Reciprocal Lattice

The reciprocal lattice is the set of wavevectors whose plane-wave phase is unchanged by every direct-lattice translation. If $\Lambda$ is the direct lattice, then the physics-normalized reciprocal lattice is

$$\Lambda^* = \left\{ \mathbf G\in\mathbb R^d \;\middle|\; e^{i\mathbf G\cdot\mathbf R}=1 \text{ for every }\mathbf R\in\Lambda \right\}.$$

This definition unifies three facts that can otherwise look unrelated:

• a lattice-periodic function has Fourier components only at reciprocal-lattice vectors;
• elastic diffraction from a perfect crystal is concentrated where the wavevector transfer equals a reciprocal vector;
• translation symmetry conserves crystal momentum only modulo a reciprocal vector.

We will construct reciprocal vectors, use them to identify diffraction conditions, and distinguish them from the finer momentum mesh of a finite periodic sample. The next article uses this lattice to construct Brillouin zones.

Required background. Crystals and Lattices supplies primitive translations, cells, motifs, and cell volume.

Helpful background. Fourier Series supplies the reciprocal-mode expansion of lattice-periodic functions.

Definition From Phase Invariance

Let the direct lattice be

$$\Lambda = \left\{ \mathbf R = \sum_{i=1}^{d}n_i\mathbf a_i \;\middle|\; n_i\in\mathbb Z \right\}.$$

A plane wave changes under $\mathbf r\mapsto\mathbf r+\mathbf R$ by

$$e^{i\mathbf q\cdot(\mathbf r+\mathbf R)} = e^{i\mathbf q\cdot\mathbf r} e^{i\mathbf q\cdot\mathbf R}.$$

It is lattice-periodic precisely when

$$e^{i\mathbf q\cdot\mathbf R}=1 \qquad \text{for all }\mathbf R\in\Lambda.$$

Since $e^{ix}=1$ exactly when $x\in2\pi\mathbb Z$, the reciprocal lattice can also be written

$$\Lambda^* = \left\{ \mathbf G \;\middle|\; \mathbf G\cdot\mathbf R \in 2\pi\mathbb Z \text{ for every }\mathbf R\in\Lambda \right\}.$$

It is enough to test the primitive translations:

$$\mathbf a_i\cdot\mathbf G \in 2\pi\mathbb Z \qquad i=1,\ldots,d.$$

If this condition holds for every $\mathbf a_i$, then it holds for every integer combination $\mathbf R$. Conversely, each $\mathbf a_i$ is itself a lattice vector, so the full definition implies the primitive conditions.

Group structure

If $\mathbf G,\mathbf G'\in\Lambda^*$, then

$$e^{i(\mathbf G+\mathbf G')\cdot\mathbf R} = e^{i\mathbf G\cdot\mathbf R} e^{i\mathbf G'\cdot\mathbf R} = 1.$$

The zero vector and additive inverses also satisfy the condition. Thus $\Lambda^*$ is itself an abelian lattice under vector addition.

The definition depends on the direct translation lattice, not directly on the atomic species or motif. For fixed primitive translations, a motif changes Fourier amplitudes and can extinguish peaks. If a new decoration breaks a translation, or identical motif elements reveal a shorter period, the full translation lattice itself must first be redetermined.

Why Fourier Analysis Produces a Lattice

Let $f(\mathbf r)$ be periodic under every $\mathbf R\in\Lambda$. For a sufficiently regular periodic function, its Fourier series can be understood through the continuum transform below. Because a nonzero periodic function is not integrable over all of space, this transform is interpreted as a distribution:

$$f(\mathbf r) = \int \frac{d^dq}{(2\pi)^d} \widetilde f(\mathbf q) e^{i\mathbf q\cdot\mathbf r}.$$

Periodicity requires

$$f(\mathbf r+\mathbf R) = f(\mathbf r).$$

Every nonzero Fourier component must therefore have

$$e^{i\mathbf q\cdot\mathbf R}=1 \qquad \text{for all }\mathbf R\in\Lambda.$$

The continuum Fourier integral reduces to a reciprocal-lattice series:

$$f(\mathbf r) = \sum_{\mathbf G\in\Lambda^*} f_{\mathbf G} e^{i\mathbf G\cdot\mathbf r}.$$

