Unit cell translation

High polymer calculations can be performed on polysaccharides. Calculation of unit cell translation vectors (15), heats of polymerization (15), and elastic moduli (16) can readily be done. The accuracy of such calculations is the same as that of equivalent molecular species. A limitation of elastic moduli calculations is that the polymer is assumed to be 100% ordered, a state not commonly found in polysaccharides. 

FIGURE 4,12. A glide plane causes a reflection across a mirror plane plus a translation of half the unit cell edge (translation = a/2) A left-handed molecule is converted to a right-handed molecule by this symmetry operation. Two such operations give the original molecule translated to the next unit cell (translation = a). 

A relatively large volume of the monoclinic unit cell translates into a considerable complexity of the diffraction pattern even in the case of a base-centered lattice, as can be seen from Figure 6.29. There are 10 reflections per degree at 20 = 50° and about 20 reflections per degree at 20 = 100°. 

The difference between monoclinic and triclinic forms arises from the layering of ribbons in adjacent sheets (within one layer, the two forms are almost isostructural). In the triclinic form, the adjacent sheets are related by a unit cell translation, and the ribbons in adjacent sheets run parallel to one another. In the monoclinic form, the adjacent sheets are related by a c glide, and ribbons in adjacent sheets run in perpendicular directions. Although this distinction is very subtle, careful analysis of CP/MAS spectra allows these differences to be recognized. 

FIGURE 3.6 In (a), the disposition of asymmetric units related by a 4j screw axis is illustrated. As with the 2i screw axis, continued application of the symmetry operator in a crystal simply generates asymmetric units in adjacent unit cells which were already present due to unit cell translations. In (b), a 61 screw axis produces six identical asymmetric units whose equivalent positions are specified according to a hexagonal coordinate system. It follows that such a symmetry axis could only be compatible with a unit cell having a hexagonal face (i.e., a hexagonal prism). 

If the coordinates of the points comprising a single asymmetric unit are known, those of the equivalent points of all the asymmetric units in the unit cell are then known, or may be generated from the space group symmetry (i.e., by equivalent positions). Any point then in any asymmetric unit anywhere in the crystal, even if thousands of unit cells away, can be found by applying the unit cell translations. The coordinates of the atoms in a single asymmetric unit, plus space group symmetry, plus the unit cell vectors, completely specify... 

From Figure 3.2, a crystal emerges as a virtually infinite array of identical unit cells that repeat in three-dimensional space in a completely periodic manner. Like a simple sine wave in one dimension, it repeats itself identically after a period of a, b, or c along each of the three axes. A crystal is in fact a three-dimensional periodic function in space, a three-dimensional wave. The period of the wave in each direction is one unit cell translation, and the value of the function at any point xj, yj, Zj within the cell, or period, is the density of electrons at that point, which we designate p(xj, yj, Zj). 

Figure 33 The P61 crystal of GFRqI. (a) A view along one sixfold screw axis of a GFRad crystal. Actually, the crystal consists of many such spirals interwoven with each other, but only one is shown for clarity, (b) A view perpendicular to the screw axis showing the 1/6 unit cell translation.

When the path length differences for X rays diffracted by atoms separated by one unit-cell translation are an integral number of wavelengths, there is reinforce-... 

The primitive orthorhombic lattice can be thought of as arising from a primitive monoclinic lattice with the added restriction that the third angle is also 90°. In that case, all the unit cell translation vectors are 90° to one another but have different lengths. In the orthorhombic system, one can construct a C-centered cell, which can also be described as an A- or B-lattice by an interchange of the orthogonal axes. In addition, there can also be an all-face-centered F-lattice structure and a body-centered I-lattice. Thus for the orthorhombic crystal system there are four unique Bravais lattices, P, I, F, and C. 

Screw Operation Screw axes are designated as w-fold rotation axes with the fraction of the unit-cell translation by the subscript g g, where n refers to -fold rotation and g refers to the fraction of unit translation. For example, a fourfold screw axis has a rotation of 90° (i.e., 360/4 = 90), and is denoted... 

Figure 15-16 Results of translating u and g basis functions through unit cell translation distances a, modulated by cos( x) with = 0 and ixja.

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