High resolution synchrotron X-ray diffraction, dielectric and Raman scattering study of a scheelite compound Li0.5Ce0.5MoO4 (LCM) revealed that it transforms to a self similar structure above 400 °C. The thermally induced isostructural phase transition (IPT), a phenomenon which has rarely been reported in the literature, is preceded by partial softening of the zone centre phonons followed by their hardening above the IPT transition temperature. The high temperature isostructural phase, which exhibits expanded lattice parameters and cell volume, nucleates and grows in the low temperature matrix over a very wide temperature range. Both the phases show nearly identical thermal expansion suggesting similarities in symmetry, unaltered coordination environments around the atoms across the transition.