Pressure scales gauges How to measure pressure adequately

Pressure scales & gauges: How to measure pressure: adequately and accurately Stefan Klotz Université P&M Curie, Paris

Summary What is pressure? Primary & secondary pressure gauges Laboratory pressure gauges (bourdon, transducers, resisitive gauges): « Fixed point » gauge Optical gauges (ruby and others) Diffraction gauges (x-rays and neutrons) Again: Primary pressure gauges: ultrasonics & shock waves Remarks and outlook: Precision in high P science & technology

What is pressure? F « P = Force/Area » P P P A DV DV DV Need more general expression for « pressure » and « deformation »

Stress and strain (in a nutshell) Strain ( « deformation » ): U = (uij) (3 x 3 matrix, symetric: 6 elements) U = a a * * a 0 * = 0 * a 0 0 Remark: Tr U = DV

s = (sij) (3 x 3 matrix, symetric: 6 elements) Stress ( « pressure » ): F s = Remarks: -p -p * * -p 0 * p 0 0 - hydrostatic pressure: sij = -dij ∙ p - hydrostatic component: p = ⅓ ∙Tr U - Relation s ↔ U : Hook’s law: s = C U * = 0

To remember: « Pressure » is a form of stress s=(sij) « Pressure » in physics means almost exclusively hydrostatic pressure: s = -dij p ex: p = -( E/ V)T B= - ( p/ ln. V)T importance of hydrostaticity in exp. studies Worthwhile to invest some time into theory of elasticity!

Primary pressure scales ( use only the definition of pressure) High pressure balances P = F/A P-range: 0 -5 kbar Accuracy: 0. 02%

High pressure balances to 3 GPa - Technically difficult - Not commercially available - High exploitation costs Accuracy: ~ 0. 1% Heydemann, J. Appl. Phys. (1967)

Secondary pressure scales = Methods which: Are more adapted to a specific P range and device Have been calibrated to a primary standard

« Bourdon gauge » « Heise » gauge, 0 -6 kbar reading precision: 2 bar - Mechanic devices - range: 0 -1 GPa - accuracy ~ 0. 3%

Pressure transducer gauges - Electronic devices: detect change of resistance, capacity with p - range: 0 -1 GPa, commercially available ~ 1 -3 k€ - accuracy ~ 0. 5%

Manganine wire gauge Manganine: alloy, 84% Cu, 14% Mn, 2% Ni ln. R/ P = +0. 023/GPa Bridgman 1911 From: http: //frustrated-electrons. ifs. hr/ - in form of coil. Small, simple, inexpensive - range: 0 -6 GPa, up to 250 °C; mainly in fluids, sometimes dynamic Ps - accuracy ~ 0. 5% - need to be prepared and aged, « handwork » needed - has non-negligible temp. dependence of resistance.

« Fixed point » gauges Use phase transitions of certain elements & compounds Usually detected resistively, sometimes volumetrically Examples: Hg liq-sol Bi I-II Tl II-III Bi III-V Pb I-II Ga. As Bi Bundy 1958 0. 7569 GPa 2. 556 GPa 3. 67 GPa 7. 69 GPa 13. 4 GPa 17. 5 -18. 5 GPa Bi

Typical application: Multianvil cells « Calibration curve » = Pressure-load relationship of a high P assembly Courtesy: BGI, Bayreuth

Optical pressure gauges = Gauges which use the pressure depencence of an optical property of a material which is next to the sample.

The ruby fluorescence gauge Ruby: Al 2 O 3: Cr 3+ ( « corundum » ) Cr 3+: ~ 0. 1 -1 at% 0 GPa 20 GPa Dl/DP = +0. 365 nm/GPa Piermarini et al. JAP 1975 K. Syassen, High Pres. Res. 28, 75 (2008)

The ruby 1986 calibration ( « Mao quasi-hydrostatic scale » ) Mao, Xu, Bell, JGR 91, 4673 (1986) - Load DAC with Cu, Ag and ruby & Ar - Measure V of Cu & Ag and determine P from shock wave Eo. S -Fit measured l(P) to a Murnaghan-function and constrain it at low P to the Piermarini coefficient A = 1904 GPa B = 7. 665

Temperature dependence - Line widths broaden considerably with T P measurements more difficult at high T Fluorescence at high P - Fluorescence weaker at high P need blue laser for very high Ps

