# watkins/fronczek - rotational symmetry 1 symmetry rotational symmetry and its graphic representation

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Watkins/Fronczek - Rotational Symmetry 1 Symmetry Rotational Symmetry and its Graphic Representation Slide 2 Watkins/Fronczek - Rotational Symmetry 2 Rotational Symmetry symmetric A pattern is symmetric if a single motif is repeated in space. A wheel is a repeating pattern of spokes; the motif is one spoke. symmetry operation Each ccw rotation through angle is a symmetry operation which produces similarity. For n equally spaced spokes, = 360 o /n is the repetition angle or "throw" of the rotation axis. 4 is the "fold" of this rotation axis. It is also called the order of the axis. 90 o 4-fold axis 22 4 Slide 3 Watkins/Fronczek - Rotational Symmetry 3 chiral The motif is the smallest part of a symmetric pattern, and can be any asymmetric chiral* object. Rotational Symmetry To produce a rotationally symmetric pattern, place the same motif on each spoke. proper rotation This pattern is produced by a proper rotation because it is a real rotation which produces similarity in the pattern. *not superimposable on its mirror image, like a right hand. line The proper rotation operator is a geometric line. 1 2 3 0,4 Slide 4 Watkins/Fronczek - Rotational Symmetry 4 One of the most important skills a student of crystallography must develop is the ability to discover the symmetry of a pattern. He or she must be able to Rotational Symmetry 1.Locate the motif; 2.name all symmetry operations which produce similarity in the pattern; 3.name the complete set of all such symmetry operations; 4.represent the set of symmetry operations in both diagramatic and mathematical terms. Slide 5 Watkins/Fronczek - Rotational Symmetry 5 Rotational Symmetry Normal crystals contain only five kinds of proper rotational symmetry: One foldIdentity 1. One fold, = 360 o (Identity) Two fold 2. Two fold, = 180 o Three fold 3. Three fold, = 120 o Four fold 4. Four fold, = 90 o Six fold 5. Six fold, = 60 o n C n 12346C 1 C 2 C 3 C 4 C 6 The proper rotation axis is a line and is denoted by the symbol n (Hermann-Maugin) or C n (Schoenflies). Thus, the five proper crystallographic rotation axes are called 1, 2, 3, 4, 6, or C 1, C 2, C 3, C 4, C 6. Note: molecules can have proper rotation axes of any value up to Slide 6 Watkins/Fronczek - Rotational Symmetry 6 Rotational Symmetry There is another, quite different way to produce a rotationally symmetric pattern: put motifs of the opposite hand on every other spoke. The imaginary operation required to do this is: Rotate Rotate the motif through angle Invert Invert the motif through a point on the rotational axis - this changes the chirality of the motif. improper rotation This roto-inversion is called an improper rotation Slide 7 Watkins/Fronczek - Rotational Symmetry 7 Rotational Symmetry Normal crystals contain only five kinds of improper rotational symmetry: One fold 1. One fold, = 360 o Two fold 2. Two fold, = 180 o Three fold 3. Three fold, = 120 o Four fold 4. Four fold, = 90 o Six fold 5. Six fold, = 60 o n 12346 The roto-inversion operator is a line and a point on the line, and is denoted by the symbol n. Thus, the five improper rotation axes are called 1, 2, 3, 4, 6. Slide 8 Watkins/Fronczek - Rotational Symmetry 8 Rotational Symmetry There is another, quite different way to produce a rotationally symmetric pattern: put motifs of the opposite hand on every other spoke. The imaginary operation could also be: Rotate Rotate the motif through angle Reflect Reflect the motif through a plane perpendicular to the rotational axis - this changes the chirality of the motif. This roto-reflection is equivalent to roto-inversion. Slide 9 Watkins/Fronczek - Rotational Symmetry 9 Rotational Symmetry Normal crystals contain only five kinds of improper rotational symmetry: One fold 1. One fold, = 360 o Two fold 2. Two fold, = 180 o Three fold 3. Three fold, = 120 o Four fold 4. Four fold, = 90 o Six fold 5. Six fold, = 60 o n 12346 The roto-reflection operator is a line and a plane perpendicular to the line, and is denoted by the symbol n. Thus, the five improper rotation axes are called 1, 2, 3, 4, 6. ~ ~~~~~ Slide 10 Watkins/Fronczek - Rotational Symmetry 10 Rotational Symmetry Normal crystals contain only five kinds of improper rotational symmetry: One fold 1. One fold, = 360 o Two fold 2. Two fold, = 180 o Three fold 3. Three fold, = 120 o Four fold 4. Four fold, = 90 o Six fold 5. Six fold, = 60 o There is equivalence between n and n ~ 12346 1 2 34 6 ~~~~~ 1 = 2 is inversion or center of symmetry i (Sch) ~ 2 = 1 is mirror m (HM) or (Sch) ~ Slide 11 Watkins/Fronczek - Rotational Symmetry 11 Rotational Symmetry Our patterns, which have 4 and 4 ( 4) symmetry looks like this: Changing the motif would change how the pattern looks, but would not change the symmetry of the pattern. We need a general representation of symmetric patterns which is independent of the motif. _ ~ Slide 12 Watkins/Fronczek - Rotational Symmetry 12 y x z Rotational Symmetry stereographic projection The stereographic projection is a graphic representation of any kind of rotation about a fixed point (point group). 