Three‐sublattice Spin Model and Quantum Rotational Band Theory     33

The iron ions are located in regular octahedral Fe³⁺O6 coordination environments, leading to negligible local spin anisotropy. The Fe³⁺ ions are linked by nearly planar pentagonal fragments {(Mo)Mo5} [55]. As shown in Figure 3-1-2, the magnetic exchange interactions between nearest-neighboring Fe³⁺ ions are transmitted via -O-Mo-O- bridges, whereas the next-nearest-neighboring superexchange interactions

Chapter 3. Geometrical Spin-frustrated Molecular Magnet {Mo72Fe30}

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go through -O-Mo-O-Mo-O- bridges and are estimated to be at least two orders of magnitude smaller by means of density functional theory calculation and thus negligible [23, 78]. Therefore I can safely assume the intramolecular exchange interactions are restricted among the nearest neighbors. The magnetic properties of {Mo72Fe30} can then be described by the isotropic Heisenberg model with a single exchange constant J, neglecting the anisotropy term [55],

B ,

ˆ ˆ ˆ

ˆ i j i

i j i

= J g

μ

< >

⋅ + ⋅

∑ ∑

H S S B S , (3.1.1)

where and are spin operators in units of ħ, B is the external field, g is the spectroscopic splitting factor, µB is the Bohr magneton, J is the exchange constant for nearest-neighbor coupling and found to be about 0.134 meV through a mean-field simulation approach, indicating antiferromagnetic exchange interaction [55].

Figure 3-1-2 Five Fe³⁺ ions (yellow) connected by a pentagonal {(Mo)Mo5} group (blue: Mo; red: O). The superexchange pathways, -O-Mo-O- between nearest neighbors 1-2 and -O-Mo-O-Mo-O- between next-nearest neighbors 1-3, are emphasized. [Picture taken from P. Kögerler, B. Tsukerblat and Achim Müller, Dalton Trans. 39 (2010) 21]

Approximate approaches must be taken to establish a spin model describing the way how spins arrange on the surface of this highly symmetric structure, because the

ˆi

S ˆ

Sj

3.1 Introduction

35

total dimension of Hilbert space of this spin system is (2S+1)³⁰ = 6³⁰, which is astronomically large so that the complete matrix diagonalization of Hamiltonian (3.1.1) is not feasible for now. Despite this difficulty, an approximate, diagonalizable effective Hamiltonian was adopted to explain the major low-temperature properties of {Mo72Fe30} [79]. The classical version of this effective Hamiltonian represents a frustrated ground state spin configuration called the “three-sublattice model”, where the 30 spins can be grouped into three sublattices of 10 spins each. All the 10 spins in each sublattice are parallel and the unit vectors of the three sublattices are co-planar with 120˚

angular difference in the zero-field limit. Here the word “co-planar” means one can always find a plane parallel to all the unit vectors of the three sublattices. The sketch map of this model is shown in Figure 3-1-1, where the vectors of the three sublattices are indicated by three colors – red, yellow and green. Obviously, there are no nearest neighbors carrying the same color.

Corresponding to this classical three-sublattice configuration, the approximate quantum model was established to account for the quantum mechanical effects. The resulting effective Hamiltonian is reduced to interactions between the three sublattices with spin operators ŜA, ŜB and ŜC, and adopted with the absence of an external magnetic field and the spin anisotropy term [23, 79, 80], which is written as

ˆeff (ˆA ˆB ˆB ˆC ˆC ˆA) 5

= J ⋅ + ⋅ + ⋅

H S S S S S S . (3.1.2) Accordingly, the energy eigenvalues of Hamiltonian (3.1.2) are given in Ref. [79]

A A B B C C meV determined from the magnetization measurements on {Mo72Fe30} [55, 79]. The degeneracies of the ground state (SA = SB = SC = 25) and the first excited state (SA = 24, SB = SC = 25 and the permutations) are (2S+1)² and 27(2S+1)², respectively [81].

The resultant low-lying energy spectrum is shown in Figure 3-1-3, where S is the total spin quantum number. From the expression of Eq. (3.1.3), it is known that the resultant S states form a parabola for one set of SA , SB and SC. The effect of an external field appears as a Zeeman term in the Hamiltonian, which lifts the degeneracies of the individual MS = -S, …, +S substates belonging to an S state.

Chapter 3. Geometrical Spin-frustrated Molecular Magnet {Mo72Fe30}

36

Spin-level-crossing will happen upon increasing the external field gradually from zero. Its positions are shown as the arrows in Figure 3-1-4. Eventually the system will saturate in S = 75, MS = 75 state at Bsat = 30|J|/(gμB) = 17.7 Tesla [23, 79, 80]. This saturation field was confirmed by the high-field magnetization at 0.46 K [55].

0 2 4 6 8 10 12

Figure 3-1-3 Low-lying section of the magnetic excitation spectrum as calculated from the quantum rotational band model.

B (T)

Figure 3-1-4 Splitting of the ground-state MS sublevels under external magnetic field.

Arrows mark where the spin-level-crossing happens.

3.1 Introduction

37

Besides the high-field magnetization measurements, other techniques were also utilized to proof the validity of the three-sublattice model for {Mo72Fe30}. According to the quantum rotational band theory, the energy difference between the two lowest ground-state levels is only |J|/5 ≈ 0.03 meV, which is very difficult to be observed experimentally. But the energy gaps between the ground state and the first excited state were directly confirmed to be ~ 0.6 meV by means of inelastic neutron scattering measurements [81]. The muon spin relaxation and ¹H nuclear magnetic resonance measurements provide further experimental determination of this first interband gap [82]. The differential susceptibility dM/dB of {Mo72Fe30} exhibits a local minimum at approximately one-third of Bsat, which is attributed to the thermal population of competing three-sublattice spin phases [61].

3.1.3   Motivation 

{Mo72Fe30} possesses a complex, yet aesthetically beautiful molecular structure. The 30 spins with antiferromagnetic coupling between nearest neighbors provide a 3D and finite analog with periodic boundary conditions of the 2D kagome lattice, which makes {Mo72Fe30} a good model to investigate the phenomenon known as geometrical spin frustration. The three-sublattice model was developed not only for icosidodecahedron but also for some other types of polyhedral spin structures. The validity of the three-sublattice spin model for {Mo72Fe30} could be of more general significance for the studies on spin frustration in a variety of spin polyhedra.

However, the aforementioned experimental support to the three-sublattice model remained indirect, as long as there were no direct observations of the microscopic spin correlations owning to the ground-state spin structure of the {Mo72Fe30} molecule.

In this chapter, an approach to the magnetic ground state of {Mo72Fe30} is attempted by a direct observation of the spin correlation function by means of polarized neutron scattering. The low-lying magnetic excitation spectrum predicted by the quantum rotational band theory is revealed using the low-temperature specific heat. The three-sublattice spin model for {Mo72Fe30} is thus strongly supported.

Chapter 3. Geometrical Spin-frustrated Molecular Magnet {Mo72Fe30}

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