GeoItalia 2011, Torino , 19-23 settembre 2011 Mauro Prencipe Dip. Scienze Mineralogiche e Petrologiche Univ. di Torino Compressibility of Pyroxenes 

Compressibility of Minerals at high pressures and temperature: the ab initio way Motivations The development of sophisticated and accurate quantum mechanical algorithms, together with the ever increasing availability of computational resources at a relatively low cost, allows for the calculation of reliable thermo-elastic parameters of minerals at conditions of simultaneous high pressure and temperature, which are in general difficult to be obtained (and measured) experimentally and, on the other hand, are important for the refinement of models of the inner Earth. Outline of the presentation Very brief and schematic review of the theory; The choice of the best Hamiltonian; The application to the estimation of the thermo-elastic parameters of some pyroxenes Compressibility of Pyroxenes 

) , ( ) ( ) , ( R x R E R x H    Hamiltonian Operator Wave Function, depending upon the electronic spatial and spin coordinates (x) and nuclear (R) Energy of the system (parametrically depending upon R) ) , ( s r x  r: space coordinates; s: spin coordinates of all the electrons A Brief Look at the Theory The key for the solution of the problem is in the time-independent Schroedinger equation, within the limit of the Born-Oppenheimer approximation (separability of electronic and nuclear motions) Compressibility of Pyroxenes 

Compressibility of Pyroxenes A Brief Look at the Theory The knowledge of the E(R) function allows for: the structure at the static equilibrium; the static pressure at any given cell volume; the effective potential determining the motion of the nuclei, and hence the frequencies of the vibrational normal modes. Static and vibrational energies, together with their dependence upon the unit cell volume, are all what we need to get theoretical estimations of thermoelastic and thermodynamic properties of crystals, within the limit of the quasi-harmonic approximation. 

Contributions to the Total Pressure By adding the vibrational contributions to the static Helmholtz free energy of a crystal: T V T V F T V P           ) , ( ) , ( Static and vibrational contributions to P can then be identified ) , ( ) ( ) ( ) , ( ) ( ) , ( T V P V P V P T V P V P T V P th zp st vib st      Vibrational pressure Athermal pressure Static pressure Zero Point pressure Thermal pressure ) , ( ) , ( ) ( ) , ( ) ( ) , ( T V TS T V U V U T V F V F T V F vib vib st vib st      where U and S are, respectively, the internal energy and the entropy, and by taking into account the general relation between P and F, one gets: Compressibility of Pyroxenes 

               j j j j j j j T st T n V h V h V E T V P      ) , ( 2 ) , ( 0 Total Pressure Static Pressure (Pst ) Zero Point Pressure (Ppz ) Thermal Pressure (Pth )                   j kT V vib vib vib T vib vib vib vib vib j Z kT T V F V F T V P T V TS T V U T V F / ) ( e , lnZ ) , ( ) , ( ) , ( ) , ( ) , (  Given the classical relation between F and P, and the statistical definition of F in terms of the partition function Z: Derivation of Fvib (V,T), with respect to V Number of phonons of the jth oscillator, at the temperature T Thermal and Zero Point Pressures: the Quasi-Harmonic approximation Compressibility of Pyroxenes 

Static Equation of State: the effect of the Hamiltonian Hamiltonian K0 K’ V0 * WC1LYP 110.3 5.6 223.4 B3LYP 102.8 5.7 228.0 PBE0 110.5 5.5 222.7 PBE 100.9 5.6 229.2 WCPBE 110.5 5.3 221.6 LDA 122.7 5.2 213.1 Diopside: Static Equation of State (third order Birch-Murnaghan; BM3) At this stage, theoretical results are not directly comparable to experimental ones, due to the lack of zero point and thermal pressures. * V0 is the volume of the primitive cell  B3LYP and PBE provide for the lowest values of the bulk modulus and a large unit cell volume;  LDA provides for the largest bulk modulus and the smallest volume;  WC1LYP, PBE0 and WCPBE provide for similar intermediate results. Compressibility of Pyroxenes 

Vibrational frequencies: the best Hamiltonian One of the ingredient for the correct estimation of zero point and thermal pressures is the vibrational spectrum. In principle, the whole phonon density of state should be know. In practice, for minerals of moderate complexity, only the spectrum at the G point of the Brillouin zone is often evaluated. Hamiltonian <|D|> (cm-1) WC1LYP 3.2 B3LYP 4.7 PBE0 6.5 PBE 18.0 WCPBE 9.7 LDA 7.3 Diopside: frequencies at the G point, at zero pressure Average absolute discrepancies (<|D|>) between the experimentally measured frequencies of the Raman active modes, and those calculated by means of different Hamiltonians. As observed for many other silicates and carbonates, the best performance is obtained by means of the hybrid HF/DFT WC1LYP Hamiltonian. PBE shows the worst performance! Compressibility of Pyroxenes 

