Unit cell constants

Figure C2.3.3. Molecular packing of SDS monohydrate viewed as projected on the ac plane. This polymoriDh crystallizes in a triclinic cell with unit cell constants a, b and c of 10.423 A, 5.662 A and 28.913 A, respectively, and with a = 86.70°, (3 = 93.44°, y = 89.55°. There are four molecules per unit cell. Adapted from figure 2 of [18].

Fichtner-Schmittler, H., Lohse, U., Engelhardt, G. et al. (1984) Unit-cell constants of zeolites stabilized by dealumination - determination of al-content from lattice-parameters, Cryst. Res. Technol., 19, Kl. 

The two structurally similar polymorphs of (R,S)-ethambutol dihydrochloride have been shown to bear an enantiotropic relationship, and can reversibly interconvert in a single crystal transformation mode [27]. It was reported that despite the identity in space group type and similarity in unit cell constants, the two forms could be distinguished on the basis of their X-ray powder diffraction and solid-state nuclear magnetic resonance properties. Interestingly, while the (polymorphic forms, the (R,S)-diastereomer was only obtained in two different polymorphs. 

When iron is heated, it undergoes a change from the bcc structure to fee at 910°C. In these forms, the unit cell constants are 363 pm and 293 pm, respectively. By what percent does the volume change during the phase transition What would be the change in density ... 

Volume of the unit cell (VcM). This may be calculated from the unit cell constants by using the following formulae. From this the density (/>) of the compound may be obtained p = Mcc /FccM, where MceH is the mass of the matter in the unit cell, that is the sum of the masses of all the atoms contained. The mass of an individual atom is calculated in grams by the ratio of the atomic molar mass and the Avogadro number. 

Examples of alternative descriptions of the unit cell In a number of cases, in order for instance, to compare a given atomic distribution and arrangement with several others, it may be useful to use different descriptions of the same structure (to refer to different, but obviously equivalent, unit cells). The transformations (of the unit cell constants and, consequently, of the coordinates of the atomic positions) are described, for the general case, for instance, in the International Tables (Hahn 2002). A few, frequently used, transformation formulae of the unit cell constants are reported here. 

A detailed example of the alternative descriptions of a given compound, both in terms of its hexagonal unit cell and of the corresponding rhombohedral primitive cell is presented in Chapter 4 the rhombohedral compound Mo6PbSx (the prototype of the family of the so-called Chevrel phases) is described and unit cell constants and atomic positions are listed for its conventional hexagonal cell and for the rhombohedral primitive cell. 

Cooper, D.A., Denkewicz, R.P., and Hertzenberg, E.P. (1993) Stable zeolite of low unit cell constant and method of making same. US Patent 5,242,577. 

The unit cell constant of the zeolites were determined by X-ray diffraction using CuKo( radiation and following ASTM procedure D-3942-80. The estimated standard deviation was - 0.01 A. The... 

In order to study he Lewis acidity of the samples, the intensity of the 1450 cm pyridine band was also measured. Sample HYUS-8 shows a high amount of Lewis centers (Fig. 4d), relative thf HYD-400 sampl (Fig. 5c). This agrees with the absence of A1 as observed by A1 MAS-NMR for HYD samples. However, chemical analysis (Table I) indicates that there is more aluminum in this sample than in that from the unit cell constant m i urements. These differences cculd be explained considering that A1 MAS-NMR does not detect octahedral EFAL because of the low symmetry of its environment (i ). If this is so, it is remarkable that this EFAL does not show Lewis acidity as measured by pyridine ad y ption. On the other hand, if indeed thej is a small amount of A1, then the EFAL should be present as Al" outside the zeolite framework. In this case it should be present as amorphous silica-alumina. 

Effect of Steam Treatment. X-ray diffraction analyses indicated that ZSM-5 retained in excess of 90% of its crystallinity after the steam treatment described in the methods section. Unit cell constant of the REY zeolite in Super-D declined from 24.658. to 24.38a due to the steam treatment. Independent measurements... 

Heavy-atom derivation of an object as large as a ribosomal particle requires the use of extremely dense and ultraheavy compounds. Examples of such compounds are a) tetrakis(acetoxy-mercuri)methane (TAMM) which was the key heavy atom derivative in the structure determination of nucleosomes and the membrane reaction center and b) an undecagold cluster in which the gold core has a diameter of 8.2 A (Fig. 14 and in and ). Several variations of this cluster, modified with different ligands, have been prepared The cluster compounds, in which all the moieties R (Fig. 14) are amine or alcohol, are soluble in the crystallization solution of SOS subunits from H. marismortui. Thus, they could be used for soaking. Crystallographic data (to 18 A resolution) show isomorphous unit cell constants with observable differences in the intensity (Fig. 15). 

