Triclinic crystal lattice

At present, isotactic polypropylene (i-PP) is commercially by far the most important system of the three modifications mentioned above. During crystallisation from the melt, i-PP is usually in the a form, which has a monoclinic crystal lattice with a Tm-value of about 160°C. The occurrence of a S form (with a hexagonal lattice and a Tm-value of about 152°C) is also possible during crystallisation under stress. Besides, a third (gamma) form with a triclinic crystal lattice is possible under exceptional circumstances [11]. 

The unit cell considered here is a primitive (P) unit cell that is, each unit cell has one lattice point. Nonprimitive cells contain two or more lattice points per unit cell. If the unit cell is centered in the (010) planes, this cell becomes a B unit cell for the (100) planes, an A cell for the (001) planes a C cell. Body-centered unit cells are designated I, and face-centered cells are called F. Regular packing of molecules into a crystal lattice often leads to symmetry relationships between the molecules. Common symmetry operations are two- or three-fold screw (rotation) axes, mirror planes, inversion centers (centers of symmetry), and rotation followed by inversion. There are 230 different ways to combine allowed symmetry operations in a crystal leading to 230 space groups.12 Not all of these are allowed for protein crystals because of amino acid asymmetry (only L-amino acids are found in proteins). Only those space groups without symmetry (triclinic) or with rotation or screw axes are allowed. However, mirror lines and inversion centers may occur in protein structures along an axis. 

The crystal descriptions become increasingly more complex as we move to the monoclinic system. Here all lattice parameters are different, and only two of the interaxial angles are orthogonal. The third angle is not 90°. There are two types of monoclinic space lattices simple monoclinic and base-centered monoclinic. The triclinic crystal, of which there is only one type, has three different lattice parameters, and none of its interaxial angles are orthogonal, though they are all equal. 

The lattice points of a triclinic crystal may be joined in various ways to form differently shaped unit cells (see p. 151). It is usually most convenient to use the cell with the shortest edges, unless there is some special feature which recommends some other direction as a unit cell edge. Donnay (1943) recommends that the shortest axis shall be called c and the longest b and that the angles a and ft shall be obtuse. 

Triclinic crystals. None of the angles of a triclinic cell are right angles in consequence, none of the axes of the reciprocal lattice are... 

If a triclinic crystal is rotated round any axis of the real cell (Fig. 93), the photograph exhibits layer lines (since the various levels of the reciprocal lattice are normal to the axis of rotation), but not row lines, since none of the points on upper or lower levels are at the same distance from the axis of rotation as corresponding points on the zero level. The indices for points on the zero level are found in the same way as for photographs of monoclinic crystals rotated round the 6 axis for the zero level of a triclinic crystal rotated round c, a net with elements a, 6, and y is constructed (Fig. 94), and distances of points from the origin are measured. The other levels, projected on to the equator, are displaced with regard to the zero level in a direction which does not lie along an equatorial reciprocal axis the simplest way of measuring values is, as before, to use the zero level network,... 

Fig. 95. Reciprocal lattice rotation diagram for triclinic crystal, constructed by measurements on Fig. 94.

For triclinic crystals the expression is so unwieldy that it is not worth while attempting to use it a graphical method based on the conception of the reciprocal lattice should be used (see pp. 154 ff). The reciprocal lattice method is also more rapid than calculation for monoclinic crystals. 

The same procedure is followed for all crystals. In dealing with photographs of monoclinic crystals oscillated round a or c, or triciinic crystals oscillated round any axis, care should be taken to use the appropriate origin for each reciprocal lattice level. (See Figs. 91 and 94.) As an example, the procedure for the first (hfcl) level of a triclinic crystal is illustrated in Fig. 98. 

Fractional Coordinates. In specifying the location of a point in a crystal lattice it is customary to employ coordinates, jt, y, z, that give the fraction of each principal vector distance (a, b, c), which define the unit cell. Thus, a point at the origin has the fractional coordinates 0,0,0 while the center of the cell has the coordinates 3,2,5. The face centers are 0, , , ,0, and , ,0 for the a, b, and c faces, respectively. It is to be emphasized that these fractional coordinates are not Cartesian except for isometric cells and are not even orthogonal for triclinic, monoclinic, or hexagonal lattices. 

Note also that in a triclinic crystal a and a are not collinear in a monoclinic crystal (b unique setting) b is parallel to b, but a and c form the obtuse angle [>, while a and c form a smaller acute angle /T given by fJ = 180 — fi. The reciprocal lattice vectors and the direct lattice vectors are a ying-yang duo of concepts, as are position space and momentum space, or space domain and time domain. Fourier transformation helps us walk across from one space to other, as convenience dictates Some problems are easy in one space, others in the space dual to it this amphoterism is frequent in physics. The directions of the direct and reciprocal lattice vectors are shown as face normals in Fig. 7.22. 

The unit cell is the smallest region of a crystal lattice that contains all the structural information about the crystal. So, the crystal lattice is visualized as a stack of multiple identical unit cells. Each unit cell has characteristic lengths and angles. There are seven types of crystal structures, each defined by the properties of its unit cell hexagonal, cubic, tetragonal, trigonal, orthorhombic, monoclinic, and triclinic. 

In the triclinic crystal system, any of the centered lattices is reduced to a primitive lattice with the smaller volume of the unit cell (rule number three). 

In the triclinic crystal system, the reduction becomes more complicated due to possible multiple choices of the basis vectors in the lattice. 

If the selected cell has M 20 (= M20) > 20 for triclinic crystal systems, or M20 > 30 i.e., M20 > 10) for monoclinic or higher symmetry crystal systems, it will be automatically refined by PI RUM, originally an interactive program, suitably modified to perform the automatic refinement of the unit-cell parameters. If more than 25 observed lines are available, the first 25 lines will be used for finding the cell, while all the lines will be involved in the refinement step. At the end of the PI RUM refinement a statistical study of the index parity of the assigned reflections is performed to detect the presence of doubled axes or of additional lattice points (A-, B-, C-, I-, R- or F-centred cell). If one of the index parity conditions is verified, an additional refinement is performed taking into account this information. 

Stated by Dana (75] were used in selection of axial lengths corresponding to the a, b, and c axes. For orthorhombic and triclinic systems, the c axis is shorter than the a axis, and the a axis is shorter than the b axis, oir c < a < b. For monoclinic systems, the c axis is shorter than the a axis dimension, or c < a. The A axial ratio is the quotient of the a axis dimension divided by the b axis dimension. In this report the A axial ratios of the compounds of the trace elements were divided by the A axial ratio of gypsum to give a number for each compound, indicative of the extent to which the crystal lattice of the compound matched that of gypsum. These numbers are listed in Table 13 in Column 6. The percent absorbed is defined as 100 minus the percent leached and is also given in Table 13. These data were fitted to Eq 3... 

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