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The Quasiharmonic Approximation

The Quasiharmonic Approximation. R. Wentzcovitch U. of Minnesota VLab Tutorial. “A simple approximate treatment of thermodynamical behavior”. Born and Huang. It treats vibrations as if they did not interact System is equivalent to a collection of independent harmonic oscillators

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The Quasiharmonic Approximation

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  1. The Quasiharmonic Approximation R. Wentzcovitch U. of Minnesota VLab Tutorial “A simple approximate treatment of thermodynamical behavior” Born and Huang • It treats vibrations as if they did not interact • System is equivalent to a collection of independent harmonic oscillators • These establish the quantum mechanical energy levels of the system • The levels are used to compute the partition function, Z, and the Helmoltz • free energy, F(T,V). From the latter, all thermodynamic functions can • be derived.

  2. Helmholtz free energy: internal energy entropy = partition function sum of Boltzman factors of all energy levels eigenvalues of energy operator For a single oscillator with freq. ωi , the energy levels are:

  3. Therefore: For a lattice of normal modes of vibration with frequencies ωi: is the vibrational free energy

  4. Complete free energy: As a solid compresses, deforms, etc…,U and ωi’s change From F(T,V,ε1,ε2,…) all the thermodynamical properties can be dirived.

  5. Notice • This (or a more complete) quantum treatment is required at “low T” • The QHA is not appropriate at “high T” because of phonon-phonon • interactions • Tlow < θDebye< Thigh < Tmelt • Phonon frequencies must be accurate (first principles) • Phonon sampling must be thorough

  6. Summation (integration) over the Brillouin Zone Ex: square BZ is the “multiplicity” of a point determined by symmetry • In general: • Compute and diagonalize the dynamical matrix at few ’s • Extract “force constants” • Recompute dynamical matrices at several points using those force constants

  7. MgSiO3 Perovskite ----- Most abundant constituent in the Earth’s lower mantle ----- Orthorhombic distorted perovskite structure (Pbnm, Z=4) ----- Its stability is important for understanding deep mantle (D” layer)

  8. Mineral sequence II Lower Mantle + + (Mg(1-x-z),Fex, Alz)(Si(1-y),Aly)O3 (Mgx,Fe(1-x))O CaSiO3

  9. Mineral sequence II Lower Mantle + (Mgx,Fe(1-x))SiO3 (Mgx,Fe(1-x))O

  10. Phonon dispersion of MgSiO3 perovskite Calc Exp Calc Exp - 0 GPa Calc:Karki, Wentzcovitch, de Gironcoli, Baroni PRB 62, 14750, 2000 Exp:Raman [Durben and Wolf 1992] Infrared [Lu et al. 1994] - 100 GPa

  11. MgSiO3-perovskite and MgO 4.8 (256) Exp.: [Ross & Hazen, 1989;Mao et al., 1991; Wang et al., 1994; Funamori et al., 1996; Chopelas, 1996; Gillet et al., 2000; Fiquet et al., 2000]

  12. Umemoto, 2005

  13. Thermal expansivity and the QHA provides an a posteriori criterion for the validity of the QHA (10-5 K-1)    MgSiO3 Karki et al, GRL (2001)

  14. Validity of the QHA

  15. Mineral sequence II Lower Mantle QHA not-valid for this mineral !! + + (Mg(1-x-z),Fex, Alz)(Si(1-y),Aly)O3 (Mgx,Fe(1-x))O CaSiO3

  16. Crystal Structures at High PT Crystal Structures at High PT

  17. Quasiharmonic Approximation (QHA) • VDoS and F(T,V) within the QHA N-th (N=3,4,5…) order isothermal (eulerian or logarithm) finite strain EoS IMPORTANT: crystal structure and phonon frequencies are uniquely related with volume !!….

  18. How to get V(P,T) • Static calculations give Vst(Pst). Vst is the volume of the optimized structure at Pst. • The free energy is obtained after phonon frequencies are calculated at each equilibrium structure, ω(Vst). • The relationship between structure and Vst or ω(Vst) is not altered by temperature after free energy calculations, only the pressure: P(V,T) = Pst + Pth. Pth is the contribution of the 2nd and 3rd terms in the rhs in the free energy formula. • Therefore: • if V(P,T) = Vstthen • Structure(P,T) = Structure(Vst) and • ω(P,T) = ω(Vst) • These are Structure(P,T) and ω(P,T) given by the statically constrained QHA . • See Carrier et al., PRB 76, 064116 (2007)

  19. High P,T Experiments x Static LDA Lattice parameters of MgSiO3 perovskite Carrier et al. PRB, 2008

  20. Stress and Strain Stress and Strain

  21. Uniform deformations (macroscopic strains) ε = dL/L L L+dL More formally 2 2 Xi – xi = uijxj X2 x2 • • εij = ½ (uij + uji) x1 1 1 X1 “Lagrangian” strains Stress σij (P,T)= 1 δF V δεij T,P

  22. Thermoelastic constant tensor CijS(T,P) kl equilibrium structure re-optimize

  23. 300 K 1000K 2000K 3000 K 4000 K cij Cij(P,T) (Oganov et al,2001) MgSiO3-pv (Wentzcovitch, Karki, Cococciono, de Gironcoli, Phys. Rev. Lett. 2004)

  24. Deviatoric Thermal Stresses Carrier et al. PRB, 2008

  25. Deviatoric thermal stresses cause T dependent X-tal structure and phonon frequencies QHA should be self-consistent! Carrier et al. PRB, 2008

  26. Anharmonicity Anharmonicity

  27. 410 km discontinuity QHA Anharmonic 2.5 MPa/K 3.5 MPa/K Mg2SiO4→ Mg2SiO4 Wu and Wentzcovitch, PRB 2009

  28. Summary • QHA is simple useful theory even for lower mantle • QHA + LDA gives excellent structural and thermodynamic properties • It is possible to obtain crystal structures at high pressures and temperatures using the QHA • Beware: thermal pressure is not isotropic • Phase boundaries appear to be affected by anharmonicity

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