How far away are accurate equations of state determinations Some issues on pressure scales and non-hydrostaticity in diamond anvil cells

Lei Liu; Yan Bi

doi:10.1016/j.mre.2016.06.002

[1] G. Shen, P. Chow, Y. Xiao, S. Sinogeikin, Y. Meng, et al., HPCAT: an integrated high pressure synchrotron facility at the advanced photon source, High Press. Res. 28 (2008) 145.

[2] L. Dubrovinsky, N. Dubrovinskaia, E. Bykova, V. Prakapenka, C. Prescher, et al., The most incompressible metal osmium at static pressures above 750 gigapascals, Nature 525 (2015) 226.

[3] S. Tateno, K. Hirose, Y. Ohishi, Y. Tatsumi, The structure of iron in Earth's inner core, Science 330 (2010) 359.

[4] K. Syassen, W.B. Holzapfel, Isothermal compression of Al and Ag to 120 kbar, J. Appl. Phys. 49 (1978) 4427.

[5] W.B. Holzapfel, Equations of state for Cu, Ag, and Au for wide ranges in temperature and pressure up to 500 GPa and above, J. Phys. Chem. Ref. Data 30 (2001) 515.

[6] K.W. Katahara, M.H. Manghnani, E.S. Fisher, Pressure derivatives of the elastic moduli of niobium and tantalum, J. Appl. Phys. 47 (1976) 434.

[7] D.J. Steiberg, Some observations regarding the pressure dependence of the bulk modulus, J. Phys. Chem. Solids 43 (1982) 1173.

[8] N.C. Holmes, J.A. Moriarty, G.R. Gathers, W.J. Nellis, The equation of state of platinum to 660 GPa (6.6 Mbar), J. Appl. Phys. 66 (1989) 2962.

[9] R.S. Hixson, J.N. Fritz, Sock compression of tungsten and molybdenum, J. Appl. Phys. 71 (1992) 1721.

[10] W.J. Nellis, J.A. Moriarty, A.C. Mitchell, M. Ross, R.G. Dandrea, et al., Metal physics at ultrahigh pressure: aluminum, copper, and lead as prototypes, Phys. Rev. Lett. 60 (1988) 1414.

[11] C.S. Zha, H.K. Mao, R.J. Hemley, Elasticity of MgO and a primary pressure scale to 55 GPa, PNAS 97 (2000) 13494.

[12] P.I. Dorogokupets, A.R. Oganov, Ruby, metals, and MgO as alternative pressure scales: a semiempirical description of shockwave, ultrasonic, X-ray, and thermochemical data at high temperatures and pressures, Phys. Rev. B 75 (2007) 024115.

[13] W.B. Holzapfel, Equations of state for Cu, Ag, and Au and problems with shock wave reduced isotherms, High Press. Res. 30 (2010) 372.

[14] K. Kunc, K. Syassen, P(V) equations of state of solids: density functional theory calculations and LDA versus GGA scaling, Phys. Rev. B 81 (2010) 134102.

[15] K. Kunc, I. Loa, K. Syassen, Equation of state and phonon frequency calculations of diamond at high pressure, Phys. Rev. B 81 (68) (2003) 094107.

[16] H.K. Mao, P.M. Bell, Specific volume measurements of Cu, Mo, Pd, and Ag and calibration of the ruby R1 fluorescence pressure gauge from 0.06 to 1 Mbar, J. Appl. Phys. 49 (1978) 3276.

[17] H.K. Mao, J. Xu, P.M. Bell, Calibration of the ruby pressure gauge to 800 kbar under quasihydrostatic conditions, J. Geophys. Res. 91 (1986) 4673.

[18] F. Datchi, R. LeToullec, P. Loubeyre, Improved calibration of the SrB4O7:Sm2t optical pressure gauge: advantages at very high pressures and high temperatures, J. Appl. Phys. 81 (1997) 3333.

[19] S.V. Raju, J.M. Zaug, B. Chen, J. Yuan, J.W. Knight, et al., Determination of the variation of the fluorescence line positions of ruby, strontium tetraborate, alexandrite, and samarium-doped yttrium aluminum garnet with pressure and temperature, J. Appl. Phys. 110 (2011) 023521.

[20] Q. Jing, Q. Wu, L. Liu, J. Xu, Y. Bi, et al., An experimental study on SrB4O7:Sm2t as a pressure sensor, J. Appl. Phys. 113 (2013) 023507.

