Optics 058 · Diffraction, Fourier optics, and computational imaging

Crystal Diffraction & Reciprocal Space

Rotate a finite cubic crystal, compute its coherent detector record, explore Ewald intersections and white-beam Laue, then derive powder rings from the same pair distances. SC / BCC / FCC systematic absences expose what a diffraction pattern can and cannot reveal.

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 $\text{valid model regime}$

Validated scientific apparatus

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 a monochromatic crystal and watch spots appear near an Ewald intersection. Switch to white-beam Laue for a wavelength band, or random powder for rings calculated from the same atom pairs. Inspect actual atom cluster, receiving face and reciprocal space separately. SC, BCC and FCC differ by basis interference; a hollow node is not the same as an allowed reflection missing the sphere. Cell edge resizes the atoms and reciprocal spacing. Gold marks an ideal reflection; cyan records are the simulated finite-cluster data. Exposure changes brightness only.

Elastic scalar kinematic scattering of equal unit point atoms in complete cubic conventional cells. Finite exact lattice sum; incoherent uniform-band Laue quadrature; exact isotropic Debye pair average with radial interpolation. Forward point-sampling detector, explicit display gain and beamstop. Ideal reciprocal geometry is separate from acquired simulated records. No phase recovery, unique structure inversion, material calibration, atomic form factors, polarization, absorption, multiple scattering, thermal motion or noise. Atoms and supports are enlarged; physical detector directions share one millimetre scale.

Physics tutorial

Audit the physics of Crystal Diffraction & Reciprocal Space

BackgroundA diffraction spot is not a miniature atom or a painted reciprocal-grid point. It appears where the outgoing direction and wavelength make many scattered amplitudes agree: $\mathbf q=(\mathbf s_{\rm out}-\mathbf s_{\rm in})/\lambda$ . The detector, cluster and Ewald geometry here use the same orientation and length scales.

Why it mattersHow can reciprocal-space spots and rings reveal an unknown real-space lattice?

Start with the essentials

Focus question: How can reciprocal-space spots and rings reveal an unknown real-space lattice?
One-sentence intuition: Add complex amplitudes from the actual finite cluster before squaring: $I=\left|\sum_j e^{2\pi i\mathbf q\cdot\mathbf r_j}\right|^2$ . A monochromatic exposure, white-beam Laue average and randomly oriented powder average are three different measurements, not interchangeable textures.

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}

This Lab uses cubic conventional cells and reciprocal frequency in cycles per angstrom. The detector scattering angle is twice the Bragg plane angle. Changing the cell edge resizes all atomic coordinates, reciprocal spacing and the recorded pattern. Angles are derived from outgoing directions, not an independent spot animation.

One finite cluster, 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 is the exact factorization of a cube of complete conventional cells, each with the same point-scatterer basis. At integer reciprocal coordinates the ratio is evaluated by its continuous limit. Increasing cell count narrows the peaks; off-node side lobes are real consequences of the finite cluster. The intensity normalization is by squared atom count, not a moving brightest pixel.

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}

All atoms have equal unit scattering amplitude. BCC permits an even index sum; FCC permits all-even or all-odd indices. A hollow reciprocal node is a basis absence, distinct from an allowed reflection that happens to miss the Ewald sphere. A finite detector pixel away from a forbidden node may still contain 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 incident direction is laboratory positive x; the Ewald centre is negative incident frequency, so the sphere passes through the reciprocal origin. A white beam supplies a nest of radii, not one monochromatic sphere with extra invented spots. Intensities from different wavelengths add incoherently. The uniform band average here uses 96–512 parameter-adaptive composite Simpson intervals and approaches the single-wavelength result at zero bandwidth.

Powder rings from pair distances, not painted circles

\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 isotropic orientation average of the same cluster. Equal pair distances are grouped exactly, including self-pairs. A radial table interpolates the detector values; the displayed profile then averages those detector records in annuli. Rotating one illustrative grain cannot alter an isotropic powder pattern. Gold ring positions are separate ideal Bragg references, not the calculated finite-cluster peaks.

What this record cannot prove

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

Spot positions and systematic absences constrain a proposed lattice, but intensity alone does not uniquely recover atomic structure. This is a forward model with equal point scatterers, not experimental material identification. Form factors, polarization, absorption, dynamical scattering, temperature and detector noise are omitted. Exposure is a declared logarithmic display gain; the central beamstop does not rewrite raw scattering values.

Common difficulties

No spot does not imply no planes

Typical misconceptionEvery reciprocal point must appear on the detector; any missing spot means the crystal is wrong.

Better mental modelFirst check basis extinction, then elastic Ewald geometry, wavelength band and the finite forward detector. These are different reasons for a missing signal; none requires painting a spot.

Run the experiment

01
Scene 1: Monochromatic rocking
Start at the monochromatic preset. Rock yaw a few degrees; inspect the atom cluster, receiving face and reciprocal space separately.
What to observe: A spot strengthens near an Ewald intersection and weakens away; increasing finite cell count narrows it.
02
Scene 2: White-beam Laue
Choose Laue. Narrow its bandwidth to zero and compare with monochromatic mode at identical settings; then rotate the crystal.
What to observe: A band of wavelengths supplies more elastic solutions; shrinking the band returns the single-wavelength record.
03
Scene 3: Powder and missing reflections
Choose powder. Switch SC, BCC and FCC; select the first missing reflection, change cell edge, and rotate the illustrative grain.
What to observe: Powder rings shift when cell edge or wavelength changes, but not when an illustrative grain rotates; basis absences distinguish the three candidate lattices.