Figure 4-15 A van der Waals Potential Energy Function. The Energy minimum is shallow and the interatomic repulsion energy is steep near the van der Waals radius. |

The interatomic distances are primarily determined by the position of the minimum in the potential energy function describing the interactions between the atoms in the crystal. The question is then, what are the sizes of the atoms and ions The extension of electron density for an atom or an ion is not rigorously defined no exact size can be [Pg.453]

The concepts. All interatomic interactions are modeled with a set of mathematical functions which, when summed over all interactions, gives the potential energy of a molecule. The potential energy functions, the PEFs, contain adjustable parameters which, for a start, are taken from similar work or are merely guessed. [Pg.177]

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. [Pg.442]

The atomistic cause of thermal expansion is often explained by the attractive and repulsive forces between atoms in a solid. The potential energy functions (force applied through a distance) for interatomic attraction, repulsion, and their sum are plotted in Figure 7.2. The base of the trough in the com- [Pg.168]

RIS theory is used to study the unperturbed dimensions of PMPS chains as a function of their stereochemical structure. The required conformational energies are obtained from semi-empirical, interatomic potential energy functions and from known results on PDMS. [Pg.229]

Lennard-Jones potential As two atoms approach one another there is the attraction due to London dispersion forces and eventually a van der Waals repulsion as the interatomic distance r gets smaller than the equilibrium distance. A well-known potential energy function to describe this behavior is the Lennard-Jones (6-12) potential (LJ). The LJ (6-12) potential represents the attractive part as r-6-dependent whereas the repulsive part is represented by an r n term. Another often used nonbonded interaction potential is the Buckingham potential which uses a similar distance dependence for the attractive part as the LJ (6-12) potential but where the repulsive part is represented by an exponential function. [Pg.757]

The Molecular Origins of Elasticity. Recall from Section 1.0.4 that atoms are held together by interatomic bonds and that there are eqnations such as Eq. (1.13) that relate the interatomic force, F, to the potential energy function between the atoms, U, and the separation distance, r [Pg.381]

In ideal circumstances, xif) properly moderates the Coulomb potential to describe the interaction between ions and atoms at all separation distances. For large distances, xif) should tend to zero, while for very small distances, should tend to unity. Such features allow a single interatomic potential energy function, (2.8), to describe the entire collision process. [Pg.16]

Solution According to Eq. (5.5), the elastic modulus, E, is proportional to the stiffness of the theoretical springs that model the bonds between atoms in the solid, Sq. According to Equation (5.1), So is in turn proportional to the second derivative of the potential energy function, U, with respect to interatomic separation distance, r. [Pg.382]

Behavior remarkably similar to that revealed by the one-dimensional model crystals is generally observed for lattice vibrations in three dimensions. Here the dynamical matrix is constructed fundamentally in the same way, based on the model used for the interatomic forces, or derivatives of the crystal s potential energy function, and the equivalent of Eq. (7) is solved for the eigenvalues and eigenvectors [2-4, 29]. Naturally, the phonon wavevector in three dimensions is a vector with three components, q = (qx, qy, qz)> and both the fiequency of the wave, co(q), and its polarization, e q), are functions [Pg.139]

Pair-additive interactions continued to be used in most materials-related simulations for over 20 years after Vineyard s work despite well-known deficiencies in their ability to model surface and bulk properties of most materials. Quantitative simulation of materials properties was therefore very limited. A breakthrough in materials-related atomistic simulation occurred in the 1980s, however, with the development of several many-body analytic potential energy functions that allow accurate quantitative predictions of structures and dynamics of materials.These methods demonstrated that even relatively simple analytic interatomic potential functions can capture many of the details of chemical bonding, provided the functional form is carefully derived from sound physical principles. [Pg.210]

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