3.5: Notation: Symmorphic 3-d Space Groups - Pmm2

Three significant characteristics of the space group are determined from its symbol.

• Point Group of the Space Group: \(mm2 = \mathcal{C}_{2v}\), which comes directly from the symbol for the essential symmetry operations.
• Crystal Class: Orthorhombic, which is determined from the point group of the space group and identifies the necessary rotational symmetry of the lattice. The unit cell involves three different and mutually perpendicular sides, so it is sufficient to specify just the lengths of each unit cell side.
• Lattice Type: Primitive, based upon the lattice designator, so that the unit cell contains one lattice point, which is typically located at cell corner(s), and, therefore, one repeating unit throughout the crystal.

Now we examine each part of the symbol \(Pmm2\) and its implication(s) for a general position, designated by the coordinates \((x,y,z) = \left( \mathbf{1} \right)\):

\(\boldsymbol{Pmm2}\): 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.

Figure 3.14

\(\boldsymbol{Pmm2}\): 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.

Figure 3.14

\(\boldsymbol{Pmm2}\): Reflection planes (red) perpendicular to \(\mathbf{a}\). The essential operation is \(\left( m_{\boldsymbol{a}} \middle| \mathbf{0} \right)\), which intersects the origin 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 \(m_{\mathbf{a}}\) 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 3.15

\(\boldsymbol{Pmm2}\): Reflection planes (red) perpendicular to \(\boldsymbol{b}\). The essential operation is \(\left( m_{\boldsymbol{b}} \middle| \mathbf{0} \right)\), which intersects the origin and generates points \(\left( \mathbf{3} \right) = \left( m_{\boldsymbol{b}} \middle| \mathbf{0} \right)\left( \mathbf{1} \right) = \left( x,\overline{y},z \right)\) and \(\left( \mathbf{4} \right) = \left( m_{\boldsymbol{b}} \middle| \mathbf{0} \right)\left( \mathbf{2} \right) = \left( \overline{x},\overline{y},z \right)\). All other parallel \(m_{\boldsymbol{b}}\) planes are generated by lattice translations along \(\boldsymbol{b}:\) \(\left( m_{\boldsymbol{b}} \middle| n_{2}\boldsymbol{b} \right) = \left( 1 \middle| n_{2}\boldsymbol{b} \right)\left( m_{\boldsymbol{b}} \middle| \mathbf{0} \right)\). &

Figure 3.16

\(\boldsymbol{Pmm2}\): 2-fold rotation axes (red) parallel to \(\boldsymbol{c}\). The essential operation is \(\left( 2_{\boldsymbol{c}} \middle| \mathbf{0} \right)\), which intersects the origin and is generated by the two orthogonal reflections: \(\left( 2_{\boldsymbol{c}} \middle| \mathbf{0} \right) = \left( m_{\boldsymbol{b}} \middle| \mathbf{0} \right)\left( m_{\boldsymbol{a}} \middle| \mathbf{0} \right)\). As a result, no additional general points occur in the unit cell. 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 3.17

As a result of this analysis, the two sets that build up the space group \(Pmm2\) are

\[\mathcal{L =}\left\{ \left( 1 \middle| n_{1}\boldsymbol{a} + n_{2}\boldsymbol{b} + n_{3}\boldsymbol{c} \right):\ n_{1},n_{2},n_{3} = \text{integers};a \neq b \neq c,\alpha = \beta = \gamma = 90{^\circ} \right\} = P(\text{ortho}), \nonumber \]

and

\[\mathcal{R =}\left\{ \left( 1 \middle| \mathbf{0} \right),\left( m_{\boldsymbol{a}} \middle| \mathbf{0} \right),\left( m_{\boldsymbol{b}} \middle| \mathbf{0} \right),\left( 2_{\boldsymbol{c}} \middle| \mathbf{0} \right) \right\} = mm2 = \mathcal{C}_{2v}. \nonumber \]

According to its multiplication table, the set\(\mathcal{\ R}\) is a group. The operation \(\left( 2_{\boldsymbol{c}} \middle| \mathbf{0} \right)\) is generated from the product of the two perpendicular mirror planes in either order:

\[\left( 2_{\boldsymbol{c}} \middle| \mathbf{0} \right) = \left( m_{\boldsymbol{a}} \middle| \mathbf{0} \right)\left( m_{\boldsymbol{b}} \middle| \mathbf{0} \right) \nonumber \]

\[\left( 2_{\boldsymbol{c}} \middle| \mathbf{0} \right) = \left( m_{\boldsymbol{b}} \middle| \mathbf{0} \right)\left( m_{\boldsymbol{a}} \middle| \mathbf{0} \right). \nonumber \]

