1. EOS of simple Solids for wide Ranges in p and T Washington 2007 Wilfried B. Holzapfel Department Physik Universität-Paderborn Germany 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 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!

2. EOS of simple Solids for wide Ranges in p and T Washington 2007 Wilfried B. Holzapfel Department Physik Universität-Paderborn Germany 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.

3. EOS of simple Solids for wide Ranges in p and T Washington 2007 Wilfried B. Holzapfel Department Physik Universität-Paderborn Germany 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)!

4. EOS of simple Solids for wide Ranges in p and T Wilfried B. Holzapfel Department Physik Universität-Paderborn Germany The thermo-physical model for the Gibbs function: Total Free-Energy: F(V,T) = E c (V) + F th (V,T) with Ground State Energy E c (V) and thermal Excitations in F th (V,T) E c (V) from one cold isotherm: p c (V) = pAP2(V,Z,V o,K o,K’ o ) F th (V,T) = F cond.el.(T,T FG (V)) + F quasi-harm.phonon (V,T) + F intr.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

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

6. An optimized pseudo-Debye-Einstein model Wilfried B. Holzapfel Department Physik Universität-Paderborn Germany 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

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

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

9. Intrinsic Anharmonicity Two Series Expansions in the C lassical F ree V olume Approach Wilfried B. Holzapfel Department Physik Universität-Paderborn Germany Washington 2007

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

11. Vibrational Grüneisen Parameter in the classical free volume approach Wilfried B. Holzapfel Department Physik Universität-Paderborn Germany Washington 2007

12. Higher Order Corrections in the Free Volume Approach Wilfried B. Holzapfel Department Physik Universität-Paderborn Germany Washington 2007

13. Heat capacity of Cu at ambient pressure Prag 2006 Experimental data +, o, x and from Ly59, Ma60, Gr72 and Ch98 Present fit of C po and C vo : solid and dashed line Wilfried B. Holzapfel Department Physik Universität-Paderborn Germany C po (T) 3R T (K)

14. Heat capacity of Ag at ambient pressure Experimental data o, x and from Mo36, MF41, and Gr72 Present fit of C po and C vo : solid and dashed line Wilfried B. Holzapfel Department Physik Universität-Paderborn Germany C po (T) 3R T (K)

15. Heat capacity of Au at ambient pressure Experimental data x and from GG52, and Gr72 Present fit of C po and C vo : solid and dashed line Wilfried B. Holzapfel Department Physik Universität-Paderborn Germany T (K) C po (T) 3R

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

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

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

19. K o (T) (GPa) T (K) 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

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 K o (T) (GPa) T (K)

21. K o (T) (GPa) 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

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 T (K) K´ o (T)

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 T (K) K´ o (T)

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 T (K) K´ o (T)

25. Hierarchy of Parameters in the Refinements 1. Fit of C po (T) at low T by Refinement of TD o and TD h 2. Fit of V o (T) at low T by Refinement of  r 3. Fit of C po (T) and V o (T) at higher T by Refinement of Anharmonicity Parameters f4 and f6 4. Minor Refinement of K’ r to improve Fit of K o (T) 5. Final Check of Step 2. and 3. Wilfried B. Holzapfel Department Physik Universität-Paderborn Germany

26. V r (cnm) K r (GPa) TD o (K) K’ r TD h (K) (r(r f6 1000xA ph 8 f4 (A(A TFeff(K) Z M f2 q Ag Cu Au Thermo-physical Parameters for

27. Discussion 1. Perfect representation of all thermo-physical data at ambient P (for “regular” Solids only!) 2. Uncertainties in  (V) and A ph (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

28. Comment on Parametric EOS If the cold isotherm p c (V) is represented by p AP2 (V,V o,K o,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 V r, K r, K’ r deviating from the thermodynamic values! Accurate representations of the present thermodynamic EOS by parametric EOS need higher order forms with “effective” parameters! Wilfried B. Holzapfel Department Physik Universität-Paderborn Germany

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

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 Prag 2006 Software and Support available on Request! Cooperation welcome! Wilfried B. Holzapfel Department Physik Universität-Paderborn Germany

35. [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) 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

36. Comparison of EOS data for Ag [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) 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

37. Comparison of EOS data for Au [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) 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

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 Washington 2007 Wilfried B. Holzapfel Department Physik Universität-Paderborn Germany 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)!

40. EOS of simple Solids for wide Ranges in p and T Washington 2007 Wilfried B. Holzapfel Department Physik Universität-Paderborn Germany 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)!

41. Fit of Ko(T) for Diamond Washington 2007 Wilfried B. Holzapfel Department Physik Universität-Paderborn Germany 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)!

42. Fit of Ko(T) for Diamond Washington 2007 Wilfried B. Holzapfel Department Physik Universität-Paderborn Germany 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)!