Crystal shape, calculation

The ciystal habit of sucrose and adipic add crystals were calculated from their intern structure and from the attachment energies of the various crystal faces. As a first attempt to indude the role of the solvent on the crystal habit, the solvent accessible areas of the faces of sucrose and adipic add and were calculated for spherical solvent probes of difierent sizes. In the sucrose system the results show that this type of calculation can qualitatively account for differences in solvent (water) adsorption hence fast growing and slow growing faces. In the adipic add system results show the presence of solvent sized receptacles that might enhance solvent interadions on various fares. The quantitative use of this type of data in crystal shape calculations could prove to be a reasonable method for incorporation of solvent effeds on calculated crystal shapes. 

An enlarged view of a crystal is shown in Fig. VII-11 assume for simplicity that the crystal is two-dimensional. Assuming equilibrium shape, calculate 711 if 710 is 275 dyn/cm. Crystal habit may be changed by selective adsorption. What percentage of reduction in the value of 710 must be effected (by, say, dye adsorption selective to the face) in order that the equilibrium crystal exhibit only (10) faces Show your calculation. 

Networks of steps, seen in STM observations of vicinal surfaces on Au and Pt (110), are analyzed. A simple model is introduced for the calculation of the free energy of the networks as function of the slope parameters, valid at low step densities. It predicts that the networks are unstable, or at least metastable, against faceting and gives an equilibrium crystal shape with sharp edges either between the (110) facet and rounded regions or between two rounded regions. Experimental observations of the equilibrium shapes of Au or Pt crystals at sufficiently low temperatures, i.e. below the deconstruction temperature of the (110) facet, could check the validity of these predictions. 

As is well known (see, for instance, Van Beijeren and Nolden, 1987), the equilibrium crystal shape is the shape that minimizes the total surface free energy at a given fixed volume. From the minimization of the free energy calculated above we can construct the equilibrium shape of the crystal around the (110) facet. This shape depends crucially... 

In a series of papers, Derby and Brown (144, 149-152) developed a detailed TCM that included the calculation of the temperature field in the melt, crystal, and crucible the location of the melt-crystal and melt-ambient surfaces and the crystal shape. The analysis is based on a finite-ele-ment-Newton method, which has been described in detail (152). The heat-transfer model included conduction in each of the phases and an idealized model for radiation from the crystal, melt, and crucible surfaces without a systematic calculation of view factors and difiuse-gray radiative exchange (153). 

Integration of a time-dependent thermal-capillary model for CZ growth (150, 152) also has illuminated the idea of dynamic stability. Derby and Brown (150) first constructed a time-dependent TCM that included the transients associated with conduction in each phase, the evolution of the crystal shape in time, and the decrease in the melt level caused by the conservation of volume. However, the model idealized radiation to be to a uniform ambient. The technique for implicit numerical integration of the transient model was built around the finite-element-Newton method used for the QSSM. Linear and nonlinear stability calculations for the solutions of the QSSM (if the batchwise transient is neglected) showed that the CZ method is dynamically stable small perturbations in the system at fixed operating parameters decayed with time, and changes in the parameters caused the process to evolve to the expected new solutions of the QSSM. The stability of the CZ process has been verified experimentally, at least... 

Figure 24. Streamlines and isotherms for the growth of silicon in a prototype Czochralski system with self-consistent calculation of interface and crystal shapes by using the quasi steady-state thermal-capillary model and the condition that the crystal radius remains constant. Calculations are for decreasing melt volume. The Grashof number (scaled with the maximum temperature difference in the melt) varies between 1.0 X 107 and 2.0 X 107 with decreasing...

Fig. 32. Peak shapes, calculated from crystal structure parameters for the heptamo-lybdate ion, Mo70246", and for two octamolybdates Mo80264 and Mo80288". The separate contributions from Mo—Mo distances (short dashes), Mo—0 distances (dotted lines), and O—0 distances (long dashes) are shown.

Absorption correction can be based on an accurate description of the crystal shape in terms of Miller planes (faces), so that the t can be calculated for each incident and diffracted beam. The total absorption is then given by equation (27), where A is the absorption factor. 

Measured density Dm Calculated density, (g cm ) Radiation type and wavelength Number and 0 range of reflections used to determine lattice parameters Linear absorption coefficient, /x Temperature of measurement, T Crystal shape, color, and size Diffractometer used Scan mode... 

The interest in multicomponent materials, in the past, has led to many attempts to relate their mechanical behaviour to that of the constituent phases (Hull, 1981). Several theoretical developments have concentrated on the study of the elastic moduli of two-component systems (Arridge, 1975 Peterlin, 1973). Specifically, the application of composite theories to relationships between elastic modulus and microstructure applies for semicrystalline polymers exhibiting distinct crystalline and amorphous phases (Andrews, 1974). Furthermore, as discussed in Chapter 4, the elastic modulus has been shown to be correlated to microhardness for lamellar PE. In addition, H has been shown to be a property that describes a semicrystalline polymer as a composite material consisting of stiff (crystals) and soft, compliant elements. Application of this concept to lamellar PE involves, however, certain difficulties. This material has a microstructure that requires specific methods of analysis involving the calculation of the volume fraction of crystallized material, crystal shape and dimensions, etc. (Balta Calleja et al, 1981). 

Even with a rather simple inductor, the numerical calculation shows that it should be possible to form a useful crystal shape (Fig. 3.9). 

There have been many papers concerning the computational studies of unidirectional solidification for solar cells, in which the growth system was imposed to be axisymmetric. However, the actual crystal shape is square, calculation of square-shaped crystals is necessary. When square crucibles are used, the configuration of the furnace becomes asymmetric, and heat transfer in the furnace consequently becomes three-dimensional. Three-dimensional (3D) global modeling is, therefore, necessary for the investigation of m-c interface shape with square crucibles [20],... 

Fig. 3. Calculated equilibrium crystal shape of anthracene (filled octagon) compared to a photograph of the epitaxial growth of an anthracene single crystal on a graphite (0001) substrate taken from Ref. [16],...

Some consequences were discussed in Section I. If a pair of chemically identical (d-d) or nonidentical molecules ( f-D) are packed together, and this packing compared with the corresponding d-l or d-L pair, each for minimum total energy, there is no relation between the pair structures to enable them to be treated in a systematic way. Each case depends on the particular molecular composition and the particular atomic non-bonded radii which collectively are responsible for the surface shape of the molecule conceived as bounded by a hard surface. For each case it is of course possible to make calculations of packing energy and of optimum structure in the way now commonly followed for the packing in molecular crystals. Such calculations have not been reported so far as we are aware. 

INITIAL INTEGRAL MASS SENSITIVITY Cf OF OSCILLATING QUARTZ CRYSTALS OF DIFFERENT RESONANCE FREQUENCY AND CRYSTAL SHAPE, EXPERIMENTALLY DETERMINED AND CALCULATED BY VARIOUS AUTHORS. 

The CTR shape is sensitive to the termination of the crystal surface. Calculations show (Fig. 5B) that CTR data are sensitive not only to the presence of the crystal termination but also to the detailed termination of the lattice. Here, we compare the scattering intensity for a semi-infinite lattice in which the outermost surface layer is ideally terminated with a bulk-like termination, or has been modified, either by its position, ds, or its scattering strength, fs. The total structure factor of the crystal with a modified surface is just the sum of individual structure factors for all atoms in the crystal (a table of commonly used structure factors is given in Appendix 3). Conceptually, this quantity can be broken into two parts, consisting of contributions from the modified surface layer, Fsurf, and from the semi-infinite substrate, Fsub -... 

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