Orthorhombic lattice system

Orthorhombic. This system has three mutually perpendicular unit translations, whose perpendicularity is rigorously required by symmetry. This can only be assured if there are three mutually perpendicular twofold axes, either all proper (222) or the combination 222 (= mm2). The lattice itself has mmm symmetry because when the centering condition is added to either of the above, it expands them from 222(Z)2) or 2wm(C2l.) to mmmiDy,). 

Now consider the orthorhombic crystal system. Simple analysis of Eq. 2.32 indicates that the following groups of reciprocal lattice points will have identical reciprocal lattice vector lengths and thus, are equivalent in terms of the corresponding Bragg angle ... 

Figure 5.14. The relationship between the lattice parameters of the hexagonal unit cell (solid and dotted lines) and the related orthorhombic unit cell (dashed lines). The unit cell parameter perpendicular to the plane of the projection is identical in both crystal systems. The smaller orthorhombic unit cell found using the DICVOL91 indexing program is indicated by the thick solid vectors (a o and c o). Open circles show lattice points and the dash-dotted vector illustrates the C-translation in the conforming orthorhombic lattice.

Replacement of Ba by Sr in the case of the La-Ba-Cu-0 system was thus favorable in decreasing aQ and increasing Tc. However the same is not true in the case of the Y-Ba-Cu-0 system. Figure 2 shows the orthorhombic lattice parameters of aQ, bo and cq for (Ba x rx)2 u3 7 f°r x=0 to 0.7. It is clearly seen that these parameters all decrease with Sr composition, x. Figure 3 shows the resistivity vs. temperature of these specimens. Quite unexpectedly from the discussions up to this point, Tc in this particular system decreases with increase in x. Although the difference in these two examples is not clear yet, it should be quite essential in understanding the mechanism of the superconductivity in the both systems. 

We now quote representative cedculations. Where the sources are chiral, here taken to be skewed dipole-quadrupoles, we consider sources at lattice points of an orthorhombic lattice. The dipole component is directed along the c orthorhombic axis with magnitude 1 em i ( 4.8 Debye) and the quadrupole moment, in the a,b,c axis system is... 

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. 

Hydrides. The thermodynamic characteristics of the Nb-H system have been investigated and the crystal structure of P-NbH, which has a face-centred orthorhombic lattice, has been determined." The crystal structure of Ta2D has been,reported." ... 

In this chapter it was shown by X-ray technique (WAXS) that the original samples of PHBV are characterized by high crystallinity. In their diffraction patterns at least 5 reflections of the orthorhombic lattice with characteristics a = 5.1 A k,b = 13.24 A and c = 5.98 A were determined that well conform with the earlier studies [11, 12]. In Fig. 2.1, the WAXS dif-fractograms are presented as asset of curves belonging to initial PHBV (7), PHBV treated by water at 40 °C (2), PHBV/SPEU blend with component ratio 40/60 (3), the same blend treated by water at 70 °C (4), and the PHBV/SPEU (60/40) blend also treated by water at 70 °C (5). The SPEU presence in the system leads to an amorphous hallo in the range 22° (cf. curves 1 and 2), its intensity is increased with the SPEU content. 

Yokouchi et al. 1973), and that the P-structure introduces from the orientation of free chains in amorphous regions between a-structure lamellar crystals that is, the development of the P-structure is not really a transformation from the a-structure (Orts et al. 1990 Iwata et al. 2003, 2005 Iwata 2005). More recently, Iwata et al. (2008) reported a 3D crystal structure of the P-structure that has lattice parameters of fl=0.528 nm, / = 0.920 nm, and c(fiber axis)=0.469 mn as an orthorhombic crystal system. 

A positive deviation from Vegard s law in the case of the orthorhombic lattice parameters [20] is mainly due to the change in the lengths of the metal-metal bonds. Inthiscase, some short-range order may appear when a given Sb atom forms with Zn, Cd, and Sb atoms. This occurs near the 50%/50% composition and may be responsible for the extrema in the dependences of various properties on the composition [13]. Thus, the system described is completely at variance with the ideal representations of so lid-solution structures. 

We need now to translate the above theoretical results into practice. First, let us start from an orthorhombic lattice with two molecules per unit cell and lower the symmetry to a supermolecular organization in which the cell can be considered as isolated and decoupled from its neighbors. This situation can be achieved in the following case (i) isotopic solid solutions (ii) molecule as a guest in a chemically different host lattice (e.g., urea clathrates) [143] (iii) liquid crystalline-like systems and (iv) systems with conformationally distorted chain ends, but with an ordered bulk [103]. The following changes of the Raman spectrum are expected (and observed) ... 

In the orthorhombic crystal system the base-centered lattice merits special attention because of different possible settings. Let the initial base-centered lattice have the setting C. The transition to base-centered lattices with settings C and A (or B) gives different results (see Appendix A). The change of setting for the transition to other types of lattice does not give new superceUs. 

In the tetragonal crystal system there are two types of Bravais lattice P and I). AU their symmetrical transformations may be obtained from the symmetrical transformations for orthorhombic lattices if one sets ni = ri2 and takes into account that base-centered and face-centered orthorhombic lattices become simple and body-centered tetragonal ones, respectively. 

A nnmber of techniques are appropriate to investigate the hierarchy of structnres formed by crystalline polymers. Crystallized polymer chains form crystal structures with lattices built up by translation of unit cells, just like crystals formed by low molar mass compounds. The space group symmetry depends on the polymer under consideration and also the conditions of the sample. For example, polyethylene usually forms a structure belonging to the orthorhombic crystal system, but at high pressures it is possible to obtain a hexagonal structure. Because it can adopt more than one crystal structure, polyethylene is said to be polymorphic. The best way to determine the crystal structure of a polymer is to perform wide-angle x-ray scattering (WAXS) experiments. WAXS on oriented polymers also provides information on the orientation of crystalline stems (chains). 

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