With a primitive cell $\mathcal C$ of measure $\Omega_{\mathrm c}$,

$$f_{\mathbf G} = \frac{1}{\Omega_{\mathrm c}} \int_{\mathcal C} d^dr\, e^{-i\mathbf G\cdot\mathbf r} f(\mathbf r).$$

The orthogonality relation is

$$\frac{1}{\Omega_{\mathrm c}} \int_{\mathcal C} d^dr\, e^{i(\mathbf G-\mathbf G')\cdot\mathbf r} = \delta_{\mathbf G,\mathbf G'}.$$

This is ordinary Fourier-series orthogonality on the quotient space $\mathbb R^d/\Lambda$. Periodic Functions and Fourier Series supplies the general analysis; the reciprocal lattice is its crystal-geometric realization.

Reciprocal Basis Vectors

Define reciprocal primitive vectors $\mathbf b_j$ by

$$\mathbf a_i\cdot\mathbf b_j = 2\pi\delta_{ij}.$$

Every reciprocal vector is then

$$\mathbf G = \sum_{j=1}^{d}m_j\mathbf b_j, \qquad m_j\in\mathbb Z.$$

Indeed,

$$\mathbf a_i\cdot\mathbf G = 2\pi m_i.$$

Conversely, any vector satisfying the phase-invariance definition has integer coefficients in this reciprocal basis.

Matrix construction

Put the direct and reciprocal vectors into column matrices:

$$\begin{aligned} A&= \begin{pmatrix} \mathbf a_1&\cdots&\mathbf a_d \end{pmatrix}, \\ B&= \begin{pmatrix} \mathbf b_1&\cdots&\mathbf b_d \end{pmatrix}. \end{aligned}$$

The defining relation becomes

$$A^{\mathsf T}B = 2\pi I_d.$$

Therefore

$$B = 2\pi A^{-\mathsf T}.$$

This is the most reliable formula for numerical work. It handles skew cells without guessing which reciprocal vector is perpendicular to which direct vector.

Three-dimensional cross products

For a right-handed direct basis,

$$\Omega_{\mathrm c} = \mathbf a_1\cdot \bigl(\mathbf a_2\times\mathbf a_3\bigr) > 0.$$

The reciprocal basis is

$$\begin{aligned} \mathbf b_1 &= 2\pi \frac{\mathbf a_2\times\mathbf a_3} {\Omega_{\mathrm c}}, \\ \mathbf b_2 &= 2\pi \frac{\mathbf a_3\times\mathbf a_1} {\Omega_{\mathrm c}}, \\ \mathbf b_3 &= 2\pi \frac{\mathbf a_1\times\mathbf a_2} {\Omega_{\mathrm c}}. \end{aligned}$$

Each $\mathbf b_i$ is perpendicular to the plane spanned by the other two direct vectors. It is not generally parallel to $\mathbf a_i$.

Two dimensions

For a right-handed two-dimensional basis embedded in the $xy$ plane,

$$\Omega_{\mathrm c} = \bigl( \mathbf a_1\times\mathbf a_2 \bigr)\cdot\hat{\mathbf z} > 0.$$

A convenient construction is

$$\begin{aligned} \mathbf b_1 &= \frac{2\pi}{\Omega_{\mathrm c}} \mathbf a_2\times\hat{\mathbf z}, \\ \mathbf b_2 &= \frac{2\pi}{\Omega_{\mathrm c}} \hat{\mathbf z}\times\mathbf a_1. \end{aligned}$$

Reversing the ordered orientation changes the corresponding signs. The invariant check is always $A^{\mathsf T}B=2\pi I$.

Direct and Reciprocal Geometry

Direct and reciprocal lattices are dual under phase pairing; panels (a) and (b) use separate length and inverse-length scales. In panel (b), $\mathbf G_{21}=2\mathbf b_1+\mathbf b_2$. In panel (c), the reciprocal point lies on the elastic Ewald circle, so $\mathbf q=\mathbf k_f-\mathbf k_i=\mathbf G$ with $|\mathbf k_i|=|\mathbf k_f|$. The circle is two-dimensional shorthand for the Ewald sphere.

The reciprocal lattice reverses length scales. A large direct-space period produces closely spaced reciprocal points, while a short direct-space period produces a large reciprocal spacing.

