The Direct (or Real) Space Lattice
A lattice is an infinite array of points in space in which every lattice point has the same environment. Because the array is periodic, every point can be reached from the origin by a vector.
Here \(\mathbf a_1\), \(\mathbf a_2\) and \(\mathbf a_3\) are non-collinear vectors pointing in three different directions. They are called the basis vectors. If the point P sits at \((u_1, u_2, u_3)\), the point Q at \((2u_1, 2u_2, 2u_3)\) is reached by \(2\mathbf t\), which also lies on the lattice.
Written compactly, with the implied summation (Einstein) convention, a repeated index is summed over:
There is an infinite number of possible choices of basis vectors. In practice, we choose them to make use of the symmetry of the lattice. Try the different basis choices in the builder: every primitive choice reaches every point with integer coefficients; a non-primitive choice leaves some points needing fractions.
Lattice Vector Builder
Walking to a Lattice Point in 3D
A triclinic lattice (no two edges equal, no right angles). Choose \(u_1, u_2, u_3\) and watch \(\mathbf t\) assemble tip-to-tail: \(u_1\) steps along \(\mathbf a_1\), then \(u_2\) along \(\mathbf a_2\), then \(u_3\) along \(\mathbf a_3\).
Essentially, Vector Algebra
Once every lattice point is a vector, the everyday questions of crystallography (which way does it point? how far apart? at what angle?) become questions about vectors.
Direction \([uvw]\)
The direction of \(\mathbf t\) is written \([uvw]\): the smallest integers proportional to the components of \(\mathbf t\). These are the Miller indices for directions.
\(\mathbf t=\mathbf a_1+2\mathbf a_2+3\mathbf a_3 \Rightarrow [123]\)
\(\mathbf t=-0.3\mathbf a_1+0.4\mathbf a_2+0.5\mathbf a_3 \Rightarrow [\bar 3 45]\)
Distance
The distance between the points \(\mathbf t_1\) and \(\mathbf t_2\):
True in both Cartesian and non-Cartesian frames of reference.
Angle
From \(\mathbf t_1\cdot\mathbf t_2 = |\mathbf t_1||\mathbf t_2|\cos\gamma\), with \(\gamma\) the angle between the vectors:
Direction-Index Reducer
Type the components of \(\mathbf t\) along \(\mathbf a_1, \mathbf a_2, \mathbf a_3\). Decimals and fractions (e.g. -0.3, 1/2) are fine.
Distance & Angle in Any Cell
In a non-Cartesian cell the dot product must remember that the basis vectors are neither unit length nor perpendicular. Expanding \(\mathbf t_1\cdot\mathbf t_2 = x_i y_j\,(\mathbf a_i\cdot\mathbf a_j)\) collects the nine products \(\mathbf a_i\cdot\mathbf a_j\) into the metric tensor \(G\), so \(\mathbf t_1\cdot\mathbf t_2 = \mathbf x^{\mathsf T} G\,\mathbf y\). The red boxes show the answer you would get by wrongly treating the axes as perpendicular, Cartesian-style.
Miller Indices for Planes
A lattice plane may be formed by any three non-collinear lattice points. How do we identify it? The recipe below works through the plane that cuts the axes at \(2/3\), \(7/4\) and \(3/5\).
Such large indices are very uncommon in practice. What matters is the principle.
Miller Index Machine
Enter the intercepts along \(\mathbf a_1, \mathbf a_2, \mathbf a_3\) (fractions, negatives or inf for a plane parallel to that axis).
Families of Directions and Planes
In a cube, [100], [010], [001] and their negatives \([\bar100]\), \([0\bar10]\), \([00\bar1]\) are symmetrically equivalent. The whole set is a family, written with angle brackets: \(\langle100\rangle\).
For a cube, the members of a family are obtained from all permutations of the indices, including negative signs. So \(\langle110\rangle\) for a cube stands for [110], [101], [011], \([\bar110]\), \([1\bar10]\), \([\bar1\bar10]\), \([\bar101]\), \([10\bar1]\), \([0\bar11]\), \([01\bar1]\), \([\bar10\bar1]\), \([0\bar1\bar1]\): twelve directions.
Similarly, in a cubic cell the planes (100), (010) and (001) (with their negatives) are symmetrically equivalent, and form the family \(\{100\}\).
The “permute everything” rule is a property of cubic symmetry. Switch the explorer to tetragonal or orthorhombic and watch the family break up.
QWhat would be the family of directions \(\langle110\rangle\) for a tetragonal unit cell?
QWhat would be the family of planes \(\{100\}\) for a tetragonal unit cell?
Family Explorer
(hkl) Names a Family of Parallel Planes
Consider a cube with basis vectors \(\mathbf a_1, \mathbf a_2, \mathbf a_3\). Click each coloured plane (or its button) and work out its Miller indices.
Symmetry-equivalent families, once more
In the cube, the green face is (010). The yellow and purple faces are (100) and (001). These are symmetrically equivalent, so the family can be written \(\{100\}\) or equally \(\{010\}\).
