2  Reciprocal lattice

2.1 The dual lattice¹

The dual lattice offers a formal Fourier-space representation of the Bravais lattice, mapping real-space periodicity onto a discrete set of vectors in reciprocal space. This mapping is a fundamental ingredient for crystal theory and provides a complementary view on periodicity.

Definition 2.1: Dual primitive vectors and lattice

Given a Bravais lattice \(\mathcal{BL}\) with primitive vectors \(\{\mathbf{t}_1, \mathbf{t}_2, \mathbf{t}_3\}\), we define the dual (reciprocal) primitive vectors \(\{\mathbf{g}_1, \mathbf{g}_2, \mathbf{g}_3\}\) by the orthonormality condition:

\[\mathbf{g}_i \cdot \mathbf{t}_j = 2\pi\,\delta_{ij}. \tag{2.1}\]

The dual (reciprocal) lattice \(\mathcal{RL}\) is the Bravais lattice generated by \(\{\mathbf{g}_1, \mathbf{g}_2, \mathbf{g}_3\}\):

\[\mathcal{RL} = \left\{\mathbf{g}_\mathbf{m} = m_1\mathbf{g}_1 + m_2\mathbf{g}_2 + m_3\mathbf{g}_3 \mid m_1,m_2,m_3 \in \mathbb{Z}\right\}. \tag{2.2}\]

It is convenient to collect the primitive vectors as columns into matrices:

\[\mathcal{T} = \begin{pmatrix} t_{1x} & t_{2x} & t_{3x} \\ t_{1y} & t_{2y} & t_{3y} \\ t_{1z} & t_{2z} & t_{3z} \end{pmatrix}, \qquad \mathcal{G} = \begin{pmatrix} g_{1x} & g_{2x} & g_{3x} \\ g_{1y} & g_{2y} & g_{3y} \\ g_{1z} & g_{2z} & g_{3z} \end{pmatrix}. \tag{2.3}\]

The duality condition Equation 2.1 then reads \(\mathcal{G}^{\top}\mathcal{T} = 2\pi\, \mathbb{I}\), so that

\[\mathcal{G} = 2\pi\cdot(\mathcal{T}^{-1})^{\!\top}. \tag{2.4}\]

In 3D the following explicit construction is possible (note the cyclic symmetry of the three expressions)

\[\begin{cases} \mathbf{g}_1 = \cfrac{2\pi}{\Omega}\,\mathbf{t}_2 \times \mathbf{t}_3,\\ \mathbf{g}_2 = \cfrac{2\pi}{\Omega}\,\mathbf{t}_3 \times \mathbf{t}_1,\\ \mathbf{g}_3 = \cfrac{2\pi}{\Omega}\,\mathbf{t}_1 \times \mathbf{t}_2. \end{cases} \tag{2.5}\]

where we recall \(\Omega = |\mathbf{t}_1\cdot(\mathbf{t}_2\times\mathbf{t}_3)|=|\det\mathcal{T}|\).

TheoremKey facts on reciprocal lattices

• The dual lattice is itself a Bravais lattice.
• The dual of the dual lattice is the original direct lattice: \(\mathcal{T}=2\pi\cdot(\mathcal{G}^{\!\top})^{-1}\).
• The volume of the dual primitive cell is \(\Omega_k = |\det\mathcal{G}| = (2\pi)^3 / \Omega\).

2.1.1 Common lattices

The cross-product formula Equation 2.5 allows us to quickly compute the dual lattice for any Bravais lattice. We present here the most important examples.

FCC direct lattice → BCC dual lattice. The conventional primitive vectors for FCC are

\[\begin{cases} \mathbf{t}_1 = a\,(0,1,1)/2,\\ \mathbf{t}_2 = a\,(1,0,1)/2,\\ \mathbf{t}_3 = a\,(1,1,0)/2, \end{cases}\]

where \(a\) is the conventional cube side. The triple product is \(\mathbf{t}_1\cdot(\mathbf{t}_2\times\mathbf{t}_3) = a^3/4\). Applying Equation 2.5 we obtain

\[\begin{cases} \mathbf{g}_1 = \cfrac{2\pi}{a}\,(-1,+1,+1),\\ \mathbf{g}_2 = \cfrac{2\pi}{a}\,(+1,-1,+1),\\ \mathbf{g}_3 = \cfrac{2\pi}{a}\,(+1,+1,-1). \end{cases}\]

These are the primitive vectors of a BCC lattice with conventional cube side \(4\pi/a\) (not \(2\pi/a\) as one might naively extrapolate from the 1D case!).

BCC direct lattice → FCC dual lattice. The conventional primitive vectors for BCC are

\[\begin{cases} \mathbf{t}_1 = a\,(-1,+1,+1)/2,\\ \mathbf{t}_2 = a\,(+1,-1,+1)/2,\\ \mathbf{t}_3 = a\,(+1,+1,-1)/2, \end{cases}\]

The dual primitive vectors are:

\[\begin{cases} \mathbf{g}_1 = \cfrac{2\pi}{a}\,(0,1,1),\\ \mathbf{g}_2 = \cfrac{2\pi}{a}\,(1,0,1),\\ \mathbf{g}_3 = \cfrac{2\pi}{a}\,(1,1,0). \end{cases}\]

These are the primitive vectors of an FCC lattice with conventional cube side \(4\pi/a\). The FCC–BCC duality is a remarkable and recurring theme in solid state physics: it determines the shape of the Brillouin zones of cubic lattices (see Section 2.3).

2D hexagonal lattice. In two dimensions, consider the hexagonal lattice with primitive vectors

\[\mathbf{t}_{1/2} = a\,(\pm1,\,\sqrt{3})/2\]

and the 2D duality condition \(\mathbf{g}_i\cdot\mathbf{t}_j = 2\pi\,\delta_{ij}\) leads to

\[\mathbf{g}_{1/2} = \frac{2\pi}{a\sqrt{3}}\,(\pm\sqrt{3},1).\]

i.e. the dual lattice is again hexagonal, with lattice constant \(4\pi/a\sqrt{3}\), and its orientation is rotated by 30° (or 90°) with respect to the original direct lattice.

2.1.2 Fourier expansions

While a plane wave expansion on a cubic lattice is rather obvious, the extension to a generic Bravais lattice might seem relatively complex, but it’s not. We first start from a simple observation.

TheoremOn plane waves and the dual lattice

A plane wave \(\exp\!\left(i\mathbf{k}\cdot\mathbf{r}\right)\) is periodic on the Bravais lattice if and only if \(\mathbf{k} \in \mathcal{RL}\).

Show proof

Let us suppose \(\mathbf{k}=x_1\mathbf{g}_1+x_2\mathbf{g}_2+x_3\mathbf{g}_3\) for generic \(x_j\in\mathbb{R}\) satisfies \(e^{i\mathbf{k}\cdot\mathbf{R}} = 1\) for any \(\mathbf{R} \in \mathcal{BL}\). If we choose \(\mathbf{R} = \mathbf{t}_j\) we deduce that \(\mathbf{k}\cdot\mathbf{t}_j=2\pi x_j\) is a multiple of \(2\pi\), thus \(x_j\) is an integer, thus \(\mathbf{k}\in\mathcal{RL}\).

