3.7: Volume A of the International Tables of Crystallography

Volume A comprehensively summarizes all structural and diffraction information about space groups that is useful for structure determination and analysis. Each 3-d space group gets at least 2 pages, such as shown for \(P4/mmm\):

Figure 3.36

The first (left) page includes:

• (i) the space group symbol in both International and Schönflies notations;
• (ii) the point group of the space group;
• (iii) the crystal system;
• (iv) a graphical display of symmetry operations using different projections of the unit cell;
• (v)a listing of essential symmetry operations and lattice origin.

The second (right) page includes:

• (vi) a list of generators, which are the fewest symmetry operations that create the entire space group when combined. This list includes three primitive lattice vectors;
• (vii) a list of Wyckoff sites, which are sets of equivalent positions in one unit cell. Each Wyckoff site is designated by its multiplicity (the number of different positions in one unit cell) and a letter, listed in reverse alphabetical order from top-to-bottom (see discussion below);
• (viii) conditions for observable intensities in diffraction experiments;
• (ix) the symmetry of certain 2-d projections (2-d plane groups).

One of these pages also contains a listing of maximal subgroups and minimal supergroups. For \(P4/mmm\), this listing occurs at the bottom of the first (left) page.

The information for the various Wyckoff sites is the most important aspect of this table concerning crystalline structure and stoichiometry. General positions \((x,y,z)\) are listed first. There are no restrictions by symmetry among the coordinates and no symmetry elements intersect these sites. Therefore, general positions have the highest multiplicity, which equals the order of the point group of the space group times the number of lattice points in one unit cell. The remaining sites, called special positions, occur with decreasing multiplicity because they fall on certain symmetry elements, i.e., rotation axes or reflection planes. At the bottom of this list is the Wyckoff “a” site, which typically has the highest point symmetry of any site in the crystal. Since \(P4/mmm\) is symmorphic, the Wyckoff site 1a has point symmetry \(4/mmm = \mathcal{D}_{4h}\). However, for some centrosymmetric nonsymmorphic space groups, the sites with highest point symmetry may not coincide with the inversion centers. These particular groups are listed using two settings for the characteristic at the origin \((0,0,0)\): either highest overall point symmetry or highest centrosymmetric point symmetry. One example is \(Fd\overline{3}m\), the space group of diamond structures. Setting #1 puts lattice points at locations with \(\overline{4}3m = \mathcal{T}_{d}\) point symmetry (order 24, no inversion, Wyckoff site 8a); setting #2 puts lattice points at locations with \(\overline{3}m = \mathcal{D}_{3d}\) point symmetry (order 16, inversion, Wyckoff site 16c).

Figure 3.37: K2PtCl4: 𝑃4/𝑚𝑚𝑚; a = 7.024 Å, c = 4.147 Å Pt (red): 1a (0, 0, 0) K (gray): 2e (0, ½, ½) Cl (green): 4j (0.2323(1), 0.2323(1), 0)

As an example of how to use the International Tables, consider K₂PtCl₄. The information listed below the figure provides all the information that is needed to generate the complete crystal structure. The crystal structure is tetragonal with overall point group symmetry \(4/mmm\ \left( \mathcal{D}_{4h} \right)\), and the unit cell shape is a = b ≠ c with angles α = β = γ = 90°. The asymmetric unit, which is listed below the lattice parameters, includes each inequivalent atom with its Wyckoff site designation and the fractional coordinates of one site. When every space group operation is applied to the asymmetric unit, the entire crystal structure is generated. For K₂PtCl₄, all atoms occupy special positions: all coordinates for the Pt and K atoms are fixed, but the 4j Cl sites \("(x,x,0)"\) allow one free parameter \("x"\), which has an experimental uncertainty. One unit cell of K₂PtCl₄ contains 1 Pt atom at \((0,0,0)\), 2 K atoms at \((0,½,½)\) and \((½,0,½)\), and 4 Cl atoms at \((0.2324,\ 0.2324,\ 0)\), \((–0.2324,–0.2324,\ 0)\), \((–0.2324,\ 0.2324,\ 0)\), and \((0.2324,–0.2324,\ 0)\), a content that is also the empirical formula. The point symmetries of each site are \(4/mmm\ \left( \mathcal{D}_{4h} \right)\) for Pt, \(mmm\ \left( \mathcal{D}_{2h} \right)\) for K, and \(m.2m\ \left( \mathcal{C}_{2v} \right)\) for Cl. Why does the point group symbol for the 4j sites contain the dot? The symmetry elements that intersect the specific site \((x,x,0)\) are \(m_{\boldsymbol{c}}\), \(m_{\boldsymbol{a} - \boldsymbol{b}}\), and \(2_{\boldsymbol{a} + \boldsymbol{b}}\), none of which have axes parallel \(\boldsymbol{a}\) or \(\boldsymbol{b}\). In the space group symbol \(P4/mmm\), the first part \("4/m.."\) has its axis along \(\boldsymbol{c}\), the second part \(".m."\) has its axes along \(\mathbf{a}\) and \(\mathbf{b}\), and the third part \("..m"\) has its axes along \(\left( \boldsymbol{a} + \boldsymbol{b} \right)\) and \(\left( \boldsymbol{a} - \boldsymbol{b} \right)\). Therefore, the dot signifies that no symmetry elements with axes parallel to \(\boldsymbol{a}\) or \(\boldsymbol{b}\) intersect these equivalent sites. Now, to obtain the complete crystal structure, the unit cell parameters provide the length scales and angles so that local structures can be determined. In K₂PtCl₄, each Pt atom is square planar coordinated by 4 Cl atoms at Pt–Cl distances of 2.310(1) Å. Each K atom is surrounded by 8 Cl atoms in a square prism (distorted cube) with a K–Cl distance of 3.240(1) Å.

