Mineraloji Lattices, Symmetry and Crystal Systems X-str --- ordered arrangement of atoms or ions Crystal --- homogeneous solid possessing 3-D internal order int. str --- reflected in external form --- regular geometric smooth plane surfaces ordered patterns --- as groups of atoms e.g. cations (Na + , Ca 2+ , Mg 2+ , Fe 3+ , etc.) anionic groups (Cl - , (SiO 4 ) 4- , (CO 3 ) 2- , (PO 4 ) 3- , (OH) - , etc.) molecules (H 2 O) v repeatedly and periodically placed on a lattice LATTICES, SYMMETRY AND CRYSTAL SYSTEMS LATTICES, SYMMETRY AND CRYSTAL SYSTEMS CRYSTALLINE SUBSTANCEperiodic repetitions along vectors OR translations of ions --- achieved in 1-D, 2-D, 3-D v vvv v row-latticesvv v plane-latticesv v space-lattices internal symmetry operations --- not occur as symmetry element TRANSLATION OPERATIONS produced by translation along single vector ions --- repeated with constant distances along a line Row-Lattices aproduced by translations along two vectors net ions --- repeated with constant distances and angles Plane-Lattices 5 unique plane-lattices a b Unit cell Unit cell a b ? Oblique Net a ? b ? ?90 o p2 p2mm Rectangular P Net a ? b ? = 90 o b a ? Rectangular C Net a ? b ? = 90 o p2mm b a Diamond Net a = b ? ? 90 o , 120 o , 60 o a 1 a 2 ? ? Hexagonal Net a 1 = a 2 ? = 60 o p6mm Square Net a 1 = a 2 ? = 90 o ? p4mm a a 1 a 2produced by translations along three vectors ions --- repeated again with constant distances and angles in 3-D 14 unique space-lattices --- as Bravais Lattices compatible with 6 symmetry systems and 32 symmetry classes principally --- five kinds of space lattices --- Bravais lattices 1. Primitive lattice (P): atoms --- at corners; found in all symmetry systems 2. Body centered lattice (I): additional atom at the centre 3. Face centered lattice (F): an atom at centre of all 6 faces 4. Top and bottom face centred lattice (C): only (001) faces --- centered 5. Rhombohedral lattice(R): primitive lattice of rhombohedral 'sub-system' of hex. sys. Space-Latticesproduced by translations along three vectors ions --- repeated again with constant distances and angles in 3-D 14 unique space-lattices --- as Bravais Lattices compatible with 6 symmetry systems and 32 symmetry classes principally --- five kinds of space lattices --- Bravais lattices 1. Primitive lattice (P): atoms --- at corners; found in all symmetry systems 2. Body centered lattice (I): additional atom at the centre 3. Face centered lattice (F): an atom at centre of all 6 faces 4. Top and bottom face centred lattice (C): only (001) faces --- centered 5. Rhombohedral lattice(R): primitive lattice of rhombohedral 'sub-system' of hex. sys. Space-Latticesproduced by translations along three vectors ions --- repeated again with constant distances and angles in 3-D 14 unique space-lattices --- as Bravais Lattices compatible with 6 symmetry systems and 32 symmetry classes principally --- five kinds of space lattices --- Bravais lattices 1. Primitive lattice (P): atoms --- at corners; found in all symmetry systems 2. Body centered lattice (I): additional atom at the centre 3. Face centered lattice (F): an atom at centre of all 6 faces 4. Top and bottom face centred lattice (C): only (001) faces --- centered 5. Rhombohedral lattice(R): primitive lattice of rhombohedral 'sub-system' of hex. sys. Space-Latticesproduced