Reciprocal-cell measure

Because $B=2\pi A^{-\mathsf T}$,

$$\Omega^* = \left|\det B\right| = \frac{(2\pi)^d}{\Omega_{\mathrm c}}.$$

Here $\Omega^*$ is the measure of any primitive reciprocal cell. The first Brillouin zone is one such cell, so the convention page denotes the same measure by $\Omega_{\mathrm{BZ}}$.

The units are inverse length to the power $d$. The product

$$\Omega_{\mathrm c}\Omega^* = (2\pi)^d$$

is basis independent.

Reciprocal metric

The direct metric is

$$g=A^{\mathsf T}A.$$

The reciprocal metric is

$$\begin{aligned} g^* &= B^{\mathsf T}B \\ &= (2\pi)^2g^{-1}. \end{aligned}$$

For integer index vector $\mathbf m$,

$$|\mathbf G_{\mathbf m}|^2 = \mathbf m^{\mathsf T} g^* \mathbf m.$$

This formula converts crystallographic indices into reciprocal-vector lengths for an arbitrary skew cell.

Change of primitive basis

If

$$A'=AM, \qquad M\in\operatorname{GL}(d,\mathbb Z),$$

then

$$B' = B M^{-\mathsf T}.$$

The new reciprocal vectors generate the same $\Lambda^*$ because $M^{-\mathsf T}$ is also an integer unimodular matrix. Individual basis vectors and integer coordinates change, but the reciprocal point set does not.

Reciprocal of the reciprocal

Applying the same $2\pi$-normalized construction to $B$ gives

$$2\pi B^{-\mathsf T} = A.$$

Thus

$$\bigl(\Lambda^*\bigr)^* = \Lambda$$

after identifying the double-dual Euclidean space with the original space.

Worked Examples

One-dimensional chain

For

$$\mathbf a_1=a\hat{\mathbf x},$$

the reciprocal primitive vector is

$$\mathbf b_1 = \frac{2\pi}{a}\hat{\mathbf x}.$$

Hence

$$G_m = \frac{2\pi m}{a}, \qquad m\in\mathbb Z.$$

The reciprocal spacing $2\pi/a$ is the angular spatial frequency of a pattern with period $a$.

Rectangular and square lattices

For

$$\mathbf a_1=a\hat{\mathbf x}, \qquad \mathbf a_2=c\hat{\mathbf y},$$

one finds

$$\mathbf b_1 = \frac{2\pi}{a}\hat{\mathbf x}, \qquad \mathbf b_2 = \frac{2\pi}{c}\hat{\mathbf y}.$$

The reciprocal of a rectangular lattice is rectangular. If $a=c$, both direct and reciprocal lattices are square, with reciprocal lattice constant $2\pi/a$.

Triangular lattice

Take

$$\begin{aligned} \mathbf a_1&=a(1,0),\\ \mathbf a_2&=a \left( \frac12,\frac{\sqrt3}{2} \right). \end{aligned}$$

Solving the duality equations gives

$$\begin{aligned} \mathbf b_1 &= \frac{2\pi}{a} \left( 1,-\frac{1}{\sqrt3} \right), \\ \mathbf b_2 &= \frac{2\pi}{a} \left( 0,\frac{2}{\sqrt3} \right). \end{aligned}$$

Both have length

$$|\mathbf b_1| = |\mathbf b_2| = \frac{4\pi}{\sqrt3a},$$

and their mutual angle is $120^\circ$. The reciprocal lattice is again triangular, rotated relative to the displayed direct basis.

Simple cubic lattice

For $\mathbf a_i=a\hat{\mathbf e}_i$,

$$\mathbf b_i = \frac{2\pi}{a}\hat{\mathbf e}_i.$$

The reciprocal lattice is simple cubic. A reciprocal vector indexed by $(h,k,l)$ is

$$\mathbf G_{hkl} = \frac{2\pi}{a} \bigl( h\hat{\mathbf x} + k\hat{\mathbf y} + l\hat{\mathbf z} \bigr).$$

Face-centered and body-centered duality

Use primitive vectors for a face-centered cubic direct lattice with conventional cubic parameter $a$:

$$\begin{aligned} \mathbf a_1&=\frac a2(0,1,1),\\ \mathbf a_2&=\frac a2(1,0,1),\\ \mathbf a_3&=\frac a2(1,1,0). \end{aligned}$$

