Single crystals symmetry

Figure 18.1 A crystal is built up from many billions of small identical units, or unit cells. These unit cells are packed against each other in three dimensions much as identical boxes are packed and stored in a warehouse. The unit cell may contain one or more than one molecule. Although the number of molecules per unit cell is always the same for all the unit cells of a single crystal, it may vary between different crystal forms of the same protein. The diagram shows in two dimensions several identical unit cells, each containing two objects packed against each other. The two objects within each unit cell are related by twofold symmetry to illustrate that each unit cell in a protein cr) stal can contain several molecules that are related by symmetry to each other. (The pattern is adapted from a Japanese stencil of unknown origin from the nineteenth century.)...

Figure 6-3. Top Structure of the T6 single crystal unit cell. The a, b, and c crystallographic axes are indicated. Molecule 1 is arbitrarily chosen, whilst the numbering of the other molecules follows the application of the factor group symmetry operations as discussed in the text. Bottom direction cosines between the molecular axes L, M, N and the orthogonal crystal coordinate system a, b, c. The a axis is orthogonal to the b monoclinic axis.

Physical Properties. According to Lagowski (Ref 32), X-ray analysis of a single crystal of nitric acid shows a monoclinic unit cell (symmetry P21/a-Cfh) the following dimensions a=16.23, b=8.57, and c=6.3lA, and 0=90°. The unit cell contains 16 molecules, and the calc d is 1.895g/cc at —41.6°... 

Since then, the vibrational spectrum of Ss has been the subject of several studies (Raman [79, 95-100], infrared [101, 102]). However, because of the large number of vibrations in the crystal it is obvious that a full assignment would only be successful if an oriented single-crystal is studied at different polarizations in order to deconvolute the crystal components with respect to their symmetry. Polarized Raman spectra of samples at about 300 K have been reported by Ozin [103] and by Arthur and Mackenzie [104]. In Figs. 2 and 3 examples of polarized Raman and FTIR spectra of a-Ss at room temperature are shown. If the sample is exposed to low temperatures the band-widths can enormously be reduced (from several wavenumbers down to less than 0.1-1 cm ) permitting further improvements in the assignment. 

Many of the polysulfides described above have been investigated by X-ray diffraction on either powders or single crystals. In all cases the more sulfur-rich anions (n>3) form unbranched chains the symmetry of which varies between Ci, C2, and Cs. According to Fig. 1 the symmetry C2 results if all torsion angles have the same sign (right-handed helix + + +... left-handed helix ----...). If the different torsion angles of the anion vary between + and... 

The density of 87O has been determined as 2.15 g cm at 25 °C and calculated from the lattice constants as 2.179 g cm at —110 °C, measured by a single-crystal X-ray diffraction analysis [1, 64, 65]. The 87O molecules are of Cl symmetry and consist of chair-hke seven-membered homocycles with the exocyclic oxygen atom in an axial position see Fig. 2. Most remarkably are the two almost planar groups 0-8-8-8 (torsion angle r=2.9°) and 8-S-8-8 (r=6.3- ). 

Sulfoxides form adducts with Lewis acids like SbCls and SnCU [80]. In the case of SsO the crystalline products SsO-SbCh [81] and (SsO)2-SnCl4 [82] have been prepared and characterized by single crystal X-ray diffraction analyses. Reaction of SsO with SbCls in CS2 at 20 °C and subsequent cooling of the solution to -50 °C resulted in orange orthorhombic crystals of SsO-SbCls in 71% yield. These molecules are of Cg symmetry see Fig.6 [81]. 

Both dialkynylated cymantrenes 16 and 17 have the same symmetry and similar NMR spectra, so that the ultimate structure elucidation had to rest upon X-ray crystallography. It was reasoned that the dialkynylcymantrene with the smaller /(H,H) coupling of the cyclopentadienyl protons should be 17, an interpretation reinforced [23 a] by the result of the single-crystal structure. 

SHG has been used to study electrode surface symmetry and order using an approach known as SH rotational anisotropy. A single-crystal electrode is rotated about its surface normal and the modulation of the SH intensity is measured as the angle (9) between the plane of incidence and a given crystal axis or direction. Figure 27.34 shows in situ SHG results for an Au(ll 1) electrode in 0.1 M NaC104 + 0.002 M NaBr, using a p-polarized beam. The results indicate the presence of two distinct onefold... 

---