3-d Space groups

(35) In 3-d, there are 230 space groups, which are best categorized according to the 7 crystal classes and the 32 crystallographic point groups. In the following, space groups are first listed according to their lattice types, and then symmorphic groups precede nonsymmorphic groups.

Triclinic System (2 space groups)

\(1\ \left( \mathcal{C}_{1} \right)\): \(P1\)

\(\overline{1}\ \left( \mathcal{C}_{i} \right)\): \(P\overline{1}\)

• No special symmetry other than possible inversion centers.
• \(P\overline{1}\) is among the most populous space groups for crystals that have been characterized.

Monoclinic System (13 space groups)

\(2\ \left( \mathcal{C}_{2} \right)\): \(P2\), \(P2_{1}\), \(C2\)

\(m\ \left( \mathcal{C}_{s} \right)\): \(Pm\), \(Pc\), \(Cm\), \(Cc\)

\(2/m\ \left( \mathcal{C}_{2h} \right)\): \(P2/m\), \(P2_{1}/m\), \(P2/c\), \(P2_{1}/c\), \(C2/m\), \(C2/c\)

• Standard convention assigns the b-axis parallel to the 2-fold axis.
• \(P2_{1}/c\) is among the most populous space groups for crystals that have been studied because it describes the pattern for effective packing of ellipsoids, which roughly model many molecular structures.

Orthorhombic System (59 space groups)

\(222\ \left( \mathcal{D}_{2} \right)\): \(P222\), \(P222_{1}\), \(P2_{1}2_{1}2\), \(P2_{1}2_{1}2_{1}\), \(C222\), \(C222_{1}\), \(F222\), \(I222\), \(I2_{1}2_{1}2_{1}\)

\(mm2\ \left( \mathcal{C}_{2v} \right)\): \(Pmm2\), \(Pmc2_{1}\), \(Pcc2\), \(Pma2\), \(Pca2_{1}\), \(Pnc2\), \(Pmn2_{1}\), \(Pba2\), \(Pna2_{1}\), \(Pnn2\), \(Cmm2\), \(Cmc2_{1}\), \(Ccc2\), \(Amm2\), \(Aem2\), \(Ama2\), \(Aea2\), \(Fmm2\), \(Fdd2\), \(Imm2\), \(Iba2\), \(Ima2\)

\(mmm\ \left( \mathcal{D}_{2h} \right)\): \(Pmmm\), \(Pnnn\), \(Pccm\), \(Pban\), \(Pmma\), \(Pnna\), \(Pmna\), \(Pcca\), \(Pbam\), \(Pccn\), \(Pbcm\), \(Pnnm\), \(Pmmn\), \(Pbcn\), \(Pbca\), \(Pnma\), \(Cmmm\), \(Cmcm\), \(Cmce\), \(Cccm\), \(Cmme\), \(Ccce\), \(Fmmm\), \(Fddd\), \(Immm\), \(Ibam\), \(Ibca\), \(Imma\)

• Assignments of a-, b-, and c-axes can be arbitrary, which leads to other equivalent space group symbols, e.g.,\(\ Pm2m\) and \(P2mm\) for \(Pmm2\).
• A standard orientation of axes is right-handed, so that the direction of a × b matches the c-direction; a left-handed orientation would have the direction of a × b along the –c-direction.
• The symbol “\(e\)” in some space groups stands for axial glide reflections along two different directions, e.g., \(Ccce\) means that a-glides and b-glides occur with respect to the c-axis.

