Surface Atlas

Structure

Begin with the periodic crystal

A surface inherits its atomic arrangement from the bulk beneath it. Start by separating the translations that repeat from the atoms carried by each repeat.

Translation vectors locate every repeat

Choose three independent primitive translation vectors, \(\mathbf a_1\), \(\mathbf a_2\), and \(\mathbf a_3\). Integer combinations of them generate every point of the Bravais lattice:

\[\mathbf R_{n_1n_2n_3}=n_1\mathbf a_1+n_2\mathbf a_2+n_3\mathbf a_3,\qquad n_i\in\mathbb Z.\]

Each \(\mathbf R\) is a lattice point with an identical environment. The vectors define the direct lattice; they need not be mutually perpendicular or equal in length. A nonprimitive conventional cell instead spans a sublattice and contains more than one lattice point. [1]

A position can be written in fractional coordinates as

\[\mathbf r=x\mathbf a_1+y\mathbf a_2+z\mathbf a_3.\]

The triplet \((x,y,z)\) has meaning only together with the stated vectors. Adding an integer to any component reaches a periodically equivalent position, so \((0,0,0)\) and \((1,0,0)\) mark equivalent points in neighbouring cells.

A lattice and a basis make the crystal

A unit cell is a region whose translations reproduce the crystal. A primitive cell contains exactly one lattice point; a conventional cell may contain several lattice points but makes the crystal symmetry easier to see. These are different coordinate descriptions of the same infinite structure, not different materials. [2]

The Bravais lattice alone specifies only equivalent points. Attach one or more atomic positions \(\boldsymbol\tau_j\), called the basis, to every lattice point to obtain all atomic positions:

\[\mathbf r_{n_1n_2n_3,j}=\mathbf R_{n_1n_2n_3}+\boldsymbol\tau_j.\]

This distinction matters at a surface: the translations determine which orientations can repeat, while the basis determines which elements and atomic layers occur along a chosen orientation.

Six structures used throughout the atlas

The coordinate lists below use explicit conventional-cell choices. SC, FCC, BCC, SH, and BCT are Bravais lattices, although centred conventional cells contain more than one lattice point. HCP instead requires a two-atom basis on a primitive hexagonal Bravais lattice.

FCC

Face-centred cubic Bravais lattice. In the conventional cubic cell:

(0, 0, 0) · (0, 1/2, 1/2) · (1/2, 0, 1/2) · (1/2, 1/2, 0)

The four occupied positions represent four lattice points per conventional cell.

BCC

Body-centred cubic Bravais lattice. In the conventional cubic cell:

(0, 0, 0) · (1/2, 1/2, 1/2)

The corner and body-centred positions represent two lattice points per conventional cell.

HCP

Hexagonal lattice plus a basis. With basal vectors separated by \(120^\circ\) and a third vector along \(c\), one common convention is:

(0, 0, 0) · (2/3, 1/3, 1/2)

The two-atom basis produces the hexagonal close-packed structure. Another vector convention may use symmetry-equivalent fractions.

SC

Simple cubic Bravais lattice. The conventional and primitive cubic cells coincide:

(0, 0, 0)

One lattice site repeats along three perpendicular translations of length \(a\).

SH

Primitive hexagonal Bravais lattice. The one-site basis is:

(0, 0, 0)

Basal triangular nets repeat directly in AA registry after the independent translation \(c\).

BCT

Body-centred tetragonal Bravais lattice. In the conventional tetragonal cell:

(0, 0, 0) · (1/2, 1/2, 1/2)

The two basal vectors have length \(a\); the perpendicular tetragonal vector has the independent length \(c\).

Calling all six “lattices” is convenient shorthand, but it hides an important distinction: HCP is a crystal structure with a two-atom basis, whereas the other five names identify Bravais lattices used here with a one-site basis. [3]

Neighbours reveal packing and layer registry

The bulk coordination number counts an atom’s nearest neighbours in the ideal infinite structure. It describes a three-dimensional neighbour shell; it is not the same as the number of surface atoms later used to construct a lateral point above the crystal.

StructureNearest-neighbour coordinationLayer and packing character
FCC12Close-packed \(\{111\}\) layers in an ABCABC sequence
BCC8Densest \(\{110\}\) planes and \(\langle111\rangle\) rows; not close packed
Ideal HCP12Close-packed basal \((0001)\) layers in an ABAB sequence
SC6Equal neighbours along the three cubic axes
SHMetric dependentAA basal layers; six basal neighbours when \(c>a\)
BCTMetric dependentBody-centred layers distorted by the chosen \(c/a\) ratio

In the stacking notation, A, B, and C label the three possible lateral registries of a close-packed layer. FCC cycles through all three; HCP alternates between two. For equal touching spheres, ideal HCP has

\[\frac{c}{a}=\sqrt{\frac{8}{3}}\approx1.633.\]

Real HCP materials can have a different \(c/a\) ratio, which splits some neighbour distances while preserving the same translational symmetry and AB stacking. Close packing and its layer sequences are geometric models, not assumptions that real atoms are rigid spheres. [4]

The bulk description now tells us where every atom is. The next step is to choose an orientation through that periodic structure without confusing the orientation with the atomic layer at which the crystal ends.

Sources

References

  1. International Union of Crystallography. Lattice. Online Dictionary of Crystallography. IUCr definition. Accessed 2026-07-13.
  2. International Union of Crystallography. Unit cell. Online Dictionary of Crystallography. IUCr definition. Accessed 2026-07-13.
  3. C. Hammond (2015). The Basics of Crystallography and Diffraction. 4th ed., Oxford University Press and IUCr. doi:10.1093/acprof:oso/9780198738671.001.0001.
  4. P. Krishna and D. Pandey (1981). Close-packed structures. IUCr Teaching Pamphlet No. 5 (electronic edition 2001). Open teaching text. Accessed 2026-07-13.