“mm2”	\((1|\mathbf{0})\)	\((m_{\boldsymbol a}|\mathbf{0})\)	\((m_{\boldsymbol b}|\mathbf{0})\)	\((2_{\boldsymbol c}|\mathbf{0})\)
\((1|\mathbf{0})\)	\((1|\mathbf{0})\)	\((m_{\boldsymbol a}|\mathbf{0})\)	\((m_{\boldsymbol b}|\mathbf{0})\)	\((2_{\boldsymbol c}|\mathbf{0})\)
\((m_{\boldsymbol a}|\mathbf{0})\)	\((m_{\boldsymbol a}|\mathbf{0})\)	\((1|\mathbf{0})\)	\((2_{\boldsymbol c}|\mathbf{0})\)	\((m_{\boldsymbol b}|\mathbf{0})\)
\((m_{\boldsymbol b}|\mathbf{0})\)	\((m_{\boldsymbol b}|\mathbf{0})\)	\((2_{\boldsymbol c}|\mathbf{0})\)	\((1|\mathbf{0})\)	\((m_{\boldsymbol a}|\mathbf{0})\)
\((2_{\boldsymbol c}|\mathbf{0})\)	\((2_{\boldsymbol c}|\mathbf{0})\)	\((m_{\boldsymbol b}|\mathbf{0})\)	\((m_{\boldsymbol a}|\mathbf{0})\)	\((1|\mathbf{0})\)

Since lattice points occur along lines where m_a and m_b intersect, these intersections coincide with the 2-fold rotation axes. Therefore, the point symmetry at each lattice point is \(mm2 = \mathcal{C}_{2v}\). Furthermore, this is the highest point symmetry anywhere in 3-d space for this space group. As the final diagram above illustrates, besides lattice points, which include \((0,0,z)\), this point symmetry occurs at \((½,0,z),(0,½,z),\) and \((½,½,z)\).

Example \(\PageIndex{1}\)

\(\ce{Ag3Sb}\) forms a crystalline structure in the space group \(Pmm2\). The accompanying figure includes the outline of a unit cell with Ag atoms in gray, Sb atoms in light green, and two sets of perpendicular reflection planes highlighted in red. The lattice constants are \(a = 4.890\ \mathring{\mathrm{A}},\ b = 3.022\ \mathring{\mathrm{A}},\ c = 5.276\ \mathring{\mathrm{A}}\). Planes through lattice points contain equal numbers of Ag and Sb atoms; planes bisecting the unit cell contain only Ag atoms.

Figure 3.18: Projection of \(\ce{Ag3Sb}\) structure along c-axis. \(\ce{Ag}\) atoms in silver; \(\ce{Sb}\) atoms in green. Reflection planes perpendicular to a- and b-axes shown in light red.

\(\boldsymbol{Imm2}\): Symmorphic space groups involving centered lattices contain screw rotations and glide reflections, although they are not included in the space group symbol. Like \(Pmm2\), the point group of this space group is \(mm2 = \mathcal{C}_{2v}\), the crystal class is orthorhombic, but the lattice type is body-centered, which means there are 2 lattice points per unit cell.

Figure 3.19

\(\boldsymbol{Imm2}\): Body-centering takes the image around \((0,0,0)\) and reproduces it around \((½,½,½)\). For example, \(\left( 1 \middle| \frac{\boldsymbol{a} + \boldsymbol{b} + \boldsymbol{c}}{2} \right)\left( \mathbf{1} \right) = (x + ½,y + ½,z + ½)\) \(= \left( \mathbf{5} \right)\). As a result, each unit cell contains 8 sites generated from the general position. &

Figure 3.20

\(\boldsymbol{Imm2}\): The point symmetry \(mm2\) is repeated around every lattice point.

Therefore, the two sets that build up the space group \(Imm2\) are

• \(\mathcal{L =}\left\{ \left( 1 \middle| \frac{n_{1}}{2}\boldsymbol{a} + \frac{n_{2}}{2}\boldsymbol{b} + \frac{n_{3}}{2}\v{c} \right):\ n_{1},n_{2},n_{3} = \text{all\ even\ or\ odd\ integers};a \neq b \neq c,\alpha = \beta = \gamma = 90{^\circ} \right\}\)
• \(\mathcal{L} = I(\text{ortho})\), and
• \(\mathcal{R =}\left\{ \left( 1 \middle| \mathbf{0} \right),\left( m_{\boldsymbol{a}} \middle| \mathbf{0} \right),\left( m_{\boldsymbol{b}} \middle| \mathbf{0} \right),\left( 2_{\boldsymbol{c}} \middle| \mathbf{0} \right) \right\} = mm2 = \mathcal{C}_{2v}\), which is a group.

