Unit cell dimensions precision

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. 

Determination of unit cell dimensions with high precision. The greatest precision in the determination of the spacings of crystal planes is attained when the angle of reflection (0) is near 90°. This is in the first place a consequence of the form of the Bragg equation... 

Determination of unit cell dimensions with high precision 193... 

The unit-cell dimensions determine the reciprocal-lattice dimensions, which in turn tell us where we must look for the data. Methods like oscillation photography require that we know precisely which reflections will fall completely and partially within a given oscillation angle so that we can collect as many reflections as possible without overlap. So we need the unit-cell dimensions in order to devise a strategy of data collection that will give us as many identifiable (by index), measurable reflections as possible. 

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. 

Compare helix (a) with (b), in which the helix has the same radius, but a longer pitch P (peak-to-peak distance). Note that the layer lines for (b) are more closely spaced. The layer-line spacing is inversely proportional to the helix pitch. The relationship is identical to that in crystals between lattice spacing and unit-cell dimensions [(Eq. (4.10)]. So precise measurement of layer-line spacing allows determination of helix pitch. 

In present-day x-ray ciystallography, powder photography is only rarely used for complete stmcture determinations. It is, however, frequently used in the precise determination of unit-cell dimensions. Its most common technical use is as an analytical tool powder patterns of thousands of ciystalline substances are known. 

Herbstein, F. H. (2000). How precise are measurements of unit-cell dimensions from single crystals Acffl Crytallogr. B, 56, 547-57. [147]... 

Each electron-density wave provides a component for summation to give the electron-density map, shown in Figure 6.11. If the electron density of a crystal could be described precisely by a single cosine wave that repeats three times in the unit cell dimension d, then the electron density has a periodicity of d/3 and the diffraction pattern will have intensity only in the third order (only one diffracted beam, 3 0 0). This is the electron-density wave that is used in the summation that gives an electron-density map if there is only one term because only one Bragg reflection is ob-... 

See Chapter 5, seetion 5.13 for details about various faetors affecting precision of the unit cell dimensions. 

We conclude this section with a simple notion it is impossible to solve the crystal structure of a material using an incorrect unit cell. Thus, proper indexing of the experimental powder diffraction pattern is of utmost importance, and in this chapter we shall consider various strategies leading to the solution of the indexing problem and how to find the most precise unit cell dimensions. 

After the powder diffraction pattern has been successfully indexed, the next step is to establish the unit cell dimensions with the highest possible precision. By combining Eqs. 5.2 and 5.3 one can see that the errors in the lattice parameters only depend on the errors in the measured Bragg angles assuming that Miller indices and A, are known exactly ... 

Ideally, the Bragg angles close to 0 = 90° (20 = 180°) should be available to claim the precision of the unit cell dimensions equivalent to that of the precision of the used wavelength. Unfortunately, measurements at 20 = 180°... 

The reliability and precision of the established unit cell dimensions are not only functions of the quality of the collected experimental data, but they also depend on the presence of measurable diffraction peaks in certain... 

Figure 5.23. The x-ray powder diffraction pattern of U3Ni6Si2 (also see Figure 5.10), schematically illustrating regions, which are critical for successful indexing and precise unit cell dimensions. The boundaries of both the low and high Bragg angle regions are diffuse and they vary from one pattern to another.

Second, is to precisely determine the unit cell dimensions without performing a semi-manual profile fitting (see Chapter 4). 

It is worth noting that in the Le Bail s decomposition, the number of free least squares variables becomes independent of the number of Bragg reflections and only background, peak shape and lattice parameters are refined during each least squares cycle. A small inconvenience of Le Bail s approach is that the unit cell dimensions should be known with a greater precision than in Pawley s method. It also takes more least squares... 

The regularity of the internal atomic arrangement in crystal is described in terms of a unit cell with specific (and measurable) dimensions. By convention the size of this repeat unit is described by edge lengths a, b, and c with angles a, p and y between them (a between b and c, P between a, and c and y between a and b). Values for each of these six parameters, with their precision, are listed in each report of a crystal structure determination. Also, by convention, the unit of measurement of unit cell dimensions and interatomic distances is an Angstrom unit (A), named for a Swedish spectroscopist (1 A = 10-8 cm, 10 nm) bond distances are of the order of 1-2 A. 

Pentasil MFI (ZSM-5) type materials exhibiting different crystallite sizes and AEL (SAPO-11) type materials have been studied. Structural investigations using X-ray diffraction techniques, developed for polycrystalline powdered samples, allow one to determine precisely the structure and crystallinity of the samples and the effect of adsorbates (e.g., p-xylene, n-hexane, etc) on the framework topology (monoclinic towards orthorhombic for MFI) and unit-cell dimensions. 

The parameters of the unit cell and the space group can be determined solely from the x-ray fiber diagram although electron diffraction recorded from single crystals of the polysaccharide are very helpful since they give much more precise information about the base plane. ) Once the unit cell dimensions are known, density measurements will tel 1 how many asymmetric units are present in the unit cell. Moreover, once the symmetry and the advance per monomer of the helix (obtained from the fiber diagram) are known, the number of chains present in the unit cell can be deduced. 

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