3.2: Screw Rotations
- Page ID
- 474766
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)A screw rotation, symbolized by \(n_{j}\ (j < n)\), is a proper \(\frac{2\pi}{n}\) ccw rotation accompanied by the fractional displacement \(\frac{j}{n}\boldsymbol{T}\) along the lattice vector \(\boldsymbol{T}\), which is parallel to the rotation axis. The subscript \(j\) can be any one of the positive integers less than \(n\). There are 11 possible screw rotations in crystalline structures: \(2_{1};3_{1},3_{2};\ 4_{1},4_{2},4_{3};\ 6_{1},6_{2},6_{3},6_{4},6_{5}\). The Seitz notation for a screw rotation \(n_{j}\) when the rotation axis passes through the origin is \(n_{m} = \left( n_{\boldsymbol{T}} \middle| \frac{j\boldsymbol{T}}{n} \right)\). As an example, consider the three possible four-fold screw rotations with their rotation axes along c:
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() \(4_{1}:\ \left( 4_{\boldsymbol{c}} \middle| \frac{\boldsymbol{c}}{4} \right)\) & \(4_{2}:\ \left( 4_{\boldsymbol{c}} \middle| \frac{2\boldsymbol{c}}{4} \right) = \left( 4_{\boldsymbol{c}} \middle| \frac{\boldsymbol{c}}{2} \right)\) & \(4_{3}:\ \left( 4_{\boldsymbol{c}} \middle| \frac{3\boldsymbol{c}}{4} \right)\)
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Each operation is a 90° ccw rotation along with one of the displacements \(\frac{\boldsymbol{c}}{4}\), \(\frac{2\boldsymbol{c}}{4} = \frac{\boldsymbol{c}}{2}\), or \(\frac{3\boldsymbol{c}}{4}\). As the diagram points out, the actions of \(4_{1}\) and \(4_{3}\) screw rotations on an object create equivalent objects at steps of \(\frac{\boldsymbol{c}}{4}\); whereas the action of \(4_{2}\) creates objects at steps of \(\frac{\boldsymbol{c}}{2}\). To see how this occurs, evaluate the sequential products of two, three, and four operations.
\({4_{1}}^{2}:\) \(\left( 4_{\boldsymbol{c}} \middle| \frac{\boldsymbol{c}}{4} \right)^{2} = \left( 4_{\boldsymbol{c}}^{2} \middle| 4_{\boldsymbol{c}}\left( \frac{\boldsymbol{c}}{4} \right) + \frac{\boldsymbol{c}}{4} \right) = \left( 2_{\boldsymbol{c}} \middle| \frac{2\boldsymbol{c}}{4} \right) = \left( 2_{\boldsymbol{c}} \middle| \frac{\boldsymbol{c}}{2} \right)\);
\({4_{2}}^{2}:\) \(\left( 4_{\boldsymbol{c}} \middle| \frac{2\boldsymbol{c}}{4} \right)^{2} = \left( 4_{\boldsymbol{c}}^{2} \middle| 4_{\boldsymbol{c}}\left( \frac{\boldsymbol{c}}{2} \right) + \frac{\boldsymbol{c}}{2} \right) = \left( 2_{\boldsymbol{c}} \middle| \boldsymbol{c} \right)\);
\({4_{3}}^{2}:\) \(\left( 4_{\boldsymbol{c}} \middle| \frac{3\boldsymbol{c}}{4} \right)^{2} = \left( 4_{\boldsymbol{c}}^{2} \middle| 4_{\boldsymbol{c}}\left( \frac{3\boldsymbol{c}}{4} \right) + \frac{3\boldsymbol{c}}{4} \right) = \left( 2_{\boldsymbol{c}} \middle| \frac{6\boldsymbol{c}}{4} \right) = \left( 2_{\boldsymbol{c}} \middle| \frac{3\boldsymbol{c}}{2} \right)\).