Conversely, if \(\mathbf{k}\in\mathcal{RL}\) then \(\mathbf{k} = m_1\mathbf{g}_1 + m_2\mathbf{g}_2 + m_3\mathbf{g}_3\) for some integers \(m_i\) and the phase factor for a generic lattice point \(\mathbf{R}=n_1\mathbf{t}_1+n_2\mathbf{t}_2+n_3\mathbf{t}_3\in\mathcal{BL}\) is equal to

\[\mathbf{k}\cdot\mathbf{R} = \sum_{i,j} m_i n_j\,\mathbf{g}_i\cdot\mathbf{t}_j = 2\pi \sum_i m_i n_i,\]

thus it is a multiple of \(2\pi\), and thus \(\exp(i\mathbf{k}\cdot\mathbf{R}) = 1\). \(\blacksquare\)

As a consequence, any function \(f(\mathbf{r})\) with the periodicity of the Bravais lattice can be expanded in a Fourier series over the dual lattice:

\[f(\mathbf{r}) = \sum_{\mathbf{g}\in\mathcal{RL}} \hat{f}_{\mathbf{g}}\, e^{i\mathbf{g}\cdot\mathbf{r}}, \tag{2.6}\]

where \(\hat{f}_{\mathbf{g}}\) are the Fourier coefficients, which can be calculated according to

\[\hat{f}_\mathbf{g} = \frac{1}{V}\int f(\mathbf{r})e^{-i\mathbf{g}\cdot\mathbf{r}}d\mathbf{r} = \frac{1}{\Omega}\int_{\mathrm{cell}} f(\mathbf{r})e^{-i\mathbf{g}\cdot\mathbf{r}}d\mathbf{r} \tag{2.7}\]

where the first integral is performed on the full crystal of volume \(V\).

CautionThis is just a basis change…

In the end, the Fourier expansion of a function periodic on a Bravais lattice can be simply seen as a basis change from a cubic coordinate system \(\mathbf{s}\in[0,1]^3\) to \(\mathbf{r}=s_1\mathbf{t}_1+s_2\mathbf{t}_2+s_3\mathbf{t}_3 = \mathcal{T}\mathbf{s}\). The Fourier expansion of a function \(f(\mathbf{s})=f(\mathcal{T}\mathbf{s})\) is obvious when expressed in terms of \(\mathbf{s}\):

\[ f_\mathbf{\nu} = \int_{[0,1]^3} f(\mathcal{T}s)e^{-2i\pi\boldsymbol{\nu}\cdot\mathbf{s}}d\mathbf{s} \]

where \(\nu\) is a vector of integer frequencies. The phase factor in the integral can be easily recast as

\[ 2\pi\,\boldsymbol{\nu}\cdot\mathbf{s}= \left(\mathcal{G}\boldsymbol{\nu}\right)\cdot\mathcal{T}\mathbf{s}= \mathbf{g}\cdot\mathbf{r}, \]

where we used \(\mathbf{g}=\mathcal{G}\boldsymbol{\nu}\). In the end, using this plus the rules for coordinate change, we obtain

\[ \begin{align} f_\mathbf{g} &= \frac{1}{|\det\mathcal{T}|}\int_{\mathrm{cell}} f(\mathbf{r})e^{-i\mathbf{g}\cdot\mathbf{r}}d\mathbf{r} = \frac{1}{\Omega}\int_{\mathrm{cell}} f(\mathbf{r})e^{-i\mathbf{g}\cdot\mathbf{r}}d\mathbf{r}\\ f(\mathbf{r}) &= f(\mathcal{T}\mathbf{s}) = \sum_{\boldsymbol{\nu}} f_{\boldsymbol{\nu}}e^{2i\pi\boldsymbol{\nu}\cdot\mathbf{s}} = \sum_{\mathbf{g}\in\mathcal{RL}} f_\mathbf{g}e^{i\mathbf{g}\cdot\mathbf{r}}. \end{align} \]

2.2 Crystal planes and Miller indices²

Lattice points can be organized in crystal planes. This comes naturally from the \(\mathcal{BL}\) nature: given any two vectors connecting nearby lattice sites, a plane of lattice sites can be generated by integer superpositions of these two vectors, which is also a subset of the \(\mathcal{BL}\).

Definition 2.2: Miller Notation

Conventionally, a crystal plane is identified by its Miller indices \((hkl)\), which can be identified as follows:

1. Find the intercepts of the plane with the three cell sides, expressed in units of the lattice parameters: \(a/h\), \(b/k\), \(c/l\).
2. Take \(h\), \(k\) and \(l\), and reduce to the smallest integers with no common divisor.
3. Miller notation \((hkl)\) is meant to be very compact: no commas (large numbers are not relevant); overline indicates negative sign; e.g. \((0\overline{1}1)\) means \(h=0\), \(k=-1\) and \(l=1\).
4. The family of symmetry-equivalent (rotated, mirrored) planes is denoted as \(\{hkl\}\).

Crystal planes are intimately connected to dual vectors \(\mathbf{g}\in\mathcal{RL}\) and the reason is simple: any such \(\mathbf{g}\) describes a plane wave \(w_{\mathbf{g}}(\mathbf{r})\) periodic on the Bravais lattice, characterized by phase planes. Since \(w_{\mathbf{g}}(\mathbf{0})=1\), one zero-phase plane must pass through the origin \(\mathbf{0}\in\mathcal{BL}\). Due to lattice periodicity, every Bravais lattice point must lie on one of these planes. As argued in the following, though, the correct identification of crystal planes requires the introduction of irreducible dual vectors.

Definition 2.3: Irreducible dual lattice vector

A dual vector \(\mathbf{g} = m_1\mathbf{g}_1 + m_2\mathbf{g}_2 + m_3\mathbf{g}_3\) is irreducible when it is not an integer multiple of a smaller lattice vector. Equivalently, we set \(\gcd(|m_1|,|m_2|,|m_3|) = 1\), i.e. \(m_1, m_2, m_3\) have no common divisor greater than 1.

An irreducible vector \(\mathbf{g}_0\) with indices \((hkl)\) is perpendicular to the family of crystal planes \((hkl)\), and the inter-plane spacing, corresponding to the wavelength of the plane wave \(e^{i{\mathbf{g}_0}\cdot\mathbf{r}}\), is

\[d = \frac{2\pi}{|\mathbf{g}_0|}. \tag{2.8}\]

Differently, a reducible vector \(\mathbf{g} = n\,\mathbf{g}_0\) corresponds to the \(n\)-th harmonic of the same family of planes: its magnitude is \(|\mathbf{g}| = 2\pi n / d_{hkl}\), and it introduces \(n-1\) fictitious “extra planes” between each pair of lattice planes. The true plane family is always determined by the irreducible part. See Figure 2.1 to play with planes and Miller indices in 2D.

import { createMillerRecip } from "./_interactive/ch02-miller-recip.js"
createMillerRecip(width)

Figure 2.1: Interactive Miller indices with direct and dual lattice. Left: direct square lattice with selected family of planes; dashed lines are extra planes from a reducible vector and the red band marks the reference plane with intercepts at \(a/h\) and \(a/k\) and distance \(d\) from the origin. Right: Red arrow is the selected irreducible \(\mathbf{g}_0\).

