In the first half of Lecture II, the complex
Consider only the purely translational symmetry operations of an infinite crystal. These operations form a group denoted
$\mathrm{\textbf{T}}$ , where$\mathrm{\textbf{T}}$ is a subgroup of the full space group$\mathrm{\textbf{G}}$ of the crystal. Denote by$R$ , the set of all symmetry operations of$\mathrm{\textbf{G}}$ that involve pure rotations only (proper or improper) as well as rotations accompanied by a translation not in$\mathrm{\textbf{T}}$ .$R$ is a subset of$\mathrm{\textbf{G}}$ called a "complex" whose elements are denoted by$\alpha$ ,$\beta$ ,$\gamma$ , ... since in general$R$ is not a group.
Consider the case wherein
$R$ is the set of all pure rotational operations (proper and improper) of$\mathrm{\textbf{G}}$ plus the glide reflection denoted$m$ and shown in the previous slide.$R$ is not a group since$mm$ is a pure translation which is not a member of$R$ .
Let $ C = {\alpha, \beta, \gamma, ...}$ be the complex consisting of all elements of
$\mathrm{\textbf{G}}$ not in$\mathrm{\textbf{T}}$ . Unlike the complex$R$ , the translation operations that accompany rotations as members of$C$ , can belong in$\mathrm{\textbf{T}}$ .
It took me too long to understand this graphically. I went over the slides, the book, and the notes I had when I was taking Math Methods. I finally realized that in the nonsymmorphic example above, both of
And here comes a sudden change.
On the other hand, the mapping of the factor group
$\mathrm{\textbf{G/T}}$ of the space group G to the constructed sub-group$\mathrm{\textbf{R}} = {\alpha_r, \beta_r, \gamma_r}$ of a symmorphic space group is one-to-one and is called an "isomorphism".
$\mathrm{\textbf{R}} = {\alpha_r, \beta_r, \gamma_r}$ are not necessarily elements of the nonsymmorphic space group and so$\mathrm{\textbf{R}}$ is not a subgroup of$\mathrm{\textbf{G}}$ . If$\alpha = \alpha_t \alpha_r$ is a glide or screw, then$\alpha_r$ (improper or proper, respectively) is not an element in$\mathrm{\textbf{G}}$ and$\alpha_t$ is not an element of$\mathrm{\textbf{T}}$ .$\mathrm{\textbf{R}}$ is still referred to as the point group of the space group$\mathrm{\textbf{G}}$ , since it contains all the information on the rotational symmetries of the space group. However, it is not a sub-group of$\mathrm{\textbf{G}}$ and contains operations that are not elements of$\mathrm{\textbf{G}}$ .
(For the diamond crystal) Let
$\mathrm{\textbf{G}}$ be the space group of diamond and$\mathrm{\textbf{T}}$ be the translational symmetry operations.$\mathrm{\textbf{T}}$ is the same for zincblende and diamond lattices. Let$C$ be the complex of operations not in$\mathrm{\textbf{T}}$ ;$C$ is different for the zincblende and diamond lattices. In particular the operation$T(1/4,1/4,1/4) i$ is in$C$ for the diamond lattice, but not in the zincblende lattice. This additional operation is in the form of$\alpha = \alpha_t \alpha_r$ . So$\mathrm{\textbf{R}}$ for the diamond lattice (which is not a subgroup because$i$ is not in$\mathrm{\textbf{G}}$ , i.e.,$\mathrm{\textbf{G}}$ is nonsymmorphic) includes all the rotations of the zincblende lattice plus$i$ .
The factor group
$\mathrm{\textbf{G/T}}$ of the diamond lattice is isomorphic to$\mathrm{\textbf{R}}$ which has in it all the elements of$\mathrm{\textbf{T_d}}$ , but with the addition of$i$ . Recall that$\mathrm{\textbf{R}}$ is still referred to as the point group of the nonsymmorphic space group of the diamond lattice because it contains all the information on the rotational symmetries of the space group; however, it contains the element$i$ which is not in the space group.