Crystal axes

Fig. 3.52. Normalized back-scattering yields of ions from Pb near the melting point, with the incident beam and scattered beam directed along <101 > crystal axes (double alignment) curve a, 295 K curve b, 506 K curve c, 561 K curve d, 600.5 K curve e, 600.8 K. Spectrum d is fitted by a sum of contributions M, from a liquid surface layer, and I, from a partially ordered transition layer [3.133].

Polarization effects are another feature of Raman spectroscopy that improves the assignment of bands and enables the determination of molecular orientation. Analysis of the polarized and non-polarized bands of isotropic phases enables determination of the symmetry of the respective vibrations. For aligned molecules in crystals or at surfaces it is possible to measure the dependence of up to six independent Raman spectra on the polarization and direction of propagation of incident and scattered light relative to the molecular or crystal axes. 

Metallurgists originally, and now materials scientists (as well as solid-state chemists) have used erystallographic methods, certainly, for the determination of the structures of intermetallic compounds, but also for such subsidiary parepistemes as the study of the orientation relationships involved in phase transformations, and the study of preferred orientations, alias texture (statistically preferential alignment of the crystal axes of the individual grains in a polycrystalline assembly) however, those who pursue such concerns are not members of the aristocracy The study of texture both by X-ray diffraction and by computer simulation has become a huge sub-subsidiary field, very recently marked by the publication of a major book (Kocks el al. 1998). 

Figure 4-12. Sketch of the relative orientations of the scxithicnyl molecules in the crystal and representation of the crystal axes a. b. and c discussed in the text. Note that the layers are slacked along the a axis.

In addition, intensity changes under increasing pressure have been observed. For example, the most intense Raman line at STP conditions is the flg component of v ( 220 cm ), but at about 2 GPa the intensity decreases in favor of the ag component of Vi ( 475 cm ) which on further compression gains more intensity (about a factor of 2 at 5 GPa) [120]. This behavior was explained by the anisotropy of the crystal s compressibihty [139] and differences in the components of the Raman tensor of the two modes [87] with respect to the crystal axes [109]. 

Figure 6.2. (I) Conventional phosphorescence spectrum of 2,3-dichloroquinoxa-line in durene at 1.6°K. (II) am-PMDR spectrum, obtained by amplitude modulation of microwave radiation that pumps the tv-t, (1.055 GHz) zf transition with the detection at the modulation frequency. Only bands whose intensities change upon microwave radiation (1.055 GHz) and thus originate from tv or rz appear in the am-PMDR spectrum. Transitions from r and rv appear with opposite sign (phase-shifted by 180°). (Hb, lie ) Polarization of the am-PMDR spectral transitions, relative to the crystal axes. The band at 0,0-490 cm-1 originates from both the r and t spin states its intensity does not change upon the 1.055-GHz saturation (no band in II) however, its polarization does rhanp. (bands in Hb and IIc ). (Reproduced with permission from M. A. El-Sayed.tt7W)...

Spectra of radicals in a dilute single crystal are obtained for various orientations, usually with the field perpendicular to one of the crystal axes. Each spectrum usually can be analyzed as if they were isotropic to obtain an effective g-value and hyperfine coupling constants for that orientation. Since the g- and hyperfine-matrix principal axes are not necessarily the same as the crystal axes, the matrices, written in the crystal axis system, usually will have off-diagonal elements. Thus, for example, if spectra are obtained for various orientations in the crystal vy-plane, the effective g-value is ... 

A sinusoidal plot of grf>2 vs.

crystal plane gives another set of Ks that depend on other combinations of the gy, eventually enough data are obtained to determine the six independent values of gy (g is a symmetric matrix so that gy = gy,). The g-matrix is then diagonalized to obtain the principal values and the transformation matrix, elements of which are the direction cosines of the g-matrix principal axes relative to the crystal axes. An analogous treatment of the effective hyperfine coupling constants leads to the principal values of the A2-matrix and the orientation of its principal axes in the crystal coordinate system. 

We have seen that spectra of dilute single crystals are analyzed in a way that gives the orientations of the g- and hyperfine-matrix principal axes relative to the crystal axes. Historically, most of the information on noncoincident matrix axes is derived from such studies. 

The experimental determination of D and E for a dilute single crystal is not trivial, even when the crystal axes are known. Durene, for example, has two molecules per unit cell with different orientations of the molecular plane. Thus for any orientation there are four resonances, two from each type of site. Sorting out the data is a challenging exercise.2... 

Isotropic samples will have no effect on the polarized light no matter how the crystal is oriented, since all crystal axes are completely equivalent. This effect is known as complete or isotropic extinction (Fig. 3a). Noncrystalline, amorphous samples will exhibit the same effect. 

The crystal axes, a, b, and c, form three adjacent edges of a parallelepiped. The smallest parallelepiped built upon the three unit translations is known as the unit cell. Although the unit cell is an imaginary construct, it has an actual shape and definite volume. The crystal... 

The unit cell is defined by the lengths (a, b, and c) of the crystal axes, and by the angles (a, f>, and y) between these. The usual convention is that a defines the angle between the b- and c-axes, p the angle between the a- and c-axes, and y the angle between the a- and 6-axes. There are seven fundamental types of primitive unit cell (whose characteristics are provided in Table 7.1), and these unit cell characteristics define the seven crystal classes. If the size of the unit cell is known (i.e., a, (i, y, a, b, and c have been determined), then the unit cell volume (V) may be used... 

Broadly speaking, chiral space groups may be divided into two classes those that contain polar axes, for example, the commonly observed space groups P2, and C2 and those that do not, such as P2,2,2,. Crystal structures belonging to the latter class contain polar directions, but these do not coincide with the crystal axes. We shall focus on chiral crystals containing polar axes, although the method can in principle be applied to all chiral crystals. 

We illustrated in Section II why conventional X-ray diffraction cannot distinguish between enantiomorphous crystal structures. It has not been generally appreciated that, in contrast to the situation for chiral crystals, the orientations of the constituent molecules in centrosymmetric crystals may be unambiguously assigned with respect to the crystal axes. Thus, in principle, absolute configuration can be assigned to chiral molecules in centrosymmetric crystals. The problem, however, is how to use this information which is lost once the crystal is dissolved. 

In terms of die original discussion of Section II, what one needs to know is the orientation of die chiral molecule in a chiral crystal relative to die crystal axes. The absolute orientation of the molecule or of a sequence of molecules in the crystal can be determined by high-resolution electron microscopy, especially in cases like rubidium tartrate or other organometallics in which die problem is to determine the relative position of die heavy metal km with respect to die... 

Globular proteins were much more difficult to prepare in an ordered form. In 1934, Bernal and Crowfoot (Hodgkin) found, that crystals were better preserved if they were kept in contact with their mother liquor sealed in thin-walled glass capillaries. By the early 1940s crystal classes and unit cell dimensions had been determined for insulin, horse haemoglobin, RNAase, pepsin, and chymotrypsin. Complete resolution of the structures required identification of the crystal axes and some knowledge of the amino acid sequence of the protein—requirements which could not be met until the 1950s. 

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