CautionCrystal directions \([hkl]\) and \(\langle hkl \rangle\)

A crystal direction is specified by a set of integers \([hkl]\) denoting the direction vector \(h\mathbf{t}_1 + k\mathbf{t}_2 + l\mathbf{t}_3\). The integers are chosen to be the smallest set with no common divisor. The family of all symmetric directions related by the point-group symmetry is denoted as \(\langle hkl \rangle\). While in cubic crystals crystal directions are perpendicular to the planes and match those of \(\mathbf{g}\), in general they are fundamentally different, as illustrated in Figure 2.2.

import { createPlanesVsDirections } from "./_interactive/ch02-planes-vs-directions.js"
createPlanesVsDirections(Math.min(width * 0.89, 720))

Figure 2.2: Crystal directions and lattice planes are not always perpendicular. The grey arrow shows the \([1,1]\) direction (\(\mathbf{t}_1+\mathbf{t}_2\)); the dashed red arrow shows the reciprocal-lattice vector \(\mathbf{G}_{11}\), which is normal to the \((1,1)\) planes. The two basis vectors are perpendicular; use the slider to change the ratio \(|\mathbf{t}_2|/|\mathbf{t}_1|\) and observe that the direction and normal coincide only for a square lattice (\(|\mathbf{t}_2|=|\mathbf{t}_1|\)).

CautionMiller-Bravais indices for hexagonal crystals

In hexagonal crystals the standard Miller notation \((hkl)\) obscures the six-fold symmetry: families of planes that are physically equivalent under a \(120°\) rotation, for instance \((100)\), \((010)\) and \((1\bar{1}0)\), carry indices that look unrelated. The Miller-Bravais scheme restores the symmetry by introducing a redundant fourth index \(i\), writing planes as \((hkil)\) with the constraint \(i=-(h+k)\). The index \(i\) is associated with a third in-plane axis \(\mathbf{t}_3 = -(\mathbf{t}_1+\mathbf{t}_2)\), which lies at \(120°\) from both \(\mathbf{t}_1\) and \(\mathbf{t}_2\). The three equivalent planes above now read \((10\bar{1}0)\), \((01\bar{1}0)\) and \((1\bar{1}00)\): they are related by cyclic permutations of the first three indices, making the symmetry manifest. See this applied to a 2D triangular lattice in Figure 2.3.

import { createMillerBravais } from "./_interactive/ch02-miller-bravais.js"
createMillerBravais(Math.min(width * 0.89, 651))

Figure 2.3: Miller-Bravais indices for a 2D hexagonal lattice. Left: direct lattice with the selected family of planes. Right: reciprocal lattice; click on a point to select the corresponding plane family. The \(c\)-axis index was dropped.

2.3 Brillouin zone³

The first Brillouin zone is the Wigner–Seitz cell of the dual lattice. It plays a central role in the theory of electrons and phonons in crystals, as it defines the key domain in \(\mathbf{k}\)-space. The shape of the first Brillouin zone is determined by the perpendicular bisector planes of the dual lattice vectors. Since the dual of a BCC lattice is FCC and vice versa, the BZ shapes are:

• Simple cubic: the BZ is a cube of side \(2\pi/a\).
• FCC direct lattice (BCC dual): the BZ is a truncated octahedron — a 14-faced polyhedron with 8 regular hexagonal and 6 square faces. It arises from the bisector planes of the 8 nearest and 6 next-nearest dual lattice vectors.
• BCC direct lattice (FCC dual): the BZ is a rhombic dodecahedron — a 12-faced rhombic polyhedron. Note that in an FCC with a lattice point at the center, the 12 nearest dual lattice vectors are placed at the middle of each edge of the cubic unit cell.

The interactive figure below shows these Brillouin zones. Select a lattice type to visualise the corresponding BZ and see how it emerges from the bisector planes.

import { renderBrillouinZones } from "./_interactive/ch02-brillouin-zones.js"
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  const container = document.createElement("div");
  container.style.cssText = "width:100%;";
  yield container;
  renderBrillouinZones(container);
}

Figure 2.4: First Brillouin zones of the three cubic Bravais lattices. Left: the truncated octahedron (FCC lattice, BCC dual). Right: the rhombic dodecahedron (BCC lattice, FCC dual). Use the dropdown to select the view.

import { renderSCHexZones } from "./_interactive/ch02-sc-hex-zones.js"
{
  const container = document.createElement("div");
  container.style.cssText = "width:100%;";
  yield container;
  renderSCHexZones(container);
}

Figure 2.5: First Brillouin zones for the simple-cubic (left) and 3D hexagonal (right) Bravais lattices. Same controls as Figure 2.4 — use the Show dropdown to switch between the bare BZ, its construction from reciprocal-lattice bisectors, or the stacking of adjacent zones.

DefinitionHigh symmetry points and lines

As visible in Figure 2.4 and Figure 2.5, high-symmetry points in the FBZ have precise names. One name is common to all lattices: the \(\Gamma\) point universally corresponds to \(\mathbf{k} = \mathbf{0}\). High-symmetry points are typically located at the edge of the FBZ, sitting along the rotational-symmetry axes inherited from the cell symmetry. The convention is to label by Greek letter the line along which a given rotation axis runs, and by capital Latin letter the point at which that line meets the FBZ boundary.

For the three cubic Bravais lattices:

Table 2.1: Main high-symmetry points in the cubic system

Symmetry line	Rotation	Direction	SC	FCC	BCC
\(\Delta\)-lines	\(C_4\)	\(\langle 100\rangle\)	\(X\)	\(X\)	\(H\)
\(\Lambda\)-lines	\(C_3\)	\(\langle 111\rangle\)	\(R\)	\(L\)	\(P\)
\(\Sigma\)-lines	\(C_2\)	\(\langle 110\rangle\)	\(M\)	\(K\)	\(N\)

High symmetry lines can also be defined for the hexagonal lattice, see Figure 2.5.

A subtlety hides in this list: not every named point is a single \(\mathbf{k}\)-state. The six \(K\) vertices of a hexagonal FBZ, for instance, split into two inequivalent classes (\(K\) and \(K'\)) under reciprocal-lattice translations alone.

CautionHow many inequivalent points?

In a 2D hexagonal Brillouin zone, out of the six corners we only have two inequivalent points called \(K\) and \(K'\). Within each class the three corners are equivalent and they differ by a reciprocal-lattice translation. Differently, there is no \(\mathbf{g}\in\mathcal{RL}\) vector connecting \(K\) to \(K'\), so the two will label genuinely distinct states of the band structure.

