Aperiodicity, structure, and dynamics in $\text{Ni}{\left(\text{CN}\right)}_2$

By combining the results of both x-ray diffraction and neutron total-scattering experiments, we show that $\text{Ni}{\left(\text{CN}\right)}_2$ exhibits long-range structural order only in two dimensions, with no true periodicity perpendicular to its gridlike layers. Reverse Monte Carlo analysis gives an experimental distinction between M–C and M–N bond lengths in a homometallic cyanide framework and identifies the vibrational modes responsible for anomalous positive and negative thermal expansion in the title compound.

Figure 1

(Color online) (a) Experimental and [(b)–(d)] calculated x-ray diffraction patterns for $\text{Ni}{\left(\text{CN}\right)}_2$ $\left[\lambda =1.0000\left(1\right)\text{Å}\right]$: (b) $P{4}_2/mmc$ model from Ref. 7; the new disordered stacking model, (c) without, and (d) with Gaussian shear distribution perpendicular to [001]. The superstructure and diffuse scattering peaks that arise from the disordered stacking along $\mathbf{c}$ are indicated by asterisks (see Appendix for derivation).

Figure 2

(Color online) (a) The four possible square-grid orientations referred to in the text, (b) an example stacking sequence, and (c) the average structure $I4/mmm$ unit cell (whose relative orientation with respect to the axes in (a) is shown in red); thermal ellipsoids were calculated from the real-space RMC fits to the 298 K neutron data (Ref. 7) and are shown at the 50% probability level. Square planar Ni atoms in green; linearly-bridged C/N atoms in purple.

Figure 3

(Color online) Experimental (top) and calculated x-ray diffraction patterns for $\text{Ni}{\left(\text{CN}\right)}_2$ using different values of the second-layer correlation variable $p$.

Figure 4

(Color online) Illustration of the sideways displacements of $\text{Ni}{\left(\text{CN}\right)}_2$ layers involved in the shear strain.

Figure 5

Calculated (top) and experimental (bottom) neutron-scattering patterns.

Figure 6

(Color online) RMC fits (red lines) to (a) $D\left(r\right)$ and (b) $QF\left(Q\right)$ neutron data (black lines) using the $P{4}_2/mmc$ (upper set in each panel; shifted by 12 units) and disordered $I4/mmm$ (bottom set in each panel) models; difference curves (data$-$fit; shifted by 4 units) shown in blue.

Figure 7

(Color online) RMC atomic positional distributions obtained by “collapsing” equilibrium RMC configurations onto a single cell of the $P{4}_2/mmc$ model: (a) and (b) are from the random stacking model, viewed down [010] and [001], respectively; (c) and (d) are for the $P{4}_2/mmc$ model viewed down [010] and [001], respectively. In both (b) and (d) only the single sheet near $z=\frac{1}{2}$ is shown.

Figure 8

(Color online) Ni–C (red) and Ni–N (blue) distributions in $\text{Ni}{\left(\text{CN}\right)}_2$ (a) from the original RMC refinements, (b) after swapping N and C atoms without further refinement, and (c) after subsequent RMC minimization.

Figure 9

(Color online) Lowest-energy dispersionless vibrational modes in $\text{Ni}{\left(\text{CN}\right)}_2$.

Figure 10

(Color online) (a) Reciprocal-space and (b) real-space unit cells in $\text{Ni}{\left(\text{CN}\right)}_2$. In both cases, the red cell corresponds to the standard crystallographic setting, and the blue cell is given to coincide with the primitive square lattice of an individual $\text{Ni}{\left(\text{CN}\right)}_2$ sheet. Columns of diffuse scattering are predicted for reciprocal-lattice vectors $\mathbf{Q}=\frac{1}{2}\left(hk\xi \right)$ ($h$, $k$ odd).