3.6: Notation: Symmorphic 3-d Space Groups - Pma2
- Page ID
- 474770
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Three characteristics of this space group are:
- Point Group of the Space Group: \(mm2 = \mathcal{C}_{2v}\), which comes from the symbol \(``ma2"\) by replacing the glide reflection \(``a"\) with a normal reflection \(``m"\). The highest point symmetry in the unit cell is a proper subgroup of \(mm2\), while the physical properties of any crystal exhibits \(mm2\) symmetry.
- Crystal Class: Orthorhombic, which is determined from the point group of the space group. The unit cell involves three different and mutually perpendicular sides.
- Lattice Type: Primitive, based upon the lattice designator, so that the unit cell contains one lattice point and, therefore, one repeating unit throughout the crystal.
Now, we examine each part of the symbol \(Pma2\) and its implication(s) for a general position, designated by the coordinates \((x,y,z) = \left( \mathbf{1} \right)\). As for \(Pmm2\), we set lattice points at the intersections of the two sets of perpendicular planes.
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\(\boldsymbol{Pma2}\): Primitive unit cell projected down c using a right-handed perspective with lattice points at the corners. The general point in the unit cell \(\left( \mathbf{1} \right) = (x,y,z)\) corresponds to the identity operation \(\left( 1 \middle| \mathbf{0} \right)\). Translational symmetry generates equivalent positions (green) associated with each lattice point.
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\(\boldsymbol{Pma2}\): Primitive unit cell projected down c using a right-handed perspective with lattice points at the corners. The general point in the unit cell \(\left( \mathbf{1} \right) = (x,y,z)\) corresponds to the identity operation \(\left( 1 \middle| \mathbf{0} \right)\). Translational symmetry generates equivalent positions (green) associated with each lattice point.
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() \(\boldsymbol{Pma2}\): Reflection planes (red) perpendicular to \(\mathbf{a}\). The essential operation is \(\left( m_{\boldsymbol{a}} \middle| \mathbf{0} \right)\), which intersects \((0,0,0)\) and generates point \(\left( \mathbf{2} \right) = \left( m_{\boldsymbol{a}} \middle| \mathbf{0} \right)\left( \mathbf{1} \right) = \left( \overline{x},y,z \right)\). All other parallel \(m_{\boldsymbol{a}}\) planes are generated by lattice translations along \(\boldsymbol{a}\): \(\left( m_{\boldsymbol{a}} \middle| n_{1}\boldsymbol{a} \right) = \left( 1 \middle| n_{1}\boldsymbol{a} \right)\left( m_{\boldsymbol{a}} \middle| \mathbf{0} \right)\). & 
Figure 3.24
\(\boldsymbol{Pma2}\): Axial glide reflection planes (dashed red) perpendicular to \(\boldsymbol{b}\). The essential operation is \(\left( m_{\boldsymbol{b}} \middle| \frac{\boldsymbol{a}}{2} \right)\), which intersects \((0,0,0)\) and generates points \(\left( \mathbf{3} \right) = \left( m_{\boldsymbol{b}} \middle| \frac{\boldsymbol{a}}{2} \right)\left( \mathbf{1} \right) = \left( x + ½,\overline{y},z \right)\) and \(\left( \mathbf{4} \right) = \left( m_{\boldsymbol{b}} \middle| \frac{\boldsymbol{a}}{2} \right)\left( \mathbf{2} \right) = \left( ½ - x,\overline{y},z \right)\). All other parallel glides planes are generated by lattice translations along \(\mathbf{b}\). & 
Figure 3.25
\(\boldsymbol{Pma2}\): 2-fold rotation axes (red) parallel to \(\boldsymbol{c}\). The essential operation is \(\left( 2_{\boldsymbol{c}} \middle| \frac{\boldsymbol{a}}{2} \right)\), which intersects the point \((¼,0,0)\) and is generated by the two reflections: \(\left( 2_{\boldsymbol{c}} \middle| \frac{\boldsymbol{a}}{2} \right) = \left( m_{\boldsymbol{b}} \middle| \frac{\boldsymbol{a}}{2} \right)\left( m_{\boldsymbol{a}} \middle| \mathbf{0} \right)\). All other parallel \(2_{\boldsymbol{c}}\) axes are generated by lattice translations \(\left( 1 \middle| n_{1}\boldsymbol{a} + n_{2}\boldsymbol{b} \right)\). & 
Figure 3.26
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The two sets that build up the space group \(Pma2\) are
\(\mathcal{L =}\left\{ \left( 1 \middle| n_{1}\boldsymbol{a} + n_{2}\boldsymbol{b} + n_{3}\boldsymbol{c} \right):\ n_{1},n_{2},n_{3} = \text{integer};a \neq b \neq c,\alpha = \beta = \gamma = 90{^\circ} \right\} = P(\text{ortho})\)
\(\mathcal{R =}\left\{ \left( 1 \middle| \mathbf{0} \right),\left( m_{\boldsymbol{a}} \middle| \mathbf{0} \right),\left( m_{\boldsymbol{b}} \middle| \frac{\boldsymbol{a}}{2} \right),\left( 2_{\boldsymbol{c}} \middle| \frac{\boldsymbol{a}}{2} \right) \right\}\).