[21] M.I. Eremets, Megabar high-pressure cells for Raman measurements, J. Raman Spectrosc. 34 (2003) 515.

[22] L. Sun, A.L. Ruoff, G. Stupian, Convenient optical pressure gauge for multipressure calibrated to 300 GPa, Appl. Phys. Lett. 86 (2005) 014103.

[23] Y. Akahama, H. Kawamura, Pressure calibration of diamond anvil Raman gauge to 310 GPa, J. Appl. Phys. 100 (2006) 043516.

[24] B.J. Baer, M.E. Chang, W.J. Evans, Raman shift of stressed diamond anvils: pressure calibration and culet geometry dependence, J. Appl. Phys. 104 (2008) 034504.

[25] T. Kawamoto, K.N. Matsukage, T. Nagai, K. Nishimura, T. Mataki, Raman spectroscopy of cubic boron nitride under high temperature and pressure conditions: a new optical pressure marker, Rev. Sci. Instrum. 75 (2004) 2451.

[26] A.F. Goncharov, J.C. Crowhurst, J.K. Dewhurst, S. Sharma, Raman spectroscopy of cubic boron nitride under extreme conditions of high pressure and temperature, Phys. Rev. B 72 (2005) 100104(R).

[27] A.F. Goncharov, S. Sinogeikin, J.C. Crowhurst, M. Ahart, D. Lakshtanov, et al., Cubic boron nitride as a primary calibrant for a high temperature pressure scale, High Press. Res. 27 (2007) 409.

[28] H. Spetzler, A. Shen, G. Shen, G. Herrmannsdoerfer, H. Schulze, et al., Ultrasonic measurements in a diamond anvil cell, Phys. Earth Planet. Inter. 98 (1996) 93.

[29] S.D. Jacobsen, H.A. Spetzler, H.P. Reichmann, J.R. Smyth, S.J. Mackwell, et al., Gigahertz ultrasonic interferometry at high P and T: new tools for obtaining a thermodynamic equation of state, J. Phys. Condens. Matter 14 (2002) 11525.

[30] N. Chigarev, P. Zinin, D. Mounier, A. Bulou, A. Zerr, et al., Laser ultrasonic measurements in a diamond anvil cell on Fe and the KBr pressure medium, J. Phys. Conf. Ser. 278 (2011) 012017.

[31] J. Lin,W. Strurhahn, J. Zhao, G. Shen, H.K. Mao, et al., Sound velocities of hot dense iron: Birch's law revisited, Science 308 (2005) 1892.

[32] F. Birch, Finite elastic strain of cubic crystal, Phys. Rev. 71 (1947) 809.

[33] F. Birch, Finite strain isotherm and velocities for single-crystal and polycrystalline NaCl at high pressure and 300 K, J. Geophys. Res. 95 (1978) 1257.

[34] P. Vinet, J. Ferrante, J.H. Rose, J.R. Smith, Compressibility of solids, J. Geophys. Res. 92 (1987) 9319.

[35] P. Vinet, J.H. Rose, J. Ferrante, J.R. Smith, Universal features of the equation of state of solids, J. Phys. Condens. Matter 1 (1989) 1941.

[36] J. Poirier, Introduction to the Physics of the Earth's Interior, Cambridge University Press, Cambridge, 2000, p. 78.

[37] G. Simmons, H. Wang, Single Crystal Elastic Constant and Calculated Aggregate Properties: A Handbook, MIT Press, Cambridge, 1971.

[38] W.B. Daniels, C.S. Smith, Pressure derivatives of the elastic constants of copper, silver and gold to 10000 Bars, Phys. Rev. 111 (1958) 713.

[39] J.C. Jamieson, J.N. Fritz, M.H. Manghnani, in: S. Akimoto, M.H. Manghnani (Eds.), High-pressure Research in Geophysics,” of Advances in Earth and Planetary Sciences vol. 12, Center for Academic Publishing, Tokyo, 1982.

[40] Y. Wang, D. Chen, X. Zhang, Calculated equation of state of Al, Cu, Ta, Mo, and W to 1000 GPa, Phys. Rev. Lett. 84 (2000) 3220.

[41] Y. Wang, R. Ahuja, B. Johansson, Reduction of shock-wave data with mean-field potential approach, Phys. Rev. B 92 (2002) 6616.

