Crystals Binding, Physics tutorial

Introduction:

The attractive electrostatic interaction between the negative charges of the electrons and positive charges of the nuclei is wholly responsible for the cohesion of solid. As the atoms come close altogether their closed electron shells will begin to overlie. The Pauli principle defines that each and every electron state can be occupied through only one electron. In order to contain overlap of closed shells, electrons have to be excited to the higher states. This costs energy and leads to the repulsive interaction between the atoms. The repulsive interaction controls for short distances between the atoms, whereas the attractive interaction dominates at big distances. The real atomic spacing in a crystal is stated by the equilibrium where the potential energy represents a minimum.

Definition:

Crystal binding is the attractive inter atomic force which hold atom altogether in the crystal.

Inter atomic forces:

Solids are the stable structures, and thus there exist interactions holding atoms in a crystal altogether. For illustration a crystal of sodium chloride is more stable than a collection of free Na and Cl atoms. This means that the Na and Cl atoms attract each other, that is, there exist an attractive inter atomic force that holds the atoms altogether. This as well means that the energy of the crystal is lower than the energy of the free atoms. The amount of energy that is needed to pull the crystal apart into the set of free atoms is termed as the cohesive energy of the crystal.

Cohesive energy = energy of free atoms - crystal energy

1852_Inter atomic forces.jpg

The magnitude of cohesive energy differs for different solids from 1 to 10 eV/atom, apart from inert gases in which the cohesive energy is of the order of 0.1eV/atom. The cohesive energy regulates the melting temperature. A usual curve for the potential energy (that is, binding energy) representing the interaction between the two atoms is as shown in the figure below. It consists of a minimum at some distance R = R0. For R > R0 the potential rises gradually, approaching 0 as R → ∞, whereas for R < Ro the potential rises very rapidly, tending to infinity at R = 0. As the system tends to encompass the lowest possible energy, it is most stable at R = R0, that is the equilibrium inter atomic distance.  The corresponding energy U0 is the cohesive energy. A typical value of the equilibrium distance is of the order of a few angstroms (e.g. 2 - 3Å), in such a way that the forces under consideration are short range. The inter atomic force is determined via the gradient of the potential energy, in such a way that:

F(R) = -∂U/∂R

Whenever we apply this to the curve in the figure above, we observe that F(R) < 0 for R > Ro. This signifies that for large separations the force is attractive, tending to pull the atoms altogether. On the other hand F(R) > 0 for R < Ro that is, the force becomes repulsive at small separations of the atoms, and tends to push the atoms separately. The repulsive and attractive forces cancel one other precisely at the point Ro that is the point of equilibrium. The attractive inter atomic forces reflect the presence of bonds between the atoms in solids, that are responsible for the stability of the crystal. There are various kinds of bonding, based on the physical origin and nature of the bonding force comprised.

However the nature of the attractive energy is dissimilar in different solids, the origin of the repulsive energy is identical in all solids and this is mainly due to the Pauli Exclusion Principle. The elementary statement of this principle is that two electrons can't occupy the similar orbital. As ions approach one other close enough, the orbits of the electrons start to overlap, that is, some electrons attempt to engage orbits already occupied through others. This is, though, forbidden through the Pauli Exclusion Principle. As an outcome, electrons are excited to unoccupied higher energy states of the atoms. Therefore, the electron overlap increases the net energy of the system and provides repulsive contribution to the interaction. The repulsive interaction is not simple to treat analytically from the first principles.  In order to make a few quantitative estimates it is frequently supposed that this interaction can be illustrated by a central field repulsive potential of the form λ exp (-r/ρ), where λ and ρ are some constants or of the form B/Rn, here n is sufficiently large and B is some constant.

