- Construct the reciprocal lattice from a direct Bravais lattice and interpret reciprocal vectors as Fourier wave vectors of lattice-periodic functions.
- Relate reciprocal-lattice vectors to Miller planes, interplanar spacings, and Brillouin-zone boundaries.
- Derive the Bragg and Laue diffraction conditions and explain their equivalence.
- Explain how the lattice determines Bragg-peak positions, while the basis determines Bragg-peak intensities through the structure factor.
- Describe how X-rays, electrons, and neutrons differ as diffraction probes.
- Interpret elastic and inelastic scattering as complementary probes of static structure and lattice dynamics.
4.1 Roadmap
- Reciprocal lattice, Fourier analysis, and Brillouin zones.
- Crystal planes, Miller indices, and the geometry of reciprocal space.
- Elastic diffraction: Bragg condition, Laue condition, and Ewald construction.
- Scattering amplitude, atomic form factor, structure factor, and systematic extinctions.
- Thermal motion, Debye-Waller reduction, and the bridge to inelastic scattering.
- Experimental methods: X-ray, electron, and neutron diffraction.
| Symbol | Meaning |
|---|---|
| \(\mathbf a_i\) | primitive translation vectors of the direct lattice |
| \(\mathbf b_i\) | primitive translation vectors of the reciprocal lattice |
| \(\mathbf R_n\) | direct-lattice vector |
| \(\mathbf G\) | reciprocal-lattice vector |
| \(h,k,l\) | integer indices of a reciprocal-lattice vector or Miller indices |
| \((hkl)\) | family of lattice planes |
| \(d_{hkl}\) | spacing of neighboring planes in the \((hkl)\) family |
| \(V_\mathrm c\) | volume of the primitive direct cell |
| \(\mathbf k,\mathbf k'\) | incident and scattered wave vectors |
| \(\Delta \mathbf k\) | scattering vector, defined here as \(\mathbf k'-\mathbf k\) |
| \(\lambda\) | wavelength of the incident radiation |
| \(\theta\) | Bragg angle, half the total scattering angle |
| \(m\) | diffraction order in Bragg’s law |
| \(\rho(\mathbf r)\) | scattering density |
| \(A(\Delta\mathbf k)\) | scattering amplitude |
| \(S_{\mathbf G}\) | structure factor of one unit cell |
| \(f_j\) | atomic form factor or scattering amplitude of atom \(j\) |
| \(\mathbf r_j\) | position of atom \(j\) inside the basis |
| \(N\) | number of illuminated unit cells |
| \(I\) | measured scattering intensity |
| \(\mathbf u(t)\) | instantaneous atomic displacement from equilibrium |
| \(\mathbf q\) | reduced wave vector of a lattice vibration |
| \(\omega_{\mathbf q}\) | angular frequency of a lattice vibration |
| \(\mu\) | absorption coefficient of a probe beam |
4.2 Why Reciprocal Space Appears in Diffraction
Diffraction
Diffraction is the phenomenon in which waves spread and form interference patterns when they encounter an obstacle or periodic structure with dimensions comparable to their wavelength.
A crystalline solid is periodic in real space, but diffraction records interference in wave-vector space. The reason is that a wave scattered from many periodically arranged atoms carries phase information. Constructive interference survives only for special wave-vector changes. Those allowed changes form the reciprocal lattice.
Diffraction methods therefore complement real-space imaging. Local imaging methods are well suited to defects, surfaces, and interfaces. Diffraction averages over a large illuminated volume and gives precise information about the ideal periodic structure: the unit-cell size, the arrangement of atoms inside the cell, and the scattering density inside the cell.
4.3 The Reciprocal Lattice
4.3.1 Definition from Periodic Plane Waves
Let the direct Bravais lattice be
A plane wave \(e^{i\mathbf k\cdot\mathbf r}\) has the periodicity of the Bravais lattice if
\[ e^{i\mathbf k\cdot(\mathbf r+\mathbf R_n)} = e^{i\mathbf k\cdot\mathbf r} \]
for every lattice vector \(\mathbf R_n\). Hence
for all \(\mathbf R_n\). The set of all wave vectors satisfying this condition is the reciprocal lattice. Its vectors are denoted by \(\mathbf G\):
Equivalently,
\[ \mathbf G\cdot\mathbf R_n = 2\pi \times \text{integer}. \]
This definition emphasizes the physical meaning of the reciprocal lattice: it is the set of wave vectors compatible with the periodicity of the direct lattice.