by translations along three vectors ions --- repeated again with constant distances and angles in 3-D 14 unique space-lattices --- as Bravais Lattices compatible with 6 symmetry systems and 32 symmetry classes principally --- five kinds of space lattices --- Bravais lattices 1. Primitive lattice (P): atoms --- at corners; found in all symmetry systems 2. Body centered lattice (I): additional atom at the centre 3. Face centered lattice (F): an atom at centre of all 6 faces 4. Top and bottom face centred lattice (C): only (001) faces --- centered 5. Rhombohedral lattice(R): primitive lattice of rhombohedral 'sub-system' of hex. sys. Space-Latticesproduced by translations along three vectors ions --- repeated again with constant distances and angles in 3-D 14 unique space-lattices --- as Bravais Lattices compatible with 6 symmetry systems and 32 symmetry classes principally --- five kinds of space lattices --- Bravais lattices 1. Primitive lattice (P): atoms --- at corners; found in all symmetry systems 2. Body centered lattice (I): additional atom at the centre 3. Face centered lattice (F): an atom at centre of all 6 faces 4. Top and bottom face centred lattice (C): only (001) faces --- centered 5. Rhombohedral lattice(R): primitive lattice of rhombohedral 'sub-system' of hex. sys. Space-Latticesa b c P Monoclinic ? = ? = 90 ? ?ß a ? b ? c a b c I = C a b P Triclinic ? ? ß ? ? a ? b ? c c c a P Orthorhombic ? = ß = ? = 90 ? a ? b ? c C FI ba 1 c P Tetragonal ? = ß = ? = 90 ? a 1 = a 2 ? c I a 2 a 1 a 3 P Isometric ? = ß = ? = 90 ? a 1 = a 2 = a 3 a 2 F I a 1 c P or C a 2 R Hexagonal Rhombohedral ? = ß = 90 ? ? = 120 ? a 1 = a 2 ? c ? = ß = ? ? 90 ? a 1 = a 2 = a 3CRYSTALLOGRAPHIC AXES AND SYMMETRY SYSTEMS 14 unique space-lattices by 3-D translation operation 6 group of X-systems --- according to Cartesian Principle Axes v intersecting in the center of the space-lattice OR X’s formed from the repetitions of them +c +a +b ? ? ß ß ? ? Axial convention: “right-hand rule” Crystal Axescrystallographic plane each two crystallographic axes generally taken parallel to intersection edges of major crystal faces a-, b-, c-axes, c-axis --- always vertical hex. sys. --- 4 axes a 1 , a 2 , a 3 (horizontal with 120° btw them), c (vertical) length of the axes from origin --- (+ or -) length of crystallographic axes + axial angles ?, ß, ? btw them variable in different symmetry Sym. Svs. Crvs. Axes Axial Angle 1. Cubic a: a: a ?=ß=?=90° (Isometric) 2. Tetragonal a: a: c ?=ß=?=90° 3. Hexagonal a: a: a: c ?=ß=90°, ?=60-120° Rhombohedral a: a: a: c ?=ß=90°, ?=60-l20° 4. Orthorhombic a: b: c ?=ß=?=90° 5. Monoclinic a: b: c ?=?=90°, ß>90° 6. Triclinic a: b: c ??ß???90°Sym. Svs. Crvs. Axes Axial Angle 1. Cubic a: a: a ?=ß=?=90° (Isometric) 2. Tetragonal a: a: c ?=ß=?=90° 3. Hexagonal a: a: a: c ?=ß=90°, ?=60-120° Rhombohedral a: a: a: c ?=ß=90°, ?=60-l20° 4. Orthorhombic a: b: c ?=ß=?=90° 5. Monoclinic a: b: c ?=?=90°, ß>90° 6. Triclinic a: b: c ??ß???90°Sym. Svs. Crvs. Axes Axial Angle 1. Cubic a: a: a ?=ß=?=90° (Isometric) 2. Tetragonal a: a: c ?=ß=?=90° 3. Hexagonal a: a: a: c ?=ß=90°, ?=60-120° Rhombohedral a: a: a: c ?=ß=90°, ?=60-l20° 4. Orthorhombic a: b: c ?=ß=?=90° 5. Monoclinic a: b: c ?=?=90°, ß>90° 6. Triclinic a: b: c ??ß???90°Sym. Svs. Crvs. Axes Axial Angle 1. Cubic a: a: a ?=ß=?