Dear all, In Marder's Condensed matter physics, it uses matrix operations to explain how to justify two different lattice systems as listed in attachment. However, I cannot understand why the two groups are equivalent if there exists a single matrix S satisfying S^{-1}RS^{-1}+S^{-1}a=R^{'}+a^{'}. Can someone help me to understand it? Thank you.
I'm sorry you are not generating any responses at the moment. Is there any additional information you can share with us? Any new findings?
Marder is saying that if there is a linear map between the two lattice systems, then they are equivalent. The original system is defined by a Rotation (R) and a translation (a). The matrix S and its inverse are performing a similarity transformation (coordinate system change) on R, and also apply it to the translation. Marder then notes if there exists one such linear transform, then there exists a family of them. Personally I found Marder too abstract for my taste, though the group theoretical approach to crystallography is very powerful. But most of the math is not very difficult - it just appears dense because of the writing style.