Solid State: One Mark Questions, Book Back Answers and Notes

About Solid State

Medium ~120 min study

A solid keeps its shape because its particles are held in fixed positions by strong attractive forces. Thermal energy can only make them vibrate about those positions. How the particles are arranged decides a great deal: whether the solid melts sharply, whether it breaks along a clean plane, whether it conducts electricity.

The chapter starts with the difference between crystalline and amorphous solids, then builds the idea of a lattice from a single repeating unit, the unit cell. From there it moves to close packing of spheres, the voids left between them, and the defects that every real crystal contains. Each step uses the same picture of atoms as hard spheres, so keep a rough sketch of a cube handy while you revise.

Questions on this unit come in two kinds. Some are direct: coordination number of a bcc lattice, number of atoms in an fcc unit cell, which defect lowers density. Others are numerical: finding density from edge length, or edge length from density. Both need the same few formulas and a clear picture of where atoms sit in a cube.

What you'll learn

• Tell crystalline and amorphous solids apart using melting behaviour, order and isotropy.
• Classify a crystalline solid as ionic, covalent, molecular or metallic from its bonding.
• Count the atoms in simple cubic, bcc and fcc unit cells using corner, face and body contributions.
• Use \(\rho = \frac{n \times M}{a^3 \times N_A}\) to find density, edge length or molar mass.
• Describe tetrahedral and octahedral voids and state how many of each form in a close-packed structure.
• Predict how Schottky, Frenkel and non-stoichiometric defects affect density, colour and conductivity.

Before you start

• The three states of matter and the idea of intermolecular forces.
• Atomic structure, molar mass and Avogadro's number.
• Ionic, covalent and metallic bonding from earlier units.

Solid State explained

From Unit Cells to Crystal Defects

Crystalline and Amorphous Solids

Every solid has a definite shape and volume. Crystalline solids such as NaCl and quartz have long-range order: the same arrangement repeats throughout the crystal. This gives them a sharp melting point, a regular shape and anisotropy (properties differ with direction). Amorphous solids such as glass and rubber have only short-range order. They soften over a range of temperature, are isotropic, and are often called supercooled liquids or pseudo solids.

Types of Crystalline Solids

Crystals are grouped by the particles present and the forces between them. Ionic solids (NaCl) are held by electrostatic attraction between cations and anions. Covalent solids (diamond, silicon carbide) are giant networks of covalently bonded atoms. Molecular solids (ice, solid CO₂, naphthalene) are held by weaker van der Waals forces or hydrogen bonds. Metallic solids are positive metal ions in a sea of mobile electrons, which is why metals conduct electricity and are malleable.

Crystal Lattice and Unit Cell

A crystal lattice is a regular three-dimensional arrangement of points, each point standing for a particle. The smallest part that repeats in all three directions to build the whole lattice is the unit cell. It is described by three edge lengths (a, b, c) and three angles (α, β, γ). Different combinations of these give the seven crystal systems, and adding body-centred, face-centred and end-centred variants gives the 14 Bravais lattices.

Density of a Unit Cell

Once you know how many atoms a unit cell holds, its density follows directly. For a cubic cell, \(\rho = \frac{n \times M}{a^3 \times N_A}\), where \(n\) is the number of atoms per unit cell, M is the molar mass, \(a\) is the edge length and N_A is Avogadro's number. Remember n = 1 for simple cubic, 2 for bcc and 4 for fcc. If a is given in pm, convert to cm (1 pm = 10^-10 cm) before cubing, so the density comes out in g cm^-3.

Packing in Crystals

Particles pack to leave as little empty space as possible. Stacking close-packed layers as ABAB gives hexagonal close packing (hcp); stacking them as ABCABC gives cubic close packing (ccp), which is the same as fcc. Both use 74% of the space. Body-centred cubic uses 68% and simple cubic only 52.4%. The gaps left behind are tetrahedral voids (surrounded by four spheres) and octahedral voids (surrounded by six). In a close-packed structure of N spheres there are N octahedral voids and 2N tetrahedral voids, and smaller ions occupy these voids in many ionic compounds.

Imperfections in Solids

Real crystals always have defects. In a Schottky defect, equal numbers of cations and anions are missing, so density falls; NaCl and CsCl show it. In a Frenkel defect, an ion (usually the smaller cation) moves to an interstitial site, so density does not change; AgBr and ZnS show it. Both keep the formula stoichiometric. Non-stoichiometric defects are different: in a metal excess defect such as NaCl heated in sodium vapour, electrons trapped in anion vacancies (F-centres) give the crystal a yellow colour. In metal deficiency defects, as in FeO, some cations are missing and others carry a higher charge. Impurity defects, such as adding SrCl₂ to NaCl, create cation vacancies. Controlled impurities in silicon produce n-type and p-type semiconductors.

Common mistakes to avoid

• Mixing up the number of atoms in a unit cell with the coordination number. In bcc there are 2 atoms per cell but the coordination number is 8.
• Using the wrong relation between atomic radius and edge length: a = 2r for simple cubic, \(\sqrt{3}\)a = 4r for bcc, \(\sqrt{2}\)a = 4r for fcc.
• Forgetting to convert pm to cm before cubing the edge length, which gives a density that is off by a huge power of ten.
• Swapping Schottky and Frenkel defects. Schottky lowers density because ions leave the crystal; Frenkel does not, because the ion only moves to an interstitial site.

Test yourself on these with the practice test, then check the worked reasoning in the solved MCQs.

Frequently asked questions

What is the main difference between crystalline and amorphous solids?

In a crystalline solid the particles repeat in a regular pattern throughout the crystal, so it melts at one sharp temperature. In an amorphous solid like glass the order is only short-range, so different parts of it need different amounts of heat and it softens gradually over a range of temperature.

How do you calculate the density of a unit cell?

Density is mass of the unit cell divided by its volume. The mass is n × M / NA, where n is the number of atoms in the cell. The volume of a cubic cell is a3. Put together, density = n × M / (a3 × NA). Use a in cm to get g cm-3.

What are Schottky and Frenkel defects?

A Schottky defect is a pair of missing ions, one cation and one anion, so the crystal stays neutral but its density decreases. It is seen in NaCl. A Frenkel defect forms when an ion, usually the smaller cation, leaves its site and sits in an interstitial position. No ion is lost, so density stays the same. AgBr shows both types.

Why is the coordination number different in the three cubic structures?

Coordination number is the number of nearest neighbours touching a particle. In simple cubic each atom touches 6 others. In bcc the body-centre atom touches the 8 corner atoms. In fcc, as in any close-packed arrangement, each atom touches 12 others.

What decides whether a semiconductor is n-type or p-type?

It depends on the impurity. Doping silicon (four valence electrons) with phosphorus or arsenic (five) leaves an extra electron, so negative electrons carry the current: n-type. Doping with boron or gallium (three) leaves an electron vacancy, or hole, which behaves like a positive carrier: p-type.

What are tetrahedral and octahedral voids?

They are the empty spaces between spheres in close packing. A tetrahedral void is enclosed by four spheres whose centres form a tetrahedron. An octahedral void is larger and is enclosed by six spheres. For N spheres in close packing there are N octahedral voids and 2N tetrahedral voids.