空间羣算符的矩阵元

本文首先通过对波失星直接乘积的分析指出,如果 ^*K″∈^*K?^*K′,则波矢峯G_k″可以有下面四种情况:A)G_k同G_k′,是G_k″的子峯,B)G_k=G_k′=G_k″,C)G_k″=G_s, D)G_s是G_k″的子峯。根据上述四种情况,分别考虑积分A=μ″^i″(k″)|f_μⁱ(k)|f_μ′^i′(k′)>的简约。并且证明,对所有四种情况,积分A的值均可由短阵U的矩阵元统一地表达出来。矩阵U是使可约表示Г′简约的转换矩阵,对四种情况,可约表示Г′分别是:A)(Г_kⁱ?Г_k′^i′)^(?),B)Гⁱ?Г_k′^i′,C)Г_k^i(s)?Г_k′^i′(s),D)(Г_k^i(s)?Г_k′^i′(s))^(?)。最后,利用准投影算符的方法,对矩阵U进行了计算,导出空间峯Wigner-Eckart定理的表式。

THE MATRIX ELEMENT OF SPACE GROUP OPERATORS

It is shown that in the direct product of two wave vector stars k?k′, if k"∈k?k′, the wave vector group, G_k", may belong to any one of the following cases: A) Both G_k and G_k′ are subgroups of G_k"; B) G_k=G_k′= G_k"; C) G_k" = G_s; D) G_s is the subgroup of G_k". The reduction of an integral A=μ″^i″(k″)|f_μⁱ(k)|f_μ′^i′(k′)>, is then studied for each of the four cases. It is found that for all cases, the value of the integral can be expressed in terms of the matrix element of a transformed matrix U by which the reducible representation has been reduced. The corresponding reducible representations Г′s are respec-tively: A) (Г_kⁱ?Г_k′^i′)^(?);B)Г_kⁱ?Г_k′^i′;C)Г_k^i(s)?Г_k′^i′(s); and D)(Г_k^i(s)?Г_k′^i′(s))^(?). The final form of the integral is then obtained by finding the explicit form of the transformation matrix by using the pseudo-projection operator.