=90° (Isometric) 2. Tetragonal a: a: c ?=ß=?=90° 3. Hexagonal a: a: a: c ?=ß=90°, ?=60-120° Rhombohedral a: a: a: c ?=ß=90°, ?=60-l20° (trigonal) 4. Orthorhombic a: b: c ?=ß=?=90° 5. Monoclinic a: b: c ?=?=90°, ß>90° 6. Triclinic a: b: c ??ß???90°Sym. Svs. Crvs. Axes Axial Angle 1. Cubic a: a: a ?=ß=?=90° (Isometric) 2. Tetragonal a: a: c ?=ß=?=90° 3. Hexagonal a: a: a: c ?=ß=90°, ?=60-120° Rhombohedral a: a: a: c ?=ß=90°, ?=60-l20° 4. Orthorhombic a: b: c ?=ß=?=90° 5. Monoclinic a: b: c ?=?=90°, ß>90° 6. Triclinic a: b: c ??ß???90°Sym. Svs. Crvs. Axes Axial Angle 1. Cubic a: a: a ?=ß=?=90° (Isometric) 2. Tetragonal a: a: c ?=ß=?=90° 3. Hexagonal a: a: a: c ?=ß=90°, ?=60-120° Rhombohedral a: a: a: c ?=ß=90°, ?=60-l20° 4. Orthorhombic a: b: c ?=ß=?=90° 5. Monoclinic a: b: c ?=?=90°, ß>90° 6. Triclinic a: b: c ??ß???90°Symmetry operations --- due to --- simple arrangement (repetation in 3D) External X-faces --- internal str OTHER SYMMETRY OPERATIONS AND SYMMETRY ELEMENTSLow of constancy of internal facial (Steno, 1669) angles btw equivalent faces of crystals of the same substance, measured at the same T, are constant e.g. quartz X --- angles btw corresponding faces --- same 120 o 120 o 120 o 120 o 120 o 120 o 120 oX created by repetition of unit cells (Å level) with internal symmetry operations v always involve translation operation internal symmetry operations --- result in external symmetry (sym. elements) v without translations both internal and external symmetry operation lattices internally across a mirror plane (m) also called reflection plane Reflection (Mirror) OperationGlide - internal symmetry operation - combination of reflection and translation - plane --- glide plane Symmetry plane - external symmetry operation - reflection operation - acts as a mirror plane (m) - all corners, edges, and planes --- repeated on both sides of mRotational Operations --- three types SYMMETRY OPERATIONS Rotation --- Internal and External --- rotation of a point or a face around axes v rotation axes --- rotation --- at each-degree turn around an axis --- same point --- repeated in X-str --- a face --- repeated on a X A Symmetrical Pattern A Symmetrical Pattern 6 6 two-fold rotationMotif Element 6 6 = the symbol for a two-fold rotation two-fold rotation6 6 first operation step second operation step two-fold rotation = the symbol for a two-fold rotationtwo-fold rotationtwo-fold rotationtwo-fold rotationtwo-fold rotationTwo-fold rotation Some familiar objects have an intrinsic symmetryTwo-fold rotation 180° rotation makes it coincident Second 180° brings the object back to its original position6 Three-fold rotation = 360 o /3 rotation to reproduce a motif in a symmetrical pattern6 step 1 step 2 step 3 Three-fold rotation = 360 o /3 rotation to reproduce a motif in a symmetrical pattern6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 2-fold 3-fold 4-fold 1-fold 6-fold 360°›1 fold axes 180°›2 fold axes 120°›3 fold axes 90°› 4 fold axes 60°› 6 fold axes- 5-, 7-fold axes --- not possible --- due to --- the principle of no empty space among unit cells Inversion - internal and external - produces an inverted object --- make object up side down - involves drawing imaginary lines from every point on object through inversion center and out an equal distances on the other side of the inversion center - also called rotary inversion (=rotoinversion) - a point / a face --- rotated around an inversion axis and inverted through the inversion center 6 6- inversion axes --- 1, 2, 3, 4, 6 --- equivalent to some other symmetry operations or combinations of them 1 ? C 2 ? m 3 ? 