IN25 · Rings reveal atomic spacing

Powder X-ray Diffractometer

How do powder rings reveal a crystal lattice? A powder contains crystallites pointing in many directions. Each allowed atomic-plane spacing sends reinforced X-rays along a cone, which an area detector cuts into a ring or arc. The angles give plane spacings; their sequence and missing reflections help distinguish candidate crystal lattices.

Move an area detector, calibrate its geometry with a known powder and integrate saved photon pixels to test one or two cubic lattices.

Turn rings into lattice spacing

FCC lattice from counts $0.400\,\mathrm{nm}$

Resolved observed peaks $6$

First observed FWHM $0.43^\circ$

Withheld reference error $0.004^\circ$

FCC lattice constant

a_{\mathrm{FCC}}/\mathrm{nm}

0.4

Second-pattern input weight

w_{\mathrm{BCC}}

0

Coherent domain size

L/\mathrm{nm}

60

RMS microstrain

\varepsilon_{\mathrm{rms}}

0.0005

Detector distance reading

D_{\mathrm{enc}}/\mathrm{mm}

100

Detector arm angle reading

\phi_{\mathrm{enc}}/{}^\circ

0

Drag the gold detector handle to change its real distance and collection angle range.

A calibrated cubic-powder exposure is ready.

Source, second lattice and hardware

Known wavelengthBCC lattice input in nanometresInstrumental Gaussian FWHM in degreesDistance encoder zero in millimetresAngle encoder zero in degreesIncident photons per exposurePhoton seed

A source or hardware change acquires fresh reference counts. Encoder poses share calibrated zeros. The detector pixel centre is assumed mechanically known.

Model assumptions

Synthetic isotropic kinematic FCC/BCC powder with equal constant atomic scatterers. Monochromatic wavelength and the fixed mechanical pixel centre are known. A separate ideal diamond standard, anchored to the NIST 640g lattice value, calibrates detector-distance and angular encoder zeros; a fresh frame at another known pose checks the result. The standard is not a reproduction of certified NIST intensities or uncertainty. Dark, flat, standard, check and sample counts are retained. The angular curve is an actual solid-angle/coverage-weighted integration of these pixels. Gaussian-equivalent size, RMS strain and instrumental widths add in quadrature; finite-crystal tails, polarization, absorption, texture, multiple scattering, wavelength dispersion and real atomic form factors are omitted. Observed widths include pixels and overlap; size and strain are not uniquely inferred. FCC/BCC candidates use positions and coupled relative line strengths within a declared cell range. Limited coverage, unresolved lines, saturation or ambiguity withhold a lattice claim. Pattern weights are not mass fractions or material certification. Displayed ray motion is slowed and representative; beam and capillary widths are enlarged.

Simulated photon observations

The curve comes from these pixels

Inspect exposure

Retained area-detector counts · logarithmic colour

0

128

255

Detector column / pixel

Colour maximum $61$ photons per pixel; red marks clipping.

Angular integration of the saved image

$I/(10^6\,\mathrm{photon\,rad^{-1}})$

$2\theta/{}^\circ$

Pixels integratedOne fitted library candidate

Peaks fitted in the observed angular profile
Peak	Position $2\theta/{}^\circ$	Observed FWHM $\beta/{}^\circ$	Signal / counting noise

Standard, coverage and immutable records

The independent check exposure uses another known detector pose. Its observed peak widths include the instrument and pixel integration. Sample widths also contain the domain-size and RMS-strain terms; overlapping peaks add further apparent width. These contributions are not uniquely separated by the displayed fit.

\beta^2=\beta_{\mathrm{inst}}^2+\left(\frac{0.886\lambda}{L\cos\theta}\right)^2+\left(4\sqrt{2\ln2}\,\varepsilon_{\mathrm{rms}}\tan\theta\right)^2

Widths in this expression are Gaussian-equivalent FWHM in radians. The finite-domain central width uses a chosen 0.886 factor; Gaussian tails are an approximation. Kept exposures preserve all five raw frames, units, masks, clipping flags, encoder poses, seeds and versions. CSV and JSON omit generating settings and can independently replay the reduction.

