International
Tables for
Crystallography
Volume H
Powder diffraction
Edited by C. J. Gilmore, J. A. Kaduk and H. Schenk

International Tables for Crystallography (2018). Vol. H, ch. 2.7, pp. 166-167

Section 2.7.14. Pressure determination

A. Katrusiaka*

aFaculty of Chemistry, Adam Mickiewicz University, Poznań, Poland
Correspondence e-mail: katran@amu.edu.pl

2.7.14. Pressure determination

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Pressure determination inside a high-pressure sample chamber is most straightforward in piston-and-cylinder devices, where the force applied to the piston and its surface area are known. Pressure is the force per unit area, with corrections for the friction between the cylinder wall and the piston (particularly significant above 1 GPa) and for the buoyancy of the piston, marginally important for all high pressures. Several types of mechanical pressure gauge are available. In the Bourdon gauge, a spiral metal tube that is pressurized inside unwinds and moves a pointer around a precise scale. Electrical resistance gauges are most often based on a manganin alloy sensor. The resistance of manganin changes at the rate of 2.4 × 10−5 GPa−1, although precise calibration depends on the alloy composition and it changes with the age of the sensor. The resistance of manganin depends only very weakly on temperature.

The most common pressure calibration method used for the DAC is the fluorescence pressure scale of the ruby R1R1 at 0.1 MPa is the reference; λ0 = 694.2 nm) and R2 (at 0.1 MPa, λR2 = 692.8 nm) lines (Forman et al., 1972[link]; Barnett et al., 1973[link]; Syassen, 2008[link]; Gao & Li, 2012[link]). Synthetic ruby with a Cr3+ concentration of 3000–5500 p.p.m., illuminated with green laser light, is commonly used. Inclusions of spinels make natural ruby unsuitable as a pressure gauge. A piece of ruby is usually crushed into small pieces and one or several small chips are placed in the DAC chamber close to the sample (Hazen & Finger, 1982[link]). Alternatively, small ruby spheres can be used for this purpose (Chervin et al., 2001[link]).

The linear pressure dependence of the ruby R1 fluorescence line was established according to the equation of state (EOS) of NaCl to 19.5 GPa:[P \, ({\rm GPa}) = 2.74 \Delta \lambda \, ({\rm nm}) ,]where Δλ = λR1 − λ0 (Piermarini et al., 1975[link]). The extension of the pressure range to 180 GPa, according to the equations of state of copper, gold and other metals, showed that P(Δλ) is quasi-linear:[P = 1904 [(\lambda/\lambda_0)^B - 1]/B ,]where B = 7.665 for quasi-hydrostatic and 5.0 for non-hydrostatic conditions (Mao et al., 1986[link], 1978[link]; Bell et al., 1986[link]).

The ruby fluorescence depends strongly on temperature (Barnett et al., 1973[link]; Vos & Schouten, 1991[link]; Yamaoka et al., 2012[link]) and a temperature change of about 6 K causes R1 line shifts equivalent to 0.1 GPa. The fluorescence-line dependence on temperature is much weaker for Sm2+:SrB4O7 (Lacam & Chateau, 1989[link]; Lacam, 1990[link]; Datchi et al., 1997[link]) and the other rare-earth-doped sensors listed in Table 2.7.2[link]. These sensors can be more sensitive to pressure than ruby, and together with ruby can be used simultaneously for both temperature and pressure calibration.

Table 2.7.2| top | pdf |
Luminescence pressure sensors, their electronic transition types (s = singlet, d = doublet) and rates of spectral shifts (after Holzapfel, 1997[link])

SensorTransitionλ0 (Å)dλ/dP (Å GPa−1)dλ/dT (× 10−2 Å K−1)(dλ/dP)/Γ (Å GPa−1)(dλ/dT)/(dλ/dP) (× 10−2 GPa K−1)
Cr3+:Al2O3 2E4A2/d 6942 3.65 (9) 6.2 (3) 4.9 17.0
Sm2+:SrB4O7 5D07F0/s 6854 2.55 −0.1 17.0 −0.4
Sm2+:BaFCl 5D07F0/s 6876 11.0 −1.6 4.8 −1.5
Sm2+:SrFCl 5D07F0/s 6903 11.2 (3) −2.36 (3) 5.8 −2.1
Eu3+:LaOCl 5D07F0/s 5787 2.5 −0.5 1.0 −2.0
Eu3+:YAG 5D07F1/d 5906 1.97 −0.5 0.7 −2.5

