Solid State for Entrance Exams: Close Packing, Voids and Band Theory

Start by identifying the unit cell

Every numerical in this topic depends on getting the lattice type right, so establish it before anything else.

Counting atoms per unit cell

Each atom is shared between neighbouring cells according to its position:

So a face-centred cubic cell has 8 × 1/8 from corners plus 6 × 1/2 from faces, giving four atoms. This counting is the basis of the density calculation:

where Z is atoms per cell and M the molar mass. Rearranged, the same equation gives Z from a measured density, which is how the lattice type is deduced experimentally — a very common numerical.

Voids and which ions occupy them

In a close-packed arrangement of N spheres there are 2N tetrahedral voids and N octahedral voids. An ionic solid is usually described as the larger ion forming the close-packed array and the smaller occupying some fraction of the voids.

Which void is used depends on the radius ratio:

The radius ratio rule is a guideline derived from hard-sphere geometry, and real structures do depart from it where bonding has significant covalent character. Saying so when a question asks about limitations is worth a mark.

Defects

The density consequence is the discriminator that questions use: Schottky removes ions from the lattice so density falls, while Frenkel merely relocates one so density is preserved.

Band theory in one page

In a solid, the atomic orbitals of very many atoms combine into bands of closely spaced levels. The highest filled band is the valence band; the lowest empty one is the conduction band. Conduction requires electrons to reach the conduction band, so the gap between the two decides the electrical behaviour.

The opposite temperature dependence of metals and semiconductors is a standard question, and the reasoning — scattering versus promotion — is what earns the marks rather than the observation itself.

Doping

Adding a group 15 element to silicon supplies an extra electron and gives an n-type semiconductor. Adding a group 13 element creates a hole and gives p-type. In both cases the dopant introduces a level inside the gap, which is why very small doping levels change conductivity so dramatically.

Frequently asked questions

Why do face-centred cubic and hexagonal close packing have the same efficiency?

Both are close-packed arrangements with coordination number 12; they differ only in the stacking sequence of the layers. Since the local packing is identical, the efficiency is the same 74%.

How do I decide the lattice type from a density numerical?

Rearrange the density equation for Z. A result near 1, 2 or 4 points to simple cubic, body-centred or face-centred respectively. Non-integer answers usually mean a units error rather than an exotic lattice.

Why do Frenkel defects not change density?

Because no ion leaves the crystal — one simply moves from a lattice site to an interstitial site. Mass and volume are both unchanged.

How much band theory is needed?

For IIT-JAM, the qualitative picture and the temperature dependence are usually enough. CSIR-NET and GATE go further into doping, carrier concentration and occasionally the Fermi level.