Rhombohedral lattices

Two modifications are known for polonium. At room temperature a-polonium is stable it has a cubic-primitive structure, every atom having an exact octahedral coordination (Fig. 2.4, p. 7). This is a rather unusual structure, but it also occurs for phosphorus and antimony at high pressures. At 54 °C a-Po is converted to /3-Po. The phase transition involves a compression in the direction of one of the body diagonals of the cubic-primitive unit cell, and the result is a rhombohedral lattice. The bond angles are 98.2°. 

The calcium ion is of such a size that it may enter 6-fold coordination to produce the rhombohedral carbonate, calcite, or it may enter 9-fold coordination to form the orthorhombic carbonate, aragonite. Cations larger than Ca2+, e.g., Sr2+, Ba2+, Pb2+, and Ra2 only form orthorhombic carbonates (at earth surface conditions) which are not, of course, isomorphous with calcite. Therefore these cations are incapable of isomorphous substitution in calcite, but may participate in isodimorphous or "forced isomorphous" substitution (21). Isodimorphous substitution occurs when an ion "adapts" to a crystal structure different from its own by occupying the lattice site of the appropriate major ion in that structure. For example, Sr2+ may substitute for Ca2 in the rhombohedral lattice of calcite even though SrC03, strontianite, forms an orthorhombic lattice. Note that the coordination of Sr2 to the carbonate groups in each of these structures is quite different. Very limited miscibility normally characterizes such substitution. 

Figure 3.4. The crystal systems and the Bravais lattices illustrated by a unit cell of each. All the points which, within a unit cell, are equivalent to each other and to the cell origin are shown. Notice that, in the primitive lattices the unit cell edges are coincident with the smallest equivalence distances. For the rhombohedral lattice, described in terms of hexagonal axis, the symbol hR is used instead of a symbol such as rP. In the construction of the so-called Pearson symbol ( 3.6.3), oS and mS will be used instead of oC and mC.

The rhombohedral lattice constants of some compounds LiMeFg and NaMeFg, isostructural with NaOsFg, were already stated by Boston and Sharp 46). The structure analysis of this type was performed somewhat later in the case of the compound LiSbFg 59). According to the investi-... 

Orthorhombic Hexagonal Trigonal (Rhombohedral) Monoclinic Tri clinic The primitive description of the rhombohedral lattice P, C, I, F P P/R P, c P is normally given the symbol R. 

In referring to any particular space-group, the symbols for the symmetry elements are put together in a way similar to that used for the point-groups. First comes a capital letter indicating whether the lattice is simple (P for primitive), body-centred (I for inner), side-centred (A, B, or C), or centred on all faces (F). The rhombohedral lattice is also described by a special letter R. Following the capital letter for the lattice type comes the symbol for the principal axis, and if there is a plane of symmetry or a glide plane perpendicular to it, the two symbols... 

Figure 16.6. Primitive unit cell of the rhombohedral lattice 3 R. The three fundamental translations a1 a2, a3 are of equal length and make equal angles with e3. Hexagonal nets in four successive layers show how the rhombohedral cell may be constructed.

Find the matrix A in terms of a and 9, and hence find an expression for 9 in terms of a. Prove that the reciprocal lattice of a rhombohedral lattice is also rhombohedral. Take bi = b, the angle between any pair of bi, b2, b3 as 3, and the angle between bi and e3 as 4>, and find expressions for b and f3 in terms of a and a. Find also the equations that determine the faces of the BZ. 

There are two allotropic forms of polonium, both primitive, with one atom per unit cell. a-Po is simple cubic (3PO) and each Po atom has six nearest neighbors (3.366 A) at the vertices of a regular octahedron. The simple cubic structure is the same as that of NaCl (Figure 3.6) with all atoms identical. The structure of (3-Po is a primitive rhombohedral lattice. Each Po atom has six nearest neighbors forming a slightly flattened trigonal antiprism (or a flattened octahedron). 

