Interatomic Forces in Solids

In many applications involving the use of interatomic potentials in physics and materials science, it is not necessary to know the precise form of the force field between the interacting particles. Even if the exact functional form of the potential energy were known, its mathematical complexity would restrict it from being used in simple analytical work, finding utility only in detailed computer calculations. Empirical atomic interactions are based on a simple analytical model, which provides a mathematically tractable, analytical expression for the pairwise interaction between two atoms or ions. 

A useful potential in modeling the condensed states of solids or liquids is the Lennard-Jones potential 

An alternative way to look at Fig. 2.1a is to recall that -dl /dr = F, where F is the force relationship between the atoms in the crystal. Note that under this convention, a positive force results when dr is negative, and a negative force results when dr is positive, which is opposite to the stress convention applied in materials science. We plot F = dF/dr in Fig. 2.1b to maintain the stress convention, wherein increasing the distance between atoms produces a positive restoring force, 

For crystalline solids, the equilibrium interatomic distance, r0, can be estimated from knowledge of lattice site separation distances and is typically expressed as some fraction of the lattice parameter ac. Aluminum forms a face-centered-cubic (fee) lattice, with lattice parameter ac = 0.405 nm. Since the densest packing direction is along the face diagonal, i.e., along the (110) direction, the equilibrium interatomic distance in Al is 4la /2 = 0 29nm. We can also calculate the distance 

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Measurements of dispersion curves provide information about the interatomic forces in solids. In fact, a dispersion curve is a function of the vibration frequency V on the wavelength X. The methods of neutron spectroscopy based on the phenomenon of diffraction of heat neutrons by crystals enables one to graph the dispersion curves of solids. The most accurate measurements of these curves are obtained by the inelastic neutron scattering using triple axis spectrometers. 

This chapter has focused on properties related to the interatomic forces in solids, in particular as they are reflected in the phonon spectrum and in the elastic constants. A phonon spectrum is often crudely characterized by a Debye temperature. The concept of a Debye temperature may give a convenient and accurate description of the vibrational spectrum, provided that the Debye temperature is properly defined. Different physical phenomena depend on different averages over the phonon spectrum and hence cannot be described by the same Debye temperature. 

The dynamics of ion surface scattering at energies exceeding several hundred electronvolts can be described by a series of binary collision approximations (BCAs) in which only the interaction of one energetic particle with a solid atom is considered at a time [25]. This model is reasonable because the interaction time for the collision is short compared witii the period of phonon frequencies in solids, and the interaction distance is shorter tlian the interatomic distances in solids. The BCA simplifies the many-body interactions between a projectile and solid atoms to a series of two-body collisions of the projectile and individual solid atoms. This can be described with results from the well known two-body central force problem [26]. 

We may summarize the above discussion by saying that a review of the crystal structures of different types of solid emphasizes the dominating role of the interatomic forces in determining the structural arrangement, and that these forces may be conveniently divided into four distinct types ... 

Here ks has the dimension of a force constant and is a certain (complicated) average over all the interatomic forces in the solid, and is the logarithmic average of the atomic masses. For instance, in TiC and in TiBj. 

Chemicals exist as gases, liquids or solids. Solids have definite shapes and volume and are held together by strong intermolecular and interatomic forces. For many substances, these forces are strong enough to maintain the atoms in definite ordered arrays, called crystals. Solids with little or no crystal structure are termed amorphous. 

Sodium atoms must be removed from the solid to form sodium gas. Energy must be supplied to do this because, as we describe in Chapter 11. interatomic forces hold the atoms together in the solid metal. The tabulated value for the enthalpy of vaporization of Na is 107.5 kJ/mol. As described in Chapter 6, at 298 K the energy of vaporization is 2.5 kJ/mol less than this ... 

