Equation of State Thermal Expansion Bulk Modulus Shear Modulus Elastic Properties

Equation of State (EoS)  = coefficient of thermal expansion (K -1 )  = compressibility K T = 1/  = isothermal bulk modulus K’ T = (  K/  P) T Ks = adiabatic bulk modulus Ks = K t (1+  T)  = Thermal Gruneisen parameter G Shear Modulus

Thermal Expansion

Volume thermal expansion Thermal expansion at zero pressure V = V 0 (1 +  T)  (T) = a 0 + a 1 T + a 2 T 2 Can be measured by XRD or mechanically. Fei Y. (1995) Mineral Physics and Crystallography, AGU Reference Shelf 2,

Four-Circle Diffractometer

Static Compression

Ultrasonics Transducers generate acoustic signal (MHz or GHz) Interferometry allows measurement of velocities to one part in Precision increases with frequency Transducer is hexagonal ZnO and generates P or S waves. But S waves don’t go beyond MHz Recently Hartmut’s group has developed a GHz S-wave generator.

Elastic Properties Stress is a second rank tensor,  ij Strain is a second-rank tensor,  ij The elastic tensor is fourth-rank c ijkl c can be represented as 6x6 matrix for cubic c 11 = c 22 = c 33, c 44 and c 12 are non-zero

Elastic Moduli Cubic (3terms) c 11, c 44, c 12 Tetragonal (6 or 7) c 11, c 33, c 44, c 66, c 12, c 13, c 23, (c 16 ) Hexagonal(5) c 11, c 33, c 44, c 12, c 13 Trigonal(6 or 7) c 11, c 33, c 44, c 12, c 13, c 14, (c 15 ) Orthorhombic: (9) c 11, c 22, c 33, c 44 c 55 c 66, c 12, c 13, c 23 Monoclinic(13) c 11, c 22, c 33, c 44, c 55, c 66, c 12, c 13, c 23, c 15, c 25, c 35, c 46