Soft phonons and structural phase transitions in La$_{1.875}$Ba$_{0.125}$CuO$_{4}$

Soft phonon behavior associated with a structural phase transition from the low-temperature-orthorhombic (LTO) phase ($Bmab$ symmetry) to the low-temperature-tetragonal (LTT) phase ($P4_{2}/ncm$ symmetry) was investigated in La$_{1.875}$Ba$_{0.125}$CuO$_{4}$ using neutron scattering. As temperature decreases, the TO-mode at $Z$-point softens and approaches to zero energy around $T_{rm d2}=62$ K, where the LTO -- LTT transition occurs. Below $T_{rm d2}$, the phonon hardens quite rapidly and its energy almost saturates below 50 K. At $T_{rm d2}$, the energy dispersion of the soft phonon along in-plane direction significantly changes while the dispersion along out-of-plane direction is almost temperature independent. Coexistence between the LTO phase and the LTT phase, seen in both the soft phonon spectra and the peak profiles of Bragg reflection, is discussed in context of the order of structural phase transitions.