Magnetic Exchange Interactions in the Molecular Nanomagnet ${\mathrm{Mn}}_{12}$

The discovery of magnetic bistability in ${\mathrm{Mn}}_{12}$ more than 20 years ago marked the birth of molecular magnetism, an extremely fertile interdisciplinary field and a powerful route to create tailored magnetic nanostructures. However, the difficulty to determine interactions in complex polycentric molecules often prevents their understanding. ${\mathrm{Mn}}_{12}$ is an outstanding example of this difficulty: although it is the forefather and most studied of all molecular nanomagnets, an unambiguous determination of even the leading magnetic exchange interactions is still lacking. Here we exploit four-dimensional inelastic neutron scattering to portray how individual spins fluctuate around the magnetic ground state, thus fixing the exchange couplings of ${\mathrm{Mn}}_{12}$ for the first time. Our results demonstrate the power of four-dimensional inelastic neutron scattering as an unrivaled tool to characterize magnetic clusters.

Figure 1

(a) Scheme of the ${\mathrm{Mn}}_{12}{\text{−}}^t\mathrm{BuAc}$ molecule, with different lines representing the relevant different exchange interactions ($J_1$: pairs 1–4, 1–3 and equivalent; $J_2$: pair 1–11 and equivalent; $J_3$: pair 1–6 and equivalent; $J_4$: pair 6–11 and equivalent; $J_4^{\prime }$: pair 11–8 and equivalent). Red circles (ions 5–11): ${\mathrm{Mn}}^{3+}$ ions ($s=2$). Blue circles (ions 1–4): ${\mathrm{Mn}}^{4+}$ ions ($s=3/2$). Seven distinct exchange constants are allowed by the $S_4$ symmetry of the molecule, but most models assume only four parameters ($J_{1-4}$ with $J_4^{\prime }=J_4$) because of similarities in some exchange paths. (b) INS spectrum collected on LET at 1.5 K, using 15.4 meV (4.2 meV in the inset) incident neutron energy. The continuous line is the corresponding simulation with the best-fit parameters (in meV) $J_1=-1.2$, $J_2=3.2$, $J_3=6.6$, $J_4=0.55$, $J_4^{\prime }=0.30$, $d=-0.315$. Eigenstates are listed in Table S1 [26]. Peak I corresponds to a transition between states $M=\pm 10$ and $M=\pm 9$ of the ground $S=10$ multiplet. The slight asymmetry of the peak is due to the instrumental resolution function of the time of flight spectrometer. Peaks II–VI are intermultiplet transitions to different excited $S=9$ multiplets. The broad peak at 3 meV is a phonon, as demonstrated by the monotonic increase as $Q^2$ of the associated form factor.

Figure 2

(a) Form factor for the intramultiplet transition $|S=10,M=\pm 10⟩\to |S=10,M=\pm 9⟩$, i.e., $S(E,\mathbf{Q})$ for $E=1.25\mathrm{meV}$ (giant-spin excitation I). The inset shows the equivalent real-space information, that is, the distribution of the static magnetization of the giant spin over the three inequivalent Mn sites, ${\mu }_n=⟨S=10,M=10|s_{zn}|S=10,M=10⟩$. The values ${\mu }_1=-1.2$(2), ${\mu }_6=1.7$(0.15), and ${\mu }_{11}=2$(0.15) are extracted directly from the form factor and compare well with polarized neutron diffraction ($-1.17$, 1.84, 1.90) [43] and NMR ($-1.3$, 1.8, 1.8) [44] data on a slightly different variant of ${\mathrm{Mn}}_{12}$. (b) $S(E,\mathbf{Q})$ as a function of $Q_x$, $Q_y$, and $E$, and integrated over the full $Q_z$ range [45]. The energy window contains the intermultiplet peaks II and III of Fig. 1.

Figure 3

(a) INS energy spectrum [same as Fig. 1]. (b) Constant-energy cuts for $S(E,\mathbf{Q})$, integrated over the full $Q_z$ range, obtained from measurements at $T=1.5\mathrm{K}$ for incident neutron energies of 4.2 meV (first column, peak I) and 15.4 meV (peaks II–VI) [45]. Each map is normalized to its maximum. (c) Corresponding simulated maps, obtained with parameters (in meV) $J_1=-1.2(1)$, $J_2=3.2(2)$, $J_3=6.6(3)$, $J_4=0.55(5)$, $J_4^{\prime }=0.30(5)$, $d=-0.315(2)$. Eigenstates are listed in Table S1 [26]. Row (d) highlights the effect of a slight variation of exchange parameters ($J_4=J_4^{\prime }=0.42\mathrm{meV}$ is assumed). Peaks IV and V are too close in energy to extract individual maps, and only their sum is addressed. (e) Precession pattern of the individual Mn spins for excitations I, II, and III. For each excitation, arrows represent the 12 vectors $(⟨s_{xn}(t)⟩,⟨s_{yn}(t)⟩)$ describing the spatial pattern of the spins preceding around $z$, after a resonant perturbation has brought a molecule from its $M=10$ ground state into a superposition state with a small component on the corresponding excited $M=9$ state. All the spins precede with the same frequency $E/h$ and dashed circular arrows indicate the direction of the spin precessions for two representative sites. Preparing the system in an initial state with opposite $M$ would induce an opposite precession of the spins. The two panels for excitation II correspond to a pair of degenerate states (Table S1 [26]). For peaks IV, V, and VI, experimental form factors are more noisy or unresolved. Their precession pattern is not directly deduced from data, and is obtained by simulations of the best-fit Hamiltonian (Fig. S5 in [26]).