Cell-dimension measurement

Diffractometer software can search for reflections, measure their precise positions, and subsequently compute unit-cell parameters. This search entails complexities we need not encounter here. Instead, I will illustrate the simplest method for determining unit-cell dimensions measuring reflection spacings from an orthorhombic crystal on a precession photograph. 

Measure the crystal density and check the unit-cell dimensions. Measure X-ray diffraction data (the native data set). 

Density. Although the polymer unit cell dimensions imply a calculated density of 1.33 g/cm at 20°C, and extrapolation of melt density data indicates a density of 1.13 g/cm at 20°C for the amorphous phase, the density actually measured is 1.15—1.26 g/cm, which indicates the presence of numerous voids in the stmcture. 

The distance of each reflection from the center of the pattern is a function of the fiber-to-film distance, as well as the unit-cell dimensions. Therefore, by measuring the positions of the reflections, it is possible to determine the unit-cell dimensions and, subsequently, index (or assign Miller indices to) all the reflections. Their intensities are measured with a microdensitometer or digitized with a scanner and then processed.8-10 After applying appropriate geometrical corrections for Lorentz and polarization effects, the observed structure amplitudes are computed. This experimental X-ray data set is crucial for the determination and refinement of molecular and packing models, and also for the adjudication of alternatives. 

Vc crystalline Va, amorphous). The densities of the pure crystalline (pc) and pure amorphous (pa) polymer must be known at the temperature and pressure used to measure p. The value of pc can be obtained from the unit cell dimensions when the crystal structure is known. The value of pa can be obtained directly for polymers that can be quenched without crystallization, polyfethylene terephtha-late) is one example. However, for most semi-crystalline polymers the value of pa is extrapolated from the variation of the specific volume of the melt with temperature [16,63]. 

Table 11.1 lists the resulting low-temperature phases calculated for this set of compounds. Where experimental data are available (marked with a star) the predicted structures are those observed at low temperatures. Inverse denotes a perovskite structure in which a large divalent ion is 12-coordinate and a smaller univalent ion 6-coordinate. Unit cell dimensions are predicted to within 1% of the measured values. 

One deduces the space group from the symmetry in the crystal s diffraction pattern and the systematic absence of specific reflections in that pattern. The crystal s cell dimensions are derived from the diffraction pattern for the crystal collected on X-ray film or measured with a diffractometer. An estimation of Z (the number of molecules per unit cell) can be carried out using a method called Vm proposed by Matthews. For most protein crystals the ratio of the unit cell volume and the molecular weight is a value around 2.15 AVOa. Calculation of Z by this method must yield a number of molecules per unit cell that is in agreement with the decided-upon space group. 

The amount of Fe203 supported on zeolite and the Si02/Al203 molar ratio (S/A ratio) of the prepared catalysts were obtained by X-ray fluorescence spectrometry (Rigaku Denki, 3080E). Specific surface areas were measured by BET method (Yuasa, QUANTACHROME). Unit cell dimension (U.D.) was determined from the diffraction angles of (642) with an X-ray powder diffractometer (Rigaku Denki, RU-200). Silicon was used as the reference. 

From a single crystal, it is possible to measure the position and intensity of the hkl reflections accurately and from this data determine not only the unit cell dimensions and space group, but also the precise atomic positions. In most cases, this can be done with speed and accuracy, and it is one of the most powerful structural techniques available to a chemist. 

The theoretical density of a crystal can be obtained from the volume of the unit cell and the mass of the unit cell contents. The results of an X-ray diffraction structure determination gives both of these data, as the unit cell dimensions are accurately measured and the type and number of formula units in the unit cell are also determined. An example of this type of calculation for FeO follows ... 

A particular crystal of FeO was found to have a unit cell dimension of 430.1 pm, a measured density of 5.728 kg m and an iron to oxygen ratio of 0.945. The unit cell volume (which is a cube) is thus (430.1 pm) =7.956x10 (pm) ... 

Even within the limitations of fixed conformation, early force fields were deficient. The predicted unit cell dimensions were too small by more than 20 percent This situation was remedied by adjusting nonbonded potentials describing the fluorine-fluorine and carbon-fluorine interactions until predicted unit cell dimensions adequately reproduced experimentally measured values. This fitting was actually carried out using structural information from several polymers and was not an ad hoc parameterization for PTFE. It should also be pointed out that reproducing PTFE cell dimensions was a process of achieving an approximately correct interchain spacing for an approximate helix since the actual crystalline structure was not known at that time. Subsequently, it has become... 

N 18.67% yel ndls (from Nitrobenz), mp 316° (dec), mp — explodes sol in dimethyl form-amide si sol in hot acet, methyl ethyl ketone glac acet acid. This compd vas prepd by Shipp (Ref 5) from 2,4,6TTr initrobenzyl chloride in methanol by heating on a steam bath with methanol contg KOH. Since this product melts at 105° higher than that.reported by Reich. et al (Refs 1 2), its identity was confirmed by mw detn, elemental analysis, and by an unambiguous independent synthesis. Its X-ray diffraction measurements were also made its unit cell dimensions reported. The product of Reich et al [mp 211°(dec), correct value 218—20°] is probably 2,4,6,2, 4, 6 -Hexanitrobibenzil... 

The x-ray diffraction powder pattern for Mg2NiH4 was indexed on the basis of a tetragonal crystal system with unit cell dimensions of a = 6.464 A and c = 7.033 A. The measured density was 2.57 g/cm3, compatible with four formula units of Mg2NiH4 per unit cell. 

Determination of the proportions of crystalline and amorphous material in partially crystalline polymers. Knowledge of the unit cell dimensions in high polymer crystals leads to a knowledge of the density of the crystalline regions. If the density of amorphous regions is also known, either by measurement of the density of an entirely amorphous specimen (if this can be obtained) or by extrapolation of the liquid density/temperature curve, it is possible to calculate, from the measured density of any partially crystalline specimen, the proportions of crystalline and amorphous material. Since the physical properties of polymer specimens are profoundly influenced by the degree of crystallinity, X-ray determinations of crystallinity are much used in such studies (see Bunn, 1957). 

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