Tetragonal System (68 space groups)

\(4\ \left( \mathcal{C}_{4} \right)\): \(P4\), \(P4_{1}\), \(P4_{2}\), \(P4_{3}\), \(I4\), \(I4_{1}\)

\(\overline{4}\ \left( \mathcal{S}_{4} \right)\): \(P\overline{4}\), \(I\overline{4}\)

\(4/m\ \left( \mathcal{C}_{4h} \right)\): \(P4/m\), \(P4_{2}/m\), \(P4/n\), \(P4_{2}/n\), \(I4/m\), \(I4_{1}/a\)

\(422\ \left( \mathcal{D}_{4} \right)\): \(P422\), \(P42_{1}2\), \(P4_{1}22\), \(P4_{1}2_{1}2\), \(P4_{2}22\), \(P4_{2}2_{1}2\), \(P4_{3}22\), \(P4_{3}2_{1}2\), \(I422\), \(I4_{1}22\)

\(4mm\ \left( \mathcal{C}_{4v} \right)\): \(P4mm\), \(P4bm\), \(P4_{2}cm\), \(P4_{2}nm\), \(P4cc\), \(P4nc\), \(P4_{2}mc\), \(P4_{2}bc\), \(I4mm\), \(I4cm\), \(I4_{1}md\), \(I4_{1}cd\)

\(\overline{4}2m\ \left( \mathcal{D}_{2d} \right)\): \(P\overline{4}2m\), \(P\overline{4}2c\), \(P\overline{4}2_{1}m\), \(P\overline{4}2_{1}c\), \(P\overline{4}m2\), \(P\overline{4}c2\), \(P\overline{4}b2\), \(P\overline{4}n2\), \(I\overline{4}2m\), \(I\overline{4}2d\), \(I\overline{4}m2\), \(I\overline{4}c2\)

\(4/mmm\ \left( \mathcal{D}_{4h} \right)\): \(P4/mmm\), \(P4/mcc\), \(P4/nbm\), \(P4/nnc\), \(P4/mbm\), \(P4/mnc\), \(P4/nmm\), \(P4/ncc\), \(P4_{2}/mmc\), \(P4_{2}/mcm\), \(P4_{2}/nbc\), \(P4_{2}/nnm\), \(P4_{2}/mbc\), \(P4_{2}/mnm\), \(P4_{2}/nmc\), \(P4_{2}/ncm\), \(I4/mmm\), \(I4/mcm\), \(I4_{1}/amd\), \(I4_{1}/acd\)

• The c-axis as parallel to the 4- or \(\overline{4}\)-axis.
• Space groups with the point group \(\mathcal{D}_{2d}\) have two distinct settings according to the orientations of the 2-fold axes and vertical mirror planes with respect to lattice vectors in the ab-plane.

Trigonal System (25 space groups)

\(3\ \left( \mathcal{C}_{3} \right)\): \(P3\), \(P3_{1}\), \(P3_{2}\), \(R3\)

\(\overline{3}\ \left( \mathcal{S}_{6} \right)\): \(P\overline{3}\), \(R\overline{3}\)

\(32\ \left( \mathcal{D}_{3} \right)\): \(P321\), \(P3_{1}21\), \(P3_{2}21\), \(P312\), \(P3_{1}12\), \(P3_{2}12\), \(R32\)

\(3m\ \left( \mathcal{C}_{3v} \right)\): \(P3m1\), \(P3c1\), \(P31m\), \(P31c\), \(R3m\), \(R3c\)

\(\overline{3}m\ \left( \mathcal{D}_{3d} \right)\): \(P\overline{3}m1\), \(P\overline{3}c1\), \(P\overline{3}1m\), \(P\overline{3}1c\), \(R\overline{3}m\), \(R\overline{3}c\)

• The c-axis as parallel to the 3- or \(\overline{3}\)-axis.
• Space groups with the point groups \(\mathcal{D}_{3d}\) and \(\mathcal{C}_{3v}\) have two distinct settings according to the orientations of the vertical mirror planes with respect to lattice vectors in the ab-plane.

Hexagonal System (27 space groups)

\(6\ \left( \mathcal{C}_{6} \right)\): \(P6\), \(P6_{1}\), \(P6_{2}\), \(P6_{3}\), \(P6_{4}\), \(P6_{5}\)

\(\overline{6}\ \left( \mathcal{C}_{3h} \right)\): \(P\overline{6}\)

\(6/m\ \left( \mathcal{C}_{6h} \right)\): \(P6/m\), \(P6_{3}/m\)

\(622\ \left( \mathcal{D}_{6} \right)\): \(P622\), \(P6_{1}22\), \(P6_{2}22\), \(P6_{3}22\), \(P6_{4}22\), \(P6_{5}22\)

\(\overline{6}m2\ \left( \mathcal{D}_{3h} \right)\): \(P\overline{6}m2\), \(P\overline{6}c2\), \(P\overline{6}2m\), \(P\overline{6}2c\)

\(6/mmm\ \left( \mathcal{D}_{6h} \right)\): \(P6/mmm\), \(P6/mcc\), \(P6_{3}/mcm\), \(P6_{3}/mmc\)

• The c-axis as parallel to the 6- or \(\overline{6}\)-axis.
• Space groups with the point group \(\mathcal{D}_{3h}\) have two distinct settings according to the orientations of the 2-fold axes and the vertical mirror planes with respect to lattice vectors in the ab-plane.
• Hexagonally closed packed (hcp) metals adopt the space group \(P6_{3}/mmc\).

