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EOS of simple Solids for wide Ranges in p and T

EOS of simple Solids for wide Ranges in p and T. Problem: The accuracy of “primary” K(V)-scales is still seriously limited by the present range and precision in measurements of K and V under high p and T. Solution: Semi-empirical EOS with theoretical and experimental input provide

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EOS of simple Solids for wide Ranges in p and T

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  1. EOS of simple Solids for wide Ranges in p and T Problem: The accuracy of “primary” K(V)-scales is still seriously limited bythe present range and precision in measurements of K and V under high p and T. Solution: Semi-empirical EOS with theoretical and experimental input provide p(V,T)-relations of many “simple” materials for the determination of p with higher accuracy from measurements of V and T. Comparison of different markers gives estimates of the accuracy! Washington 2007 Wilfried B. Holzapfel Department Physik Universität-Paderborn Germany <holzapfel@physik.upb.de>

  2. EOS of simple Solids for wide Ranges in p and T EOS of “simple” or “regular” solids are well understood theoretically! All thermo-physical data including the EOS must be modeled by the same thermodynamic potential (the same Gibbs function). “Cold” (0 K) isotherms can be determined from a priory theory or from semi-empirical effective potential forms (APL). Thermal contributions are accurately modeled by thermodynamics with experimental input from ambient pressure and theoretical support for the volume dependence of the intrinsic anharmonicity. Washington 2007 Wilfried B. Holzapfel Department Physik Universität-Paderborn Germany <holzapfel@physik.upb.de>

  3. EOS of simple Solids for wide Ranges in p and T The approach: 1) Set up a thermodynamic model for the solid with anharmonicity! 2) Use Kr and Kr’ values from ultrasonic or Brillouin measurements for one (cold) reference isotherm! 3) Use ambient pressure data of Vo(T), αo(T),Cpo(T) and Ko(T) to determine thermo-physical parameters (θo, γo, ...) of the model! 4) Refine Kr’ as best fitting Ko(T)! Washington 2007 Wilfried B. Holzapfel Department Physik Universität-Paderborn Germany <holzapfel@physik.upb.de>

  4. EOS of simple Solids for wide Ranges in p and T Thethermo-physical model for the Gibbs function: Total Free-Energy: F(V,T) = Ec(V) + Fth (V,T) with Ground State Energy Ec(V) and thermal Excitations in Fth(V,T) Ec(V) from one cold isotherm: pc(V) = pAP2(V,Z,Vo,Ko,K’o) Fth(V,T) = Fcond.el.(T,TFG(V)) + Fquasi-harm.phonon(V,T) + Fintr.anharm.(V,T) Quasi-harmonic Phonons are modeled with an optimized pseudo-Debye-Einstein approximation: opDE, Intrinsic Anharmonicity with a Modified Mean Field approach: MMF Washington 2007 Wilfried B. Holzapfel Department Physik Universität-Paderborn Germany <holzapfel@physik.upb.de>

  5. Effective potential forms for cold isotherm Mie EOS (Mi3) Effective Rydberg EOS (ER2) Adapted Polynomial EOS (AP2) Thomas-Fermi-limit is modeled by co! Washington 2007 Wilfried B. Holzapfel Department Physik Universität-Paderborn Germany <holzapfel@physik.upb.de>

  6. An optimized pseudo-Debye-Einstein model Cold Isotherm: The Mie-Grüneisen approach for the internal energy gives : with An optimized pseudo-Debye-Einstein model for quasi-harmonic phonons with g=0.068 a=0.0434 and is conveniently used here! Washington 2007 Wilfried B. Holzapfel Department Physik Universität-Paderborn Germany <holzapfel@physik.upb.de>

  7. Intrinsic Anharmonicity Classical Free Volume Approach with Modified Mean Field Potential g(r,V) J.G. Kirkwood, J. Chem. Phys. 18, 380 (1950) Y. Wang, Phys. Rev. B 61, R11863 (2000) Prag 2006 Wilfried B. Holzapfel Department Physik Universität-Paderborn Germany <holzapfel@physik.upb.de>

