Crystal Structure and Crystallography of Materials Chapter 10: Space Group 1

Space Groups: Point groups: Combination of rotation and reflection (inversion) with perpendicular translation Space groups: Combination of rotation and reflection (inversion) with parallel translation Rotation and parallel translation – screw operation Reflection and parallel translation – glide operation

Combination of a rotation and a translation: Remember: Combination of a rotation and a perpendicular transition. A’ A B P2 P1 P3 3 1 2 α Theorem 1. A rotation about an axis A through angle α, followed by a transition T, perpendicular to the axis, is equivalent to a rotation through the same angle α, in the same sense, but about an axis B located on the bisector of AA’ and at a distance (T/2) cot (α/2) from AA’ Fig.2. Combination of a rotation, Aα, with a translation t┴

Combination of a rotation and a parallel translation: Q U α V T11 PV : Aα VQ : T11 PV · VQ = Aα · T11 = PQ (screw rotation) Here, the rotation and translation are permutable operations ∴ PV : VQ = PU : UQ P Theorem 2. A rotation about an axis, and a translation parallel to an axis, are permutable operations, and the combination is equivalent to a screw motion. Aα · T11 = T11 · Aα = Aα ,T11

Bα,T11 Aα’ Aα Combination of a rotation and a general translation: Any general translation can be resolved into its components perpendicular and parallel to a rotation axis. Bα,T11 Aα’ Aα When the repetition scheme contains a rotation and a general translation, that repetition scheme also contains a screw motion.

Importance of Screw Axis: To every point group appearance, there are several possible space patterns Angular relations for each point group axis carry over into the corresponding space groups. → Thus, the angular relationships between different point group axes are maintained in the corresponding space groups. Screw axis and pure rotation axis of the same order n, repeat a translation in the same way. n = 1, 2, 3, 4, 6

Possible values of screw operations: The most general requirement on the translation, or pitch, component τ, of a screw must be as follows: After n screw operations have been concluded, the accumulation of rotations (each through angle α) has reached 2π, and the accumulated pitch intervals, namely nτ, has attained some translation of the lattice in that direction, i.e., mt. In short, nτ = mt (m, n = integer) So that the possible pitch values are since every feature of the crystal is repeated by the translation of the lattice, anyway.

The permissible screw axis and their designations: The screws 31 and 32, 41 and 43, 61 and 65, 62 and 64 are otherwise the same except that they are right-handed and left-handed.

Derivation of the parallel-axial space groups: Principles of derivation: Deriving the space groups correspond to those point groups in which the only symmetry element is a single rotation axis. -> 20 of these space groups -> characterized by sets of parallel axes called parallel-axial space groups To derive the space groups corresponding with the point groups 1, 2, 3, 4, and 6, each of the isogonal screws for a specific value of n must be combined, in turn, with the translations of each of the lattices permitted to that value of n. * Primitive lattice : primitive translations corresponding only to vectors between points at the cell vertices. * Nonprimitive lattice : in addition, translations corresponding to vectors from points at the cell vertices to one or more points within the cell.

Derivation of the parallel-axial space groups: Principles of derivation: Deriving the space groups correspond to those point groups in which the only symmetry element is a single rotation axis. -> 20 of these space groups -> characterized by sets of parallel axes called parallel-axial space groups To derive the space groups corresponding with the point groups 1, 2, 3, 4, and 6, each of the isogonal screws for a specific value of n must be combined, in turn, with the translations of each of the lattices permitted to that value of n. * Primitive lattice : primitive translations corresponding only to vectors between points at the cell vertices. * Nonprimitive lattice : in addition, translations corresponding to vectors from points at the cell vertices to one or more points within the cell.

(1) Space groups isogonal with point group 1: Derivation of the parallel-axial space groups: To see how these translations combine with the general screw operation, Aα,τ · T = Bα,τ+T11 New screw operation arising at location B with a translation component increased by an amount equal to the parallel component, T11, of the translation translation screw operation (1) Space groups isogonal with point group 1: P1: P1 space group

(2) Space groups isogonal with point group 2: Derivation of the parallel-axial space groups: (2) Space groups isogonal with point group 2: Obtained by combining one of the axes 2 or 21 with the several nonequivalent translations of a cell of either a P or an I lattice. First, consider combining either 2 or 21 with the translations of P. A B C D T’” T” T’ The rotation of a 2-fold axis at A, combined with the translations, because 2-fold rotations to arise at B,C,D. note