Cubic System (36 space groups)

\(23\ \left( \mathcal{T} \right)\): \(P23\), \(P2_{1}3\), \(F23\), \(I23\), \(I2_{1}3\)

\(m\overline{3}\ \left( \mathcal{T}_{h} \right)\): \(Pm\overline{3}\), \(Pn\overline{3}\), \(Pa\overline{3}\), \(Fm\overline{3}\), \(Fd\overline{3}\), \(Im\overline{3}\), \(Ia\overline{3}\)

\(\overline{4}3m\ \left( \mathcal{T}_{d} \right)\): \(P\overline{4}3m\), \(P\overline{4}3n\), \(F\overline{4}3m\), \(F\overline{4}3c\), \(I\overline{4}3m\), \(I\overline{4}3d\)

\(432\ \left( \mathcal{O} \right)\): \(P432\), \(P4_{1}32\), \(P4_{2}32\), \(P4_{3}32\), \(F432\), \(F4_{1}32\), \(I432\), \(I4_{1}32\)

\(m\overline{3}m\ \left( \mathcal{O}_{h} \right)\): \(Pm\overline{3}m\), \(Pn\overline{3}n\), \(Pm\overline{3}n\), \(Pn\overline{3}m\), \(Fm\overline{3}m\), \(Fm\overline{3}c\), \(Fd\overline{3}m\),

\(Fd\overline{3}c\), \(Im\overline{3}m\), \(Ia\overline{3}d\)

• Cubic closed packed (ccp) metals adopt the space group \(Fm\overline{3}m\).
• Body-centered cubic (bcc) metals adopt the space group \(Im\overline{3}m\).
• The diamond structure (C, Si, Ge, and Sn) adopts the space group \(Fd\overline{3}m\).

The characteristics of the crystallographic point groups influence the characteristics of the corresponding space groups. Among the 230 3-d space groups, 92 are centrosymmetric and 138 are noncentrosymmetric. Among the 138 noncentrosymmetric space groups, there are 65 chiral groups (no improper rotations) and 68 polar groups (no fixed origin). These symmetry characteristics influence the properties of crystals, e.g., chiral crystals can rotate plane polarized light and polar crystals can exhibit spontaneous electrical polarization (ferroelectricity).

Deriving the 230 3-d Space Groups

Space groups are derived systematically from the 32 crystallographic point groups and the corresponding Bravais lattices for each crystal system. The derivations can be accomplished either geometrically, by examining the spatial relationships among symmetry operations, or arithmetically, by considering the allowed combinations of symmetry operations using appropriate lattice algebra. This section illustrates the algebraic approach to obtain the 13 monoclinic space groups.

The general derivation procedure involves determining the rotation-displacement operations of the set of essential symmetry operations for each crystallographic point group that are compatible with translational periodicity of the lattice. The distinct solutions to this problem identify the allowed space groups for each crystallographic point group. This approach can yield redundant solutions arising from specific characteristics of each crystal class. The specific procedure is:

1. Listing the generators \(\left( R \middle| \alpha\boldsymbol{a} + \beta\boldsymbol{b} + \gamma\boldsymbol{c} \right) \equiv \left( R \middle| \alpha\ \beta\ \gamma \right)\) of the group. The short-hand expression is meant to be spatially efficient. The displacement parameters are set by addition modulo 1, which means that \(0 \leq \alpha,\beta,\gamma < 1\). For example, all primitive lattice vectors are \(\left( 1 \middle| 0\ 0\ 0 \right)\); C-centered lattices have lattice vectors \(\left( 1 \middle| 0\ 0\ 0 \right)\) and \(\left( 1 \middle| ½\ ½\ 0 \right)\); and a two-fold screw axis along \(\boldsymbol{b}\) and intersecting the origin is \(\left( 2_{010} \middle| 0\ ½\ 0 \right)\).
2. Identifying the point group presentation from the generators. This step is equivalent to obtaining the multiplication table for the point group of the space group. Each expression involves a product of operations that provide arithmetical constraints on the displacement parameters \(\alpha\ \beta\ \gamma\), whose solutions become the basis for determining the space groups.
3. Applying the presentation equations to each of the allowed lattice types within the crystal class. The solutions of these equations give the allowed values of the displacement parameters \(\alpha\ \beta\ \gamma\) in the generators.
4. Constructing a table that lists all possible solutions. In some crystal classes, every possible solution does not necessarily lead to a distinct space group, especially for lower symmetry crystal classes for which there are equivalent ways of assigning the unit cell parameters.
5. Summarizing the distinct space groups for each crystallographic point group.

