Structural properties of potassium-ammonia liquids in graphite.

Abstract

The in-plane x-ray diffuse scattering ${I}_{\ensuremath{\perp}}$(q) from the metal-ammonia K-${\mathrm{NH}}_{3}$ liquid layers in the ternary graphite intercalation compound K(${\mathrm{NH}}_{3}$${)}_{4.33}$${\mathrm{C}}_{24}$ has been measured at room temperature and yields a liquid structure factor which is dominated by a relatively sharp peak at q=${q}_{\ensuremath{\perp}}$=0.88 A${\r{}}^{\mathrm{\ensuremath{-}}1}$ and a broad, more intense peak at ${q}_{\ensuremath{\perp}}$=2.00 A${\r{}}^{\mathrm{\ensuremath{-}}1}$. We have also measured a ${c}^{\mathrm{*}}$ scan I(${q}_{\ensuremath{\perp}}$=0.88 ) which is symmetric and monotonically decreasing about the position ${q}_{\ensuremath{\perp}}$=0.88 =0.This ${c}^{\mathrm{*}}$ scan shows that the K?(hy${\mathrm{NH}}_{3}$ liquid layers in different graphite host galleries are uncorrelated and consequently that the liquid is two dimensional (2D). The structure of this 2D liquid has been modeled using computer-generated planar distributions of ``hard'' ${\mathrm{K}}^{+}$ and ${\mathrm{NH}}_{3}$ discs. In each model, the ${K}^{+}$ discs had fixed identical radii, but the ${\mathrm{NH}}_{3}$ discs had radii which were randomly selected from a probability distribution that reflects the distribution in the tilt angle between the threefold axis of the ${\mathrm{NH}}_{3}$ molecule and the c axis of the graphite host.In addition, each model structure was N-fold coordinated, 0\ensuremath{\le}N\ensuremath{\le}4, such that every ${\mathrm{K}}^{+}$ ion was bound to exactly N ${\mathrm{NH}}_{3}$ molecules in a symmetric planar cluster. These rigid clusters and the remaining bare ``spacer'' ${\mathrm{NH}}_{3}$ molecules were assembled in such a way as to maximize the density of the resultant distribution while simultaneously satisfying the observed stoichiometry of the specimen. Theoretical expressions for the in-plane and ${c}^{\mathrm{*}}$ contributions to the diffuse scattering from a 2D liquid in the galleries of a layer structure such as graphite were deduced and applied to the computer-generated models with fourfold, threefold, and zerofold coordination. Wave-vector-dependent scattering factors for both the ${\mathrm{K}}^{+}$ and ${\mathrm{NH}}_{3}$ discs were incorporated in the calculations which involved 2165 ${\mathrm{NH}}_{3}$ particles and 500 ${\mathrm{K}}^{+}$ particles. The calculated diffraction patterns became independent of the size of the models when the total number of particles exceeded approximately 200. We found that the in-plane diffraction pattern calculated from the fourfold coordinated structure gave a very good semiquantitative fit to the observed diffraction pattern, whereas the zerofold and threefold structures gave inferior fits.The fourfold-coordinated structure is shown to be the planar analogue of the sixfold (octahedral) coordination that is found in bulk (3D) K?(hy${\mathrm{NH}}_{3}$ metal-ammonia solutions. The ${\mathrm{K}}^{+\mathrm{?}}$(hy${\mathrm{K}}^{+}$, ${\mathrm{NH}}_{3\mathrm{?}}$(hy${\mathrm{NH}}_{3}$, and ${\mathrm{K}}^{+\mathrm{?}}$(hy${\mathrm{NH}}_{3}$ pair contributions to ${I}_{\ensuremath{\perp}}$(q) have been calculated from our structural models and reveal several subtle connections between features in the structure and corresponding shapes in the diffraction pattern. In particular, the ${\mathrm{K}}^{+\mathrm{?}}$(hy${\mathrm{NH}}_{3}$ contribution is dominated by a Bessel function ${J}_{0}$(q(${R}_{\mathrm{mn}}$=2.89 A\r{})), where 2.89 A\r{} is the predominant K?(hy${\mathrm{NH}}_{3}$ nearest-neighbor distance. The full particle-particle radial distribution functions (RDF's) and the partial ${K}^{+?}$(${\mathrm{hyK}}^{+}$ RDF have been determined for each of the structural models we constructed and from the former the average areal densities, ${\ensuremath{\rho}}_{0}$, have been deduced.Only the fourfold-coordinated structure yields a density, ${\ensuremath{\rho}}_{0}$=0.10 particles/A${\r{}}^{2}$ which is compatible with the experimentally determined density of our specimen 0.09\ifmmode\pm\else\textpm\fi{}0.01 particles/A${\r{}}^{2}$. The particle-particle RDF's also show that the short-range structure of the K?(hy${\mathrm{NH}}_{3}$ liquid is rhombohedral with a rhombohedral angle \ensuremath{\alpha} between 60\ifmmode^\circ\else\textdegree\fi{} and 90\ifmmode^\circ\else\textdegree\fi{}. The ${\mathrm{K}}^{+\mathrm{?}}$(hy${\mathrm{K}}^{+}$ RDF calculated from the fourfold-coordinated structure shows pronounced long-range multineighbor ${\mathrm{K}}^{+\mathrm{?}}$(hy${\mathrm{K}}^{+}$ correlations out to 24 A\r{}, whereas the corresponding RDF's for the threefold- and zerofold-coordinated structures show, respectively, only near-neighbor and short-range multineighbor correlations.