The dual lattice, as a construction

Every lattice has a second lattice attached to it, and the usual introduction gives it as a formula — divide by the cell area, rotate by a quarter turn — which is correct and explains nothing. The construction underneath is geometric and takes one sentence: each family of lattice rows contributes one point, perpendicular to the rows, at the inverse of their spacing.

Doing it that way makes the properties fall out rather than having to be remembered, and it makes the object’s role obvious: a diffraction experiment measures directions in which rows of atoms reinforce, so the natural bookkeeping for an experiment is a lattice of directions-with-spacings, which is exactly what the construction produces.

Rows, and how they are named #

A family of lattice rows is a set of parallel lines that between them contain every lattice point, evenly spaced. Every lattice has infinitely many such families, and they are named by how the first line away from the origin cuts the axes.

The family labelled $(h\,k)$ is the one whose lines cut the first axis into $h$ parts and the second into $k$. So $(1\,0)$ is the family running along the second axis, one cell apart; $(1\,1)$ runs diagonally and is spaced more closely; $(2\,1)$ more closely still. Larger indices mean lines packed tighter together, which is the first thing the construction inverts.

Those labels are the Miller indices, and in three dimensions they are the language everything in crystallography is written in. The convention that they name reciprocals of intercepts rather than intercepts is what makes the labelling work for families parallel to an axis, where the intercept would be infinite and the index is simply zero.

The construction, and why it inverts #

Take the family $(h\,k)$ and measure the perpendicular distance $d$ between neighbouring lines. Draw a vector perpendicular to the family with length $1/d$. That vector is the family’s contribution to the dual lattice, and every point of the dual lattice arises this way from exactly one family.

The inversion is not a convention; it is what makes the object useful. Closely spaced rows — high indices — give long dual vectors, and widely spaced rows give short ones. So a crystal with a long axis produces a dual lattice with closely spaced points along that direction, and a diffraction pattern is stretched the opposite way to the crystal that made it.

The claim that the geometric construction agrees with the algebraic definition $\mathbf{b}_i \cdot \mathbf{a}_j = \delta_{ij}$ is not obvious and is checked rather than asserted. For each family drawn, the figure computes the spacing from the geometry, builds the vector of length $1/d$ perpendicular to the rows, computes $h\mathbf{b}_1 + k\mathbf{b}_2$ from the reciprocal basis, and requires the two to agree to within a rounding error. Two routes to the same point, sharing only the original lattice.

What the construction makes obvious #

Four properties, each of which is a memorised fact in the formula presentation and a consequence here.

The dual of the dual is the original. Rows in the dual lattice are perpendicular to vectors in the original, with spacings that invert back. Applying the construction twice returns where it started, exactly.

The dual has the same point symmetry. Any operation mapping the lattice to itself permutes its families of rows and preserves spacings, so it permutes the dual points and preserves their lengths. The two lattices have the same holohedry — which is why the five plane lattice types are a classification of both at once.

Long becomes short. Stated above, and the reason a needle-shaped cell gives a plate-shaped diffraction pattern.

A dense direction is a strong direction. Families with small indices have widely spaced, densely populated rows, and those are the reflections a crystal scatters most strongly. The correlation between low indices and strong reflections that every crystallographer relies on is a statement about how many atoms lie in a row.

The spacings, worked out #

The construction is worth running on numbers once, because the pattern in them is the whole of what a diffraction pattern shows.

Take a rectangular lattice with axes of length $a$ and $b$. The family $(1\,0)$ consists of lines parallel to the second axis, one $a$ apart, so $d_{10} = a$ and its dual vector has length $1/a$. The family $(0\,1)$ gives $1/b$. The family $(1\,1)$ runs diagonally, and its spacing works out as

$$d_{11} = \frac{1}{\sqrt{1/a^2 + 1/b^2}}$$

which is smaller than either $a$ or $b$ — closer-packed rows — so its dual vector is longer than either of the first two. The general formula for a rectangular lattice is $1/d_{hk}^2 = h^2/a^2 + k^2/b^2$, and reading it as Pythagoras in the dual lattice is the whole content: the dual vector of $(h\,k)$ is $h$ steps along one dual axis and $k$ along the other.

Two things follow immediately and are worth having.

Indices grow, spacings shrink. The families available at any given resolution are those with $1/d$ below a limit, which is a disc in the dual lattice. That disc is what an experiment can reach, and improving the resolution of a measurement means enlarging it.

