27 December 2013


Bonding in Solids

          A major problem of solid state physics is to account for the origin of binding forces which bind the atoms in a molecule and also the forces which bind the molecules in a solid. It is observed that in nature there are four types of forces such as gravitational, electrical, nuclear and weak forces. Gravitational forces bind the star and planets in the universe. Electrical forces are the responsible in binding the atoms and molecules giving different solid structures. The nuclear forces do not play any part in the formation of solids because these forces bind the nucleons. The type of bonding between the atoms determines the different physical properties of solids.

          In general, the chemical bonds that occur in solids are classified as        1) Primary bonds and 2) Secondary bonds.

The primary atomic bonds are formed when the interaction forces are relatively strong. The primary bonds are classified as 1) Ionic bonds 2) Covalent bonds and 3) Metallic bonds.

          The secondary bonds are classified into two groups. 1) Permanent dipole bonds : These bonds are formed between molecules that possess permanent dipoles. 2) Fluctuating dipole bonds : These bonds take place among atoms as a result of an asymmetrical distribution of electron densities around their nuclei. The bonding is very weak and is termed fluctuating since the electrons density continuously changes with time.

1. Ionic – Alkali halides, Alkaline oxide, Sodium Chloride, Epsom salts(Magnesium sulfate heptahydrate) etc.
2. Covalent (Homopolar bonded) – Diamond, Silicon etc.
3. Metallic – Various metals and alloys
4. Molecular – In gases like He, Ne, A, Kr, Xe, O2, H2
5. Hydrogen Bonded – Ice, KH2Po4, Compound which have water of crystallization, and some fluorides.

          A rigid line of demarcation can not be drawn between five types of bonding because several crystals can not be put with precision into any one of these categories. For example, in CuO the bonds are partly ionic and it also behaves electrically like the covalent semiconductors. 

Types of Bonding

Ionic Bond:

          The ionic bond is formed between two atoms when one atom transfers an electron to another acceptor atom to give ions of opposite charge. The bonds are formed as a result of strong Coulomb forces. Examples for ionic substances are NaCl, KBr, MgO, etc.
Ionic bonding occurs between electropositive atoms [elements on the left side of the periodic table] and electronegative atoms [elements on the right side of the periodic table]. In the formation of NaCl crystal, each Na atom readily gives its valence electron [due to low ionization energy] to Cl atom and the Cl atom takes electron [due to high electron affinity] from Na atom. As a result, Na+ and Cl ions are formed because of opposite charge; these ions attract strongly and form a bond called an ionic bond. During this process, energy is released. This reaction is represented as:    Na + Cl Na+ + Cl  NaCl
Properties of ionic solids:
1. Ionic crystals are rigid, so they possess high melting and boiling points.
2. Pure and dry ionic compounds are insulators, whereas in solution they conduct electricity due to the moment of charged ions. Their electrical conductivity is much smaller than that of metals at room temperature. But the conductivity increases with increasing temperature.
3. They are easily soluble in polar solvents like water [because of high dielectric constant] and insoluble in non-polar solvents like benzene [C6H6], carbon tetrachloride [CCl4], etc. [because of very low dielectric constant].
4. When subjected to stresses, they cleave (break) along certain planes of crystal.
Covalent Bond:
The covalent bond is formed between atoms which share electrons present in the outer most shells.
          It is known that the word ‘covalent’ describes the sharing of valence electrons by adjacent atoms. Hydrogen molecule is a good example of covalent bonding. The two protons share two electrons of opposite spins. The electrons lie between the hydrogen atoms the major part of time. To a first approximation, the covalent bond in a hydrogen molecule can be considered to arise from the attraction of positive ions to the intervening pair of electrons with opposite spins as shown below.    
H. + H. → H:H (Hydrogen molecule)
In these cases, two p-electrons are shared between the atoms. The above examples have single bonds. But many elements form multiple bonds.
          The most important covalent bonded material is carbon. The carbon is the basic element for many organic materials like polymeric materials. Carbon can form two covalent bonds. Sulphur, Selenium, Tellurium etc., form two bonds; Phosphorous, nitrogen, arsenic, antimony can form three bonds.
Characteristics of Covalent bonds:
1) Covalent bond is a strong bond. Cohesive energies of 6 to 12 eV per atom are typical of covalent crystals, which is more than the usual cohesive energies in ionic crystals. All covalent crystals are hard, have high melting points, and are insoluble in all ordinary liquids.
2) Covalent bonds are strongly directional i.e., stereospecific.
3) The other important characteristic property of the covalent bond is its saturability. Saturability means that each atom can form covalent bonds only with a limited number of its neighbours.
4) The conductivity of covalent crystals varies over a wide range. Some crystals are insulators (diamond) and some are semiconductors (Ge). The conductivity increases with the increase of temperature.
5) The optical properties of covalent crystals are characterized by high refractive index and high dielectric constant. Covalent crystals are transparent to long-wavelength radiation but opaque to shorter wavelengths.