Vander Waals (Inter atomic) bonding:

This kind of binding is represented through solid noble gas crystals. The outermost electron shell is fully filled and the electron distribution is spherically symmetric. Each and every atom is neutral and consists of no permanent dipole moment. The attractive forces between the atoms occur from the fluctuations in the electron distribution. Such give an instantaneous fluctuating dipole moment in the atom. Its interaction by the induced dipole moments in the neighboring atom leads to a weak interaction. The electron distribution in inert gases is extremely close to that in free atoms. The noble gases like neon (Ne), argon (Ar), krypton (kr) and xenon (Xe) are characterized through filled electron shells and a spherical distribution of electronic clouds in the free atoms. In crystal, the inert gas atoms pack altogether within the cubic fcc structure. Let take two inert gas atoms (1 and 2) separated by distance R. The average charge distribution in a single atom is spherically symmetric that means that the average dipole moment of atom 1 is zero: <d1> = 0. Here the brackets represent the time average of the dipole moment. Though, at any moment of time there might be a non-zero dipole moment caused due to fluctuations of the electronic charge distribution. We represent this dipole moment by d1. From electrostatics consideration, this dipole moment generates an electric field that induces a dipole moment on atom 2. This dipole moment is proportional to the electric field that is in its turn proportional to the d1/R3 so that:

d2 α E α d1/R3

The dipole moments of the two atoms interact by one other. The energy is thus decreased due to this interaction. The energy of the interaction is proportional to the product of the dipole moments and inversely proportional to the cube of the distance between the atoms, so that:

-d1d2/R3 α - d12/R6

and that the coupling between the two dipoles, one caused due to the fluctuation, and the other induced through the electric field generated by the first, results in the attractive force that is termed as the Van der Waals force. The time averaged potential is determined via the average value of <d12> that is not vanish, even although <d1> is zero.

U α - <d12>/R6

The respective potential reduces as R6 reduces by the separation between the atoms. Van der Waals bonding is relatively weak; the respective cohesive energy is of the order of 0.1eV/atom.  This attractive interaction illustrated by the above equation holds only for a relatively huge separation between the atoms. At small separations an extremely strong repulsive forces cause due to the overlap of the inner electronic shells begin to dominate. It appears that for inert gases this repulsive interaction can be fitted quite well through the potential of the form B/R12 where 'B' is a positive constant. By combining this with the above equation we get the total potential energy of two atoms at separation 'R' that can be written as:

U = 4ε [(σ/R)12 - (σ/R)6]

Here 4εσ6 ≡ A and 4εσ26 ≡ B. This potential is termed as Lennard-Jones potential.

Ionic bonding:

The ionic bond results from the electrostatic interaction of oppositely charged ions. Consider sodium chloride as an illustration. In the crystalline state, each and every Na atom loses its single valence electron to a neighboring Cl atom, generating Na+ and Cl- ions that have filled electronic shells. As an outcome an ionic crystal is formed having positive and negative ions coupled through a strong electrostatic interaction.

Na + 5.1 eV (Ionization energy) → Na+ e-

e- + Cl → Cl- + 3.6 eV (electron affinity)

Na+ + Cl- → NaCl + 7.9 eV (electrostatic energy)

The cohesive energy with respect to neutral atoms can be computed as 7.9 eV - 5.1 eV + 3.6 eV that is, Na + cl → NaCl + 6.4 eV (cohesive energy). The structure of NaCl is two interpenetrating fcc lattices of Na+ and Cl+ ions as represented in the figure below.

191_Structure of NaCl.jpg

Therefore each Na+ ion is surrounded through 6 Cl- ions and vice-versa. This structure recommends that there is a strong attractive Coulombic force between the nearest-neighbors ions that is responsible for the ionic bonding. To compute binding energy we require comprising Coulomb interactions by all atoms in the solid. As well we require taking into account the repulsive energy that we suppose to be exponential. Therefore the interaction among the two atoms i and j in a lattice is represented by:

Uij = λe(-rij/ρ) ± q2/rij

Here rij is the distance between the two atoms, 'q' is the electric charge on the atom, the (+) sign is taken for the like charges and the (-) sign for the unlike charges. The net energy of the crystal is the sum over i and j in such a way that:

U = (1/2) Σi,j Uij = N Σj (λe-rij/ρ ± q2/rij)