Plane wave
A plane wave is a wave whose surfaces of constant phase are infinite, parallel planes, with the wave propagating perpendicular to these planes.
4.3.2 Reciprocal Primitive Vectors
The reciprocal primitive vectors are defined by
For the primitive direct-cell volume
\[ V_\mathrm c=\mathbf a_1\cdot(\mathbf a_2\times\mathbf a_3), \]
one obtains
Every reciprocal-lattice vector can be written as
The reciprocal lattice is itself a Bravais lattice, and the reciprocal lattice of the reciprocal lattice is the original direct lattice. If the direct primitive cell has volume \(V_\mathrm c\), the reciprocal primitive cell has volume \((2\pi)^3/V_\mathrm c\).
4.3.3 Fourier Interpretation
If a function has the periodicity of the Bravais lattice,
\[ f(\mathbf r+\mathbf R_n)=f(\mathbf r), \]
then its Fourier expansion contains only reciprocal-lattice vectors:
This is why reciprocal space is not just a geometric construction. It is the natural Fourier space for any lattice-periodic quantity, such as an ideal electron density or scattering density.
4.3.4 Examples: Simple Cubic, fcc, and bcc
For a simple cubic lattice,
\[ \mathbf a_1=a\hat{\mathbf x}, \qquad \mathbf a_2=a\hat{\mathbf y}, \qquad \mathbf a_3=a\hat{\mathbf z}, \]
so
The reciprocal lattice of a simple cubic lattice is therefore again simple cubic, with reciprocal lattice constant \(2\pi/a\).
For cubic centered lattices, the important result is:
This fact is central for interpreting Brillouin zones and diffraction selection rules in cubic crystals.
4.4 Crystal Planes and Reciprocal Vectors
4.4.1 Reciprocal Vectors as Plane Normals
For a family of lattice planes labeled by Miller indices \((hkl)\), the associated reciprocal-lattice vector is
The vector \(\mathbf G_{hkl}\) is normal to the corresponding plane family. The shortest reciprocal vector perpendicular to that family has magnitude
Thus reciprocal space converts a family of parallel real-space planes into a single normal vector.
4.4.2 Simple Cubic Plane Spacing
For the simple cubic lattice,
\[ \mathbf G_{hkl} = \frac{2\pi}{a} \left( h\hat{\mathbf x} +k\hat{\mathbf y} +l\hat{\mathbf z} \right), \]
and therefore
Larger Miller indices correspond to shorter interplanar spacings.
4.5 Brillouin Zones
The first Brillouin zone is the Wigner-Seitz cell of the reciprocal lattice.
To construct it, choose a reciprocal-lattice point as the origin, connect it to neighboring reciprocal-lattice points, and draw perpendicular bisecting planes. These planes are called Bragg planes because wave vectors ending on them satisfy the elastic diffraction condition.
Higher Brillouin zones are defined similarly: the \(n\)th zone consists of the region reached from the origin after crossing exactly \(n-1\) Bragg planes.
For a bcc direct lattice, the reciprocal lattice is fcc, so the first Brillouin zone is the Wigner-Seitz cell of an fcc lattice. For an fcc direct lattice, the reciprocal lattice is bcc, so the first Brillouin zone is the Wigner-Seitz cell of a bcc lattice.
4.6 Elastic Diffraction from a Periodic Crystal
4.6.1 Bragg’s Direct-Space Picture
Consider radiation of wavelength \(\lambda\) incident on a family of lattice planes separated by \(d_{hkl}\). If the beam makes an angle \(\theta\) with the planes, the path difference between waves reflected from neighboring planes is
\[ 2d_{hkl}\sin\theta . \]
Constructive interference requires this path difference to be an integer multiple of the wavelength:
This is Bragg’s law. Since typical interplanar spacings are of order angstroms, the probe wavelength must also be of that order. Visible light is therefore not suitable for resolving ordinary atomic crystal spacings by Bragg diffraction.
