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Unformatted text preview: Physics 927 E.Y.Tsymbal 1 Section 3: Crystal Binding Interatomic forces Solids are stable structures, and therefore there exist interactions holding atoms in a crystal together. For example a crystal of sodium chloride is more stable than a collection of free Na and Cl atoms. This implies that the Na and Cl atoms attract each other, i.e. there exist an attractive interatomic force, which holds the atoms together. This also implies that the energy of the crystal is lower than the energy of the free atoms. The amount of energy which is required to pull the crystal apart into a set of free atoms is called the cohesive energy of the crystal. Cohesive energy = energy of free atoms – crystal energy Magnitude of the cohesive energy varies for different solids from 1 to 10 eV/atom, except inert gases in which the cohesive energy is of the order of 0.1eV/atom (see table 1 in Kittel). The cohesive energy controls the melting temperature (compare table 1 and table 2 in Kittel). Fig.1 A typical curve for the potential energy (binding energy) representing the interaction between two atoms is shown in Fig.1. It has a minimum at some distance R = R . For R > R the potential increases gradually, approaching 0 as R →∞ , while for R < R the potential increases very rapidly, tending to infinity at R =0. Since the system tends to have the lowest possible energy, it is most stable at R=R , which is the equilibrium interatomic distance. The corresponding energy U is the cohesive energy. A typical value of the equilibrium distance is of the order of a few angstroms (e.g. 2-3Å), so that the forces under consideration are short range. The interatomic force is determined by the gradient of the potential energy, so that ( ) U F R R ∂ = − ∂ . (3.1) If we apply this to the curve in Fig.1, we see that F ( R )<0 for R > R . This means that for large separations the force is attractive , tending to pull the atoms together. On the other, hand F ( R )>0 for R < R , i.e. the force becomes repulsive at small separations of the atoms, and tends to push the atoms apart. The repulsive and attractive forces cancel each other exactly at the point R , which is the point of equilibrium. U R R 0 U 0 repulsive energy attractive energy cohesive energy Physics 927 E.Y.Tsymbal 2 The attractive interatomic forces reflect the presence of bonds between atoms in solids, which are responsible for the stability of the crystal. There are several types of bonding , depending on the physical origin and nature of the bonding force involved. The four main types are: Van der Waals (or molecular ) bonding, ionic bonding, covalent bonding and metallic bonding. Although the nature of the attractive energy is different in different solids, the origin of the repulsive energy is similar in all solids. The origin of the repulsive force is mainly due to the Pauli exclusion principle . The elementary statement of this principle is that two electrons cannot occupy the same orbital. As ions approach each other close enough, the orbits of the electrons begin to overlap, i.e. orbital....
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