As visible in Figure 2.6, the redundancy of a point at the edge of a FBZ can be identified in two alternative ways: (i) one can look at which other points are connected by a reciprocal lattice vector; (ii) one can check how many nearby cells share the same point.

import { renderHexEquivalence } from "./_interactive/ch02-hex-equivalence.js"
{
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  container.style.cssText = "width:100%;";
  yield container;
  renderHexEquivalence(container);
}

Figure 2.6: Equivalent and inequivalent points of the 2D hexagonal Brillouin zone.

2.4 Diffraction methods⁴

Periodicity in crystal structure can be revealed by a set of diffraction methods, provided the wavelength of the coherent incident radiation is comparable to the lattice spacing (\(\sim 1\,\text{Å}\)). Three types of probes are commonly used, each with a different energy–wavelength relationship and peculiar characteristics:

DefinitionX-ray photons

For massless photons, \(\lambda = hc/E\), giving

\[\lambda\,[\text{Å}] = \frac{12398.5}{E\,[\text{eV}]}. \tag{2.9}\]

A typical laboratory source is the Cu K\(\alpha\) line at \(E = 8.04\,\text{keV}\), yielding \(\lambda = 1.54\,\text{Å}\). Synchrotron sources provide tunable X-rays from a few eV to \(100\,{\mathrm{keV}}\). Photons are not charged and weakly interacting, thus they are ideal to investigate crystalline structures by diffraction effects deriving from interference in first-order elastic scattering.

DefinitionElectrons

For massive particles with kinetic energy \(E\), the de Broglie wavelength is \(\lambda = h/\sqrt{2mE}\). For non-relativistic electrons

\[\lambda\,[\text{Å}] = \frac{12.264}{\sqrt{E\,[\text{eV}]}}. \tag{2.10}\]

Differently from photons, electrons are charged and strongly interacting and are thus typically used to investigate surfaces. A typical method is LEED (Low-Energy Electron Diffraction): electrons in this method are required to have a relatively low energy, in fact \(\lambda = 1.0\,\text{Å}\) corresponds to about \(E = 150\,\text{eV}\). Further notable methods are: RHEED (Reflection High-Energy Electron Diffraction), typically used as a monitoring method in epitaxy, it requires higher energies due to the grazing angle configuration it normally implements; TEM (transmission electron microscopy) diffraction is also possible, but uses much higher energies (100–300 keV) and requires extremely thin samples.

DefinitionThermal neutrons

The same dispersion relation also holds for neutrons, which however have a much heavier mass. As a result, typical relevant energies fall close to room-temperature thermal energy \(k_BT\). The de Broglie wavelength is:

\[\lambda\,[\text{Å}] = \frac{0.2860}{\sqrt{E\,[\text{eV}]}}. \tag{2.11}\]

At room temperature (\(T = 300\,\text{K}\), \(E \approx 25.9\,\text{meV}\)), \(\lambda \approx 1.78\,\text{Å}\), which is perfectly suited for crystallography. Unlike X-rays, neutrons scatter off nuclei (not electrons), making them sensitive to light atoms and to isotopic contrast. Neutrons are again neutral and weakly interacting, and thus highly penetrating. While photons are mostly interacting with the electrons in the crystal, neutrons interact by strong force with the nuclei and, thanks to their magnetic dipole, can be used to investigate magnetic ordering as well. Finally, nuclei are much more localized than electrons, implying that very high order diffraction processes are possible (see the Fourier discussion at the end of the chapter).

import { renderProbeCalculator } from "./_interactive/ch02-probes.js"
{
  const container = document.createElement("div");
  container.style.cssText = "width:100%;";
  yield container;
  renderProbeCalculator(container);
}

Figure 2.7: Energy–wavelength converter for diffraction probes. The conversion uses Equation 2.9 for photons, Equation 2.10 for electrons, and Equation 2.11 for neutrons.

Summary of key diffraction probes: interaction mechanism and sensitivity.

Probe	Interaction	Scatters off	Sensitive to
X-ray photon	Electromagnetic	Electrons	charge density
Electron	Coulomb	Electrons + nuclei	charge distribution; surfaces
Neutron	Nuclear / magnetic	Nuclei + magnetic moments	Nuclear positions, magnetic ordering

CautionWhat else are we missing

Investigation methods for crystalline structures and solids go well beyond the diffraction phenomena here discussed. A few key ones are mentioned here:

• Inelastic scattering. We assumed elastic (Rayleigh) scattering where the photon (or particle) changes direction but not energy. In reality several inelastic channels exist:

  • Compton scattering. The X-ray photon transfers part of its energy to a loosely-bound electron and emerges with a longer wavelength \(\lambda_f = \lambda_i + h(1-\cos\theta)/(m_ec)\). Compton scattering produces a diffuse incoherent background and becomes progressively important at higher photon energies.
  • Phonon scattering. The probe exchanges energy with lattice vibrations. This is exploited in inelastic neutron scattering and inelastic X-ray scattering (IXS) to map phonon dispersion relations, but it weakens and broadens Bragg peaks (captured phenomenologically by the Debye-Waller factor). At smaller photon energies, this process is also the basis of Raman and Brillouin spectroscopy.
  • Plasmon and single-particle excitations are relevant mostly for electron probes (EELS, Electron Energy Loss Spectroscopy), where the fast electron can excite collective (plasmon) or single-particle transitions in the solid.

• Multiple scattering and dynamical diffraction. Bragg’s law and the first-order interaction work well for weak scattering, but fail for strong scatterers. Electrons interact with matter much more strongly than X-rays, so higher order effects typically need to be taken into account.

• Spectroscopic and combined techniques. Many powerful methods combine diffraction geometry with energy analysis:

  • PES (Photoemission Spectroscopy): high-energy photons can eject electrons from the solid; measuring their kinetic energy reveals binding energies and hence chemical composition and electronic structure. In ARPES (Angle-Resolved PES) the electron momentum is also measured, directly giving access to the band dispersions \(E_n(\mathbf{k})\).
  • EDX / EDS (Energy-Dispersive X-ray Spectroscopy): a focused electron beam (in SEM or TEM) excites characteristic X-ray emission from the sample, providing element-specific chemical analysis with a spatial resolution that strongly depends on the beam energy and is typically on the scale of one micron.
  • Auger electron spectroscopy (AES): after core-hole creation (by electrons or X-rays) the atom can de-excite by emitting an Auger electron instead of a photon. As in the case of EDX, the Auger energy is element-specific. Differently from EDX, the short mean free path of the emitted electron (\(\sim 0.5\)–\(3\) nm) makes AES extremely surface-sensitive.
  • XAS (X-ray Absorption Spectroscopy): measures the X-ray absorption coefficient as a function of photon energy near an absorption edge. Features captured by this method can give access to local structural information (bond lengths, coordination numbers).

From here on we will focus on X-ray diffraction as the method of choice to study the crystalline structure. Diffraction conditions can be formulated in two equivalent ways, as discussed in the following.