4.6.2 Laue’s Reciprocal-Space Picture
Now describe diffraction as scattering from a three-dimensional lattice of point scatterers. Let the incident wave vector be \(\mathbf k\) and the scattered wave vector be \(\mathbf k'\). For elastic scattering,
\[ |\mathbf k|=|\mathbf k'|=\frac{2\pi}{\lambda}. \]
For constructive interference from all equivalent lattice points, the phase change between waves scattered from points separated by \(\mathbf R_n\) must be an integer multiple of \(2\pi\):
\[ e^{i(\mathbf k'-\mathbf k)\cdot\mathbf R_n}=1 . \]
Comparison with the definition of the reciprocal lattice gives the Laue condition:
The reciprocal lattice therefore determines the possible elastic diffraction peaks.
Some texts define the scattering vector with the opposite sign. Here we use \(\Delta\mathbf k=\mathbf k'-\mathbf k\). The measured intensity is unchanged by reversing the sign convention.
4.6.3 Ewald Construction
The Ewald construction gives a geometric way to see when the Laue condition is satisfied.
Draw the incident vector \(\mathbf k\) so that its tip ends at a reciprocal-lattice point. Then draw a sphere of radius \(|\mathbf k|\) centered at the tail of \(\mathbf k\). If another reciprocal-lattice point lies on the sphere, then an elastically scattered beam exists in the direction \(\mathbf k'\) from the sphere center to that reciprocal-lattice point.
Changing the wavelength changes the radius of the Ewald sphere. Rotating the crystal rotates the reciprocal lattice. Both operations are practical ways to bring reciprocal-lattice points onto the Ewald sphere.
4.6.4 Equivalence of Bragg and Laue Conditions
For elastic scattering, the triangle formed by \(\mathbf k\), \(\mathbf k'\), and \(\mathbf G=\mathbf k'-\mathbf k\) gives
\[ |\mathbf G|=2|\mathbf k|\sin\theta . \]
Using \(|\mathbf k|=2\pi/\lambda\) and \(|\mathbf G|=m 2\pi/d_{hkl}\) gives
\[ m\frac{2\pi}{d_{hkl}} = 2\frac{2\pi}{\lambda}\sin\theta . \]
Thus
Bragg’s law and the Laue condition are therefore the same diffraction condition expressed in real space and reciprocal space.
4.6.5 Bragg Planes and Brillouin-Zone Boundaries
Starting from \(\mathbf k'=\mathbf k+\mathbf G\) and using elastic scattering,
\[ |\mathbf k'|^2=|\mathbf k|^2 . \]
Therefore
\[ |\mathbf k+\mathbf G|^2=|\mathbf k|^2, \]
or
\[ 2\mathbf k\cdot\mathbf G+|\mathbf G|^2=0 . \]
Since \(-\mathbf G\) is also a reciprocal-lattice vector, this can be written as
This is the equation of the plane halfway between the origin and the reciprocal-lattice point \(\mathbf G\). Hence a wave vector from the center of a Brillouin zone to a zone boundary satisfies the Laue condition.
4.7 General Scattering Amplitude and the Phase Problem
4.7.1 Scattering Density
In the kinematic approximation, each point in the sample scatters once, and the scattered waves add coherently. This approximation is usually adequate for X-rays and neutrons, while electron diffraction often requires multiple-scattering corrections.
Let \(\rho(\mathbf r)\) be the scattering density. The scattered amplitude is proportional to the Fourier transform of \(\rho\):
The measured intensity is
Diffraction therefore measures the squared magnitude of a Fourier component of the scattering density.
4.7.2 The Phase Problem
If the complex amplitude \(A(\Delta\mathbf k)\) were measured, the scattering density could be reconstructed directly by inverse Fourier transformation. Standard diffraction experiments measure intensity, not phase. They therefore determine \(|A|^2\), but not the complex phase of \(A\).
This is the phase problem. In practice, one proposes a structural model, calculates its diffraction intensities, compares them with the measured data, and refines the structural parameters until agreement is achieved.
4.7.3 Autocorrelation View
Because the intensity is the squared magnitude of a Fourier transform, it can also be interpreted as the Fourier transform of the autocorrelation of the scattering density:
\[ I(\Delta\mathbf k) \propto \int AC(\mathbf r) e^{-i\Delta\mathbf k\cdot\mathbf r} d^3r , \]
where
\[ AC(\mathbf r) = \int \rho(\mathbf r') \rho(\mathbf r'+\mathbf r) d^3r' . \]
The autocorrelation has maxima at displacement vectors connecting pairs of atoms. This interpretation helps explain why diffraction is sensitive to relative atomic positions, even though phases are not directly measured.
