advanced25 min studyLesson 491 in Physics

Bravais lattices and crystal bases

Separate translational geometry from the atomic motif, distinguish primitive and conventional cells, and count atoms without confusing lattice points with basis sites.

Before you begin

On this page

The question for today

What you’ll understand

Construct crystal positions from integer translations and a basis.
Calculate primitive-cell area or volume with a determinant.
Distinguish an enlarged conventional cell from a genuine change of translation symmetry.

A crystal combines a repeating translation pattern with a motif attached to every translation point. Keeping these ideas separate makes diffraction, mode counting, and lattice dynamics much easier. A lattice point is a geometric marker, not necessarily an atom, and a visually convenient cube need not be primitive. We will construct positions explicitly and use cell volume to check atomic counts before moving to reciprocal space.

Translations and motifs answer different questions

A Bravais lattice consists of all integer combinations of independent primitive translation vectors. Every lattice point has the same environment under those translations. A crystal adds a basis: specified atoms at offsets tau relative to each lattice point. The full atomic positions are lattice translations plus basis offsets. Chemical identity belongs to the motif. A translation that exchanges two different atomic species generally is not a symmetry of the crystal even when a drawing of unlabelled points seems unchanged.

Primitive vectors are not unique. Integer changes of basis with determinant plus or minus one generate the same lattice and preserve primitive-cell volume. The parallelepiped spanned by such vectors tiles space without gaps and contains one lattice point after boundary sharing is counted. A Wigner–Seitz cell is another primitive choice, constructed from points closer to one lattice point than to any other. It need not be a parallelepiped, but it has the same volume and represents the same translational information.

\mathbf R=n_1\mathbf a_1+n_2\mathbf a_2+n_3\mathbf a_3,\quad n_i\in\mathbb Z,\qquad \mathbf r_{\ell s}=\mathbf R_\ell+\boldsymbol\tau_s

The vectors a are primitive translations with length units, and tau_s specifies the position and species of basis atom s. A three-dimensional basis with r atoms gives r atoms per primitive cell.↔ Scroll wide equations horizontally.

Count with volume and boundary sharing

The primitive volume is the absolute scalar triple product of the primitive vectors; in two dimensions it is the absolute determinant of the two translation vectors. If a primitive basis has r atoms, the number density is r divided by that volume. This check is independent of how the cell is drawn. An atom on a cell boundary contributes only its shared fraction when counting one cell, while an interior atom contributes fully. Forgetting sharing overcounts a crystal even if every visible corner has been labelled correctly.

For an identical-atom body-centred cubic crystal, the conventional cube contains eight shared corners and one interior atom, giving two atoms. Its body-centred translation pattern is itself Bravais and has one atom per primitive cell of half the cube volume. Describing the same arrangement using a simple-cubic conventional grid with a two-point motif is useful bookkeeping, but that grid omits some true primitive translations. If the corner and body-centre species differ, those extra translations no longer preserve chemistry and the primitive lattice can instead be simple cubic.

v_{\mathrm p}=|\mathbf a_1\cdot(\mathbf a_2\times\mathbf a_3)|,\qquad n_{\mathrm{atom}}=\frac{r}{v_{\mathrm p}},\qquad v_{\mathrm p}^{\mathrm{bcc}}=\frac{a_{\mathrm c}^3}{2}

The first and last volumes have cubic-length units. Here a_c is the conventional cubic edge, while r counts atoms in a true primitive motif; do not combine a conventional volume with a primitive count.↔ Scroll wide equations horizontally.

A larger cell changes labels, not the solid

One may deliberately choose a supercell containing several primitive cells. This increases the number of basis atoms and decreases the reciprocal-space zone volume by the same factor. Later, phonon branches will fold when expressed in that smaller zone, but the number of physical displacement degrees of freedom remains unchanged. Cell choice is therefore partly a representation choice. It should not be confused with an actual structural distortion or chemical ordering that destroys a previously valid translation symmetry.