Model and primary sources

Isotropic kinematic FCC/BCC powders have constant equal atomic scatterers. The source is monochromatic; the mechanical pixel centre is fixed and known. The detector is a finite 153.6-millimetre square plane, with 256 pixels on each side. Polarization, absorption, texture, real atomic form factors, multiple scattering and wavelength spread are omitted. The synthetic diamond standard adopts only the NIST 640g lattice anchor. Pattern scales describe this restricted intensity model; they are not mass composition or certified material labels. Line centres are included below 85 degrees; the inverse uses covered angles from 5 to 85 degrees. Counts use a Gaussian approximation to Poisson draws above a mean of 40. Missing coverage, saturated pixels, unresolved component lines, incompatible references and competing lattice candidates withhold an inference.

Start with the essentials

Focus question: How do powder rings reveal a crystal lattice?
One-sentence intuition: Calibrate the geometry, integrate the observed pixels and compare the complete sequence of allowed and missing lines.

Core mathematical model

The extra path selects an angle

2d\sin\theta=\lambda

The displayed scattering angle is twice the angle between the incident beam and a reflecting plane. Here the reflection order is included in the plane indices.

A cubic cell links several spacings

d_{hkl}=\frac{a}{\sqrt{h^2+k^2+l^2}}

Each set of integer plane indices supplies a spacing. Several observed angles must agree on the same cubic cell edge.

Atoms within a cell can cancel a line

F_{hkl}=\sum_j f_j\exp[2\pi i(hx_j+ky_j+lz_j)]

The complex amplitudes from the declared basis atoms add before their magnitude is squared. This produces the FCC and BCC missing-reflection rules.

Several widths enter one observed peak

\beta_{\mathrm{sample}}^2=\beta_{\mathrm{inst}}^2+\beta_{\mathrm{size}}^2+\beta_{\mathrm{strain}}^2

This Gaussian-equivalent teaching model uses separate size and RMS-strain terms. Pixel integration and overlapping lines further affect measured widths; the fit does not uniquely recover domain size or strain.

Common difficulties

A larger ring can mean a longer camera distance

Typical misconceptionA change in ring radius always means the atomic spacing changed.

Better mental modelPixel radius also depends on detector distance, tilt and wavelength. The raw standard supplies the geometry corrections.

Peak width contains several effects

Typical misconceptionThe inverse of any peak width gives a unique crystallite size.

Better mental modelInstrument width, finite pixel size, RMS strain and line overlap can all broaden it. This Lab keeps those effects explicit and reports observed widths.

A library candidate has a limited meaning

Typical misconceptionMatching a few lines certifies a real material and its mass fraction.

Better mental modelThis library contains synthetic equal-scatterer FCC and BCC bases. It tests lattice candidates under those assumptions; real phase identification needs material-dependent intensities and additional evidence.

Run the experiment

01
Follow one exposure
Start with One cubic powder and inspect the image below the apparatus.
What to observe: The angular curve integrates those same photon pixels. Its resolved peaks support one FCC candidate.
02
Move the actual panel
Drag the gold detector handle while watching the rings, then choose Detector face.
What to observe: The ring positions on the panel and its covered angular range change. Geometry zeros calibrated by the standard transfer between known encoder poses.
03
Add a second lattice
Choose Two lattices overlap. Compare the observed peak table with the two reported candidates.
What to observe: Some lines overlap. Candidate intensities share one scale per phase; a larger-cell alias predicts extra low-angle lines that the pixels can reject.
04
Compare broadening
Compare Small domains broaden rings with Strain blurs high angles.
What to observe: Size and RMS strain have different angular dependence. At strong overlap the restricted candidate fit is withheld, although measured peaks remain visible.
05
Keep a measurement and test a limit
Keep exposure, change a setting, then select the earlier exposure. Export that selection as CSV or JSON. Try The detector misses lines.
What to observe: The earlier raw image and references stay fixed. Limited field of view leaves too few independent lines to report a lattice.

Powder X-ray Diffractometer

The curve comes from these pixels

Read spacing from angles, not ring radius

Core mathematical model

The extra path selects an angle

A cubic cell links several spacings

Atoms within a cell can cancel a line

Several widths enter one observed peak

Common difficulties

A larger ring can mean a longer camera distance

Peak width contains several effects

A library candidate has a limited meaning

Run the experiment

Follow one exposure

Move the actual panel

Add a second lattice

Compare broadening

Keep a measurement and test a limit

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