Reciprocal Space in One Page
A quick reminder of the reciprocal basis and the key result from Lecture 14. Everything in this lecture builds on these three lines.
A second basis
So \(\mathbf a_1^*\perp\mathbf a_2,\mathbf a_3\); \(\mathbf a_2^*\perp\mathbf a_3,\mathbf a_1\); \(\mathbf a_3^*\perp\mathbf a_1,\mathbf a_2\), whether or not the cell has right angles.
Cross products over the volume
\(V\) is the cell volume. If the \(\mathbf a_i\) have dimension \([L]\), the \(\mathbf a_i^*\) have dimension \([L^{-1}]\): hence reciprocal.
\(\mathbf g\) is normal to \((hkl)\)
For \(\mathbf g = h\mathbf a_1^*+k\mathbf a_2^*+l\mathbf a_3^*\) and \(\mathbf r = x\mathbf a_1+y\mathbf a_2+z\mathbf a_3\). The components of \(\mathbf g\) are the Miller indices of the planes it is perpendicular to.
(hkl) Names a Family of Parallel Planes
Consider a cube with basis vectors \(\mathbf a_1, \mathbf a_2, \mathbf a_3\). Click each coloured plane (or its button) and work out its Miller indices.
Symmetry-equivalent families, once more
In the cube, the green face is (010). The yellow and purple faces are (100) and (001). These are symmetrically equivalent, so the family can be written \(\{100\}\) or equally \(\{010\}\).
For a cube, the members are obtained by permuting the indices and their signs: \(\{100\}\) = (100), (010), (001), \((\bar100)\), \((0\bar10)\), \((00\bar1)\).
For a tetragonal lattice (\(c \ne a = b\)), \(\{100\}\) has only (100), (010), \((\bar100)\) and \((0\bar10)\). (001) and \((00\bar1)\) are not symmetrically equivalent to the others. You can check this in the Family Explorer of Lecture 14.
Two parallel planes, one \((hkl)\)
A plane \((hkl)\) cuts the axes at \(1/h, 1/k, 1/l\), so its equation is \(hx+ky+lz=1\). The parallel plane through the origin is \(hx+ky+lz=0\). Both have Miller indices \((hkl)\), and the spacing between these nearest parallel neighbours is \(d_{hkl}\): equivalently, the distance of the plane with intercepts \(1/h, 1/k, 1/l\) from the origin.
One Plane, Two Families
The grey plane cuts \(\mathbf a_2\) at \(\tfrac12\): intercepts \((\infty, \tfrac12, \infty)\) give Miller indices \((020)\). The next \((020)\) plane in the \(+\mathbf a_2\) direction is… the green \((010)\) plane! The same plane can belong to two sets of planes with different Miller indices.
QWhat is the interplanar spacing for planes with Miller indices (010)?
QAnd for (020)?
Every \(\mathbf r \perp \mathbf g\) Lies in One Plane
Describe a vector \(\mathbf r\) in the real frame and a vector \(\mathbf g\) in the reciprocal frame:
with \(h, k, l\) mutually prime integers, and \(\mathbf r\perp\mathbf g\). Then \(\mathbf r\cdot\mathbf g = 0 \Rightarrow hx+ky+lz=0\), because \(\mathbf a_i\cdot\mathbf a_j^*=\delta_{ij}\).
So, for a fixed \(\mathbf g\), the set of all \(\mathbf r\) with real \(x, y, z\) satisfying \(\mathbf r\cdot\mathbf g=0\) forms a plane, the plane \(hx+ky+lz=0\). The family of planes parallel to it has Miller indices \((hkl)\).
Sweeping Out the Plane
Random real-space vectors \(\mathbf r\) (purple) are generated subject only to \(\mathbf r\cdot\mathbf g=0\). Each leaves a dot at its tip. The dots fill a plane perpendicular to \(\mathbf g\) (red), in a triclinic cell.
\(d_{hkl} = 1/|\mathbf g_{hkl}|\)
The interplanar spacing \(d_{hkl}\) is the perpendicular distance from the origin to the plane with intercepts \(1/h, 1/k, 1/l\) on the real basis vectors. Project any vector from the origin to that plane onto the unit normal.