The monoclinic crystal class is uniaxial and is generated by 2-fold proper or improper rotations with axes along \(\boldsymbol{b}\). These space groups are determined from 3 point groups and 2 lattice types, primitive and base-centered. For the direction of the 2-fold axes, allowed base-centered lattices are A- and C-types; C-centering is the standard setting. Furthermore, there are 4 equivalent ways to assign the unit cell vectors that maintain a right-handed coordinate system: \(\boldsymbol{abc}\), \(\overline{\boldsymbol{c}}\boldsymbol{ba}\), \(\left( \boldsymbol{a} + \boldsymbol{c} \right)\boldsymbol{bc}\), and \(\overline{\boldsymbol{c}}\boldsymbol{b}(\boldsymbol{a} + \boldsymbol{c})\).

Point Group \(\mathcal{C}_{\mathbf{2}}\mathbf{= 2}\)

This group has order 2, containing a 2-fold rotation axis.

• Generator: \(\left( 2_{010} \middle| 0\ \beta\ 0 \right)\): set the \(2_{010}\) axes to intersect lattice points.
• Presentation: \(\left( 2_{010} \middle| 0\ \beta\ 0 \right)^{2} = \left( 1 \middle| 0\ 2\beta\ 0 \right)\) = lattice translation: sets allowed values for \(\beta\).
• Solutions: Determine possible values of one parameter: \(\beta\).

P-lattice: \(\left( 1 \middle| 0\ 2\beta\ 0 \right) = \left( 1 \middle| 0\ 0\ 0 \right)\). \(\beta = 0,½\).

C-lattice: \(\left( 1 \middle| 0\ 2\beta\ 0 \right) = \left( 1 \middle| 0\ 0\ 0 \right)\). \(\beta = 0,½\).

Each lattice type has two possible solutions:

@ >p(- 8) * >p(- 8) * >p(- 8) * >p(- 8) * >p(- 8) * @

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\[\mathbf{\beta} \nonumber \]

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\[\mathbf{\beta} \nonumber \]

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() & Primitive & \(\left( 2_{010} \middle| 0\ 0\ 0 \right)\) & \(2\)-fold \((0\ y\ 0)\) & \(P2\)
& C-centered & \(\left( 2_{010} \middle| 0\ 0\ 0 \right)\)

\(\left( 2_{010} \middle| ½\ ½\ 0 \right)\) & \(2\)-fold \((0\ y\ 0)\)

\(2_{1}\)-screw \((¼\ y\ 0)\) & \(C2\)
& Primitive & \(\left( 2_{010} \middle| 0\ ½\ 0 \right)\) & \(2_{1}\)-screw \((0\ y\ 0)\) & \(P2_{1}\)
& C-centered & \(\left( 2_{010} \middle| 0\ ½\ 0 \right)\)

\(\left( 2_{010} \middle| ½\ 0\ 0 \right)\) & \(2_{1}\)-screw \((0\ y\ 0)\)

\(2\)-fold \((¼\ y\ 0)\) & \(C2_{1} = C2\)
()

C-centering generates \(2_{1}\)-screw axes from \(2\)-fold rotation axes along \(\mathbf{b}\) and vice versa. Therefore, the two possible solutions for the C-centered monoclinic lattice give the same space groups and differ only by the positions of the two types of rotation axes.

There are 3 distinct space groups: \(P2,\ P2_{1},\) and \(C2\).

Point Group \(\mathcal{C}_{\boldsymbol{s}}\boldsymbol{= m}\)

This group has order 2, containing a reflection plane.

• Generator: \(\left( m_{010} \middle| \alpha\ 0\ \gamma \right)\): set the \(m_{010}\) planes to intersect lattice points.
• Presentation: \(\left( m_{010} \middle| \alpha\ 0\ \gamma \right)^{2} = \left( 1 \middle| 2\alpha\ 0\ 2\gamma \right)\) = lattice translation: sets allowed values for \(\alpha,\ \gamma\).
• Solutions: Determine possible values of two parameters: \(\alpha\) and \(\gamma\).

P-lattice: \(\left( 1 \middle| 2\alpha\ 0\ 2\gamma \right) = \left( 1 \middle| 0\ 0\ 0 \right)\). \(\alpha = 0,½;\ \gamma = 0,½\).

C-lattice: \(\left( 1 \middle| 2\alpha\ 0\ 2\gamma \right) = \left( 1 \middle| 0\ 0\ 0 \right)\). \(\alpha = 0,½;\ \gamma = 0,½\).

Each lattice type has four possible solutions because there is no relationship between \(\alpha\) and \(\gamma\).

