Probability density distribution crystal

The aim of crystallization is to separate the observed component into higher quality crystals. The crystal size and probability density distribution of its size become very important factors for the product or the following processes. Although multi-phase mixing is fairly common in industries, there have been few investigations on the mixing performance of operations/equipment. In crystallization operation, the assumption of MSMPR has been used to design a crystallizer without a detailed discussion. Therefore, the assumption of MSMPR must be studied quantitatively. 

Figure 5.5 Data of crystal size probability density distribution and fitted PSD curve based on new PSD function.

Particle-in-a-box states for an electron in a 20 Pd atom linear chain assembled on a single crystal NIAI surface. The left set of curves shows the predictions of the one-dimensional parti-cle-in-a-box model, the center set of images is the 2D probability density distribution, and the right of curves is a line scan of the probability density distribution taken along the center of the chain. The chain was assembled and the probability densities measured using a scanning tunneling microscope. 

With the Fourier difference method, data obtained from single-crystal neutron diffraction provide a full view of the probability density of the H(D) atoms. For this purpose, once the crystal structure has been determined, Bragg peak intensities can be calculated for an ideal crystal in which the scattering cross-section of the H(D) atoms of the methyl groups is set to zero. The difference from the original pattern contains specific information on the methyl H(D) atoms. Further Fourier back-transformation gives the probability density distribution in direct space (see Figure 8.16). 

Figure 8.17 Measured probability density distributions for the three inequivalent methyl groups in the manganese diacetate tetrahydrate crystal at 300 K. Labels A, B and C refer to those in Figures 8.15 and 8.16. Reproduced with permission of the International Union of Crystallography.

Fig. 1-2. Energy distribution of electrons near the Fermi level, cf> in metal crystals c = electron energy f(.i) s distribution function (probability density) ZXe) = electron state density, = occupied...

Both the number and weight basis probability density functions of final product crystals were found to be expressed by a %2-function, under the assumption that the CSD obtained by continuous crystallizer is controlled predominantly by RTD of crystals in crystallizer, and that the CSD thus expressed exhibits the linear relationships on Rosin-Rammler chart in the range of about 10-90 % of the cumulative residue distribution. 

The temporal evolution of spatial correlations of both similar and dissimilar particles for d = 1 is shown in Fig. 6.15 (a) and (b) for both the symmetric, Da = Dft, and asymmetric, Da = 0 cases. What is striking, first of all, is rapid growth of the non-Poisson density fluctuations of similar particles e.g., for Dt/r = 104 the probability density to find a pair of close (r ro) A (or B) particles, XA(ro,t), by a factor of 7 exceeds that for a random distribution. This property could be used as a good aggregation criterion in the study of reactions between actual defects in solids, e.g., in ionic crystals, where concentrations of monomer, dimer and tetramer F centres (1 to 3 electrons trapped by anion vacancies which are 1 to 3nn, respectively) could be easily measured by means of the optical absorption [22], Namely in this manner non-Poissonian clustering of F centres was observed in KC1 crystals X-irradiated for a very long time at 4 K [23],... 

Here, P pb) is the probability (density) for finding the committor ps in the ensemble g = g. If this distribution is peaked around pb = 0.5, the constraint ensemble g = g is located on the separatrix and coincides with the transition state ensemble. In this case, g is a good reaction coordinate, at least in the neighborhood of the separatrix. This is illustrated in Fig. 11a. Other possible scenarios for the underlying free energy landscape result in different committor distributions, and are also illustrated in Fig. 11. The committor distribution can thus be used to estimate how far a postulated reaction coordinate is removed from the correct reaction coordinate. An application of this methodology can be found in [33], where the reaction coordinate of the crystallization of a Lennard-Jones fluid has been resolved by analysis of committor distributions. 

Present knowledge of the details of the conformation of proteins is based almost exclusively on results of studies of protein crystals by x-ray diffraction. Protein crystals contain anywhere from 20 to 80% solvent (1 ) (dilute buffer, often containing a high molarity of salt or organic precipitant). While some solvent molecules can be discerned as discrete maxima of the electron density distribution calculated from the x-ray results, the majority of the solvent molecules cannot be located in this manner most of the solvent appears to be very mobile and to have a fluctuating structure perhaps similar to that of liquid water. Many additional distinct locations near which a solvent molecule is present during much of the time have been identified in the course of crystallographic refinement of several small proteins (2,3,4,5, 6), but in all cases the description of solvent structure in the crystal is incomplete probably because only a statistical description is inherently appropriate. 

Although the crystal structure of cytochrome bi from Hansenula anomala is presently not known, its three-dimensional structure is probably similar to that of Saccharomyces cerevisiae which has been determined at 2.4 A resolution. The authors calculated that the distance between the heme iron and G of the FMN radical is equal to 15 A by using the crystal coordinates and the spin density distribution of an anionic flavin radical deduced from ENDOR studies. ... 

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