Triclinic unit cells, 421 table

As is obvious from columns 2 and 3 in Table 5.24, different indexing algorithms result in different choices of the unit cell for the same lattice and, therefore, unit cell reduction is especially important to compare the results in triclinic symmetry. The unit cell dimensions, reduced using the WLepage program, are listed in Table 5.24 in columns 6-8. Obviously, all of them are represented by the same unit cell, except the incorrect solution shown in row 2. The triclinic unit cell was confirmed by a single crystal diffraction experiment, as shown in row 6. 

Figure 9.7 shows the seven primitive space lattices (unit cells). The variables a, b, c, a, and y, are free, viz. they can have whatever value which define a 3-D object which can be multiplied to produce the macroscopic crystal. In all but the triclinic unit cell some variables are correlated (the axis) or restricted (the angles). Those unit cells which have no axis correlations are of lower symmetry (Fig. 9.7, left), triclinic being the least symmetrical. In higher symmetry space groups (Fig. 9.7, right) one or more correlations between the variable exist (see Table 9.1). The unit cells will differentiate from each other by the correlations of the six variables. Table 9.1 gives the definitions for each variable of the seven unit cells. 

Table 1 shows the monomer repeats for centrosymmetric chains with a range of values of Xj and X2, where it is seen that c/2 does not vary by more than 0.2A. We are now considering the packing of the poly(MDI-BDO) chains in the proposed triclinic unit cell, and will report on this in due course. 

Structure type B has been determined from three-dimensional singlecrystal diffraction intensity data (27). The parameters of the 22 independent atoms in the triclinic unit cell have been determined from 4708 observations and refined to a residual i = 5.8%, which corresponds to an average e. s. d. of 0.007 A for the main cation-oxygen distances. Final atomic parameters are given in Table 16. 

MPDI has a triclinic unit cell and is significantly less crystalline than PPTA (Table 13.1). Savinov [14] proposed that crystallinity depends on the conditions of polymer precipitation from solution. Precipitation of polymer in water leads to a noncrystalline material while precipitation in water containing some solvent leads to a crystalline form. Krasnov [15] showed that increased fiber orientation leads to higher crystallinity. SVM, the Russian... 

TABLE 6 The preparative conditions of PTT and the corresponding lattice constants of the triclinic unit cell... 

Table 1.3 Results of an IsoQuest search using the compound PhsSiCI [CSD refcode (Nov. 2006 version) BARNUD], listed according to decreasing similarity. The structure of BARNUD is reported in the nonstandard space group P2iA). The triclinic unit cells have half the volume of the monoclinic unit cells...

The B-form of soaps has a bilayer structure with parallel chains (Lomer, 1952). The chains are tilted 50-52 C, and the dimensions of the triclinic unit cell are given in Table 8.18. The long spacings (d(OOl)) fit thecurve6.80 + 1.95 n (A), where n is the number of carbon atoms. 

Table 8.18 Dimensions of the triclinic unit cell of the B-form of potassium soaps (Lomer, 1952)...

Seven crystal systems as described in Table 3.2 occur in the 32 point groups that can be assigned to protein crystals. For crystals with symmetry higher than triclinic, particles within the cell are repeated as a consequence of symmetry operations. The number of asymmetric units within the unit cell is related but not necessarily equal to the number of molecules in a unit cell, depending on how the molecules are related by symmetry operations. From the symmetry in the X-ray diffraction pattern and the systematic absence of specific reflections in the pattern, it is possible to deduce the space group to which the crystal belongs. 

Tabic l.l gives those crystal data for the C,S polymorphs that have been obtained using single crystal methods. The literature contains additional unit cell data, based only on powder diffraction evidence. Some of these may be equivalent to ones in Table 1.1, since the unit ceil of a monoclinic or triclinic crystal can be defined in different ways, but some are certainly incorrect. Because only the stronger reflections are recorded, and for other reasons, it is not possible to determine the unit cells of these complex structures reliably by powder methods. The unit cells of the T, Mj and R forms are superficially somewhat different, but all three are geometrically related transformation matrices have been given (12,HI). 

Atoms can also reside on centers of inversion. Since there is no inversion center in the space group Cmm2, which was considered in Table 1.18, we turn our attention to the distribution of the centers of inversion in the unit cell that belongs to the triclinic space group symmetry P1 Figure 1.45). 

The correct unit cell may be identified by associating indices to n inter-planar distances, where n depends on the lattice symmetry. In accordance with Table 7.3, the minimum values of n are n=l for the cubic system, n = 2 for tetragonal and hexagonal crystals, = 3, 4, 6 for orthorhombic, monoclinic and triclinic systems, respectively. 

This author proposed a method for estimating the unit-cell volume (Vest) directly from the powder diffraction data, via d and N, where d is the value of the Mh observed line (i.e. ii N = 20 and 20 is the value of the 20th observed line, Fest 13.39dio). In the triclinic system Q(hk ) is a complicated function of the direct cell parameters (see Table 7.2) then the algorithm is applied in Q space by using Equation (8). DICVOL91 is highly sensitive to the quality of the data. 

Crystals of tetracene and pentacene are triclinic, with two molecules per unit cell. The dimensions of the unit cell in the directions of the a and b axes are similar to those of naphthalene and anthracene the dimension in the direction of the c-axis increases proportionally to the number of benzene rings in the molecule. In all four crystal structures the long axes of both molecules in the unit cell are approximately parallel to the c crystal axis. However, in the triclinic structures of tetracene and pentacene, the only crystal symmetry operation is inversion. The results of calculation for tetracene crystals are shown in Table 3.6. Analogous results for pentacene can be found in (60), (61). The experimental data of the splitting cm-1 for the io-o transition in the tetracene crystal are in the interval 600-700 cm-1. For the /q o transition in a crystal of pentacene the... 

There are seven crystal systems, listed in Table 1, that result from the possible symmetry of the crystal lattice (24). For example, if the crystal lattice describes a cubic unit cell, rotations of 90° or 120° or 180° about appropriate directions will give a lattice indistinguishable from the original this can be verified by examination ofa cube. The unit cell conditions (a = b = c, a=P=y=90°) follow from the lattice symmetry. If, however, a = b=t c and neither a nor P nor y= 90°, then the symmetry of the crystal lattice is low (triclinic). 

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