The insertion of a certain amount of chlorine atoms into the unit cells of polyethylene and of isotactic polypropylene has been observed by PLATfe, Tran Kheu, and Shibaev (75) by chlorination of the pure homopolymers. The unit cell constants of the two pure homopolymers increase with increasing amount of bound chlorine. The amorphisation of polyethylene is reached with a chlorine content of about 50%. 

Distance least squares (DLS), a method developed by Meier and Vill-iger (1) for generating model structures (DLS models) of prescribed symmetry and optimum interatomic distances, can supply atomic coordinates which closely approach the values obtained by extensive structure refinement. DLS makes use of the available information on interatomic distances, bond angles, and other geometric features. It is primarily based on the fact that the number of crystallographically non-equivalent interatomic distances exceeds the number of coordinates in framework-type structures. A general DLS program is available (8) which allows any combination of prescribed parameters (interatomic distances, ratios of distances, unit cell constants etc). In addition, subsidiary conditions (as discussed in Refs. 1 and 8) can also be prescribed. 

Structure refinements of the AH 200 and AH 300 samples were conducted in the same way. Unit cell constants, final atomic parameters, and R indexes are given in Table I. (Observed and calculated structure factors are available from the authors.) Interatomic distances and angles are given in Table II. Estimated errors on the population and position of the cations may in some cases be greatly underestimated especially for atoms with low occupancy factors. 

R.W.G. Wyckoff,. Crystal Structures, Vols. 1-6, 2nd ed., Wiley, New York, 1963-1968. The most comprehensive coverage of crystal structures with fine figures, space groups, unit cell constants and atom coordinates. Vols. 1-4, inorganic compounds and Vols. 5 and 6, organic compounds. 

Parameters were calculated from the reported positional coordinates and the crystallographic unit cell constants and are listed in Table 11 for selected complexes. For complexes which lacked crystallographic C3 symmetry the C3 axis was defined as follows. The midpoints of the lines connecting the ligating atoms of each bidentate chelate were calculated and the plane containing these three midpoints constructed. The C3 axis is defined as the normal to this plane which contains the metal ion. The parameters 3, (i and were then calculated.0)... 

An alternative grid, shown in red, defines an alternative set of planes that also contains all equivalent atoms, and under different conditions, also satisfies Bragg s equation. Such planes, which may be constructed in an endless number of ways, all have one property in common - they make rational intercepts on the axes of the unit cell. The fractional intercepts are defined as a/h, h/k, c/l in terms of the unit cell constants and the integer Miller indices, hkl. 

Table 1. The unit cell constants and volume of YxZr1.xMn111FenCOpV0Crq (m+n+o+p+q=2) alloys...

The x-ray diffraction patterns were made with a Picker diffractometer, using Ni-filtered CuK radiation and glass-slide mounts. To derive accurate unit cell constants, slow-scan x-ray patterns were internally standardized with NaCl, KC1, or Si. The integrated intensities were obtained by measuring the area under each peak with a planimeter. 

Both symmetry and unit cell constants are similar to the corresponding values of K2NiF4 type compounds, as is shown in Table II. Indeed, we have found that a reasonable structure can be derived for Sr2Pd03 which is closely related to the K2NiF4 structure. In this proposed structure, the space group D2h25-Zmmm is assumed and the atoms are placed as follows ... 

Takaki Y., Taniguchi T., Nakata K. and Yamaguchi H., Program for Finding the Unit-cell Constants and the Space Group from X-ray Powder Diffraction Data - The Case Where Approximate Unit-cell Contants are Known, J.Ceram. Soc. Jpn., 97 (1989)pp.763-766. 

Nonstoichiometry is a pervasive aspect of oxide chemistry, particularly where the cation can assume two or more valences or aliovalent cation substitutions are facile. These can be classified into three rather broad ranges. Class 1 includes systems where the nonstoichiometry approaches or exceeds that which caimot be detected by classical methods of chemical analysis (i.e., less than 1 part in 1000) but may manifest itself in dramatic changes in electrical or optical properties. Class n includes systems where the nonstoichiometry is of the order of several mole % and readily discernible by chemical analysis, density measurements, or X-ray diffraction measurements of unit cell constants. Class 111 are those systems with broad ranges of nonstoichiometry such as the alkah metal tungsten bronzes. 

The determination of previously unknown unit cell constants (viz., a, b, c, a, (i, y) from the various interlayer spacing values obtained from the powder diffraction profile... 

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