Long-range three-dimensional magnetic structures of the spin $S$=1 hexamer cluster fedotovite-like $A_2{\mathrm{Cu}}_3\mathrm{O}$(${\mathrm{SO}}_4{)}_3$ ($A_2$=${\mathrm{K}}_2$, NaK, ${\mathrm{Na}}_2$): A neutron diffraction study

The crystal and magnetic structures of the spin $S=1$ hexamer cluster fedotovite-like $A_2{\mathrm{Cu}}_3\mathrm{O}{\left({\mathrm{SO}}_4\right)}_3$ ($A_2={\mathrm{K}}_2$, NaK, ${\mathrm{Na}}_2$) were studied by neutron powder diffraction at temperatures 1.6–290 K. The crystal structures in all compounds are well refined in the monoclinic space group C2/c. The basic magnetic units of the compounds are copper hexamers, which are coupled by weak superexchange interactions giving rise to three-dimensional long-range magnetic order below $3.0<T_N<4.7$ K. We have found that for $A_2={\mathrm{K}}_2$ and NaK the propagation vector of the magnetic structure is $\mathit{k}=[0,0,0]$, and the coupling of the Cu hexamers is ferromagnetic (FM) along the $ab$ diagonal and antiferromagnetic along the $bc$ diagonal. In contrast, for $A=\mathrm{Na}$ the propagation vector is $\mathit{k}=[0,1,0]$, and the Cu hexamers are coupled antiferromagnetically (AFM) along the $ab$ diagonal. The hexamers are formed by three Cu pairs arranged along the $b$ axis. The calculated spin expectation values $\langle s\rangle$ for the simplest symmetric spin Hamiltonian (obtained from inelastic neutron spectroscopy) of the isolated hexamers in the mean field amounted to $\langle s\rangle =3/8$ for the side spins Cu1 and Cu2 and $\langle s\rangle =1/4$ for Cu3 in the middle. The Cu spins are FM coupled in pairs and AFM between neighboring pairs. The experimental magnetic moments of the ${\mathrm{Cu}}^{2+}$ ions turn out to be not completely collinear due to spin frustrations within the weak interhexamer interactions. The sizes of magnetic moments of Cu in the hexamers determined from the diffraction data are in fair agreement with the calculated values.

Figure 1

Two Cu hexamers out of four in the unit cell are shown for the $A_2=\mathrm{NaK}$ sample. For better visibility, some atoms in hexamers are extended to neighboring cells. The hexamers are related by inversion. Two other hexamers, which are not shown, are related by C-centering translation ($1/2,1/2,0$). The bond lengths in Å are indicated for the bottom hexamer. $J_i$ ($i=1–7$) show the exchange coupling constants used in Sec. 4 and in Eq. (1).

Figure 2

The Rietveld refinement pattern and the difference plot between observed and calculated intensities of the neutron diffraction data for the sample $A_2{\mathrm{Cu}}_3\mathrm{O}{\left({\mathrm{SO}}_4\right)}_3$ ($A_2=\mathrm{NaK}$) at $T=1.6$ K measured at HRPT with the wavelength $\lambda =2.45$ Å. The row of tick marks shows the Bragg peak positions. The difference between observed and calculated intensities is shown by the dotted blue line.

Figure 3

The crystal structure in approximately one unit cell in the sample with $A=\mathrm{Na}$. Six hexamers can be seen; polyatomic anions (${\mathrm{SO}}_4{)}^{2-}$ are shown as tetrahedra. The distances between tetramers are indicated. The connectivity between the hexamers is shown in more detail in Fig. 12.

Figure 4

The Rietveld refinement pattern and the difference plot between observed and calculated intensities of the neutron diffraction data for the sample $A_2{\mathrm{Cu}}_3\mathrm{O}{\left({\mathrm{SO}}_4\right)}_3$ ($A_2=\mathrm{NaK}$) at $T=6$ K at CNCS/SNS with the wavelength $\lambda =4.96$ Å. All structure parameters were fixed by the values obtained from HRPT data (Fig. 2). The row of tick marks shows the Bragg peak positions. The difference between observed and calculated intensities is shown by the dotted blue line.

