The Effect of Exchange Interaction on Phase Transition Behavior of Magnetic Nanotubes

In this paper, the phase diagram of the Blume-Capel model under exchange interaction between lattice points in different positions in magnetic nanotube lattice is studied with effective field theory. The research shows that the interaction between nearest neighbors and the lattice field strength strongly influences the system’s three critical points and phase transition temperature. Under certain conditions, there are three critical points in the system. That is, the relationship between the crucial points is linear from the second-order phase transition to the first-order phase transition. First-order phase transition replaces second-order phase transition in the system’s phase transition. The findings demonstrate that exchange interaction significantly affects the system’s phase transition, and they offer a theoretical framework for both industrial production and experimental study.


Introduction
Since Blume-Capel (BC) model was created [1,2] , the magnetothermal properties and phase transition properties of BC models on various lattices have been studied [3][4][5] .Canko's scientific research team explored the phase transition characteristics of single-spin [6] (S=1) system, mixed-spin system [7][8] (S=1/2 and S=1, S=1/2 and S=3/2), and found that the lattice field significantly affected the magnetothermal properties and phase transition of the system, especially the negative crystal field.At the same time, the group also studied the hysteresis behavior of the BC model on nanotubes and gave the hysteresis loop of the system by studying the magnetization of the outer shell and inner shell of nanotubes [9] .When discussing the lattice magnetic susceptibility of nanotubes, it is found that the magnetic susceptibility is significantly affected by the exchange interaction between atoms in the inner and outer layers that are closest to one another [10] .Li et al. researched the magnetization and phase transition behavior of nanotube lattice under the action of diluted crystal field, staggered crystal field, and isotropic crystal field, and obtained the magnetic properties and phase transition behavior different from the system in stable crystal field [11] .Li et al. studied the magnetothermal properties of the nanotube lattice under the action of diluting crystal field and found that the system showed relatively rich magnetization and thermodynamic properties under the action of diluting crystal field [12] .At the same time, effective field theory was used to examine the phase transition behavior of mixed spin systems on nanotubes, and the reentrant phenomena were discovered [13] .Under certain conditions, the system's phase transition can be modified, and there are three critical points in the facies diagram [14] .According to Li et al.'s research, the relationship between a system's phase transition behavior and its value probability, crystal field strength ratio, crystal field parameters, temperature, and the shell is closest to neighbor exchange interaction [15] .It is recognized that the impact of recent neighbor exchange interactions on the spin-1 nanotube system's triple critical point and phase transition has not been examined.Effective field theory is adopted to investigate the phase transition behavior of the BC model in nanotube systems with different nearest-neighbor exchange interaction intensities, and the phase transitions of nanotube systems under various conditions are obtained.This allows for the determination of how the behavior of the phase transition in the nanotube system changes when the nearest neighbor exchange interaction is different.

Model and Method
Figure 1 is a schematic diagram of nanotubes: a three-dimensional schematic diagram and a transverse cross-sectional schematic diagram [11] .The blue circle, green square, and red triangle are used to distinguish the magnetic atoms on different lattice points, indicating the magnetic atoms with coordination numbers of 5, 6, and 7 respectively.The spin of each magnetic atom is 1, and the nearest neighbor exchange interaction is shown in the picture by the connecting lines, namely  ,  , and  [12- 13] .
Figure 1 The nanotube schematic illustration The Hamiltonian expression of the Blume-Capel model of the nanotube system is where  is taken as ;  , J, and  represent the exchange interaction nearest neighbor atom of the outer shell, the nearest neighbor atom of the inner shell, nearest neighbor atom spin of the outer shell and inner shell, respectively;  is a symbol for the crystal field's force on the lattice point [11][12][13] .
According to Kaneyoshi et al.'s research [16][17][18] , the self-consistent equations of and of lattice magnetization of the outer shell and  of lattice magnetization of the inner shell can be obtained as follows.
in the formula is: , where  .

