§1 — Why X-ray diffraction for pressure determination?
At synchrotron beamlines, sample pressure in a DAC is routinely determined from the measured lattice parameter of a pressure standard co-loaded with the sample. The lattice parameter shrinks as pressure increases: by measuring it with X-ray powder or single-crystal diffraction and applying the known equation of state (EOS) of the standard, one obtains the local pressure with an accuracy of ±0.5–2%. This approach is independent of optical access to the sample and remains valid at ultra-high pressures where ruby fluorescence becomes unreliable (>150 GPa). It also allows pressure to be measured in situ at high temperature (e.g. laser-heated DAC), where the ruby scale requires large thermal corrections.

§2 — The 3rd-order Birch-Murnaghan EOS
The most widely used EOS in high-pressure physics is the 3rd-order Birch-Murnaghan formulation (Birch 1978, JGR 83:1257 [15]), derived from finite-strain theory. It expresses pressure as a function of the normalized volume V/V₀: P = (3K₀/2)·[(V₀/V)^7/3 − (V₀/V)^5/3] · {1 + (3/4)(K₀′−4) · [(V₀/V)^2/3 − 1]} where K₀ is the isothermal bulk modulus at ambient pressure, K₀′ = dK/dP its pressure derivative, and V₀ the unit-cell volume at ambient conditions. For cubic crystals, V₀ = a₀³. These three parameters uniquely define the P–V curve of each material. The XRD Gauges tab calculates P directly from the measured lattice parameter a; the EOS BM3 tab accepts any V/V₀ or V value and draws the full P–V curve.

§3 — Preset materials: sources and parameters
Gold Au (fcc, a₀ = 4.0782 Å, K₀ = 167 GPa, K₀′ = 6.0, V₀ = 67.85 ų)
The most widely used secondary pressure standard above 30 GPa. These parameters come from Dewaele et al. (2004, Phys. Rev. B 70:094112 [29]), confirmed in Dewaele & Loubeyre (2007, HPR 27:419 [30]) after correction for non-hydrostatic stress; K₀ = 167 GPa matches the ultrasonic value and K₀′ = 6.0 the modern high-temperature gold scale. Au is preferred for its strong XRD signal (high Z) and chemical inertness. Caution: Au is soft and can introduce deviatoric stress artefacts in the sample.

Platinum Pt (fcc, a₀ = 3.9231 Å, K₀ = 277 GPa, K₀′ = 5.2, V₀ = 60.38 ų)
Parameters from Dewaele et al. (2008, Phys. Rev. B 78:104102 [3]); K₀ = 277 GPa is the established ultrasonic value (Macfarlane et al. 1965). Pt is harder than Au and introduces less deviatoric stress. It serves as a primary scale for the diamond Raman gauge calibration (Akahama & Kawamura 2010). Valid to megabar pressures.

MgO (rocksalt B1, K₀ = 160.2 GPa, K₀′ = 4.03, V₀ = 74.71 ų)
From Zha, Mao & Hemley (2000, PNAS 97:13494 [28]). MgO is transparent, non-reactive, and serves as a primary pressure scale derived from Brillouin elasticity measurements. It is the preferred scale for laser-heated DAC experiments because it does not absorb the laser and remains stable above 100 GPa.

NaCl B1 (cubic F, a₀ = 5.6402 Å, K₀ = 24.7 GPa, K₀′ = 5.5)
Third-order BM parameters approximating the Decker (1971, J. Appl. Phys. 42:3239 [16]). The B1 (rocksalt) phase is stable below ~29 GPa. NaCl is commonly co-loaded with the sample as both a pressure medium (quasi-hydrostatic below 26 GPa) and a pressure standard. Its very low bulk modulus makes it sensitive to small pressure changes at low P.

NaCl B2 (CsCl-type, K₀ = 30.69 GPa, K₀′ = 4.33, V₀ = 41.35 ų — BM3 fit)
NaCl transforms from the B1 to the B2 structure near 29 GPa. This gauge is used above the transition, where the B1 gauge no longer applies. The B2 phase does not exist at ambient pressure, so V₀ = 41.35 ų (a₀ ≈ 3.458 Å) is an extrapolated value. Parameters: 3rd-order Birch-Murnaghan fit by Fei et al. (2007, PNAS 104:9182); their Vinet fit of the same data gives K₀ = 26.86 GPa, K₀′ = 5.25.

Iron Fe ε (hcp, K₀ = 163.4 GPa, K₀′ = 5.38, V₀ = 22.43 ų — Vinet fit)
The hexagonal ε-phase of iron is stable above ~13 GPa. Vinet-fit parameters (17–197 GPa, He medium) from Dewaele et al. (2006, Phys. Rev. Lett. 97:215504 [31]). Fe is studied as an Earth's inner-core analogue; its EOS is central to geophysics. Using Fe itself as its own gauge (via the BM3 calculator) allows pressure estimation without an external standard.

Tungsten W (bcc, a₀ ≈ 3.165 Å, K₀ = 310 GPa, K₀′ = 4.0, V₀ = 31.71 ų)
Parameters from Dewaele et al. (2004, PRB 70:094112 [29]). W has the highest bulk modulus of the metallic standards, making it the least compressible (smallest V/V₀ change per GPa) and therefore the most precise at very high pressures. It is chemically inert and gives sharp diffraction peaks.

Aluminium Al (fcc, K₀ = 72.8 GPa, K₀′ = 4.6, V₀ = 66.43 ų)
From Dewaele et al. (2004, PRB 70:094112 [29]). Al is used as a low-to-medium pressure standard (to ~50 GPa). Its low Z gives a weak XRD signal but it is chemically compatible with many sample environments.

Copper Cu (fcc, K₀ = 133 GPa, K₀′ = 5.3, V₀ = 47.24 ų)
From Dewaele & Loubeyre (2007, HPR 27:419 [30]) after non-hydrostatic correction. Cu was historically used to calibrate the ruby pressure scale (Mao et al. 1986). It remains a standard reference, particularly for soft X-ray diffraction beamlines.

§4 — How to use this page
XRD Gauges tab: enter the measured lattice parameter a (Å) of your standard. Pressure is calculated immediately from the BM3 EOS with the built-in parameters.
EOS BM3 tab: select a preset or enter custom K₀, K₀′, V₀. Enter either V/V₀ directly or the absolute volume V (ų) — the other field updates automatically. The P–V curve is plotted interactively; click on the curve to read P at any V/V₀.

Key references for EOS parameters:
[15] Birch F. (1978) J. Geophys. Res. 83:1257 — 3rd-order Birch-Murnaghan EOS formulation
[16] Decker D.L. (1971) J. Appl. Phys. 42:3239 — NaCl B1/B2 EOS
[28] Zha C.S., Mao H.K. & Hemley R.J. (2000) PNAS 97:13494 — MgO primary pressure scale
[29] Dewaele A., Loubeyre P. & Mezouar M. (2004) Phys. Rev. B 70:094112 — Au, Pt, W, Al, Cu EOS in helium PTM
[30] Dewaele A. & Loubeyre P. (2007) High Press. Res. 27:419 — Non-hydrostatic correction; Au, Cu EOS
[3] Dewaele A. et al. (2008) Phys. Rev. B 78:104102 — Pt EOS to megabar; ruby calibration
[31] Dewaele A. et al. (2006) Phys. Rev. Lett. 97:215504 — Fe ε-phase EOS