) J ) with equal prob-ability, where J is chosen as 13.74 K for { Mo Fe } and 32.63 K for { Mo Cr } as determined by high-temperature susceptibility measurements ; 2) For the n th system, the individual values of the classical ex-change constant are allowed to assume any value in theinterval ((1 − ρ ) J n , (1 + ρ ) J n ) with equal probability.For each molecule the two parameters τ, ρ characteriz-ing these rectangular probability distributions were de-termined so as to provide an optimal fit with our exper-imental data for ∂M/∂H versus H .In Fig. 4 we present our results for ∂M/∂H versus H for { Mo Fe } and { Mo Cr } . Note the excellentagreement between the experimental data and the sim-ulational results obtained using our multiple- J model(solid curve). In the case of { Mo Fe } , the opti-mal choices of the parameters τ, ρ were τ = 0 .
15 and ρ = 0 .
40, whereas for { Mo Cr } these were τ = 0 and ρ = 0 . τ and ρ , respec-tively, to complementary effects which only in combina-tion lead to the observed properties of both molecules.The value of τ controls the variation in the values ofthe mean exchange constant per molecule , which leadsto variations in the value of the saturation field H s andhence in the value of the minimum in ∂M/∂H versus H at H s /
3. By averaging over those variations one findsthat ∂M/∂H versus H starts to decrease at a much lowervalue of H than predicted by the single- J model and si-multaneously finds that the minimum at H s / H dependenceof ∂M/∂H in the low T - low H regime, because each FIG. 4: Measured differential susceptibility ∂M/∂H versus H , shown in red, for { Mo Fe } for T = 0.42 K and 60 mK(inset) and for { Mo Cr } for T = 1.3 K (inset: 0.5 K)and simulational results (solid black curve) using a multiple- J model for the optimal choice of the probability distributionparameters as given in the text. molecule is still characterized by a single exchange con-stant. Introducing a second distribution, characterizedby the parameter ρ , leads to a variation in the values ofthe exchange constant within a molecule with the effectthat the corner-sharing spin triangles are of the scalene-type rather than equilateral-type. This gives rise to thenon-analytic behavior of the magnetization at H = 0 for0 K and hence to the characteristic effects we have foundin the low H - low T regime. In our simulations we havestudied a very large range of choices of parameter pairs.The optimal choice for { Mo Fe } can be narrowed to τ = 0 . ± .
02 and ρ = 0 . ± .
02. For { Mo Cr } wefind ρ = 0 . ± .
02, however τ can be chosen in the range0 to 0.2 without any observable effect. This is due to thefact that for { Mo Cr } magnetization measurementsabove the saturation field ( H s = 60 T) are not achiev-able at the present time. In the case of { Mo Fe } , forwhich H s = 17.7 T, the availability of magnetization dataabove the saturation field allows for a greatly reduced un-certainty in the value of τ . FIG. 5: Simulational results for ∂M/∂H versus T based onthe multiple- J model using the optimal distribution parame-ters for { Mo Fe } and for { Mo Cr } as given in the text.Results for the corresponding single- J model calculations areshown as red dashed curves. Shown in Fig. 5 are our simulational results for ∂M/∂H versus T for several different values of H usingthe probability distribution with optimal parameters ap-propriate for { Mo Fe } and { Mo Cr } . These results(black curves) are strikingly similar to the experimentalcurves seen in Fig. 1; the corresponding curves (shownin red) for the single- J model, for the same choice of themean value J , are essentially indistinguishable from oneanother. This again strongly supports the existence ofthe multiple- J scenario. Note that, in the experimentas well as in the simulations, with increasing tempera-ture the curves for different field values rapidly convergeand become indistinguishable from one another. Also,for increasing temperature, the results for the multiple- J model merge with those of the single- J model, as ex-pected, since the average exchange constant across theensemble, J , is chosen to equal to the exchange con-stant of the single- J model. Finally, it remains to beseen whether the sharp rise in the curves of the upperpanel of Fig. 5 below 200 mK is an experimental featurein { Mo Fe } or merely an artifact of the multiple- J model based on a rectangular probability distribution. V. SUMMARY AND DISCUSSION
In this article, we have presented our experimentaldata for the differential susceptibility of the pair of mag-netic molecules { Mo Fe } and { Mo Cr } as a func-tion of magnetic field and temperature. Below 5 K thesedata are strikingly different from what can be providedusing a classical Heisenberg model with a single value ofthe nearest-neighbor exchange constant (single- J model).We have achieved excellent agreement with our experi-mental data upon adopting a classical Heisenberg modelwhere the 60 nearest-neighbor interactions are not identi-cal; instead, the values of the exchange constants are de-scribed by a two-parameter probability distribution witha mean value as determined from experimental ∂M/∂H data above 30 K using the single- J model. Above 5 Kthe single- J model provides a satisfactory description ofeach molecule.Since the