@ >p(- 8) * >p(- 8) * >p(- 8) * >p(- 8) * >p(- 8) * @

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\[\mathbf{\alpha\ \ \gamma} \nonumber \]

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Lattice Type

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\[\mathbf{\alpha\ \ \gamma} \nonumber \]

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Space Group

() & Primitive & \(\left( m_{010} \middle| 0\ 0\ 0 \right)\) & mirror \((y = 0)\) & \(Pm\)
& C-centered & \(\left( m_{010} \middle| 0\ 0\ 0 \right)\)

\(\left( m_{010} \middle| ½\ ½\ 0 \right)\) & mirror \((y = 0)\)

\(a\)-glide \((y = ¼)\) & \(Cm\)
& Primitive & \(\left( m_{010} \middle| 0\ 0\ ½ \right)\) & \(c\)-glide \((y = 0)\) & \(Pc\)
& C-centered & \(\left( m_{010} \middle| 0\ 0\ ½ \right)\)

\(\left( m_{010} \middle| ½\ ½\ ½ \right)\) & \(c\)-glide \((y = 0)\)

\(n\)-glide \((y = ¼)\) & \(Cc\)
& Primitive & \(\left( m_{010} \middle| ½\ 0\ 0 \right)\) & \(a\)-glide \((y = 0)\) & \(Pa = Pc\) (lattice vectors: \(\boldsymbol{c}\overline{\boldsymbol{b}}\boldsymbol{a}\))
& C-centered & \(\left( m_{010} \middle| ½\ 0\ 0 \right)\)

\(\left( m_{010} \middle| 0\ ½\ 0 \right)\) & \(a\)-glide \((y = 0)\)

mirror \((y = ¼)\) & \(Ca = Cm\)
& Primitive & \(\left( m_{010} \middle| ½\ 0\ ½ \right)\) & \(n\)-glide \((y = 0)\) & \(Pn = Pc\) (lattice vectors: \(\boldsymbol{c}\overline{\boldsymbol{b}}\left( \boldsymbol{a} + \boldsymbol{c} \right)\))
& C-centered & \(\left( m_{010} \middle| ½\ 0\ ½ \right)\)

\(\left( m_{010} \middle| 0\ ½\ ½ \right)\) & \(n\)-glide \((y = 0)\)

\(c\)-glide \((y = ¼)\) & \(Cn = Cc\)
()

For primitive lattices, two of four possible solutions are redundant arising from the different equivalent assignments of unit cell settings (lattice vectors).

C-centering generates \(a\)-glides and \(n\)-glides, respectively, from mirrors and \(c\)-glides, and vice versa. Therefore, two of the four possible solutions are redundant and differ only by the positions of the two types of reflection planes.

There are 4 distinct space groups: \(Pm,\ Pc,Cm\) and \(Cc\).

Point Group \(\mathcal{C}_{\mathbf{2}\boldsymbol{h}}\boldsymbol{= 2/m}\): The holohedral and centrosymmetric group has order 4 with 2-fold rotation axes and orthogonal reflection planes.

• Generators: \(\left( 2_{010} \middle| 0\ \beta_{1}\ 0 \right)\): set the \(2_{010}\) axes to intersect lattice points; \(\left( m_{010} \middle| \alpha_{2}\ 0\ \gamma_{2} \right)\): set the \(m_{010}\) planes to intersect lattice points.

• Presentation: \(\left( 2_{010} \middle| 0\ \beta_{1}\ 0 \right)^{2} = \left( 1 \middle| 0\ 2\beta_{1}\ 0 \right)\) = lattice translation: sets allowed values for β₁.
  \(\left( m_{010} \middle| \alpha_{2}\ 0\ \gamma_{2} \right)^{2} = \left( 1 \middle| 2\alpha_{2}\ 0\ 2\gamma_{2} \right)\) = lattice translation: sets allowed values for α₂, γ₂.

  \(\left( \left( m_{010} \middle| \alpha_{2}\ 0\ \gamma_{2} \right)\left( 2_{010} \middle| 0\ \beta_{1}\ 0 \right) \right)^{2} = \left( \overline{1} \middle| \alpha_{2}\ \left( - \beta_{1} \right)\ \gamma_{2} \right)^{2} = \left( 1 \middle| 0\ 0\ 0 \right)\): no other constraints.

• Solutions: Determine possible values of three parameters: β₁, α₂, γ₂.

  P-lattice: \(\left( 1 \middle| 0\ 2\beta_{1}\ 0 \right) = \left( 1 \middle| 0\ 0\ 0 \right)\); β₁ = 0, ½;

  \(\left( 1 \middle| 2\alpha_{2}\ 0\ 2\gamma_{2} \right) = \left( 1 \middle| 0\ 0\ 0 \right)\). α₂ = 0, ½; γ₂ = 0, ½.

