Incommensurate modulations of Bi-III and Sb-II

Single-crystal diffraction data from the incommensurate host-guest composite structure of Bi-III—stable between 2.8 and $7.7\mathrm{GPa}$—are found to contain very weak satellite reflections arising from structural modulations attributed to interactions between the host and guest structures. The principal effects of the modulation are a quasipairing of the guest atoms along their chains and some reduction in the closest approach of the host and guest atoms. It is suggested that these modulations play a key role in the stability of the structure. A similar modulated structure fits powder-diffraction data from Sb-II.

Figure 1

(Color online) Incommensurate host-guest structure of Bi-III and Sb-II as viewed down the tetragonal $c$ axis. The host atoms are shown as light spheres, and the guest atom chains as dark spheres.

Figure 2

Powder-diffraction profile from Sb-II at $10.3\mathrm{GPa}$. Tick marks beneath the main profile mark the calculated position of reflections from the basic composite structure. The inset shows additional weak satellite reflections, and the tick marks mark the calculated positions of (a) the basic composite and (b) the satellite reflections. Asterisks mark the two satellite reflections detected in the previous study (Ref. 1).

Figure 3

Displacements (in fractional units) of the Bi1 host atoms and Bi2 guest atoms away from their unmodulated positions, as determined from the MEM refinement, as a function of $x_{14}$ and $x_{24}$ (Ref. 21), respectively. $x_{14}$ can be considered as the fractional coordinate of the host atoms in the guest unit cell, given by $x_{14}=z\left(\mathrm{Bi}1\right)c_H∕c_G$, and $x_{24}$ as the fractional coordinate of the guest atoms in the host unit cell, given by $x_{24}=z\left(\mathrm{Bi}2\right)c_G∕c_H$. The displacements shown apply (a) to the host atoms at (0.1532, $-0.3468$, $z=0,1,2,\dots$) and, with the sign of the $x$ and $y$ deviations reversed, to the host atoms at ($-0.1532$, $-0.3468$, $z=\frac{1}{2},\frac{3}{2},\frac{5}{2},\dots$), and (b) to the guest atoms at (0,0, $z=0,1,2,\dots$) and at ($\frac{1}{2},\frac{1}{2}$, $z=\frac{1}{2},\frac{3}{2},\frac{5}{2},\dots$). (The $x_{14}$ values for the seven layers of Bi1 atoms shown in Fig. 4 are 0.000, 0.654, 0.309, 0.964, 0.618, 0.273, and 0.927 from bottom to top. The $x_{24}$ values for the five guest atoms in Fig. 4 can be read directly as 0.000, 0.764, 0.528, 0.292, and 0.056.)

Figure 4

Part of the Bi-III structure plotted over the range $-0.5<x<0.5$, $-0.5<y<0.5$, and $0⩽z<3.2$, where $z$ is referred to the host unit cell. Host atoms (light spheres) and guest atoms (dark spheres) are all shown in their unmodulated positions. Some atoms are labeled with their $z$ coordinates relative to the host unit cell. The host atoms referred to in Fig. 3 are those furthest to the left in each layer. The host atoms form squares in the $x\text{−}y$ plane of side $d$, oriented at angle $\theta$ around $z$. Arrows on the guest atoms show (much exaggerated) the relative magnitudes and directions of the modulation displacements derived from Fig. 3. The values of these displacements $\Delta z$ (Å) are tabulated, with also the modulations $\Delta d$ (Å) in $d$ and $\Delta \theta$ (°) in $\theta$ for the host atoms.