Projecting \(\mathbf t\) on \(\hat{\mathbf n}\)
QWhat will be the d-spacing if the Miller indices are \((nh\ nk\ nl)\)?
Measuring \(d_{hkl}\)
A Plane in Real Space Is a Point in Reciprocal Space
A whole family of lattice planes \((hkl)\) collapses to one reciprocal lattice point \(hkl\): the tip of \(\mathbf g_{hkl}\). And \(\mathbf a_1^*\perp\mathbf a_2,\mathbf a_3\); \(\mathbf a_2^*\perp\mathbf a_3,\mathbf a_1\); \(\mathbf a_3^*\perp\mathbf a_1,\mathbf a_2\), whether or not the basis vectors are orthogonal. Here in 2D, looking down \(\mathbf a_3\).
After Fig. 7, Chapter 2 (Geometry of Crystals) of Elements of X-Ray Diffraction by Cullity and Stock, which shows crystal lattices and the corresponding reciprocal lattices for a cubic and a hexagonal system (there, \(\mathbf b_1, \mathbf b_2, \mathbf b_3\) denote the reciprocal basis vectors).
Interplanar Spacings for Different Crystal Systems
Evaluating \(1/d_{hkl}^2 = |\mathbf g_{hkl}|^2 = \mathbf g\cdot\mathbf g\) for each cell geometry gives these expressions. Click a row to load it into the calculator.
| Crystal system | Interplanar spacing |
|---|---|
| Cubic | \[d_{hkl}=\frac{1}{\sqrt{\dfrac{h^2}{a^2}+\dfrac{k^2}{a^2}+\dfrac{l^2}{a^2}}}=\frac{a}{\sqrt{h^2+k^2+l^2}}\] |
| Tetragonal | \[d_{hkl}=\frac{1}{\sqrt{\dfrac{h^2}{a^2}+\dfrac{k^2}{a^2}+\dfrac{l^2}{c^2}}}\] |
| Orthorhombic | \[d_{hkl}=\frac{1}{\sqrt{\dfrac{h^2}{a^2}+\dfrac{k^2}{b^2}+\dfrac{l^2}{c^2}}}\] |
| General form valid for all crystal systems with cell edges \(a, b, c\) and angles \(\alpha, \beta, \gamma\) |
\[\frac{1}{d_{hkl}^2}=\frac{1}{V^2}\Big[h^2b^2c^2\sin^2\alpha+k^2a^2c^2\sin^2\beta+l^2a^2b^2\sin^2\gamma+2hkabc^2(\cos\alpha\cos\beta-\cos\gamma)\]
\[\qquad+\,2kla^2bc(\cos\beta\cos\gamma-\cos\alpha)+2hlab^2c(\cos\gamma\cos\alpha-\cos\beta)\Big]\]
where \(V\) is the unit-cell volume:
\[V=(\mathbf a\times\mathbf b)\cdot\mathbf c = abc\sqrt{1-\cos^2\alpha-\cos^2\beta-\cos^2\gamma+2\cos\alpha\cos\beta\cos\gamma}\] |
d-Spacing Calculator
Choose a system; only the independent lattice parameters are editable. \(d\) is computed from the general formula and cross-checked as \(1/|\mathbf g|\).
Each stick is one set of equivalent planes placed at \(2\theta = 2\sin^{-1}(\lambda/2d)\) (Bragg's law, coming up in the diffraction lectures). Height = number of \((hkl)\) with that spacing, not a diffracted intensity; systematic absences are ignored. Hover a stick, click to select it.
X-rays Are Electromagnetic Waves
X-rays are the same kind of thing as visible light, only with a much shorter wavelength: roughly 0.01 to 10 nm (X-ray diffraction typically uses 0.05–0.25 nm, e.g. Cu Kα = 0.154 nm). That is the size of the gaps between atoms, and this coincidence is the reason X-rays can “see” crystal structure.
An Electromagnetic Wave in Flight
The electric field E and the magnetic field B oscillate in planes at right angles to each other, and both are at right angles to the direction the wave travels, which is the direction of \(\mathbf E\times\mathbf B\).
Why X-rays, and not light?
