Exchange interactions and high-energy spin states in ${\mathrm{Mn}}_{12}$-acetate

We perform inelastic neutron scattering measurements on the molecular nanomagnet ${\mathrm{Mn}}_{12}$-acetate to measure the excitation spectrum up to $45\mathrm{meV}$ $\left(500\mathrm{K}\right)$. We isolate magnetic excitations in two groups at $5–6.5\mathrm{meV}$ $(60–75\mathrm{K})$ and $8–10.5\mathrm{meV}$ $(95–120\mathrm{K})$, with higher levels appearing only at $27\mathrm{meV}\left(310\mathrm{K}\right)$ and $31\mathrm{meV}\left(360\mathrm{K}\right)$. From a detailed characterization of the transition peaks we show that all of the low-energy modes appear to be separate $S=9$ excitations above the $S=10$ ground state, with the peak at $27\mathrm{meV}\left(310\mathrm{K}\right)$ corresponding to the first $S=11$ excitation. We consider a general model for the four exchange interaction parameters of the molecule. The static susceptibility is computed by high-temperature series expansion and the energy spectrum, matrix elements, and ground-state spin configuration by exact diagonalization. The theoretical results are matched with experimental observation by inclusion of cluster anisotropy parameters, revealing strong constraints on possible parameter sets. We conclude that only a model with dominant exchange couplings $J_1\sim J_2\sim 5.5\mathrm{meV}\left(65\mathrm{K}\right)$ and small couplings $J_3\sim J_4\sim 0.6\mathrm{meV}\left(7\mathrm{K}\right)$ is consistent with the experimental data.

Figure 1

(Color online) Structure of ${\mathrm{Mn}}_{12}$-acetate viewed along the $c$ axis. Large solid spheres represent ${\mathrm{Mn}}^{3+}$ ions (outer ring) and ${\mathrm{Mn}}^{4+}$ ions (inner core). All $\mathrm{Mn}$ atoms have a distorted octahedral coordination geometry.

Figure 2

Schematic representation of the magnetic exchange paths in ${\mathrm{Mn}}_{12}$-acetate as first proposed by Sessoli et al. (Ref. 16) $J_1$ couples ${\mathrm{Mn}}^{3+}$ and ${\mathrm{Mn}}^{4+}$ ions $\left(d=2.771\mathrm{\AA }\right)$, $J_{2a,b}$ corresponds to ${\mathrm{Mn}}^{3+}∕{\mathrm{Mn}}^{4+}$ ($d=3.45\mathrm{\AA }$), $J_3$ to ${\mathrm{Mn}}^{4+}∕{\mathrm{Mn}}^{4+}$ $\left(d=2.817\mathrm{\AA }\right)$, and $J_{4a,b}$ to nearest-neighbor ${\mathrm{Mn}}^{3+}$ ions ($d=3.33$ and $3.41\mathrm{\AA }$).

Figure 3

INS spectra obtained at $T=8\mathrm{K}$ on MARI at (a) $E_i=50\mathrm{meV}$, (b) $E_i=17\mathrm{meV}$, and (c) $E_i=12\mathrm{meV}$. The $Q$ range is restricted to $1⩽Q⩽2{\mathrm{\AA }}^{-1}$ in panels (a) and (b) and to $0.25⩽Q⩽1.25{\mathrm{\AA }}^{-1}$ for panel (c). Transitions are labeled as discussed in the text. In panel (c), (GS1) is a transition within the $S=10$ ground-state multiplet and (Ph) denotes a low-energy phonon mode.

Figure 4

(a) Inelastic spectra at 1.5, 10, and $20\mathrm{K}$ obtained on FOCUS at an incident wavelength $\lambda =3.1\mathrm{\AA }$. At $1.5\mathrm{K}$, transitions (Ia) and (Ib) are observed at $\hslash \omega =5.3\mathrm{meV}$ and $\hslash \omega =6.2\mathrm{meV}$, respectively. The data taken at $1.5\mathrm{K}$ are reproduced as a solid line and shifted upwards to facilitate comparison with the $20\mathrm{K}$ data. (b) Inelastic spectra at $T=1.5$, 20, and $50\mathrm{K}$ obtained on IN4 at $\lambda =2.2\mathrm{\AA }$.

Figure 5

$Q$ dependence at $1.5\mathrm{K}$ of peaks (Ia) and (Ib) obtained from the inelastic spectra shown in Fig. 4a. Data from both FOCUS and IN4 are shown after normalization of the maximum intensity at $Q\approx 1.45{\mathrm{\AA }}^{-1}$.

Figure 6

(a) INS spectra obtained on MARI at $E_i=12\mathrm{meV}$ and $T=8\mathrm{K}$ for different values of $Q$. (b) INS spectra from IN4 at $\lambda =2.2\mathrm{\AA }$ and $T=1.5\mathrm{K}$ for different values of $Q$.

Figure 7

(a) Detail of the low-energy levels inferred from our INS data (see text). After subtraction of the anisotropy shift, the lowest $S=9$ states appear at 4.0 and $4.9\mathrm{meV}$ above the $S=10$ ground state. (b) Energy levels observed by INS over the full range investigated. The spin values and anisotropy parameters of the higher levels are not known precisely.

Figure 8

Static magnetic susceptibility measured with an applied field of $0.1\mathrm{T}$ (points) and the best fit obtained by high-temperature series expansion (solid lines). The parameter set used is given in Eq. (10). The solid lines are obtained from different approximations to the eighth-order high-temperature series, and represent the error estimate for the extrapolation procedure.

Figure 9

Suitability of parameter sets giving a ground state of spin $S=10$ for fitting the high-temperature susceptibility data: (a) as a function of $J_1$ and $J_2$ for all values of $J_3$ and $J_4$ and (b) as a function of $J_3$ and $J_4$ for all values of $J_1$ and $J_2$. Darker shades indicate superior agreement with $\chi \left(T\right)$.

Figure 10

Calculated magnetization curve for a ${\mathrm{Mn}}_{12}$-acetate cluster with exchange parameters given in Eq. (10).

Figure 11

Comparison with INS data of the location of calculated magnetic excitations, obtained using the optimal parameter set (10). An anisotropy correction is applied by using Eq. (3) with a single $D$ value for all (ground and excited) states of each total spin $S=9$, 10, or 11. The experimental data was obtained on MARI at (a) low energy transfer [Figs. 3b, 3c] and (b) high-energy transfer [Fig. 3a].

Figure 12

Comparison between magnetic excitations calculated by exact diagonalization (ED) and within a spin-wave approximation using the parameter set of Eq. (10). No uniaxial anisotropy is included. Higher $S=9$ and 10 excitations not included within the three spin-wave branches enter only at the highest energy of the $S=11$ branch.