Interatomic potentials hardness

Themiodynamic stability requires a repulsive core m the interatomic potential of atoms and molecules, which is a manifestation of the Pauli exclusion principle operating at short distances. This means that the Coulomb and dipole interaction potentials between charged and uncharged real atoms or molecules must be supplemented by a hard core or other repulsive interactions. Examples are as follows. 

Statistical mechanical theory and computer simulations provide a link between the equation of state and the interatomic potential energy functions. A fluid-solid transition at high density has been inferred from computer simulations of hard spheres. A vapour-liquid phase transition also appears when an attractive component is present hr the interatomic potential (e.g. atoms interacting tlirough a Leimard-Jones potential) provided the temperature lies below T, the critical temperature for this transition. This is illustrated in figure A2.3.2 where the critical point is a point of inflexion of tire critical isothemr in the P - Vplane. 

We will describe integral equation approximations for the two-particle correlation fiinctions. There is no single approximation that is equally good for all interatomic potentials in the 3D world, but the solutions for a few important models can be obtained analytically. These include the Percus-Yevick (PY) approximation [27, 28] for hard spheres and the mean spherical (MS) approximation for charged hard spheres, for hard spheres with point dipoles and for atoms interacting with a Yukawa potential. Numerical solutions for other approximations, such as the hypemetted chain (EfNC) approximation for charged systems, are readily obtained by fast Fourier transfonn methods... 

We will define the degree of normalized hardness of an interatomic potential by... 

Hope et al. (116) presented a combined volumetric sorption and theoretical study of the sorption of Kr in silicalite. The theoretical calculation was based on a potential model related to that of Sanders et al. (117), which includes electrostatic terms and a simple bond-bending formalism for the portion of the framework (120 atoms) that is allowed to relax during the simulations. In contrast to the potential developed by Sanders et al., these calculations employed hard, unpolarizable oxygen ions. Polarizability was, however, included in the description of the Kr atoms. Intermolecular potential terms accounting for the interaction of Kr atoms with the zeolite oxygen atoms were derived from fitting experimental results characterizing the interatomic potentials of rare gas mixtures. In contrast to the situation for hydrocarbons, there are few direct empirical data to aid parameterization, but the use of Ne-Kr potentials is reasonable, because Ne is isoelectronic with O2-. 

Parenthetically, we would like to use this opportunity to correct a misunderstanding that is common in enzymatic literature. Highly anharmonic potentials do not necessarily exclude harmonic dynamics An example is water the interatomic potential is extremely anharmonic (hard spheres), but water supports harmonic waves (sound). The resolution of the paradox is that the variable that describes sound waves (density) is not the variable that enters the anharmonic interatomic potentials, so it is possible for equilibrium fluctuations, like sound, to have harmonic dynamics. 

If the projectile and target atoms interact like colliding billiard balls (elastic hard-spheres), the interatomic potential that represents this condition is called a hard-sphere potential. For a hard-sphere potential, the power-law cross-section parameter m in (4.19) is equal to 0. Derive the total cross-section, a (It), for a hard-sphere potential. 

The second and fourth Kinchin-Pease assumptions, which treat the colliding particles as hard-spheres (Assumption 2) and ignore electronic stopping (Assumption 4) result in an overestimate of (Nd(E)) by (7.3). By correctly accounting for electronic stopping and using a realistic interatomic potential to describe the atomic interactions, the Kinchin-Pease damage function is modified to... 

It is remarkable that the LCP model based on hard contacts works so well. Although it could possibly be improved by using soft contacts based on some appropriate interatomic potential, this would unnecessarily complicate what appears to be a simple but useful model. 

The shapes of the interatomic potential curves are approximations chosen for mathematical convenience. Such potential functions are generally used in discussions on a variety of properties of molecules and lattices optical absorption and luminescence, laser action, infrared spectroscopy, melting, thermal expansion coefficients, surface chemistry, shock wave processes, compressibility, hardness, physisorption and chemisorption rates, electrostriction, and piezoelectricity. The lattice energies and the vibration frequencies of ionic solids are well accounted for by such potentials. On heating, the atoms acquire a higher vibrational energy and an increasing vibrational amplitude until their amplitude is 10-15% of the interatomic distance, at which point the solid melts. 

The melting point of a solid and its response to shock waves depends on the form and depth of the interatomic potential. Its compressibility is simply related to the Born repulsion parameter. The mechanical hardness is a function of the bulk modulus (inverse compressibility), which in turn correlates strongly with the volume density of the chemical bonding energy, i.e., the bonding energy per unit volume the smaller the atoms and the stronger the atomic bond, the harder the solid. In the box on p. 34, a simple relation is derived between the bulk modulus and the Bom repulsion parameter. 

M. Stoneham, J. Harding, and A. Harker, MRS Bull., 29 February 1996. The Shell Model and Interatomic Potentials for Ceramics. 

The simplest interatomic potential is the hard-sphere potential, that can be characterised as... 

Find the value of the Lennard-Jones representation of the interatomic potential function of argon at interatomic distances equal to each of the effective hard-sphere diameters of argon at different temperatures in Table A. 15 of Appendix A. Explain the temperature dependence of your values. 

This section was called reahstic interatomic potentials for good reason. We did improve upon the hard-sphere model by adding a qualitative correction, namely, the long-range attraction, and by having a more reasonable, somewhat softer repulsion. But the potentials that we have so far discussed stiU have a major limitation they were taken to depend only on the distance R between the centers of the two interacting molecules. We were still discussing particles without internal chemical structure. The anisotropic shape of molecules means that a molecule does not look the same from all possible directions of approach. 

Figure 4.10 The phase shift S/ computed for a realistic interatomic potential vs. /. The computation is for the realistic value A= kx = 00, meaning that many partial waves contribute to the scattering a is the range of the potential and k= p/ft is the wave vector). Note the steep variation of the phase shift at lower k that is due to the repulsive core of the potential (the initial decline is with a slope of n/2, which is what we expect for a hard-sphere scattering). The stationary point occurs at the glory impact parameter, /g= kbg.

Fig. 4.2 (a) The full curve shows the normalized pair potential, / , versus the normalized interatomic distance, R/Rh, for the degree of normalized hardness, = 2 corresponding to X = 2. The two dashed curves show the repulsive and attractive contributions respectively. The shaded region delineates the hard-core potential with = 1 corresponding to X = oo. The two vertical arrows mark the equilibrium nearest-neighbour distances for = 1 and respectively, (b) The normalized pair potential, / , versus the normalized interatomic distance for different values of the degree of normalized hardness, . Note that = 0 corresponds to a totally soft potential, ah = 1 to a totally hard potential. 

According to Ferry12 the free-volume per cm of substance, i. e. the fractional free-volume /, is hard to define exactly and should be regarded as merely a useful semi quantitative concept. Specifically, the thermal expansion coefficients of liquids for the most part reflect the increase in fractional free-volume only a small part is connected with the anharmonic dependence of potential energy or interatomic and intermoleeular distances. 

---