Refinement of the cell parameters

Once the crystal system of a given phase is known, each interplanar distance measured can be associated with a set of Miller indices. These indices express the relation between these distances and the cell parameters. 

From an approximate value for the cell parameter, the interplanar distances are calculated and associated with the distances measured. The discrepancy between the experimental and the calculated distances is then minimized by slightly varying the values of the cell parameters [MAS 96b, PAW 81]. The system is eomprised of six unknowns at the most (the six cell parameters) and of as many equations as there are diffraction peaks for the phase in question. Thus, the parameter values are obtained by gradual refinement, usually from least square calculations. 

Measuring the positions of the peaks can be done with peak by peak fitting, but also by fitting the whole diffraction pattern [PAW 81, TOR 94], The advantage of the latter method is significant, of course, when the phase s crystal symmetry is low, since the pattern will then be comprised of many partially overlapping peaks. 

Laboratory systems equipped with movements accurate to the ten-thousandth of a degree and that use parallel geometry can achieve an uncertainty in the range of 10 ppm. Very high resolution systems associated with synchrotron sources can achieve an uncertainty close to 1 ppm [HAR 90], We have to point out, however, that the tests conducted on a same sample by several laboratories generally lead to much greater errors. The differences between the measured values are often in the range of 100 ppm [PAR 60, MAS 96b]. 

We saw in the historical introduction that the study of crystal structure, that is the determination of the nature and the positions of the atoms inside the crystal cell, quickly became an essential application of X-ray diffraction. W.H. and W.L. Bragg, for example, were awarded the Nobel Prize for their works on the determination of the crystal stracture of several simple phases. Throughout the 20 century, the determination of crystal structure was one of the major driving forces in developing the study of condensed matter by X-ray diffraction with regard to both the improvements made to the instraments and the implementation of more efficient methods of data processing. 

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X-Ray Measurements. A Picker automatic four-circle diffractometer, equipped with a hne focus Mo anode tube, used for data collection. Twelve high-angle reflections (using Mo Koc, ( = 0.709261 A) radiation, at a take-off angle of 2 j were used for a least-squares refinement of the cell parameters. Data were collected and treated as described in a recent article.Three standard reflections 040,080, and 600 were monitored every 200 reflections and showed no decay in intensity during the course of data collection. 

The nature of the crystalline phase and the determination of unit cell volume were made from X-ray diffractograms obtained by Siemens D-5000 diffractometer working with CuKa radiation. For less crystalline samples, ultrasound treatment was used to separate the crystalline phase from the amorphous phase. The refinements of the cell parameters were made with a home-made programme using the values of 35 reflections 20, in the range 1O°<20<5O , measured accurately with the FIT (SOCABIM-Siemens) programme for the decomposition of peaks. 

Note The arrays on the right-hand side are the components of the variance-covariance matrix from each least-squares refinement. For the two cell parameters, the matrix entries are for the refinement of the cell parameter cubed. ... 

Identification of the crystal system and refinement of the cell parameters... 

Note finally that the range of the study has to be as wide as possible. In particular, taking into acconnt peaks diffracted at high angles improves the refinement of the cell parameters and of the thermal agitation factors. 

The variation of the unit cell parameters versus temperature is reported in Figure 2. For the as-synthesized sample, at room temperature, the cell parameter are a=7.5675, b=l 8.1187, c=26.0605 A and the cell volume is 3573.2 A3. In the first step of heating (T <120 °C) only small variation of the cell parameters are shown. The volume variation is mostly due to the c parameter shortening, since it is the most subjected to temperature induced modifications. Between 120 and 360 °C a remains almost constant, c decreases of 0.1%, while b slightly increase-up to 215 °C- and subsequently regains its initial value. The combination of these variations leads an inflection in the volume contraction, slowing down its decrease. Above 390 °C the cell volume remains almost constant and only minor variations in the parameters are observed. The final values obtained after the refinement at 715 °C accounted a variation of -0.25, +0.07, -0.77 and -0.95% for a, b, c and V respectively. The minor variation of the cell parameters above 450 °C indicates that at this temperature the dehydration process is almost fulfilled. The TG curve in flowing air shows that the total mass variation of the as-synthesized phase is 15.8%. Dehydration process is almost fulfilled at about 500 °C above this temperature only... 