The reciprocal primitive vectors are

$$\begin{aligned} \mathbf b_1&=\frac{2\pi}{a}(-1,1,1),\\ \mathbf b_2&=\frac{2\pi}{a}(1,-1,1),\\ \mathbf b_3&=\frac{2\pi}{a}(1,1,-1). \end{aligned}$$

These generate a body-centered cubic reciprocal lattice with conventional reciprocal parameter $4\pi/a$. Conversely, the reciprocal of a body-centered cubic direct lattice is face-centered cubic.

The statement concerns primitive translation lattices. If one starts from a nonprimitive conventional cubic cell, extra Fourier-grid points appear and centering factors extinguish the ones that are not in the true reciprocal lattice.

Lattice Planes and Miller Indices

For integer indices $(h,k,l)$, define

$$\mathbf G_{hkl} = h\mathbf b_1 + k\mathbf b_2 + l\mathbf b_3.$$

The surfaces

$$\mathbf G_{hkl}\cdot\mathbf r = 2\pi n, \qquad n\in\mathbb Z,$$

are parallel planes normal to $\mathbf G_{hkl}$. Every direct-lattice point lies on one of these planes because

$$\mathbf G_{hkl}\cdot\mathbf R = 2\pi \bigl( hn_1+kn_2+ln_3 \bigr).$$

For coprime $h,k,l$, adjacent planes have separation

$$d_{hkl} = \frac{2\pi}{|\mathbf G_{hkl}|}.$$

To see this, move by $\Delta\mathbf r=d\,\widehat{\mathbf G}$ between adjacent planes:

$$\mathbf G\cdot\Delta\mathbf r = |\mathbf G|d = 2\pi.$$

For a simple-cubic lattice,

$$d_{hkl} = \frac{a} {\sqrt{h^2+k^2+l^2}}.$$

Miller indices depend on the chosen crystallographic basis, while the physical plane normal and spacing do not. A bar over an index denotes a negative integer, such as $(1\bar10)$.

The 2π Convention

Two reciprocal-space conventions are standard.

Context	Duality relation	Plane-wave phase
condensed-matter physics	$\mathbf a_i\cdot\mathbf b_j=2\pi\delta_{ij}$	$e^{i\mathbf G\cdot\mathbf r}$
much of crystallography	$\mathbf a_i\cdot\mathbf a_j^*=\delta_{ij}$	$e^{2\pi i\mathbf g\cdot\mathbf r}$

They are related by

$$\mathbf b_i = 2\pi\mathbf a_i^*, \qquad \mathbf G = 2\pi\mathbf g.$$

In fractional coordinates $\mathbf r=A\boldsymbol\xi$,

$$e^{i\mathbf G_{hkl}\cdot\mathbf r} = e^{2\pi i \left( h\xi_1+k\xi_2+l\xi_3 \right)}.$$

Here we use the physics convention. A quoted reciprocal coordinate in inverse ångströms must distinguish cycles per length from radians per length; the factor $2\pi$ connects them. The conventions reference collects the related Fourier and state-normalization choices.

Diffraction Condition

Suppose a probe scatters from an ideal crystal. Let $\mathbf k_i$ and $\mathbf k_f$ be incident and outgoing wavevectors, and define

$$\mathbf q = \mathbf k_f-\mathbf k_i.$$

With this sign convention, $\hbar\mathbf q$ is the probe’s momentum change; momentum conservation assigns $-\hbar\mathbf q$ to the sample when no other reservoir carries momentum. Some diffraction texts instead define the sample transfer as $\mathbf Q=\mathbf k_i-\mathbf k_f=-\mathbf q$. The sign choice does not change the reciprocal-lattice selection rule because $\Lambda^*$ is closed under $\mathbf G\mapsto-\mathbf G$. A scattering amplitude from equivalent cells contains the lattice sum

$$S_\Lambda(\mathbf q) = \sum_{\mathbf R\in\Lambda} e^{-i\mathbf q\cdot\mathbf R}.$$

For an infinite lattice, Poisson summation gives

$$\sum_{\mathbf R\in\Lambda} e^{-i\mathbf q\cdot\mathbf R} = \frac{(2\pi)^d}{\Omega_{\mathrm c}} \sum_{\mathbf G\in\Lambda^*} \delta^{(d)} \bigl( \mathbf q-\mathbf G \bigr).$$

Thus the coherent lattice contribution is concentrated at

$$\mathbf q=\mathbf G.$$

Because $\Lambda^*$ is closed under sign reversal, a source using $\mathbf q=\mathbf k_i-\mathbf k_f$ obtains the equivalent condition $\mathbf q=\mathbf G$ with relabeled $\mathbf G$.