In this formula 1/2 is due to the fact that each and every pair of interactions must be counted only once. The second equality outcomes from the fact in the NaCl structure the sum over j doesn't based on whether the reference ion 'i' is negative or positive, which provides the total number of atoms. The latter divided by two gives the number of molecules 'N', comprised of a positive and a negative ion. For simplicity, we assume that the repulsive interaction is non-zero only for the nearest neighbors (as it drops down very fast with the distance between atoms). In this case we get:

U = N (zλe-R/ρ - αq2/R)

Here 'R' is the distance between the nearest neighbors, 'z' is the number of the nearest neighbors and 'α' is the Madelung constant:

α = Σj≠i (±1)/Pij

Here pij is defined through rij ≡ pijR. The value of the Madelung constant plays a significant role in the theory of ionic crystals. In common it is not possible to calculate the Madelung constant analytically. A powerful process for calculation of lattice sums was developed through Ewald that is termed as Ewald summation.

Illustration: A one-dimensional lattice of ions of alternating sign as described in the figure given below:

2406_Lattice of ions of alternating sign.jpg

In this case:

α = 2[1 - (1/2) + (1/3) - (1/4) + (1/5) + ....] = 2ln2

Here, we took into account the logarithm expansion into sequence ln (1 + x) = n=1Σ (-1)n-1 Xn/n

Therefore the Madelung constant for 1-dimensional chain is α = 2 ln2.

In (3-D) three dimensions computation of the series is much more difficult and can't be performed so easy.

Covalent bonding:

The covalent bond is a different significant kind of bond that exits in numerous solids. The covalent bond among the two atoms is generally formed through two electrons, one from each atom participating in the bond. The electrons forming the bond tend to be partially localized in the area among the two atoms joined via the bond. Generally the covalent bond is strong: for illustration, it is the bond that couples carbon atoms in diamond. The covalent bond is as well responsible for the binding of silicon and germanium crystals. In a two-atomic molecule (that is, one electron per atom) the energy levels are divided into a binding and an antibinding one. The two electrons are shared among the two atoms and fill the lowest, binding, molecular orbital. In solid the energy levels are no longer discrete however the binding and antibinding levels become broad energy bands. The structure of covalent crystals is found out by the direction of the bonds, they have frequently fewer nearest neighbor atoms (that is, lower coordination number). 

Compounds where the atoms have different number of valence electrons represent a mixture of ionic and covalent binding. Example: GaAs consists of 3 valence electrons and As consists of 5. On an average we contain 4 electrons per atom that can be shared in tetrahedral bonds by neighboring atoms. Though if the bonds are to be symmetrical the Ga will be negatively charged and As positively charged. Therefore partial ionic binding can't be avoided in this and identical cases.

Metallic bonding:

Metals are characterized via a high electrical conductivity that implies that a huge number of electrons in a metal are free to move. The electrons capable to move all through the crystal are termed as the conductions electrons. Generally the valence electrons in atoms become the conduction electrons in solids. The major characteristic of the metallic bond is the lowering of the energy of the valence electrons in metal as compared to the free atoms. Beneath, a few qualitative arguments are given to elucidate this fact. According to the Heisenberg uncertainty principle the indefiniteness in coordinate and in the momentum are associated to each other in such a way that Δx Δp = h/2π. In a free atom the valence electrons are limited through a relatively small volume. Thus, Δp is relatively large that makes the kinetic energy of the valence electrons in a free atom large. On the other hand, in the crystalline state the electrons are free to move all through the entire crystal, the volume of which is large. Thus the kinetic energy of the electrons is greatly reduced, that leads to reducing the net energy of the system in the solid. This method is the source of the metallic bonding. Figuratively speaking, the negatively charged free electrons in a metal serve as glue which holds positively charged ions altogether. The metallic bond is rather weaker than the ionic and covalent bond. For example the melting temperature of metallic sodium is around 400o which is smaller than 1100o in Nacl and around 400o in diamond. However, this kind of bond must be regarded as strong. In transition metals such as Fe, Ni, Ti, Co the method of metallic bonding is more complex. This is due to the fact that in addition to s electrons that behave like free electrons we contain 3d electrons that are more localized. Therefore the d electrons tend to make covalent bonds by nearest neighbors. The d electrons are generally strongly hybridized having s electrons making the picture of bonding much more complex.

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