Data processing crystal lattice determination

Although the condensation of isolated ions to form a crystal lattice is not a chemically important process, the lattice energies may be used, in conjunction with other data, to determine other quantities of greater interest. 

The dhki values for any crystal can be calculated from knowledge of the lattice parameters (see Chapter 2). The Bragg equation, applied to diffraction data, results in a list of dhk, values for a compound. It is possible, by putting these to data sets together, to determine the size of the unit cell of the material producing the diffraction pattern. In effect, this means allocating a value hkl to each diffracted beam, a process called indexing the diffraction pattern. 

We have discussed the diffraction of X-rays as though each atom in the crystal lattice were a point from which the X-rays are scattered. In fact, the principal scattering is from electrons, which are distributed over the entire unit cell. The scattering from the different parts of the unit cell interferes constructively and destructively in ways that are determined by the electron density in the unit cell. Analysis of the relative intensities of the different diffracted beams allows in some cases for the reconstruction of the electron density as a function of position in the unit cell. This is a complicated process, which we do not describe. The first such structure determinations were done before the advent of programmable computers, with many hours of hand calculation. Present-day calculations are done automatically by computer programs, using intensity data taken with automated computer-driven diffractometers. 

In spite of these caveats, there is intense activity in the application of these methods to polymorphic systems and considerable progress has been made. Two general approaches to the use of these methods in the study of polymorphism may be distinguished. In the first, the methods are utilized to compute the energies of the known crystal structures of polymorphs to evaluate lattice energies and determine the relative stabilities of different modifications. By comparison with experimental thermodynamic data, this approach can be used to evaluate the methods and force fields employed. The ofher principal application has been in fhe generation of possible crystal structures for a substance whose crystal structure is not known, or which for experimental reasons has resisted determination. Such a process implies a certain ability to predict the crystal structure of a system. However, the intrinsically approximate energies of different polymorphs, the nature of force fields, and the inherent imprecision and inaccuracy of the computational method still limit the efificacy of such an approach (Lommerse et al. 2000). Nevertheless, in combination with other physical data, in particular the experimental X-ray powder diffraction pattern, these computational methods provide a potentially powerful approach to structure determination. The first approach is the one applicable to the study of conformational polymorphs. The second is discussed in more detail at the end of this chapter. 

Therefore, if distorted region is assumed to be a sphere with similar dimensions as the lattice constants of polyethylene crystals, the relaxation time t can be estimated from the value of D determined from the data of the decay reaction. The rdaxation time is considered to be a time constant of the molecular motion causing a slight distortion associated with the diffusion of the free radical. Thus, the relaxation time of the molecular motion associated with the decay reaction can be estimated. In order to validate this procedure, the diffusion constant was estimated from the known relaxation time obtained in dynamic mechanical studies of polyethylene within the temperature region of the so-called relaxation process in a crystalline phase and... 

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