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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 coefficient 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.
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Keywords:
- BaTiO3 nanoparticles /
- Bessel function /
- ferroelectrics /
- critical dimension
[1] Hou Z W, Kang A G, Ma W Q, Zhao X L 2014 Chin. Phys. B 23 117701
[2] Sharma V, Pilania G, Rossetti G A, Jr Slenes K, Ramprasad R 2013 Phys. Rev. B 87 134109
[3] Zhou Z, Lin Y R, Tang H X, Henry A Sodano 2013 Nanotechnology 24 095602
[4] Sakuma T, Yoshida H 2009 Mater. Trans. 50 229
[5] Cui L, Xu Q, Han Z Y, Xu X 2012 Chin. Phys. Lett. 29 037701
[6] Astefanoaei, Dumitr, Stancu 2013 Chin. Phys. B 22 128102
[7] Wang Y L, Zhang P C, Liu H R, Liu B T, Fu G S 2011 Acta Phys. Sin. 60 077702 (in Chinese) [王英龙, 张鹏程, 刘虹让, 刘保亭, 傅广生 2011 60 077702]
[8] Hou Z W, Kang A G, Ma W Q, Zhao X L 2014 Journal of Synthetic Crystals 43 2219 (in Chinese) [侯志文, 康爱国, 马维青, 赵晓龙 2014 人工晶体学报 43 2219]
[9] Mark McNeal P, Sei-Joo Jang, Robert Newnham E 1998 J. Appl. Phys. 83 3288
[10] Wu H, Zhan Y G, Xing H Z, Shen W Z 2009 Solid State Commun. 149 802
[11] Gao K, Wei M M, Qu Z P, Fu Q, Bao X H 2013 Chinese Journal of Catalysis 34 889
[12] CAI M Q, DU Y, HUANG B Y 2009 Trans. Nonferrous Met. Soc. China 19 1634
[13] Xing Y h, Liang H, Li X L, Si L Q 2009 Particuology 7 414
[14] Cohen R E, Krakauer H 1992 Ferroelectrics 136 65
[15] Cui L, Lu T Q, Sun P N, Xue H J 2010 Chin. Phys. B 19 077701
[16] Cohen R E 1992 Nature 136 95
[17] Astefanoaei I, Dumitru I, Stancu Al 2013 Chin. Phys. B 22 128102
[18] Zhong W L, Wang Y G, Zhang P L, Qu B D 1994 Phys. Rev. B 50 698
[19] Xue W D, Chen S Y, Yang C, Li Y R 2005 Acta Phys. Sin. 54 857 (in Chinese) [薛卫东, 陈召勇, 杨春, 李言荣 2005 54 857]
[20] Zhang H T, Kang A G, Yang B G, Xue H, Guo X L 2013 Journal of Synthetic Crystals 42 1848 (in Chinese) [张海涛, 康爱国, 杨北革, 薛 辉, 郭小龙 2013 人工晶体学报 42 1848]
[21] Zhang X Y, Li L J, Huang Y C 2014 Acta Phys. Sin. 63 190301 (in Chinese) [章新友, Li L J, 黄永畅 2014 63 190301]
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[1] Hou Z W, Kang A G, Ma W Q, Zhao X L 2014 Chin. Phys. B 23 117701
[2] Sharma V, Pilania G, Rossetti G A, Jr Slenes K, Ramprasad R 2013 Phys. Rev. B 87 134109
[3] Zhou Z, Lin Y R, Tang H X, Henry A Sodano 2013 Nanotechnology 24 095602
[4] Sakuma T, Yoshida H 2009 Mater. Trans. 50 229
[5] Cui L, Xu Q, Han Z Y, Xu X 2012 Chin. Phys. Lett. 29 037701
[6] Astefanoaei, Dumitr, Stancu 2013 Chin. Phys. B 22 128102
[7] Wang Y L, Zhang P C, Liu H R, Liu B T, Fu G S 2011 Acta Phys. Sin. 60 077702 (in Chinese) [王英龙, 张鹏程, 刘虹让, 刘保亭, 傅广生 2011 60 077702]
[8] Hou Z W, Kang A G, Ma W Q, Zhao X L 2014 Journal of Synthetic Crystals 43 2219 (in Chinese) [侯志文, 康爱国, 马维青, 赵晓龙 2014 人工晶体学报 43 2219]
[9] Mark McNeal P, Sei-Joo Jang, Robert Newnham E 1998 J. Appl. Phys. 83 3288
[10] Wu H, Zhan Y G, Xing H Z, Shen W Z 2009 Solid State Commun. 149 802
[11] Gao K, Wei M M, Qu Z P, Fu Q, Bao X H 2013 Chinese Journal of Catalysis 34 889
[12] CAI M Q, DU Y, HUANG B Y 2009 Trans. Nonferrous Met. Soc. China 19 1634
[13] Xing Y h, Liang H, Li X L, Si L Q 2009 Particuology 7 414
[14] Cohen R E, Krakauer H 1992 Ferroelectrics 136 65
[15] Cui L, Lu T Q, Sun P N, Xue H J 2010 Chin. Phys. B 19 077701
[16] Cohen R E 1992 Nature 136 95
[17] Astefanoaei I, Dumitru I, Stancu Al 2013 Chin. Phys. B 22 128102
[18] Zhong W L, Wang Y G, Zhang P L, Qu B D 1994 Phys. Rev. B 50 698
[19] Xue W D, Chen S Y, Yang C, Li Y R 2005 Acta Phys. Sin. 54 857 (in Chinese) [薛卫东, 陈召勇, 杨春, 李言荣 2005 54 857]
[20] Zhang H T, Kang A G, Yang B G, Xue H, Guo X L 2013 Journal of Synthetic Crystals 42 1848 (in Chinese) [张海涛, 康爱国, 杨北革, 薛 辉, 郭小龙 2013 人工晶体学报 42 1848]
[21] Zhang X Y, Li L J, Huang Y C 2014 Acta Phys. Sin. 63 190301 (in Chinese) [章新友, Li L J, 黄永畅 2014 63 190301]
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