JEE Chemistry The Solid State Notes Set 06

Classification of Solids

Solids keep a fixed shape and volume because their particles vibrate about fixed positions and are held by strong forces. They are first split by the order in the arrangement of particles, and then by the kind of force that holds the particles together.

Amorphous solids are also called pseudo solids or super-cooled liquids. One substance can crystallise in different forms depending on conditions; this is polymorphism.

Unit Cell and Crystal Systems

A lattice is the regular three-dimensional array of points representing the constituent particles. The unit cell is the smallest block that, repeated in all directions without rotation, builds the whole lattice. It is described by six parameters: the edges a, b, c and the angles alpha, beta, gamma. A primitive cell has points only at corners; a centred cell has extra points inside (body-centred), on faces (face-centred) or on a pair of opposite faces (end-centred).

Together there are 14 Bravais lattices. The contribution of a lattice point to one cube is 1/8 for a corner, 1/4 for an edge centre, 1/2 for a face centre and 1 for the body centre.

Density of a Cubic Crystal

The mass of one unit cell is \( Z\times M/N_A \) and its volume is \( a^3 \), so

Use a in cm and M in g/mol to get g/cm\(^3\). Nearest-neighbour distances are a (simple cubic), \( \sqrt{3}a/2 \) (bcc) and \( a/\sqrt{2} \) (fcc).

Close Packing

Spheres in a plane touch six others in the closest 2D packing (hexagonal). A second layer sits over half the hollows of the first. The third layer can then go over the hollows that lie above the first layer (ABAB..., hexagonal close packing, hcp) or over the remaining hollows (ABCABC..., cubic close packing, ccp or fcc). Both have coordination number 12 and a packing efficiency of 74.05%, i.e. 26% empty space.

For hcp: the cell has 6 atoms, the base has side \( a=2r \), the height is \( c=2\sqrt{2/3}\,a=4\sqrt{2/3}\,r \), and the volume is \( 24\sqrt{2}\,r^3 \). Spacing between adjacent layers is \( \sqrt{8/3}\,r \).

Voids in Close Packing

Between close-packed spheres there are two kinds of hole. For N spheres there are N octahedral voids (six neighbours) and 2N tetrahedral voids (four neighbours). A void is characterised by the largest sphere that fits, expressed as a radius ratio \( r_{void}/r_{sphere} \).

In an fcc cell there are 4 octahedral voids (body centre plus the 12 edge centres, each shared by 4 cells) and 8 tetrahedral voids (inside the cell). In hcp there are 6 octahedral and 12 tetrahedral voids per cell.

Ionic Structures

In ionic crystals the larger ion (usually the anion) forms the lattice and the smaller ion fills a void. The radius ratio \( r_+/r_- \) predicts the coordination number: below 0.225 gives 3, up to 0.414 gives 4, up to 0.732 gives 6 and above that 8.

Higher pressure increases the coordination number (NaCl type changes to CsCl type, for example); higher temperature has the opposite tendency. In NaCl the edge is \( a=2(r_++r_-) \); in CsCl the body diagonal gives \( \sqrt{3}a=2(r_++r_-) \).

Worked Examples

Example 1. Copper is fcc with a = 3.61 Å. Z = 4, so \( \rho=\dfrac{4\times63.5}{6.022\times10^{23}\times(3.61\times10^{-8})^3}=8.96 \) g/cm\(^3\). Radius \( =a/(2\sqrt{2})=1.276 \) Å.

Example 2. Potassium is bcc with a = 5.2 Å. Z = 2, so \( \rho=\dfrac{2\times39.1}{6.022\times10^{23}\times(5.2\times10^{-8})^3}=0.92 \) g/cm\(^3\). Radius \( =\sqrt{3}a/4=2.25 \) Å.

Example 3. In a crystal the B atoms are in a ccp lattice and the A atoms occupy half of the octahedral voids. Per 4 B atoms there are 4 octahedral voids, so 2 A atoms are present and the formula is AB2.

Example 4. FeO is found as \( \ce{Fe_{0.93}O} \). Let x be the fraction of iron present as Fe(III). Charge balance: \( 3x+2(0.93-x)=2 \), so \( x=0.14 \), and the percentage of Fe(III) in total iron is \( 0.14/0.93=15.05\% \).

Imperfections in Solids

Schottky and Frenkel defects are stoichiometric; metal-excess and metal-deficiency defects change the formula ratio. Doping a crystal of a divalent impurity such as CdCl2 into AgCl creates cation vacancies equal to the number of Cd2+ ions added.

Electrical and Magnetic Properties

Conductors have conductivity about 10 to the power 7 ohm-1 m-1, insulators 10 to the power -20 to -10 and semiconductors in between. In metals the conduction band overlaps the valence band; in semiconductors the gap is small; in insulators it is large. Semiconductor conductivity rises with temperature while a metal's falls.

Both are electrically neutral. Compounds of groups 13-15 (GaAs) and 12-16 (ZnS, CdS) are also semiconductors.