For a cube, the members are obtained by permuting the indices and their signs: \(\{100\}\) = (100), (010), (001), \((\bar100)\), \((0\bar10)\), \((00\bar1)\).
For a tetragonal lattice (\(c \ne a = b\)), \(\{100\}\) has only (100), (010), \((\bar100)\) and \((0\bar10)\). (001) and \((00\bar1)\) are not symmetrically equivalent to the others. You can check this in the Family Explorer above.
Two parallel planes, one \((hkl)\)
A plane \((hkl)\) cuts the axes at \(1/h, 1/k, 1/l\), so its equation is \(hx+ky+lz=1\). The parallel plane through the origin is \(hx+ky+lz=0\). Both have Miller indices \((hkl)\), and the spacing between these nearest parallel neighbours is \(d_{hkl}\): equivalently, the distance of the plane with intercepts \(1/h, 1/k, 1/l\) from the origin.
One Plane, Two Families
The grey plane cuts \(\mathbf a_2\) at \(\tfrac12\): intercepts \((\infty, \tfrac12, \infty)\) give Miller indices \((020)\). The next \((020)\) plane in the \(+\mathbf a_2\) direction is… the green \((010)\) plane! The same plane can belong to two sets of planes with different Miller indices.
QWhat is the interplanar spacing for planes with Miller indices (010)?
QAnd for (020)?
Directions and Planes in the Hexagonal System
With three axes, directions and planes that are obviously equivalent by the 6-fold symmetry get indices that are not permutations of each other. A fourth basal axis fixes this.
Three Indices ↔ Four Indices
The 3-index system is the Miller indices system; the 4-index system is the Miller–Bravais indices system. Because the three basal axes satisfy \(\mathbf a_1+\mathbf a_2+\mathbf a_3=\mathbf 0\), one of the four indices is always redundant: \(t = -(u+v)\).
Directions
Afterwards, multiply or divide by a common factor to get the smallest integers.
Planes
Simply drop the third index \(t\). (For planes the four indices are often written \((hkil)\) with \(i=-(h+k)\).)
Hexagonal Index Converter
Reciprocal Space
Miller indices \((hkl)\) are obtained by taking reciprocals of intercepts along the real-lattice basis vectors. What if \(h\), \(k\) and \(l\) were themselves the components of a vector, in a different basis?
- Reciprocal basis vectors \(\mathbf a_i^*\) defined by \(\mathbf a_i\cdot\mathbf a_j^*=\delta_{ij}\)
- Why \(\mathbf g = h\mathbf a_1^*+k\mathbf a_2^*+l\mathbf a_3^*\) is normal to the plane (hkl)
- Families of parallel planes and the spacing \(d_{hkl}=1/|\mathbf g_{hkl}|\)
- A plane in real space is a point in reciprocal space; \(d_{hkl}\) for every crystal system
Defining the Reciprocal Basis
If the real lattice is described by the basis \(\mathbf a_i\) (meaning \(\mathbf a_1, \mathbf a_2, \mathbf a_3\); bold letters are vectors, with a magnitude and a direction), consider another basis \(\mathbf a_j^*\) chosen so that
\(\delta_{ij}\) is the Kronecker delta.
In matrix form, the nine dot products make the identity matrix:
Each tile is one dot product \(\mathbf a_i\cdot\mathbf a_j^*\): rows \(i\), columns \(j\).
Constructing \(\mathbf a_i^*\)
\(\mathbf a_1^*\) stands on the face spanned by \(\mathbf a_2\) and \(\mathbf a_3\)
\(\mathbf a_1^*\) is normal to the face containing \(\mathbf a_2\) and \(\mathbf a_3\); it is generally not parallel to \(\mathbf a_1\) unless the cell has right angles.
Reciprocal Basis Builder
Deform the real cell (dark arrows) and watch the reciprocal basis (teal arrows) respond. Pick one \(\mathbf a_i^*\) to see the face it is perpendicular to. Lengths are drawn so that 1 unit of \(\mathbf a\) ↔ 1 unit\(^{-1}\) of \(\mathbf a^*\).
Lattice Planes and Reciprocal Space
Take any lattice point written in the reciprocal frame of reference, and any vector \(\mathbf r\) of real space that is perpendicular to it. The condition \(\mathbf r\cdot\mathbf g=0\) turns out to be the equation of a lattice plane.
From \(\mathbf r\cdot\mathbf g = 0\) to \((hkl)\)
Normal Finder
The grey arrow is the lattice direction \([hkl] = h\mathbf a_1+k\mathbf a_2+l\mathbf a_3\). It is tempting to assume that \([hkl]\) is perpendicular to \((hkl)\), but that holds only in cubic cells. The true normal is \(\mathbf g_{hkl}\).