The body-centering lattice translations, when multiplied with the essential reflections and rotations, generate additional symmetry operations noted in red:

Figure 3.21: See accompanying text above figure.

• a diagonal glide plane \(n_{\boldsymbol{a}}\) oriented perpendicular to \(\boldsymbol{a}\) and intersecting the point \((¼,0,0)\). \[\left( 1 \middle| \frac{\boldsymbol{a} + \boldsymbol{b} + \boldsymbol{c}}{2} \right)\left( m_{\boldsymbol{a}} \middle| \mathbf{0} \right) = \left( m_{\boldsymbol{a}} \middle| \frac{\boldsymbol{a} + \boldsymbol{b} + \boldsymbol{c}}{2} \right) = \left( 1 \middle| \frac{\boldsymbol{a}}{4} \right)\left( m_{\boldsymbol{a}} \middle| \frac{\boldsymbol{b} + \boldsymbol{c}}{2} \right)\left( 1 \middle| \frac{- \boldsymbol{a}}{4} \right) \nonumber \]
• a diagonal glide plane \(n_{\boldsymbol{b}}\) oriented perpendicular to \(\boldsymbol{b}\) and intersecting the point \((0,¼,0)\). \[\left( 1 \middle| \frac{\boldsymbol{a} + \boldsymbol{b} + \boldsymbol{c}}{2} \right)\left( m_{\boldsymbol{b}} \middle| \mathbf{0} \right) = \left( m_{\boldsymbol{b}} \middle| \frac{\boldsymbol{a} + \boldsymbol{b} + \boldsymbol{c}}{2} \right) = \left( 1 \middle| \frac{\boldsymbol{b}}{4} \right)\left( m_{\boldsymbol{b}} \middle| \frac{\boldsymbol{a} + \boldsymbol{c}}{2} \right)\left( 1 \middle| \frac{- \boldsymbol{b}}{4} \right) \nonumber \]
• a two-fold screw axis \(2_{1}\) parallel to \(\boldsymbol{c}\) and intersecting the point \((¼,¼,0)\). \[\left( 1 \middle| \frac{\boldsymbol{a} + \boldsymbol{b} + \boldsymbol{c}}{2} \right)\left( 2_{\boldsymbol{c}} \middle| \mathbf{0} \right) = \left( 2_{\boldsymbol{c}} \middle| \frac{\boldsymbol{a} + \boldsymbol{b} + \boldsymbol{c}}{2} \right) = \left( 1 \middle| \frac{\boldsymbol{a} + \boldsymbol{b}}{4} \right)\left( 2_{\boldsymbol{c}} \middle| \frac{\boldsymbol{c}}{2} \right)\left( 1 \middle| \frac{\mathbf{-}\left( \boldsymbol{a} + \boldsymbol{b} \right)}{4} \right) \nonumber \]

Therefore, lattice centering generates glide reflections and screw rotations to symmorphic space groups. In \(Imm2\), there are diagonal glide reflections perpendicular to \(\boldsymbol{a}\) and \(\boldsymbol{b}\), as well as two-fold screw axes along \(\boldsymbol{c}\). Therefore, another possible symbol for this space group could be \(Inn2_{1}\). However, there is a hierarchy of symbolism for the International notation: straightforward rotations and reflections take precedent over any screw rotations and glide reflections, respectively.

Example \(\PageIndex{2}\)

\(\ce{NaNO2}\) forms a crystalline structure in the space group \(Imm2\). The accompanying figure shows the outline of a unit cell with Na atoms in gray, N atoms in blue, O atoms in red, two sets of perpendicular reflections highlighted in gray, and two sets of perpendicular diagonal glide planes highlighted in red. The lattice constants are \(a = 3.512\ \mathring{\mathrm{A}},\ b = 6.148\ \mathring{\mathrm{A}},\ c = 5.17\ \mathring{\mathrm{A}}\). All atoms lie on \(m_{a}\) reflection planes as Na⁺ cations and NO₂^– anions.

Figure 3.22: Projection of \(\ce{NaNO2}\) structure along c-axis. Na atoms in silver; N atoms in blue; O atoms in red. Reflection planes perpendicular to a shown in gray and diagonal glide planes perpendicular to b shown in light red.