Optics 058 · Diffraction, Fourier optics, and computational imaging

Crystal Diffraction & Reciprocal Space

How can the spots and rings in a diffraction pattern reveal a crystal’s lattice? Spot spacing gives the lattice-plane spacing, and the pattern's symmetry gives the lattice symmetry. Each spot marks a reciprocal-lattice point touching the Ewald sphere. A powder holds every orientation at once, so spots spread into rings. The reflections that go missing tell simple, body-centred and face-centred cubic apart. The pattern tests a model but cannot rebuild the structure uniquely.

Rotate a finite cubic crystal and compute its coherent detector record. Explore Ewald-sphere intersections and white-beam Laue patterns, then derive powder rings from the same pair distances. Systematic absences in simple, body-centred and face-centred cubic lattices show what a diffraction pattern can and cannot tell you.

Cell edge / selected plane spacing

a,d_{hkl}

$0.50$

Selected basis intensity / atom count

|F/n_b|^2,N_{\rm at}

$50\%$

Ideal scattering angle / Ewald mismatch

2\theta_B,\Delta_E

$0.00\pi$

Model regime $\text{valid model regime}$

Model checked against analytic results

Crystal Diffraction & Reciprocal Space

Conventional cell edge (not arbitrary plane spacing)

a

4\,\mathring{\mathrm A}

Crystal yaw

\varphi

11.10^{\circ}

Crystal tilt

\chi

0^{\circ}

Wavelength / Laue central wavelength

\lambda_0

1.54\,\mathring{\mathrm A}

Complete cells per axis

N

7

Lattice: SC / BCC / FCC

\ell

0

Mode: mono / Laue / powder

m

0

Laue relative half-bandwidth

b

0.35

Physical crystal-to-detector distance

D

120\,\mathrm{mm}

Display gain exponent (raw data unchanged)

g

3

Selected reflection (indices on buttons below)

j

0

Rock the crystal in monochromatic light and watch spots appear near an Ewald-sphere intersection. Switch to white-beam Laue to use a band of wavelengths, or to random powder to get rings calculated from the same atom pairs. Inspect the atom cluster, the detector face and reciprocal space separately. Simple, body-centred and face-centred cubic lattices differ because of interference within the basis; a hollow node is not the same as an allowed reflection that misses the sphere. The cell edge rescales the atoms and the reciprocal spacing. Gold marks an ideal reflection; cyan records are the simulated data from the finite cluster. Exposure changes brightness only.

Scattering is elastic, scalar and kinematic, from identical unit point atoms in complete conventional cubic cells. The lattice sum is exact over a finite crystal; Laue patterns integrate incoherently over a uniform wavelength band; powder rings use the exact isotropic Debye pair average with radial interpolation. The forward detector takes point samples, and the display gain and beamstop are shown explicitly. The ideal reciprocal geometry is kept separate from the simulated records. The Lab does not recover phases or invert a unique structure, and it leaves out material calibration, atomic form factors, polarization, absorption, multiple scattering, thermal motion and noise. Atoms and supports are enlarged; real detector directions share one millimetre scale.

Physics tutorial

Audit the physics of Crystal Diffraction & Reciprocal Space

BackgroundA diffraction spot is not a tiny picture of an atom. It appears in an outgoing direction where, at the given wavelength, the waves scattered by many atoms arrive in phase: $\mathbf q=(\mathbf s_{\rm out}-\mathbf s_{\rm in})/\lambda$ . The detector, the atom cluster and the Ewald construction share the same orientation and length scales.

Why it mattersHow can the spots and rings in a diffraction pattern reveal a crystal’s lattice?

Start with the essentials

Focus question: How can the spots and rings in a diffraction pattern reveal a crystal’s lattice?
One-sentence intuition: Add the complex amplitudes from every atom in the finite cluster, then square: $I=\left|\sum_j e^{2\pi i\mathbf q\cdot\mathbf r_j}\right|^2$ . A single-wavelength exposure, a white-beam Laue average and an average over randomly oriented powder grains are three different measurements.

Core mathematical model

Cell edge is not plane spacing

\begin{aligned}\mathbf g_{hkl}&=R\,(h,k,l)/a\\d_{hkl}&=a/\sqrt{h^2+k^2+l^2}\\2d_{hkl}\sin\theta_B&=\lambda\end{aligned}

The Lab uses cubic conventional cells and gives reciprocal frequency in cycles per angstrom. The scattering angle at the detector is twice the Bragg angle to the planes. Changing the cell edge rescales every atomic coordinate, the reciprocal spacing and the recorded pattern together. The angles come from the outgoing directions.

