BCC, FCC & HCP — Crystal Structures of Metals

1. The 14 Bravais Lattices

A crystal is an infinite periodic arrangement of atoms. The underlying periodicity is described by a Bravais lattice — a set of points R = n₁a₁ + n₂a₂ + n₃a₃ where a₁, a₂, a₃ are the primitive lattice vectors and n₁, n₂, n₃ are integers.

Auguste Bravais proved in 1848 that there are exactly 14 distinct lattice types in three dimensions (grouped into 7 crystal systems: cubic, tetragonal, orthorhombic, monoclinic, triclinic, trigonal, and hexagonal). The three cubic Bravais lattices are:

• Simple cubic (SC): Atoms only at cube corners. Rarely found in metals (Po is the notable exception) because the packing is very inefficient (APF = 52%).
• Body-centred cubic (BCC): Corners plus one atom in the centre of each cube.
• Face-centred cubic (FCC): Corners plus one atom at the centre of each face.

The hexagonal close-packed (HCP) structure is not a Bravais lattice itself but a hexagonal lattice with a two-atom basis — two atoms per lattice point, positioned at (0,0,0) and (⅓, ⅔, ½) in fractional coordinates.

2. Body-Centred Cubic (BCC)

In the BCC structure, atoms occupy the corners of a cube and one atom sits at the geometric centre. The conventional unit cell contains 2 atoms (8 × ⅛ corner + 1 body centre).

Geometry and Atomic Packing Factor

Atoms in BCC touch along the body diagonal. If R is the atomic radius and a is the lattice parameter:

The coordination number (number of nearest neighbours) in BCC is 8 — the body-centre atom touches all 8 corners, and each corner atom touches the body-centre atoms of all 8 surrounding cubes.

There are also 6 next-nearest neighbours at distance a (the face centres of the cube), only 15% further than the nearest neighbours. This near-equal spacing of 1st and 2nd shells influences BCC properties.

BCC Metals

Common BCC metals: Fe (α-iron, below 912°C), W (tungsten — highest melting point of all metals), Mo, Cr, V, Nb, Ta, K, Na, Li.

3. Face-Centred Cubic (FCC)

In FCC, atoms occupy all 8 cube corners plus the centres of all 6 faces. The conventional unit cell contains 4 atoms (8 × ⅛ + 6 × ½).

Geometry and Atomic Packing Factor

FCC atoms touch along the face diagonal:

FCC is one of the two close-packed structures (along with HCP), achieving the maximum possible packing fraction for equal spheres (proved by the Kepler conjecture, solved by Hales in 2005).

The coordination number in FCC is 12: each atom is surrounded by 12 equidistant nearest neighbours (4 in the same layer, 4 above, 4 below).

The FCC structure can also be viewed as ABCABC... stacking of close-packed {111} planes. The three layers A, B, C each consist of atoms in a triangular array, offset so each layer nestles in the hollows of the one below.

FCC Metals

Common FCC metals: Cu (copper), Al (aluminium), Ni, Ag (silver), Au (gold), Pt, Pb, γ-Fe (iron above 912°C), Ca, Sr.

FCC metals are generally soft and ductile because the {111} close-packed planes — 4 distinct families of which exist — provide many slip systems for dislocation motion, enabling plastic deformation without fracture.

4. Hexagonal Close-Packed (HCP)

HCP also achieves the 74% maximum packing efficiency (APF ≈ 0.7405) but via a different stacking sequence: ABABAB... rather than ABCABC. Close-packed planes alternate between two positions A and B only, never visiting a third position C.

Unit Cell and Ideal c/a Ratio

The HCP unit cell is a right hexagonal prism with basal plane hexagonal lattice parameter a and height c. It contains 2 atoms. The ideal close-packed geometry requires:

When c/a differs from ideal, nearest-neighbour distances within the basal plane and between planes are unequal, reducing the effective coordination number and affecting mechanical anisotropy.

HCP Metals

Common HCP metals: Mg, Ti (α-phase), Zn, Co (α-phase), Cd, Zr (α-phase), Be, Ru, Os.

5. Comparison Table

6. Miller Indices: Planes and Directions

Crystallographic planes and directions are identified using Miller indices (hkl) for planes and [hkl] for directions. The procedure for planes:

1. Find the intercepts of the plane on the three crystal axes in units of the lattice parameter
2. Take the reciprocals of these intercepts
3. Clear fractions to obtain the smallest integers h, k, l
4. Enclose in parentheses: (hkl). Negative intercepts denoted with overbars: (h̄kl)

For HCP, a four-index notation (hkil) is used, where i = −(h+k), to make the symmetry-equivalent planes more obvious in the hexagonal system.

7. Slip Systems and Plastic Deformation

Plastic deformation in metals occurs primarily by dislocation glide: a line defect (dislocation) moves through the crystal on a slip plane in a slip direction. The combination of a slip plane and slip direction is a slip system.

Slip is energetically easiest on close-packed planes in close-packed directions, because the atoms are densest, the inter-planar spacing is largest (lowest energy to shear), and the Burgers vector (lattice displacement per slip step) is smallest.

FCC Slip Systems

FCC has 4 close-packed {111} planes, each with 3 close-packed <110> directions:

BCC Slip Systems

BCC has no true close-packed planes, but the closest are {110}, {112}, and {123} planes. The slip direction is always <111> (body diagonal, the close-packed direction):

HCP Slip Systems

HCP has only one close-packed plane family (the basal plane, {0001}) with three <11̄20> directions, giving just 3 basal slip systems. This is below the 5 independent systems required by the Von Mises criterion for arbitrary shape change, which is why HCP metals (Mg, Zn, Ti) are much less ductile than FCC metals. Additional slip on prismatic {101̄0} and pyramidal {101̄1} planes can activate at high temperatures.

8. Why Metals Choose Their Structure

Crystal structure is determined by quantum mechanics — the electronic band structure of each element determines which arrangement of atoms minimises the total energy (kinetic energy of electrons + ion-electron attraction + ion-ion repulsion + electron-electron repulsion). Simple rules of thumb:

• FCC preference: Elements with broad, unfilled d-bands (Cu, Ag, Au, Ni, Pd, Pt) tend to FCC. The electronic band structure stabilises close-packed ABCABC stacking. Al, with only s and p electrons, also adopts FCC due to the Madelung energy minimisation in a face-centred arrangement.
• BCC preference: Elements at the beginning or end of the transition metal series (Group 1, 2, 5, 6) often prefer BCC. The half-filled d-band or directional d-orbital bonding can stabilise the less densely packed BCC structure. Fe-BCC is stabilised by magnetic exchange interactions (ferromagnetism lowers BCC energy below FCC above the Curie temperature-structure coupling).
• HCP preference: Elements with nearly filled or half-filled d-bands (Ti, Zr, Mg, Co, Ru) often adopt HCP. The energy difference between HCP and FCC is typically very small (<10 meV/atom), explaining why many metals transform between the two with temperature (allotropic transformations): Co transforms HCP→FCC at 417°C, Ti transforms HCP→BCC at 882°C.

Alloying can shift crystal structure preferences: adding carbon to iron expands the FCC (austenite) stability range, which is why steelmakers heat treat steel above 912°C to dissolve carbon in the FCC phase before quenching.