Lattice Dynamics

1. Lattice Dynamics U. Milan Short Course

2. Thermal Expansion • As temperature changes, density changes • Thermodynamics • Relates this change to changes in other properties • Cannot tell the magnitude or even the sign! • Why positive alpha? • What value vs. P,T,X? • Macroscopic to Microscopic • Thermodynamics to Statistical Mechanics

3. Interatomic Forces • Ambient Structure • Minimum • Bulk Modulus • Curvature • Thermal Expansivity • Beyond harmonic • Molecules • Solids

4. One Dimensional Lattice

5. One Dimensional Lattice • Periodicity reflects that of the lattice • Brillouin zone center, k=0: =0 • Brillouin zone edge, k=/a: =maximum • All information in first Brillouin zone Frequency,  1st Brillouin Zone 2(K/m)1/2 0 Wavevector, k

6. One Dimensional Lattice k=/a=2/; =2a k0; 

9. Acoustic Velocities • k0 • w=2(K/m)1/2ka/2 • w/k=dw/dk=a(K/m)1/2 • Acoustic Velocity • v=a(K/m)1/2 • Three dimensions • Wavevector, ki • Polarization vector, pi • For each ki, 3 acoustic branches • ki pi longitudinal (P) wave • kipi transverse (S) waves (2)

10. Polyatomic Lattice a • Unit cell doubled • Brillouin Zone Halved • Acoustic Branches folded • New, finite frequency mode at k=0 • Optic Branch New Brillouin Zone

11. Fumagalli et al. (2001) EPSL General Lattice • Number of modes = 3N organized into 3Z branches • Z= number of atoms in unit cell • 3 Acoustic branches • 3Z-3 optic branches • Experimental Probes • Optic zone center • Raman • Infrared • Acoustic near zone center • Brillouin • Full phonon spectrum • Inelastic neutron scattering Quartz

12. MgSiO3 Perovskite Movie

13. Internal Energy Energy • Sum over all vibrational modes • Energy of each mode depends on • Frequency • Population • Frequency • Temperature n=3 n=2 n=1 n=0 Displacement

14. Heat Capacity • or • CV=3R per mol of atoms (Dulong-Petit)

15. Thermal Pressure 1 Energy • Compression Increases • Vibrational frequencies • Vibrational energy •  Thermal pressure Displacement

16. Thermal Pressure 2 Thermal Pressure Bulk Modulus Volume dependence of  Thermal pressure

17. Interatomic Forces • Ambient Structure • Minimum • Bulk Modulus • Curvature • Thermal Expansivity • Beyond harmonic • Molecules • Solids

18. Fundamental Thermodynamic Relation Helmholtz free energy as a function of volume and temperature Complete information of equilibrium states/properties Divide into purely volume dependent “cold” part and a thermal part Recall we already have an expression for the “cold” part

19. Cold part • Start from fundamental relation • Helmholtz free energy • F=F(V,T,Ni) • Isotherm, fixed composition • F=F(V) • Taylor series expansion • Expansion variable must be V or function of V • F = af2 + bf3 + … • f = f(V) Eulerian finite strain • a = 9K0V0

20. Thermal part Not easily evaluated, need to know all vibrational frequencies at all pressures This information not available for ANY mantle mineral! What to do? Frequencies only appear in sums Thermodynamics insensitive to details of distribution of frequencies Assume i=E, all i, where E is a characteristic frequency of the material

21. Comparison to experiment Characteristic (Debye) frequencies Anorthite 522 cm-1 Forsterite 562 cm-1 Corundum 647 cm-1