- Distinguish crystalline solids from amorphous solids by the range and regularity of atomic order.
- Explain how a unit cell and its lattice vectors reproduce a crystal without leaving voids.
- Relate the seven crystal systems and fourteen Bravais lattices to the three cubic lattices.
- Determine atoms per unit cell, coordination number, close-packed direction, atomic radius, and atomic packing fraction for SC, BCC, and FCC lattices.
- Compare packing efficiency among square and hexagonal arrangements in two dimensions and among the three cubic lattices in three dimensions.
- Electronic structure
- Molecular orbital theory
- Multi-electron atoms
- Types of chemical bonding
Order and Arrangement in Solids
Solid-state classification depends on both bonding and the spatial organization of atoms.
Solid-state chemistry builds on electronic structure, molecular orbital theory, multi-electron atoms, and the different forms of bonding. Classifying a solid involves two central questions: how its atoms interact and how those atoms are positioned. Here the emphasis is on atomic arrangement.
| Solid type | Atomic arrangement |
|---|---|
| Crystalline solid | A regular motif repeats across very long distances. |
| Amorphous solid, or glass | Atoms are disordered, or any order persists for only a few bond lengths. |
Efficient packing helps explain why ordered patterns matter. A plant can place more seeds on a limited surface by using a regular arrangement instead of a random one. In the seventeenth century, Robert Hooke similarly compared ways of stacking cannonballs, considering both stability and how many would fit within a given area or volume. His observations of salt crystals led him to consider comparable packing at the atomic scale.
Finding the Repeating Pattern
Repetition is easy to recognize in an M.C. Escher pattern of light and dark ducks. A region containing the repeating motif can be translated sideways or upward to recreate the design. Repeating these regions covers the plane completely.
Lattice Vectors and Two-Dimensional Packing
Translations define the repeating lattice, while geometry determines how much of each cell can be occupied.
The lattice vectors work like directions for a stamp: place the same contents at one lattice point, translate by a vector, and repeat. The lattice specifies repetition; the repeated contents could be cannonballs, ducks, circles, or atoms.
Changing the lattice changes the limit. A two-dimensional hexagonal lattice reaches about 91% area coverage, compared with about 78% for the square lattice. Atoms likewise seek efficient packing within the restrictions imposed by their bonding.
Why is circle packing less efficient on a square lattice than on a hexagonal lattice, and how do the remaining void shapes differ?
The square arrangement leaves larger diamond-shaped spaces. Shifting rows into a hexagonal arrangement produces smaller triangular spaces, increasing the occupied fraction from about 78% to about 91%.
From Crystal Systems to Cubic Bravais Lattices
Three-dimensional lattices are classified first by cell geometry and then by distinct lattice-point arrangements.
There are seven unique crystal systems in three dimensions. They are distinguished by the lengths of three lattice vectors, a, b, and c, and by their relative angles, α, β, and γ. Together these systems describe the available ways to fill three-dimensional space without voids.
Across the seven systems, fourteen distinct Bravais lattices enumerate the possible lattice-point arrangements. Exactly three have cubic symmetry: simple cubic (SC), body-centered cubic (BCC), and face-centered cubic (FCC).
Simple Cubic Structure
SC contains one atom per unit cell and has the lowest packing fraction of the cubic lattices.
A simple cubic cell places atoms only at its eight corners. Because each corner belongs to eight neighboring cells, the cell receives one eighth of each corner atom.
- Coordination number: 6, with neighbors above, below, left, right, in front, and behind.
- Close-packed direction: the cube edge.
- Radius at maximum packing: r = a/2.
This arrangement uses only about 52% of the available volume. At standard conditions, polonium is the sole element identified with the simple cubic structure, a result associated with relativistic behavior of its electrons.
Body-Centered Cubic Structure
BCC adds a complete central atom and packs more efficiently than SC.
A body-centered cubic cell contains atoms at all eight corners and one full atom at its center. The corner portions combine into one atom, and the unshared central atom supplies another.
- Coordination number: 8, arranged as four nearest neighbors above and four below.
- Close-packed direction: the body diagonal.
- Body-diagonal length: a√3.
At 68%, BCC uses space more effectively than SC. Many metals adopt BCC, including the alkali metals.
Face-Centered Cubic Structure
FCC places atoms on corners and faces, producing the densest of the three cubic arrangements.
A face-centered cubic cell has atoms at its eight corners and at the center of each of its six faces. Corner atoms contribute one eighth each, while every face-centered atom is shared by two cells and contributes one half.
- Coordination number: 12, with four nearest neighbors in the same layer, four above, and four below.
- Close-packed direction: the face diagonal.
- Face-diagonal length: a√2.
FCC reaches 74%, the greatest possible packing fraction for equal spheres, and is also called cubic closest packing. Copper, silver, gold, and nickel are examples of metals with this structure.
How is coordination number connected to packing efficiency, and why does FCC have twelve nearest neighbors while SC has six?
Greater packing efficiency places more atoms in immediate contact around each atom. SC touches along cube edges and provides six nearest neighbors, whereas the denser FCC arrangement touches along face diagonals and surrounds each atom with twelve.
Connecting the Three Cubic Lattices
Atoms per cell and the direction of first contact determine each cubic lattice's packing fraction.
| Lattice | Atoms per cell | Coordination number | Close-packed direction | Radius in terms of a | APF |
|---|---|---|---|---|---|
| SC | 1 | 6 | Cube edge, a | a/2 | π/6 ≈ 52% |
| BCC | 2 | 8 | Body diagonal, a√3 | a√3/4 | π√3/8 ≈ 68% |
| FCC | 4 | 12 | Face diagonal, a√2 | a√2/4 | π√2/6 ≈ 74% |
The low 52% value helps explain why simple cubic is rare. BCC and FCC are common among metals because their denser arrangements allow more neighboring contacts. A periodic table can indicate an element's most stable crystal structure at room temperature and pressure, although temperature or pressure can allow an element to adopt another symmetry.
These cubic structures establish a basis for studying sphere-stacking sequences and for connecting crystal geometry with materials such as metals and ceramics.
Common misconceptions
Watch out: A repeating pattern must have only one visibly obvious unit-cell choice.
Instead: More than one translated region may reproduce the same pattern; a valid choice must repeat to cover space without gaps.
Watch out: Atoms in BCC first touch along the cube edges, as they do in SC.
Instead: BCC atoms first meet along the body diagonal. Edge contact is the close-packed direction only for SC.
Watch out: Because FCC is densely packed, it must contain an atom at the cube's center.
Instead: FCC has atoms at corners and face centers but none at the body center; BCC is the lattice with a central atom.
Lesson recap
- Crystalline solids display extensive repeating order; amorphous solids have little or only short-range order.
- A unit cell and its lattice vectors generate the repeating crystal by translation.
- Seven crystal systems contain fourteen Bravais lattices, including the three cubic forms SC, BCC, and FCC.
- The close-packed direction supplies r in terms of a, allowing APF to be calculated from occupied atomic volume divided by cell volume.
- Packing efficiency rises from SC at 52%, through BCC at 68%, to FCC at 74%. Their coordination numbers increase from 6 to 8 to 12.