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The nanometer scale lattice deformation brought about by the dopants in hightemperature superconducting cuprate La$_{2-x}$Sr$_x$CuO$_4$(x=0.08) wasinvestigated by measuring the associated X-ray diffuse scattering aroundmultiple Bragg peaks. A characteristic diffuse scattering pattern was observed,which can be well described by continuum elastic theory. With the fitted dipoleforce parameters, the acoustic type lattice deformation pattern wasre-constructed and found to be of similar size to lattice thermal vibration at7 K. Our results address the long-term concern of dopant introduced locallattice inhomogeneity, and show that the associated nanometer scale latticedeformation is marginal and cannot, alone, be responsible for the patchedvariation in the spectral gaps observed with scanning tunneling microscopy inthe cuprates.

Quantitative characterization of the nanoscale local lattice strain induced by Sr dopants in La$_{1.92}$Sr$_{0.08}$CuO$_4$