  8. Intrinsic Anharmonicity Classical Free Volume Approach with Modified Mean Field Potential g(r,V) J.G. Kirkwood, J. Chem. Phys. 18, 380 (1950) Y. Wang, Phys. Rev. B 61, R11863 (2000) Prag 2006 Wilfried B. Holzapfel Department Physik Universität-Paderborn Germany <holzapfel@physik.upb.de>

  9. Intrinsic Anharmonicity Two Series Expansions in the Classical Free Volume Approach Washington 2007 Wilfried B. Holzapfel Department Physik Universität-Paderborn Germany <holzapfel@physik.upb.de>

  10. Vibrational Free-Energy in the Classical Free-Volume Approach with linear contributions from and only. Mie-Grüneisen Approximation implies Washington 2007 Wilfried B. Holzapfel Department Physik Universität-Paderborn Germany <holzapfel@physik.upb.de>

  11. Vibrational Grüneisen Parameter in the classical free volume approach Washington 2007 Wilfried B. Holzapfel Department Physik Universität-Paderborn Germany <holzapfel@physik.upb.de>

  12. Higher Order Corrections in the Free Volume Approach Washington 2007 Wilfried B. Holzapfel Department Physik Universität-Paderborn Germany <holzapfel@physik.upb.de>

  13. Heat capacity of Cu at ambient pressure Experimental data +, o, x and • from Ly59, Ma60, Gr72 and Ch98 Present fit of Cpo and Cvo: solid and dashed line Cpo(T) 3R T (K) Prag 2006 Wilfried B. Holzapfel Department Physik Universität-Paderborn Germany <holzapfel@physik.upb.de>

  14. Heat capacity of Ag at ambient pressure Experimental data o, x and • from Mo36, MF41, and Gr72 Present fit of Cpo and Cvo: solid and dashed line Cpo(T) 3R T (K) Wilfried B. Holzapfel Department Physik Universität-Paderborn Germany <holzapfel@physik.upb.de>

  15. Heat capacity of Au at ambient pressure Experimental data x and • from GG52, and Gr72 Present fit of Cpo and Cvo: solid and dashed line Cpo(T) 3R T (K) Wilfried B. Holzapfel Department Physik Universität-Paderborn Germany <holzapfel@physik.upb.de>

  16. Relative atomic volume of Cu at ambient pressure Experimental data • from TK75 and present fit as solid line Vo(T) Vr )Vo(T) Vr x 10000 _ 0 200 400 600 800 1000 1200 T (K)

  17. Relative atomic volume of Ag at ambient pressure Experimental data • from TK75 and present fit as solid line Vo(T) Vr )Vo(T) Vr x 10000 0 200 400 600 800 1000 1200 T (K)

  18. Relative atomic volume of Au at ambient pressure Experimental data • from TK75 and present fit as solid line Vo(T) Vr )Vo(T) Vr x 10000 0 200 400 600 800 1000 1200 T (K)

  19. Isothermal bulk modulus of Cu at ambient pressure Experimental data o from CH66 and • from VT79 Present fit with and without anharmonic contributions: solid and dashed line Ko(T) (GPa) T (K)

  20. Isothermal bulk modulus of Ag at ambient pressure Experimental data • from CH66 and o from BV81 Present fit with and without anharmonic contributions: solid and dashed line Ko(T) (GPa) T (K)

  21. Isothermal bulk modulus of Au at ambient pressure Experimental data • BV81 and o from CH66 Present fit with and without anharmonic contributions: solid and dashed line Ko(T) (GPa) T (K)

  22. Pressure derivative of the isothermal bulk modulus for Cu at ambient pressure Temperature dependent data • from VT79 with additional data for 300 K. Present fit with anharmonic contributions: solid line K´o(T) T (K)

  23. Pressure derivative of the isothermal bulk modulus for Ag at ambient pressure Temperature dependent data • from BV81 with additional data for 300 K. Present fit with anharmonic contributions: solid line K´o(T) T (K)

  24. Pressure derivative of the isothermal bulk modulus for Au at ambient pressure Temperature dependent data • from BV81 with additional data for 300 K. Present fit with anharmonic contributions: solid line K´o(T) T (K)