[42] Y. Wang, Theoretical Studies of Thermodynamic Properties of Condensed Matter Under High Temperature and High Pressure, Ph.D thesis, Uppsala University, Uppsala, 2004.

[43] A.D. Chijioke, W.J. Nellis, A. Soldatov, I.F. Silvera, The ruby pressure standard to 150 GPa, J. Appl. Phys. 98 (2005) 114905.

[44] K. Jin, X. Li, Q. Wu, H. Geng, L. Cai, et al., The pressure-volumetemperature equation of state of MgO derived from shock Hugoniot data and its application as a pressure scale, J. Appl. Phys. 107 (2010) 113518.

[45] K. Jin, Q. Wu, H. Geng, X. Li, L. Cai, et al., Pressure-volume-temperature equation of state of Au and Pt up to 300 GPa and 3000 K: Internally consistent pressure scales, High Press. Res. 31 (2011) 560.

[46] Q. Wu, Studies on Equation of State and Gru¨neisen Parameter for Metals at High Pressures and Temperatures, Ph.D. thesis, China Academy of Engineering Physics, 2004.

[47] A.L. Ruoff, R.C. Lincoln, Y.C. Chen, High-pressure calibration with a new absolute-pressure gauge, J. Appl. Phys. 22 (1973) 310.

[48] B. Li, J. Kung, T. Uchida, Y. Wang, Pressure calibration to 20 GPa by simultaneous use of ultrasonic and X-ray techniques, J. Appl. Phys. 98 (2005) 013521.

[49] J. Xu, E. Huang, Primary pressure scaleeA proposal, J. Geol. Soc. China 41 (1998) 199.

[50] K. Syassen, Ruby under pressure, High Press. Res. 28 (2008) 75.

[51] K.K. Zhuravlev, A.F. Goncharov, S.N. Tkachev, P. Dera, V.B. Prakapenka, Vibrational, elastic, and structural properties of cubic silicon carbide under pressure up to 75 GPa: Implication for a primary pressure scale, J. Appl. Phys. 113 (2013) 113503.

[52] R.A. Forman, G.J. Piermarini, J.D. Barnett, S. Block, Pressure measurement made by the utilization of ruby sharp-line luminescence, Science 176 (1972) 284.

[53] I.V. Aleksandrov, A.F. Goncharov, A.N. Zisman, S.M. Stishov, Diamond at high pressures: Raman scattering, equation of state, and high pressure scale, Sov. Phys. JETP 66 (1987) 384.

[54] W.B. Holzapfel, Refinement of the ruby luminescence pressure scale, J. Appl. Phys. 93 (2003) 1813.

[55] K. Kunc, I. Loa, K. Syassen, Diamond under pressure: Ab-initio calculations of the equation of state and optical phonon frequency revisited, High Press. Res. 24 (2004) 101.

[56] A. Dewaele, P. Loubeyre, M. Mezouar, Equations of state of six metals above 94 GPa, Phys. Rev. B 70 (2004) 094112.

[57] A. Dewaele, M. Torrent, P. Loubeyre, M. Mezouar, Compression curves of transition metals in the Mbar range: experiments and projector augmented-wave calculations, Phys. Rev. B 78 (2008) 104112.

[58] A.D. Chijioke, W.J. Nellis, I.F. Silvera, High-pressure equations of state of Al, Cu, Ta, and W, J. Appl. Phys. 98 (2005) 073526.

[59] L. Liu, Y. Bi, J. Xu, Ruby fluorescence pressure scale: revisited, Chin. Phys. B 22 (2013) 056201.

[60] S.D. Jacobsen, C.M. Holl, K.A. Adams, R.A. Fischer, E.S. Martin, et al., Compression of single-crystal magnesium oxide to 118 GPa and a ruby pressure gauge for helium pressure media, Am. Mineral. 93 (2008) 1823.

[61] Y. Bi, J. Xu, A Corrected Absolute MgO Pressure Scale up to 65 GPa and the Accuracy of Pressure Determination, unpublished, 2011.

[62] S. Sinogeikin, J. Bass, V. Prakapenka, D. Lakshtanov, G. Shen, et al., Brillouin spectrometer interfaced with synchrotron radiation for simultaneous X-ray density and acoustic velocity measurements, Rev. Sci. Instrum. 77 (2006) 103905.