Bravais showed in 1850 that all three-dimensional lattices can be classified into 14 distinct types, namely the fourteen Bravais lattices, the unit cells of which are displayed in Fig. 9.2.3. Primitive lattices are given the symbol P. The symbol C denotes a C face centered lattice which has additional lattice points at the centers of a pair of opposite faces defined by the a and b axes likewise the symbol A or B describes a lattice centered at the corresponding A or B face. When the lattice has all faces centered, the symbol F is used. The symbol I is applicable when an additional lattice point is located at the center of the unit cell. The symbol R is used for a rhombohedral lattice, which is based on a rhombohedral unit cell (with a = b = c and a = ft = y 90°) in the older literature. Nowadays the rhombohedral lattice is generally referred to as a hexagonal unit cell that has additional lattice points at (2/3,1 /3, /s) and (V3,2/3,2/3) in the conventional obverse setting, or ( /3,2/3, ) and (2/3, /3,2/3) in the alternative reverse setting. In Fig. 9.2.3 both the primitive rhombohedral (.R) and obverse triple hexagonal (HR) unit cells are shown for the rhombohedral lattice. 

As a concrete example, consider a rhombohedral lattice and the relationship between the primitive rhombohedral unit cell (in the conventional obverse setting) and the associated triple-sized hexagonal unit cell, as indicated in Fig. 9.2.5. 

Relationship between rhombohedral (obverse setting) and hexagonal unit cells for a rhombohedral lattice. Note that in the right figure, lattice points at z = 0,1/3, and 2/3 are differentiated by circles of increasingly darker circumferences, and the lattice point atz = 1 is indicated by a filled circle, which obscures the lattice point at the origin. 

Fig. 1. Plot of the rhombohedral lattice parameters aR of a variety of binary and ternary Carbonates of calcite structure (e.g. Ca-M, Ca-M-M, Mg-M, M-M where M,M = Mn, Fe, Co, Cd, etc.) against the mean cation radius.

Primitive rhombohedral lattices, i.e. when a = b = c and a = p = y 90° are nearly always treated in the hexagonal basis with rhombohedral (R) lattice centering. In a primitive... 

Why then do the centered lattices appear in the list of the fourteen Bravais lattices If the two cells in Fig. 2-7 describe the same set of lattice points, as they do, why not eliminate the cubic cell and let the rhombohedral cell serve instead The answer is that this cell is a particular rhombohedral cell with an axial angle a of 60°. In the general rhombohedral lattice no restriction is placed on the angle a the result is a lattice of points with a single 3-fold symmetry axis. When a becomes equal to 60°, the lattice has four 3-fold axes, and this symmetry places it in the cubic system. The general rhombohedral cell is still needed. 

Fig. A4-1 Rhombohedral and hexagonal unit cells in a rhombohedral lattice.

Since a rhombohedral lattice may be referred to hexagonal axes, it follows that the powder pattern of a rhombohedral substance can be indexed on a hexagonal Hull-Davey or Bunn chart. How then can we recognize the true nature of the lattice From the equations given above, it follows that... 

In sodium azide a transition occurs at 19 °C (9) and also on application of 1 kbar pressure 37a), in which the rhombohedral lattice transforms by a shearing motion of the azide ion layers to form a monoclinic unit ceU (9). The latter is isostructural with the unit cell of lithium azide shown in Fig. 2 b. Among the tetragonal rubidium, cesium and thallous azides a high temperature transformation in the range 151 °C to 315 °C to a cubic structure takes place (77), while at —40 °C a transition to an orthorhombic structure has been recently established for thallous azide 38). In the range 4 to 6 kbar, Pistorius 39) has observed pressure induced polymorphs of rubidium, cesium and thallous azides which are expected to be isostructural with the low temperature phase in thallous azide. 

The pseudo threefold symmetry results in a rhombohedral lattice involving only complexes of identical chirality. The resulting network has large voids which are filled by a second network of opposite chirality and molecules of water. 

Fig. 14.33. Clinographic projection of the framework of quinol molecules in the structure of quinol clathrates. The plane hexagons represent rings of hydroxyl groups united by hydrogen bonds and the long lines represent the HO—OH axes of the quinol molecules. The framework extends indefinitely in three dimensions and the hexagons lie at the points of a rhombohedral lattice.

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