In the context of elastic deformation two parameters, known as stress and strain respectively, are very relevant. Stress is an internal distributed force which is the resultant of all the interatomic forces that come into play during deformation. In the case of the solid bar loaded axially in tension, let the cross sectional area normal to the axial direction be A0. From a macroscopic point of view the stress may be considered to be uniformly distributed on any plane normal to the axis and to be given by o A0 where o is known as the normal stress. The stress has to balance the applied load, F, and one must, therefore have o Aq = F or o = F/Aq. The units of stress are those of force per unit area, i.e., newtons per square... 

In ordinary solids such as crystalline or amorphous glassy materials, an externally applied force changes the distance between neighboring atoms, resulting in interatomic or intermolecular forces. In these materials, the distance between two atoms should only be altered by no more than a fraction of an angstrom if the deformation is to be recoverable. At higher deformations, the atoms slide past each other, and either flow takes place or the material fractures. The response of rubbers on the other hand is almost entirely intramolecular [4,5]. 

Obviously, we need to start from microscopic (classic and quantum) models. These models require some knowledge about the nature of the interatomic (or interionic) bonding forces in our sohd and whether or not the valence electrons are free to move inside the solid. 

Derived for gases at high d, in terms of interatomic forces, using statistical mechanics. The atom in a dense gas was considered similar to that in a liq or cryst, subject to multiple collisions at all times (Ref 1). This method was later extended to liqs (Ref 2) and solids (Refs 3 4), and was used by Murgai and others for the calcn of the expl properties of TNT and PETN (Refs 5 6)... 

INTERATOMIC FORCE CONSTANTS IN PERIODIC SOLIDS FROM DENSITY FUNCTIONAL PERTURBATION THEORY... 

Interatomic Force Constants (IFCs) are the proportionality coefficients between the displacements of atoms from their equilibrium positions and the forces they induce on other atoms (or themselves). Their knowledge allows to build vibrational eigenfrequencies and eigenvectors of solids. This paper describes IFCs for different solids (SiC>2-quartz, SiC>2-stishovite, BaTiC>3, Si) obtained within the Local-Density Approximation to Density-Functional Theory. An efficient variation-perturbation approach has been used to extract the linear response of wavefunctions and density to atomic displacements. In mixed ionic-covalent solids, like SiC>2 or BaTiC>3, the careful treatment of the long-range IFCs is mandatory for a correct description of the eigenfrequencies. 

In the present paper, a particular technique for the computation of the Interatomic Force Constants of periodic solids in the framework of Density-Functional Theory (DFT) will be described, as well as some of its applications. 

In the description of the technique, the particular aspects that make it different of other schemes aimed at the computation of IFCs in solids or molecules [1-9] will be emphasized. These aspects are connected to the central use of a variational principle in order to find the changes in the wavefunctions due to atomic displacements, on one hand, and to find the change in electronic energy due to the changes in wavefunctions, on the other hand. Some technical details, related with the presence of relatively long-ranged interatomic force constants, caused by... 

Interatomic Force Constants in ionic and covalent solids. 

In the metals, the same type of interatomic force acts between atom of different metals that acts between atoms of a single element. We have already stated that for this reason liquid solutions of many metals with each other exist in wide ranges of composition. There, are many other cases in which two substances ordinarily solid at room temperature are soluble in each other when liquefied. Thus, a great variety of molten ionic crystals are soluble in each other. And among the silicates and other substances held by valence bonds, the liquid phase permits a wide range of compositions. This is familiar from the glasses, which can have a continuous variability of composition and which can then supercool to essentially solid form, still with quite arbitrary compositions, and yet perfectly homogeneous structure. 

The lattice energy based on the Born model of a crystal is still frequently used in simulations [14]. Applications include defect formation and migration in ionic solids [44,45],phase transitions [46,47] and, in particular, crystal structure prediction whether in a systematic way [38] or from a SA or GA approach [ 1,48]. For modelling closest-packed ionic structures with interatomic force fields, typically only the total lattice energy (per unit cell) created by the two body potential,... 

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