  Under the framework of thermodynamics, the phase transition of the black hole
is a general issue in general relativity. In this work, the phase transition of
charged black holes is discussed carefully. The metric tensor of thermodynamics
is redefined in the charged black hole, based on the Ruppeiner geometry. With
the well-defined metric tensor of thermodynamics, the scalar curvature of the
charged black hole is obtained. It is indicated that the scalar curvature is
diverged and infinite when the mass M or charge Q are set by some values, and
it is shown that the charged black hole suffers from a phase transition. At the
same time, there is a phase transition from small mass to large mass or from
small to high charged state. It is shown that the phase transition of a charged
black hole is a common and general process and this work is meaningful for the
construction of microscopic states of black holes.