4.8 Lattice, Basis, and Structure Factor
4.8.1 Factorization of the Scattering Amplitude
A crystal structure can be viewed as a Bravais lattice decorated by a basis. Let the basis contain atoms at positions \(\mathbf r_j\). Then the total scattering density may be regarded as a convolution of
- a lattice of delta functions at \(\mathbf R_n\),
- a basis of atoms at \(\mathbf r_j\),
- and the scattering density of each atom.
Using the Fourier transform of this convolution, the scattering amplitude factorizes into a lattice part and a basis part.
For a finite crystal with \(N\) illuminated cells,
The lattice sum selects \(\Delta\mathbf k=\mathbf G\). For allowed Bragg peaks,
The structure factor is
Here \(\rho_B(\mathbf r)\) is the scattering density of the basis inside one unit cell.
4.8.2 Atomic Form Factor
The basis scattering density can be decomposed into atomic scattering densities. This gives
where the atomic form factor is
Thus the structure factor contains two ingredients:
- the atomic form factors \(f_j\), which depend on the type of atom and the probe;
- phase factors \(e^{-i\mathbf G\cdot\mathbf r_j}\), which depend on where the atoms sit inside the unit cell.
For a spherical atomic charge density, the atomic form factor depends only on \(|\mathbf G|\):
\[ f(G) = \int_0^{R_A} 4\pi r^2\rho_A(r) \frac{\sin(Gr)}{Gr} dr . \]
In the forward-scattering limit, \(G\to 0\), one obtains
For X-ray diffraction, this means that the scattering strength of an atom at small scattering angle is mainly controlled by its number of electrons. Light elements are therefore difficult to detect next to heavy elements by X-ray diffraction.
4.8.3 What Peak Positions and Intensities Tell Us
The location of a Bragg peak determines a reciprocal-lattice vector and hence the unit-cell geometry. The intensity of that peak is controlled by the structure factor and hence by the basis.
More precisely, measured peak positions determine lattice parameters and symmetry, while measured intensities constrain the atomic positions and scattering strengths inside the unit cell.
4.9 Examples of Structure Factors and Systematic Extinctions
4.9.1 Fractional Coordinates
Write a basis position as
\[ \mathbf r_j = u_j\mathbf a_1+v_j\mathbf a_2+w_j\mathbf a_3 . \]
Then
\[ \mathbf G\cdot\mathbf r_j =2\pi(hu_j+kv_j+lw_j), \]
and the structure factor becomes
4.9.2 CsCl Structure
For the CsCl structure, take atoms at
\[ (0,0,0), \qquad \left(\frac12,\frac12,\frac12\right). \]
Then
\[ S_{hkl} = f_1+f_2e^{-i\pi(h+k+l)}. \]
Hence
Odd reflections are not necessarily absent, because Cs and Cl have different scattering factors. Their intensity is reduced when the two scattering factors are similar.
4.9.3 bcc Lattice
A bcc lattice can be treated as a simple cubic lattice with identical atoms at
\[ (0,0,0), \qquad \left(\frac12,\frac12,\frac12\right). \]
Now \(f_1=f_2=f\), so
Thus bcc diffraction has systematic extinctions for \(h+k+l\) odd.
4.9.4 fcc Lattice
Using the conventional cubic cell, the fcc positions are
\[ (0,0,0), \qquad \left(0,\frac12,\frac12\right), \qquad \left(\frac12,0,\frac12\right), \qquad \left(\frac12,\frac12,0\right). \]
For identical atoms,
\[ S_{hkl} = f \left[ 1 + e^{-i\pi(k+l)} + e^{-i\pi(h+l)} + e^{-i\pi(h+k)} \right]. \]
Therefore
So fcc reflections occur only when all Miller indices have the same parity.
Systematic extinctions are not experimental failures. They are interference effects caused by the basis.
