Rhombohedral symmetry

Structural Analysis A logical initial choice for the rhombohedral space group is R3c (which is the highest symmetry rhombohedral sub-group of Fm3c). However, for ease of comparison, the face-centered rhombohedral space group R,3c is used with a ... 

So-called ternary or mixed oxide systems based on the rock salt structure are well established. Thus LiFe02 exists both in a disordered form in which Li and Fe are randomly distributed on the metal sites, and as an ordered variant with tetragonal symmetry. Rhombohedral distortions are found for LiNi02, LiV02, and NaFe02 in which the cations are ordered on different sublattices. The HCP counterpart of the sodium chloride structure, the so-called Nickel Arsenide stucture (5ee Chalcogenides Solid-state Chewistr, is found only for the heavier members of the Oxygen family. 

For structures crystallizing in the lower symmetry rhombohedral Laue group, in addition to obverse/reverse twinning, the twofold axis parallel to a may act as a further twin law (matrix 01010000—1). In this case the twirmed reflection data file will contain up to four contributions to each observed intensity. Reflections that are only present for the obverse setting contain the two components kh — I and hkl with batch numbers —3 and 1, while reflections with I = 3n contain the four components —k — h — l,kh — l,—h — kl, and hkl with the batch numbers assigned as -4, -3, -2, and 1 (see the second example in 7.8.3). 

Triclinic Rhombohedral Three unequal axes intersecting obliquely Two equal axes making equal angle with each other No planes or axes of symmetry a b c y 90°... 

Also known for some time is a phase transition at low temperature (111K), observed in studies with various methods (NQR, elasticity measurement by ultrasound, Raman spectrometry) 112 temperature-dependent neutron diffraction showed the phase transition to be caused by an antiphase rotation of adjacent anions around the threefold axis ([111] in the cubic cell) and to lower the symmetry from cubic to rhombohedral (Ric). As shown by inelastic neutron scattering, this phase transition is driven by a low-frequency rotatory soft mode (0.288 THz 9.61 cm / 298 K) 113 a more recent NQR study revealed a small hysteresis and hence first-order character of this transition.114 This rhombohedral structure is adopted by Rb2Hg(CN)4 already at room temperature (rav(Hg—C) 218.6, rav(C—N) 114.0 pm for two independent cyano groups), and the analogous phase transition to the cubic structure occurs at 398 K.115... 

Yet another type of canal structure has been reported for the urea inclusion compound of 1,4-dichlorobutane 51). Even though the canals are pseudo-hexagonal in dimension, there is a significant difference in their symmetry. The host lattice is orthorhombic, space group Pbcn. The difference lies in the directions of the six pseudo-3j helices of host molecules around the walls of each canal the sequence is cyc/o-RRRLLL, as opposed to cyc/o-RRRRRR in the hexagonal inclusion compounds and cvc/o-RLRLRL in the rhombohedral. This orthorhombic host structure probably occurs also in the urea inclusion compounds with 1,5-dichloropentane and 1,6-dibromohexane 51). 

There are two other phases that are similar to (3- and (3"-alumina but are built from spinel blocks that are six close-packed oxygen layers thick. The material (3" -alumina is the analog of (3-alumina, with the spinel blocks related by 180° mirror-plane (hexagonal) symmetry, while the phase (3""-alumina, the analog of (3"-alumina, has blocks related by 120° rotation and rhombohedral symmetry. 

Cl 6c +(u,u,u +m, + m, + m + w, + w, +m) w= This structure has a rhombohedral symmetry but is usually described in terms of a triple-volume hexagonal cell, which makes comparison with the idealized Cdl2 structure simpler. The idealized unit cell is adequate for the purposes of this book. In this representation the anion layers are in cubic closest packing. .. ABC ABC ABC... The metal and nonmetal stacking sequence is... 

Rhombic prism lattice, 8 114t Rhombohedral structure, of ferroelectric crystals, 11 95, 96 Rhombohedron lattice, 8 114t Rhomboidal symmetry, 8 114t Rhone-Poulenc process, 24 482, 485 Rhovanil extra pure vanillin, 25 548t, 549-550... 

Bookin and Drits [98-100] have also described the large number of possible six-layer poly types, some of which have rhombohedral symmetry (6R) and the remainder hexagonal symmetry (6H). 

The individual satelhte transitions cannot be resolved above Tc due to exchange of Ti ions between off-center sites. Discrimination between tetragonal, orthorhombic, and rhombohedral breaking of the cubic symmetry can be obtained from the angular dependence of the second moments M2 of the satelhte background [9]. 

Measurements of NMR for Ti, Ti [33], and Sr [34,35] were carried out for STO 16 and STO 18-96. Ti and Sr nuclear magnetic resonance spectra provide direct evidence for Ti disorder even in the cubic phase and show that the ferroelectric transition at Tc = 25 K occurs in two steps. Below 70 K, rhomb ohedral polar clusters are formed in the tetragonal matrix. These clusters subsequently grow in concentration, freeze out, and percolate, leading to an inhomogeneous ferroelectric state below Tc. This shows that the elusive ferroelectric transition in STO 18 is indeed connected with local symmetry lowering and impHes the existence of an order-disorder component in addition to the displacive soft mode [33-35]. Rhombohedral clusters, Ti disorder, and a two-component state are found in the so-called quantum paraelectric... 

Another zinc-utilizing enzyme is carbonate/dehydratase C (Kannan et al., 1972). Here, the zinc is firmly bound by three histidyl side chains and a water molecule or a hydroxyl ion (Fig. 27). The coordination is that of a distorted tetrahedron. Metals such as Cu(II), Co(Il), and Mn(ll) bind at the same site as zinc. Hg(II) also binds near, but not precisely at, this site (Kannan et al., 1972). Horse liver alcohol dehydrogenase (Schneider et al., 1983) contains two zinc sites, one catalytic and one noncatalytic. X-Ray studies showed that the catalytic Zn(II), bound tetrahedrally to two cysteines, one histidine, and water (or hydroxyl), can be replaced by Co(II) and that the tetrahedral geometry is maintained. This is also true with Ni(Il). Insulin also binds zinc (Adams etai, 1969 Bordas etal., 1983) and forms rhombohedral 2Zn insulin crystals. The coordination of the zinc consists of three symmetry-related histidines (from BIO) and three symmetry-related water molecules. These give an octahedral complex... 

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