Effect of the Acoustic Modes Compressibility of Pyroxenes The evaluation of the vibrational contributions to the total pressure by using the G point frequencies only or, in other words, the neglection of phonon dispersion, is a generally acceptable approximation, at least for crystals having relatively large unit cells. In the G point approximation, the low frequencies acoustic modes, which reduce to pure translations of the crystal at G, are completely neglected and the way to include their contribution to the vibrational pressure should be devised. This can be done through a supercell approach. Phonon dispersion: an example The Y1- Longitudinal Optical mode at 152 cm-1 in diopside The same mode occurs at 170 cm-1 in jadeite, and at 159 cm-1 in (ordered P2/n) omphacite 

Y Au Ag Y1+ Y1- LA LO Г Phonon Dispersion in Diopside Avoided intersections and modes mixing: an example Having the same symmetry, the two L1 dashed lines, respectively connecting the ungerade (u) and the gerade (g) phonon modes, cannot intersect. The avoided intersection produces the real dispersion curves (red curves), by mixing the g and u modes. +1/4 +1/4 -1/4 -1/4 X Y Au Translation along Y Y1- ν: 152 cm-1 Ag ν: 141 cm-1 Y1+ ν: 130 cm-1 Г Y BZ Compressibility of Pyroxenes 

T (K) 0 200 400 600 800 1000 K 0 (GPa) 90 95 100 105 110 115 Acoustic contribution included No acoustic contribution Diopside (WC1LYP Hamiltonian) 1 GPa 4 GPa Bulk Modulus: Effect of the Acoustic Modes dK/dT=-0.014 GPa/K dK/dT=-0.010 GPa/K T=300K K0 =106.2 GPa K’=5.5 K0 =107.4 GPa K’=5.5 T=1000K K0 = 96.6 GPa K’=5.4 K0 =100.8 GPa K’=5.4 T=0K K0 =109.7 GPa K’=5.4 K0 =109.7 GPa K’=5.4 T No acoustic modes Correct results Compressibility of Pyroxenes Experimental dK/dT = -0.012 GPa/K 

T (K) 0 200 400 600 800 1000 K-1) 0 1e-5 2e-5 3e-5 4e-5 P=0GPa P=5GPa P=10GPa Thermal expansion of diopside As the temperature increases, the pressure of phonon gas inside the crystal also increases; to keep the equilibrium with the external pressure, the static pressure of the crystal must decrease: this is the ultimate reason of thermal expansion. Compressibility of Pyroxenes  (K-1) at T=300K, and P=0 Gpa: Calc. (WC1LYP) 2.9·10-5 Exp. 3.1(3)·10-5 

Compressibility of Pyroxenes K0 (Gpa) K’ dK/dT (Gpa/K)  (K-1)* Diopside WC1LYP 106.2 5.5 -0.014 2.9·10-5 EXP. 106(1) [1] 6.1(5) [1] -0.012 [2] 3.1(3)·10-5 [3] Omphacite (ordered P2/n) WC1LYP 112.4 5.1 -0.017 2.4·10-5 EXP. 116.6(3) [4] 6.0(6) [4] Jadeite WC1LYP 130.7 3.5 -0.011 1.8·10-5 EXP. 134.0(7)[5] 4.4[5] 2.4·10-5 [6] *  is evaluated at T=300 K and P=0 GPa [1] Gavrilenko et. al (2010) Am Mineral, 95, 608; [2] Li & Neuville (2010) PEPI, 183, 398; [3] Finger & Ohashi (1976) Am Mineral, 61, 303; [4] Pavese et al. (2001) Phys Chem Minerals, 28,9; [5] Nestola et al. (2006) Phys Chem Minerals, 33, 417; [6] Cameron et al. (1973) Am Mineral, 58, 594. A Comparison with the Experiments 

Compressibility of Pyroxenes Conclusions and Future Work In the case of diopside, where calculations are extended to include acoustic modes, results reproduce the best experimental measurements available. Estimations of the bulk modulus up to 1000K can, at present, be considered reliable within 2-3 GPa. The bulk moduli of jadeite and ordered omphacite, at T=300K, appear to be underestimated of about 4 GPa with respect to the experimental data. The low value for K’ in jadeite is to be noted. Especially for jadeite, inclusion of the acoustic modes in the calculation must be performed, but it is unlikely that it will have a so large impact in increasing K’. 

Compressibility of Pyroxenes Thank you for your attention!