Computing Crystal Properties

There has not been as much progress computing the properties of crystals as for molecular calculations. One property that is often computed is the bulk modulus. It is an indication of the hardness of the material. 

It may be desirable to predict which crystal structure is most stable in order to predict the products formed under thermodynamic conditions. This is a very difficult task. As of yet, no completely automated way to try all possible crystal structures formed from a particular collection of elements (analogous to a molecular conformation search) has been devised. Even if such an effort were attempted, the amount of computer power necessary would be enormous. Such studies usually test a collection of likely structures, which is by no means infal- 

FIGURE 34.3 Crystal orbital overlap plot for CoNb4Si. 

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To a chemist concerned Tyith the synthesis of new high-explosive compounds the ability to compute detonation properties (detonation pressure, energy, and velocity as well as product composition) from a given molecular structure and the known or estimated crystal density is a problem of the utmost importance. The calculated properties could be meaningful in the decision as to whether it is worth the effort to attempt a new and complex synthesis. One reason behind the recent development of detonation-properties programs for use on high-speed computers has been to supply this desired information. One such program, the ruby code,1 has recently been made available to a number of laboratories, the authors included. 

Crystal properties of si, sll, and sH are given in Table 2.2a. Table 2.2b lists the atomic coordinates for these structures, which will enable the advanced reader to generate computer models of the hydrate crystals. The contrast of si and sll structures is obtained by linking the basic 512 cavity in two different ways to achieve fourfold hydrogen bonds. All modes of associating pentagonal... 

Molecules in Crystals. - Theoretical work on crystalline structures is not generally covered in this review, but papers predominantly concerned with crystal properties sometimes include new computations on the properties of the constituent molecules. 

Figure 2.23 shows the diffraction spectrum of a powder sample of calcium phosphate after subtracting background. With assistance of a computer, we can identify the peak positions in the spectrum and search for a possible match between the spectrum and a PDF data file. Additional chemical information is often used to help in the search process. For example, this specimen contents Ca, P and O. The computer quickly searches for a compound containing Ca, P and O. It finds a match between the diffraction spectrum of a sample with data for hydroxyapatite (Figure 2.24). There are two important parameters in a standard data file shown in Figure 2.24 the position of diffraction (20) and relative intensities of peaks (j ), or int-f in the PDF. I is the peak intensity with the maximum value in a spectrum. The highest int-f value is 999 which should be read as 0.999 in the relative intensity. The PDF may also list the corresponding d-spacing of peaks, which are the true crystal properties.

At the beginning of the 1980s quanffim chemical methods and computer hardware had developed to a stage that the computation of properties depending on PESs of systems larger than two atoms could be contemplated. Examples are thermodynamic properties, such as virial coefficients [11,103] and moments of collision-induced infrared spectral densities [104,105]. The computation of spectroscopic properties of van der Waals molecules came into reach [106-111] and also of molecular crystals [112]. 

Three applications will be considered to illustrate the scope of the computational techniques described in this chapter. These are (1) the calculation of the effect of temperature on framework aluminosilicate structures, showing the predictive capabilities of energy minimization (2) simulation of elasticity at applied temperature and pressure, illustrating that one of the advantages of energy minimization techniques is that crystal properties other than structure can be calculated reliably and (3) calculation of the phase relationship between different MgSiC>3 pyroxenes which demonstrates the range and scope of current techniques. 

Advances in the calculation of crystal properties by band calculations have paralleled the development of localized electron computations. Band theory... 

The electron gas model is an alternative, non-empirical, method for the calculation of crystal properties. The electron gas model has the advantages that it is more computationally efficient than the periodic Hartree-Fock or Kohn-Sham calculations, and does not have the ambiguity of the force field calculations. In the electron gas model the crystal is assumed to be made up of ions The total crystal energy is obtained as the interaction energy of the ions in the crystal plus the self-energy of the ions. The self-energies of the ions are obtained by accurate Hartree-Fock or Kohn-Sham methods, and the interaction energy is obtained approximately with density functionals. 

Blaha P, Schwarz K, Sorantin P, Trickey SB (1990) Comput Phys Common 59 399 Blaha P, Schwarz K, Madsen GKH, Kvasnicka D, Luitz J (2001) An augmented plane wave plus local orbitals program for calculating crystal properties. Vienna University of Technology, Austria. ISBN 3-9501031-1-2... 

The currently-accepted method for averaging single crystal constants to compute bulk properties was proposed by Hill (1952). Based on energy arguments, he proved that Voigt s method provided the upper bound and Reuss s method the lower bound such that the arithmetic mean of their averages would yield more accurate results than either individual average. This is expressed as... 

This chapter includes a discussion of only a small portion of the theoretical work on liquid crystals. Much of what has been done is based on the simple models described here, embellishing them by adding additional interactions or calculating additional quantities. Clearly the task of describing the liquid crystal phase is a difficult one, yet mar of these models present a simple picture for the origin of the most important liquid crystal properties. The use of new analytical techniques and the ability to use more powerful computers continue to refine our understanding of liquid crystals and the reasons behind their complex behaviour. 

Special attention to these polymers is defined by their specific feature, which is orientation in the melt, mostly associated with the intense development in computer technologies. Owing to this property such polymers are devoted to the Tamily of liquid-crystal polymers. The liquid-crystal properties are also observed for PAI with an uneven number of CH2-groups [1]. It should be noted that polyalkanimide (PA-12), discussed in [2 -14], also displays liquid-crystal properties under definite processing modes. 

The consequences of the computed defect properties for polymer crystal behavior have been discussed in the literature cited and will not be gone into here. We suffice to conclude that it is possible to devise a computational strategy that allows the energy minimization of chain assemblies large enought to permit realistic and practical simulation of the energies and packings of defects in polymer crystals. 

Crystal can compute a number of properties, such as Mulliken population analysis, electron density, multipoles. X-ray structure factors, electrostatic potential, band structures, Fermi contact densities, hyperfine tensors, DOS, electron momentum distribution, and Compton profiles. 

Measurement of Residual Stress and Strain. The displacement of the 2 -value of a particular line in a diffraction pattern from its nominal, nonstressed position gives a measure of the amount of stress retained in the crystaUites during the crystallization process. Thus metals prepared in certain ways (eg, cold rolling) have stress in their polycrystalline form. Strain is a function of peak width, but the peak shape is different than that due to crystaUite size. Usually the two properties, crystaUite size and strain, are deterrnined together by a computer program. 

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