Crystal energies

Analyses of large sets of structures have shown that the packing energy correlates reasonably with the molecular volume and also with the number of valence electrons in the molecule [19]. In another approach, neural network techniques have been used to identify a correlation between molecular composition and sublimation enthalpy based on 60 compounds [20]. The correlation is based on a 3-parameter model involving the number of carbon atoms C, the number of hydrogen bond donors HBD and hydrogen bond acceptors HBA (Eq. (8-2)). 

The empirical correlations do not differentiate between configurational isomers or polymorphs, but they are useful in order to estimate the overall expected values of PE. As can be seen from Table 8.2, the values derived from the multilinear regression analysis (MLRA) are at average 28% ( ) lower than the PEs calculated by the force field, which indicates a likely overestimation of polar forces by the chosen charge model. In contrast, the values for the least polar molecule listed in Table 8.2, P.B.16, are almost identical. 

For structure correlation purposes, the crystal energy must be calculated for a large number of crystals of sometimes very complex molecules. It is therefore vital that the method of calculation be relatively simple. For molecular crystals, this can be accomplished by using the so-called atom-atom scheme, pioneered by Kitaigorodski and his school and reviewed by the same author in more recent times [13] it is a two-body approximation, quite suitable for many applications. If the indices i and k denote two atoms in two different molecules of the cluster, the interaction energy between them is written as  

R being the interatomic distance, and A, B, C, D, m, n... being empirical parameters. 

The nature of the repulsive force that arises when two closed-shell systems approach each other can be described in many ways, one of which is as follows. At larger distances the electron clouds of two atoms attract each other by dispersion forces, while at short distances the Pauli exclusion principle drives electrons with the same quantum numbers away from the space they are trying to share, so that a local deshielding of the nuclei occurs, and repulsion arises. Note that the Pauli principle does not imply forces , but only the purely quantum mechanical effect of mutual electron avoidance. At equilibrium, which results from a balance between dispersion attraction and nuclear repulsion forces, there is usually a net gain in energy. The parametric potential must therefore consist of a repulsive (the exponential) and an attractive (mostly, m = 6) term. 

The equilibrium distance is called Rq. It should not be confused with the packing radius, called R, which is a radius for each atomic species such that interatomic distances below the sum of the corresponding radii never or hardly ever occur - it is thus essentially an experimental quantity, based on observations of thousands of crystal structures. Nor should Rq be confused with R/, which is the distance from the nucleus at which the electron density falls below a selected threshold (say, 0.02 e A ). These latter quantities may be numerically similar, and one may even be used in place of another for convenience (we use here / w for Rf, for example) but they are nevertheless conceptually distinct. 

The physical meaning of Equation (12.5) is better understood (for m = 6, D = 0, the commonly occurring values) in the following form [14]  

---

We may recall that in the NaCl crystal structure each positive ion is surrounded by six negative ions, while each negative ion is surrounded by six positives as a result the crystal can be broken up into its component ions only if work is done equal to the crystal energy. In a dilute solution wc find a tendency toward a somewhat similar situation each positive ion is surrounded by a cloud of negative charge, while at the same time each negative ion is surrounded by a cloud of positive charge ... 

The principles described in the following six sections have been deduced in part from the empirical study of known crystal structures and in part from considerations of stability involving the crystal energy. 

In the case of crystals containing highly charged cations the most important terms in the expression for the crystal energy are those representing... 

A set of principles governing the structure of complex ionic crystals, based upon the assumption of a coordinated arrangement of anions about each cation at the comers of an approximately regular polyhedron, is formulated with the aid of considerations based upon the crystal energy. Included in the set is a new electrostatic principle which is of wide application and considerable power. 

The theoretical treatment of the properties of ionic crystals and molecules has been carried farther than that of other types of atomic aggregates. The Bom theory of crystal energy permits the calculation to within... 

