ABC notation

The structure of layered compounds often can be described by the ABC notation, shown in conjunction with the sites between close-packed layers in Fig. 7.1. Given a close-packed layer of atoms, with atoms at positions we call A, there are three possible places to put the atoms of the next layer directly above the first layer (also in the A positions), or fitted into the interstices of the first layer, either in B or C positions. In this notation, a trigonal prism sandwich is denoted AbA, and an octahedral sandwich is AbC, where the uppercase letters denote the layers of chalcogen or oxygen, and the lowercase letters the layers of transition metal atoms. 

Perhaps the most well known of the lithium intercalation compounds is Li jTiSj. Both Li (for x < 1) and Ti are octahedrally coordinated by S (Fig. 7.1) in the ABC notation, the structure is AbC(b)AbC, where the letter in parentheses denote lithium atoms. This structure is also called the IT form, because of its trigonal (T) symmetry and the single layer per unit cell. The electrochemical behaviour of Li in TiSj is described below in connection with staging. 

The stacking sequence of these polytypes can be described by the ABC notation, where A, B and C represent the three sites available in one sublattice. For example (...ABCABC...) and (...ABABAB...) stand for cubic (3C or p-SiC) and simple hexagonal (wurtzite) respectively. The intermixing of these two simple forms can also occur, generating a large number of... 

R ABC notation J Z % of hexagon -ality Space group No. of atoms per unit cell... 

The first column of Table 14.3 gives the response notation (or, equivalently, the factor combination). The next eight columns list the eight factor effects of the model the three main effects (A, B, and C), the three two-factor interactions (AB, AC, and BC), the single three-factor interaction (ABC), and the single offset term (MEAN, analogous to PJ in the equivalent linear model). 

In angular momentum theory a very important role is played by the invariants obtained while summing the products of the Wigner (or Clebsch-Gordan) coefficients over all projection parameters. Such quantities are called 7-coefficients or 3ny-coefficients. They are invariant under rotations of the coordinate system. A j-coefficient has 3n parameters (n = 1,2,3,...), that is why the notation 3nj-coefficient is widely used. The value n = 1 leads to the trivial case of the triangular condition abc, defined in Chapter 5 after formula (5.25). For n = 2,3,4,... we have 67 -, 9j-, 12j-,. .. coefficients, respectively. 3nj-coefficients (n > 2) may be also defined as sums of 67-coefficients. There are also algebraic expressions for 3nj-coefficients. Thus, 6j-coefficient may be defined by the formula... 

Copper(I) nitride, Cu3N, crystals are also cubic, OPm3m, a0 = 3.814 A, with one molecule in the unit cell. Each N has an octahedral arrangement of six Cu atoms. Cu atoms have linear bonds to two N atoms and eight nearest Cu neighbors (see Figure 5.24). These are two sets of four planar Cu atoms bonded to the N atoms. The N atoms occupy sites for a simple cubic cell. To accommodate Cu and N in the P system, Cu atoms occupy P sites in an ABC sequence with N atoms in O sites. The P sites are three-quarters occupied and O sites are one-quarter occupied to give the notation 3 2P3/4Oi/4. 

F ions using CrystalMaker shows that they are in close-packed layers in an ABC sequence. Bi atoms occupy one-fifth of octahedral sites. The BiFf, octahedra share F ions along the c direction. The notation is 3-2P01/s(t). 

Mn2Au5 is monoclinic with two molecules per unit cell, a = 9.188, c = 6.479 A, and (3 = 97.56°. It is a superstructure of face-centered subcells. In Figure 9.17a we can see the face-centered subunit (dotted lines) and the smaller body-centered subunits (solid lines). There are close-packed layers parallel to 111 planes of the pseudocell. The close-packed layers are in an ABC sequence with 21 layers repeating (Figure 9.17b). The notation is 3 7P(m)(2 5). 

Ga3Pt5 is orthorhombic with two molecules per unit cell, a = 8.031, b = 7.440, and c = 3.948 A. The unit cell corresponds to a block of four distorted fee pseudocells. Figure 9.18 shows the close-packed sequence is ABC. The figure shows locations of one point in each layer B and C. Each layer is occupied in the 3 5 ratio. In the horizontal rows one row is filled by Pt and the next three are one-half occupied by Pt. In rows parallel to the other cell faces one row is filled by Pt and the next is one-quarter filled by Pt. The 3 5 ratio is maintained for both sets of layers. There are 12 layers repeating, giving the notation 3 4P(3 5)(o). Each atom has CN 12. The closest Ga—Pt distance is 2.59 A. 

Kyanite is triclinic, C(, PI, a = 7.1262, b = 7.852, c = 5.724 A, a = 89.99°, (3 = 101.11°, and y = 106.03°, with four molecules per cell. The packing sequence is ABC. Between the first two P layers each T layer is one-fifth occupied by Si only and the octahedral layer is one-fifth occupied by Al. Between the next two P layers, only octahedral sites are three-fifths occupied by Al. This alternating pattern continues, (see Figure 10.17). The notation is 3 6PTi/50i/5T1/5P03/5(tri). Kyanite is 14% more dense than andalusite and 11.5% more dense than sillimanite. 

ABC positions, but the layers are partially (one-half) filled. The notation is 3Pi/2(m). 

Here the notation (cab), etc., means that the indices (abc) in the preceding expression have to be permuted to (cab). The general fourth- and fifth-order equations are given by King and Komornicki (1986a). 

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