Neutron powder diffraction measurements, combined with magnetization and resistivity data, have been carried out in the doped perovskite La$_{1-x}$Ca$_x$MnO$_3$ ($x=0.47$, 0.50, and 0.53) to elucidate the structural, magnetic, and electronic properties of the system around the composition corresponding to an equal number of Mn3+ and Mn4+. At room temperature all three samples are paramagnetic and single phase, with crystallographic symmetry Pnma. The samples then all become ferromagnetic (FM) at $T_Capprox 265$ K. At $sim 230$ K, however, a second distinct crystallographic phase (denoted A-II) begins to form. Initially the intrinsic widths of the peaks are quite large, but they narrow as the temperature decreases and the phase fraction increases, indicating microscopic coexistence. The fraction of the sample that exhibits the A-II phase increases with decreasing temperature and also increases with increasing Ca doping, but the transition never goes to completion to the lowest temperatures measured (5 K) and the two phases therefore coexist in this temperature-composition regime. Phase A-II orders antiferromagnetically (AFM) below a N{e}el temperature $T_N approx 160$ K, with the CE-type magnetic structure. Resistivity measurements show that this phase is a conductor, while the CE phase is insulating. Application of magnetic fields up to 9 T progressively inhibits the formation of the A-II phase, but this suppression is path dependent, being much stronger for example if the sample is field-cooled compared to zero-field cooling and then applying the field. The H-T phase diagram obtained from the diffraction measurements is in good agreement with the results of magnetization and resistivity.
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