4.10 Thermal Motion, Inelastic Scattering, and Debye-Waller Reduction
4.10.1 Why Atoms Cannot Be Treated as Perfectly Static
So far we assumed a rigid lattice. Real atoms vibrate about their equilibrium positions. At \(T>0\), thermal vibrations are present; even at \(T=0\), quantum zero-point motion remains.
The instantaneous position of an atom can be written as
where \(\mathbf u(t)\) is the displacement from the equilibrium position.
Inserting this into the phase factor gives
\[ e^{-i\Delta\mathbf k\cdot\mathbf u(t)} \simeq 1 -i\Delta\mathbf k\cdot\mathbf u(t) -\frac12[\Delta\mathbf k\cdot\mathbf u(t)]^2+\cdots . \]
The constant term gives elastic Bragg scattering. The linear term gives one-phonon inelastic scattering. The quadratic and higher terms lead to multiphonon contributions and to the reduction of elastic intensity.
4.10.2 Inelastic Scattering Selection Rules
A lattice vibration may be represented as a superposition of waves with wave vector \(\mathbf q\) and frequency \(\omega_{\mathbf q}\). The inelastic scattering condition becomes
The energy condition is
Equivalently, in quantum language, the probe exchanges energy \(\hbar\omega_{\mathbf q}\) and crystal momentum \(\hbar\mathbf q\) with a lattice vibration, while \(\hbar\mathbf G\) may be transferred to the crystal as a whole.
This is the conceptual bridge from diffraction to phonon spectroscopy.
4.10.3 Debye-Waller Factor
Thermal motion does not mainly broaden the elastic Bragg peaks. Instead, it reduces their elastic intensity and transfers spectral weight into diffuse inelastic background.
For isotropic displacements, averaging the elastic structure factor over vibrations gives approximately
\[ \langle S_{\mathbf G}\rangle \simeq S_{\mathbf G}^\text{stat} \exp\left[ -\frac16G^2\langle u^2\rangle \right]. \]
Since intensity is proportional to the squared amplitude,
This exponential reduction is the Debye-Waller factor.
In a classical harmonic estimate,
\[ \langle u^2\rangle = \frac{3k_BT}{M\omega^2}, \]
so
The elastic intensity decreases with increasing temperature and with increasing reciprocal-vector magnitude.
4.11 Experimental Probes and Methods
4.11.1 Wavelength Requirement
Diffraction requires a wavelength comparable to lattice spacings. For matter waves, the de Broglie relation gives
For a nonrelativistic particle of mass \(M\) and kinetic energy \(E\),
The useful wavelength range for crystal diffraction is typically \(\lambda\sim 1,\text{\AA}\).
A second practical requirement is sufficient penetration depth. If a beam passes through thickness \(d\), its intensity is attenuated approximately as
\[ I=I_0e^{-\mu d}. \]
For bulk crystal structure analysis, the absorption length should be comparable to or larger than typical crystal dimensions.
4.11.2 X-rays
X-rays are produced in laboratory tubes by bombarding metal targets, such as Cu or Mo, with energetic electrons. The emitted spectrum contains a continuous bremsstrahlung background and characteristic lines. Common characteristic wavelengths are
\[ \lambda_{\mathrm{Cu},K\alpha_1}=1.541,\text{\AA}, \qquad \lambda_{\mathrm{Mo},K\alpha_1}=0.709,\text{\AA}. \]
X-rays are well suited to bulk structural analysis because their absorption depths are often in the millimeter to centimeter range. Their main limitation is that X-ray scattering strength scales strongly with electron number, so light atoms are difficult to detect next to heavy atoms.
4.11.3 Electrons
Electrons scatter strongly through Coulomb interactions with electrons and nuclei. Low-energy electrons with angstrom-scale wavelengths have short penetration depths and are especially useful for surface diffraction methods such as LEED and RHEED.
High-energy electrons in transmission electron microscopes have much shorter wavelengths, can be focused into very small probes, and can analyze small sample regions. Because electrons scatter strongly, samples must usually be very thin to reduce multiple scattering.
4.11.4 Neutrons
Thermal neutrons naturally have wavelengths of order angstroms and energies comparable to room-temperature thermal energies. They interact mainly with nuclei rather than electron clouds. Their scattering strength varies strongly between elements and even between isotopes, rather than increasing smoothly with atomic number.
