24-spin clusters in the mineral boleite ${\mathrm{KPb}}_{26}{\mathrm{Ag}}_9{\mathrm{Cu}}_{24}{\mathrm{Cl}}_{62}{\left(\mathrm{OH}\right)}_{48}$

The crystal structure of the mineral boleite contains clusters of 24 $S=1/2{\mathrm{Cu}}^{2+}$ ions that have the shape of a truncated cube formed of eight trimers connected by edges. Susceptibility measurements and exact diagonalization calculations suggest that there are strong antiferromagnetic intratrimer interactions, such that effective $S=1/2$ degrees of freedom emerge on the trimers below $T\lesssim 100$ K. Weaker intertrimer interactions lead to the formation of a singlet ground state for these effective spins at $T\lesssim 5$ K. The clusters in boleite offer a situation similar to single molecule magnetism, accessible to both experiment and numerics, in which the interplay of quantum spins, geometric frustration, spin entanglement, and mesoscopic system size can be studied.

Figure 1

The full unit cell of mineral boleite contains one formula unit of ${\mathrm{KPb}}_{26}{\mathrm{Ag}}_9{\mathrm{Cu}}_{24}{\mathrm{Cl}}_{62}\left({\mathrm{OH}}_{48}\right)$ (a). Within this unit cell is a cluster of ${\mathrm{Cu}}^{2+}$ with the form of a truncated cube (b). The two closest magnetic interactions ($J_1$ and $J_2$) are mediated by different Cu-O configurations, as described in the text (c).

Figure 2

The molar magnetic susceptibility (a) and inverse susceptibility (b) of fifteen boleite crystal pieces. The inset of (a) shows the two types of behavior: Type I crystals have a weak maximum, while type II do not. The inverse susceptibility in panel (b) shows two Curie-Weiss laws (dashed lines), as fitted to data from sample K. A fraction of the measured data points are shown with enlarged symbols to highlight the different crystals. The symbol shape indicates the instrument used to measure the susceptibility, as given in Appendix pp1.

Figure 3

The real part of magnetic $ac$ susceptibility from crystal P as a function of field at two different temperatures. The sharp peak can be fitted by an expression for a simple paramagnet [Eq. (2)].

Figure 4

Comparison of the lowest 50 excitation energies of the boleite system, as predicted by Lanczos (RLexact) and projectional model calculations (a). The susceptibility of crystal K compared to the trimer, projectional, and the combined CPT model, with $J_1=16.8$ meV and $J_2=2.3$ meV, with inset showing the low temperature behavior (b).

Figure 5

The susceptibility of crystal K with (intrinsic K) and without (raw K) a paramagnetic contribution of 1.4 paramagnetic spins per unit cell subtracted. The resulting corrected data is compared to the projectional, trimer, and combined CPT model, $J_1=17.2$ meV and $J_2=3.2$ meV, with the inset showing low temperature behavior.

Figure 6

The estimated intrinsic susceptibility of each type I (a) or type II (b) crystal measured using the MPMS (triangles) and PPMS (circles) at PSI, as found by fitting a combination of CPT model and Curie tail and subtracting the tail from the raw data (only every second data point is shown). The resulting data is plotted together with the relevant parametrization of the CPT model (lines). The best fit values for each crystal are given in Table 1.

Figure 7

The effective magnetic moment of crystal K calculated from the measured susceptibility [i.e., $\mu \left(T\right)=\sqrt{(3k/N{\mu }_B^2)\chi T}]$ with (intrinsic) and without (raw) contribution of 1.3 paramagnetic spins per unit cell subtracted. The corrected $\mu \left(T\right)$ is plotted together with the trimer, projectional, and CPT model with $J_1=17.2$ meV and $J_2=3.2$ meV.

Figure 8

Magnetization of boleite crystal C4 as function of applied magnetic field at $T=1.8$ K. A straight line $\left[f\right(x)=ax]$ and a paramagnetic model, described in Eq. (1), with 0.8 paramagnetic spin per unit cell are superimposed on the crystal data (a). The paramagnetic model is subtracted from the raw crystal data and the resulting intrinsic magnetization is plotted together with the projectional model for $J_2=3.3$ meV (b). (In these figures only every third experimental data point is shown.)

Figure 9

Representative x-ray Laue pictures from crystals B and K.

Figure 10

Rietveld refinements of the boleite structure against synchrotron x-ray powder diffraction data collected for scrapings of sample C (a) and H (b). Both refinements have boleite and silicon phases with peak positions indicated by ticks [no silicon peaks are within the range of panel (b)]. Although the boleite structure describes the data from sample C well, the unindexed peaks in the low angle part of the diffraction pattern for sample H shows that at least one other unidentifiable material is present.

Figure 11

Linear correlation of the measured and calculated structure factors from the refinement of the boleite structure against single crystal neutron diffraction data measured using sample (b) at $T=6$ K.

Figure 12

Inelastic neutron scattering from boleite measured at IN4. The sample was continuously rotated through 360 degrees while measuring at $T=1.5$ K (a) and $T=250$ K (b). The data has been normalized with the Bose population factor. The color scales are in arbitrary units. Solid black lines mark the position of 1D cuts through the data and the dashed lines show the integration boundaries. 1D cuts at $q=3.0\pm 0.4\AA$ for $T=1.5$ K (blue symbols) and $T=250$ K (red symbols) (also with Bose factor normalization), plotted together with the difference between the two $\mathrm{\Delta }I=I_{1.5\text{K}}-I_{250\text{K}}$ (green symbols, with scale on right axis) (c).