2.4.1 Bragg and von Laue condition

Diffraction occurs at specific conditions where scattered photons from the atoms undergo constructive interference. Let us first consider a family of parallel crystal planes with spacing \(d\) and assume they act as mirrors. An X-ray beam incident at angle \(\theta\) (measured from the atomic plane, not the normal direction as often done in optical configurations) is reflected by successive planes. Constructive interference occurs when the path difference between reflections from adjacent planes equals an integer number of wavelengths:

\[2d\sin\theta = n\lambda, \qquad n \in \mathbb{Z}. \tag{2.12}\]

This is the Bragg condition. It provides a simple geometric picture of diffraction: the crystal acts as a set of mirrors, one for each family of planes. See Figure 2.8.

import { renderBragg } from "./_interactive/ch02-bragg.js"
{
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  container.style.cssText = "width:100%;";
  yield container;
  renderBragg(container);
}

Figure 2.8: Interactive Bragg diffraction geometry. Two parallel X-ray beams (blue) reflect from adjacent crystal planes separated by spacing d. The extra path traveled by the lower beam (orange segments) equals 2d sin θ. Use the slider to vary θ and observe when the Bragg condition 2d sin θ = nλ is satisfied.

A more general formulation uses the concept of transferred momentum (or scattering vector). If \(\mathbf{k}_i\) and \(\mathbf{k}_f\) are the wave vectors of the incident and scattered beams (with \(|\mathbf{k}_i| = |\mathbf{k}_f| = 2\pi/\lambda\) for elastic scattering), the scattering vector is

\[\mathbf{Q} = \mathbf{k}_i - \mathbf{k}_f. \tag{2.13}\]

The condition for constructive interference from a periodic structure is \(\mathbf{Q} \in \mathcal{RL}\), i.e. the scattering vector must be a dual lattice vector. This is the von Laue condition, which is fully equivalent to the Bragg condition, as shown below. The von Laue condition will also be derived in the next section where we analyze the scattering from multiple scattering centers arranged in a periodic pattern.

TheoremEquivalence between Bragg and von Laue conditions

The two formulations of Bragg and von Laue for the diffraction conditions are completely equivalent, i.e.

\[\mathbf{Q} \in \mathcal{RL} \iff 2d\sin\theta = n\lambda\]

Show proof

From von Laue to Bragg. Elastic scattering gives \(|\mathbf{k}_f|=|\mathbf{k}_i|\equiv k = 2\pi/\lambda\). The scattering vector \(\mathbf{Q}=\mathbf{k}_i-\mathbf{k}_f\) satisfies

\[|\mathbf{Q}|^2 = |\mathbf{k}_f|^2 + |\mathbf{k}_i|^2 - 2\mathbf{k}_f\cdot\mathbf{k}_i = 2k^2(1-\cos 2\theta) = 4k^2\sin^2\theta,\]

so \(|\mathbf{Q}| = 2k\sin\theta\). The von Laue condition requires \(\mathbf{Q} = \mathbf{g}_{hkl}\) for some reciprocal lattice vector. Writing \(\mathbf{g}_{hkl} = n\,\mathbf{g}_0\) where \(\mathbf{g}_0\) is the shortest reciprocal vector in that direction (with \(|\mathbf{g}_0| = 2\pi/d_{hkl}\)), we obtain

\[2k\sin\theta = n\,|\mathbf{g}_0| = n\,\frac{2\pi}{d_{hkl}}.\]

Substituting \(k = 2\pi/\lambda\):

\[2\,\frac{2\pi}{\lambda}\sin\theta = n\,\frac{2\pi}{d_{hkl}} \quad\Longrightarrow\quad 2d_{hkl}\sin\theta = n\lambda,\]

which is exactly the Bragg condition.

From Bragg to von Laue. Conversely, the Bragg condition for the \((hkl)\) family of planes gives a scattering vector of magnitude \(|\mathbf{Q}| = 2k\sin\theta = n\cdot 2\pi/d_{hkl}\), directed along the plane normal \(\hat{\mathbf{n}}_{hkl}\). Since \(\hat{\mathbf{n}}_{hkl} = \mathbf{g}_0/|\mathbf{g}_0|\), we have \(\mathbf{Q} = n\,\mathbf{g}_0 = \mathbf{g}_{hkl}\), which is a reciprocal lattice vector. \(\blacksquare\)

2.4.2 Form and structure factor

When an electromagnetic wave of frequency \(\omega\) impinges on a free charged particle of charge \(q\) and mass \(m_q\), the oscillating electric field drives the particle into oscillation. The accelerating charge re-radiates electromagnetic energy according to Thomson scattering. Assuming the charge is located at a given position \(\mathbf{r}_q\) and the impinging electromagnetic field with wave vector \(\mathbf{k}_i\) and frequency \(\omega\) is equal to

\[\mathbf{E}(\mathbf{r},t) = \hat{\mathbf{e}}_i\,E_0\,e^{i(\mathbf{k}_i\cdot\mathbf{r}-\omega t)}\]

where \(\hat{\mathbf{e}}_i\) is the incident polarization and \(E_0\) the magnitude of the electric field. The non-relativistic radiated field by an oscillating charge, when observed at a relative position \(\mathbf{R}=\hat{\mathbf{n}}R=\mathbf{r}-\mathbf{r}_q\) is

\[\mathbf{E}_d(\mathbf{r},t;\mathbf{r}_q) = -\cfrac{q}{4\pi\varepsilon_0c^2R}\,\hat{\mathbf{n}}\times\left[\hat{\mathbf{n}}\times\mathbf{a}(t-R/c)\right]\]

where \(\mathbf{a}(t) = q\mathbf{E}(\mathbf{r}_q,t)/m_q\) is the acceleration of the charge located in \(\mathbf{r}_q\). The propagation delay turns into a phase delay \(\omega R/c = \mathbf{k}_f\cdot\mathbf{R} = \mathbf{k}_f\cdot\left(\mathbf{r}-\mathbf{r}_q\right)\), and, finally, considering all the phase delays we have

\[\mathbf{E}_d(\mathbf{r},t;\mathbf{r}_q) = -\cfrac{q^2E_0}{4\pi\varepsilon_0 m_qc^2R}\hat{\mathbf{n}}\times\left[\hat{\mathbf{n}}\times\hat{\mathbf{e}}_i\right]\,e^{i\mathbf{Q}\cdot\mathbf{r}_q}\,e^{i(\mathbf{k}_f\cdot\mathbf{r}-\omega t)}\]

where \(\mathbf{Q}=\mathbf{k}_i-\mathbf{k}_f\) is the transferred momentum. From a given observation point \(\mathbf{r}\) and in the limit of large \(R\), all scatterers will share the same \(\mathbf{k}_f\) and we can expect

\[\left|\mathbf{E}_d\right|^2 \propto \left|\sum_{\mathbf{r}_q} e^{i\mathbf{Q}\cdot\mathbf{r}_q}\right|^2 \to \left|\int n(\mathbf{r})e^{i\mathbf{Q}\cdot\mathbf{r}}d^3r\right|^2.\]

This demonstrates the validity of the von Laue condition, since the transform of the periodic \(n(\mathbf{r})\) is ideally (see also the Fourier discussion at the end of the chapter) non-zero only when \(\mathbf{Q}\in\mathcal{RL}\). Note also that the diffraction intensity is proportional to \(1/m_q^2\): since the ions have a mass which is at least 1836 times larger (in hydrogen, much worse in any other case), we learn that X-ray scattering is overwhelmingly sensitive only to electrons.

