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1、河北工业大学 硕士学位论文 三方、六方晶系晶体本征铁弹相变研究 姓名：赵华萍 申请学位级别：硕士 专业：理论物理 指导教师：周国香 2010-12 河北工业大学硕士学位论文 i 三方、六方三方、六方晶系晶体晶系晶体本征本征铁弹相变研究铁弹相变研究 摘摘 要要 本征铁弹相变属于结构相变的一种，该相变的唯一序参量是自发应变。本征铁弹相变 的研究对新型晶体材料的特性、开发及应用具有指导意义。本课题主要应用 MATLAB 软 件模拟三方、六方晶系晶体的三维空间慢度曲面，应用软模理论分析三方、六方晶系晶体 的慢度曲面，然后结合居里原理讨论晶体相变前后的对称性变化。本文研究的内容具体阐 述如下： 一、应用
2、 MATLAB 软件编程求解三方、六方晶系晶体的慢度曲面方程 三方、六方晶系晶体的慢度曲面方程即为其 Christoffel 方程的特征方程。弹性波只有 在三方、六方晶系晶体的特殊方向或平面内传播时，慢度曲面方程才能够因式分解。对于 三方晶系32,3 ,3m m晶类，弹性波在 YZ 平面，X、Y、Z 轴上传播时，其慢度曲面方程才可 以因式分解；对于3,3晶类，弹性波在 YZ 平面，X、Y、Z 轴上传播时，经过坐标转换其 慢度曲面方程也可以因式分解。 六方晶系6,6,6,622,6,62 ,6mmmmmmm晶类， 弹性波只有 在 XY 平面内传播时，其慢度曲面方程才可以因式分解。要求解弹性波在晶
3、体内任意平面 或者方向上传播时慢度曲面方程的解，通过 MATLAB 编程可以实现。本文通过 MATLAB 编程求解三方、六方晶系晶体的慢度曲面方程，求解出晶体在任意方向上的慢度。 二、应用 MATLAB 软件绘制出六方晶系晶体的三维慢度曲面并确定出慢度最大方向； 分析六方晶系晶体本征铁弹相变与其弹性劲度系数和自发应变的相互关系。 应用晶体的弹性劲度系数矩阵的雅可比顺序主子式的值均大于零的方法，计算出六方 晶系晶体的稳定性条件。满足稳定性条件的前提下，全面考虑弹性劲度系数之间的关系， 模拟出六方晶系晶体存在的各种慢度曲面嵌套关系，并计算出最大慢度值的方向，即最易 发生本征铁弹相变的方向。经过六方
4、晶系晶体的慢度曲面嵌套关系分析，铁弹相变与弹性 劲度系数 44 c和 1112 cc有关，产生的自发应变为 12456 ,SS S S S。最后模拟出弹性劲度系数 44 c和 1112 cc趋于零的情况下，六方晶系晶体的慢度曲面的嵌套关系。 三、求解三方晶系晶体的稳定性条件，应用逆向思维分析三方晶系晶体的本征铁弹相 变。 应用三方晶系晶体的弹性劲度系数矩阵的雅可比顺序主子式的值均大于零的方法，计 算出三方晶系各晶类满足的稳定性条件。根据三方晶系各晶类的弹性劲度系数矩阵，得知 三方、六方晶系晶体本征铁弹相变研究 ii 应力和应变的关系后，计算出晶体产生自发应变时弹性劲度系数之间的关系。满足晶体稳
5、 定性条件的前提下，根据该关系模拟三方晶系晶体的慢度曲面存在的嵌套关系和三方晶系 晶体模拟晶体的慢度最大值得出三方晶系晶体的铁弹相变发生的情况。 三方晶系32,3 ,3m m 晶类自发应变有 12456 ,SS S S S， 其本征铁弹相变主要与弹性劲度系数 11121444 ,cccc有关。 3,3晶类的自发应变主要有 12456 ,SSSSS，发生本征铁弹相变主要与弹性劲度系数 1112144425 ,ccccc有关。 四、应用居里原理确定晶系晶体铁弹相的对称性。 六方晶系晶体产生可能的自发应变为 12456 ,SS S S S，三方晶系晶体产生可能的自发 应变为 12456 ,SS S
6、S S。按照自发应变的种类，结合晶体点群的极射赤平投影图，分别讨 论出每种可能的自发应变之下的可能的本征铁弹相变，应用居里原理确定出晶体在一定条 件下发生的唯一相变。 关键词：关键词：铁弹相变，三方晶系晶体，六方晶系晶体，慢度曲面，自发应变 河北工业大学硕士学位论文 iii STUDY ON FERROELASTIC PHASE TRANSITION OF TRIGONAL SYSTEM AND HEXAGONAL SYSTEM ABSTRACT Ferroelastic phase transition is one kind of construction transition. Spon
7、taneous strain is the only order parameter of ferroelastic phase transition. The study of ferroelastic phase transition has very important significance to properties and application of crystal. The slowness surfaces of trigonal and hexagonal system crystal have been drawn by MATLAB. The ferroelastic
8、 phase transitions have been analyzed by theory of soft mode. Finally, the symmetry of crystals has been studied by using the principle of Curie. The mainly work is summarized as follows. In the first part, the equations of slowness surface of trigonal and hexagonal system are solved. The characteri
9、stic equations of Christoffel equation of trigonal and hexagonal system are also the equation of slowness surface of them. When the elastic wave transmitted in the particular plane or direction, the equation of slowness surface can be solved. For example, the equation of slowness surface of trigonal
10、 can be solved, when the elastic wave transmitted in the YZ plane and along the direction of X、Y or Z axis. For the hexagonal system, the equations were factored when the elastic wave transmitted in the XY plane. By using MATLAB, the equations of slowness surface can be solved whatever the elastic w
11、ave transmitted in any plane or along any direction. In the second part, the stability conditions of hexagonal system have been obtained by math method. Then the three dimensional slowness surfaces and the maximum slowness were gained by MATLAB under the stability conditions. In the question of simu
12、lating, the relation among the elastic coefficients must be all considered. By analyzing the nesting relationship of the slowness, it is found that the ferroelastic transition of hexagonal system related with the elastic stiffness coefficients 44 cand 1112 cc. The spontaneous strains 12456 ,SS S S S
13、 would happen, when the ferroelastic transition occurred. At last, the slowness surfaces of hexagonal system have been drawn, when 44 0cand 1112 0cc. Thirdly, the stability conditions of trigonal system can be calculated, when the Jacobi order-main sub determinant of elastic coefficient matrix are d
14、efinitely positive. According to the matrix of trigonal system, the relation between stress and strain can be known. So, the relation of spontaneous strains and elastic stiffness coefficients can be observed. Then, the slowness 三方、六方晶系晶体本征铁弹相变研究 iv surfaces of different nesting relation have been dr
15、awn. In the other word, the nesting relation of ferroelastic transition was given. According analyzing, the spontaneous strains contain 12456 ,SS S S S of the 32,3 ,3m m crystal classes. Ferroelastic transition of the32,3 ,3m m classes of crystal is referring to the elastic stiffness coefficients 11
16、121444 ,cccc. The ferroelastic transition of 3,3crystal classes build the relation with the elastic stiffness coefficients 1112144425 ,ccccc, when the spontaneous strain 12456 ,SS S S S happens. Finally, the symmetry of the proper ferroelastic trigonal and hexagonal system is analyzed respectively by using Curie principle. When the proper ferroelastic phase transitions of trigonal system occur, the spontaneous strains 12456 ,SS S S S appear. The p
