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

 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 Summary

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

Stress and strain (in a nutshell) Strain (« deformation »):U = (u ij ) (3x3 matrix, symetric: 6 elements) U =    * * * = 0 U = 0 0  * * 0 0  0 00  00 Tr U =  V Remark:

Stress (« pressure »):  = (  ij ) (3x3 matrix, symetric: 6 elements)  = -p * * * = 0  = 0 0  * * 0 0  0 00  00 - hydrostatic pressure:  ij = -  ij ∙ p - hydrostatic component: p = ⅓ ∙Tr U - Relation  ↔ U : Hook’s law:  = C U Remarks: F

To remember:  « Pressure » in physics means almost exclusively hydrostatic pressure:  = -  ij p ex: p = -(  E/  V) T B= - (  p/  lnV) T  importance of hydrostaticity in exp. studies  « Pressure » is a form of stress  =(  ij )  Worthwhile to invest some time into theory of elasticity!

Primary pressure scales (  use only the definition of pressure) High pressure balances P-range: 0-5 kbar Accuracy: 0.02% P = F/A

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 » - Mechanic devices - range: 0-1 GPa - accuracy ~ 0.3% « Heise » gauge, 0-6 kbar reading precision: 2 bar

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 From: Manganine: alloy, 84% Cu, 14% Mn, 2% Ni  lnR/  P = /GPa - 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. Bridgman 1911

« Fixed point » gauges Use phase transitions of certain elements & compounds Usually detected resistively, sometimes volumetrically Examples: Hg liq-sol GPa Bi I-II2.556 GPa Tl II-III3.67 GPa Bi III-V7.69 GPa Pb I-II13.4 GPa GaAs GPa Bi Bundy 1958

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

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+ : ~ at% K. Syassen, High Pres. Res. 28, 75 (2008) 0 GPa20 GPa  /  P = nm/GPa Piermarini et al. JAP 1975

The ruby 1986 calibration (« Mao quasi-hydrostatic scale ») - Load DAC with Cu, Ag and ruby & Ar - Measure V of Cu & Ag and determine P from shock wave EoS -Fit measured l(P) to a Murnaghan-function and constrain it at low P to the Piermarini coefficient Mao, Xu, Bell, JGR 91, 4673 (1986) A = 1904 GPa B = 7.665

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

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

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 Syassen, High Pres. Res. 28, 75 (2008)

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

The « diamond edge » optical scale Akahama & Kawamura, J. Appl. Phys 96, 3748 (2004) 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 »

= 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) Diffraction pressure gauges

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

Brown’s 1999 NaCl scale M. Brown, J. Appl. Phys., 1999 P Brown -P decker (GPa) P Brown -P Decker /P  3% Probably more accurate than Decker!

Mao, Xu, Bell, JGR 91, 4673 (1986 ) General observations Initial slope forced to be coherent with Decker scale  Limited to 0-35 GPa (B1-B2 transition)  Decker less frequently used  « The Decker scale is the mother of the Ruby scale »

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

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

l Ultrasonic measurements  v 2 = C 11 Example: cubic system along [100] B = (C 11 +2C 12 )/3 Problem: gives adiabatic B: « B S » B T = B S /(1+  T)  T  1% at 300 K measure speed v of a pulse: Precision: ! Feasible, but accuracy limited by precision in Grüneisen-parameter 

 Shock wave measurements PEPE UpUp E0E0 USUS compressed (shocked) material uncompressed Material: P=0 U: speed s: shock front, p: particle shock front « Rankine-Hugoniot equations » Measure speeds  get P! But: Need to be « reduced » to T=const!

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

Outlook I: Accuracy in everyday life Temperature: 0.1 K / 300K ~ 0.01 % Mass/weight:1 g / 1 kg = 0.1% Length: 1 mm/1 m = 0.1 % Time: 1 sec / 1 day = % Pressure: 0.1 bar / 2 bar 5 % (tyre) 10 hPa/1000 hPa 1% (atm. pressure) Science, 3-10 GPa: 3-5% !!

Outlook II: High pressure metrology: A boring subject? K. Syassen, High Pressure Research (2008) ~ 4500 citation in total (2008) 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 (1975) Mao, Bell, Shaner, Steinberg, J. Appl. Phys. 49, 3276 (1978) Mao, Xu & Bell, J. Geophys. Res. 91, 4673 (1986)

Thank you!