Why a finite cluster gives broadened peaks

\begin{aligned}J(\mathbf q)&=\frac{|F(aR^{\mathsf T}\mathbf q)|^2}{n_b^2}\\&\quad\prod_{\alpha=1}^{3}\left[\frac{\sin(\pi N t_\alpha)}{N\sin(\pi t_\alpha)}\right]^2\\\mathbf t&=aR^{\mathsf T}\mathbf q\end{aligned}

The formula factors the sum exactly for a cube of complete conventional cells, each holding the same basis of point scatterers. At integer reciprocal coordinates the ratio takes its continuous limit. More cells give narrower peaks, and the side lobes between nodes are a real effect of the finite cluster. Intensity is divided by the square of the atom count, not by whichever pixel happens to be brightest.

Basis interference creates systematic absences

\begin{aligned}F_{\rm SC}&=1\\F_{\rm BCC}&=1+(-1)^{h+k+l}\\F_{\rm FCC}&=1+(-1)^{h+k}\\&\quad+(-1)^{h+l}+(-1)^{k+l}\end{aligned}

Every atom scatters with the same amplitude, equal to one. BCC allows reflections whose indices add up to an even number; FCC allows indices that are all even or all odd. A hollow reciprocal node marks an absence caused by the basis. That differs from an allowed reflection that simply misses the Ewald sphere. A detector pixel near, but not on, a forbidden node can still pick up side-lobe intensity.

Ewald sphere and white-beam Laue

\begin{aligned}|\mathbf s_{\rm in}/\lambda+\mathbf g|&=1/\lambda\\\lambda_{\mathbf g}&=-\frac{2\mathbf s_{\rm in}\cdot\mathbf g}{|\mathbf g|^2}\\\overline J_{\rm Laue}&=\int w(\lambda)J_\lambda\,\mathrm d\lambda\end{aligned}

The beam comes in along lab positive x. The Ewald sphere is centred at minus the incident frequency vector, so it passes through the reciprocal origin. A white beam gives a nested set of spheres with different radii instead of a single one. Intensities from different wavelengths add incoherently. The average over the uniform band uses composite Simpson integration with 96 to 512 intervals, adapted to the parameters, and it approaches the single-wavelength result at zero bandwidth.

Powder rings from interatomic pair distances

\begin{aligned}J_{\rm powder}(q)&=\frac1{N_{\rm at}^2}\sum_{i,j}\frac{\sin(2\pi q r_{ij})}{2\pi q r_{ij}}\\q&=\frac{2\sin\theta}{\lambda}\\r_{\rm detector}&=D\tan(2\theta)\end{aligned}

The Debye sum is the exact average of the same cluster over all orientations. Pairs at equal distances are grouped exactly, self-pairs included. Detector values are interpolated from a radial table, and the displayed profile averages those detector values over rings. Rotating the one illustrative grain cannot change an isotropic powder pattern. The gold ring positions are separate ideal Bragg references, not the computed peaks of the finite cluster.

What this record cannot prove

J=|A|^2\quad\not\Rightarrow\quad\arg A

Spot positions and systematic absences narrow down a proposed lattice, but intensity alone cannot uniquely recover the atomic structure. This is a forward model with identical point scatterers; it does not identify a real material. Form factors, polarization, absorption, dynamical scattering, temperature and detector noise are left out. The exposure setting is a stated logarithmic display gain, and the central beamstop leaves the raw scattering values unchanged.

Common difficulties

A missing spot does not mean the planes are missing

Typical misconceptionEvery reciprocal-lattice point should show up on the detector, so any missing spot means the crystal model is wrong.

Better mental modelFirst check for extinction by the basis. Then check the elastic Ewald geometry, the wavelength band and the limited size of the forward detector. Each is a different reason why a signal can be missing.

Run the experiment

01
Scene 1: Monochromatic rocking
Start with the monochromatic preset. Rock the yaw back and forth by a few degrees, and look at the atom cluster, the detector face and reciprocal space one at a time.
What to observe: A spot brightens as its node nears the Ewald sphere and fades as the node moves away. More cells make the spot narrower.
02
Scene 2: White-beam Laue
Choose Laue. Narrow the bandwidth to zero and compare with monochromatic mode at the same settings. Then rotate the crystal.
What to observe: A band of wavelengths lets more reflections meet the elastic condition. Shrinking the band brings back the single-wavelength record.
03
Scene 3: Powder and missing reflections
Choose powder. Switch between SC, BCC and FCC, select the first missing reflection, change the cell edge and rotate the illustrative grain.
What to observe: Powder rings shift when the cell edge or the wavelength changes, but not when the illustrative grain rotates. The absences caused by the basis tell the three candidate lattices apart.

Crystal Diffraction & Reciprocal Space

Audit the physics of Crystal Diffraction & Reciprocal Space

Core mathematical model

Cell edge is not plane spacing

Why a finite cluster gives broadened peaks

Basis interference creates systematic absences

Ewald sphere and white-beam Laue

Powder rings from interatomic pair distances

What this record cannot prove

Common difficulties

A missing spot does not mean the planes are missing

Run the experiment

Scene 1: Monochromatic rocking

Scene 2: White-beam Laue

Scene 3: Powder and missing reflections

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