用二次型的正定性判断晶体相稳定性

The Judgement for Crystal Stability of Phase in Positive Definite Quadratic Form

二次型理论有着十分广泛的应用.由二次型的定义出发,给出对于二次型是正定的和与其一一对应的正定矩阵的所有顺序主子式大于零是等价的,并给出理论证明.把稳定相的判据归结为二次型的正定问题,通过求吉布斯自由能函数的一阶偏导等于零可求得自由能函数的极小值及序参量的二阶偏导的Jacobi顺序主子式都大于零得到相稳定条件.又借助Poincare截面,通过解稳定不动点所满足的方程和久期方程,从而得到稳定相温区.

The theory of quadratic form is of very extensive application. In this paper, we begin with the definition of the quadratic form, and then according to the quadratic form, we propose the lemma which is listed as follows： The quadratic form which is positive is equivalent to its corresponding positive matrix whose ordered-main subdeterminants are all larger than zero. The judgement of stability phase can be generalized by the problem of positive definite quadratic form. The minimal value of the Gibbs free energy can be obtained by the first order derivative of free energy equal to zero and the stability condition of phase can be reached when the Jacobi ordered-main subdeterminant of the second order de- tivative of the order parameter is definitely positive. By dint of Poincare cross-section, the temperature zones of the stability phase can be obtained by solving the equation which the stable and immovable points are met and the secular equation.