Lattice instabilities and the effect of copper-oxygen-sheet distortions on superconductivity in doped<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant="normal">La</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant="normal">CuO</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math>

Synchrotron x-ray and neutron-diffraction measurements demonstrate that ${\mathrm{Nd}}^{3+}$ substitution at the ${\mathrm{La}}^{3+}$ site in both metallic and insulating ${\mathrm{La}}_{2\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{Sr}}_{\mathit{x}}$${\mathrm{CuO}}_{4}$ induces low-temperature structural instabilities which reorient the tilting of the copper-oxygen octahedra. These structural transformations strongly affect superconductivity. In addition, there is a bicritical point involving phases of Bmab, Pccn, and P${4}_{2}$/ncm space-group symmetries in the structural phase diagram of ${\mathrm{La}}_{1.6\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{Nd}}_{0.4}$${\mathrm{Sr}}_{\mathit{x}}$${\mathrm{CuO}}_{4}$ at x\ensuremath{\sim}0.15. There are also anomalies in the structural transformation temperatures and the superconducting transition temperatures when x\ensuremath{\sim}1/8, providing evidence for an electronic instability at this doping level.