Research Results
By varying the field strength parameter, it is possible to reach the temperature equivalent to zero magnetization under specific circumstances.The lattice field strength is increased to establish the firstorder phase transition temperature of the system.Using Origin Graph drawing software, the system's phase diagram is created under the nearest neighbor exchange interaction and lattice field, as shown in Figure 2-Figure 4.
The phase diagram of this system is shown in Figure 2. When J 2 /J=1.0 and J 1 /J are different, as shown in the picture, the space is divided into two areas by the phase transition temperature curve (phase transition line) for the system with crystal field parameters: paramagnetic region (Figure 2 PM region) and ferromagnetic region (Figure 2 FM region).The first-order phase change line and second-order phase change line of the system are represented by the imaginary curve and the real curve in the phase diagram respectively, and the black solid dot is the three critical points of the system.The research shows that when J 1 /J is 2.5, 1.5, 0.5, 0.1, and 0.01, the three critical points of the system are (-4.969,3.08), (-3.422, 2.22), (-2.515, 1.60), (-2.356, 1.45) and (-2.328, 1.47), respectively.Figure 2 illustrates that the nearest neighbor exchange interaction between the lattice points of the outer shell layer has a significant influence on the system's first-and second-order phase transition temperatures, and the stronger the intensity is, the more obvious the three critical points of the system are.When J 1 /J>0.1 and the exchange interaction is different, the phase transition temperature is obviously different.However, when J 1 /J<0.1, the phase transition behavior is basically the same, and the linear relationship of the three critical points is  0.61823836 0.03633279.
Figure 2 Transition of the system's phases, when J 1 /J is different and J 2 /J=1.0.
The phase diagram of the system at J 1 /J=1.0 and J 2 /J is shown in Figure 3.According to this research, the closest neighbor exchange interaction J 2 /J is 2.5, 1.5, 0.5, 0.1, and 0.01 respectively, and the three critical points of the system are (-4.405,2.75), (-3.373, 2.18), (-2.358, 1.51), (-1.9622,1.19)and (-1.8744,1.14)respectively.As can be seen from Figure 3, the three crucial points of the system become more visible as the closest neighbor exchange contact between the lattice points of the outer shell and the inner shell is greater.There are three linear critical points in the system, and the linear regression equation is  0.64216991 0.04055667 .Comparing Figure 3 with Figure 2, it has been discovered that when the strength of the closest neighbor exchange interaction (J 1 /J) between the outer shell's lattice points is roughly equivalent to the strength of the interaction between the outer shell's and the inner shell's lattice points (J 2 /J), the intensity of negative crystal field decreases when the system has three critical points, and the corresponding first-order phase transition temperature decreases; When J 2 /J is small and the exchange interaction is different, the difference of phase transition temperature is also obvious.The phase transition of the system is affected differently by the closest neighbor exchange contact between lattice points in various placements of magnetic nanotubes.
Figure 3 Transition of the system's phases, when the J2/J is different and J1/J=1.0.The system's phase diagram is shown in Figure 4 when J 1 /J=J 2 /J=1.0.The research shows that when J 1 =J 2 =J is 2.5, 1.5, 0.5, and 0.1 respectively, the three critical points of the system are (-7.150,4.80), (-4.293, 2.83), (-1.433, 0.93) and (-0.291, 0.1) respectively.Figure 4 illustrates that the stronger the exchange interaction is, the more visible the three critical points of the system are when the closest neighbor exchange interaction among the outer shell lattice, the nearest neighbor exchange interaction within the outer shell lattice, and the inner shell lattice is the same.The system's three critical points are linear, and the linear regression equation is  0.6753333 0.04302839.Comparing Figure 4 with Figure 3 and Figure 2, it is found that when the exchange interaction is weak, the three systemic key points vanish.In the system, only the second-order phase change takes place, for example, J 1 =J 2 =J=0.01.

Conclusion
In this research, the spin-1 Blume-Capel model's behavior during phase transitions under the nearest neighbor exchange interaction between lattice points in different positions in the magnetic nanotube lattice is investigated by adopting effective field theory.The research shows that the three critical points, the system's first-and second-order phase transition temperatures are closely related to the nearest neighbor exchange interaction and lattice field strength between lattice points in different positions.
Many factors compete with each other, which makes the system show a complex phase transition phenomenon: under certain conditions, the phase diagram of the system features three critical points, which are linear and are located between the second-order and first-order phase transitions.The findings demonstrate that exchange interaction significantly influences the system's phase transition.Studying phase transition temperature (temperature when the ferromagnetic phase changes to the paramagnetic phase) under different exchange interaction intensities provides a certain theoretical basis for industrial production and experimental research.