icosidodecahedron structure consists ofcorner-sharing triangles, it is not surprising that theHeisenberg model of independent classical isosceles spintriangles provides a simple yet instructive model in thatit exhibits the main qualitative features of our experi-mental data. We note here that a similar approach hasbeen employed successfully for various two-dimensionalspin systems on triangular lattices . For example, in thecase of manganese tricyanomethanide the so-called ‘rowmodel’ based on connected isosceles triangles provides theexplanation for an unusual magnetic-field dependence ofthe spin ordering . In the context of independent clas-sical isosceles spin triangles one can figuratively describethe effect of multiple exchange constants as modifying thespin frustration from the standard 120 ◦ angular separa-tion between spin vectors of the equilateral spin triangle.The operational consequence is that the magnetization,for T = 0, of an isosceles spin triangle is a non-analytic function of magnetic field for H = 0, and this is mani-fested in ∂M/∂H being a highly sensitive function of itsarguments for small H and T .The existence of a distribution of nearest-neighbor ex-change constants can be expected to be responsible for asignificant lifting of degeneracies of magnetic energy lev-els. To be specific, the quantum rotational band model ,which is a solvable alternative to the nearest-neighborsingle- J quantum Heisenberg model, predicts a discretespectrum of energy levels, many of which have a veryhigh degeneracy due to large multiplicity factors. Per-turbing this model Hamiltonian by using a distributionof J -values would remove a major fraction of these de-generacies. The lifting of level degeneracies could providea reasonable explanation for three long-standing puz-zling issues concerning these magnetic molecules: Thefirst issue is the very broad peak (maximum at 0.6 meV)that has been observed by inelastic neutron scattering on { Mo Fe } at 65 mK. In order to qualitatively repro-duce the observed peak using the rotational band model,it was necessary in Ref. 21 to perform the calculationsupon assigning a large energy width (0.3 meV) for theindividual energy levels. We suggest that the source ofthis large energy width might be the lifting of the major-ity of degeneracies associated with a single- J model.A second important consequence of the splitting ofhighly degenerate levels would be that the moleculescould exhibit classical characteristics down to very lowtemperatures. This would provide a very reasonable ex-planation for the surprising fact that our simulational re-sults based on the classical Heisenberg Hamiltonian areso successful in describing { Mo Cr } , despite the factthat the Cr III ions have a small spin (3/2). Stated differ-ently, with the lifting of degeneracies and the fanning outof energy levels the effective temperature for the crossoverfrom classical to quantum behavior can be anticipated tobe considerably lower than that expected a priori for thesingle- J model.Third, the failure of efforts to observe magnetiza-tion steps, in measurements of magnetization versus H ,in the mK temperature range in both { Mo Fe } and { Mo Cr } could also be attributed to the removal of de-generacies of the magnetic energy levels. The occurrenceof magnetization steps at low temperatures is associatedwith the field-induced crossing of successive energy levelsof the lowest rotational band. However, a discrete levelassociated with total spin quantum number S has multi-plicity 2 S +1 [total degeneracy (2 S +1) ]. If this degener-acy is lifted there will be a multitude of level crossings atslightly different field values and thus give rise to blurredeffects down to lower temperatures than would otherwisebe expected.Given the finite-spin values of the Fe III and Cr
III ions,is it possible to explain the present experimental find-ings based on a quantum
Heisenberg model that adoptsa common, single value of the exchange constant for allof the nearest-neighbor interactions? We strongly doubtthat this is possible, for we have seen, albeit with a clas- sical
Heisenberg model, that it is the spread in valuesof the nearest-neighbor exchange constant that fuels thesensitive dependence of ∂M/∂H on T , or equivalentlythe non-analytic behavior of magnetization on H in thelow H - low T regime.Basing our simulations on a probability distributionfor the nearest-neighbor exchange interaction has led toexcellent agreement with the detailed features of our ex-perimental data including the sensitive dependence of ∂M/∂H on T and H . One can attribute the failureof the single- J model to the combined effect of a largenumber of diverse perturbing mechanisms. The effectsof impurities, variations in the exchange-coupling geome-try, weak magnetic exchange interactions of more-distantneighbors, Dzyaloshinsky-Moriya and dipole-dipole in-teractions in these magnetic molecules are some of themany effects that are excluded when one uses an ide-alized single- J model. On the other hand, it is at thisstage an extremely difficult, essentially impossible taskto realistically quantify the effects of the diverse mecha-nisms. A theoretical description based on a Heisenbergmodel where the nearest-neighbor exchange constant ischosen using a