Metallic Bond:

In metallic crystals, the metallic bond arises when all of the atoms share all of the valence electrons. The valence electrons of the atoms comprising a metal are common to the entire aggregate, so that a kind of “gas” of free electrons pervades it. The crystal is held together by the electrostatic attraction between the negative electron gas and the positive metal ions. Metals tend to have high melting points and boiling points suggesting strong bonds between atoms. The best example for a metallic crystal is sodium(melting point 97.8C). Elements having only one electron in the outermost orbit are metallic in nature (like sodium, potassium etc.) while elements (like Magnesium, Aluminium etc.) having 2 or 3 electrons in the outermost orbit are just metallic and the elements (like tin, lead etc.) having 4 electrons in the outermost orbit becomes much less in metallic character. When valency electrons becomes 6 or 7, the element loose their metallic character.  

   The cohesion (i.e., the ability to remain a solid) of the metallic crystal results from a combination of forces:
(i) the attraction of the electron cloud for the ion cores
(ii) the mutual repulsion of the electrons
(iii) the mutual repulsion of the ion cores

Characteristics of Metallic bonds:

1) They have electrical and thermal conductivity. The conduction of electricity in metals is a property of valence electrons. The electrons have a mobility within the solid which comes from the overlap of the atomic state functions.

2) The metals are opaque to all electromagnetic radiations from very low frequency to the middle UV, where they become transparent. Thus metals have high optical reflection and absorption coefficients.

3) The cohesive energy is defined as the amount of work required to separate the atoms in the solid into isolated neutral atoms. Metals are about midway in the energy scale for solids. They are much more tightly bonded than solid He, H, Ne etc., and less tightly bonded than the covalent crystal like diamond Ge and Si.
Actual binding is quite low as for alkali metals which have low melting an boiling points. Sometimes metallic crystals have low melting point as Tungsten.

Hydrogen bonds:
      It was recognized that a hydrogen atom can be attracted simultaneously to two different atoms, thereby, showing a coordination number of two. Thus the hydrogen atom serves as a bridge between the two species and can be considered as the basis of a bond between the two atoms. This is the hydrogen bond. This configuration is represented as X-H…Y where X is called ‘donor’ while Y is called ‘acceptor’. The weaker of the two bonds (shown by dotted line) is the hydrogen bond while the other (shown by a full line) is a strong covalent bond. H2O, NH3 and HF are examples of hydrogen bonded crystals. These bonds are stronger than vander Waals bonds but weaker than ionic or covalent bonds.

Crystal Structure

Crystal Lattice:

The array of points in three dimensions such that the environment about any one point is identical with that about any other point is called Crystal lattice or Space lattice.

Crystal lattices are used to describe the geometrical symmetry within any crystal.


The distances between the successive lattice points along crystal axes are called primitives. They are represented by abc.

