Modelling space group method

The lattice gas model of Bell et al. [33] neither gave any detailed mechanism of the orientational ordering nor separated the contributions of the headgroup and the acyl chain. Lavis et al. [34] discussed Ref. 33 critically and concluded that the sharp kink point in the isotherm at transition was an artifact of the mean field approximation used. An improved correspondence to experimental data was claimed by the use of the real-space renormalization group method [35]. The same authors returned to the problem [35] and concluded that in addition to the orientation of the molecules, chain melting had to be included in a model which could interpret the phase transitions. 

The first approach, taking the advantage of the BCH formula, was initiated hy Jeziorski and Monkhorst [23] and, so far, it has been intensively developed within Paldus s group [5,51-55] who formulated an orthogonally spin-adapted Hilbert space MR CC method for a special case of a two-dimensional model space spanned by closed-shell-type reference configurations. The unknown cluster amplitudes are obtained by the solution of the Bloch equation [45-49]... 

Both methods can easily be derived from a very simple model. Consider a unit cell which contains two XY molecules which are equivalent in the group-theoretical sense, i.e. they are transformed into one-another by the operations of the group of the unit cell (this group is the factor group of the space group 10 and is isomorphous with one of the 32 crystallographic point groups).8 ... 

The whole-pattern-fitting structure-refinement method can give correct, refined structural information for linear polymers. Use of generalized coordinates, in which accurately known structural information can be introduced as known parameters, makes the least-squares fitting problem tractable. Excellent fits of x-ray diffraction data for isotactic polypropylene permitted the selection of the correct space group and a preferred model. 

The crystal structure analysis (by two-dimensional Fourier methods) was facilitated by the fact that the crystal space group requires the molecule to have symmetry 222, the asymmetric crystal unit consisting of one-quarter of the chemical molecule. If there were no distortions from a regular planar model with a trigonal arrangement of bonds... 

The space group was assigned to P2i, which is monoclinic for Y Cs2. The experimental data were analyzed in an iterative way of a combination of Rietveld analysis (Rietveld, 1969) and the maximum entropy method (MEM) (Bricogne, 1988 Collins, 1982). The MEM can produce an election density distribution map from a set of X-ray structure factors without using any structural model. By the MEM analysis (Kumazawa et al., 1993 Sakata and Sato, 1990), the Rj becomes as low as 1.5% for Y Cs2-... 

A total of 1463 raw intensity data were collected. Inspection of the azimuthal scan data showed a variation of /mia//max = 0.82 for the average curve. An empirical correction based on the observed variation was applied to the data as a first approximation. The structure was solved by Patterson methods in space group P. Refinement and elucidation of additional atoms proceeded via standard least-squares and Fourier techniques. Examination of the triclinic model demonstrated the correct monoclinic space group, P2Jc (the apparent absence of hOl, l 2n has been found), and refinement continued in (hat group... 

In addition to reciprocal and direct space techniques considered in the previous sections, a large variety of approaches may be employed to create a model of the crystal structure in direct space. One of these, i.e. the geometrical method, has been implicitly employed in section 6.9, where the location of a single La atom in the unit cell was established from a simple analysis of the unit cell dimensions and from the availability of low multiplicity sites in the space group symmetry P6/mmm. Here we consider a more complex example, i.e. the solution of several crystal structures occurring in the series of Gd5(SixGei x)4 alloys. These examples illustrate the power of the powder diffraction method in detecting subtle details of the... 

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