Screws Glides and Space Groups Part 1 Dr

Screws, Glides, and Space Groups, Part 1 Dr. Stephen Crabtree November 2, 2018

Derived from Plane Groups p 2 mg Mirror Planes Glide Planes p 4 mm Mirror Planes Glide Planes

Space Groups • There are 230 space groups – We will not be going over many of them – We will be discussing their derivation • The shorthand notation for space groups describes their internal structure, positions of components – Mirror Planes – Rotation Axes – Glide Planes – Screw Axes Combine reflection and translation Combine rotation and translation • The internal symmetry is controlled by the lattice structures and types – Remember that there are only 14 Bravais Lattices

Space Groups

Space Groups

Space Groups

Screw Axes • Rotational symmetry of elements along the axis length to full translation (t) • Notably important handedness and frequency of enantiomorphic figures around an axis • Screw handedness rotates from the top-down Right-handed (cloclwise) Neutral Left-handed (counter-clockwise)

Screw Axes • Rotational symmetry of elements along the vertical axis • Numbering sequence starts with n 0 , just a pure vertical translation • Subscript from 1 to (n-1), from the most right-handed to the most lefthanded 41 42 43

Screw Axes • 41 = ¼ t right for one motif per 90° step in rotation • 42 = 2/4 = ½ t for two motifs per 90° step in rotation • 43 = ¼ t left for one motif per 90° step in rotation • 41 and 43 are enantiomorphic pairs of screw axes 41 42 43

Screw Axes • Rotational symmetry of elements along a vertical axis • Numbering sequence starts with n 0 , just a pure vertical translation, then proceeds from 1 to (n-1), from the most right-handed to the most left-handed • Neutral Screw Axes: – 21 , 42 , 63 • Enantiomorphic Pairs: – – 31 and 32 – R vs. L single motif 41 and 43 – R vs. L single motif 61 and 65 – R vs. L single motif 62 and 64 – R vs. L paired motif Rotoinversion axes are NOT replaceable by screw axes

Glide Planes • Motif repeated by combining a mirror reflection and a translation (m + t) • Related along a glide line in 2 D for ½ translation • Primary glide planes relate to fundamental crystallographic axes – a, b, and c

a, b, and c Glide Planes • Primary glide planes relate to fundamental crystallographic axes – Parallel Two Axes – Intersect One Axis • Axial glide planes restrict the translation vector to ½ unit cell translation along one of the unit cell axes parallel to the reflection plane

a, b, and c Glide Planes Glide Direction in terms of parallel axis Glide Plane defined by Miller Index of axis normal to plane Two a-glide planes – Axis (100) relative to [010] and [001]

a, b, and c Glide Planes Glide Direction in terms of parallel axis Glide Plane defined by Miller Index of axis normal to plane Two b-glide planes – Axis (010) relative to [100] and [001]

a, b, and c Glide Planes Glide Direction in terms of parallel axis Glide Plane defined by Miller Index of axis normal to plane Two c-glide planes – Axis (001) relative to [100] and [010]

n Glide Planes • Diagonal glide planes relate to 2 (or rarely 3) fundamental axes – Parallel One Axis – Intersect Two Axes • Represented as: – a/2 + b/2 – a/2 + c/2 – b/2 + c/2 – a/2 + b/2 + c/2 **

n Glide Planes • Diagonal glide planes relate to 2 (or rarely 3) fundamental axes – Parallel One Axis – Intersect Two Axes • Which n-glide is pictured at left? – Plane [100] – Plane [010] – Plane [001]

d Glide Planes – effectively lines • Diamond glide planes relate to pairs or trios of fundamental axes – Parallel None – Intersect All Three Axes • Step along line is ¼ t: – a/4 + b/4 – a/4 + c/4 – b/4 + c/4 – a/4 + b/4 + c/4 **

d Glide Planes • Diamond glide planes relate to pairs of fundamental axes – Parallel None – Intersect All Three Axes • Which d-glide is pictured at left? – Plane [100] – Plane [010] – Plane [001]

Several Glides Together in a Space Group Often Complemented by Screw Axes

Taking a closer look shows that the symmetry operators in this space group (which are mostly glides and screws) do not intersect at the origin, but are displaced. This is true of many of the nonsymmorphic groups.