3-fold+C 6 ? 3/mScrew - internal - combination of rotation and translation - axis --- screw axis --- 2-, 3-, 4-, 6- fold screw axes - rotation direction --- clockwise or anticlockwise - as an external symmetry operation --- called symmetry axes --- both rotation and rotoinversions - acts as a rotations axis / inversion axis (A)center of symmetry - 3rd kind of symmetry element - an inversion operation with 360° rotation - equivalent to 1 - exists in most of the symmetry classes - in modern crystallography --- not mentioned separately - symmetry of X’s --- described in terms of symmetry plane, rotation and inversion axis2-D Symmetry combining a 2-fold rotation axis with a mirror2-D Symmetry combining a 2-fold rotation axis with a mirror Step 1: reflect (could do either step first)combining a 2-fold rotation axis with a mirror Step 1: reflect Step 2: rotate (everything)2-D Symmetry combining a 2-fold rotation axis with a mirror Step 1: reflect Step 2: rotate (everything)2-D Symmetry combining a 2-fold rotation axis with a mirror Step 1: reflect Step 2: rotate (everything)2-D Symmetry combining a 2-fold rotation axis with a mirror The result is Point Group 2mm “2mm” indicates 2 mirrors The mirrors are different2-D Symmetry Now try combining a 4-fold rotation axis with a mirror2-D Symmetry Now try combining a 4-fold rotation axis with a mirror Step 1: reflect2-D Symmetry Now try combining a 4-fold rotation axis with a mirror Step 1: reflect Step 2: rotate 12-D Symmetry Now try combining a 4-fold rotation axis with a mirror Step 1: reflect Step 2: rotate 22-D Symmetry Now try combining a 4-fold rotation axis with a mirror Step 1: reflect Step 2: rotate 32-D Symmetry Now try combining a 4-fold rotation axis with a mirror Any other elements?2-D Symmetry Now try combining a 4-fold rotation axis with a mirror Any other elements? Yes, two more mirrors2-D Symmetry Now try combining a 4-fold rotation axis with a mirror Any other elements? Yes, two more mirrors Point group name??2-D Symmetry Now try combining a 4-fold rotation axis with a mirror Any other elements? Yes, two more mirrors Point group name?? 4mm Why not 4mmmm?2-D Symmetry 3-fold rotation axis with a mirror creates point group 3m Why not 3mmm?2-D Symmetry 6-fold rotation axis with a mirror creates point group 6mm3-D Symmetry New 3-D Symmetry Elements 4. Rotoinversion a. 1-fold rotoinversion ( 1 )3-D Symmetry New 3-D Symmetry Elements 4. Rotoinversion a. 1-fold rotoinversion ( 1 ) Step 1: rotate 360/1 (identity)3-D Symmetry New 3-D Symmetry Elements 4. Rotoinversion a. 1-fold rotoinversion ( 1 ) Step 1: rotate 360/1 (identity) Step 2: invert This is the same as i, so not a new operation3-D Symmetry New Symmetry Elements 4. Rotoinversion b. 2-fold rotoinversion ( 2 ) Step 1: rotate 360/2 Note: this is a temporary step, the intermediate motif element does not exist in the final pattern3-D Symmetry New Symmetry Elements 4. Rotoinversion b. 2-fold rotoinversion ( 2 ) Step 1: rotate 360/2 Step 2: invert3-D Symmetry New Symmetry Elements 4. Rotoinversion b. 2-fold rotoinversion ( 2 ) The result:3-D Symmetry New Symmetry Elements 4. Rotoinversion b. 2-fold rotoinversion ( 2 ) This is the same as m, so not a new operation3-D Symmetry New Symmetry Elements 4. Rotoinversion c. 3-fold rotoinversion ( 3 )3-D Symmetry New Symmetry Elements 4. Rotoinversion c. 3-fold rotoinversion ( 3 ) Step 1: rotate 360 o /3 Again, this is a temporary step, the intermediate motif element does not exist in the final pattern 13-D Symmetry New Symmetry Elements 4. Rotoinversion c. 3-fold rotoinversion ( 3 ) Step 2: invert through center3-D Symmetry New Symmetry Elements 4. Rotoinversion c. 3-fold rotoinversion ( 3 ) Completion of the first sequence 1 23-D Symmetry New Symmetry Elements 4. Rotoinversion c. 3-fold rotoinversion ( 3 ) Rotate another 360/33-D Symmetry New Symmetry Elements 4. Rotoinversion c. 3-fold rotoinversion ( 3 ) Invert through center3-D Symmetry New Symmetry Elements 4. Rotoinversion c. 3-fold rotoinversion ( 3 ) Complete second step to create face 3 1 2 33-D Symmetry New Symmetry Elements 4. Rotoinversion c. 3-fold rotoinversion ( 3 ) Third step creates face 4 (3 › (1) › 4) 1 2 3 43-D Symmetry New Symmetry Elements 4. Rotoinversion c. 3-fold rotoinversion ( 3 ) Fourth step creates face 5 (4 › (2) › 5) 1 2 53-D Symmetry New Symmetry Elements 4. Rotoinversion c. 3-fold rotoinversion ( 3 ) Fifth step creates face 6 (5 › (3) › 6) Sixth step returns to face 1 1 6 53-D Symmetry New Symmetry Elements 4. Rotoinversion c. 3-fold rotoinversion ( 3 ) This is unique 1 6 5 2 3 43-D Symmetry New Symmetry Elements 4. Rotoinversion d. 4-fold rotoinversion ( 4 )3-D Symmetry New Symmetry Elements 4. Rotoinversion d. 4-fold rotoinversion ( 4 )3-D Symmetry New Symmetry Elements 4. Rotoinversion d. 4-fold rotoinversion ( 4 ) 1: Rotate 360/43-D Symmetry New Symmetry Elements 4. Rotoinversion d. 4-fold rotoinversion ( 4 ) 1: Rotate 360/4 2: Invert3-D Symmetry New Symmetry Elements 4. Rotoinversion d. 4-fold rotoinversion ( 4 ) 1: Rotate 360/4 2: Invert3-D Symmetry New Symmetry Elements 4. Rotoinversion d. 4-fold rotoinversion ( 4 ) 3: Rotate 360/43-D Symmetry New Symmetry Elements 4. Rotoinversion d. 4-fold rotoinversion ( 4 ) 3: Rotate 360/4 4: Invert3-D Symmetry New Symmetry Elements 4. Rotoinversion d. 4-fold rotoinversion ( 4 ) 3: Rotate 360/4 4: Invert3-D Symmetry New Symmetry Elements 4. Rotoinversion d. 4-fold rotoinversion ( 4 ) 5: Rotate 360/43-D Symmetry New Symmetry Elements 4. Rotoinversion d. 4-fold rotoinversion ( 4 ) 5: Rotate 360/4 6: Invert3-D Symmetry New Symmetry Elements 4. Rotoinversion d. 4-fold rotoinversion ( 4 ) This is also a unique operation3-D Symmetry New Symmetry Elements 4. Rotoinversion d. 4-fold rotoinversion ( 4 ) A more fundamental representative of the pattern3-D Symmetry New Symmetry Elements 4. Rotoinversion e. 6-fold rotoinversion ( 6 ) Begin with this framework:3-D Symmetry New Symmetry Elements 4. Rotoinversion e. 6-fold rotoinversion ( 6 ) 13-D Symmetry 1 New Symmetry Elements 4. Rotoinversion e. 6-fold rotoinversion ( 6 )3-D Symmetry 1 2 New Symmetry Elements 4. Rotoinversion e. 6-fold rotoinversion ( 6 )3-D Symmetry 1 2 New Symmetry Elements 4. Rotoinversion e. 6-fold rotoinversion ( 6 )3-D Symmetry 1 3 2 New Symmetry Elements 4. Rotoinversion e. 6-fold rotoinversion ( 6 )3-D Symmetry 1 3 2 New Symmetry Elements 4. Rotoinversion e. 