This work behaves like an extension of the Safaryan theory. A new analytical description has been proposed based on the vibrations of crystal planes. LiMO3 is formed by a succession of the plans in the following sequence: Li, M, O3, etc. In the N doped LiMO3 crystal, we supposed that the LiNMO3 system is formed by a succession of the plans of the sequence: Li, N, M, O, ect. These four crystalline plans vibrate in the direction of trigonal c-axis where each plan consists the ions Li+, Nx+, M5+ or O2-. The dynamic solution of the vibrations of these planes shows that one of two optical branches describes the soft mode at the ferroelectric transition in LiNMO3. From dispersion equation, we have show the bond between the soft mode frequency, the Curie temperature, the charges, the masses of ions and the distance between these ions. The substitution mechanism of the doped compositions N in LiMO3 crystal is discussed. So, the mechanism of phase transition due to thermal expansion of crystal is described.This work behaves like an extension of the Safaryan theory. A new analytical description has been proposed based on the vibrations of crystal planes. LiMO3 is formed by a succession of the plans in the following sequence: Li, M, O3, etc. In the N doped LiMO3 crystal, we supposed that the LiNMO3 system is formed by a succession of the plans of the sequence: Li, N, M, O, ect. These four crystalline plans vibrate in the direction of trigonal c-axis where each plan consists the ions Li+, Nx+, M5+ or O2-. The dynamic solution of the vibrations of these planes shows that one of two optical branches describes the soft mode at the ferroelectric transition in LiNMO3. From dispersion equation, we have show the bond between the soft mode frequency, the Curie temperature, the charges, the masses of ions and the distance between these ions. The substitution mechanism of the doped compositions N in LiMO3 crystal is discussed. So, the mechanism of phase transition due to thermal expansion of crystal is described
