  A series of high-quality La2-xSrxCuO4 (LSCO) superconductor crystals in the
under-doping region with x = 0.063, 0.07, 0.09, 0.10, 0.111 and 0.125 has been
successfully prepared by traveling-solvent floating-zone (TSFZ) technique. The
crystals are large and free of sub-grains and foreign phases. The high crystal
quality has been revealed by double-crystal x-ray rocking curves and Rutherford
backscattering spectrometry combined with ion-beam channeling effect. We find
that the evolutions of the superconducting transition width and volume fraction
as a function of carrier concentration exhibit interesting anomalies in the
vicinity of some magic number doping levels such as x =1/4^2 (=0.0625) and
1/3^2 (=0.111). We argue that these behaviours are of intrinsic electronic
origin.
