Lattice Parameter Determination

Mineralogical Composition (in wt %) of Several Natural Zeolite Rocks Employed to Illustrate Some Properties and Applications of These Materials 

Notes Sample identification (label deposit name, location). HC Castillas, Havana, Cuba CMT Tasajeras, Villa Clara, Cuba C1-C6 Camaguey, Cuba MP Palmarito, Santiago de Cuba, Cuba AD Aguas Prietas, Sonora, Mexico CZ Nizni Harabovec, Slovakia GR Dzegvi, Georgia. 

Relations between Interplanar Spacing, Miller Indexes, and Lattice Parameters 

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Indexings and Lattice Parameter Determination. From a powder pattern of a single component it is possible to determine the indices of many reflections. From this information and the 20-values for the reflections, it is possible to determine the unit cell parameters. As with single crystals this information can then be used to identify the material by searching the NIST Crystal Data File (see "SmaU Molecule Single Stmcture Determination" above). 

In Table 2 there are given data pertinent to the lattice-parameter determination, as well as intensity data used in the determination of atomic positional parameters. 

Lattice parameter determination, diffractometers in, 26 428 Lattice polymers, fullerene, 12 250, 251 Lattice-type inclusion compounds,... 

As a final example, we may mention the NaCl-type phases formed in the V-0 systems. The V01 x phase is homogeneous in the composition range 42-57 at.% O. Lattice parameter determination in combination with density measurements evidenced that, in the structure, vacancies occur in both V and O sublattices through the entire range of composition. At the stoichiometric composition VO there are = 15% of sites vacant in each sublattice. 

Identification of unknown crystal structures and determination of phase fields by X-rays can be problematical if the characteristic patterns of the various phases are quite similar, for example in some b.c.c. A2-based ordered phases in noble-metal-based alloys. However, in many cases the characteristic patterns of the phases can be quite different and, even if the exact structure is not known, phase fields can still be well established. Exact determination of phase boundaries is possible using lattice-parameter determination and this is a well-established method for identifying solvus lines for terminal solid solutions. The technique simply requires that the lattice parameter of the phase is measured as a function of composition across the phase boimdary. The lattice parameter varies across the single-phase field but in the two-phase field becomes constant. Figure 4.12 shows such a phase-boundary determination for the HfC(i i) phase where results at various temperatures were used to define the phase boundary as a fimction of temperature (Rudy 1969). As can be seen, the position of is defined exactly and the method can be used to identify phase fields across the whole composition range. 

Figure 1. Lattice parameter versus the duration of intercalation pressure near 0.1 MPa room temperature. The error of the fullerite lattice parameter determined hy powder x-ray diffractometry was 0,02 %.

Figure 2. Temperature dependences of pure fullerite C60 cubic lattice parameter (o) [8] and fullerite C 60 intercalated with helium (A).The error of the pure and intercalated fullerite lattice parameter determination was 0,02 %.

Indeed, lattice parameters of both the copper and the zinc oxide were found to depend on the catalyst composition. The lattice extension of copper was attributed to alpha brass formation upon partial reduction of zine oxide, and an attempt was made to correlate the lattice constant of copper with the decomposition rate of methanol to methyl formate. Furthermore, the decomposition rate of methanol to carbon monoxide was found to correlate with the changes of lattice constant of zinc oxide. Although such correlations did not establish the cause of the promotion in the absence of surface-area measurements and of correlations of specific activities, the changes of lattice parameters determined by Frolich et al. are real and indicate for the first time that the interaction of catalyst components can result in observable changes of bulk properties of the individual phases. Frolich et al. did not offer an interpretation of the observed changes in lattice parameters of zinc oxide. Yet these changes accompany the formation of an active catalyst, and much of this review will be devoted to the origin, physicochemical nature, and catalytic activity of the active phase in the zinc oxide-copper catalysts. 

Berger, H. Systematic errors in precision lattice-parameter determination of single crystals caused by asymmetric line profiles. J. Appl. Cryst. 19, 34-38 (1986). 

P. W. Stephens (private communication). In K3C60, the lattice parameter determined by x-ray diffraction changes from 14.265 A at 300 K to 14.168 A at 11 K, corresponding to a reduction in the nearest-neighbor carbon-carbon distance of 0.07 A. 

Figure 73. The observed and calculated diffraction patterns of LaNi4 g5Sno,i5. The scattered intensity was calculated using scale factor, instrumental and lattice parameters determined during Rietveld refinement, and guessed overall atomic displacement parameter fi = 0.5 A. All notations are identical to Figure 7.2.

The coordinates of individual atoms listed in Table 6.34 were used as a starting point together with the background, peak shape, zero shift and lattice parameters determined from Le Bail s full pattern decomposition (file... 

