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Lattice Parameters and Vectors

Expressing Directions in a Crystal

Because lattice vectors are neither necessarily orthogonal nor of equal length, directions in this coordinate system (lattice directions) are written using square brackets \([\,]\) as \([u\,v\,w]\), called direction indices. The notation \([u\,v\,w]\) denotes the vector \(u\mathbf{a}+v\mathbf{b}+w\mathbf{c}\) — equivalently, the coordinates in the vector space spanned by \(\mathbf{a}, \mathbf{b}, \mathbf{c}\) (the basis vectors).

While \(u, v, w\) may be any real numbers, when they are integers the vector connects two lattice points and is necessarily a translation vector (e.g., \([100]\) is \(\mathbf{a}\) itself; \([211]\) is \(2\mathbf{a}+\mathbf{b}+\mathbf{c}\)). Negative values are preferably written with a bar over the digit rather than a minus sign¹ (e.g., \([1\bar{1}1]\)).

When the crystal’s symmetry makes several directions crystallographically equivalent — for example when \([100]\) and \([010]\) are equivalent — angle brackets \(\langle\,\rangle\) are used: \(\langle100\rangle\) denotes not a single vector but the family of all equivalent directions. Angle brackets are used only in the context of a specific symmetry (crystal system or point group).

Another concept for expressing directions and distances in a crystal is the “lattice plane,” explained in “4.1. Crystal Planes and Diffraction.”

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Crystal Systems

A crystal system is one classification of crystals, grouped by the shape of unit cell compatible with the crystal’s non-translational symmetry. Understanding this classification precisely requires knowledge of space groups, so the full discussion is deferred; for now simply read on knowing that 3D crystals are classified into the following systems.

Crystal system	Triclinic	Monoclinic	Orthorhombic	Trigonal / Hexagonal	Tetragonal	Cubic
Description	No two edges are perpendicular	One edge (\(b\)) is perpendicular to the other two	All three edges are mutually perpendicular	Two edges (\(a,b\)) meet at 120° and are equal in length; the third (\(c\)) is perpendicular to them	All three edges are perpendicular; two (\(a,b\)) are equal	All three edges are perpendicular and equal
Diagram
Unique symmetry element2	\(1\) or \(\bar{1}\)	\(2\) or \(m\ (=\bar{2})\)	\(2\) or \(m\ (=\bar{2})\) in three directions	\(3\) or \(6\)	\(4\) or \(\bar{4}\)	\(3\) in four directions

Each crystal system has a unique symmetry element (rotation axis or rotoinversion axis) that characterises it; its direction is called the principal axis. When more than one unique element exists, those after the first are called secondary axes. This is discussed further in “2.3. Notation for Point Groups and Space Groups.”

Here are a few noteworthy observations about crystal systems.

Why Is There No “Diclinic” System?

Why does a “triclinic” and “monoclinic” system exist, but not a “diclinic”? A “diclinic” system, taken literally, would have \(\alpha=90°\) but \(\beta\) and \(\gamma\) not equal to \(90°\). Such a parallelepiped might seem to have higher symmetry than triclinic, but careful examination shows that the only operation leaving it invariant is the identity or inversion — no rotation or rotoinversion of order ≥ 2 is allowed. In terms of symmetry, it is the same as triclinic. Even if \(\alpha\) happens to equal \(90°\), the crystal remains triclinic.

Difference Between Trigonal and Hexagonal Systems

Why must these two systems, which share the same unit-cell shape, be distinguished? See the “6.1. Topics in Trigonal/Hexagonal Crystal Systems” page for details.

Tetragonal/Cubic Systems and 4-fold Rotation

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Primitive and Non-Primitive Lattices

Note: throughout this section, open circles ○ represent lattice points — they do not imply the presence of actual material such as atoms. What matters is the direction and distance of the translation vectors connecting lattice points.

Although a non-primitive lattice is chosen deliberately to be large, making it arbitrarily large defeats the purpose. There are exactly four meaningful types of non-primitive lattice:

1. Body-centred lattice (\(I\)): a lattice point exists at the centre of the unit cell.
2. Base-centred lattice (\(A, B, C\)): lattice points exist at the centres of one pair of opposite faces.
3. Face-centred lattice (\(F\)): lattice points exist at the centres of all six faces.
4. Rhombohedral lattice (\(R\)): lattice points exist at positions that trisect the body diagonal of the unit cell (trigonal system only).

Let us now look at the geometric relationship between each non-primitive lattice and a primitive lattice.

Body-centred lattice (\(I\)⁴)

A body-centred lattice contains 2 lattice points per unit cell. In the figure on the left, 9 lattice points are drawn, but each corner lattice point contributes only 1/8 to this cell. Note also that lattice points do not necessarily coincide with atomic positions. Since the body-centred lattice is deliberately chosen to be large, it can be converted to a primitive lattice: by arranging four body-centred cells and choosing the lattice indicated by the bold black lines on the right, a primitive lattice of half the volume is obtained.