A useful construction procedure is to identify translations that preserve the entire decorated structure, choose an independent primitive set, list one representative of each basis atom, and verify the density. The basis offsets may lie outside a chosen cell and be shifted back by lattice translations without changing the crystal. Double-counting two offsets that differ by a translation, or treating unlike species as interchangeable, breaks this procedure. The distinction will return when an apparently missing diffraction peak turns out to reflect a smaller primitive cell or chemical contrast.

Work it through

Build an oblique two-dimensional crystal

A two-dimensional crystal has primitive vectors $\mathbf a_1=(2,0)\,\mathrm{\mathring A}$ and $\mathbf a_2=(1,3)\,\mathrm{\mathring A}$ . Its two-atom basis is at (0,0) and $(0.5,1)\,\mathrm{\mathring A}$ . Find the cell area, areal atomic density, and second atom in cell $(n_1,n_2)=(2,-1)$ .

The determinant is $2\times3-0\times1=6\,\mathrm{\mathring A^2}$ . Obliqueness changes the cell shape but is already included in this area calculation.
There are two atoms per primitive cell, so the areal density is $2/6=1/3\,\mathrm{\mathring A^{-2}}$ . The count refers to the motif, not the number of primitive vectors.
The cell translation is $2(2,0)-(1,3)=(3,-3)\,\mathrm{\mathring A}$ . Adding the second offset gives $(3.5,-2)\,\mathrm{\mathring A}$ . Translating the motif preserves its internal separation and species labels.

The result

The primitive area is 6 square ångströms, density is one-third atom per square ångström, and the requested atom is at (3.5, −2) ångströms.

The living notebook

Explore the model

Interactive lab

Interactive lab · matrix

Cell shape and area are separate properties

Reset and set BOTH Diagonal scale a = 1 and Shear b = 0, then Play or Step to inspect the selected transform. For a fresh run, Reset and set a = 1 and b = 1; predict the same determinant 1 despite shear. Reset geometry starts at identity; determinant readouts always describe the selected final matrix while playback interpolates toward it.

Interpolated geometryOriginal unit squareSelected final transform

Area scale · det A: 1.4
Singular values · σmax, σmin: 1.597, 0.876
Condition number · κ₂(A): 1.823

Diagonal scale a1.4Shear b0.6

t = 0 s

The selected matrix is A = [[a,b],[0,1]]. Playback interpolates geometry between the identity and A; readouts always describe A. Its determinant is a and its singular values come from AᵀA. Positive a makes every supported matrix invertible. This two-dimensional family is not a general eigensolver or a learned network. Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forThe triangular matrix family illustrates planar cell deformation. It does not construct a Bravais lattice, identify a primitive basis, or count atoms.

Pause & reason

Check your understanding

Interactive checks need JavaScript. You can still read the explanation.

Read the checkpoint explanation

Primitivity counts lattice points, not necessarily atoms. A multi-atom motif can occupy a primitive cell, and its vectors need not be orthogonal or form a cube.

Your turn

Try it yourself

An identical-atom bcc solid has conventional cubic edge $a_c=4\,\mathrm{\mathring A}$ . Count atoms in that cube, find the primitive volume, and calculate its atomic number density.

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

Each corner is shared by eight cubes, while the central atom belongs to one cube. The primitive bcc motif contains one atom.

Compare with the solution

The conventional count is $8(1/8)+1=2$ atoms. Its volume is $4^3=64\,\mathrm{\mathring A^3}$ , so the primitive volume is $32\,\mathrm{\mathring A^3}$ . The density is $1/32\,\mathrm{\mathring A^{-3}}$ , equivalently $2/64$ from the conventional count. Using one atom with the full conventional volume would underestimate density by a factor of two. Chemical inequivalence of the central and corner atoms would change the primitive translation analysis.

Keep the idea

Take it with you

A Bravais lattice encodes translations; the basis encodes the repeated atomic motif.
Primitive volume and motif count must be used consistently when computing density.
Cell enlargement changes representation, while lost translations change physical symmetry.

Read further

Original sources and deeper explanations to follow at your own pace.

Optional progress, saved in this browser when available. Revisit any lesson at any time.

Lesson not marked complete.