Obverse setting

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. 

D rows one reflection (family reflection) out of N occurs for subfamily A polytypes [presence criterion Ic = Nhl3) mod M), + for the obverse setting of the family... 

Fig. 7.7 Reciprocal space plots of the / = 0 layer Qi down and k to the right) of a rhombohedral crystal in obverse setting (A), reverse setting (B) and the superposition of both settings (C).

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). 

However, comparing the mean intensity of all reflections (5.5) with the mean intensity of the reflections that should be absent in the obverse setting (1.9) and reverse setting (3.5), respectively, and inspection of reciprocal space plots (Figure 7.13) give a first indication of obverse/reverse twiiming. 

The monoclinic crystals now are listed with the b axis as the unique axis, but prior to 1940, another popular "setting" used c as the unique axis. Of the 230 space groups, 7 have two choices of unit cell, a primitive rhombohedral one (R) and, for convenience, a nonprimitive hexagonal one (H), with three times the volume of the rhombohedral cell. The 3x3 transformation matrices from rhombohedral (obverse, or positive, or direct) cipbj, Cr to hexagonal axes aH, bur Ch and vice versa are shown in the caption to Fig. 7.17. 

Relationship between the rhombohedral axes aR, bR, cR (obverse or positive, or direct setting) and the hexagonal axes aH, bH, cH [4, pp. 20-21). 

Tplytyp. The 3Tpolytype is an orthogonal subfamily A polytype, for which the six orientations of the structural model are equivalent. They can be grouped into two sets of odd or even parity, corresponding to obverse and reverse setting of the family structure... 

The obverse side of this coin is not to be neglected. Since every atomic absorption user will at one time or another wish to determine a considerable number of samples in succession, the instrument should be capable of a fast response. At the fastest speed setting (lowest time constant), the meter needle or recorder pen should come to a balance in less than two seconds. 

For structures that crystallize in rhombohedral space groups, a twofold axis parallel to the threefold axis (matrix -100 0-10 OOlinthe hexagonal setting) or parallel to a-b (matrix 0-10 -100 00-1 in the hexagonal setting) as twin law produces a so-called obverse/reverse twin (see Figure 7.7). 

Figure 1-1 Projections of p-quartz (a) and a-quartz (b) and (c) along the c-axis. Both obverse (b) and reverse (c) settings are shown. The double helix structure of p-quartz is shown in (d).

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