Diffraction playground
This sheet: https://xraytools.com/pxrdcalc?playground=true
Page: https://xraytools.com/pxrdcalc
Change the crystal, watch the diffraction
Keep a reference and change one thing. Predict what will move or disappear, then try it.
Model crystal: a cubic cell with a silicon site at (0, 0, 0) and another at (½, ½, ½). This is a teaching model, not the structure of silicon.
Frozen reference: a = 5.64 Å; centre occupancy = 1; wavelength = 1.54 Å.
Changed: a = 5.64 Å; centre occupancy = 1; wavelength = 1.54 Å.
Reference above the baseline; changed below. Both halves show positive intensities, each normalised to its own strongest line = 100 over its calculated angle range. Hover over a line for its values; the table below has the same values for touch and keyboard use.
The chart shows the window both calculations cover. On the d axis, matching spacings align even when the wavelengths differ. Relative heights can still differ because the Lorentz–polarisation factor depends on angle. Sticks show coincident reflections; no peak width, background, absorption or diffuse scattering is simulated.
Compare the numbers
| State | Reflections | 2θ / ° | d / Å | Relative intensity |
|---|---|---|---|---|
| Reference | 100 | 15.694 | 5.6400 | 0.000 |
| Reference | 110 | 22.265 | 3.9881 | 100.000 |
| Reference | 111 | 27.356 | 3.2563 | 0.000 |
| Reference | 200 | 31.692 | 2.8200 | 18.650 |
| Reference | 210 | 35.550 | 2.5223 | 0.000 |
| Reference | 211 | 39.074 | 2.3025 | 40.355 |
| Reference | 220 | 45.430 | 1.9940 | 12.849 |
| Reference | 300 (6) + 221 (24) | 48.356 | 1.8800 | 0.000 |
| Reference | 310 | 51.155 | 1.7835 | 17.945 |
| Reference | 311 | 53.847 | 1.7005 | 0.000 |
| Reference | 222 | 56.450 | 1.6281 | 4.434 |
| Reference | 320 | 58.977 | 1.5643 | 0.000 |
| Reference | 321 | 61.438 | 1.5074 | 20.586 |
| Reference | 400 | 66.200 | 1.4100 | 2.059 |
| Reference | 410 (24) + 322 (24) | 68.514 | 1.3679 | 0.000 |
| Reference | 411 (24) + 330 (12) | 70.792 | 1.3294 | 10.159 |
| Reference | 331 | 73.039 | 1.2939 | 0.000 |
| Reference | 420 | 75.260 | 1.2611 | 5.709 |
| Reference | 421 | 77.458 | 1.2307 | 0.000 |
| Reference | 332 | 79.637 | 1.2025 | 4.923 |
| Reference | 422 | 83.954 | 1.1513 | 4.338 |
| Reference | 500 (6) + 430 (24) | 86.098 | 1.1280 | 0.000 |
| Reference | 510 (24) + 431 (48) | 88.237 | 1.1061 | 11.711 |
| Changed | 100 | 15.694 | 5.6400 | 0.000 |
| Changed | 110 | 22.265 | 3.9881 | 100.000 |
| Changed | 111 | 27.356 | 3.2563 | 0.000 |
| Changed | 200 | 31.692 | 2.8200 | 18.650 |
| Changed | 210 | 35.550 | 2.5223 | 0.000 |
| Changed | 211 | 39.074 | 2.3025 | 40.355 |
| Changed | 220 | 45.430 | 1.9940 | 12.849 |
| Changed | 300 (6) + 221 (24) | 48.356 | 1.8800 | 0.000 |
| Changed | 310 | 51.155 | 1.7835 | 17.945 |
| Changed | 311 | 53.847 | 1.7005 | 0.000 |
| Changed | 222 | 56.450 | 1.6281 | 4.434 |
| Changed | 320 | 58.977 | 1.5643 | 0.000 |
| Changed | 321 | 61.438 | 1.5074 | 20.586 |
| Changed | 400 | 66.200 | 1.4100 | 2.059 |
| Changed | 410 (24) + 322 (24) | 68.514 | 1.3679 | 0.000 |
| Changed | 411 (24) + 330 (12) | 70.792 | 1.3294 | 10.159 |
| Changed | 331 | 73.039 | 1.2939 | 0.000 |
| Changed | 420 | 75.260 | 1.2611 | 5.709 |
| Changed | 421 | 77.458 | 1.2307 | 0.000 |
| Changed | 332 | 79.637 | 1.2025 | 4.923 |
| Changed | 422 | 83.954 | 1.1513 | 4.338 |
| Changed | 500 (6) + 430 (24) | 86.098 | 1.1280 | 0.000 |
| Changed | 510 (24) + 431 (48) | 88.237 | 1.1061 | 11.711 |
Three experiments
1. Expand the cell
Do the lines move to larger or smaller angles?
Why it happens
Larger spacings put each reflection at a smaller angle for the same wavelength. Switch to d-spacing to see the spacings increase.
2. Empty the centre
Will existing reflection positions move, or will extra lines appear?
Why it happens
Keeping the conventional cell fixed keeps the d-spacing of each hkl fixed. Removing the centre atom releases the odd h + k + l reflections from cancellation. The fully occupied model is body-centred; the empty-centre model is primitive.
3. Shorten the wavelength
Does the crystal change when the diffraction angles change?
Why it happens
The same hkl moves to a smaller angle, but its spacing has not changed. Switch to d-spacing: corresponding lines align within the common window.
Fractional occupancy is the average fraction of occupied centre sites. This model does not describe vacancy ordering. It uses the same atomic scattering factors, structure-factor sum and powder intensity calculation as the full HKL calculator.
Where this comes from
- X-ray scattering factors computed from numerical Hartree-Fock wave functions
D. T. Cromer and J. B. Mann, Acta Cryst. A 1968, 24, 321–324 · doi:10.1107/S0567739468000550
The atomic scattering factors used for both model sites. The powder intensities sum the squared structure factors of coincident reflections, with the unpolarised Bragg–Brentano Lorentz–polarisation factor. Displacement parameters are zero in this model.