The infinite-lattice identity is distributional. To calculate an intensity, begin with a finite sample rather than square a delta distribution. For a chain of $N$ equally spaced scatterers,

$$\begin{aligned} S_N(q)&=\sum_{j=0}^{N-1}e^{-iqaj} =e^{-iqa(N-1)/2}\frac{\sin(Nqa/2)}{\sin(qa/2)},\\ I_N(q)&\propto |S_N(q)|^2. \end{aligned}$$

At a reciprocal point $q=2\pi m/a$, the removable limit is $S_N=N$. The peak height is proportional to $N^2$, while the first zeros lie at offsets $\pm2\pi/(Na)$. Increasing the sample size makes the coherent peak taller and narrower. Its integrated weight over a reciprocal period grows as $N$, consistent with the Fourier orthogonality sum rule. Disorder, strain, domains and instrumental resolution can change this ideal shape; thermal motion also redistributes intensity into diffuse and inelastic scattering.

Ewald construction

For elastic scattering,

$$|\mathbf k_i| = |\mathbf k_f| = k.$$

Place the reciprocal-lattice origin at the tip of $\mathbf k_i$ when the tail of $\mathbf k_i$ is at the Ewald-sphere center. A reciprocal point produces an elastic reflection when it lies on the sphere of radius $k$. The vector from the reciprocal origin to that point is

$$\mathbf G = \mathbf k_f-\mathbf k_i.$$

The construction encodes both energy conservation and the reciprocal-lattice selection rule.

Bragg’s law

Let $\theta$ be the angle between the incident beam and a family of planes with spacing $d$. The scattering angle is $2\theta$, so elastic geometry gives

$$|\mathbf q| = 2k\sin\theta.$$

For the $n$th reciprocal harmonic normal to those planes,

$$|\mathbf G_n| = n\frac{2\pi}{d}.$$

Using $k=2\pi/\lambda$ and $\mathbf q=\mathbf G_n$ yields

$$2d\sin\theta = n\lambda.$$

The Laue condition $\mathbf q=\mathbf G$ and Bragg’s law are not competing theories. They are the same elastic interference condition written in vector and plane-spacing language.

Lattice Peaks Versus Motif Intensities

Let the ideal density be assembled from motif functions:

$$\rho(\mathbf r) = \sum_{\mathbf R,\alpha} f_\alpha \bigl( \mathbf r-\mathbf R-\boldsymbol\tau_\alpha \bigr).$$

Its Fourier transform factorizes:

$$\widetilde\rho(\mathbf q) = \underbrace{ \sum_{\mathbf R} e^{-i\mathbf q\cdot\mathbf R} }_{\text{lattice factor}} \underbrace{ \sum_\alpha f_\alpha(\mathbf q) e^{-i\mathbf q\cdot\boldsymbol\tau_\alpha} }_{F(\mathbf q)}.$$

The density factorization is exact for the repeated motif above. In the kinematic, single-scattering approximation, it also gives the corresponding amplitude factorization: the lattice factor selects peak positions and $F(\mathbf G)$ controls motif interference. Strong multiple scattering requires a dynamical diffraction treatment. The measured intensity is proportional to $|F(\mathbf G)|^2$ only after the probe coupling, polarization, kinematics, multiplicity, absorption, motion, and instrument response are handled.

Centering extinction example

Represent a body-centered cubic lattice using a conventional simple-cubic cell with motif positions

$$\boldsymbol\tau_1=\mathbf0, \qquad \boldsymbol\tau_2 = \frac a2(1,1,1).$$

For identical scatterers,

$$\begin{aligned} F_{hkl} &\propto 1+ e^{-i\pi(h+k+l)} \\ &= 1+(-1)^{h+k+l}. \end{aligned}$$

The conventional-grid reflection vanishes when $h+k+l$ is odd. The surviving points, with even index sum, form the face-centered reciprocal lattice expected from the primitive bcc translation group.

