  We obtain the exact solution to the Dirac equation with the Poschl-Teller
double ring-shaped Coulomb (PTDRSC) potential for any spin-orbit quantum number
K. The relativistic scattering amplitude for spin 1/2 particles in the field of
this potential has been studied. The wave functions are expressed in terms of
the hyper-geometric series of the continuous states on the K/2Pi scale. A
formula for the phase shifts has also been found. In the nonrelativistic
limits, our solution to the Dirac system converges to that of the Schr\"odinger
one. At the high temperature, the partition function is calculated in order to
study the behavior of some thermodynamic properties.
