Unit cell reduction

Dealumination processes are usually used in conjunction with production of the acid form of zeolite Y for many catalytic apphcations, and zeolites A and X are in most cases no longer used in acid catalytic applications because the high amount of aluminum in their frameworks makes them difficult to stabilize using various dealumination techniques. Successful dealumination and at least partial annealing of defects, resulting from movement of silicon cations to the aluminum vacancies, can be assessed by measurement of the reduction of the unit cell size of the zeolite. This unit cell reduction is a consequence of the relative ionic radii of AF (0.54 A) and Si + (0.40A). 

There are two broadly accepted methods of unit cell reduction. One of them was introduced by Delaunay and then applied to a transformation of a... 

The unit cell reduction using Delaunay-Ito method can be easily automated as is done in the ITO indexing computer code, which is discussed in section 5.11. The Delaunay-Ito reduced unit cell, however, may not be the one with the shortest possible vectors, although the latter is conventionally defined as a standard reduced unit cell. 

As is obvious from columns 2 and 3 in Table 5.24, different indexing algorithms result in different choices of the unit cell for the same lattice and, therefore, unit cell reduction is especially important to compare the results in triclinic symmetry. The unit cell dimensions, reduced using the WLepage program, are listed in Table 5.24 in columns 6-8. Obviously, all of them are represented by the same unit cell, except the incorrect solution shown in row 2. The triclinic unit cell was confirmed by a single crystal diffraction experiment, as shown in row 6. 

Here Pyj is the structure factor for the (hkl) diffiaction peak and is related to the atomic arrangements in the material. Specifically, Fjjj is the Fourier transform of the positions of the atoms in one unit cell. Each atom is weighted by its form factor, which is equal to its atomic number Z for small 26, but which decreases as 2d increases. Thus, XRD is more sensitive to high-Z materials, and for low-Z materials, neutron or electron diffraction may be more suitable. The faaor e (called the Debye-Waller factor) accounts for the reduction in intensity due to the disorder in the crystal, and the diffracting volume V depends on p and on the film thickness. For epitaxial thin films and films with preferred orientations, the integrated intensity depends on the orientation of the specimen. 

The symmetry of the structure we are looking for is imposed on the field 0(r) by building up the field inside a unit cubic cell of a smaller polyhedron, replicating it by reflections, translations, and rotations. Such a procedure not only guarantees that the field has the required symmetry but also enables substantial reduction of independent variables 0/ the function F (f)ij k )- For example, structures having the symmetry of the simple cubic phase are built of quadrirectangular tetrahedron replicated by reflection. The faces of the tetrahedron lie in the planes of mirror symmetry. The volume of the tetrahedron is 1 /48 of the unit cell volume. 

Characterization of the samples by TGA and CHN analysis shows that the template was effectively removed (C < 0.2 wt%). Small-angle X-ray scattering data of the calcined solid shows a reduction in the unit cell due to thermal shrinkage, while the values for the Fenton samples coincide with the starting precursor. Our approach therefore completely preserves the unit cell corresponding to the diameter of the micelles contained in the mesophase. 

Translationengleiche subgroups have an unaltered translation lattice, i.e. the translation vectors and therefore the size of the primitive unit cells of group and subgroup coincide. The symmetry reduction in this case is accomplished by the loss of other symmetry operations, for example by the reduction of the multiplicity of symmetry axes. This implies a transition to a different crystal class. The example on the right in Fig. 18.1 shows how a fourfold rotation axis is converted to a twofold rotation axis when four symmetry-equivalent atoms are replaced by two pairs of different atoms the translation vectors are not affected. 

The group-subgroup relation of the symmetry reduction from diamond to zinc blende is shown in Fig. 18.3. Some comments concerning the terminology have been included. In both structures the atoms have identical coordinates and site symmetries. The unit cell of diamond contains eight C atoms in symmetry-equivalent positions (Wyckoff position 8a). With the symmetry reduction the atomic positions split to two independent positions (4a and 4c) which are occupied in zinc blende by zinc and sulfur atoms. The space groups are translationengleiche the dimensions of the unit cells correspond to each other. The index of the symmetry reduction is 2 exactly half of all symmetry operations is lost. This includes the inversion centers which in diamond are present in the centers of the C-C bonds. 

Given the apparent relationship between covalence and contraction of the unit cell volume described previously, it should be possible to relate Rv to the reduction in magnetic moment found by resonance and neutron diffraction. In this we are limited to the cations Mn2+, Fe2+, Co2+, and Ni2+ in octahedral coordination. 

With further increase of the concentration (in p, phase range for H cLii cNb03) many new bands were observed. The fact that the low concentration boundary of the P phases is approximately x = 0.5 leads to the assumption for some kind of ordering of Li and as reported. On one hand, it can be assumed that the protons form a (nearly) ordered sub-lattice. Such a structure would have a phonon spectrum different from that of a pure LiNbOs, see . On the other hand, the PE probably leads to a reduction of the crystal symmetry, i.e. due to the incorporation of H, the two Li sites in the unit cell may become non-equivalent. In such case, the symmetry would be reduced from Csv to C3. As a result, the number of molecules per unit cell would remain the same, but new bands would appear in the vibration spectrum. 

At room temperature, akaganeite is, like lepidocrocite, paramagnetic. It becomes antiferromagnetic below the Neel temperature of 290 K (Murad, 1988). The value of Tn and the strength of the magnetic interactions are variable and depend upon synthesis conditions (i.e. the temperature and the length of the hydrolysis period). These influence the amount of interstitial water in the compound, which in turn induces spin reduction. Tn decreases linearly to 250 K as the H20/unit cell rises to 0.02 mol mor (Chambaere De Grave, 1984 a). 

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