Unit cell, accurate dimensions

X-ray diffraction allows the dimensions of the unit cell to be accurately measured. If the structure type of the material is known, the ideal cell contents are also known. Thus, the unit cell of a crystal of composition M2O3 that adopts the corundum structure contains 12 M atoms and 18 O atoms (Supplementary Material, SI). This readily allows the theoretical density of a solid to be calculated. The weights of all of the atoms in the cell are added, and this is divided by the cell volume. 

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 ... 

Sharp photographs with a low background intensity are produced by such cameras. Focusing cameras, however, are used less for identification than for special purposes such as the accurate determination o unit cell dimensions. Further information on this subject will be found on p. 193. 

This value, based on the experimental density, is to be considered the more reliable, since it is based on a measured property of CsCl, while the ionic radii are based on averages over many different compounds. Unit cell dimensions can be measured accurately by x-ray diffraction and from them the theoretical density can be calculated. The measured density is usually lower because most samples that are large enough to measure are not perfect single crystals and contain empty spaces in the form of grain boundaries and various crystalline imperfections. 

Although useful, the unit cell parameter is not always the most accurate measure of acid-site density. For example, studies by Bezman (47) showed that HC catalysts based on zeolite-Y materials with similar unit-cell dimensions had significantly... 

The dimensions of the unit cells of 2,4-diphenyl- and 2,4,5-triphenyl-dithiolium perchlorate and of the simple isotrithione have been determined (Section II, C, 2). Accurate molecular data, however, are lacking.126... 

The object of this experiment is to determine the crystal structure of a solid substance from x-ray powder diffraction patterns. This involves determination of the symmetry classification (cubic, hexagonal, etc.), the type of crystal lattice (simple, body-centered, or face-centered), the dimensions of the unit cell, the number of atoms or ions of each kind in the unit cell, and the position of every atom or ion in the unit cell. Owing to inherent limitations of the powder method, only substances in the cubic system can be easily characterized in this way, and a cubic material will be studied in the present experiment. However, the recent introduction of more accurate experimental techniques and sophisticated computer programs make it possible to refine and determine the structnres of crystals of low syimnetiy from powder diffraction data alone. 

As menhoned above, the unit cell of a crystalline specimen can be determined by SAED. However, the accuracy of the unit cell dimensions obtained is largely dependent upon the calibrahon of camera length, which is a funchon of specimen position and the conditions of the microscope. The procedure is complicated, and in most cases the unit cell parameters determined by SAED are less accurate than those obtained from XRD or neutron diffraction. Owing to the mulhple scattering problem, determination of space group by SAED is less reliable. Consequenhy, HRTEM is not an ideal technique for final determination of crystal structure. 

Although the diffraction data available from RUF4 are of low quality, probably as a result of the small crystallite size, consequent on the method of preparation, some firm structural conclusions can be drawn. The structure is of the same type as VF4 derived by Becker and Muller from single-crystal studies. The two Ru atoms in the monoclinic unit cell are at 0,0, 0 and /z, /z, /z, the Ru-Ru distances then being simply determined by the unit Cell dimensions, which are accurately defined. From this, each Ru atom is seen to have four close Ru atom neighbors in a roughly square arrangement in the same plane. This Ru-Ru nearest-... 

A crystal-structure determination of AuFg was first undertaken by Asprey, Jack, Kruse, and Maitland, but the absence of satisfactory single-crystal results prevented the completion of their structure analysis. Recently, we have prepared small single crystals of excellent quality for X-ray diffraction work and have completed the AuFg structure analysis. We have employed the accurate unit-cell dimensions of Asprey et al and their value for the measured density of AuFj. 

Unit cell dimensions are obtained from measurements of 29 values of several Bragg reflections for which the indices h, k, and I are known. Values of 26 are measured as accurately as possible, and, since the wavelength A. of the radiation used is known, a value of dhu may be found by Bragg s Law, Equation 3.2 (Chapter 3). The value of is related to the unit cell dimensions and, if 26 values are measured for several reflections, values of the unit cell dimensions may be derived. The selected group of reflections chosen to do these calculations should contain a distribution of Miller indices and they should have relatively high 29 values ... 

Using all Bragg peaks which have been indexed and the associated observed Bragg angles, more accurate unit cell dimensions and, if applicable, systematic experimental errors, e.g. sample displacement, sample transparency, or zero shift, which are described in section 2.8.2, Chapter 2, should be refined by means of a least squares technique (see section 5.13, below). 

Figure 6.38. The results of Le Bail s full pattern decomposition of the powder diffraction pattern of GdsGe. The discrepancies between the observed and calculated profiles are small and all residuals are low, indicating that the unit cell dimensions are accurately determined and that the chosen peak shape function (pseudo-Voigt) is a good choice for this experiment.

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