Their special field of investigation dealt with the electrical and thermal properties of metals. More recently considerable attention has been paid to the question of the nature of the interatomic forces in metals, which are significant for properties such as density, compressibility, crystal energy, and hardness and it has been found possible to treat this problem in a reasonably satisfactory way for the case of the alkali metals, with a single valence electron per atom.8... 

This qualitative description of the interactions in the metal is compatible with quantum mechanical treatments which have been given the problem,6 and it leads to an understanding of such properties as the ratio of about 1.5 of crystal energy of alkali metals to bond energy of their diatomic molecules (the increase being the contribution of the resonance energy), and the increase in interatomic distance by about 15 percent from the diatomic molecule to the crystal. 

The source of Au was most likely the gold crucible generally used for crystal growing. A comparative analysis of the crystal energy dispersive spectrometry using a SEM (q.v.) identified the primary elements, plus a very weak peak for Au and no indication of Cr and Fe traces. This demonstrates the advantage of PIXE over SEM in terms of sensitivity for trace element detection. 

Since polymorphs differ from one another in their crystal energies, the more energetic ones will seek to revert to the most stable (and the least energetic) crystal form. When several polymorphs and solvates (substances that incorporate solvent in a stoichiometric fashion into the crystal lattice) are present, the conditions under which they may interconvert can become quite complex, as is true of fluprednisolone [58]. 

As radiation interacts in the scintillation crystal, energy is transferred to bound electrons of the crystal s atoms. If the energy that is transferred is greater than the ionization energy, the electron enters the conduction band and is free from the binding forces of the parent atom. 

Gatlin [1.30] measured not only Tg. for mannitol and Na-cefazolin by DSC, but also the dependence of the exothermic crystallization energy from the rewarming rate (Fig. 1.47). 

The crystallization energy, extrapolated for the warming rate zero is calculated for mannitol (13.5 kJ/Mol) and forNa-cefazolin (39,1 kJ/Mol). These data agree with measurements by other methods. The activation energies are generated with certain assumptions to be 335 kJ/Mol for mannitol and 260 kJ/Mol for Na-cefazolin. DeLuca L1.31J derived at with slightly different data at a warming speed of 0.625 °C/min, he found 16.3 kJ/Mol for mannitol and 41.8 kJ/Mol for Na-cefazolin. 

Fig. 1.47. Crystallization energy of sodium cefazolin as a function of the warming rate, measured by DSC (Fig. 2 from [1.301).

The product is an infinite plate, which is cooled from one site only, and the energy flows only perpendicular to its infinite expansion. The crystallization energy flows from the crystallization zone, through the already frozen ice, through the container bottom to a shelf and into the cooling brine. 

In our opinion, the use of and calculations for one-particle Green s functions are uniquely suitable for solid-state systems periodic in any number of dimensions. When faithfully implemented, it satisfies all criteria above. Green s functions offer analytically compact and physically rich tools for representing many properties for extended, periodic systems. They satisfy powerful and elegant relations for quantities such as density of states, lifetimes for excitations, dielectric functions, photo-emission and absorption spectra, total crystal energies, and many more. 

Experimental STDs from CORCEMA-ST method STDs Crystal Energy SICO... 

When there is no weak bonding at all, one returns within the frame of the boundary-layer theory. In this case, however, the chemisorbed particles do not produce any levels in the crystal energy spectrum. 

Among three-dimensional framework hosts forming intercalation compounds, zeolites (see Section 1.5) have attracted considerable attention because of their technical importance (Derouane, 1982 Dyer, 1984). Intracrystalline voids in anhydrous zeolites provide a strongly polar environment that can be filled with polar or ionic species to increase the crystal energy. Treatment of zeolites (e.g. Na-Y zeolite) with vapours of sodium results in the formation of a red product consisting of Na " intercalated into the zeolite cavities (Thomas, 1983). The locations of these clusters inside the zeolites are not yet known, but they seem to be buried within the rather inaccessible cavities inside zeolites, with the result that ordinary solvents do not reach them. Such intercalation of ionic species in zeolites may have implications in nuclear-waste treatment and storage. 

---