Quantized antiferromagnetic spin waves in the molecular Heisenberg ring ${\text{CsFe}}_8$

We report on inelastic neutron-scattering (INS) measurements on the molecular spin ring ${\text{CsFe}}_8$, in which eight spin-5/2 Fe(III) ions are coupled by nearest-neighbor antiferromagnetic Heisenberg interaction. We have recorded INS data on a nondeuterated powder sample up to high energies at the time-of-flight spectrometers FOCUS at PSI and MARI at ISIS, which clearly show the excitation of spin waves in the ring. Due to the small number of spin sites, the spin-wave dispersion relation is not continuous but quantized. Furthermore, the system exhibits a gap between the ground state and the first excited state. We have modeled our data using exact diagonalization of a Heisenberg-exchange Hamiltonian together with a small single-ion anisotropy term. Due to the molecule’s symmetry, only two parameters $J$ and $D$ are needed to obtain excellent agreement with the data. The results can be well described within the framework of the rotational-band model as well as antiferromagnetic spin-wave theories.

Figure 1

(Color online) Crystal structure of ${\text{CsFe}}_8$ [with Fe as dark gray spheres (red online) and H atoms omitted].

Figure 2

INS spectra taken on FOCUS at a temperature of $T=1.4\text{K}$ and wavelengths $\lambda =2.3\text{Å}$ (main panel) and $3.8\text{Å}$ (inset).

Figure 3

(Color online) Temperature dependence of the INS spectra as recorded on MARI for incident neutron energies of 25 meV (main panel) and 15 meV (inset). Intensity was summed over all detector banks.

Figure 4

(Color online) $S\left(Q,w\right)$ plots for the temperatures $T=5\text{K}$ (upper plots) and 58 K (lower plots) for the two incident neutron energies $E_i=15\text{meV}$ (left) and 25 meV (right) as measured on MARI. The intensity is color coded $\left(\text{blue}=\text{low},\text{red}=\text{high}\text{intensity}\right)$; the color scale is identical for each of the four plots.

Figure 5

(Color online) $Q$ slices for (a) the $E_i=25\text{meV}$ and (b) the $E_i=15\text{meV}$ data at $T=5\text{K}$. In panel (a), the lowest (black), middle (red), and upper (green) curves correspond to $Q∊\left[1.0,2.5\right[$, [2.5,4.0[, and $\left[4.0,5.5\right[{\text{Å}}^{-1}$, and in panel (b) to $Q∊\left[1.0,2.0\right[$, [2.0,3.0[, and $\left[3.0,4.0\right]{\text{Å}}^{-1}$. The curves have been offset in order to improve visibility. The gray curves represent the Bose-factor corrected 58 K data as discussed in the text. The solid lines at the bottom are calculated INS spectra obtained with the best-fit parameters. The spikes (red) mark the exact position and relative scattering strengths of the individual INS transitions, the solid (black) curve was obtained by convolution with a Gaussian accounting for the experimental resolution.

Figure 6

(Color online) (a) $S\left(Q,\omega \right)$ plot of the $E_i=25\text{meV}$ MARI data at $T=5\text{K}$ after subtraction of the phonon background as described in the text. (b) $S\left(Q,\omega \right)$ plot $\left(T=5\text{K}\right)$ as calculated from the spin Hamiltonian (3) with the parameters $J=-1.79\text{meV}$ and $D=-0.05\text{meV}$ (a Gaussian linewidth of 0.72 meV was used).

Figure 7

(Color online) (a) Dependence of the INS spectrum on $J$, which is varied by a maximum of $\pm 10\mathrm{\%}$ around a center value of $J=-1.79\text{meV}$ (red curve). The anisotropy is $D=-0.05\text{meV}$. (b) Dependence of the INS spectrum on $D$, which is varied by a maximum $\pm 50\mathrm{\%}$ around a center value of $D=-0.05\text{meV}$ (red curve). The coupling is $J=-1.79\text{meV}$. (c) Dependence of the INS spectrum on a modulation of the $J$ values along the ring. For all curves the anisotropy is $D=-0.05\text{meV}$. Red (solid) curve: no modulation, $J_1=J_2=-1.79\text{meV}$. Blue (short dashed) curve: $J_1=-2.14\text{meV}$ and $J_2=-1.45\text{meV}$. Black (long dashed) curve: $J_1=-2.48\text{meV}$ and $J_2=-1.10\text{meV}$.

Figure 8

(Color online) Calculated low-energy spectrum of Hamiltonian (3) with $J=-1.79\text{meV}$ and $D=-0.05\text{meV}$. Only states in the sector $S\le 2$ are plotted. The energy of the ground state was set to zero. Due to the anisotropy term, the degeneracy in $M$ of each $S$ multiplet is partially lifted. The relevant INS transitions are plotted as solid green (dashed red) arrows, marking transitions to the $E$ band ($L$ band).

Figure 9

(Color online) Spin-wave excitation energies as calculated by exact numerical diagonalization (stars) normalized to $\left|J\right|$ and the different SWTs (solid, dashed, and dotted lines) as discussed in the text. The abbreviations are LSWT: linear SWT, ISWT: interacting SWT, IMSWT: full-diagonalization interacting modified SWT, $\text{LSWT}+{\Delta }_c$: linear SWT plus classical gap ${\Delta }_c$.