  With the energy surface of the nucleus in U(5) symmetry being analyzed in the
framework of thermodynamics, the vibration and rotation phase diagram in terms
of the angular momentum and deformation parameter is given. Together with
examining the energy spectrum, we propose a theoretical approach to describe
the vibrational to axially rotational phase transition along the yrast line. By
analyzing the available experimental data we show that the vibrational to
rotational shape phase transition along the yrast line takes place in many
nuclei.