  25. Hierarchy of Parameters in the Refinements 1. Fit of Cpo(T) at low T by Refinement of TDo and TDh 2. Fit of Vo(T) at low T by Refinement of r 3. Fit of Cpo(T) and Vo(T) at higher T by Refinement of Anharmonicity Parameters f4 and f6 4. Minor Refinement of K’r to improve Fit of Ko(T) 5. Final Check of Step 2. and 3. Wilfried B. Holzapfel Department Physik Universität-Paderborn Germany <holzapfel@physik.upb.de>

  26. Thermo-physical Parameters for Cu Ag Au Z M TFeff(K) Vr(cnm) Kr(GPa) K’r TDo(K) TDh(K) (r 1000xAph f6 f2 8 f4 q (A

  27. Discussion 1. Perfect representation of all thermo-physical data at ambient P (for “regular” Solids only!) 2. Uncertainties in (V) and Aph(V)are reduced by MMF-calculation (No experimental data give better constraints!) 3. Reliable basis for the calculation of thermo-physical data at any P 4. Perfect agreement with Shock Wave Data Wilfried B. Holzapfel Department Physik Universität-Paderborn Germany <holzapfel@physik.upb.de>

  28. Comment on Parametric EOS If the cold isotherm pc(V) is represented bypAP2(V,Vo,Ko,K’o) (or by any other second order parametric EOS) no other isotherm is perfectly represented by the same form even with best fitted “effective values” for Vr, Kr, K’r deviating from the thermodynamic values! Accurate representations of thepresent thermodynamic EOS by parametric EOS need higher order forms with “effective” parameters! Wilfried B. Holzapfel Department Physik Universität-Paderborn Germany <holzapfel@physik.upb.de>

  29. Difference between thermodynamic and parametric EOS for Cu a) pAP2 b) pAP3 0 K 500 K 1000 K 0 K 500 K 1000 K 0 K 500 K HHS 01: 500 K 1000 K c) pAP2 + pAP3 with data point from HHS-01 for 500K

  30. Differences between thermodynamic and parametric EOS using pAP2+pAP3 for Ag and Au Data from HHS 01 for 500K are given for comparison 0 K 500 K 1000 K HHS 01: 500 K 0 K 500 K 1000 K HHS 01: 500 K

  31. Reference data for the parametric EOS of Cu T Vr Kr K’r K’reff c3

  32. Reference data for parametric EOS of Ag T Vr Kr K’r K’reff c3

  33. Reference data for parametric EOS of Au T Vr Kr K’r K’reff c3

  34. EOS for Solids with Mean-Field Anharmonicity Software and Support available on Request! Cooperation welcome! Prag 2006 Wilfried B. Holzapfel Department Physik Universität-Paderborn Germany <holzapfel@physik.upb.de>

  35. Comparison of EOS data for Cu Relative deviations of different EOS data for Cu at 300 K with respect to an AP2 form using Ko=132.2 GPa and K’o=5.40 used by HH2002 [WR57]J.M. Walsh, M.H. Rice, R.G. McQueen, and F.L. Yarger, Phys. Rev. 108 196 (1957) [KK72] R.N. Keeler, and G.C. Kennedy, American Institut of Physics Handbook, 4938, ed. D.E. Gray, New York: McGraw Hill (1972) [MB78] H.K. Mao, P.M. Bell, J.W. Shaner, and D.J. Steinberg, J. Appl. Phys. 49 3276 (1978) [AM85] R.C.Albers, A.K. McMahan, and J.E. Miller, Phys. Rev. B31 3435 (1985) [AB87] L.V. Al'tshuler, S.E. Brusnikin, and E.A. Kuz'menkov, J. Appl. Mech. and Tech. Phys. 28 129 (1987) [NM88] W.J. Nellis, J.A. Moriarty, A.C. Mitchell, M. Ross, R.G. Dandrea, N.W. Ashcroft, N.C. Holmes, and G.R. Gathers, Phys. Rev. Lett. 60, 1414 (1988) [Mo95] J.A. Moriarty, High Pressure Res. 13 343 (1995) [WC000] Yi Wang, Dongquan Chen, and Xinwei Zhang, Phys. Rev. Lett. 84 3220-3223 (2000) SESAME (provided by D.Young with permission)