Each of two successive \(4_{j}\) operations is a 180° rotation accompanied by different displacements parallel to \(\boldsymbol{c}\). Notice that the four-fold rotation does not affect \(\mathbf{c}\), i.e., \(4_{c}\boldsymbol{c} = \boldsymbol{c}\). Due to translational periodicity along the \(\boldsymbol{c}\)-axis, if \(\left( n_{\mathbf{c}}^{j} \middle| \mathbf{c} \right)\) is an operation, then so is \(\left( n_{\boldsymbol{c}}^{j} \middle| \boldsymbol{0} \right)\). Therefore, the \(4_{2}\) screw rotation includes a proper 2-fold rotation \(\left( 2_{\boldsymbol{c}} \middle| \boldsymbol{0} \right)\), which is shown in the middle diagram. Likewise, because two successive \(4_{3}\) screw rotations generate \(\left( 2_{\boldsymbol{c}} \middle| \frac{3\boldsymbol{c}}{2} \right)\), then so is \(\left( 2_{\boldsymbol{c}} \middle| \frac{\boldsymbol{c}}{2} \right)\), as seen in the right-hand diagram. Now, three successive \(4_{j}\) operations yield
\({4_{1}}^{3}:\) \(\left( 4_{\boldsymbol{c}} \middle| \frac{\boldsymbol{c}}{4} \right)^{3} = \left( 4_{\boldsymbol{c}}^{3} \middle| \frac{3\boldsymbol{c}}{4} \right)\);
\({4_{2}}^{3}:\) \(\left( 4_{\boldsymbol{c}} \middle| \frac{2\boldsymbol{c}}{4} \right)^{3} = \left( 4_{\boldsymbol{c}}^{3} \middle| \frac{6\boldsymbol{c}}{4} \right) \rightarrow \left( 4_{\boldsymbol{c}}^{3} \middle| \frac{2\boldsymbol{c}}{4} \right) = \left( 4_{\boldsymbol{c}}^{3} \middle| \frac{\boldsymbol{c}}{2} \right)\);
\({4_{3}}^{3}:\) \(\left( 4_{\boldsymbol{c}} \middle| \frac{3\boldsymbol{c}}{4} \right)^{3} = \left( 4_{\boldsymbol{c}}^{3} \middle| 3 \cdot \frac{3\boldsymbol{c}}{4} \right) \rightarrow \left( 4_{\boldsymbol{c}}^{3} \middle| \frac{\boldsymbol{c}}{4} \right)\);
and four consecutive operations yield
\({4_{1}}^{4}:\) \(\left( 4_{\boldsymbol{c}} \middle| \frac{\boldsymbol{c}}{4} \right)^{4} = \left( 4_{\boldsymbol{c}}^{4} \middle| \frac{4\boldsymbol{c}}{4} \right) = \left( 1 \middle| \boldsymbol{c} \right) =\) lattice translation by \(\boldsymbol{c}\);
\({4_{2}}^{4}:\) \(\left( 4_{\boldsymbol{c}} \middle| \frac{2\boldsymbol{c}}{4} \right)^{4} = \left( 4_{\boldsymbol{c}}^{4} \middle| \frac{4\boldsymbol{c}}{2} \right) = \left( 1 \middle| 2\boldsymbol{c} \right) =\) lattice translation by \(2\boldsymbol{c}\);
\({4_{3}}^{4}:\) \(\left( 4_{\boldsymbol{c}} \middle| \frac{3\boldsymbol{c}}{4} \right)^{4} = \left( 4_{\boldsymbol{c}}^{4} \middle| 4 \cdot \frac{3\boldsymbol{c}}{4} \right) = \left( 1 \middle| 3\boldsymbol{c} \right) =\) lattice translation by \(3\boldsymbol{c}\);
which are some multiple of the lattice translation \(\boldsymbol{c}\). According to this analysis and the diagrams, the \(4_{1}\) and \(4_{3}\) screw rotations are enantiomorphic because they are related by a mirror plane parallel to the rotation axes. If the \(4_{1}\) operation is described as a right-handed screw, then the \(4_{3}\) operation is a left-handed screw. In general, \(n_{j}\) and \(n_{n - j}\) screw rotations form an enantiomorphic pair.
The general Seitz symbol for a screw rotation \(n_{j}\) is \(\left( n_{\boldsymbol{T}} \middle| \frac{j\boldsymbol{T}}{n} + \boldsymbol{\tau}_{\bot\boldsymbol{T}} \right)\) in which \(\boldsymbol{\tau}_{\bot\boldsymbol{T}}\) is a displacement perpendicular to the rotation axis and will set the axis relative to the origin.