Thus some apparent extinctions remove the extra grid points introduced by a nonprimitive cell; others result from interference within the actual motif. Structure factors develops the correlation-function description of scattering.

Momentum Conservation Modulo a Reciprocal Vector

Consider a periodic potential

$$V(\mathbf r) = \sum_{\mathbf G} V_{\mathbf G} e^{i\mathbf G\cdot\mathbf r}.$$

Between continuum plane waves,

$$\langle\mathbf k'|V|\mathbf k\rangle \propto \sum_{\mathbf G} V_{\mathbf G} \delta^{(d)} \bigl( \mathbf k'-\mathbf k-\mathbf G \bigr).$$

The periodic potential couples wavevectors satisfying

$$\mathbf k' = \mathbf k+\mathbf G.$$

This is the simplest appearance of momentum conservation modulo the reciprocal lattice. A lattice translation cannot distinguish $\mathbf k$ from $\mathbf k+\mathbf G$ because

$$e^{i(\mathbf k+\mathbf G)\cdot\mathbf R} = e^{i\mathbf k\cdot\mathbf R}.$$

Bloch theory turns these equivalence classes into crystal-momentum labels.

Many-particle selection rule

A translation-invariant lattice interaction can scatter modes subject to

$$\mathbf k_1+\mathbf k_2 = \mathbf k_3+\mathbf k_4+\mathbf G.$$

The $\mathbf G=\mathbf0$ representation is often called a normal process. A process represented with nonzero $\mathbf G$ is called Umklapp after all external labels have been placed in a chosen Brillouin zone.

That classification depends on the zone representatives, but its transport consequences are real. Umklapp allows the crystal-momentum carried by a selected low-energy sector to change by a reciprocal vector, so a current need not be protected as if continuous mechanical momentum were conserved within that sector.

Wavevector is not mechanical momentum

$\hbar\mathbf k$ is the generator label associated with discrete translations. It is not generally the expectation value of the mechanical momentum operator. In a full isolated system, actual momentum accounting includes ions, fields, boundaries, and the apparatus. “The lattice absorbs $\hbar\mathbf G$” is useful shorthand only within a stated effective description.

Reciprocal Lattice Versus Finite Momentum Mesh

The reciprocal lattice and the allowed $\mathbf k$ points of a finite periodic sample are different sets.

For $N_i$ cells along $\mathbf a_i$, Born–von Karman boundary conditions require

$$e^{i\mathbf k\cdot N_i\mathbf a_i} = 1.$$

Allowed wavevectors can be chosen as

$$\mathbf k_{\mathbf m} = \sum_{i=1}^{d} \frac{m_i}{N_i} \mathbf b_i, \qquad m_i=0,\ldots,N_i-1.$$

There are

$$N_{\mathrm c} = \prod_iN_i$$

inequivalent points modulo $\Lambda^*$. Their spacing shrinks as the sample grows. By contrast, the reciprocal vectors $\mathbf G=\sum_i\ell_i\mathbf b_i$ are fixed by the primitive direct lattice and identify equivalent translation characters.

The finite orthogonality relation is

$$\frac{1}{N_{\mathrm c}} \sum_{\mathbf R} e^{i(\mathbf k-\mathbf k')\cdot\mathbf R} = \delta_{\mathbf k,\mathbf k'\bmod\Lambda^*}.$$

For a supercell matrix $S$ with

$$A_{\mathrm s}=AS,$$

the supercell reciprocal basis is

$$B_{\mathrm s} = B S^{-\mathsf T}.$$

The larger real-space supercell produces a finer reciprocal grid. The finite $\mathbf k$ mesh can be viewed as representatives of the quotient between the supercell reciprocal lattice and the primitive reciprocal lattice.

Constructing a reciprocal lattice numerically

With direct vectors stored as the columns of $A$, solve the linear system $A^{\mathsf T}B=2\pi I$. Check that the cell-volume product is $(2\pi)^d$. Sorting reciprocal points by $\mathbf m^{\mathsf T}B^{\mathsf T}B\mathbf m$ then identifies the shortest vectors without assuming an orthogonal cell.

