The thermal conductivity \ensuremath{\kappa} and the sound velocity in ${\mathrm{La}}_{2}$${\mathrm{CuO}}_{4}$ are found to be significantly decreased by Nd, Zn, and Ni doping, reflecting the lattice instability of this compound. The rapid decrease of \ensuremath{\kappa} with x in ${\mathrm{La}}_{2\mathrm{\ensuremath{-}}\mathit{x}}$${\mathit{M}}_{\mathit{x}}$${\mathrm{CuO}}_{4}$ (M=Sr,Ba) in a small-x region that was previously reported by us is understood to be the result of an enhancement of the lattice instability induced by M doping, just as in the Nd-, Zn-, and Ni-doped insulating systems. The y dependence of the transition temperature from the LTO1 (low-temperature orthorhombic 1) to LTT (low-temperature tetragonal) phase in ${\mathrm{La}}_{1.875\mathrm{\ensuremath{-}}\mathit{y}}$${\mathit{R}}_{\mathit{y}}$${\mathrm{Sr}}_{0.125}$${\mathrm{CuO}}_{4}$ (R=Nd,Sm,Eu,Gd) indicates that the larger the difference of the ionic radii between ${\mathrm{La}}^{3+}$ and ${\mathit{R}}^{3+}$ ions, the more easily the transition to the LTT phase is induced. We propose that the LTT region exists around the ${\mathit{R}}^{3+}$ ion even in the LTO1 phase and extends with the decrease of temperature and the structural transition to the LTT phase is induced when the LTT region becomes connected in a whole area of the crystal. We also point out the importance of the local lattice distortion for the Cu-site-substitution effects on ${\mathrm{La}}_{2\mathrm{\ensuremath{-}}\mathit{x}}$${\mathit{M}}_{\mathit{x}}$${\mathrm{CuO}}_{4}$.
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