According to the multiplication table, the set\(\mathcal{\ \ R}\) is not a group because the products highlighted in light green are not members of \(\mathcal{R}\). Nevertheless, they are members of the space group \(\mathcal{R}\bigotimes_{}^{}\mathcal{L}\). Also, the perpendicular reflection planes intersect at lattice points, but the resulting 2-fold axes do not. Therefore, the point symmetry at each lattice point is \(m\ = \ \mathcal{C}_{s}\). Points coincident with the 2-fold axes have point symmetry \(2\ = \ \mathcal{C}_{2}\). Because both of these groups are proper subgroups of \(mm2\) with 2, the origin point (lattice points) could be assigned to either one of these settings.
| () “ma2” | \((1|\boldsymbol 0)\) | \((m_{\boldsymbol{a}}|\boldsymbol 0)\) | \((m_{\boldsymbol{b}}|{\boldsymbol a\over 2})\) | \((2_{\boldsymbol{c}}|{\boldsymbol a\over 2})\) |
|---|---|---|---|---|
| () () “ma2” | \((1|\boldsymbol 0)\) | \((m_{\boldsymbol{a}}|\boldsymbol 0)\) | \((m_{\boldsymbol{b}}|{\boldsymbol a\over 2})\) | \((2_{\boldsymbol{c}}|{\boldsymbol a\over 2})\) |
| () \((1|\boldsymbol 0)\) | \((1|\boldsymbol 0)\) | \((m_{\boldsymbol{a}}|\boldsymbol 0)\) | \((m_{\boldsymbol{b}}|{\boldsymbol a\over 2})\) | \((2_{\boldsymbol{c}}|{\boldsymbol a\over 2})\) |
| \((m_{\boldsymbol{a}}|\boldsymbol 0)\) | \((m_{\boldsymbol{a}}|\boldsymbol 0)\) | \((1|\boldsymbol 0)\) | \((2_{\boldsymbol{c}}|-{\boldsymbol a\over 2})\) | \((m_{\boldsymbol{b}}|-{\boldsymbol a\over 2})\) |
| \((m_{\boldsymbol{b}}|{\boldsymbol a\over 2})\) | \((m_{\boldsymbol{b}}|{\boldsymbol a\over 2})\) | \((2_{\boldsymbol{c}}|{\boldsymbol a\over 2})\) | \((1|\boldsymbol a)\) | \((m_{\boldsymbol{a}}|\boldsymbol a)\) |
| \((2_{\boldsymbol{c}}|{\boldsymbol a\over 2})\) | \((2_{\boldsymbol{c}}|{\boldsymbol a\over 2})\) | \((m_{\boldsymbol{b}}|{\boldsymbol a\over 2})\) | \((m_{\boldsymbol{a}}|\boldsymbol 0)\) | \((1|\boldsymbol 0)\) |
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TlF3 forms a crystalline structure in the space group Pma2. The accompanying figure shows the outline of a unit cell with Tl atoms in gray, F atoms in light green. Normal mirror planes perpendicular to the \(\mathbf{a}\) are highlighted in red; a axial glide planes perpendicular to \(\mathbf{b}\) are shown in magenta. These two sets of planes do not intersect at lattice points, which occur along the intersections of 2c axes and ab glide planes. The lattice constants are \(a = 5.175\ \mathring{\mathrm{A}},\ b = 6.092\ \mathring{\mathrm{A}},\ c = 5.488\ \mathring{\mathrm{A}}\). All atoms lie on the \(m_{\boldsymbol{a}}\) reflection planes.
