Phase estimation

Step 11. At this point a computer program refines the atomic parameters of the atoms that were assigned labels. The atomic parameters consist of the three position parameters x,j, and for each atom. Also one or six atomic displacement parameters that describe how the atom is "smeared" (due to thermal motion or disorder) are refined for each atom. The atomic parameters are varied so that the calculated reflection intensities are made to be as nearly equal as possible to the observed intensities. During this process, estimated phase angles are obtained for all of the reflections whose intensities were measured. A new three-dimensional electron density map is calculated using these calculated phase angles and the observed intensities. There is less false detail in this map than in the first map. 

To start the design, the peak eurrent should have been estimated during the proeeeding blaek box estimation phase. The eore style and material should have also been seleeted (refer to Appendix D). This time an air gap is neeessary for reliable operation. 

With the chemical structure of PbTX-1 finally known and coordinates for the molecule available from the dimethyl acetal structure, we wanted to return to the natural product crystal structure. From the similarities in unit cells, we assumed that the structures were nearly isomorphous. Structures that are isomorphous are crystallographically similar in all respects, except where they differ chemically. The difference between the derivative structure in space group C2 and the natural product structure in P2. (a subgroup of C2) was that the C-centering translational symmetry was obeyed by most, but not all atoms in the natural product crystal. We proceeded from the beginning with direct methods, using the known orientation of the PbTX-1 dimethyl acetal skeleton (assuming isomorphism) to estimate phase... 

Bradshaw J., Bachu S., et al. Discussion paper on C02 storage capacity estimation (Phase 1) A taskforce for review and development of standards with regards to storage capacity measurment. 2005 CSLF-T 15. 

Fig. 10.7. Phase diagram for a homopolymer of chain length r = 8onal0xl0xl0 simple cubic lattice of coordination number z = 6. Filled circles give the reduced temperature, T and mean volume fraction, () of the three runs performed. Arrows from the run points indicate the range of densities sampled for each simulation. The thick continuous line is the estimated phase coexistence curve. Reprinted by permission from [6], 2000IOP Publishing Ltd...

Sine fitting methods for estimating phase and modulation... 

With some further assumptions, it is possible to use single frequency FLIM data to fit a two-component model, and calculate the relative concentration of each species, in each pixel [16], To simplify the analysis, we will assume that in each pixel of the sample we have a mixture of two components with single exponential decay kinetics. We assume that the unknown fluorescence lifetimes, iq and r2, are invariant in the sample. In each pixel, the relative concentrations of species may be different and are unknown. We first seek to estimate the two spatially invariant lifetimes, iq and t2. We make a transformation of the estimated phase-shifts and demodulations as follows ... 

Figure 2.3 suggests a simple strategy to recover the fluorescence lifetimes t, and t2 Given estimated phase shifts and demodulations,... 

J. S. Chikos, W. E. Acree Jr., and J. F. Liebman, in Computational Thermochemistry, K. K. Irikura and D. J. Frurip, Eds., ACS Symposium Series 677, American Chemical Society, Washington, DC, 1998, pp. 63-91. Estimating Phase-Change Enthalpies and Entropies. 

Table 3. Estimated phase error 8 from imaging a 2.5 cm object at various distances using 1 THz radiation. The baseline b is an estimate of the physical size of the imaging array. Note that for distances of 5-10m, the size of the imaging array is 6-12 cm suggesting that a hand-held unit...

Two other common methods are used to estimate phases in protein crystallography molecular replacement (MR), which uses the... 

The Athena Visual Studio code and the output with LISTS = 1 are shown below, The results are similar to those of Example C.4, except that the number of function calls for the estimation phase is reduced from 15 to 8. 

This method does not require powder samples. Thus, it is convenient for estimating phase contents in polycrystalline solids. 

DM are ab initio crystal structure solution techniques able to estimate phases directly from structure factor magnitudes. These methods started in 1948. With the advent of modern computers and the use of sophisticated mathematical approaches, they are able to definitively solve, in practice, the phase problem for small molecules. Protein structures, up to say 2000 non-hydrogen atoms in the asymmetric unit, are today solved ab initio, provided single-crystal diffraction data with resolution better than 1.2 A are available. 

The discrimination phase of the method runs in parallel with the estimation phase. It tries to find out whether the change experienced is in the autoregressive parameters or in the constant term (level) of the auto-correlated process variable. The parameter estimates from the estimation phase are used to estimate the level parameter (Eq. 2.45). If the alternate hypothesis is accepted, the change experienced involves variation... 

To be suitable for commercial applications, a sorbent should have a high selectivity to enable sharp separations, high capacity to minimize the amount of sorbent needed, and the capability of being regenerated for reuse. These properties depend upon the dynamic equilibrium distribution of the solute between the fluid and the solid surface. Unlike vapor-liquid and liquid-liquid equilibria where theory is often applied to estimate phase distribution of the solute, no acceptable theory has been developed to predict solid-sorbent equilibria. Thus, it is necessary to obtain... 

Meulbroek et al. have created an innovative computational framework to estimate phase equilibria, equation of state properties and compositionally dependent viscosity for the modelling of reservoir processes. This toolkit has been designed with the reservoir geochemist in mind and therefore is created as a set of modules that can be utilized within standard spreadsheet applications. 

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