Theoretical Investigation On First-order Phase Transition Of Ising Model

The transverse Ising model with crystal field, transverse Ising metamagnet and mixed spin Ising model with different crystal fields are investigated in this dissertation. The global phase diagrams including second-order ordered-disordered phase transition, first-order ordered-disordered phase transtion and first-order ordered-ordered phase transition of the models mentioned above are obtained. The dissertation studies emphatically on the effect of the crystal field, transverse field and the magnetic field on the first-order phase transtion. An effective method to calculate the Gibbs free energy within effective-field theory with correlations (EFT) is presented. By using EFT and mean-field approximation (MFA), the Ising models above are studied of camparing. The main results are as follows:1. The phase transiton of the single-spin Ising model in the transverse and crystal fieldRecently, the high spin Ising models with crystal-fields, i.e. Blume-Capel model, have attracted attention of many scientists.The studies of the transverse Ising model are also very popular. So the phase transitions of three high spin value Ising models (spin-1, spin3/2, spin-2) with transverse and crystal field are studied using EFT and MFA. The basic equations of the phase transition, the longitudinal magnetization, the transverse magnetization, the internal energy, the specific heat and the free energy are derived. The effect of the crystal and transvese field on the groud-state and the finite-temperature phase transition of the single-spin Ising model are studied systematically. Some meaningful conclusions stemming from the the competition between the crystal field and the transverse field are obtained. When the transverse field is small, the first-order phase transition hasppens at low temperature only when the anisotropy constant of every ion is bigger than its threshold value. As the transverse field increases, that is the increasing of the quantum effect, the upper and lower limits of the cystal field’s absolute values that can make the system under the first-order phase trasnsition decreases closing to the second power with the increasing of the transverse field. Because the lower limit decreases slowly, as the transverse field bigger than the threshold value, the crystal interval of the first-order phase transtion becomes zero, i. e. there is no first-order phase transition in the system. Only in the Ising system in which the spin is bigger than one first-order ordered-ordered phase transtion happen on the ground-state.2. The phase transition in the transverse Ising metamagnet with an external magnetic fieldThe studies of the metamagnet are very popular recently. We have studied the phase transitions in the transverse Ising metamagnet with an external magnetic field by using the EFT. The basic equations of the phase transition, the longitudinal magnetization, the transverse magnetization, the internal energy, the specific heat and the free energy are derived. The effect of the magnetic and transvese field on the groud-state and the finite-temperature phase transition of the metamagnet are studied systematically. The results show that the first-order phase transition fades away as the transverse field increases. In addition, an interesting phenomenon is found. When the thermodynamic properties are studied, the specific heat is found to have two peaks, i.e. at the position of lower Curie temperature there is a second specific heat peak. Further studying indicates that at the temperature position of the second peak, the magnetization with respect to temperature changed sharply (abnormal second-order phase transtion), i.e. there are two peaks of the magnetization with respect to temperature, one is at the position of the normal first-order phase transition, the other is at abnormal second-order phase transition position. The abnormal second-order phase transtion lines are plot in the phase diagram. The effect of the transverse field on the abnormal second-order phase transtion is similar to the first-order phase transition. The abnormal second-order vanished when the first-order vanished.3. The phase transition in the mixed spin Ising model with different crystal fieldsBy using MFA, two mixed spin Ising models (mixed spin-3/2 and spin-2, mixed spin-3/2 and spin-5/2) have been studied. The basic equations of the phase transition, magnetization, the specific heat and the free energy are derived. The effect of the two different crystal fields on the groud-state and the finite-temperature phase transition of the mixed spin Ising model are studied systematically. The phase transition temperature decreased with the crystal fields increasing. For the mixed spin-3/2 and spin-2 Ising model, when the crystal field DB is big, the system is in the disordered state. In the finite-temperature condition, when DA is small and DB is bigger than the threshold value (DB/z’J> 0.466), the system happens first-order ordered-disordered phase transtion. In the low temperature area, there are first-order ordered-ordered phase transtions, and the phase diagrams have four different topological structures. For the mixed spin-3/2 and spin-5/2 Ising model, the disordered state does not occur in the ground-state, only six ordered states occur. In the finite-temperature condition, the system only happens second-order ordered-disordered phase transition. When the crystal field is big, the phase transition temperature tend to a nonzero constant value, kBT/z’J= 0.25.In this dissertation, an effective method to calculate the Gibbs free energy within EFT is presented. The first-order phase transition of Ising model has been studied systematically. According to the low temperature results in good agreement with the ground-state, it has been proven that the EFT free energy equation, proposed first in this dissertation, is correct. The problem of studying the first-order phase transition of Ising model by EFT is sloved.