Triclinic Space Groups •

Triclinic Space Groups •

Monoclinic Space Groups • Available Lattices: P, C – a ≠ b ≠ c – a = b = 90° ≠ g • Point Groups: 2, m, 2/m • Plane Groups: – Oblique – Rectangular – Centered Rectangular

Monoclinic Class 2 Space Groups • Available Lattices: P, C • Point Group: 2 – Rotation Axis: 2 – Screw Axis: 21 • Derived Groups: – P 2, P 21, C 2, – Not C 21 May not have screw axes in C -lattice because rotation is the B-axis

Monoclinic Class 2 Space Groups • P 2: Simple 2 -fold rotating of elements – Just translation in three dimensions of monoclinic P-lattice – Elements alternately showing front or back faces in each repetition of b-axis View along c-axis View Perpendicular to b-axis

Monoclinic Class 2 Space Groups • P 21: Screw 2 -fold rotating of elements – Screw axis at center of monoclinic P-lattice – Results in near and far repetitions (on face and midway through lattice) View along b-axis View along c-axis View Perpendicular to b-axis

Monoclinic Class 2 Space Groups • C 2: Simple 2 -fold with screw-rotation of elements – 2 -fold along b-axis in Monoclinic B-lattice, screws at half-steps between repetitions of b-axis – Results in near and far repetitions on b-faces View along b-axis View along c-axis View Perpendicular to b-axis

Monoclinic Class 2 Space Groups P 21 C 2

Monoclinic Class 2 Space Groups • Available Lattices: P, C • Point Group: m – Mirror Plane: m – Glide Plane: c • Derived Groups: – Pm, Pc, Cm, Cc

Monoclinic Class m Space Groups • Pm: Simple mirroring of elements – Mirror at center of monoclinic P-lattice – Results in top and bottom reflections View along b-axis View Perpendicular to b-axis

Monoclinic Class m Space Groups • Pc: Glides relating elements from top to bottom – Glide Plane at center of monoclinic P-lattice – Results in near and far reflections (on front and midway through lattice) View along b-axis View Perpendicular to b-axis

Monoclinic Class m Space Groups • Cm: Mirroring of elements in C-centered lattice – Mirror at center of monoclinic B-lattice; Glides at ¼, ¾ planes – Motif repeated relative to face-centered node – Reflections near, far, and intermediate View along b-axis View Perpendicular to b-axis View along c-axis

Monoclinic Class m Space Groups • Cc: Glides of elements relative to face-centered axis – Glide at center of monoclinic C-lattice, and parallel positions – Motif repeated relative to face-centered node – Results in near and far glide motifs , View along b-axis View Perpendicular to b-axis View along c-axis

Monoclinic Class m Space Groups Pm ↑ Pc → Cm ↑ Cc → ,

Monoclinic Class 2/m Space Groups • Available Lattices: P, C • Point Group: 2/m – – Rotation Axis: 2 Screw Axis: 21 Mirror Plane: m Glide Plane: c • Derived Groups: – – – P 2/m, P 21/m C 2/m P 2/c, P 21/c C 2/c Not C 21/m, C 21/c • Primary axis is B-axis, not C

Monoclinic Class 2/m Space Groups • P 2/m: – Mirror at center of monoclinic P-lattice, and top and bottom – 2 -fold axes at corners, midpoints parallel to b-axis View along b-axis View Perpendicular to b-axis

Monoclinic Class 2/m Space Groups • P 21/m: – Mirror at ¼ and ¾ elevations in monoclinic P-lattice – 21 screw axes at corners, midpoints parallel to b-axis View along b-axis View Perpendicular to b-axis

Monoclinic Class 2/m Space Groups • C 2/m: – – Mirror at center of monoclinic B-lattice, and top and bottom Glides at ¼ and ¾ elevations within monoclinic B-lattice 2 -fold axes at corners, midpoints parallel to b-axis 21 screws between 2 -fold axes View along b-axis View along c-axis View Perpendicular to b-axis

Monoclinic Class 2/m Space Groups • P 2/c: – Glide at center of monoclinic P-lattice, and top and bottom – 2 -fold axes at ¼ and ¾ points along edges and mid-line relative to c-axis, parallel to b-axis View along b-axis View Perpendicular to b-axis

Monoclinic Class 2/m Space Groups • P 21/c: – Mirror at ¼ and ¾ elevations in of monoclinic P-lattice – 21 screw axes at ¼ and ¾ points along edges and mid-line relative to c-axis, parallel to b-axis View along b-axis View Perpendicular to b-axis

Monoclinic Class 2/m Space Groups • C 2/c: – n and c Glide Planes at 0, ¼, ½, ¾, and 1 plane positions – Rotation axes at edges and midpoints at ¼ and ¾ – Incidental 21 screw axes between 2 -folds View along b-axis View Perpendicular to b-axis

Monoclinic Class 2/m Space Groups P 21/m P 2/c P 21/c C 2/m C 2/c

Monoclinic Space Groups – Summary • Minimal flexibility due to limited symmetry in system – All lattices are P or B; All planes are mirrors or b-glides – Class 2: P 2, P 21, C 2 • Single rotation or screw axis, in P or B lattice – Class m: Pm, Pc, Cm, Cc • Single mirror or b-glide in P or B lattice – Class 2/m: P 2/m, P 21/m, C 2/m, P 2/c, P 21/c, C 2/c • By far the most symmetry, but still pairing limited options together

Why does this matter? ? ?