The calibration of most pressure gauges is based on comparisons of theoretical and shock-wave data (Holzapfel, 1997[link]), so the derived equations of state are used with an accuracy of about 5% up to 1 TPa. The EOS recommended by Holzapfel (1997[link]) is[P = [3K_0 (1 - x)/x^5] \exp [c_0 (1 - x)] , \eqno(2.7.1)]where x = a/a0 = (V/V0)1/3, c0 = 3(K0′ − 3)/2, a is the unit-cell dimension, V is the unit-cell volume, a0R and V0R are the reference parameters under ambient conditions (Table 2.7.3[link]), α0R is the thermal expansion coefficient, K0R is the bulk modulus, and K0′ = dK0/dP. dK0/dT = −3α0RK0δαR, where δαR = ∂(lnα)/∂(lnV)TR. Pressure can be computed by assuming constant K0′ and linear temperature relations for T close to or higher than TR = 300 K:[\eqalign{ a_0 (T) & = a_{0R} [1 + \alpha_{0R} (T - T_R)] , \cr V_0 (T) & = V_{0R} [1+ 3 \alpha_{0R} (T - T_R )] , \cr K_0 (T) & = K_{0R} [1 - \alpha_{0R} \delta_{\alpha R} (T - T_R)] .}]

Table 2.7.3| top | pdf |
Parameters recommended for pressure determination by EOS measurements

Various face-centred cubic (f.c.c.) and body-centred cubic (b.c.c.) metals are used as calibrants, with the EOS given by equation (2.7.1)[link] and the reference temperature TR = 300 K (after Holzapfel, 1997[link]).

Metala0R (pm)K0R (GPa)K0Rα0R (× 106 K)δαR
Al 404.98 (1) 72.5 (4) 4.8 (2) 23.0 (4) 5.5 (11)
Cu 361.55 (1) 133.2 (2) 5.4 (2) 16.6 (3) 6.1 (6)
Ag 408.62 (1) 101.0 (2) 6.2 (2) 19.2 (4) 7.1 (6)
Au 407.84 (1) 166.7 (2) 6.3 (2) 14.2 (2) 7.2 (6)
Pd 388.99 (1) 189 (3) 5.3 (2) 11.6 (4) 6.0 (11)
Pt 392.32 (1) 277 (5) 5.2 (2) 8.9 (4) 5.9 (l1)
Mo 314.73 (1) 261 (5) 4.5 (5) 5.0 (4) 5.2 (14)
W 316.47 (1) 308 (2) 4.0 (2) 4.5 (4) 4.7 (11)

The details of the parameterization are explained by Holzapfel (1991[link], 1994[link]) and listed for the simple face-centred (f.c.c.) and body-centred (b.c.c.) cubic metals in Table 2.7.3[link] (Holzapfel, 1997[link]). Powders of these metals can be mixed with the sample and its pressure can be calibrated according to the unit-cell dimension of the standard. The well known compressibilities of NaCl, CaF2 and MgO, in the form of either powders or single crystals, are also often used as internal pressure standards (Dorfman et al., 2010[link], 2012[link]; Dorogokupets & Dewaele, 2007[link]).

An independent pressure assessment can be obtained from standard materials undergoing pressure-induced phase transitions (Table 2.7.4[link]). This method is limited to just a few pressure points (Holzapfel, 1997[link]), but they can provide a useful verification of other pressure gauges.

Table 2.7.4| top | pdf |
Pressure fixed points at ambient temperature (after Holzapfel, 1997[link]; Hall, 1980[link])

P (GPa)Element transition
0.7569 (2) Hg freezing at 273 K
1.2 (1) Hg freezing at 298 K
2.55 (6) Bi I–II at 298 K
3.67 (3) Tl h.c.p.–f.c.c.
2.40 (10) Cs I–II
4.25 (1) Cs II–III
4.30 (1) Cs III–IV
5.5 (1) Ba I–II
7.7 (2) Bi III–IV
9.4 (3) Sn I–II
12.3 (5) Ba II–III
13.4 (6) Pb I–II
Onset of forward transition.
Centre of hysteresis.

Other methods of pressure calibration are still being developed. For example, it has been shown that very high pressure can be determined from the Raman shift of strained diamond-anvil culets (Akahama & Kawamura, 2004[link]). The strong piezochromic effect of visible colour changes in soft coordination polymers allows pressure calibration without spectrometers. These changes can proceed gradually (Andrzejewski & Katrusiak, 2017a[link]) and abruptly at phase transitions (Andrzejewski & Katrusiak, 2017b[link]). Another method of pressure calibration is based on the luminescence lifetime of lanthanide nanocrystals (Runowski et al., 2017[link]).

References

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