Base-centred lattice (\(A, B, C\))

A base-centred lattice contains 2 lattice points per unit cell. As shown on the left, lattice points exist at the centres of one pair of opposing faces. By placing two base-centred lattices side by side and choosing the lattice shown by bold black lines on the right, a primitive lattice of half the volume is obtained.

The three symbols \(A, B, C\) indicate which pair of faces is centred, in relation to the axes \(a, b, c\). For example, base-centred \(A\) means that lattice points exist at \(b/2+c/2\).

Face-centred lattice (\(F\))

A face-centred lattice contains 4 lattice points per unit cell. As shown on the left, lattice points exist at the centres of all six faces. Choosing the lattice shown by bold black lines on the right gives a primitive cell of 1/4 the volume.

Rhombohedral lattice (\(R\))

The rhombohedral lattice is used for certain trigonal crystals, but its meaning varies by context. “Rhombohedral lattice” alone often means a lattice with equal edge lengths and equal angles (\(a=b=c,\;\alpha=\beta=\gamma\)), but here it refers to a crystal lattice in the rhombohedral lattice system⁵ described with hexagonal axes (\(a=b,\;\alpha=\beta=90°,\;\gamma=120°\)) — also called the rhombohedral-hexagonal lattice.

As shown on the left, lattice points exist at positions trisecting the body diagonal of the unit cell, giving 3 lattice points per cell. Like the other non-primitive lattices, it can be converted to a primitive lattice: because the trigonal system always has a 3-fold rotation axis, arranging three rhombohedral-hexagonal lattices at 120° to each other and connecting the central lattice points together with those at heights 1/3 and 2/3 yields a primitive lattice (the primitive rhombohedral setting) of 1/3 the volume. For more on the geometry of the rhombohedral lattice, see “6.1. Topics in Trigonal/Hexagonal Crystal Systems.”

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Bravais Lattices⁶

How many distinct types of crystal — where atoms exist only at lattice points — can be classified by symmetry (i.e., how many non-isomorphic groups are there)? This is the concept of Bravais lattices⁷. Put differently, starting from the combination of all crystal systems with all types of centring, those for which a smaller unit cell preserves the symmetry or which become equivalent by a change of basis vectors are removed. A detailed discussion is deferred to “5.1. Classification of Space Groups“; for now, simply note that there are 5 Bravais lattices in 2D and 14 in 3D. Below, the 14 three-dimensional Bravais lattices are listed by crystal system.

Triclinic

Only the primitive lattice exists for triclinic. The only non-translational symmetry operations are \(1\) or \(\bar{1}\), so choosing a body-centred or face-centred cell offers no advantage. (In special circumstances, such as describing phase transitions without changing the cell shape, non-primitive cells may be used intentionally.)

Monoclinic

Monoclinic has primitive and base-centred lattices, normally with \(b\) as the principal axis. Body-centred and face-centred monoclinic can always be converted to base-centred by an appropriate axis transformation and are therefore not classified separately. (Alternative axis choices are discussed in “6.3. Choice of Axes and Axis Transformation.”)

Orthorhombic

All four lattice types — primitive, base-centred, body-centred, and face-centred — exist for orthorhombic.

Trigonal and Hexagonal

With \(c\) as the principal axis, trigonal and hexagonal have primitive and rhombohedral lattices. See also “6.1. Topics in Trigonal/Hexagonal Crystal Systems” for the geometry of the rhombohedral lattice.

Tetragonal

With \(c\) as the principal axis, tetragonal has primitive and body-centred lattices. A base-centred tetragonal lattice can always be recast as a primitive tetragonal lattice and is not classified separately; similarly, a face-centred tetragonal lattice is equivalent to body-centred tetragonal.

Cubic

Cubic has primitive, body-centred, and face-centred lattices. A base-centred cubic lattice is incompatible with the 3-fold rotation in the \([111]\) direction and therefore does not exist.

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1. This notation superficially resembles the rotoinversion symbol, but the two are completely different — beware. ↩︎
2. Only rotations and rotoinversions are listed here, but glide planes and screw axes can also be unique elements. A glide plane is a \(m\) element (\(\bar{2}\)) with added translation, so it is treated equivalently to \(m\). Screw axes correspond directly to their rotation order, so for example three \(2_1\) screw axes in three directions indicates an orthorhombic crystal. ↩︎
3. Without non-primitive lattices, a crystal with a 2-fold rotation axis might not be expressible with a rectangular unit cell, greatly complicating the analysis of its structure and properties. ↩︎
4. The symbol comes from the German innenzentriert (inner-centred). In English the natural choice would be “B” for “body-centred,” but “B” is already used for base-centred. ↩︎
5. For the lattice system, see “5.1. Classification of Space Groups.” ↩︎
6. Named after Auguste Bravais, a French crystallographer of the 19th century. ↩︎
7. The open circles ○ in the figures may be thought of as atoms rather than lattice points if one wishes, but the importance of the Bravais lattice concept lies not in “crystals with atoms only at lattice points” per se, but in the classification (Bravais arithmetic class) that assigns every crystal to a Bravais lattice type. The detailed discussion appears in “5.1. Classification of Space Groups.” ↩︎