1.Start with a sphere and a 3-d coordinate system. 2.Place the rotation axis (proper or improper) of highest order along z. 3.The point of an improper rotation is at the center of the sphere. 4.Look down the rotation axis at the x-y (equatorial) plane. n or n Slide 13 Watkins/Fronczek - Rotational Symmetry 13 Rotational Symmetry 1.Place the chiral motif on the surface of the sphere. 2.Project each motif onto the equatorial plane. equatorial plane x y Slide 14 Watkins/Fronczek - Rotational Symmetry 14 Rotational Symmetry equatorial plane x y equipoints The symbols are called equipoints because in a symmetry pattern, each one is equivalent to all the others by symmetry. Slide 15 Watkins/Fronczek - Rotational Symmetry 15 Rotational Symmetry equatorial plane x y plane group point group If all equipoints lie on one side of the equatorial plane, the pattern belongs to a 2-D plane group. Otherwise, the pattern belongs to a 3-D point group. Slide 16 Watkins/Fronczek - Rotational Symmetry 16 Rotational Symmetry 11 2 1C 1 1 or C 1 2C 2 2 or C 2 One symmetry operator Two symmetry operations one equipoint Slide 17 Watkins/Fronczek - Rotational Symmetry 17 Rotational Symmetry 1 2 1 C i 1 or C i 2 1 2 m C s = m or C s inversion center horizontal mirror ( h ) Two symops Two equipoints Two symops Two equipoints Slide 18 Watkins/Fronczek - Rotational Symmetry 18 Rotational Symmetry 1 3 2 3C 3 3 or C 3 3 S 6 3 or S 6 1 3 5 4 6 2 3 symops 3 equipoints 6 symops 6 equipoints Slide 19 Watkins/Fronczek - Rotational Symmetry 19 Rotational Symmetry 11 3 4C 4 4 or C 4 2 3 4 4 S 4 4 or S 4 2 4 Slide 20 Watkins/Fronczek - Rotational Symmetry 20 Rotational Symmetry 1 5 3 6C 6 6 or C 6 6 C 3h 6 or C 3h 6 4 2 1 4 5 2 3 6 Slide 21 Watkins/Fronczek - Rotational Symmetry 21 Rotational Symmetry 1, 2, 3, 4, 6 1, 2, 3, 4, 6crystallographic point groupsThe 10 symmetry patterns 1, 2, 3, 4, 6 and 1, 2, 3, 4, 6 are called crystallographic point groups because these are patterns found in crystals. There are 22 other patterns (combinations of proper and improper rotations) also found in crystals, for a total of 32 crystal classes. The general nomenclature for these (and other) patterns is as follows: Slide 22 Watkins/Fronczek - Rotational Symmetry 22 Rotational Symmetry C n one n-fold proper rotation axis only (the primary axis) 1 2 3 4 6 (Hermann-Maugin) C 1 C 2 C 3 C 4 C 6 (Schoenflies) The primary axis is oriented along the z- direction of the stereographic projection. Crystallographic Point Groups Cyclic Groups Non-xtal objects can have n from 1 to . n Slide 23 Watkins/Fronczek - Rotational Symmetry 23 Rotational Symmetry C nh one n-fold proper rotation axis and one horizontal mirror. m 2/m 3/m 4/m 6/m C 1h C s C 2h C 3h C 4h C 6h The proper rotation axis is along z; the improper rotation axis is also along z (the mirror is in the equatorial plane). Crystallographic Point Groups Cyclic Groups n 2 Slide 24 Watkins/Fronczek - Rotational Symmetry 24 Rotational Symmetry C nv one n-fold proper rotation axis and n vertical mirrors. m mm2 m3 mm4 mm6 C 1v C s C 2v C 3v C 4v C 6v The primary axis is along z, the n mirror planes are perpendicular to the equatorial (x-y) plane (2 secondary axes). Crystallographic Point Groups Cyclic Groups n 2 Slide 25 Watkins/Fronczek - Rotational Symmetry 25 Rotational Symmetry D n one n-fold proper rotation axis and n secondary 2-fold axes (dihedral 2-folds). 2 222 23 224 226 D 1 C 2 D 2 D 3 D 4 D 6 The primary rotation axis is along z, with n 2-fold secondary axes in the equatorial plane perpendicular to the primary axis. Crystallographic Point Groups Dihedral Groups n 2 Slide 26 Watkins/Fronczek - Rotational Symmetry 26 D nh one primary n-fold proper rotation axis, n dihedral (secondary) 2-folds, one horizontal mirror, and n vertical mirrors coincident with the secondary 2-folds. Rotational Symmetry Crystallographic Point Groups Dihedral Groups mm2 2 2 2 mmm 2 3 m 2 2 4 mmm 2 2 6 mmm D 1h C 2v D 2h D 3h D 4h D 6h n 2 2 2 Slide 27 Watkins/Fronczek - Rotational Symmetry 27 Rotational Symmetry D nd one primary n-fold proper rotation axis, n dihedral secondary 2-folds, n dihedral mirrors bisecting the secondary 2-folds 2/m 42m 62m D 1d C 2h D 2d D 3d (D 4d D 6d ) Crystallographic Point Groups Dihedral Groups n 2 2 Slide 28 Watkins/Fronczek - Rotational Symmetry 28 S n one n-fold improper (roto-reflection) axis only. m, 1, 3/m, 4, 3 S 1 = C s, S 2 = C i, S 3 = C 3h, S 4, S 6 Rotational Symmetry Crystallographic Point Groups S Groups n Slide 29 Watkins/Fronczek - Rotational Symmetry 29 Illustration of differences between cyclic, dihedral, and S-type groups (from Wikipedia) Slide 30 Watkins/Fronczek - Rotational Symmetry 30 Rotational Symmetry octahedrontetrahedron The five cubic point groups are based on the two geometric solids which can be derived fro

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