To learn about a repeating pattern, a wave has to be about as long as the repeat. Ocean waves slide right past a thin pole but get scattered by a row of piers. In the same way, visible light (400–700 nm) is a thousand times longer than the gap between atoms (about 0.1–0.5 nm) and simply does not notice the individual atoms.
X-rays, with \(\lambda \sim 0.1\) nm, are just the right size. Slide the wavelength in the explorer and compare the wave with a row of atoms drawn to the same scale.
Wavelength Explorer
Writing a Wave Down
A few lines of algebra turn the picture of a travelling wave into an expression, and show that the natural “address” of a wave is a vector in reciprocal space.
The electric field of a wave travelling along \(x\) can be written as
where \(A\) is the amplitude, \(\nu\) is the frequency of oscillation, and \(k = 1/\lambda\) is the inverse of the wavelength. (The magnetic field can be written the same way, but we will not need it.)
Mathematically it is much easier to handle waves with Euler’s formula, \(e^{i\phi} = \cos\phi + i\sin\phi\). Since \(\nu = c/\lambda = kc\),
Only the real part of this is physical: it is the shadow that a rotating arrow (a phasor) casts on the real axis. The panel on the right shows exactly this.
For diffraction we care about how the wave varies in space, not in time, so we drop the time part:
Phasor → Wave
The arrow of length \(A\) spins at angle \(\phi = 2\pi(kx_0-\nu t)\). Its projection on the real (horizontal) axis is \(A\cos\phi\): the value of the physical field at the marked point \(x_0\). Drag across the wave to move \(x_0\).
Beyond One Dimension: \(E(\mathbf r) = A\,e^{2\pi i\,\mathbf k\cdot\mathbf r}\)
In three dimensions \(kx\) becomes the dot product \(\mathbf k\cdot\mathbf r\), where \(\mathbf r\) is a real-space position vector. The field has the same value everywhere that \(\mathbf k\cdot\mathbf r\) is the same. In particular, \(E\) is the same as at the origin wherever \(\mathbf k\cdot\mathbf r = 0\), that is, wherever \(\mathbf r\) is perpendicular to \(\mathbf k\). These surfaces of equal phase are flat wavefronts, perpendicular to \(\mathbf k\).
- Coloured background = the wave seen from above, like ocean waves from a plane. Red/pink = crest (field positive), blue = trough (negative), white = near zero.
- Thin red lines = crests. They slide along \(\mathbf k\) as time passes; that motion is just the wave travelling.
- Red arrow \(\mathbf k\) = direction of travel. Its length is \(1/\lambda\): a longer arrow means crests closer together. Crests are always perpendicular to it.
- Dashed blue line = the wavefront through the origin, where \(\mathbf k\cdot\mathbf r = 0\). It stays fixed as a reference.
- Black dots = atoms of a square lattice (spacing \(a\)). Blue arrow \(\mathbf r\) points from the origin to the atom you tapped.
- Green dots = atoms where \(\mathbf k\cdot\mathbf r\) is a whole number. The wave there always does exactly what it does at the origin.
- Drag the red \(\mathbf k\) arrow. The crests turn to stay perpendicular to \(\mathbf k\). Shorten the arrow and the crests spread apart (longer \(\lambda\)).
- Tap an atom on the dashed line. The readout shows \(\mathbf k\cdot\mathbf r = 0\): that atom is perpendicular to \(\mathbf k\) and has the same field as the origin.
- Tap any other atom and read the phase. A value like 1.30 means it is 0.30 of a cycle (108°) out of step with the origin; 2.00 means perfectly in step.
- Press “Snap k to a reciprocal vector”. Now every atom turns green, because \(\mathbf g\cdot\mathbf r = hn_1 + kn_2\) is always an integer. The constant-phase lines run along rows of atoms, the \((hk)\) planes. Snap to \(\mathbf g_{10}\) for vertical planes, or drag near (1,1) and snap for diagonal \((11)\) planes.
When every atom feels the wave in the same phase, every atom re-radiates in step and the scattered waves add up to a strong diffracted beam. For a random \(\mathbf k\), the atoms are out of step and their contributions cancel. That is why reciprocal lattice vectors decide where diffraction happens.
