Local Structure of La$_{2-x}$Sr$_x$CuO$_4$ by Pulsed Neutron PDF Analysis

La$_{2-x}$Sr$_x$CuO$_4$ (LSCO) has orthorhombic symmetry at $x$ = 0, and orthorhombicity decreases continuously with $x$ up to $x$ = 0.21, where a structural transition to tetragonal symmetry occurs. But the high-energy oxygen LO phonon softening appears to occur suddenly near $x$ = 0.06 where superconductivity sets in. Through the high-resolution pulsed neutron pair-density function (PDF) analysis we show that the medium-range structure changes significantly near this composition. The PDF was determined using the NPDF of LANSCE, LANL. Because of the high Q resolution of this spectrometer the PDF can be determined up to 30 $nm$, connecting the local structure smoothly to the average structure. We found that the PDF peaks in the short-range up to 1 $nm$ change little with $x$, and those in the long-range over 3 $nm$ follow the average structure. However, in the intermediate range from 1 to 3 $nm$ some PDF peaks change non-linearly around $x$ = 0.06, and appear to correspond to the insulator-superconductor transition. Possible implications of these changes in relation to nano-scale charge inhomogeneity will be discussed.