The spacings alone do not determine the indices. Two different families can have the same spacing — on a square lattice, $(5\,0)$ and $(4\,3)$ both give $1/d^2 = 25/a^2$ — and no measurement of spacing alone can separate them. That coincidence is the seed of everything a powder pattern loses.

The dual of a centred lattice #

The construction has one consequence that surprises people who met the formula first, and it is a good test of whether the idea has been absorbed.

The dual of a centred rectangular lattice is a centred rectangular lattice — but the centring is in the other setting. Working through the families explains it without any calculation: a centred lattice’s rows include families that the underlying rectangular lattice does not have, because the centring points sit between the rectangular rows and halve their spacing. Halved spacings mean doubled dual vectors, so the dual acquires points at twice the distance in those directions and lacks the ones the uncentred dual would have had.

Which is the same statement as the systematic absence rule, read geometrically. The reflections with $h + k$ odd are missing from a centred lattice’s diffraction not because something cancels but because those dual points are not in the dual lattice at all. The structure-factor calculation and the row-spacing construction agree, and they are independent arguments.

In three dimensions the same reasoning gives the standard pairing that catches everybody once: the dual of a face-centred cubic lattice is body-centred cubic, and the dual of body-centred is face-centred. Nothing has to be memorised if the families are counted.

What the exactness rests on #

The agreement between construction and definition is exact, and it rests on a fact about the plane that is worth naming because it fails in general.

The construction assumes that the perpendicular direction to a family of rows is well defined and that “distance” means the ordinary Euclidean distance. Both are properties of the metric, not of the lattice — a lattice is a set of integer combinations and knows nothing about lengths until a metric is imposed. So the dual lattice as constructed here is really the dual of a lattice-with-a-metric, and changing the metric changes it.

That is not a pedantic point. In crystallography the metric is the cell geometry, and the whole business of refining cell parameters is the business of pinning it down. Two crystals with identical lattices in the abstract sense and different cell parameters have different reciprocal lattices, and it is the reciprocal lattice that an experiment measures.

There is a second, purely algebraic dual — the set of linear functionals taking integer values on the lattice — which needs no metric at all and is the one a number theorist means. In the presence of a metric the two coincide under the identification of a functional with the vector that represents it, and where this site says “dual” it means the metric one, because that is the one a diffraction pattern shows.

The generalisation #

The construction is dimension-agnostic and its three-dimensional form is the working tool.

A family of lattice planes with Miller indices $(h\,k\,l)$ has a spacing $d_{hkl}$, and contributes a reciprocal vector of length $1/d_{hkl}$ perpendicular to the planes. Bragg’s law, $\lambda = 2 d \sin\theta$, is then a statement about that length, and the whole of diffraction geometry becomes a construction in reciprocal space — the Ewald sphere, which turns the question “which reflections are accessible at this wavelength” into “which reciprocal points lie on a sphere of radius $1/\lambda$”.

The convention question that appears here is a genuine nuisance and worth stating. Physics generally defines the reciprocal basis with a factor of $2\pi$, so that $\mathbf{b}_i \cdot \mathbf{a}_j = 2\pi\delta_{ij}$ and plane waves come out as $e^{i\mathbf{k}\cdot\mathbf{r}}$ without extra factors. Crystallography generally omits it, so that reciprocal lengths are literally inverse spacings and Bragg’s law has no $\pi$ in it. Neither is wrong; formulae copied between the two conventions are, routinely. This site uses the crystallographic convention, without the $2\pi$, which is why every reciprocal length on these pages is an inverse distance and can be read straight off a spacing.

Building it the other way round #

A useful exercise, and one the figures here support directly: run the construction backwards.

Given a set of dual points — which is what a diffraction pattern is — the original lattice is recovered by the same procedure applied to them. Each dual point names a direction and a spacing; a family of rows perpendicular to it, spaced at the inverse, is a family of the original; and the original lattice is the intersection of all those families.

That is how a crystal’s cell is determined in practice, and stating it this way makes clear what could go wrong. The recovered lattice is only as good as the set of dual points supplied, and a set that is missing points — because they were systematically absent, or too weak to see, or outside the resolution limit — recovers a lattice too small: a sublattice of the true one, with a cell that is a multiple of the right one. Indexing software’s characteristic failure is exactly this, and the symptom is a cell that explains every observed reflection and predicts many that are never seen.

The surprising part #

The dual lattice is a lattice of directions with spacings, and directions with spacings are not points. Treating them as points is a choice that turns out to be extraordinarily productive, and it is worth noticing how odd it is.