CautionAngular dependence: Thomson differential cross-section

The angular dependence of a radiating dipole is null in the direction of the oscillation axis, but the oscillation axis depends on the input polarization. Typical diffraction lobes are given assuming an average in the input and output polarization; this yields the so-called Thomson differential cross-section

\[\left(\cfrac{d\sigma}{d\Omega}\right)_{Th}^{\mathrm{(unpol)}} = r_0^2\,\cfrac{1+\cos^2\theta_s}{2}\]

where \(\theta_s\) is the scattering angle (equal to \(2\theta\) in the Bragg configuration). The key parameter controlling the scattering amplitude is the electromagnetic radius \(r_0=e^2/4\pi\varepsilon_0m_qc^2=2.82\,{\mathrm{fm}}\).

When two or more charges coherently scatter the same plane wave, in the far-field limit \(\hat{\mathbf{n}}\) will be constant and we can expect an interferometric sum

\[\cfrac{d\sigma}{d\Omega} = \left(\cfrac{d\sigma}{d\Omega}\right)_{\mathrm{Th}}\left|\sum_{\mathbf{r}_q} e^{i\mathbf{Q}\cdot\mathbf{r}_q}\right|^2 \to \left(\cfrac{d\sigma}{d\Omega}\right)_{\mathrm{Th}}\left|\int n(\mathbf{r})e^{i\mathbf{Q}\cdot\mathbf{r}}d^3r\right|^2 \tag{2.14}\]

where we have taken a limit to the continuum and \(n(\mathbf{r})\) is the electron density.

Definition 2.4: Form and structure factors

The Fourier transform of the density \(n(\mathbf{r})\) is called form factor

\[F(\mathbf{Q}) = \int n(\mathbf{r})\, e^{i\mathbf{Q}\cdot\mathbf{r}}\, d^3r, \tag{2.15}\]

and is directly connected to the quantity appearing in Equation 2.14

\[S(\mathbf{Q}) \propto |F(\mathbf{Q})|^2. \tag{2.16}\]

which is called structure factor. Since all complex phases are lost in \(S(\mathbf{Q})\), reconstructing \(n(\mathbf{r})\) is less obvious than performing an inverse Fourier transform. This is the well-known phase problem of crystallography.

For a crystal with a basis of \(N_b\) atoms, the electron density can be decomposed as:

\[n(\mathbf{r}) = \sum_{\mathbf{t}\in\mathcal{BL}} \sum_{\alpha=1}^{N_b} n_\alpha(\mathbf{r} - \mathbf{t} - \mathbf{d}_\alpha),\]

where \(n_\alpha(\mathbf{r})\) is the electron density of atom \(\alpha\), and \(\mathbf{t}\) runs over the Bravais lattice. Substituting into Equation 2.15:

\[F(\mathbf{Q}) = \underbrace{\sum_{\mathbf{t}} e^{i\mathbf{Q}\cdot\mathbf{t}}}_{\text{lattice sum}} \;\times\; \sum_{\alpha=1}^{N_b} \underbrace{\left[\int n_\alpha(\mathbf{r}')\, e^{i\mathbf{Q}\cdot\mathbf{r}'}\, d^3r'\right]}_{f_\alpha(\mathbf{Q})} e^{i\mathbf{Q}\cdot\mathbf{d}_\alpha}.\]

The sum is nonzero only when \(\mathbf{Q}\in\mathcal{RL}\), in which case it equals \(N\) (number of unit cells), and thus

\[F(\mathbf{Q}) = N \sum_{\alpha=1}^{N_b} f_\alpha(\mathbf{Q})\, e^{i\mathbf{Q}\cdot\mathbf{d}_\alpha}, \tag{2.17}\]

where

\[f_\alpha(\mathbf{Q}) = \int n_\alpha(\mathbf{r})\, e^{i\mathbf{Q}\cdot\mathbf{r}}\, d^3r \tag{2.18}\]

is the atomic form factor of atom \(\alpha\). Here we assumed an independent atom model: the electron density of each atom is taken to be that of an isolated atom, neglecting bonding-induced redistribution of charge. Under the additional assumption that isolated atoms have spherically symmetric \(n_\alpha(\mathbf{r})\), \(f_\alpha\) depends only on \(|\mathbf{Q}|\). Note also that \(f_\alpha(0) = Z_\alpha\), the atomic number (total electron count).

The structure factor can vanish for certain reciprocal lattice vectors, causing the corresponding Bragg reflections to disappear (extinction rules).

Definition 2.5: Basis form factor

The basis form factor (or geometric structure factor) collects all information about the internal structure of the unit cell. It basically stems from the constructive/destructive interference from the different sublattices:

\[F_{\mathrm{basis}}(\mathbf{Q}) = \sum_{\alpha=1}^{N_b} f_\alpha(\mathbf{Q})\, e^{i\mathbf{Q}\cdot\mathbf{d}_\alpha}, \tag{2.19}\]

so that \(F(\mathbf{Q}) = N\, F_{\mathrm{basis}}(\mathbf{Q})\) when evaluated at a reciprocal lattice vector. This factor encodes the relative positions \(\mathbf{d}_\alpha\) and scattering strengths \(f_\alpha\) of the atoms within the cell. Because \(F_{\mathrm{basis}}\) is a coherent sum of complex exponentials, it can vanish for certain \(\mathbf{Q}\) even when the individual \(f_\alpha\) are nonzero — giving rise to extinction rules, or forbidden reflections, that are diagnostic of the basis geometry.

CautionExample: extinction rules for BCC

Let us for a moment improperly consider a BCC pattern as a simple cubic lattice plus basis at \(\mathbf{d}_1 = 0\), \(\mathbf{d}_2 = a(1,1,1)/2\), assuming identical atoms (\(f_1 = f_2 = f\)). The form factor for a vector \(\mathbf{Q}_{hkl}=2\pi(h,k,l)/a\) is

\[F(\mathbf{Q}_{hkl}) = f\bigl(1 + e^{i\pi(h+k+l)}\bigr) = \begin{cases} 2f & h+k+l \text{ even} \\ 0 & h+k+l \text{ odd.} \end{cases}\]

Since \(S = |F|^2\), reflections with \(h+k+l\) odd have \(S=0\) and are forbidden: this is an FCC pattern. Similarly, starting from an FCC pattern and improperly considering it a simple cubic plus a basis, only reflections with \(h,k,l\) all even or all odd survive. This correctly reproduces the duality BCC \(\leftrightarrows\) FCC.