Integer coordinates are often safer identifiers than rounded Cartesian coordinates. If the basis is nearly singular, small coordinate errors can cause large reciprocal-vector errors; check the conditioning and units before interpreting the result as an extreme physical anisotropy.

Exercises

Exercise 1: basis covariance

Let $A'=AM$ with $M\in\operatorname{GL}(d,\mathbb Z)$. Derive the reciprocal transformation and the transformation of integer reciprocal coordinates.

Solution

The new reciprocal matrix is

$$\begin{aligned} B' &= 2\pi(A')^{-\mathsf T} \\ &= 2\pi(AM)^{-\mathsf T} \\ &= B M^{-\mathsf T}. \end{aligned}$$

For the same geometric reciprocal vector,

$$\mathbf G = B\mathbf m = B'\mathbf m' = B M^{-\mathsf T}\mathbf m'.$$

Therefore

$$\mathbf m' = M^{\mathsf T}\mathbf m.$$

Because $M$ is unimodular, $\mathbf m'$ remains integral. The Cartesian vector and reciprocal lattice are unchanged even though their indexed coordinates differ.

Exercise 2: finite-lattice orthogonality

For a one-dimensional ring with $N$ cells, $R_n=na$ and $k_m=2\pi m/(Na)$. Show

$$\frac1N \sum_{n=0}^{N-1} e^{i(k_m-k_{m'})R_n} = \delta_{m,m'\bmod N}.$$

Solution

The sum is

$$\frac1N \sum_{n=0}^{N-1} \exp \left[ \frac{2\pi i}{N} (m-m')n \right].$$

If $m-m'$ is divisible by $N$, every term is one and the result is one. Otherwise it is a geometric series with ratio

$$z = e^{2\pi i(m-m')/N} \neq1$$

but $z^N=1$. Hence

$$\sum_{n=0}^{N-1}z^n = \frac{1-z^N}{1-z} = 0.$$

This finite Fourier orthogonality explains both the $N$ distinct momentum labels and their equivalence modulo $2\pi/a$.

Exercise 3: finite coherent peak weight

Show that $\int_{-\pi/a}^{\pi/a}|S_N(q)|^2\,dq=2\pi N/a$, even though the peak height is $N^2$.

Solution

Expand $|S_N|^2=\sum_{j,l}e^{-iqa(j-l)}$. Integration over one reciprocal period annihilates every term with $j\ne l$ and gives $2\pi/a$ for each of the $N$ diagonal terms. The narrow peaks concentrate a total weight proportional to $N$; their growing height does not violate that count.

Exercise 4: changing the motif changes the translations

Replace the identical scatterers of the body-centered example by distinct species with amplitudes $f_A$ and $f_B$. Find the even- and odd-index amplitudes. Is the decorated structure still a body-centered Bravais lattice?

Solution

$F_{hkl}=f_A+(-1)^{h+k+l}f_B$. Even sums give $f_A+f_B$; odd sums give $f_A-f_B$, which need not vanish. The half-body-diagonal shift exchanges different species and is no longer a pure symmetry translation. The translation lattice of this decorated structure is simple cubic with a two-species motif, as in the ideal CsCl geometry. Equal scattering amplitudes at one probe condition can hide that distinction; equal amplitudes alone do not make the species physically identical.

Continue with Brillouin zones to select inequivalent wavevectors, or X-ray scattering to connect the geometric condition to a particular probe.

References

• J. Als-Nielsen and D. McMorrow, Elements of Modern X-Ray Physics, 2nd ed. (Wiley, 2011), Chapters 1–3.
• N. W. Ashcroft and N. D. Mermin, Solid State Physics (Holt, Rinehart and Winston, 1976), Chapters 5–7.
• International Union of Crystallography, International Tables for Crystallography, especially Volumes A and C.
• B. Souvignier, “A general introduction to space groups,” International Tables for Crystallography A, Section 1.3.2.5 (2016), doi:10.1107/97809553602060000921. The International Tables use the no-$2\pi$ crystallographic reciprocal basis.
• G. L. Squires, Introduction to the Theory of Thermal Neutron Scattering, 3rd ed. (Cambridge University Press, 2012), Chapters 1–3.