Green does not mean “on a red crest”. The crests keep sliding past the green atoms. Green means “always in step with the origin”. Press Pause to see crests sometimes on the atom rows and sometimes between them; the green atoms still share one phase.
Two different k’s. Bold \(\mathbf k\) is the wave vector; the italic \(k\) in \(\mathbf g_{hk}\) is a Miller index. They are unrelated: it is only a clash of letters.
What Is Diffraction?
Diffraction is what happens when waves (for us, EM waves) interact with a periodic, crystalline object: the waves leave in only a few special directions, producing spots of high and low intensity.
Red Laser Pointer Through a Fine Mesh
In class we shine a red laser pointer (\(\lambda \approx 640\)–\(660\) nm) through a fine woven mesh. The mesh is a 2D “crystal” with a spacing of a few micrometres, comparable to the wavelength of red light, so it diffracts the light into a grid of spots on the wall. X-rays on a real crystal do exactly the same, just 10 000 times smaller.
Make the mesh finer (smaller a) and watch the spots spread further apart: the pattern on the wall behaves like the inverse of the mesh. For a spot of order \((h,k)\): \(\sin\alpha_x = h\lambda/a\), \(\sin\alpha_y = k\lambda/b\).
2D Real Lattices and Their Diffraction Patterns
Pick a real (direct) lattice on the left. The pattern on the right is computed by adding up the waves scattered by every point, which is the same as taking a Fourier transform of the lattice.
How Do Atoms Scatter X-rays?
The X-ray doesn’t bounce off the atom like a ball. It shakes the atom’s electrons, and the shaking electrons send out new X-rays of their own.
The electric field of the incoming X-ray pushes on the electrons in each atom. Because the field swings back and forth, the electrons are driven to oscillate at the same frequency as the X-ray.
Every accelerating or decelerating charge emits EM waves. An oscillating electron is always accelerating, so it acts like a tiny radio antenna.
The emitted wave has the frequency of the electron’s oscillation, which is the frequency of the incident X-ray. So each atom re-emits X-rays of the same wavelength, spreading out in all directions (spherical waves).
An Electron as a Tiny Antenna
Arrows on the left: the incident E-field. The electron (blue) is pushed up and down in step with it and sends out circular wavefronts. Change \(\lambda\): the emitted rings always have the same spacing as the incoming crests.
A Single Plane of Atoms
An X-ray beam falls on one plane of atoms. Every atom emits a spherical wave. The picture below is a true sum of all those wavelets, recomputed every frame. Where crests meet crests, the waves add up (constructive interference); where crests meet troughs, they cancel.
?Why are the scattered waves coherent?
Imagine carrying a detector far away and walking in a half-circle above the row of atoms, noting how bright the X-rays are in each direction. The graph is a record of that walk.
- Centre dot = the whole row of atoms. From far away the row shrinks to a point, so every direction is measured from there.
- Grey baseline = the plane of atoms. 0° = skimming along the plane to the right, 90° = straight up, 180° = skimming to the left. Angles are measured from the plane, like \(\theta\).
- Distance from the centre = brightness. The outer arc (“max”) means all N wavelets arrive exactly in step. The inner arc is half that, and the centre itself means complete cancellation.
- Green shape = brightness in every direction. Each bulge that reaches the outer arc is a lobe, that is, a scattered beam. Each lobe is labelled with its angle.
- Red arrow = the incoming X-rays, drawn only for reference. It is not a lobe.
For an exit angle \(\beta\), the wavelets from two neighbouring atoms (spacing \(s\)) differ in path by \(s(\cos\theta - \cos\beta)\). A beam forms wherever this is a whole number \(n\) of wavelengths:
- \(n = 0\) gives \(\beta = \theta\), the mirror beam. It is always there, whatever the spacing or wavelength.
- \(n = \pm1, \pm2, \dots\) give extra diffracted beams, but only if the angle they need lies between 0° and 180°.
- Small spacing (\(s < \lambda/(1+\cos\theta)\)): only \(n = 0\) fits, so there is one lobe.
- Default settings (\(\theta = 35^\circ\), \(s = 0.8\lambda\)): \(n = 0\) and \(n = +1\) fit, so there are two lobes, at 35° and about 116°. That is why you usually see two.