  36. Comparison of EOS data for Ag Relative deviations of different EOS data for Ag at 300 K with respect to an AP2 form using Ko=101.1 GPa and K’o=6.15 used by HH2002 [WR57]J.M. Walsh, M.H. Rice, R.G. McQueen, and F.L. Yarger, Phys. Rev. 108 196 (1957) [KK72] R.N. Keeler, and G.C. Kennedy, American Institut of Physics Handbook, 4938, ed. D.E. Gray, New York: McGraw Hill (1972) [MB78] H.K. Mao, P.M. Bell, J.W. Shaner, and D.J. Steinberg, J. Appl. Phys. 49 3276 (1978) [AB87] L.V. Al'tshuler, S.E. Brusnikin, and E.A. Kuz'menkov, J. Appl. Mech. and Tech. Phys. 28 129 (1987)

  37. Comparison of EOS data for Au Relative deviations of different EOS data for Au at 300 K with respect to an AP2 form using Ko=166.7 GPa and K’o=6.20 used by HH2002 [KK72] R.N. Keeler, and G.C. Kennedy, American Institut of Physics Handbook, 4938, ed. D.E. Gray, New York: McGraw Hill (1972) [JF82] J.C. Jamieson, J.N. Fritz, and M.H. Manghnani in: High Pressure Research in Geophysics, ed. S. Akimoto, M.H. Manghnani Center for Acad. Public., Tokyo (1992) [AB87] L.V. Al'tshuler, S.E. Brusnikin, and E.A. Kuz'menkov, J. Appl. Mech. and Tech. Phys. 28 129 (1987) [GN92] B.K. Godwal, A.Ng, R. Jeanloz, High Pressure Res. 10 7501 (1992) [AI89] O.L. Anderson, D.G. Isaak, S. Yamamoto, J. Appl. Phys. 65 1534 (1989)

  38. Relative deviations of different EOS data for Au at 1000 K with respect to the MoDE2 model used by HH2002. Deviations with respect to their effective AP2 form are shown by the thin line.

  39. EOS parameters for diamond from different fits of theoretical E(V)-data with a fixed best value for Vor The approach: 1) Set up a thermodynamic model for the solid with anharmonicity! 2) Use Kr and Kr’ values from ultrasonic or Brillouin measurements for one (cold) reference isotherm! 3) Use ambient pressure data of Vo(T), αo(T),Cpo(T) and Ko(T) to determine thermo-physical parameters (θo, γo, ...) of the model! 4) Refine Kr’ as best fitting Ko(T)! Washington 2007 Wilfried B. Holzapfel Department Physik Universität-Paderborn Germany <holzapfel@physik.upb.de>

  40. EOS of simple Solids for wide Ranges in p and T The approach: 1) Set up a thermodynamic model for the solid with anharmonicity! 2) Use Kr and Kr’ values from ultrasonic or Brillouin measurements for one (cold) reference isotherm! 3) Use ambient pressure data of Vo(T), αo(T),Cpo(T) and Ko(T) to determine thermo-physical parameters (θo, γo, ...) of the model! 4) Refine Kr’ as best fitting Ko(T)! Washington 2007 Wilfried B. Holzapfel Department Physik Universität-Paderborn Germany <holzapfel@physik.upb.de>

  41. Fit of Ko(T) for Diamond The approach: 1) Set up a thermodynamic model for the solid with anharmonicity! 2) Use Kr and Kr’ values from ultrasonic or Brillouin measurements for one (cold) reference isotherm! 3) Use ambient pressure data of Vo(T), αo(T),Cpo(T) and Ko(T) to determine thermo-physical parameters (θo, γo, ...) of the model! 4) Refine Kr’ as best fitting Ko(T)! Washington 2007 Wilfried B. Holzapfel Department Physik Universität-Paderborn Germany <holzapfel@physik.upb.de>

  42. Fit of Ko(T) for Diamond The approach: 1) Set up a thermodynamic model for the solid with anharmonicity! 2) Use Kr and Kr’ values from ultrasonic or Brillouin measurements for one (cold) reference isotherm! 3) Use ambient pressure data of Vo(T), αo(T),Cpo(T) and Ko(T) to determine thermo-physical parameters (θo, γo, ...) of the model! 4) Refine Kr’ as best fitting Ko(T)! Washington 2007 Wilfried B. Holzapfel Department Physik Universität-Paderborn Germany <holzapfel@physik.upb.de>

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