Orthorhombic crystals, diffraction patterns

In contrast to single-crystal work, a fiber-diffraction pattern contains much fewer reflections going up to about 3 A resolution. This is a major drawback and it arises either as a result of accidental overlap of reflections that have the same / value and the same Bragg angle 0, or because of systematic superposition of hkl and its counterparts (-h-kl, h-kl, and -hkl, as in an orthorhombic system, for example). Sometimes, two or more adjacent reflections might be too close to separate analytically. Under such circumstances, these reflections have to be considered individually in structure-factor calculation and compounded properly for comparison with the observed composite reflection. Unobserved reflections that are too weak to see are assigned threshold values, based on the lowest measured intensities. Nevertheless, the number of available X-ray data is far fewer than the number of atomic coordinates in a repeat of the helix. Thus, X-ray data alone is inadequate to solve a fiber structure. 

Fig. 7. A typical X-ray diffraction pattern of the Fepr protein fromZJ. vulgaris (Hil-denborough). The pattern was recorded on station 9.6 at the Synchrotron Radiation Source at the CCLRC Daresbury Laboratory using a wavelength 0.87 A and a MAR-Research image-plate detector system with a crystal-to-detector distance of 220 nun. X-ray data clearly extend to a resolution of 1.5 A, or even higher. The crystal system is orthorhombic, spacegroup P2i2i2i with unit cell dimensions, a = 63.87, b = 65.01, c = 153.49 A. The unit cell contains four molecules of 60 kDa moleculEu- weight with a corresponding solvent content of approximately 48%.

As already shown in Figure 2.29a, in the ideal limit-ordered model of form I of sPP, right- and left-handed twofold helical chains alternate along the a and b axes of the orthorhombic unit cell.59 In the electron diffraction patterns of single crystals grown at low temperatures, reported by Lovinger et al.,145 the... 

A possible economically attractive alternative would be the production of acrylic acid in a single step process starting from the cheaper base material propane. In the nineteen nineties the Mitsubishi Chemical cooperation published a MoVTeNb-oxide, which could directly oxidise propane to acrylic acid in one step [6], Own preparations of this material yielded a highly crystalline substance. Careful analysis of single crystal electron diffraction patterns revealed that the MoVTeNb-oxide consists of two crystalline phases- a hexagonal so called K-Phase and an orthorhombic I-phase, which is the actual active catalyst phase, as could be shown by preparing the pure phases and testing them separately. 

If no external evidence is available, it is still possible to determine the unit cell dimensions of crystals of low symmetry from powder diffraction patterns, provided that sharp patterns with high resolution are avail able. Hesse (1948) and Lipson (1949) have used numerical methods successfully for orthorhombic crystals. (Sec also Henry, Lipson, and Wooster, 1951 Bunn 1955.) Ito (1950) has devised a method which in principle will lead to a possible unit cell for a crystal of any symmetry. It may not be the true unit cell appropriate to the crystal symmetry, but when a possible cell satisfying all the diffraction peaks on a powder pattern lias been obtained by Ito s method, the true unit cell can be obtained by a reduction process first devised by Delaunay (1933). Ito applies the reduction process to the reciprocal lattice (see p. 185), but International Tables (1952) recommend that the procedure should be applied to the direct space lattice. 

A previous examination of a synthetic calcium mordenite 15) revealed an orthorhombic cell. A synthetic strontium mordenite 16) had a C-centered orthorhombic cell although Kerr 12) reported that a few crystals giving electron diffraction patterns corresponding approximately to the body-centered structure Immm) have been synthesized hydro-thermally from aluminosilicate gels containing strontium similar to those gels which yielded a strontium-mordenite. ... 

Many mineral and synthetic samples examined by Bennett and Gard 17,18) gave typical C-centered orthorhombic diffraction patterns and with few exceptions had streaks in the hOl section, indicating an incomplete c-glide plane. A few crystals gave electron diffraction patterns having diffuse maxima in the hOl streaks, which could be interpreted as representing an I-centered mordenite structure. 

The Klebsiella K8 polysaccharide has another modification, in which the molecule has a four-fold screw symmetry and packs tetragonally. Since the intensity distribution in the diffraction pattern of the tetragonal form is very similar to that in the orthorhombic form, we assumed (as a first approximation) that the molecule has a four-fold helical (4 or 4 ) symmetry, the same in both crystal forms. 

In situ X-ray examination of crystallizing polyethylene, at high temperature and pressure, then confirmed this proposal in detail, showing that the wide-angle diffraction pattern changed abruptly with the optical texture [ 10]. That corresponding to the spherulitic texture was of the usual orthorhombic form while the new intermediate phase had two-dimensional hexagonal symmetry, with an increased cross-sectional area per chain, but without... 