- Large spacing (e.g. \(s = 2.5\lambda\), \(\theta = 80^\circ\)): \(n = -2\) to \(+2\) fit, so there are five lobes.
This is the same equation as an optical diffraction grating.
- Is one lobe the incident beam? No. Every lobe is scattered (diffracted) X-rays. The incident beam is the red arrow, arriving from \(180^\circ - \theta\).
- Where is the beam that goes straight through? It goes below the plane. The graph shows only the half above the plane, the reflection side.
- Small wiggles between lobes are partial cancellation, because the row has only N atoms. There are \(N-2\) of them between neighbouring beams, and they shrink as N grows.
- N = 1 gives a full half-circle because a single atom has nothing to interfere with, so it scatters equally everywhere.
- More atoms make narrower lobes. With many atoms, even a small change in angle puts the far-apart atoms out of step, so the beam survives only in a very narrow range. A real crystal plane, with millions of atoms, gives extremely sharp beams.
Scattering by a Single Electron: the Thomson Equation
An accelerating charge re-radiates at the frequency it is driven at. How much it radiates in a given direction depends on the angle between that direction and the electron’s acceleration.
Let the X-ray beam travel along \(+x\) and place an observer at P, a distance \(r\) from the electron, in the \(x\)–\(z\) plane, at the scattering angle \(2\theta\) from the beam. If \(\alpha\) is the angle between the line of sight O→P and the electron’s acceleration (which is parallel to \(\mathbf E\)), the scattered intensity is given by the Thomson equation:
Unpolarised beam. A laboratory X-ray tube gives an unpolarised beam: \(\mathbf E\) points in a random direction in the \(y\)–\(z\) plane. On average half the intensity is in each component, \(I_{0y} = I_{0z} = I_0/2\).
- y\(\mathbf E\parallel y\): P lies in the \(x\)–\(z\) plane, so O→P is perpendicular to \(y\) for every \(2\theta\). Then \(\alpha = \pi/2\) and \(I_y(P) = I_{0y}\,K/r^2\).
- z\(\mathbf E\parallel z\): O→P makes \(2\theta\) with \(x\), so \(\alpha = \pi/2-2\theta\) with \(z\) and \(\sin^2\alpha = \cos^2 2\theta\). Then \(I_z(P) = I_{0z}\,(K/r^2)\cos^2 2\theta\).
The two components are incoherent, so their intensities simply add:
The angular term in brackets is the polarization factor.
Where Is \(\alpha\)?
The shaded shapes are the radiation patterns \(\propto\sin^2\alpha\) of an electron driven along \(z\) (left) and along \(y\), out of the page (right). The observer P sits at \(2\theta\) in the \(x\)–\(z\) plane. Watch \(\alpha\) change for the \(z\)-component while the \(y\)-component always radiates fully.
Polarization Factor: Angular Dependence
Coherent vs Incoherent Scattering of X-rays
Recall: the scattered waves interfere because they are coherent. Why? Not all scattering is.
Thomson scattering
- The E-field drives the electron, which re-emits at the same frequency.
- Every electron shifts the phase by exactly \(\pi\).
- Phase relations are preserved, so waves from different atoms can interfere: diffraction.
Compton scattering
- The photon acts as a particle: a billiard-like collision that transfers momentum.
- The photon loses energy, so its wavelength increases.
- Random phase change: no diffraction information, only a smooth background.
Scattering by a Single Atom: the Atomic Scattering Factor
- θ=0No path difference between the electrons, so all \(Z\) scatter in phase: \(f = Z\).
- θ≠0Path differences inside the electron cloud cause partial cancellation, so \(f\) falls as \(\sin\theta/\lambda\) grows.
- ZHeavier atoms (more electrons) scatter more strongly.
\(f\) Falls with \(\sin\theta/\lambda\)
Parametrised form used by De Graef and McHenry (Eq. 12.3), with tabulated coefficients \(a_i, b_i\) for each element.
Did It Sink In? - 15-16
Test your mastery of families of planes, \(d_{hkl}=1/|\mathbf g_{hkl}|\), X-rays as waves, diffraction, Thomson scattering, the polarization factor and the atomic scattering factor.