  C-lattice: \(\left( 1 \middle| 0\ 2\beta_{1}\ 0 \right) = \left( 1 \middle| 0\ 0\ 0 \right)\); β₁ = 0, ½;

  \(\left( 1 \middle| 2\alpha_{2}\ 0\ 2\gamma_{2} \right) = \left( 1 \middle| 0\ 0\ 0 \right)\). α₂ = 0, ½; γ₂ = 0, ½.

Each lattice type has eight possible solutions.

@ >p(- 8) * >p(- 8) * >p(- 8) * >p(- 8) * >p(- 8) * @

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\[\mathbf{\beta}_{\mathbf{1}}\mathbf{\ \ }\mathbf{\alpha}_{\mathbf{2}}\mathbf{\ \ }\mathbf{\gamma}_{\mathbf{2}} \nonumber \]

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Space Group

() ()

\[\mathbf{\beta}_{\mathbf{1}}\mathbf{\ \ }\mathbf{\alpha}_{\mathbf{2}}\mathbf{\ \ }\mathbf{\gamma}_{\mathbf{2}} \nonumber \]

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Lattice Type

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Operations

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Space Group

() & Primitive & \(\left( 2_{010} \middle| 0\ 0\ 0 \right)\)

\(\left( m_{010} \middle| 0\ 0\ 0 \right)\) & 2-fold \((0\ y\ 0)\)

mirror \((y = 0)\) & \(P2/m\)
& C-centered & \(\left( 2_{010} \middle| 0\ 0\ 0 \right)\)

\(\left( 2_{010} \middle| ½\ ½\ 0 \right)\)

\(\left( m_{010} \middle| 0\ 0\ 0 \right)\)

\(\left( m_{010} \middle| ½\ ½\ 0 \right)\) & 2-fold \((0\ y\ 0)\)

\(2_{1}\)-screw \((¼\ y\ 0)\)

mirror \((y = 0)\)

\(a\)-glide \((y = ¼)\) & \(C2/m\)
& Primitive & \(\left( 2_{010} \middle| 0\ ½\ 0 \right)\)

\(\left( m_{010} \middle| 0\ 0\ 0 \right)\) & \(2_{1}\)-screw \((0\ y\ 0)\)

mirror \((y = 0)\) & \(P2_{1}/m\)
& C-centered & \(\left( 2_{010} \middle| 0\ ½\ 0 \right)\)

\(\left( 2_{010} \middle| ½\ 0\ 0 \right)\)

\(\left( m_{010} \middle| 0\ 0\ 0 \right)\)

\(\left( m_{010} \middle| ½\ ½\ 0 \right)\) & \(2_{1}\)-screw \((0\ y\ 0)\)

2-fold \((¼\ y\ 0)\)

mirror \((y = 0)\)

\(a\)-glide \((y = ¼)\) & \(C2_{1}/m = C2/m\)
& Primitive & \(\left( 2_{010} \middle| 0\ 0\ 0 \right)\)

\(\left( m_{010} \middle| 0\ 0\ ½ \right)\) & \(2\)-fold \((0\ y\ 0)\)

\(c\)-glide \((y = 0)\) & \(P2/c\)
& C-centered & \(\left( 2_{010} \middle| 0\ 0\ 0 \right)\)

\(\left( 2_{010} \middle| ½\ ½\ 0 \right)\)

\(\left( m_{010} \middle| 0\ 0\ ½ \right)\)

\(\left( m_{010} \middle| ½\ ½\ ½ \right)\) & 2-fold \((0\ y\ 0)\)

\(2_{1}\)-screw \((¼\ y\ 0)\)

\(c\)-glide \((y = 0)\)

\(n\)-glide \((y = ¼)\) & \(C2/c\)
& Primitive & \(\left( 2_{010} \middle| 0\ ½\ 0 \right)\)

\(\left( m_{010} \middle| 0\ 0\ ½ \right)\) & \(2_{1}\)-screw \((0\ y\ 0)\)

\(c\)-glide \((y = 0)\) & \(P2_{1}/c\)
& C-centered & \(\left( 2_{010} \middle| 0\ ½\ 0 \right)\)

\(\left( 2_{010} \middle| ½\ 0\ 0 \right)\)

\(\left( m_{010} \middle| 0\ 0\ ½ \right)\)

\(\left( m_{010} \middle| ½\ ½\ ½ \right)\) & \(2_{1}\)-screw \((0\ y\ 0)\)

2-fold \((¼\ y\ 0)\)

\(c\)-glide \((y = 0)\)

\(n\)-glide \((y = ¼)\) & \(C2_{1}/c = C2/c\)
& Primitive & \(\left( 2_{010} \middle| 0\ 0\ 0 \right)\)

\(\left( m_{010} \middle| ½\ 0\ 0 \right)\) & \(2\)-fold \((0\ y\ 0)\)