\(\boldsymbol{Ima2}\): Let’s examine how lattice centering affects nonsymmorphic space groups. The point group of this space group is \(mm2 = \mathcal{C}_{2v}\), the crystal class is orthorhombic, and the lattice type is body-centered, so that there are 2 lattice points per unit cell.
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() \(\boldsymbol{Ima2}\): Body-centering takes the image associated with \((0,0,0)\) and reproduces it around \((½,½,½)\). As a result, each unit cell contains 8 sites generated from the general position. & \(\boldsymbol{Ima2}\): The point symmetry \(mm2\) is repeated around every lattice point.
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Therefore, the two sets that build up the space group \(Ima2\) are
\(\mathcal{L =}\left\{ \left( 1 \middle| \frac{n_{1}}{2}\boldsymbol{a} + \frac{n_{2}}{2}\boldsymbol{b} + \frac{n_{3}}{2}\boldsymbol{c} \right):\ n_{1},n_{2},n_{3} = \text{all\ even\ or\ odd\ integers};a \neq b \neq c,\alpha = \beta = \gamma = 90{^\circ} \right\}\)
\(\mathcal{L} = I(\text{ortho})\), and
\(\mathcal{R =}\left\{ \left( 1 \middle| \mathbf{0} \right),\left( m_{\boldsymbol{a}} \middle| \mathbf{0} \right),\left( m_{\boldsymbol{b}} \middle| \frac{\boldsymbol{a}}{2} \right),\left( 2_{\boldsymbol{c}} \middle| \frac{\boldsymbol{a}}{2} \right) \right\}\), which is not a group.
The body-centering lattice translations, when multiplied with the essential reflections and rotations, generate additional symmetry operations:
\(\left( 1 \middle| \frac{\boldsymbol{a} + \boldsymbol{b} + \boldsymbol{c}}{2} \right)\left( m_{\boldsymbol{a}} \middle| \mathbf{0} \right) = \left( m_{\boldsymbol{a}} \middle| \frac{\boldsymbol{a} + \boldsymbol{b} + \boldsymbol{c}}{2} \right) = \left( 1 \middle| \frac{\boldsymbol{a}}{4} \right)\left( m_{\boldsymbol{a}} \middle| \frac{\boldsymbol{b} + \boldsymbol{c}}{2} \right)\left( 1 \middle| \frac{- \boldsymbol{a}}{4} \right) =\) a diagonal glide plane \(n_{\boldsymbol{a}}\) oriented perpendicular to \(\boldsymbol{a}\) and intersecting the point \((¼,0,0)\).
\(\left( 1 \middle| \frac{- \boldsymbol{a} + \boldsymbol{b} + \boldsymbol{c}}{2} \right)\left( m_{\boldsymbol{b}} \middle| \frac{\boldsymbol{a}}{2} \right) = \left( m_{\boldsymbol{b}} \middle| \frac{\boldsymbol{b} + \boldsymbol{c}}{2} \right) = \left( 1 \middle| \frac{\boldsymbol{b}}{4} \right)\left( m_{\boldsymbol{b}} \middle| \frac{\boldsymbol{c}}{2} \right)\left( 1 \middle| \frac{- \boldsymbol{b}}{4} \right) =\) an axial glide plane \(c_{\boldsymbol{b}}\) along \(\boldsymbol{c}\) oriented perpendicular to \(\boldsymbol{b}\) and intersecting the point \((0,¼,0)\).
\(\left( 1 \middle| \frac{- \boldsymbol{a} + \boldsymbol{b} + \boldsymbol{c}}{2} \right)\left( 2_{\boldsymbol{c}} \middle| \frac{\boldsymbol{a}}{2} \right) = \left( 2_{\boldsymbol{c}} \middle| \frac{\boldsymbol{b} + \boldsymbol{c}}{2} \right) = \left( 1 \middle| \frac{\boldsymbol{b}}{4} \right)\left( 2_{\boldsymbol{c}} \middle| \frac{\boldsymbol{c}}{2} \right)\left( 1 \middle| \frac{\boldsymbol{- b}}{4} \right) =\) a two-fold screw axis \(2_{1}\) parallel to \(\boldsymbol{c}\) and intersecting the point \((0,¼,0)\).
