A new ring-shaped non-harmonic oscillator potential is proposed. The precise hound solution of Dirac equation with the potential is gained when the scalar potential is equal to the vector potential. The angular equation and radial equation are obtained through the variable separation method. The results indicate that the normalized angle wave function can be expressed with the generalized associated-Legendre polynomial, and the normalized radial wave function can be expressed with confluent hypergeometric function. And then the precise energy spectrum equations are obtained. The ground state and several low excited states of the system are solved. And those results are compared with the non-relativistic effect energy level in Phys. Lett. A 340 (2005) 94. The positive energy states of system are discussed and the conclusions are made properly.

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Taking TiCu alloy as an example, based on the measurement of thermal expansion coefficient, the harmonic vibration cpefficient and anharmonic vibration coefficient of atoms were calculated. With the help of Collins model, the author worked out the thermodynamic function and the solubility curve under the atomic anharmonic vibration. And, the influence of the anharmonic vibration was also discussed.

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