Keywords: 
• 钛酸钡纳米颗粒 /
• 贝塞尔方程 /
• 铁电体 /
• 临界尺寸

Abstract: The expression of the total polarization intensity of BaTiO3 nanoparticles based on Euler-Lagrange equation for ferroelectric particles and the form of the series solution of Bessel function are obtained, the influences of coe?cient on the total polarization intensity are analyzed. According to the expression for the total polarization intensity, the ferroelectricity of BaTiO3 nanoparticles with different size below 100 nm is simulated and analyzed by MATLAB. Based on experimental data, the effect of grain size on the ferroelectricity is discussed by searching for the numerical value of the solution, and the critical grain size of BaTiO3 nanoparticles with ferroelectricity is predicted to be 6 nm subsequently;or, more specifically, based on the Euler-Lagrange equation of ferroelectric particle’s total free energy, and according to the boundary condition, the equation is given in spherical coordinates, and the transformed equation has the form and characteristics of the Bessel equation, Therefore, it can be discussed according to the characteri-stics of the Bessel equation. It is considered that it has the series solution, and according to the form of deduced series solution, at the same time, under reasonable conditions, the change of polarization is simulated. By combining with the boundary conditions, the total expression of polarization of nanoparticles may be obtained. It contains some factors, first of all, it is analyzed on the whole, and the effect the factors have on the total expression of polarization of the nanoparticles is analyzed. The factors that directly affect, or indirectly affect the polarization intensity, and thus influence the trend of polarization intensity change is analyzed and identified. Then, the effect of each factor on the dielectric constant is simulated by adjusting the factors, and the numerical solution obtained is consistent with the experimental data, so the predictive value can be obtained.