Elastic, anelasatic and nonlinear acoustic properties of high $T_c$ superconducting materials

The temperature and pressure dependence of longitudinal and shear ultrasonic waves have been measured in a wide range of the cuprate high $T_c$ superconductors: $La_2_xSr_xCuO_4$, $YBa_2Cu_3O_7_x$ and $GdBa_2Cu_3O_7$ in both their orthorhombic and tetragonal forms, $Bi_2Sr_2CaCu_2O_8_+_y$ (the $Bi$2212 phase), $Bi_2Sr_2Ca_2Cu_3O_1_0_+_y$ (the $Bi$2223 phase), the electron doped material $Nd_2_xCe_xCuO_4_y$. For $La__x2Sr_xCuO_4$ pronounced acoustic mode softening occurs below room temperature. In the case of $Nd_1_._8_5Ce_o_._1_5CuO_4_$.y there are no indications of such behaviour. However a change of slope found at about 220K in the temperature dependence of the longitudinal mode velocity is consistent with coupling of this mode to the ionic volume change induced by enhancement of the 4+ contribution of the intermediate valence cerium ion. Ultrasonic attenuation and wave velocities in the orthorhombic (superconducting) and the tetragna1 (non-superconducting) forms of $GdBa_2Cu_3O_7_x$_ are compared. Elastic response in both forms is found to be dominated by conventional anharmonic and anelastic phenomena: no effects that can be specifically attributed to the formation òf the superconducting state have been observed. To shed further light on outstanding problems associated with the magnitude of the bulk modulus and its pressure dependence in high $T_c$ superconductors, a comparative study has been made of the effects of hydrostatic pressure on the elastic properties. The effects of porosity on the elastic properties have been taken into account by use of wave scattering theory in a porous medium. Very large values of $(\delta \beta/ \delta \ P)=o$ and $(\delta \ C_1_1/\delta \ P)_p=o$ are intrinsic properties of these materials. The true bulk modulus at atmospheric pressure is the one $B_0$ derived from ultrasonic measurement rather than that $B^T(P)$ obtained from $x$-ray experiments at very high pressures. The apparent discrepancy between $B_0$ and $B^T(P)$, a common characteristic of these cuprates, can be resolved by taking the contribution of $(\delta \ B/\delta \ P)=o$ to the high pressure bulk modulus into account in $B^T(P)$