§1 — What are d-spacings and why do they matter?
In any crystalline solid, atoms are periodically arranged along families of planes, each separated by a characteristic distance called the
interplanar spacing or
d-spacing, denoted d
hkl and expressed in Ångströms. The Miller indices (h, k, l) label each family of planes. When X-rays of wavelength λ hit these planes at a grazing angle θ, constructive interference occurs at angles satisfying
Bragg’s law, derived experimentally by W.H. Bragg and W.L. Bragg in 1913 [
Bragg & Bragg 1913, Proc. R. Soc. A 88:428]:
2d · sinθ = nλ (n = 1 for the fundamental reflection)
Measuring 2θ and knowing λ directly gives d. In a DAC experiment, increasing pressure compresses the unit cell and shrinks all d-spacings.
Tracking d vs P gives the equation of state — one of the central measurements in high-pressure physics. This is what drives the daily use of d-spacing conversions at every synchrotron beamline.
§2 — d-spacing formulae for each crystal system
The formula relating d
hkl to the lattice parameters depends on the crystal symmetry [
International Tables for Crystallography, Vol. A, IUCr 2016]:
| System | Parameters | 1/d²hkl |
|---|
| Cubic | a | (h²+k²+l²) / a² |
| Hexagonal | a, c | (4/3)·(h²+hk+k²)/a² + l²/c² |
| Tetragonal | a, c | (h²+k²)/a² + l²/c² |
| Orthorhombic | a, b, c | h²/a² + k²/b² + l²/c² |
Most high-pressure standards are cubic (Au, Pt, NaCl B1, MgO, W) or hexagonal (ε-Fe, Re, hcp-He). The cubic formula reduces to d = a/√(h²+k²+l²), which makes quick mental arithmetic straightforward at the beamline.
§3 — X-ray sources and wavelengths used in this calculator
The choice of wavelength drives everything: a shorter λ shifts peaks to smaller 2θ, requires a longer detector distance, but gives better access to high-angle reflections and better transmission through thick diamond anvils.
| Source | λ (Å) | Energy | Typical use |
|---|
| Cu Kα₁ | 1.5406 | 8.05 keV | Lab diffractometer; useful to ~30 GPa in DAC |
| Mo Kα₁ | 0.7107 | 17.5 keV | Lab source; ~100 GPa, better diamond penetration |
| Synchrotron | 0.4133 | 30 keV | ESRF ID27 typical; benchmark EOS datasets |
| Synchrotron | 0.3100 | 40 keV | Wide angular access / thick gaskets |
| Synchrotron | 0.1771 | 70 keV | PETRA-III, APS; multi-megabar >400 GPa |
Cu/Mo Kα: Int. Tables Vol. C, IUCr. Benchmark metal EOS at 30 keV: Dewaele et al. 2004. Hard-X-ray multi-megabar (70 keV): Eremets et al. 2023, Nat. Commun. 14:907.
§4 — Three practical uses in a DAC experiment
This calculator is not a replacement for full data-reduction software — it is a
real-time sanity check at the beamline or at the lab bench.
•
Live verification: you observe a 2θ peak and want to confirm it matches your pressure standard (Au, Pt, NaCl…) at the expected pressure. Enter 2θ and λ, read d directly.
•
Peak position prediction: you know a from your EOS and want to predict where (hkl) will appear in 2θ before the experiment, to set detector geometry or flag peak overlaps.
•
Cross-instrument transfer: switch between Cu Kα, Mo Kα, and synchrotron energies to compare datasets or plan measurements on a different instrument.
§5 — Worked example — NaCl B1 at 10 GPa
NaCl B1 (cubic,
Fm¯3m) has a
0 = 5.6402 Å at ambient pressure (JCPDS 5-0628) and compresses to a ≈ 5.52 Å at 10 GPa. The (200) reflection gives:
d200 = 5.52 / √(4) = 2.760 Å
With Mo Kα (λ = 0.7107 Å): 2θ = 2 · arcsin(0.7107 / (2 × 2.760)) = 14.79°
With synchrotron 30 keV (λ = 0.4133 Å): 2θ = 8.59°
Enter these values in the calculator to verify. The NaCl B1 → B2 (CsCl-type) transition occurs near 27–30 GPa at room temperature [
Hsieh 2021, Sci. Rep.], so NaCl is a reliable soft pressure medium and internal standard up to that pressure.
§6 — References
Bragg W.H. & Bragg W.L. (1913) — The reflection of X-rays by crystals.
Proc. R. Soc. A 88:428 — original Bragg’s law derivation
International Tables for Crystallography Vol. A (IUCr 2016) — d-spacing formulae for all crystal systems and space groups
Dewaele A., Loubeyre P., Mezouar M. (2004) — Equations of state of six metals above 94 GPa.
Phys. Rev. B 70:094112
Dewaele A. et al. (2008) — Compression curves of transition metals; He pressure medium.
Phys. Rev. B 78:104102
Eremets M.I. et al. (2023) — Universal diamond edge Raman scale to 0.5 TPa.
Nat. Commun. 14:907
Hsieh W.-P. (2021) — High-pressure thermal conductivity and velocity of NaCl B1/B2.
Sci. Rep. 11:21321