Every \(\mathbf r \perp \mathbf g\) Lies in One Plane
Describe a vector \(\mathbf r\) in the real frame and a vector \(\mathbf g\) in the reciprocal frame:
with \(h, k, l\) mutually prime integers, and \(\mathbf r\perp\mathbf g\). Then \(\mathbf r\cdot\mathbf g = 0 \Rightarrow hx+ky+lz=0\), because \(\mathbf a_i\cdot\mathbf a_j^*=\delta_{ij}\).
So, for a fixed \(\mathbf g\), the set of all \(\mathbf r\) with real \(x, y, z\) satisfying \(\mathbf r\cdot\mathbf g=0\) forms a plane, the plane \(hx+ky+lz=0\). The family of planes parallel to it has Miller indices \((hkl)\).
Sweeping Out the Plane
Random real-space vectors \(\mathbf r\) (purple) are generated subject only to \(\mathbf r\cdot\mathbf g=0\). Each leaves a dot at its tip. The dots fill a plane perpendicular to \(\mathbf g\) (red), in a triclinic cell.
\(d_{hkl} = 1/|\mathbf g_{hkl}|\)
The interplanar spacing \(d_{hkl}\) is the perpendicular distance from the origin to the plane with intercepts \(1/h, 1/k, 1/l\) on the real basis vectors. Project any vector from the origin to that plane onto the unit normal.
Projecting \(\mathbf t\) on \(\hat{\mathbf n}\)
QWhat will be the d-spacing if the Miller indices are \((nh\ nk\ nl)\)?
Measuring \(d_{hkl}\)
A Plane in Real Space Is a Point in Reciprocal Space
A whole family of lattice planes \((hkl)\) collapses to one reciprocal lattice point \(hkl\): the tip of \(\mathbf g_{hkl}\). And \(\mathbf a_1^*\perp\mathbf a_2,\mathbf a_3\); \(\mathbf a_2^*\perp\mathbf a_3,\mathbf a_1\); \(\mathbf a_3^*\perp\mathbf a_1,\mathbf a_2\), whether or not the basis vectors are orthogonal. Here in 2D, looking down \(\mathbf a_3\).
After Fig. 7, Chapter 2 (Geometry of Crystals) of Elements of X-Ray Diffraction by Cullity and Stock, which shows crystal lattices and the corresponding reciprocal lattices for a cubic and a hexagonal system (there, \(\mathbf b_1, \mathbf b_2, \mathbf b_3\) denote the reciprocal basis vectors).
Interplanar Spacings for Different Crystal Systems
Evaluating \(1/d_{hkl}^2 = |\mathbf g_{hkl}|^2 = \mathbf g\cdot\mathbf g\) for each cell geometry gives these expressions. Click a row to load it into the calculator.
| Crystal system | Interplanar spacing |
|---|---|
| Cubic | \[d_{hkl}=\frac{1}{\sqrt{\dfrac{h^2}{a^2}+\dfrac{k^2}{a^2}+\dfrac{l^2}{a^2}}}=\frac{a}{\sqrt{h^2+k^2+l^2}}\] |
| Tetragonal | \[d_{hkl}=\frac{1}{\sqrt{\dfrac{h^2}{a^2}+\dfrac{k^2}{a^2}+\dfrac{l^2}{c^2}}}\] |
| Orthorhombic | \[d_{hkl}=\frac{1}{\sqrt{\dfrac{h^2}{a^2}+\dfrac{k^2}{b^2}+\dfrac{l^2}{c^2}}}\] |
| General form valid for all crystal systems with cell edges \(a, b, c\) and angles \(\alpha, \beta, \gamma\) |
\[\frac{1}{d_{hkl}^2}=\frac{1}{V^2}\Big[h^2b^2c^2\sin^2\alpha+k^2a^2c^2\sin^2\beta+l^2a^2b^2\sin^2\gamma+2hkabc^2(\cos\alpha\cos\beta-\cos\gamma)\]
\[\qquad+\,2kla^2bc(\cos\beta\cos\gamma-\cos\alpha)+2hlab^2c(\cos\gamma\cos\alpha-\cos\beta)\Big]\]
where \(V\) is the unit-cell volume:
\[V=(\mathbf a\times\mathbf b)\cdot\mathbf c = abc\sqrt{1-\cos^2\alpha-\cos^2\beta-\cos^2\gamma+2\cos\alpha\cos\beta\cos\gamma}\] |
d-Spacing Calculator
Choose a system; only the independent lattice parameters are editable. \(d\) is computed from the general formula and cross-checked as \(1/|\mathbf g|\).
Each stick is one set of equivalent planes placed at \(2\theta = 2\sin^{-1}(\lambda/2d)\) (Bragg's law, coming up in the diffraction lectures). Height = number of symmetry-equivalent \((hkl)\) with that spacing, not a diffracted intensity; systematic absences are ignored. Hover a stick, click to select it.
Did It Sink In? - 14
Test your mastery of lattice vectors, Miller and Miller–Bravais indices, families of directions and planes, the reciprocal basis, and interplanar spacings.