[63] F. Datchi, A. Dewaele, P. Loubeyre, R. Letoullec, Y. LeGodec, et al., Optical pressure sensors for high-pressure-high-temperature studies in a diamond anvil cell, High Press. Res. 27 (2007) 447.

[64] C.S. Zha, W.A. Bassett, Internal resistive heating in diamond anvil cell for in situ X-ray diffraction and Raman scattering, Rev. Sci. Instrum. 74 (2003) 1255.

[65] J. Xu, H.K. Mao, P.M. Bell, Position-sensitive x-ray diffraction: hydrostatic compressibility of argon, tantalum, and copper 769 kbar, High Temperature-High Press. 16 (1984) 495.

[66] J. Xu, E. Huang, J. Lin, L. Xu, Raman study at high pressure and the thermodynamic properties of corundum: application of Kieffer's model, Am. Mineral. 80 (1995) 1157.

[67] L.S. Dubrovinsky, N.A. Dubrovinskaia, T. Le Bihan, Aggregate sound velocities and acoustic Gru¨neisen parameter of iron up to 300 GPa and 1200 K, PNAS 98 (2001) 9484.

[68] T. Kenichi, Evaluation of the hydrostaticity of a helium-pressure medium with powder X-ray diffraction techniques, J. Appl. Phys. 89 (2001) 662.

[69] R.J. Angel, M. Bujak, J. Zhao, G.D. Gatta, S.D. Jacobsen, Effective hydrostaticity limits pf pressure media for high-pressure crystallographic studies, J. Appl. Crystallogr. 40 (2007) 26.

[70] Q. Jing, Y. Bi, Q. Wu, F. Jing, Z. Wang, et al., Yield strength of molybdenum at high pressures, Rev. Sci. Instrum. 78 (2007) 073906.

[71] A.K. Singh, The lattice strains in specimen (cubic system) compressed nonhydrostatically in an opposed anvil device, J. Appl. Phys. 73 (1993) 4278.

[72] A.K. Singh, C. Balasigh, The lattice strains in a specimen (hexagonal system) compressed nonhydrostatically in an opposed anvil high pressure setup, J. Appl. Phys. 75 (1994) 4956.

[73] A.K. Singh, C. Balasigh, H.K. Mao, R.J. Hemley, J. Shu, Analysis of lattice strains measured under nonhydrostatic pressure, J. Appl. Phys. 83 (1998) 7567.

[74] A.K. Singh, T. Kenichi, Measurement and analysis of nonhydrostatic lattice strain component in niobium to 145 GPa under various fluid pressure-transmitting media, J. Appl. Phys. 90 (2001) 3269.

[75] T. Uchida, N. Funamori, T. Yagi, Lattice strains in crystals under uniaxial stress field, J. Appl. Phys. 80 (1996) 739.

[76] D. He, T.S. Duffy, X-ray diffraction study of the static strength of tungsten to 69 GPa, Phys. Rev. B 73 (2006) 134106.

[77] T.S. Duffy, G. Shen, J. Shu, H.K. Mao, R.J. Hemley, et al., Elasticity, shear strength, and equation of state of molybdenum and gold from xray diffraction under nonhydrostatic compression to 24 GPa, J. Appl. Phys. 86 (1999) 6729.

[78] R.J. Hemley, H.K. Mao, G. Shen, J. Badro, P. Gillet, et al., X-ray imaging of stress and strain of diamond, iron, and tungsten at megabar pressures, Science 276 (1997) 1242.

[79] A.K. Singh, H.K. Mao, J. Shu, R. Hemley, Estimation of single-crystal elastic moduli from polycrystalline X-ray diffraction at high pressure: application to FeO and iron, Phys. Rev. Lett. 80 (1998) 2157.

[80] H.K. Mao, J. Shu, G. Shen, R.J. Hemley, B. Li, et al., Elasticity and rheology of iron above 220 GPa and the nature of the Earth's inner core, Nature 396 (1998) 741.

[81] A.K. Singh, Analysis of nonhydrostatic high-pressure diffraction data (cubic system): assessment of various assumptions in the theory, J. Appl. Phys. 106 (2009) 043514.

[82] S. Karato, Theory of lattice strain in a material undergoing plastic deformation: Basic formulation and applications to a cubic crystal, Phys. Rev. B 79 (2009) 214106.

