Symmetry. Motif: the fundamental part of a symmetric design that, when repeated, creates the whole pattern

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1 Symmetry Motif: the fundamental part of a symmetric design that, when repeated, creates the whole pattern Operation: some act that reproduces the motif to create the pattern Element: an operation located at a particular point in space

2 CRYSTALLOGRAPHY FOR MINERALOGIST External Shape of Crystals reflects Internal Structure External Shape is best described by Symmetry Symmetry Repetitive arrangement of features (faces, corners and edges) of a crystal around imaginary lines, points or planes Reflects internal ordering of atoms in the mineral structure

3 Symmetry Elements Rotation Operation Two-fold rotation = 360 o /2 rotation to reproduce a motif in a symmetrical pattern 6 6 Motif 6 Element first operation step = the symbol for a two-fold rotation second operation step 6

4 Reflection (m) Reflection across a mirror plane reproduces a motif m = symbol for a mirror plane m

5 Rotational Symmetry Rotation of x degrees with respect to a line called a rotation axis leaves the image or shape unchanged If an object looks the same after rotation of 360o/n, that object is said to have n-fold rotational symmetry or an n-fold axis Called n-fold, because it takes n rotations to return to its original position Only certain angles ( -folds ) of rotational symmetry are possible in minerals Rotational Symmetry in Minerals Name Short-hand Angle Symbol

6 5-fold, 7-fold and other symmetries are not possible because one cannot fill space with 5 - sided OR 7- sided

7 Inversion (i) For every point or face on one side of the center of symmetry, there is similar point or face at an equal distance on the opposite side of the center 6 Inversion centre inversion through a center to reproduce a motif in a symmetrical pattern i = symbol for an inversion center 6

8 The Cube and Octahedron are simple, common Isometric Forms

9 2-D Symmetry We now have 6 unique 2-D symmetry operations: m Rotations are congruent operations reproductions are identical Inversion and reflection are enantiomorphic operations reproductions are opposite-handed

10 2-D Symmetry Combinations of symmetry elements are also possible To create a complete analysis of symmetry about a point in space, we must try all possible combinations of these symmetry elements In the interest of clarity and ease of illustration, See more 2-D examples

11 2-fold rotation Step -1: reflect Step -3: Reflect & rotate Step -4: Mirror plane Second m required The result is Point Group 2mm

12 4-fold rotation with m 6- fold rotation with m The result is Point Group 4mm The result is Point Group 6mm 3-fold rotation axis with a mirror creates point group 3m & 6 fold rotation axis with mirror creates point group 6mm

13 3-D Symmetry Rotoinversion a) 1-fold rotoinversion ( 1 ) Step 1: rotate 360 o /1 Step 2: invert This is the same as i, so not a new operation

14 2-fold rotoinversion ( 2 ) Step 1: 2-fold 360/2 Step 2: invert m The result: This is the same as m, so not a new operation

15 c. 3-fold rotoinversion ( 3 ) = 3fold + i 1 Step 1: rotate 360 o /3 1 2 Step 2: invert through center Completion of the sequence

16 4-fold rotoinversion ( 4 ) Rotate 360/4 Invert

17 Rotoinversion 4-fold operation ( 4 ) 4-fold 4-fold rotoinversion ( 4 ) This is also a unique operation Note: 2 faces (or pairs of faces) on top and 2 faces on the bottom off-set by 90 o 4

18 Rotoinversion ( 6 ) This is the same as a 3-fold rotation axis perpendicular to a mirror plane Rotate 360 o /6 = 60 o

19 Types of symmetry possible in Minerals 1, 2, 3, 4, 6 : proper rotations m : mirror planes 1 or i : center of symmetry or inversion 3 : bar 3 rotoinversion 4 : bar 4 rotoinversion 6 : bar 6 rotoinversion PLUS OTHER COMBINATION OF ROTATION, MIRROR These can be combined in 32 ways to make crystal shapes