In \(Ima2\), there are diagonal glide reflections perpendicular to \(\boldsymbol{a}\), axial glide reflections along \(\boldsymbol{c}\) and perpendicular to \(\boldsymbol{b}\), as well as two-fold screw axes along \(\boldsymbol{c}\). Therefore, another possible symbol for this space group could be \(Inc2_{1}\).
(SrCa)MnGaO54 forms a crystalline structure in the space group \(Ima2\). The accompanying figure shows the structure with Sr/Ca atoms in gray, Mn atoms in purple, Ga atoms in blue, and O atoms in red. Perpendicular to \(\boldsymbol{a}\) are the mirrors (gray) and diagonal glide (magenta) reflections; perpendicular to \(\boldsymbol{b}\) are the axial a-glide (blue) and axial c-glide (red) reflections. Lattice points occur along lines where the diagonal na planes and axial ab planes intersect – these lines correspond to the 2-fold rotation axes so that the point symmetry at each lattice point is 2 = \(\mathcal{C}_{2}\). Ga atoms sit only on \(m_{\boldsymbol{a}}\) planes; Mn atoms sit at the intersections of \(n_{\boldsymbol{a}}\) and \(a_{\boldsymbol{b}}\) glide planes.
(32) PRACTICE QUESTION 1: Extracting Information from the Space Group Symbol.
Fill in the blanks for each of the following space group symbols.
\(\boldsymbol{C2/m:}\) Point Group = __________ Crystal Class = __________
Lattice Type = __________ Highest Point Symmetry = __________
\(\boldsymbol{I4/mmm:}\) Point Group = __________ Crystal Class = __________
Lattice Type = __________ Highest Point Symmetry = __________
\(\boldsymbol{Fm}\overline{\mathbf{3}}\boldsymbol{m:}\) Point Group = __________ Crystal Class = __________
Lattice Type = __________ Highest Point Symmetry = __________
\(\boldsymbol{P}\mathbf{2}_{\mathbf{1}}\boldsymbol{/c:}\) Point Group = __________ Crystal Class = __________
Lattice Type = __________ Highest Point Symmetry = __________
\(\boldsymbol{P}\mathbf{6}_{\mathbf{3}}\boldsymbol{/mmc:}\) Point Group = __________ Crystal Class = __________
Lattice Type = __________ Highest Point Symmetry = __________
\(\boldsymbol{Fd}\overline{\mathbf{3}}\boldsymbol{m:}\) Point Group = __________ Crystal Class = __________
Lattice Type = __________ Highest Point Symmetry = __________
AnswerS:
\(\boldsymbol{C2/m:}\) Point Group = \(\boldsymbol{2/m\ }\left( \mathcal{C}_{\boldsymbol{2h}} \right)\) Crystal Class = Monoclinic
Lattice Type = Base-centered Highest Point Symmetry = \(\boldsymbol{2/m\ }\left( \mathcal{C}_{\boldsymbol{2h}} \right)\)
\(\boldsymbol{I4/mmm:}\) Point Group = \(\boldsymbol{4/mmm\ }\left( \mathcal{D}_{\boldsymbol{4h}} \right)\) Crystal Class = Tetragonal
Lattice Type = Body-centered Highest Point Symmetry = \(\boldsymbol{4/mmm\ }\left( \mathcal{D}_{\boldsymbol{4h}} \right)\)
\(\boldsymbol{Fm}\overline{\mathbf{3}}\boldsymbol{m:}\) Point Group = \(\boldsymbol{m}\overline{\mathbf{3}}\boldsymbol{m\ }\left( \mathcal{O}_{\boldsymbol{h}} \right)\) Crystal Class = Cubic
Lattice Type = Face-centered Highest Point Symmetry = \(\boldsymbol{m}\overline{\mathbf{3}}\boldsymbol{m\ }\left( \mathcal{O}_{\boldsymbol{h}} \right)\)
(Adopted by FCC metals such as Cu, Al, and Pb)
\(\boldsymbol{P}\mathbf{2}_{\mathbf{1}}\boldsymbol{/c:}\) Point Group = \(\boldsymbol{2/m\ }\left( \mathcal{C}_{\boldsymbol{2h}} \right)\) Crystal Class = Monoclinic
Lattice Type = Primitive Highest Point Symmetry = \(\overline{\mathbf{1}}\mathbf{\ }\left( \mathcal{C}_{\boldsymbol{i}} \right)\)
(Most common space group, along with \(P\overline{1}\))
\(\boldsymbol{P}\mathbf{6}_{\mathbf{3}}\boldsymbol{/mmc:}\) Point Group = \(\boldsymbol{6/mmm\ }\left( \mathcal{D}_{\boldsymbol{6h}} \right)\) Crystal Class = Hexagonal