Practical aspects S. Klotz, unpublished ruby Ruby: easy to get, single crystalline, low. Z, unexpensive, strong fluorescence! Can work with very small rubies: ~ 5 mm (prefer ruby spheres) laser fiber spectro Optical set-up simple and compact, relatively cheap (~ 5 k€)

Syassen, High Pres. Res. 28, 75 (2008) How precise is the ruby scale? Below ~ 30 GPa, the Mao 1986 scale is accurate to ~ 1%. At 1 -2 Mbar, it underestimates P by probably ~ 5 -10% Many suggestions for revisons, but no general consensus

Other fluorescence gauges Ruby: Moderate dl/dp Large dl/d. T Strontium borate Matlockite Chen et al. , High Press. Res. 7, 73 (1991)

The « diamond edge » optical scale Rule of thumb: (distance of the two edges in cm-1)/2 = P in GPa « Use the diamond edge when you have nothing else to measure P » Akahama & Kawamura, J. Appl. Phys 96, 3748 (2004)

Diffraction pressure gauges = Gauges which use the pressure dependence of the lattice parameter (unit cell volume) of a material in close contact with the sample. Usually measured by a diffraction experiment (x-ray & neutrons)

The Decker Na. Cl scale (1971) • “Table”, semi-empirical (interatom. pot. + exp. input params) • P-range: 0 -300 kbar • Accuracy: 0 -5 % (!? ) Decker, J. Appl. Phys. 42, 3239 (1971)

Brown’s 1999 Na. Cl scale PBrown-Pdecker (GPa) M. Brown, J. Appl. Phys. , 1999 PBrown-PDecker/P 3% Probably more accurate than Decker!

General observations « The Decker scale is the mother of the Ruby scale » Mao, Xu, Bell, JGR 91, 4673 (1986 ) Initial slope forced to be coherent with Decker scale Limited to 0 -35 GPa (B 1 -B 2 transition) Decker less frequently used

Other (more recent) diffraction gauges: metals Dewaele et al. , PRB 2004 Anzellini et al. , JAP 2014 Dewaele & Takemura, PRB 2008 R Re PRuby (GPa) No tables: Take B 0, B 0’, V 0 & plug into Eo. S form: « Birch Murnaghan Eo. S » « Vinet-Rydberg Eo. S » X = (V/V 0)

Primary standards at P>3 GPa? « Integration of bulk modulus » Compress a sample and: - Determine V by diffraction (precision : 10 -4) - Determine simultaniously compressibility by some technique - Integrate

Ultrasonic measurements Example: cubic system along [100] r v 2 = C 11 l B = (C 11+2 C 12)/3 measure speed v of a pulse: Precision: 10 -4! Problem: gives adiabatic B: « BS » BT = BS/(1+ag. T) ag. T 1% at 300 K Feasible, but accuracy limited by precision in Grüneisen-parameter g !

Shock wave measurements shock front r P E Up compressed (shocked) material US r 0 E 0 uncompressed Material: P=0 U: speed s: shock front, p: particle Measure speeds get P! « Rankine-Hugoniot equations » But: Need to be « reduced » to T=const!

Mao, Xu, Bell, JGR 91, 4673 (1986 ) Shock-wave reduced Eo. S data To remember: Shock waves data provide (in theory) a primary P gauge in the Mbar range

Outlook I: Accuracy in everyday life Mass/weight: 1 g / 1 kg = 0. 1% Temperature: 0. 1 K / 300 K ~ 0. 01 % Length: 1 mm/1 m = 0. 1 % Time: 1 sec / 1 day = 0. 001 % Pressure: 0. 1 bar / 2 bar 5 % (tyre) 10 h. Pa/1000 h. Pa 1% (atm. pressure) Science, 3 -10 GPa: 3 -5% !!

Outlook II: High pressure metrology: A boring subject? Forman, Piermarini, Barnett & Block, Science 176, 285 (1972) Piermarini, Block & Barnett, J. Appl. Phys. 44, 5377 (1973) Barnett, Block & Piermarini, RSI 44, 1 (1973) Piermarini, Block, Barnett & Forman, J. Appl. Phys. 2774 (1975) Mao, Bell, Shaner, Steinberg, J. Appl. Phys. 49, 3276 (1978) Mao, Xu & Bell, J. Geophys. Res. 91, 4673 (1986) ~ 4500 citation in total (2008) K. Syassen, High Pressure Research (2008)

Thank you!