Triclinic lattices

These 14 Bravais Lattices are unique in themselves. If we arrange the crystal systems in terms of symmetry, the cube has the highest symmetry and the triclinic lattice, the lowest symmetry, as we showed above. The same hierarchy is maintained in 2.2.4. as in Table 2-1. The symbols used by convention in 2.2.4. to denote the type of lattice present are... 

Let us now develop systematically the 14 lattices shown in Figure 11.11. Clearly, if we impose no special requirements on the set of defining vectors (Figure 11.12a), namely, a b c, and a p y, we have the 3D analog of our 2D oblique lattice. It is called a triclinic lattice. As with the 2D oblique lattice, there is no unique.way to choose the vector set, but normally one would choose the three shortest vectors. Even if one angle happens to be 90° or two of the three vectors happen to be equal, the lattice is still triclinic because these special relations do not enhance its symmetry. The triclinic lattice has inversion centers as its only symmetry elements. Moreover, a triclinic lattice is necessarily primitive, since if any additional points were introduced at the center of the cell or at any of the face centers, we would have to redefine our vector set in order to include them in a true lattice. 

A 3D lattice can be built up by stacking 2D lattices. If a 2D lattice is defined by two translation vectors, t, and t2, we need to introduce a third translation vector, t3, that defines the stacking pattern. For example, if we stack a set of (identical) oblique lattices (defined by t, and t2) employing a vector t3 that is not orthogonal to the 2D lattice planes, we generate the triclinic lattice, while if we require t3 to be orthogonal to the 2D lattice planes and connect each plane with a point in the nearest neighboring plane we get the primitive monoclinic lattice. 

For the triclinic lattice there is no symmetry other than the inherent inversion symmetry already noted. 

Where are the inversion centers in a triclinic lattice How many distinct ones are there ... 

FIGURE 3.13 Example of a point lattice, where the base vectors, a, b, and c are nonorthogonal and create a completely general triclinic lattice in which there are no angles of 90°. The lattice nonetheless is periodic in all directions and has a value of zero except at the specified points. 

Later, Nagata et al. on the basis of own X-ray diffraction studies claimed that the triclinic lattice parameters reported by Wang et al. can be transformed into the orthorhombic parameters of a-Ss at that pressure... 

DMSO = dimethylsulfoxide (32). Its structure consists of oxamato-bridged Mn(II)Cu(II) linear chains running along the fe-axis of a triclinic lattice with an intrachain Mn—Cu distance of 5.387 A (see Fig. 

TTF-DETCNQ crystallizes in a triclinic lattice with each acceptor molecule surrounded by four donor molecules and vice versa /lo, 11/. In this respect, TTF-DETCNQ resembles TTF-TNAP /12/ and HMTSF-TCNQ /13/, which also have the "four-nearest-neighbours" structure and not the usual "herring-bone" with, only two nearest neighbours. 

In the triclinic system, there are no restrictions on the magnitudes of the lengths of the unit cell axes or on their interaxial angles. One can therefore always take a triclinic lattice and center it, to produce a new lattice that will be compatible with the conditions of the triclinic crystal system. However there is nothing new about this lattice, since a smaller primitive cell can be determined with the same complete arbitrariness of the cell edges and angles. Thus for the triclinic crystal system there can be only one Bravais lattice, the primitive or P-lattice. 

It must be noted that the metric is imposed by the symmetry of the crystal. A triclinic lattice at a given temperature and pressure could have a monoclinic metric by coincidence but still not be monoclinic. 

By wide-angle X-ray diffraction, four different crystal forms have been identified (3,4). These polymorphs are referred to as the a, y and "smectic" forms and their unit cell structures have been determined to be monoclinic, hexagonal, triclinic and pseudohexagonal, respectively. The relative amounts of these phases are very sensitive to the conditions of crystallization. In the process of slow cooling and isothermal crystallization, the iPP melt crystallizes into the monoclinic, hexagonal and more rarely triclinic lattice. Quenching the melt into the liquid nitrogen produces a "smectic" phase. 

The scientists of the pre-X-ray period postulated that any macroscopic crystal was built up by repetition of a fundamental structural unit composed of atoms, molecules, or ions, called the unit crystal lattice or space group. This unit crystal lattice has the same geometric shape as the macroscopic crystal. This line of reasoning led to the 14 basic arrangements of atoms in space, called space lattices. Among these are the familiar simple cubic, hexagonal, and triclinic lattices. 

If the fibres are uniaxial, there is still some symmetry in the material, and the number of parameters needed to describe the elastic behaviour is smaller than 21, the value for a triclinic lattice. If the fibres are directed, but their positions in space are irregular or arranged on a hexagonal lattice, the material is transversally isotropic i. e., its properties are the same in all directions perpendicular to the fibre direction. In this case, there are five independent elastic constants (see section 2.4.6). If the fibres are uniaxial and arranged on a rectangular lattice, the material is orthotropic, and the number of independent elastic components is nine (see section 2.4.5). 

V cell volume (the orthorhombic, monoclinic or triclinic lattice cell is defined by the... 

The point symmetry group of a triclinic lattice p (Fig. 2.2) consists of only inversion in the coordinates origin. 

Several perovskite compounds have been proposed as pigments (Table 12.1). They are both oxide and non-oxide phases with crystal symmetry spanning from ideal cubic perovskite to increasingly distorted hexagonal, tetragonal, and orthorhombic types, down to peculiar cases of monoclinic and triclinic lattices. 

Syndiotactic polypropylene, sPP, exists in three different conformations a stable two-fold helix, a planar zigzag both with orthorhombic lattice and an intermediate conformation with a triclinic lattice. The crystallographic data of sPP are reported in Table 3. The stable two-fold helix structure is called the high-tem-perature orthorhombic form. Three different unit cells for this form are proposed [78-85]. 

Chatani el al. [87] observed the triclinic lattice in cold draw sPP samples quenched in ice water. The samples were subjected to vapor absorption for several days the solvents used were benzene, toluene and p-xylene. The triclinic lattice transforms into the helical form by annealing above 50°C. 

In a triclinic lattice none of the axes is perpendicular to another, making the relationships more complex. Table 4.4 gives the pertinent relationships. 

It is difficult to get more primitive than the tetragonal or triclinic lattice, and the base-centered monoclinic lattice (Figure 4.12g) rounds out the 14 Bravais lattices. 

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