probability distribution provides a rel-atively simple, phenomenological platform for compro-mising between the need for microscopic realism versuspractical limitations. Ultimately it is significant that atwo-parameter probability description can actually pro-vide the level of agreement that we have found. Finally,we remark that other choices of probability distributionscan be expected to perform equally well.As one example of the complications in assessing theplethora of perturbing mechanisms, we consider the vari-ation in the intramolecular distances between nearest-neighbor magnetic ions. A geometric analysis utiliz-ing existing low-temperature single crystal X-ray struc-ture data for { Mo Fe } molecules shows that the sub-structure of the magnetic ions is close to an ideal I h -symmetric geometry. There is a standard deviation of0.04525 ˚A (0.70%) and a maximum deviation of 1.4%from the average Fe-Fe distance of 6.4493 ˚A for all 60Fe-Fe nearest-neighbor distances. Besides the distancesbetween the spin centers, geometric variations within thepolyoxomolybdate exchange ligand are observed. In par-ticular the O-Mo-O bond angle variations should affectthe total orbital overlap and thus the exchange energydue to the spatially anisotropic character of the (unoccu-pied) Mo(4d) orbitals. For { Mo Fe } , an angular range103 . ◦ − . ◦ is observed, which in part is caused bycrystallographic disorder of Mo positions. To assess theinfluence of various geometric parameters involved in thesuperexchange pathways between two nearest-neighborspin centers, both binding to a pentagonal diamagnetic[Mo VI O (H O) ] − = { Mo } fragment, we performedsystematic Density Functional Theory-Broken Symme-try calculations on a model system in which two s = 1/2 [V IV O(H O) ] groups are coordinated to sucha { Mo } fragment in a nearest-neighbor (1,2) configu-ration. The fragment is augmented by an additional Zn II (H O) group binding in a (1,3,5) configuration forcharge neutrality. The geometry of this model systemwas adjusted to match the actual configurations occur-ring in { Mo Fe } . We find that J in such a model sys-tem can deviate by up to ±
8% from the average value J .Given the similarity between VO , Cr III , and Fe
III inthe Keplerate systems, namely that the magnetic orbitalscause a nearly isotropic spin density distribution, we ex-pect that the variations in the relative values of J span avery similar interval for { Mo Fe } and { Mo Cr } . Asthe intra-molecular variation in the values of the nearest-neighbor exchange constants implied by the optimal val-ues (given in Sec. IV) of the parameter ρ are significantlylarger, we suggest that this is due to numerous other per-turbing mechanisms, some of which we listed above.We also note that other attempts to explain limitedfeatures of ∂M/∂H , specifically the broadening of theminimum versus H for { Mo Fe } , have been consideredin the literature. One attempt assumed an elevated spintemperature during the pulsed field measurements, how-ever this could be ruled out since a subsequent steady-field measurement reproduced the results obtained bythe pulsed-field technique. Second, in a simulationalstudy based on classical Monte Carlo calculations, ef-fects of magnetic anisotropies, Dzyaloshinsky-Moriya,and dipole-dipole interactions have been considered. However, our own comprehensive simulational studies ofthese same mechanisms have shown that they give rise toonly very minor corrections on the width of the minimumin ∂M/∂H versus H for any reasonable choices of modelparameters.Finally, we suggest the additional possibility that inthese molecules the variation of the exchange inter-action is spontaneously generated so as to lower thesystem’s magnetoelastic energy. Such behavior hasbeen observed experimentally and described theoreti-cally for a variety of antiferromagnetic oxide pyrochlorecompounds . The pyrochlore lattice consists ofcorner-sharing tetrahedra and exhibits geometric frus-tration. In this regard one can understand this struc-ture as the three-dimensional ‘cousin’ of the corner-sharing triangle type structures realized in { Mo Fe } and { Mo Cr } .In any event, it is highly satisfying that the frustratedmagnetic molecules { Mo Fe } and { Mo Cr } , osten-sibly zero-dimensional systems, are a source of novel andintriguing magnetic behavior. Acknowledgments
Research performed by C.S. at the Applied SciencesUniversity Bielefeld was supported by an institutionalgrant. Work at the Ames Laboratory was supportedby the Department of Energy-Basic Energy Sciences un-der Contract No. DE-AC02-07CH11358. R.P. acknowl-edges financial support from the Alfred P. Sloan Foun-dation. A.M. thanks the Deutsche Forschungsgemein-schaft, the Fonds der Chemischen Industrie, and the Eu-ropean Union for financial support. We thank H. No-jiri for sharing experimental data with us and for help-ful discussions. We also thank the thousands of volun-teers participating in the public resource computing facil- ity, Spinhenge@home [http://spin.fh-bielefeld.de]. Thelarge-scale Monte Carlo simulations necessary for thepresent research were made possible due to the availabil-ity of their personal computers. ∗ Electronic address: [email protected] O. Kahn,
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