Fig. 15 Diffraction patterns of ethylene-1-octene copolymer (5.2 mol %) shown from 100 °C to 25 °C while cooling at 10 °C/min recorded during crystallization from melt at 3.8 kbar. The open-orthorhombic phase appears at 80 °C, intensity and position of this reflection remains unchanged. The open-orthorhombic phase is followed by the incoming of the (100) monoclinic reflection concomitant with a shift to higher angles and drop in the intensity of the (110) dense-orthorhombic reflection. The X-ray wavelength used for these experiments is 0.744 A...

X-Ray Studies. Lines in the x-ray, powder diffraction pattern of the terpolymer were similar to those found in polyethylene (PE). Since the terpolymer contains a majority of ethylene segments, we assumed that it crystallized in an orthorhombic unit cell as polyethylene does. The lattice parameters for the terpolymer were calculated to be a = 7.76 A, b = 5.06 A, and c = 2.56 A. These values are to be compared with the dimensions of PE unit cell a — 7.40 A, b = 4.93 A, and c = 2.54 A (8). Thus x-ray data suggest that the terpolymer crystallizes with an expanded PE unit cell. 

Peter Paul Ewald (1888-1985) uses reciprocal lattice vectors to interpret the diffraction patterns by orthorhombic crystals and later generalized the approach to any crystal class. 

The diffraction pattern of a crystal has its own syimnetry (known as Lane syrmnetry), related to the symmetry of the stmcture, thns in the absence of systematic errors (particnlarly absorption), reflections with different, bnt related, indices should have equal intensities. According to Friedel s law, the diffraction pattern of any crystal has a center of inversion, whether the crystal itself is centrosymmetric or not, that is, reflections with indices hkl and hkl ( Friedel equivalents ) are equal. Therefore Lane symmetry is equal to the point-group symmetry of a crystal plus the inversion center (if it is not already present). There are 11 Lane symmetry classes. For example, if a crystal is monoclinic (fi 90°), then I hkl) = I hkl) = I hkl) = I hkl) I hkl). For an orthorhombic crystal, reflections hkl, hkl, hkl, hid and their Friedel equivalents are equal. If by chance a monochnic crystal has f 90°, it can be mistaken for an orthorhombic, but Lane symmetry will show the error. 

The diffraction pattern of the scattered x-rays, including intensities, can be calculated from the electron density distribution of the sample by means of Fourier transforms. Equation 1 shows the calculation of the structure factors F(hkl) for an orthorhombic unit cell of dimensions axbxc from the electron density p(x,y,z) and the exponential phase-factor term. The integers h, k, and / specify the order of the diffraction peaks and are related to the Miller indices that specify the reflecting planes within the crystal producing the diffraction spots (vide infra). An observed intensity I is calculated from I = F hkl)F hkl) = F hkl) where... 

Fig. 5.3. Protein structure determination by X-ray diffraction. A. Crystals of porcine heart aconitase composed of 754 amino acids. The orthorhombic crystals shown are about 0.5 mm in the longest dimension. B. Film showing the diffraction pattern obtained from the above crystal. These data were used to obtain a 2.7 A resolution structure shown in two representations in panels C and D. Panel C shows the tracing of the protein backbone, with the small molecule (in red and yellow) in the central region depicting the iron-containing cofactor of the enzyme. Panel D shows the space-filling representation. (Courtesy of Dr Arthur H. Robbins, Miles Pharmaceuticals Inc. For details see A.H.Robbins and C.D.Stout (1989). Proteins Structure, Function, and Genetics 5, 289 312.)...

The similarity of the two crystal structures leads to very similar X-ray powder diffraction patterns (Fig. 9.3), reminiscent of the situation in terephthalic acid (Section 4.4). Careful inspection reveals that the orthorhombic modification can be distinguished by a peak (the 511 reflection in the Golovina et al. (1994) cell) at 29 = 27.24°, while the monoclinic modification can be distinguished peak (from 211) at 20 = 19.26°. 

Any symmetry in the intensities in the diffraction pattern other than that implied by Friedel s Law is called Laue symmetry (because it can be displayed on Laue X-ray diffraction photographs of an appropriately-aligned crystal, see Figure 4.16). Friedel s Law implies that there is a center of symmetry in the diffraction pattern. Therefore the Laue symmetry-displayed by the diffraction pattern is the point-group symmetry of the crystal with an additional center of symmetry (if this does not already exist). If a crystal is monoclinic then, the intensities I[hkl) and I hkl) are the same, although I hkl) does not equal I[hkl). Orthorhombic... 

Laue symmetry of an orthorhombic crystal (see Figure 4.17). Thus, Laue symmetry, not unit cell dimensions, give a measure of the crystal system. As was stressed earlier, it is the symmetry of the diffraction pattern that tells us that a crystal (lattice) is orthorhombic, not the fact that a = /9 = 7 = 90°. 

---