20 The 32 3-D Point Groups Every 3-D pattern must conform to one of them. This includes every crystal, and every point within a crystal Increasing Rotational Symmetry Rotation axis only Rotoinversion axis only 1 (= i ) 2 (= m) (= 3/m) Combination of rotation axes One rotation axis mirror 2/m 3/m (= 6) 4/m 6/m One rotation axis mirror 2mm 3m 4mm 6mm Rotoinversion with rotation and mirror 3 2/m 4 2/m 6 2/m Three rotation axes and mirrors 2/m 2/m 2/m 4/m 2/m 2/m 6/m 2/m 2/m Additional Isometric patterns /m 3 2/m 2/m 3 43m

21 32 3-D Point Groups Regrouped by Crystal System (H & M symbol) Crystal System No Center Center Triclinic 1 1 Monoclinic 2, 2 (= m) 2/m Orthorhombic 222, 2mm 2/m 2/m 2/m Tetragonal 4, 4, 422, 4mm, 42m 4/m, 4/m 2/m 2/m Hexagonal 3, 32, 3m 3, 3 2/m 6, 6, 622, 6mm, 62m 6/m, 6/m 2/m 2/m Isometric 23, 432, 43m 2/m 3, 4/m 3 2/m

22 3-D Symmetry The 32 3-D Point Groups After Bloss, Crystallography and Crystal Chemistry. MSA

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24 Crystal Axes +c +a β γ α +b Axial convention: right-hand rule

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29 FORMS The term form is used to indicate general outward appearance In Crystallography external shape is denoted by the word habit, where as form is used in a special and restricted sense. Thus forms consists of group of crystal faces, all of which have the same relation to element of symmetry and display the same chemical and physical properties The number of faces that belongs to a form is determined by the symmetry of the crystal class. Miller Indices enclosed in parenthesis as (hkl) or (010) indicate crystal face Miller Indices enclosed in braces as {hkl} or {010} indicate form symbols In each crystal there is form, faces of which intersect all the crystallographic axes at different lengths general form {hkl}. All other forms are called as special forms The concept of a general form can also be related to the symmetry elements of a specific crystal class. An (hkl) face will not be parallel or perpendicular to a single crystal symmetry elements regardless of the crystal class, where as special form consists of faces that are parallel or perpendicular any Symmetry elements in the crystal class. Nomenclature of crystallographic forms was initially proposed by Groth, 1895 and later modified by A. F. Rogers in 1935 It recognizes 48 forms of which 32 are the general forms of 32 crystal classes. 10 are special, closed forms of isometric system and 6 special open forms (prisms- hex and tetragonal) Different scheme : Fedrov Institute of Leningrad, 1925 : 47 instead of 48 forms (Dihedrons)

30 NON ISOMETRIC FORMS

31 NON - ISOMETRIC FORMS ISOMETRIC FORMS MALFORMATION OF CRYSTAL FACES ISOMETRIC FORMS

32 Pedion (Monohedron) : A single face comprising form Pinacoid (Parallehedron) : An open form made up of two parallel faces Dome (Dihedron) : Two non parallel faces symmetric w.r.t a mirror plane (m). Sphenoid (Dihedron) : Two non parallel faces symmetric w.r.t a 2-fold rotation axis Prism : An open form composed of 3,4,6,8 or 12 faces, all of which are parallel to the same axis Pyramid : An open form composed of 3,4,6,8 or 12 non parallel faces that meet at a point Dipyramid : A closed form having 6,8,12,16 or 24 faces. Trapezohedron : A closed form that has 6,8 or 12 faces in all, with 3,4 or 6 upper faces offset with 3,4 or 6 faces lower faces. These faces are the result of 3,4 or 6 fold axis combined with perpendicular 2-fold axes. There is Isometric trapezohedron (tetragon-trisoctahedron) 24 face form Scalenohedron : A closed form with 8 or 12 faces grouped in symmetrical pairs. In the tetragonal scalenohedron (rhombic scalenohedron) pairs of upper faces are related by an axis of 4-fold rotoinverstion to pairs of lower faces. The 12 faces of hexagonal scalenohedron display three pairs of upper faces and three pairs of lower faces in alternating positions. The pairs are related by the centre of symmetry, coexist with a 3-fold axis of rotoinverstion. Rhombohedron : A closed form composed of six faces of which three faces at the top alternates with three faces at the bottom, the two sets of faces being offset by 60 o. Only seen in point groups Disphenoid : A closed form consisting of two upper faces that alternate with two lower faces, offset by 90 o.

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