CautionExample: forbidden reflections in silicon (diamond structure)

Silicon crystallises in the diamond structure: an FCC lattice with a two-atom basis at \(\mathbf{d}_1 = \mathbf{0}\) and \(\mathbf{d}_2 = a(1,1,1)/4\). Since both atoms are identical (\(f_1 = f_2 = f\)), the form factor is

\[F(\mathbf{Q}_{hkl}) = f\left(1 + e^{i\frac{\pi}{2}(h+k+l)}\right).\]

However, the underlying FCC lattice already imposes the condition that \(h,k,l\) must be all even or all odd (otherwise the reflection is forbidden by the FCC extinction rule). Among the surviving reflections, the diamond basis introduces further cancellations:

\[ F(\mathbf{Q}_{hkl}) = \begin{cases} 2f & h+k+l = 4n \quad \text{(e.g. 400, 220)} \\ (1\pm i)f & h+k+l = 4n\pm 1 \quad \text{(e.g. 111, 311)} \\ 0 & h+k+l = 4n+2 \quad \text{(e.g. 200, 222).} \end{cases}\]

The reflections with \(h+k+l = 4n+2\) (all even indices, but with \(h+k+l\) not a multiple of 4) are absent. For example, the \((222)\) reflection is forbidden in silicon, while \((111)\) and \((220)\) are allowed. These extra extinctions are a direct fingerprint of the diamond structure and can be used to distinguish it from a simple FCC arrangement.

2.5 Quasicrystals

The restriction theorem forbids 5-fold rotational symmetry in any periodic crystal. Yet in 1982, Dan Shechtman observed 10-fold symmetric diffraction from a rapidly quenched Al–Mn alloy (published in 1984). These materials, called quasicrystals, possess long-range orientational order but lack translational periodicity (thus the theorem does not apply).

A 2D analogue is the Penrose tiling (Figure 2.9), built from two rhombus shapes — thick (72°/108°) and thin (36°/144°) — arranged according to local matching rules that enforce aperiodicity. The tiling exhibits global 5-fold rotational symmetry and covers the plane without gaps, yet no finite patch repeats periodically. The right panel of Figure 2.9 shows the structure factor \(S(\mathbf{Q})\) of the tiling vertices. Despite the absence of periodicity, diffraction displays sharp peaks with 10-fold symmetry. This proves that long-range orientational order alone is sufficient to produce Bragg-like peaks, even without periodicity.

Shechtman’s discovery was awarded the Nobel Prize in Chemistry in 2011 and forced the International Union of Crystallography to redefine crystal as “any solid having an essentially discrete diffraction diagram”, a definition broad enough to encompass both periodic and quasiperiodic structures.

import { renderQuasicrystal } from "./_interactive/ch02-quasicrystal.js"
{
  const W = Math.min(width, 740);
  const container = document.createElement("div");
  container.style.cssText = "width:100%; max-width:" + W + "px; position:relative; display:flex; flex-direction:column;";
  yield container;
  renderQuasicrystal(container);
}

Figure 2.9: Penrose P3 tiling and its diffraction pattern. Left: aperiodic tiling of thick rhombi (amber, 72°/108°) and thin rhombi (teal, 36°/144°). Right: structure factor \(S(\mathbf{Q})\), computed from the vertex positions. Use the slider to change the recursive Robinson subdivision.

CautionFibonacci

The deep connection between a quasicrystal and an ordinary lattice becomes most transparent in one dimension via the cut-and-project method. Start from a 2D square lattice and cut it with a straight line whose slope is the irrational number \(1/\varphi\), where \(\varphi = (1+\sqrt{5})/2\) is the golden ratio. A strip of perpendicular width equal to the projection of the unit cell onto the direction orthogonal to the line (i.e. \(\cos\theta + \sin\theta\) with \(\tan\theta = 1/\varphi\)) selects one and only one lattice point per “column” of the cut. Projecting each selected point perpendicularly onto the line produces a 1D point pattern. The spacings between consecutive projected points take only two values \(L = \cos\theta\) and \(S = \sin\theta\).

import { renderFibonacci } from "./_interactive/ch02-fibonacci.js"
{
  const container = document.createElement("div");
  container.style.cssText = "width:100%;";
  yield container;
  renderFibonacci(container);
}

Figure 2.10: 1D Fibonacci quasicrystal. The square lattice (gray dots) is rotated so that the cutting line lies horizontally; the blue strip of perpendicular width \(\cos\theta + \sin\theta\) (with \(\tan\theta = 1/\varphi\)) selects one lattice point per “column” of the cut (filled blue dots). Projection onto the line yields a 1D pattern with gaps \(L = \cos\theta\) (blue) or \(S = \sin\theta\) (orange).

What is the link to the Fibonacci sequence?

The sequence of L and S is not random: it can be generated by the substitution rule

\[L \to LS, \qquad S \to L.\]

Starting from a single word \(L\) and iterating,

\[ \begin{align} S_1 &=L\\ S_2 &=LS \\ S_3 &= S_2S_1 = LSL\\ S_4 &= S_3S_2 = LSLLS\\ S_5 &= S_4S_3 = LSLLSLSL\\ S_6 &= S_5S_4 = LSLLSLSLLSLLS\\ S_7 &= S_6S_5 = LSLLSLSLLSLLSLSL{\mathbf{LSLSL}}\\ S_8 &= S_7S_6 = LSLLSLSLLSLLSLSL{\mathbf{LSLSLLSLLSLSLLSLLS}}\\ S_9 &= S_8S_7 = [\dots]LSLSLLSLLSLSLLSLLSLSLLSLSLLSLLS\dots \end{align} \]

Note in the last line we have the first (of many) occurrence of the sequence reported in Figure 2.10. The length of the \(n\)-th word in the sequence is \(1,\,2,\,3,\,5,\,8,\,13,\,21,\,34,\ldots\). These are the Fibonacci numbers \(F_n\) defined by \(F_1 = F_2 = 1\) and the recursive rule \(F_{n+1} = F_n + F_{n-1}\).

The Fibonacci chain is the cleanest example of a quasicrystal. The Penrose tiling of Figure 2.9 is its 2D analogue, the icosahedral quasicrystals observed by Shechtman are its 3D analogues, and in each of these cases the construction can proceed by cutting a higher-dimensional periodic lattice with an irrationally-oriented subspace.

In conclusion…

ImportantTake home messages…

At the end of this chapter you should know…

• Reciprocal lattice. Definition and construction from the direct lattice: what is the connection with Fourier expansion? What is the dual of the common lattices (e.g. of FCC)?
• Planes and Miller indices. Connection between families of lattice planes and \(\mathcal{RL}\) vectors: how are Miller indices defined and used? what is an irreducible dual vector?
• First Brillouin zone. Definition and construction, high-symmetry points of common FBZs, counting of inequivalent points: why is the first BZ used?
• Diffraction. Laue condition and Bragg formulation (why are they equivalent?), form and structure factors: where do extinction rules come from?