\(a\)-glide \((y = 0)\) & \(P2/a = P2/c\)

(lattice vectors: \(\boldsymbol{c}\overline{\boldsymbol{b}}\boldsymbol{a}\))
& C-centered & \(\left( 2_{010} \middle| 0\ 0\ 0 \right)\)

\(\left( 2_{010} \middle| ½\ ½\ 0 \right)\)

\(\left( m_{010} \middle| ½\ 0\ 0 \right)\)

\(\left( m_{010} \middle| 0\ ½\ 0 \right)\) & 2-fold \((0\ y\ 0)\)

\(2_{1}\)-screw \((¼\ y\ 0)\)

\(a\)-glide \((y = 0)\)

mirror \((y = ¼)\) & \(C2/a = C2/m\)
& Primitive & \(\left( 2_{010} \middle| 0\ ½\ 0 \right)\)

\(\left( m_{010} \middle| ½\ 0\ 0 \right)\) & \(2_{1}\)-screw \((0\ y\ 0)\)

\(a\)-glide \((y = 0)\) & \(P2_{1}/a = P2_{1}/c\)

(lattice vectors: \(\boldsymbol{c}\overline{\boldsymbol{b}}\boldsymbol{a}\))
& C-centered & \(\left( 2_{010} \middle| 0\ ½\ 0 \right)\)

\(\left( 2_{010} \middle| ½\ 0\ 0 \right)\)

\(\left( m_{010} \middle| ½\ 0\ 0 \right)\)

\(\left( m_{010} \middle| 0\ ½\ 0 \right)\) & \(2_{1}\)-screw \((0\ y\ 0)\)

2-fold \((¼\ y\ 0)\)

\(a\)-glide \((y = 0)\)

mirror \((y = ¼)\) & \(C2_{1}/a = C2/m\)
& Primitive & \(\left( 2_{010} \middle| 0\ 0\ 0 \right)\)

\(\left( m_{010} \middle| ½\ 0\ ½ \right)\) & \(2\)-fold \((0\ y\ 0)\)

\(n\)-glide \((y = 0)\) & \(P2/n = P2/c\)

(lattice vectors: \(\boldsymbol{c}\overline{\boldsymbol{b}}\left( \boldsymbol{a} + \boldsymbol{c} \right)\))
& C-centered & \(\left( 2_{010} \middle| 0\ 0\ 0 \right)\)

\(\left( 2_{010} \middle| ½\ ½\ 0 \right)\)

\(\left( m_{010} \middle| ½\ 0\ ½ \right)\)

\(\left( m_{010} \middle| 0\ ½\ ½ \right)\) & 2-fold \((0\ y\ 0)\)

\(2_{1}\)-screw \((¼\ y\ 0)\)

\(n\)-glide \((y = 0)\)

\(c\)-glide \((y = ¼)\) & \(C2/n = C2/c\)
& Primitive & \(\left( 2_{010} \middle| 0\ ½\ 0 \right)\)

\(\left( m_{010} \middle| ½\ 0\ ½ \right)\) & \(2_{1}\)-screw \((0\ y\ 0)\)

\(n\)-glide \((y = 0)\) & \(P2_{1}/n = P2_{1}/c\)

(lattice vectors: \(\boldsymbol{c}\overline{\boldsymbol{b}}\left( \boldsymbol{a} + \boldsymbol{c} \right)\))
& C-centered & \(\left( 2_{010} \middle| 0\ ½\ 0 \right)\)

\(\left( 2_{010} \middle| ½\ 0\ 0 \right)\)

\(\left( m_{010} \middle| ½\ 0\ ½ \right)\)

\(\left( m_{010} \middle| 0\ ½\ ½ \right)\) & \(2_{1}\)-screw \((0\ y\ 0)\)

2-fold \((¼\ y\ 0)\)

\(n\)-glide \((y = 0)\)

\(c\)-glide \((y = ¼)\) & \(C2_{1}/n = C2/c\)
()

For primitive lattices, four of the eight possible space groups are redundant solutions arising from the different equivalent assignments of unit cell settings (lattice vectors).

C-centering generates \(a\)-glides and \(n\)-glides, respectively, from mirrors and \(c\)-glides, and vice versa. Therefore, two of the eight possible solutions are redundant and differ only by the positions of the two types of reflection planes.

There are 6 distinct space groups: \(P2/m,\ P2_{1}/m,P2/c,P2_{1}/c,C2/m\) and \(C2/c\).

Summary

There are 13 monoclinic space groups. The arithmetic solution strategy generates other equivalent solutions. For primitive lattices, these other solutions arise because of the different equivalent ways to assign lattice vectors while retaining a right-handed perspective. For C-centered lattices, equivalence arises from the types of rotation-translation operations generated by C-centering.