[83] P.A. Turner, C.N. Tome, A study of residual-stresses in zircaloy-2 with rod texture, Acta Metall. Mater. 42 (1994) 4143.

[84] B. Clausen, T. Lorentzen, T. Leffers, Self-consistent modeling of the plastic-deformation of fcc polycrystals and its implications for diffraction measurements of internal-stresses, Acta Mater. 46 (1998) 3087.

[85] D.J. Weidner, L. Li, M. Davis, J.H. Chen, Effect of plasticity on elasticmodulus measurements, Geophys. Res. Lett. 31 (2004) 6621.

[86] L. Li, D.J. Weidner, J.H. Chen, M.T. Vaughan, M. Davis, et al., X-ray strain analysis at high pressure: effect of plastic deformation in MgO, J. Appl. Phys. 95 (2004) 8357.

[87] S. Merkel, C. Tome, H.R. Wenk, Modeling analysis of the influence of plasticity on high pressure deformation of hcp. Co, Phys. Rev. B 79 (2009) 064110.

[88] H.K. Mao, J. Badro, J. Shu, R.J. Hemley, A.K. Singh, Strength, anisotropy, and preferred orientation of solid argon at high pressures, J. Phys. Condens. Matter 18 (2006) S963.

[89] E. Menendez-Proupin, A.K. Singh, Ab initio calculations of elastic properties of compressed Pt, Phys. Rev. B 76 (2007) 054117.

[90] K.W. Katahara, M.H. Manghnani, E.S. Fisher, Pressure derivatives of the elastic moduli of bcc titanium-vanadium-chromium, niobium-molybdenum and tantalum-tungsten alloys, J. Phys. F 9 (1979) 773.

[91] L. Liu, H.X. Song, H.Y. Geng, Y. Bi, J. Xu, Compressive behaviors of bcc bismuth up to 55 GPa, Phys. Status Solidi B 250 (2013) 1398.

[92] L. Liu, H.X. Song, Z.G. Wang, H.Y. Geng, Q. Jing, et al., Strength and equation of state of fluorite phase CeO2 under high pressures, J. Appl. Phys. 112 (2012) 013532.

[93] J.I. Langford, X-ray powder diffraction studies of vitromet samples, J. Appl. Crystallogr. 4 (1971) 159.

[94] A.K. Singh, H.P. Liermann, Y. Akahama, H. Kawamura, Aluminum as a pressure-transmitting medium cum pressure standard for X-ray diffraction experiments to 200 GPa with diamond anvil cell, J. Appl. Phys. 101 (2007) 123526.

[95] A.K. Singh, H.P. Liermann, S.K. Saxena, Strength of magnesium oxide under high pressure: evidence for the grain-size dependence, Solid State Commun. 132 (2004) 795.

[96] A.K. Singh, H.P. Liermann, S.K. Saxena, H.K. Mao, U. Devi, Nonhydrostatic compression of gold powder to 60 GPa in a diamond anvil cell: estimation of compressive strength from X-ray diffraction data, J. Phys. Condens. Matter 18 (2006) S969.

[97] A.K. Singh, X-ray diffraction from solids under nonhydrostatic compression-some recent studies, J. Phys. Chem. Solids 65 (2004) 1589.

[98] Z. Wang, S.K. Saxena, V. Pischedda, H.P. Liermann, C.S. Zha, In situ X-ray diffraction study of the pressure-induced phase transformation in nanocrystalline CeO2, Phys. Rev. B 64 (2001) 012102.

[99] L. Gerward, J.S. Olsen, L. Petit, G. Valitheeswaran, V. Kanchana, et al., Bulk modulus of CeO2 and PrO2-an experimental and theoretical study, J. Alloys Compd. 400 (2005) 56.

[100] M.I. Eremets, High Pressure Experimental Methods, Oxford University Press, Oxford, 1996, p. 761.

[101] B. Chen, A.E. Gleason, J.Y. Yan, K.J. Koski, S. Clark, et al., Elasticity, strength, and refractive index of argon at high pressure, Phys. Rev. B 81 (2010) 144110.

[102] H. Marquardt, S. Speziale, A. Gleason, S. Sinogeikin, I. Kantor, et al., Brillouin scattering and X-ray diffraction of solid argon to 65 GPa and 700 K: Shear strength of argon at HP/HT, J. Appl. Phys. 114 (2013) 093517.