AI 中文总结
该研究将笛卡尔四元数相关矩阵推广为交叉度量张量，确定六大晶系对应的交叉度量张量，阐释了单位晶体学四元数的几何表示及虚数的几何意义。
AI 中文摘要
两个笛卡尔四元数的乘积可表示为基于4×4矩阵的二次型，该矩阵由符号1、i、j、k构成，其中i、j、k是哈密顿引入的虚数。我们将该矩阵推广至非笛卡尔基，并证明其由度量张量与交叉张量构成，将其命名为交叉度量张量，其符号为s、a、b、c，即基本晶体学四元数。我们确定了六大晶系对应的交叉度量张量，还证明任意单位晶体学四元数可由无穷多对定向平面几何表示，这些平面沿四元数的矢量分量相交，且两平面间的夹角为旋转半角，四元数的组合遵循直观的源-目标规则。基本四元数a、b、c分别由沿a、b、c轴相交的一对垂直定向平面几何表示，还引入了互补四元数a'、b'、c'，它们是ma、mb、mc平面对。基本四元数a、b、c遵循虚数平方的哈密顿规则，互补四元数遵循双积与三积的哈密顿规则。对于笛卡尔基，四元数i、j、k作为晶体学四元数的特例，由立方体mx、my、mz的一对垂直平面构成，这些平面分别沿x、y、z轴相交，故基本四元数与互补四元数相等，即i=i'、j=j'、k=k'，这解释了为何仅用三个四元数i、j、k即可同时满足平方、双积与三积的哈密顿规则。
英文摘要
The product of two Cartesian quaternions can be written as a quadratic form based on a 4x4 matrix made of the symbols 1,i,j,k where i,j,k are imaginary numbers introduced by Hamilton. We generalized this matrix to non-Cartesian bases, and showed that the matrix is made of the metric and the cross tensors. We called it crossmetric tensor. Its symbols are s,a,b,c; they are the elementary crystallographic quaternions. We determined the crossmetric tensors for the six crystal families. We also showed that any unit crystallographic quaternion can be geometrically represented by an infinity of pairs of oriented planes intersecting along the vectorial component of the quaternion such that the angle between them is the semiangle of the rotation. The composition of quaternions follows the intuitive source-target rule. The elementary quaternions a,b,c are geometrically represented by pair of perpendicular and oriented planes intersecting along the axis a,b,c, respectively. Other complementary quaternions noted a',b',c' were also introduced. They are the pairs of planes ma, mb, mc. The elementary quaternions a,b,c follow Hamilton rules on the squares of imaginary numbers. The complementary quaternions follow Hamilton rules on the bi and tri-products. For Cartesian basis, the quaternions i,j,k appear as a specific case of crystallographic quaternions. They are formed by the pairs of perpendicular faces of the cube mx, my, mz. Since these planes intersect along the x, y and z axis, respectively, the elementary and complementary quaternions are equal, i.e. i=i',j=j',k=k', which explains why the square, bi and tri-product Hamilton rules are satisfied all together with only three quaternions i,j,k.
Comments18 pages, 4 figures, 1 appendix