Test your understanding

As in the previous chapter, this final section collects two interactive tools to test your understanding of diffraction: the Ewald construction and a Fourier-transform playground.

Ewald sphere construction

The Ewald sphere provides a geometric visualisation of the diffraction condition. The recipe to obtain the diffraction beams for a given configuration is the following:

1. Draw the incident wave vector \(\mathbf{k}_i\) ending at \(\Gamma\).
2. Draw a sphere of radius \(|\mathbf{k}_i| = 2\pi/\lambda\) centred at the start of \(\mathbf{k}_i\).
3. Any dual lattice point on the sphere satisfies von Laue \(\mathbf{Q} = \mathbf{g}\)

In practice, to probe multiple reflections one can:

• Rotate the crystal so that different reciprocal lattice points sweep through the Ewald sphere. This is the basis of single-crystal X-ray diffraction.
• Modify the energy: a similar sweep can be obtained by changing the size of the Ewald sphere.
• Use a powder sample so that all orientations are averaged. The resulting diffraction pattern is a set of concentric rings, each ring corresponding to a specific \(d\)-spacing. Although the angular information is lost, crystal phases can be identified from tabulated \(d\)-spacing databases.

DefinitionMinimum wavevector for diffraction

For the Ewald sphere to intersect any reciprocal lattice point (other than the origin), its radius must be at least half the shortest reciprocal lattice vector:

\[|\mathbf{k}_i| \ge \frac{1}{2}\,|\mathbf{g}_{\min}| \qquad\Longleftrightarrow\qquad \lambda \le 2\,d_{\max} \tag{2.20}\]

where \(d_{\max}\) is the largest inter-plane spacing. This is just \(2d\sin\theta = \lambda\) at the limiting angle \(\theta = 90°\). For a cubic lattice with parameter \(a\), \(d_{\max} = a\) (for SC) or \(a/\sqrt{3}\) (for FCC), so X-rays (\(\lambda \sim 1\,\mathring{\mathrm{A}}\)) satisfy the condition easily, whereas visible light (\(\lambda \sim 5000\,\mathring{\mathrm{A}}\)) does not — which is why crystal diffraction requires short wavelengths.

import { createEwaldSphere } from "./_interactive/ch02-ewald.js"
createEwaldSphere(width)

Figure 2.11: Interactive Ewald sphere construction. Left: real-space view, drag the orange handle to rotate the crystal; diffracted beams (red) appear when the Bragg condition is satisfied. Right: reciprocal-space view, the Ewald circle (blue dashed) intersects dual lattice points. Hover on the diffraction vectors to highlight the active family plane.

On Fourier transform

Here we analyze how the diffraction spots depend on the actual crystal parameters. This is a Fourier transform: the name “reciprocal space” is not accidental and features that are large in direct space become small in reciprocal space, and vice versa. This duality has several concrete consequences:

1. Peak width. The finite size of the crystal yielding coherent scattering determines the width of the single diffraction peaks. A larger coherent volume produces sharper peaks.

2. Atomic size. The size of electronic clouds within each atom sets the rate at which the diffraction peak intensities decay with increasing \(|\mathbf{Q}|\). Sharp (smooth) electron density yields many (fewer) harmonic components. Thermal vibration also yields a similar effect (Debye-Waller factor).

3. Basis. When the unit cell contains more than one atom, \(F(\mathbf{Q})\) becomes modulated: some peaks are enhanced, others suppressed, and some may vanish entirely (extinction rules).

These mechanisms are simple, but they might be non-intuitive: play with Figure 2.12 to build intuition.

Caution1D analytical model of the diffraction pattern

Observations above can be made quantitative in a simple 1D model. Consider a crystal of \(N\) atoms at positions \(t_n = na\) (\(n = 0, \dots, N-1\)), each with a Gaussian electron density of width \(\sigma\). The total density is

\[\rho(x) = \sum_{n=0}^{N-1} f(x - na), \qquad f(x) = \frac{1}{\sqrt{2\pi}\sigma}\,e^{-x^2/2\sigma^2}.\]

By the convolution theorem, \(\tilde\rho(k) = \tilde{f}(k)\cdot\tilde{L}(k)\) where \(\tilde{f}\) is the atomic form factor and \(\tilde{L}\) is the lattice sum.

1. Infinite lattice of point scatterers (\(N\to\infty\), \(\sigma\to 0\)): the density is a Dirac comb \(\rho_\infty(x) = \sum_{n}\delta(x-na)\), and its Fourier transform is also a comb:

\[\tilde\rho_\infty(k) = \frac{2\pi}{a}\sum_m \delta\!\left(k - \frac{2\pi m}{a}\right). \tag{2.21}\]

This is the idealized diffraction pattern: sharp peaks at \(k_m = 2\pi m/a\), all with equal intensity.

2. Extended atoms (finite \(\sigma\)): the atomic form factor acts as an envelope:

\[\tilde{f}(k) = e^{-\sigma^2 k^2/2}. \tag{2.22}\]

The peak positions are unchanged, but their intensities decay as a Gaussian in \(k\): larger atoms (\(\sigma\) large) suppress high-order reflections more strongly.

3. Finite crystal (\(N\) finite): the lattice sum becomes

\[\tilde{L}(k) = \sum_{n=0}^{N-1} e^{-ikna} = \frac{\sin(Nka/2)}{\sin(ka/2)}\,e^{-i(N-1)ka/2}. \tag{2.23}\]

Each peak acquires a finite width \(\Delta k \approx 2\pi/(Na) = 2\pi/L\) which is inversely proportional to the total crystal size \(L = Na\). The complete diffraction intensity is therefore

\[I(k) \propto \left|\tilde{f}(k)\right|^2 \cdot \frac{\sin^2(Nka/2)}{\sin^2(ka/2)}, \tag{2.24}\]

which separates the three roles cleanly: peak positions from the lattice periodicity \(a\), peak widths from the crystal size \(L\), and the intensity envelope from the atomic form factor \(\tilde{f}(k)\).

import { renderFourier } from "./_interactive/ch02-fourier.js"
{
  const container = document.createElement("div");
  container.style.cssText = "width:100%;";
  yield container;
  renderFourier(container);
}

Figure 2.12: Fourier transform of a periodic train of Gaussians. Top: the atom chain — a finite crystal of \(N\) unit cells with an optional second basis atom at \(a/3\), whose size (and weight in the density) is set by the Basis slider. Middle: the electron density \(n_e(x)\), a Gaussian peak of width \(\sigma\) on each atom. Bottom: magnitude of the Fourier transform \(|F(k)|\), showing peaks at multiples of \(2\pi/a\).

Grosso, Giuseppe, and Giuseppe Pastori Parravicini. 2014. Solid State Physics. 2nd ed. Academic Press.

---

1. Grosso and Pastori Parravicini (2014, sec. 2.4).

2. Grosso and Pastori Parravicini (2014, sec. 2.4).

3. Grosso and Pastori Parravicini (2014, sec. 2.5).

4. Grosso and Pastori Parravicini (2014, secs. 10.1–2).