Lattice Type = Primitive Highest Point Symmetry = \(\overline{\mathbf{3}}\boldsymbol{m\ }\left( \mathcal{D}_{\boldsymbol{3d}} \right)\)
Highest Point Symmetry = \(\overline{\mathbf{6}}\boldsymbol{m2\ }\left( \mathcal{D}_{\boldsymbol{3h}} \right)\)
(Adopted by HCP metals such as Mg, Ti, and Gd)
\(\boldsymbol{Fd}\overline{\mathbf{3}}\boldsymbol{m:}\) Point Group = \(\boldsymbol{m}\overline{\mathbf{3}}\boldsymbol{m\ }\left( \mathcal{O}_{\boldsymbol{h}} \right)\) Crystal Class = Cubic
Lattice Type = Face-centered Highest Point Symmetry = \(\overline{\mathbf{4}}\mathbf{3}\boldsymbol{m\ }\left( \mathcal{T}_{\boldsymbol{d}} \right)\)
Highest Point Symmetry = \(\overline{\mathbf{3}}\boldsymbol{m\ }\left( \mathcal{D}_{\boldsymbol{3d}} \right)\)
(Space group of diamond-type structures such as C, Si, and Ge)
There are two settings for the origin because the site of highest point symmetry does not coincide with an inversion center.
(33) PRACTICE QUESTION 2:
The nickel-carbon plane in CeNiC2 consists of Ni atoms and C2-dimers that form [NiC2] triangles and 7-membered rings of atoms (see right).
Determine the space group symbol for this 2-d structure by
- (a) identifying rotational symmetry points or reflection lines, if they exist;
- (b) identifying a unit cell and lattice type, placing lattice points at centers of inversion, if they occur;
- (c) determining the set of essential symmetry operations and writing their Seitz symbols;
- (d) assigning the space group.
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Rotational Symmetry: There are no \(C_{n}\) rotation axes but there are reflection planes (red), which bisect the C2-dimers.
Unit Cell: Lattice points can be selected arbitrarily because there are no Cn axes, so we choose the Ni sites. The smallest unit cell is a rhombus with no symmetry restrictions on the angle: a = b, γ ≠ 90° or 120°.
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Rotational Symmetry: There are no \(C_{n}\) rotation axes but there are reflection planes (red), which bisect the C2-dimers.
Unit Cell: Lattice points can be selected arbitrarily because there are no Cn axes, so we choose the Ni sites. The smallest unit cell is a rhombus with no symmetry restrictions on the angle: a = b, γ ≠ 90° or 120°.
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Figure 3.34 & Lattice Type: The primitive unit cell describes the centered-rectangular lattice (c), so a revised unit cell with two lattice points can be identified with a rectangular shape: a ≠ b, γ = 90°.
Centered-rectangular:
\(\mathcal{L =} c = \left\{ \frac{n_{1}}{2}\boldsymbol{a} + \frac{n_{2}}{2}\boldsymbol{b};\ n_{1},n_{2} = \text{both\ even\ or\ both\ odd\ integers} \right\}\)
Essential Symmetry Operations: Reflection planes intersect lattice points, and they are oriented (arbitrarily) perpendicular to \(\mathbf{a}\) of the rectangular unit cell.
Base-centering with reflections generates glide planes:
\(\small{\left( 1 \middle| \frac{\boldsymbol{a} + \boldsymbol{b}}{2} \right)\left( m_{\boldsymbol{a}} \middle| \mathbf{0} \right) = \left( m_{\boldsymbol{a}} \middle| \frac{\boldsymbol{a} + \boldsymbol{b}}{2} \right) = \left( 1 \middle| \frac{\boldsymbol{a}}{4} \right)\left( m_{\boldsymbol{a}} \middle| \frac{\boldsymbol{b}}{2} \right)\left( 1 \middle| \frac{- \boldsymbol{a}}{4} \right)}\)
which shows a \(b\)-glide intersecting the point \((¼,0)\). Lattice vectors along \(\mathbf{a}\) generate other \(b\)-glide reflections.
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