Crystal minimum lattice energy

The lattice energy will be a minimum when the crystal is at equilibrium (i.e., when the internuclear distance is at the equilibrium value of Fq). If we minimize the lattice energy (see Box), we get ... 

Because of the orientational freedom, plastic crystals usually crystallize in cubic structures (Table 4.2). It is significant that cubic structures are adopted even when the molecular symmetry is incompatible with the cubic crystal symmetry. For example, t-butyl chloride in the plastic crystalline state has a fee structure even though the isolated molecule has a three-fold rotation axis which is incompatible with the cubic structure. Such apparent discrepancies between the lattice symmetry and molecular symmetry provide clear indications of the rotational disorder in the plastic crystalline state. It should, however, be remarked that molecular rotation in plastic crystals is rarely free rather it appears that there is more than one minimum potential energy configuration which allows the molecules to tumble rapidly from one orientation to another, the different orientations being random in the plastic crystal. 

For the description of die temperature and stress-related behavior of the crystal we used die method of consistent quasi-hannonic lattice dynamics (CLD), which permits die determination of the equilibrium crystal structure of minimum free energy. The techniques of lattice dynamics are well developed, and an explanation of CLD and its application to the calculation of the minimum free-energy crystal structure and properties of poly(ethylene) has already been presented. ... 

Equation (4) expresses G as a function of temperature and state of applied stress (pressure) (o. Pa), (/(a) is given by the force field for the set of lattice constants a, Vt is the unit cell volume at temperature T, and Oj and are the components of the stress and strain tensors, respectively (in Voigt notation). The equilibrium crystal structure at a specified temperature and stress is determined by minimizing G(r, a) with respect to die lattice parameters, atomic positions, and shell positions, and yields simultaneously the crystal structure and polarization of minimum free energy. 

Figure 2. The minimum potential energy, tm, of a helium atom interacting with the 100 face of an argon crystal is plotted as a function of the position of the helium atom relative to the surface lattice...

The value of the exponent n can be deduced from the compressibility of the crystal values used in calculating lattice energies are 7, 9,10, and 12 for ions with Ne, Ar, Kr, and Xe configurations. The value of B is calculated in the following way. At equilibrium the energy is a minimum, that is, the attractive and repulsive forces balance one another, therefore... 

Given that strontium oxide crystallizes with the same structure as calcium oxide, use data tables to estimate the minimum cation/anion distance and hence determine the lattice energy for SrO. 

Table 10 shows MPA results for the crystal structures of pyrimidine and s-tetrazine. " ° In the MPA calculations, all parameters not fixed by the observed space group symmetry were varied simultaneously to find the minimum of the lattice energy. The monopole charges were multiplied by a scale factor of 0.83 so that they approximately reproduce experimental dipole moments. In the MPA calculations foreshortened C —H bond lengths were used, and monopole values were adjusted to retain the nonforeshortened bond dipole value. 

This formalism has been used to adapt THBREL, a crystal structure relaxation program which has been widely used in modelling inorganic solids, to model rigid molecules whose electrostatic interactions are described by a DMA model (Willock et al., 1995). The program uses the Cartesian coordinates of the centres of mass of each molecule, rotations around the molecule fixed axes and the six strain matrix elements as variables, so the relaxation of the crystal structure to a minimum in the lattice energy is not restricted to specific space groups. 

Further improvements in our model potentials and simulation methods will therefore undoubtedly increase the detailed accuracy of molecular crystal structure predictions and will be required for crystal structures that correspond to weakly defined minima. However, for a routine transferable scheme, the addition of a realistic ab initio based electrostatic model clearly improves the range of molecules where a minimum in the lattice energy is close to the observed structure. The use of a theoretically derived, rather than an empirical potential, also increases confidence in the extrapolation of the potential to regions sampled in hypothetical crystal structures. 

---