(2) Space groups isogonal with point group 2: Derivation of the parallel-axial space groups: (2) Space groups isogonal with point group 2: Now consider the translation components of the screws. → Every translation of the cell listed for 2p in Table 4 has a C component of either zero or unity Thus all axes arising at the points B, C, and D have the same translation component as the axis placed at A → 2 and 21. P2 P21

(2) Space groups isogonal with point group 2: Derivation of the parallel-axial space groups: (2) Space groups isogonal with point group 2: Now consider the translation components of the screws. → Every translation of the cell listed for 2p in Table 4 has a C component of either zero or unity Thus all axes arising at the points B, C, and D have the same translation component as the axis placed at A → 2 and 21. P2 P21

Derivation of the parallel-axial space groups: Next, consider the combination of I with 2, designated I2, and the combination of I with 21, designated I21. All the translations along the cell edges and face diagonals of I are the same as those of P. On the other hand, the I lattice has cell translations along the cell diagonal with components along C of C/2 which results in Note that I2 = I21

→ therefore, 3P, 3P1 and 3P2 exists. Space groups isogonal with point group 3: Obtained by combining 3, 31, or 32 with the translations of lattice P or R. lattice P (consider combining the axes with P) → all the translations of P have components on C of zero or unity. Since locations B’ and C”, are identical, and B’’ and C are the equivalent, it is evident that the operations 1, B2π/3,τ, and B4π/3,τ exists at B’ and C’. → meaning that, if a screw is placed at A, the translations of the lattice require identical screws at B’ and C’ → therefore, 3P, 3P1 and 3P2 exists.

Space groups isogonal with point group 3:

where, D’ and E” are identical, and D” and E’ are equivalent. Space groups isogonal with point group 3: 3R: translations are the same as those P with the addition of the two translations T’” and T”” A E’ D’ T”” D” E” 2/3 T’” 1/3 where, D’ and E” are identical, and D” and E’ are equivalent.

Space groups isogonal with point group 3: R3, R31, R32 define the same space group ⇒ R3 The full cell outlines a hexagonal cell; broken lines outline a rhombohedral cell. Space groups with point group 3: P3 P31 P32 R3

Space groups isogonal with point group 4: Can be found by combining one of the axes 4, 41, 42, or 43 with the translations of lattice P or I. Powers of Bπ/2,τ C’ A B’” B” B’ C” T” T’ C’”

Designation of space group Space groups isogonal with point group 4: Where, locations B’, C”, and B’” are equivalent (identical) → to have the powers of the operations of the screw Bπ/2,τ At point B”, the operations of the screw Bπ,2τ appears. Screw placed at A τ for screw at A Components of screws Screws at Designation of space group B’π/2,τ B”π,2τ B’ B” 4 π/2, 0 π, 0 2 P4 41 c/4 π/2, c/4 π, c/2 21 P41 42 c/2 π/2, c/2 π, c P42 43 3c/4 π/2, 3c/4 π, 3c/2 P43 Results of 4, 41, 42, or 43 with P

Space groups isogonal with point group 4:

Combinations of 4-fold screws with the nonequivalent translations of lattice 4I: Necessary to derive further operations at D’, D”, and D’” due to T’” → c component is c/2. D’ A B” T’” D” D’” 1/2 T’

Designation of space group Combinations of 4-fold screws with the nonequivalent translations of lattice 4I: Screw placed at A τ for screw at A Components of screws Screws at Designation of space group B”π/2,τ D”π,2τ+c/2 B” D” 4 π/2, 0 π, c/2 42 2 I4 41 c/4 π/2, c/4 π, c 43 21 I41 c/2 π/2, c/2 π, 3c/2 I42=I4 3c/4 π/2, 3c/4 π, 2c P43=I41

Space groups isogonal with point group 6: Obtained by combining one of the axis 6, 61, 62, 63, 64, or 65 with the one possible lattice P. A B’’’’’ B”” B’” B” B’ T The nonequivalent translations in the cell project as T and their components parallel to the axis A are zero or unity.

Designation of space group Space groups isogonal with point group 6: Screw placed at A τ for screw at A Components of screws Screws at Designation of space group B”2π/3,2τ B’”π,3τ B’ B’” 6 2π/3, 0 π, 0 3 2 P6 61 c/6 2π/3, c/3 π, c/2 31 21 P61 62 c/3 2π/3, 2c/3 π, c 32 P62 63 c/2 2π/3, c π, 3c/2 P63 64 2c/3 2π/3, 4c/3 π, 2c P64 65 5c/6 2π/3, 5c/3 π, 5c/2 P65

Space groups isogonal with point group 6:

Space groups isogonal with point group 6: