Indexing known unit cell

Expected Bragg peak positions, shall be computed for all 

See section 4.4 in Chapter 4 and http //www.icdd.com/ for more information about the Powder Diffraction File. 

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). 

Using the improved unit cell dimensions obtained in step 3, the process is repeated from step 1 until all observed Bragg peaks have been indexed. The indexing is considered complete when index assignments and the refined lattice parameters remain unchanged after the last iteration in comparison with the same from the previous cycle. 

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The direction and the periodicity of each cosine wave are given by its index u = hkl), the amplitude of the cosine wave is 2 F(u), proportional to the structure factor amplitude F(u). More importantly, the positions of the maxima and minima of the cosine wave (in relation to the unit cell origin) are determined by the structure factor phase ( )(u). If both the amplitudes F(u) and the phases ( )(u) of the structure factors for all reflections u are known, the potential cp(r) can be obtained by adding a series of such cosine waves. 

The crude molecular image seen in the F0 map, which is obtained from the original indexed intensity data (IFobsI) and the first phase estimates (a calc), serves now as a model of the desired structure. A crude electron density function is devised to describe the unit-cell contents as well as they can be observed in the first map. Then the function is modified to make it more realistic in the light of known properties of proteins and water in crystals. This process is called, depending on the exact details of procedure, density modification, solvent leveling, or solvent flattening. 

A 3D crystal has its atoms arranged such that many different planes can be drawn through them. It is convenient to be able to describe these planes in a systematic way and Fig. 4 shows how this is done. It illustrates a 2D example, but the same principle applies to the third dimension. The crystal lattice can be defined in terms of vectors a and b, which have a defined length and angle between them (it is c in the third dimension). The box defined by a and b (and c for 3D) is known as the unit cell. The dashed lines in Fig. 4A show one set of lines that can be drawn through the 2D lattice (they would be planes in 3D). It can be seen that these lines chop a into 1 piece and b into 1 piece, so these are called the 11 lines. The lines in B, however, chop a into 2 pieces, but still chop b into 1 piece, so these are the 21 lines. If the lines are parallel to an axis as in C, then they do not chop that axis into any pieces so, in C, the lines chopping a into 1 piece and which are parallel to b are the 10 lines. This is a simple rule. The numbers that are generated are known as the Miller indices of the plane. Note that if the structure in Fig. 6.4 was a 3D crystal viewed down the c axis, the lines would be planes. In these cases, the third Miller index would be zero (i.e., the planes would be the 110 planes in A, the 210 planes in B, and the 100... 

The powder pattern of etodolac is shown in Figure 3, and a summary of the observed scattering angles, d-spacings, and relative intensities is shown in Table 1. Since the unit cell parameters of etodolac are known [9], it was possible to index the observed lines to the PbCa and these assignments are also found in Table 1. 

Toraya s WPPD approach is quite similar to the Rietveld method it requires knowledge of the chemical composition of the individual phases (mass absorption coefficients of phases of the sample), and their unit cell parameters from indexing. The benefit of this method is that it does not require the structural model required by the Rietveld method. Furthermore, if the quality of the crystallographic structure is poor and contains disordered pharmaceutical or poorly refined solvent molecules, quantification by the WPPD approach will be unbiased by an inadequate structural model, in contrast to the Rietveld method. If an appropriate internal standard of known quantity is introduced to the sample, the method can be applied to determine the amorphous phase composition as well as the crystalline components.9 The Rietveld method uses structural-based parameters such as atomic coordinates and atomic site occupancies are required for the calculation of the structure factor, in addition to the parameters refined by the WPPD method of Toraya. The additional complexity of the Rietveld method affords a greater amount of information to be extracted from the data set, due to the increased number of refinable parameters. Furthermore, the method is commonly referred to as a standardless method, since the structural model serves the role of a standard crystalline phase. It is generally best to minimize the effect of preferred orientation through sample preparation. In certain instances models of its influence on the powder pattern can be used to improve the refinement.12... 

MOt 78. A distinct phase of relatively narrow composition range occurs as Pr0178. The splitting of the strong lines in the powder pattern can be accounted for in terms of a face-centered rhombohedral unit cell. The true unit cell for this phase is not known. Bevan (3) reports a rhomobohedral phase of narrow composition range at this stoichiometry (y phase). It was indexed with a hexagonal unit cell. 

The size of the crystallite perpendicular to the planes indexed at 16 = 20° is 404 A. If the indices are known, say the peak was the 111 reflection, the perpendicular is along the unit cell diagonal. 

Thus, a fast experiment is routinely suitable for evaluation of the specimen and phase identification, i.e. qualitative analysis. When needed, it should be followed by a weekend experiment for a complete structural determination. An overnight experiment is required for indexing and accurate lattice parameters refinement, and a weekend-long experiment is needed for crystal structure determination and refinement. In some instances, e.g. when a specimen has exceptional quality and its crystal structure is known or very simple, all relevant parameters can be determined using data collected in an overnight experiment. Similarly, fast experiment(s) may be suitable for unit cell determination in addition to phase identification. In any case, one should use his/her own judgment and experience to assess both the suitability of the experimental data and the reliability of the result. 

Indexing of powder diffraction data when unit cell dimensions are known with certain accuracy includes ... 

The zone search indexing method does not require an assumption about the crystal system and therefore, it results in a primitive lattice in most cases. When the lattice is confirmed by the subsequent indexing of all observed Bragg peaks, it shall be converted to one of the 14 standard Bravais lattices. The latter is achieved in a process known as the reduction of the unit cell. 

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 ... 

The shape and size of the unit cell are deduced from the angular positions of the diffraction lines. An assumption is first made as to which of the seven crystal systems the unknown structure belongs to, and then, on the basis of this assumption, the correct Miller indices are assigned to each reflection. This step is called indexing the pattern and is possible only when the correct choice of crystal system has been made. Once this is done, the shape of the unit cell is known (from the crystal system), and its size is calculable from the positions and Miller indices of the diffraction lines. 

Whereas within the family of the cubic Prussian blue analogs a large number of lattice constants have been determined, little attention has been devoted so far to polymeric cyanides not belonging to the cubic system. It must be emphasized, however, that polynuclear cyanides having unit cell symmetries other than cubic are by no means rare exceptions. Hexacyanometalates(III) of Zn2+ and Cd2+ are obtained not only in a cubic modification but also as samples with complicated and not yet resolved X-ray patterns of definitely lower symmetry than cubic (55). The exact conditions for obtaining either modification are not yet known in detail. The hexacyanoferrates(II), -ruthenates(II), and -osmates(II) of Mn2+ and several modifications of the corresponding Co 2+ salts show very complicated X-ray powder patterns which cannot be indexed in the cubic system (55). Preliminary spectroscopic studies show the presence of nearly octahedral M C6-units in these compounds, too. 

When the unit cell parameters are known, it is fairly easy to index the powder pattern. But it becomes trickier and time consuming if one has to index a new, unknown and never indexed... 

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