Translation Vector:

The position of a lattice point relative to any arbitrary lattice point is represented by a vector called translation vector. i.e., T=n1a+n2b+n3c

Unit Cell:

The parallelepiped whose edges are represented by primitives is called a unit cell. The simplest repeating unit in a crystal is called a unit cell. Each unit cell is defined in terms of lattice points – the points in space about which the particles are free to vibrate in a crystal.


The unit assembly of a crystal is called basis. i.e., a single atom in atomic crystal and a single molecule in a molecular crystal form the basis.

Crystal Structure:

If a basis is placed at each point of space lattice, the arrangement forms crystal structure. i.e., crystal structure = space lattice + basis

Bravais lattices:

Bravais in 1850 made a suggestion that in each variety of crystal a representative unit cell can be isolated. This unit cell may be a group of atoms or ions or molecules. The crystal is imagined to be made up of a large number of such unit cells repeated in three dimensions along the three reference axes. Bravais showed that crystals could be divided into 14 unit cells, which meet the following criteria.

* The unit cell is the simplest repeating unit in the crystal.

* Opposite faces of a unit cell are parallel.

* The edge of the unit cell connects equivalent points.

       A space lattice is a three dimensional array of points such that the neighboring points of each lattice point surround in an identical way. In the simplest crystals there is a single atom at each lattice point. Usually an ssembly of two or more atoms occupy each lattice point.

       The fourteen Bravais lattices are shown in figure. Seven sets of axes a, b and c are sufficient to construct the 14 Bravais lattice. These are referred to as the seven crystal systems. The axes a, b, c and the anglesa, b, g are called the lattice parameters.

Number of lattices
Lattice symbols
Unit cell characteristics
P (or) sc
I (or) bcc
F (or) fcc
(a = b = c, a = b = g = 90°)
P, I
(a = b ¹ c, a = b = g = 90°)
(a ¹ b ¹ c, a = b = g = 90°)
(a = b ¹ c, a = b = 90°,       g = 120°)
(a = b = c, a = b = g ¹ 90°)
P, C
(a ¹ b ¹ c, a = g = 90° ¹ b)
(a ¹ b ¹ c, a ¹ b ¹ g)

4 Types unit cells (P, I, F & C) + 7 systems = 14 lattices.

P – Primitive; I – Body Centred; F – Face Centred; C – Side Centred

Miller Indices

Miller indices form a notation system in crystallography for planes in crystal lattices. The lattice planes in a crystal are specified by three numbers. This method was first introduced by William Hallowes Miller (W.H.Miller) in 1839. Hence these numbers are called Miller Indices.

Miller indices are the set of integers in the ratio of the reciprocals of the fractional intercepts which a lattice plane makes with the crystallographic axes.

* Miller indices are used to specify directions and planes.

* These directions and planes could be in lattices or in crystals.

* The number of indices will match with the dimension of the lattice or the crystal.

* Example in 1D there will be 1 index and 2D there will be two indices etc.

Notation Summary:

* (h,k,l) represents a point – note the exclusive use of commas

* Negative numbers/directions are denoted with a bar on top of the number

* [hkl] represents a direction

* represents a family of directions

* (hkl) represents a plane

* {hkl} represents a family of planes

Miller indices for directions:

* A vector r passing from the origin to a lattice point can be written as
r = r1 a + r2 b + r3 c

Where a, b, c → basic vectors and miller indices → (r1r2r3)

* Fractions in (r1r2r3) are eliminated by multiplying all components by their common denominator.

* Example [1, ¾, ½] will be expressed as [432]

Miller indices for planes:

In order to determine the miller indices of a plane, the intercepts made by the plane with the three reference axes are measured.

Let a/h, b/k and c/l be the actual intercepts made by the plane. The fractional intercepts are  . The ration of the reciprocals of the fractional intercepts is h:k:l. Hence the Miller indices for this plane are (hkl).

The miller indices do not define a particular plane only. They define a set of planes parallel to each other.