HAR90] HART M., CERNIK R.J., PARRISH W., TORAYA H., Lattice parameter determination for powders using synchrotron radiation , J. Appl Cryst, vol. 23, p. 286-291,1990. 

MAS 96b] MASCIOCCHIN., ARTIOLI G., Lattice parameters determination from powder diffraction data results from a round robin mo QcC, Powder Diffraction, vol. 11, no. 3, p. 253-258,1996. 

MgCu 2-type, the high-angle lines were too diffused for a lattice parameter determination. 

The X-ray diffraction technique is the most commonly used method to determine the lattice parameters for SiC. Taylor and Jones [1] used this technique to perform some of the most complete lattice parameter determinations. The samples appeared to be very pure but the N2 contamination was not measured which can lead to an uncertainty in the fourth decimal place. A detailed measurement of the lattice constants for 3C and 6H polytypes as a function of temperature can be found in [1]. 

Figure 6.8 Ceria lattice parameters determined from time-resolved XRD patterns for a 2.4 wt% Au-CeOj catalyst under different gases at 500°C pure He, 5% CO in He, 5% CO and 3% HjO in He, again 5% CO in He, and finally 5% Oj in He. ...

Early in 1989, Kuznetsov et al. [40] suggested that nearly all alloys between Pt and Sn are possible. However, the situation is complicated because Pt and Sn also form the intermetallic phases PtaSn, PtSn, Pt2Sn3, PtSna and PtSoj. These intermetallic phases are identified by definite crystalline structures as revealed by X-ray diffraction. Therefore, according to Radmilovic et al. [41] lattice parameters determined from DRX may be the result of mixtines of different phases containing Pt and Sn. 

Magnetostrictive effects in Er are clearly visible in the temperature variation of the lattice parameters determined in the neutron diffraction study of Haben-schuss et al. (1974). The appearance of the c-axis modulated structure below Tv = 84.4 K has little effect on the lattice constants (fig. 6.35), but below the basal plane ordering temperature Ts 52.4 K, the c-axis expands in the manner of the exchange magnetostriction found in the spiral phases of Tb and Dy. Ferromagnetic alignment of the c-axis moment at Tc = 18 K is marked by a... 

The Ui Y FeioSi2 system exists over the whole concentration range (Andreev et al. 1992c) and the lattice parameters determined at room temperature are presented in fig. 57 (lower panel). The smooth monotonic increase in both lattice parameters with increasing... 

Independent of the lattice symmetry, a linear dependence of the lattice parameters (determined by the least-squares fit to the interplanar spacing of selected reflections in the XRD pattern [45] or by the more accurate full-profile fitting analysis [46]) on the Ti content has been found (Fig. 5). The equation relating the unit cell volume to the Ti content (Table 4) is particularly usefiil for determining the real framework composition directly from XRD analysis by comparing this with the Ti content resulting from elemental analysis, the amount of possible extra-framework Ti species can be estimated [46]. 

It should be noted that the dimensions of lattice parameters, determined by x-ray diffraction, have been commonly used to establish the purity of the crystalline phase. Extensive studies of this kind have been carried out with polyethylene copolymers.(21,88-94) The basic assumption is made that the expansion of the lattice reflects the inclusion of the co-unit. However, Bunn has pointed out that this interpretation is not unique.(95) The crystallite thicknesses of such copolymers are relatively small, being less than 100 A, depending on the composition.(74) The strain that develops in the thick interfacial region of such thin crystallites could easily cause the lattice expansion. Hence, the analysis of lattice parameters does not necessarily yield definitive information with respect to the issue of interest. In some cases this analysis has led to incorrect conclusions. 

Figure 2.3. High-resolution scans over characteristic spinel reflections of some catalyst precursors using the Guinier transmission geometry and monochromated Co radiation. The graph indicates the dependence of the spinel lattice parameter determined from the dependence of the spinel lattice parameter determined from the (440) reflection on the aluminum content built into the lattice. Note that all catalyst samples contain nominally the same amount of aluminum. Lattice parameters above the line indicate the presence of an excess of calcium besides aluminum. The data Ml-3 are pure alumina spinel samples.

XRD is one of the most useful techniques for material characterization [28, 29]. It is a rapid, non-destructive and simple technique that provides structural information statistically averaged over an extended spatial region. The sample can be a single crystal of a few micrometers size or polycrystalline of a few centimeters in diameter, loose powder or consolidated agglomerates. Powder XRD (PXRD) technique is used to characterize samples in the form of loose powder or aggregates of finely divided material. It covers various investigations like identification of phases of unknown materials, determination of crystallinity, lattice parameter determination, lattice strains, high-temperature studies and in recent years determination of crystal structure ab initio from PXRD data. 

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