02 Atomic Bonding and Crystal Structures

How atoms bond

Atoms in solids arrange themselves to lower their energy. The balance of attraction and repulsion between atoms or ions helps determine their stable separation, crystal structure, and many material properties. Bonding is commonly classified as ionic, covalent, metallic, or secondary, although real materials can combine more than one kind.

Bond types and their effects

Ionic bonding is the electrostatic attraction between oppositely charged ions, commonly formed when electrons transfer between atoms. Because this attraction acts in all directions, an ionic solid forms an extended, electrically neutral lattice rather than a collection of separate ion pairs. Ionic crystals are often hard and brittle. They typically conduct electricity when molten or dissolved, when their ions can move, but not as solids.

Covalent bonding involves shared electrons concentrated between particular atoms. These bonds are directional and favor specific arrangements. In network solids such as diamond and silicon carbide, covalent bonds extend through the crystal; these materials are often hard and have high melting temperatures.

Metallic bonding arises from attraction between positive atomic cores and valence electrons delocalized through the solid. Mobile electrons help explain metals’ electrical and thermal conductivity. Metallic bonding also allows layers of atoms to shift without the strong repulsion that would result from bringing like-charged ions together, helping many metals deform without shattering.

Secondary interactions, including van der Waals forces and hydrogen bonding, are generally weaker than primary bonds. They can be important between molecules, between polymer chains, and between layers in materials such as graphite. These bonding categories are useful models rather than rigid boundaries; many ceramics, for example, have both ionic and covalent character.

Crystal order and unit cells

A crystal has long-range periodic order. It can be described as a repeating lattice, a regular array of points, together with a basis: the atom or group of atoms associated with each point. A Unit cell is a volume that repeats by translation to build the crystal. A primitive cell is the smallest cell that represents the lattice; a conventional cell may be larger when it makes the symmetry easier to see.

The cell edge lengths are $a$, $b$, and $c$, and its interaxial angles are $\alpha$, $\beta$, and $\gamma$. Atoms on cell boundaries are shared with neighboring cells, so each contributes only a fraction to any one cell. Each corner atom of a cubic cell contributes $\frac{1}{8}$ of an atom, while each face-centered atom contributes $\frac{1}{2}$.

For example, a Body-centered cubic (BCC) conventional cell contains $2$ atoms: $8$ corners each contribute $\frac{1}{8}$, and there is one interior atom. A Face-centered cubic (FCC) cell contains $4$ atoms: $8$ corners each contribute $\frac{1}{8}$, and $6$ faces each contribute $\frac{1}{2}$. Coordination number counts an atom’s nearest neighbors; BCC has coordination number $8$, while FCC has coordination number $12$.

Directions and planes in crystals

Miller indices describe directions and plane orientations relative to a crystal’s axes. The notation distinguishes one direction, written $[u\ v\ w]$, from a family of symmetry-equivalent directions, written $\langle u\ v\ w\rangle$. One plane orientation is written $(h\ k\ l)$, while a family of symmetry-equivalent planes is written $\{h\ k\ l\}$.

To index a direction, express its vector components along the cell axes, clear fractions, and reduce the result to the smallest integer ratio. For example, a vector with components $(a,a,0)$ has direction $[1\ 1\ 0]$, usually written $[110]$.

To index a plane, find its intercepts with the axes in units of the cell edges, take their reciprocals, and clear fractions to obtain integers. An intercept at infinity means the plane is parallel to that axis and gives an index of $0$. Thus, a plane that intercepts the $a$ and $b$ axes at one cell length and is parallel to $c$ has indices $(110)$. A negative index is written with a bar, as in $(\bar{1}10)$. If a plane passes through the origin, use a parallel plane and determine its intercepts instead.

In a cubic crystal, $[hkl]$ is perpendicular to $(hkl)$. This shortcut does not generally hold in non-cubic crystals.

Common structures in metals

Many metals adopt Body-centered cubic (BCC), Face-centered cubic (FCC), or Hexagonal close-packed (HCP) structures. FCC and HCP are close-packed arrangements, with an atomic packing factor of about $0.74$, the maximum for equal spheres. Their close-packed layers differ in stacking: FCC follows $\mathrm{ABCABC}\ldots$, while HCP follows $\mathrm{ABAB}\ldots$. BCC is not close-packed and has a packing factor of about $0.68$.

• BCC: The conventional cell contains $2$ atoms and has coordination number $8$. Examples include $\alpha$-iron and tungsten.

• FCC: The conventional cell contains $4$ atoms and has coordination number $12$. Examples include copper and aluminum.

• HCP: The conventional cell contains $6$ atoms and has coordination number $12$. It has a hexagonal cell and $\mathrm{ABAB}\ldots$ layer stacking. Examples include magnesium and $\alpha$-titanium.

A metal’s structure can change with temperature or pressure. Iron, for example, has more than one crystal structure, so the same material can have different structural forms under different conditions.

Common structures in ceramics

Ceramic crystal structures must accommodate charge neutrality and the relative sizes of cations and anions. These factors help determine which sites ions occupy and how many oppositely charged neighbors surround each ion. Ceramic bonding is often mixed ionic–covalent, and many structures can be understood as one ion arrangement with the other ion type occupying suitable interstitial sites.

• Rock-salt structure: Two interpenetrating sublattices; each ion has six nearest neighbors of opposite charge, giving coordination $6{:}6$. Examples include $\mathrm{NaCl}$ and $\mathrm{MgO}$.

• Cesium chloride structure: Each ion has eight nearest neighbors of opposite charge, giving coordination $8{:}8$. Although its conventional cell looks body-centered, $\mathrm{CsCl}$ is not a BCC Bravais lattice: the corner and body-center positions represent different ions.

• Zinc blende (sphalerite) structure: Each ion has four oppositely charged nearest neighbors in a tetrahedral arrangement, giving coordination $4{:}4$. $\mathrm{ZnS}$ is a common example. Wurtzite has the same fourfold local coordination but a different, hexagonal stacking arrangement.

• Fluorite structure: In $\mathrm{CaF_2}$, each calcium ion has eight fluorine neighbors, while each fluorine ion has four calcium neighbors.

• Perovskite structure: A broad ceramic structure family with formula $\mathrm{ABO_3}$; barium titanate, $\mathrm{BaTiO_3}$, is a familiar example.

A chemical formula alone does not specify a crystal structure: different compounds can adopt different arrangements, and different arrangements can share similar local coordination.

Connecting bonding and structure

Bonding describes the forces that hold atoms or ions together, while crystal structure describes their repeating arrangement. Unit cells provide a compact description of periodic order. Miller indices identify directions and planes, while coordination and stacking distinguish common structures. Metals commonly use BCC, FCC, or HCP arrangements; ceramic structures are more varied and are shaped by charge balance, ion sizes, and bonding character.