Why does this matter? ? ? • The internal structure and arrangement of molecules within minerals must abide by space groups Example: Clinopyroxene (Ca, Fe, Mg)2 Si 2 O 6 Monoclinic C 2/c 2/m Single-chain inosilicate Arrangement of octahedral and tetrahedral sites

Why does this matter? ? ? • Internal mineral structures fit into space groups • Ex: Clinopyroxene – Monoclinic C 2/c – c-face centered lattice – Glide plane ll c, – Alternating 2 and 21 axes along b axis

Why does this matter? ? ? • The internal structure and arrangement of molecules within minerals must abide by space groups Example: Clinopyroxene (Ca, Fe, Mg)2 Si 2 O 6 Monoclinic C 2/c 2/m Single-chain inosilicate Arrangement of octahedral and tetrahedral sites

Orthorhombic 222 Space Groups • Available Lattices: – P, C, F, I • Plane Groups: – Rectangular – Centered Rectangular • Rotation Axes – 2, 21 • Available Space Groups – – P 222, P 2221, P 212121 C 2221, C 222 F 222 I 222, I 212121

Orthorhombic mm 2 Space Groups • Available Lattices: – P, C, A, F, I • Plane Groups: – Rectangular – Centered Rectangular • Rotation Axes: 2, 21 • Reflection Planes: m, c, b, a, n, d • Available Space Groups – Pmm 2, Pnn 2, Pcc 2, Pma 2, Pba 2, Pnc 2, Pmc 21, Pmn 21, Pca 21, Pna 21 – Cmm 2, Cmc 21, Ccc 2 – Amm 2, Abm 21, Ama 2, Aba 2 – Fmm 2, Fdd 2 – Imm 2, Iba 2, Ima 2

Orthorhombic 2/m 2/m Space Groups • Available Lattices: P, C, F, I • Plane Groups: – Rectangular – Centered Rectangular • Rotation Axes: 2, 21 • Reflection Planes: m, c, b, a, n, d • Available Space Groups – Implicit 2 or 21 axes wherever two planes (glide or mirror) intersect at 90° – Pmmm, Pmmn, Pnnm, Pnnn Pmma, Pmna, Pnma, Pnna Pbam, Pban Pbcm, Pbcn, Pbca Pccm, Pccn, Pcca – Cmmm, Cmma, Cmcm Cccm, Ccca, Cmca – Fmmm, Fddd – Immm, Imma, Ibam, Ibca

Orthorhombic Space Groups - Summary • P space lattice allows the most flexibility of elements – May be all identical: • Ex: P 222, P 212121, Pmm 2, Pnn 2, Pcc 2, Pmmm, Pnnn – May be all different if axes and mirrors are both present – Cannot have glides as corresponding axis – no Pa**, P*b*, or P**c • Primary axis symbolized in order – a, b, c • Ex: Pma 2, Pba 2, Pnc 2, Pmc 21, Pmn 21, Pca 21, Pna 21 • Ex: Pmma, Pmna, Pnma, Pnna, Pbam, Pban, Pbcm, Pbcn, Pbca, Pccm, Pccn, Pcca

Orthorhombic Space Groups - Summary • A and C space lattices available, not B-lattice – No n-planes or d-planes – disallowed due to single-face nodes – May be more variable, but within limits: • • • Ex: C 2221, C 222 – only corresponding c-axis is screw Ex: Cmm 2, Cmc 21, Ccc 2 – only screw if different reflection planes Ex: Amm 2, Abm 21, Ama 2, Aba 2 Ex: Cmmm, Cmma, Cmcm Ex: Cccm, Ccca, Cmca

Orthorhombic Space Groups - Summary • F-Lattice – All faces have a central node – enforces identical behavior • Ex: F 222, Fmm 2, Fdd 2, Fmmm, Fddd • I-Lattice – All faces relate to a single node – enforces identical behavior • Ex: I 222, I 212121, Imm 2, Ima 2, Iba 2, Immm, Imma, Ibam, Ibca

Why does this matter? ? ? • The internal structure and arrangement of molecules within minerals must abide by space groups Example: Topaz Al 2(Si. O 4)(F, OH)2 Orthorhombic Pbnm mmm 2/m 2/m Complex Nesosilicate Arrangement of polyhedra sites

Why does this matter? ? ? • Internal mineral structures fit into space groups • Ex: Topaz – Orthorhombic Pbnm – b-glide plane perpendicular to a-axis – n-glide plane perpendicular to b-axis – Simple mirror plane perpendicular to c-axis

Space Groups • There are 230 space groups – Typically most-symmetric have more space groups – More symmetry elements more space groups • The shorthand notation for space groups describes their internal structure, positions of components – Mirror Planes – Rotation Axes – Glide Planes – Screw Axes Combine reflection and translation Combine rotation and translation • The internal symmetry is controlled by the lattice structures and types – Remember that there are only 14 Bravais Lattices