Neutrons are therefore useful for locating light atoms, distinguishing neighboring elements in the periodic table, and probing magnetic structures because neutrons carry a magnetic moment.
4.11.5 X-ray Diffractometry Methods
The Bragg condition contains both the wavelength and the angle. Different experimental methods satisfy it in different ways.
Laue method. A single crystal is illuminated by a continuous X-ray spectrum at fixed orientation. Each set of planes selects the wavelength that satisfies Bragg’s law. The result is a spot pattern useful for determining crystal orientation and symmetry.
Rotating-crystal method. Monochromatic radiation is used, and the crystal orientation is varied. In an \(\omega\)-\(2\theta\) scan, the sample and detector angles are coordinated so that Bragg peaks are recorded as the condition is met.
Debye-Scherrer or powder method. Monochromatic radiation is incident on a powder sample containing many small crystallites in all orientations. For each set of planes, some crystallites satisfy Bragg’s law, producing cones of scattered intensity that appear as rings or peaks on a detector.
4.12 Optional / Further Reading: Mössbauer Effect
The same ideas of elastic scattering, recoil, and phonon exchange also underlie the Mössbauer effect: recoil-free emission and absorption of gamma radiation by nuclei embedded in a crystal.
For a free atom emitting a gamma quantum, recoil shifts the emission energy and prevents resonant absorption by an identical atom. In a crystal, the recoil may be taken up by the whole lattice, or equivalently the process can occur without creating a phonon. This produces a sharp zero-phonon line. The probability of recoil-free emission is closely related to the Debye-Waller factor.
This topic is not needed for the main diffraction analysis, but it illustrates how lattice vibrations, elastic scattering, and quantized excitations are connected.
4.13 Figure Placeholders
4.14 Bridge to the Next Lecture
This lecture established reciprocal space as both a mathematical and an experimental language. The reciprocal lattice describes the Fourier components of a periodic crystal, and diffraction makes those components visible through Bragg peaks. Elastic diffraction determines the static crystal structure: lattice geometry from peak positions and basis information from peak intensities.
The next step is to allow the atoms to move. Once atomic displacements become time dependent, scattering is no longer purely elastic. The momentum transfer can be written as a reciprocal-lattice vector plus a reduced wave vector, and the energy transfer gives the frequency of a lattice vibration. This is the route from diffraction to phonon dispersion and, ultimately, to the thermal properties of solids.
4.15 Take-Home Messages
- The reciprocal lattice is the set of wave vectors compatible with the translational periodicity of the direct lattice.
- Lattice-periodic functions have Fourier components only at reciprocal-lattice vectors.
- Reciprocal-lattice vectors encode both the normals and spacings of crystal planes.
- Bragg’s law and the Laue condition are equivalent descriptions of elastic diffraction.
- The Ewald construction gives a geometric picture of which reciprocal-lattice points are sampled in an experiment.
- Bragg-peak positions determine the lattice, while Bragg-peak intensities constrain the basis.
- Systematic extinctions are interference effects caused by the positions and identities of atoms in the unit cell.
- Thermal motion reduces elastic Bragg intensities and produces inelastic scattering.
4.16 Problem Set
Reciprocal Lattice. Starting from real-space FCC and BCC lattices, compute the corresponding reciprocal lattice vectors.
From Laue to Bragg. Starting from the condition for constructive interference from lattice points separated by \(\mathbf R_n\), derive the Laue condition \[ \Delta\mathbf k=\mathbf G. \] Then show that elastic scattering gives Bragg’s law.
Structure Factor of CsCl and bcc. For atoms at \((0,0,0)\) and \((1/2,1/2,1/2)\), compute the structre factor \(S_{hkl}\). Discuss the difference between CsCl, where the two atoms have different form factors, and bcc, where they are identical.
fcc Extinction Rule. Use the conventional fcc positions \[ (0,0,0), \quad \left(0,\frac12,\frac12\right), \quad \left(\frac12,0,\frac12\right), \quad \left(\frac12,\frac12,0\right) \] to derive the fcc selection rule.
Diffraction Probes and Thermal Motion. Answer the following:
- Why are X-rays poor at locating light atoms next to heavy atoms?
- Why are neutrons often useful for locating hydrogen or distinguishing neighboring elements?
- What happens to elastic Bragg intensity when temperature increases?