A family of rows has no location — it fills the plane — so the dual point representing it is not anywhere in particular. What the construction does is take an object with no position and give it a position anyway, by using the only two numbers the family has: a direction and a spacing. Nothing forces this to be consistent, and the reason it is consistent is that combining two families in the natural way corresponds to adding their vectors, so the set of families inherits the structure of a lattice.

The connection worth carrying is that this is the same move as the Fourier transform, which takes a function of position and returns a function of frequency — also not a place. A periodic pattern’s Fourier transform is supported exactly on its dual lattice, and the construction on this page is what that statement looks like when it is done with a ruler instead of an integral. Two descriptions of one fact, and the geometric one came first by about a century.

What the dual is not #

Three misreadings, each common enough to be worth naming.

It is not a lattice of atoms. Nothing sits at a dual point. The dual lattice is a bookkeeping device for families of rows, and a diffraction pattern is a map of which of those families a crystal was able to reinforce in. Reading a diffraction photograph as a picture of the structure — spots as atoms — is the single most persistent beginner’s error in the subject, and it is encouraged by how much a diffraction pattern looks like a lattice of dots.

It is not determined by the lattice type alone. Two square lattices with different cell sizes have differently scaled duals, and the scaling is what carries the length information. The symmetry of the dual is fixed by the lattice type; its dimensions are not.

It is not a picture of the pattern’s symmetry. The dual lattice depends on the translations only. Two patterns with the same lattice and quite different groups — p1 and p4m on a square lattice, say — have identical dual lattices, and what distinguishes them is the intensity at each dual point rather than the positions. Where the dual points are is the lattice; how bright they are is the structure; which are missing is the glides and the centring.

That third one is the useful summary of what a diffraction experiment separates. Geometry, intensity, absence: three channels, carrying three different things, and each read by a different part of the argument.

Who built it #

The reciprocal lattice as an explicit construction is Ewald’s, from 1913 — his sphere construction and the reciprocal lattice were developed together, in the two years after von Laue’s diffraction experiment, and were the tools that turned diffraction from a phenomenon into a method.

The underlying idea is older and comes from Gibbs, who introduced reciprocal systems of vectors in his lectures on vector analysis in the 1880s for reasons having nothing to do with crystals. Bravais had already been describing families of lattice planes by their indices in the 1840s, and the notation crystallography uses is Miller’s, from an 1839 textbook on mineralogy.

So the object arrived in three pieces from three subjects — the indices from mineralogy, the vector algebra from thermodynamics, the geometry from the first diffraction experiments — and the construction on this page is the sequence in which they fit together rather than the sequence in which they were found.

Where the ladder goes next #

The experimental meaning of the construction is the reciprocal lattice as a place where a crystal scatters, and the information it withholds is the phase.

The description ambiguity it inherits is the choice of cell, removed by reduction.

And the reading that makes an experiment interpretable is systematic absences, where the dual points that are missing say more than the ones that are present.

What the pictures here cannot show. The families of rows drawn on the left of each figure are five out of infinitely many, chosen because they fit legibly. The dual lattice is what all of them produce, and a drawing of five families producing five points is an illustration of a construction rather than a demonstration that it generates a lattice. That it does is a consequence of the arithmetic, checked family by family and not visible in the picture.

The one number the two lattices share #

Four properties fall out of the construction and there is a fifth that is arithmetic rather than geometric, and it is the one every later calculation uses.

The cell of the dual lattice has the reciprocal area of the original’s. In three dimensions, the reciprocal volume; in n dimensions, the reciprocal of the n-dimensional volume. The relation is exact and it holds for every lattice whatever its shape.

The reason is the definition read as a matrix identity. The dual basis satisfies bᵢ · aⱼ = δᵢⱼ, which says the matrix of dual vectors is the inverse transpose of the matrix of direct ones — and a determinant of an inverse is the reciprocal of the determinant.

That gives the density statement in its most useful form. The number of dual points per unit area is the area of the direct cell, so a large cell has a dense dual and a small one a sparse dual. A crystal with a big unit cell has closely spaced reflections, and the count of them inside a fixed sphere is the volume times the sphere’s — an identity whose whole content is this one.

And it is the check that catches a description reported in the wrong space. A recovered cell whose volume is the reciprocal of the expected one has been reported in reciprocal space when direct space was wanted, or the other way round — an error that changes every number in a report and that this single comparison detects immediately.

The volume is also the only quantity a change of basis cannot touch, in either lattice, so it is the one number on which two descriptions of one crystal must agree exactly.