Physics · Molecular physics and thermodynamics · Grade 9

Crystalline and amorphous solids

Why is table salt cubic, and why do diamond and graphite differ although both are carbon?

The situation

If you look at small grains of table salt through a magnifying glass, you will see that almost all of them are little cubes. Diamond is the hardest natural substance, while graphite is so soft that it rubs off a pencil onto paper — yet both consist only of carbon. Ice melts at exactly 0 °C, while glass and resin soften gradually when heated. The secret lies in how the atoms are arranged: in crystals they form a strictly ordered lattice. How can this order, invisible to the eye, be measured?

Topic

Crystalline and amorphous solids. Table salt, diamond and the graphite lattice; why glass is not a crystal

Key ideas

Crystal lattice
Melting point
Bragg's law

Formulas

k = \frac{N_{c e l l} \cdot \frac{4}{3} π r^{3}}{V_{c e l l}}

Packing fraction: simple cubic 52 %, body-centred 68 %, face-centred and hexagonal close-packed 74 %.

Q = λ m

Heat of fusion: during melting the temperature of the crystal does not change. Naphthalene: $λ \approx 1.5 \cdot 10^{5}$ J/kg, $t_{m} = 80$ °C.

2 d \sin θ = n λ

Bragg condition: X-rays reflected from parallel atomic planes reinforce each other.

d_{h k l} = \frac{a}{\sqrt{h^{2} + k^{2} + l^{2}}}

Distance between $(h k l)$ planes in a cubic lattice; $h, k, l$ are the Miller indices.

Γ = ν e^{- E / k T}

Hopping frequency of a vacancy: diffusion in a solid depends exponentially on temperature.

Theory

Crystalline and amorphous solids

In a crystal the atoms are arranged in an order that repeats over long distances — in a lattice; in an amorphous solid (glass, resin, rosin, many polymers) there is only short-range order, just as in a liquid. That is why a crystal melts at a definite temperature and its properties depend on direction (anisotropy), whereas an amorphous solid softens gradually and is isotropic. Polymers are molecules with long chains, often partly crystalline. Liquid crystals flow, but their molecules are lined up in one direction — liquid crystal displays (LCDs) are based on this.

Lattice, unit cell and Bravais lattices

A lattice is a set of repeating points in space; if one or several atoms (the basis) are placed at each point, a crystal results. The smallest repeating piece is the unit cell; the positions of atoms are written as fractions of the cell edges (crystallographic coordinates), for example $(\frac{1}{2}, \frac{1}{2}, \frac{1}{2})$ . By symmetry there are 7 crystal systems and 14 Bravais lattices (cubic: simple, body-centred and face-centred; tetragonal, orthorhombic, hexagonal, trigonal, monoclinic, triclinic). A cube has 3 axes of fourfold symmetry, 4 of threefold and 6 of twofold symmetry, 9 planes of symmetry and a centre of symmetry. Different lattices of the same substance are called polymorphism: diamond and graphite, or the change of iron from BCC to FCC at 912 °C.

Types of chemical bonding

In an ionic bond (NaCl) the Na⁺ and Cl⁻ ions attract each other electrostatically, and each ion is surrounded by 6 ions of opposite charge. In a covalent bond (diamond) each carbon atom is bonded to its 4 neighbours by shared electron pairs — very hard. In a metallic bond (Cu, Fe, Mg) the ions are packed closely in a “gas” of free electrons (coordination number 8–12) — such metals are malleable and conduct heat well. The van der Waals bond is weak: it holds the layers of graphite together, which is why the layers slide easily. The hydrogen bond forms the ice crystal. The binding energy is the energy needed to separate a crystal into individual atoms: about 7.4 eV per atom in diamond, 3.5 eV in copper, 3.2 eV in table salt; the van der Waals bond between graphite layers, however, is only about 0.05 eV per atom (shown for each lattice in the results window).

Melting, solidification and crystallisation

When a crystal reaches its melting temperature, the heat supplied goes into breaking up the lattice and the temperature stays constant (the horizontal part of the graph), $Q = λ m$ . On cooling, crystallisation begins at the same temperature, and that heat is released again. An amorphous substance has no such point: as it is heated its viscosity gradually decreases (shown as $\lg η$ on the HUD), and it softens and starts to flow. Some crystals turn straight into vapour without melting — sublimation: naphthalene, iodine, dry ice (this is the source of the smell of naphthalene). The model: in a water bath there are 20 g of naphthalene and 20 g of resin of the rosin type; the $c$ and viscosity of the resin are approximate. The melting of ice and water is covered in the “Melting, boiling and evaporation” experiment.

Defects, dislocations and diffusion

A real crystal is not perfect. Point defects: a vacancy (an empty site), an interstitial atom, a foreign atom. The fraction of vacancies is $\sim e^{- E_{f} / k T}$ (for copper $E_{f} \approx 1.3$ eV), and their hops move atoms around — this is how diffusion in solids takes place; at room temperature it almost stops. A dislocation is the edge of an extra half-plane inside the crystal; when it moves, parts of the crystal shift by one atomic step. To shift a whole plane at once would need $τ \approx G / 2 π$ (for copper ≈ 7.6 GPa), whereas a dislocation moves at megapascals — this is the origin of the plasticity of metals. A single crystal has one lattice throughout its volume; a polycrystal consists of grains with different orientations and is isotropic on average. For example, the Young’s modulus of a copper single crystal is ≈ 67 GPa along [100] and ≈ 191 GPa along [111].

X-ray diffraction and the reciprocal lattice

The wavelength of X-rays (≈ 0.1 nm) is close to the distance between atoms, so a crystal acts as a diffraction grating for them. Rays reflected from a family of parallel $(h k l)$ planes reinforce each other when $2 d \sin θ = n λ$ . The Miller indices are the reciprocals of the intercepts of a plane on the axes, reduced to whole numbers. The type of lattice determines which peaks are present: in FCC $h, k, l$ are all even or all odd, in BCC $h + k + l$ is even. The vectors of the reciprocal lattice are $| 𝐆 | = 2 π / d_{h k l}$ ; the Bragg condition is equivalent to $𝐤^{'} - 𝐤 = 𝐆$ . The Wigner–Seitz cell of the reciprocal lattice — the Brillouin zone — explains the energy bands of electrons. At a surface the atoms are left without neighbours: because the surface energy differs from one face to another, a crystal grows with its own flat faces (salt grows as cubes). If a crystal is reduced to a few nanometres in one, two or three directions, a nanostructure results: a thin layer is a quantum well (2D), a thin wire is a quantum wire (1D), a tiny grain is a quantum dot (0D); in them the energy of the electron is quantised (more in the “Quantum dots and the potential well” experiment).

Did you know?

A layer of graphite one atom thick — graphene — was isolated in 2004; for this work Andre Geim and Konstantin Novoselov received the 2010 Nobel Prize in Physics.
The diffraction of X-rays in a crystal was discovered in 1912 by Max von Laue; using this phenomenon, William Henry Bragg and his son William Lawrence Bragg created a method of determining crystal structure and received the Nobel Prize in 1915.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

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