Figure 5

The raw neutron diffraction patterns for the sample $A_2{\mathrm{Cu}}_3\mathrm{O}{\left({\mathrm{SO}}_4\right)}_3$ ($A=\mathrm{K}$) at $T=1.6$ K (blue dotted line and open symbols), 6 K (red dashed line and closed symbols), and difference pattern containing purely magnetic contribution (black line and closed symbols), measured at CNCS/SNS with the wavelength $\lambda =4.96$ Å. A constant 50 has been added to the difference pattern. Note, the $y$ axis has logarithmic scale.

Figure 6

The model-free Le Bail fit of the difference pattern for the sample $A_2{\mathrm{Cu}}_3\mathrm{O}{\left({\mathrm{SO}}_4\right)}_3$ ($A=\mathrm{K}$) containing purely magnetic contribution, measured at CNCS/SNS with the wavelength $\lambda =4.96$ Å. The row of tick marks shows the Bragg peak positions. The difference between the observed and calculated intensities is shown by the dotted blue line. The reliability factors [15] $R_p, R_{wp}, R_{\exp }, {\chi }^2$ are 6.15, 5.88, 4.65, 1.60.

Figure 7

The magnetic structures formed by the hexamers in MSGs (top) C2'/c and (bottom) P2_1/c.1'_C[C2/c] (UNI symbol [23]) or C_P2'/c (OG symbol). The number of atoms is 24 in one unit cell, but some atoms are shown outside the unit cell to better represent the four hexamers. The hexamers related by inversion in C2'/c settings along $bc$ diagonal (those that are closest and that are arranged approximately in vertical direction in the figure) are coupled AFM. Two other hexamers related by C-centering translation ($1/2,1/2,0$) are coupled FM for the samples with $A_2={\mathrm{K}}_2$ and NaK (left and right top structures), but AFM coupled for the sample with $A=\mathrm{Na}$ (bottom). The side spin pairs Cu1 and Cu2 are in green and red; the central Cu3 pair is in blue.

Figure 8

The fit to C2'/c model of the difference pattern for the sample $A_2{\mathrm{Cu}}_3\mathrm{O}{\left({\mathrm{SO}}_4\right)}_3$ ($A_2=\mathrm{NaK}$) containing purely magnetic contribution, measured at CNCS/SNS with the wavelength $\lambda =4.96$ Å. The row of tick marks shows the Bragg peak positions. The difference between the observed and calculated intensities is shown by the dotted blue line.

Figure 9

The fit to C2'/c model of the difference pattern for the sample $A_2{\mathrm{Cu}}_3\mathrm{O}{\left({\mathrm{SO}}_4\right)}_3$ ($A=\mathrm{K}$) containing purely magnetic contribution, measured at CNCS/SNS with the wavelength $\lambda =4.96$ Å. The row of tick marks shows the Bragg peak positions. The difference between the observed and calculated intensities is shown by the dotted blue line. The reliability factors [15] $R_p, R_{wp}, R_{\exp }, {\chi }^2$ are 6.45, 6.04, 4.65, 1.69.

Figure 10

The fit to the model P2_1/c.1'_C[C2/c] [basis transformation from parent (1,0,0), (0,1,0), (0,0,1); $1/4,1/4,0$] of the difference pattern for the sample $A_2{\mathrm{Cu}}_3\mathrm{O}{\left({\mathrm{SO}}_4\right)}_3$ ($A=\mathrm{Na}$) containing purely magnetic contribution, measured at CNCS/SNS with the wavelength $\lambda =4.96$ Å. The row of tick marks shows the Bragg peak positions. The difference between the observed and calculated intensities is shown by the dotted blue line.

Figure 11

The magnetic structure in the sample with $A=\mathrm{K}$ (C2'/c) in two projections. The $ac$ and $bc$ layers are FM stacked along the $b$- and $a$-axis directions, respectively. The side spin pairs Cu1 and Cu2 are in green and red; the central Cu3 pair is in blue.

Figure 12

The magnetic structure in the sample with $A=\mathrm{K}$ (C2'/c) showing the connectivity between hexamers. Each hexamer unit is outlined by dashed green box. The side spin pairs Cu1 and Cu2 are in green and red; the central Cu3 pair is in blue. The black arrows indicate the bonds between side hexamer spins Cu1 and Cu2 of different hexamers. Three bonds are labeled in the notation from Ref. [9]. Five hexamers are shown with the transforming symmetry operations indicated by the blue dashed lines. The hexamers that are AFM coupled are related by inversion $-$1'. The ones that are FM coupled are related by C-centering ($1/2,1/2,0$) or just translation (0,1,0).