The crystal structures of the starting alloy and its hydride were characterized by Cu Ka X-ray diffraction using Thermo Ariel diffractometer. The Rietveld refinement of diffraction profiles was performed in RIETAN 97 program. An accuracy of determination of the cell parameters was 0.001 - 0.005 A. 

The refinement method is the same as the one used in peak by peak fitting and the calculated profiles are chosen in the same way. Usually, the diagram is simultaneously refined over a large angular range, in order to significantly restrict variations in peak positions. Note that, with this approach, the values of the cell parameters can be directly determined. 

Ffe. 4.2 a XRD patterns of AgBii cSb Se2 solid solution samples, b Variation of the cell parameters (refined from the power patterns) with the Sb content (x)... 

The calculated intensities depend on several parameters which may be refined and defined these include the structural characteristics (the cell parameters, see Chapter 3) of the phase, its quantity in a multiphase sample, etc. 

The isomorphous replacement method requires attachment of heavy atoms to protein molecules in the crystal. In this method, atoms of high atomic number are attached to the protein, and the coordinates of these heavy atoms in the unit cell are determined. The X-ray diffraction pattern of both the native protein and its heavy atom derivative(s) are determined. Application of the so-called Patterson function determines the heavy atom coordinates. Following the refinement of heavy atom parameters, the calculation of protein phase angles proceeds. In the final step the electron density of the protein is calculated. 

Mallinson et al. (1988) have performed an analysis of a set of static theoretical structure factors based on a wave function of the octahedral, high-spin hexa-aquairon(II) ion by Newton and coworkers (Jafri et al. 1980, Logan et al. 1984). To simulate the crystal field, the occupancy of the orbitals was modified to represent a low-spin complex with preferential occupancy of the t2g orbitals, rather than the more even distribution found in the high-spin complex. The complex ion (Fig. 10.14) was centered at the corners of a cubic unit cell with a = 10.000 A and space group Pm3. Refinement of the 1375 static structure factors (sin 8/X < 1.2 A 1) gave an agreement factor of R = 4.35% for the spherical-atom model with variable positional parameters (Table 10.12). Addition of three anharmonic thermal... 

Structure refinements of the AH 200 and AH 300 samples were conducted in the same way. Unit cell constants, final atomic parameters, and R indexes are given in Table I. (Observed and calculated structure factors are available from the authors.) Interatomic distances and angles are given in Table II. Estimated errors on the population and position of the cations may in some cases be greatly underestimated especially for atoms with low occupancy factors. 

In reexamining the structure (6) we utilized intensity data for 45 observed reflections in deproteinized lobster tendon (Homarus americanus). The unit cell contains disaccharide sections of two chains, and the model was refined in terms of the following parameters ... 

Abstract. Three independent determinations have been made of the crystalline structure of the a-phase of poly (tetra-methylene terephthalate). The data on which these determinations have been based are used to asses the contributions to the uncertainties of the structural parameters caused by errors in the unit cell parameters, structure factors, and bond parameters. The effects of differences in the model from which refinement is started are also assessed. The major contribution to uncertainty arises from errors in the structure factors (the "R-factor" between structure factor sets from two different laboratories can be greater than 20%) but errors in bond parameters also make a sizeable contribution. Hamilton s test indicates that one of the structure factor sets used in this study is less inaccurate than the other two and using this the best model satisfying all the other data is estimated together with the uncertainties in its parameters. 

The Rietveld refinements of the low temperature data, 250 - 5 K, converged successfully with use of the room-temperature, I2/m structural model. Table 1. The occupations of the oxygen and Cr/Nb cation sites were fixed at the values obtained from the room-temperature refinement. When occupancy refinement was attempted with the neutron data their least squares shifts were hi ly correlated with those of the thermal parameters. The thermal parameters for all the atoms decreased smoothly with decreasing temperature. Magnetic reflections or magnetic intensity added to existing reflections were not observed at any temperature. Unit cell parameters, atomic positions, occupational and thermal parameters, bond distances and bond angles as a function of temperature are listed in Table 1. 

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