6-fold rotoinversion ( 6 )3-D Symmetry 1 3 4 2 New Symmetry Elements 4. Rotoinversion e. 6-fold rotoinversion ( 6 )3-D Symmetry 1 2 3 4 New Symmetry Elements 4. Rotoinversion e. 6-fold rotoinversion ( 6 )3-D Symmetry 1 2 3 4 5 New Symmetry Elements 4. Rotoinversion e. 6-fold rotoinversion ( 6 )3-D Symmetry 1 2 3 4 5 New Symmetry Elements 4. Rotoinversion e. 6-fold rotoinversion ( 6 )3-D Symmetry 1 2 3 4 5 6 New Symmetry Elements 4. Rotoinversion e. 6-fold rotoinversion ( 6 )3-D Symmetry New Symmetry Elements 4. Rotoinversion e. 6-fold rotoinversion ( 6 ) Note: this is the same as a 3-fold rotation axis perpendicular to a mirror plane (combinations of elements follows) Top ViewR.Axis Symbol Rotation Angle I.Axis (Symbol) Equivalence 1-fold 360° 1 ? C 2-fold A 2 180°? A 2 2 ? m 3-fold A 3 120°? A 3 3 ? 3+C 4-fold A 4 90°? A 4 6-fold A 6 60°? A 6 6 ? 3/mSYMMETRY CLASSES --- based on the external forms of X’s --- 32 symmetry classes --- grouped into 6 symmetry systems --- normal class of each system --- max # of symmetry elements --- max # of faces --- subclasses --- some of symmetry elements --- missing --- name of classes particularly in cubic, hexagonal and rhombohedral systems --- general or special forms HABIT --- general shape of X --- expressed by X form (e.g. cubic, octahedral, prismatic etc.) by particular terms --- equant, acicular, nodular, colloform etc. --- controlled by environment (P, T, chem. comp) by impuritiesShape --- external shape of minerals --- morphology --- planes on X --- in various stages a. all of the planes are present›euhedral or idiomorphic, b. some planes are present›subhedral or hypidiomorphic, c. no planes are present›anhedral or xenomorphic. --- aggregate of anhedral crystals --- massive / granular - crystalline seen by naked eye - microcrystalline seen under microscope only - cryptocrystalline detected only by X-rays or electron microscope Size --- related to - surface energy, critical size - crystallization conditions T, P, conc. of els (e.g. minor or trace elements acting as catalysts) cooling rate presence of volatiles for transportation of ions EXTERNAL FORMS (MORPHOLOGY)Ideal and Distorted Crystals ideal crystal X with equally well-developed X-faces distorted crystal --- X with unequant faces (some --- larger than others) --- peculiar shapes --- curved faces and barrel shape --- symmetry --- not be obvious --- on distorted faces --- luster -- be slightly lost due to - rapid growth in certain directions - irregular cavities due to leaching by chemical reagents - abrasion during transportation - periodical growth of two faces - lineation on X-faces (striations) (eg. Tour, Q, Py) distorted gold octahedranTWINNING - symmetrical intergrowth of two (or more) crystals of same substance - twinned aggregate - related by a new symmetry element (twin element/twin operations) twin plane --- reflection by a mirror plane twin axis --- rotation about a crystal direction common to both, generally 180° composition surface --- a surface on which 2 individuals --- united repeated (multiple) twins three or more individuals twinned polysynthetic twin - if all successive comp. surf. --- parallel cyclic twin - if all successive composition surfaces --- not parallel simple twins (contact twins) composition plane --- if comp. surface --- a plane penetration twins --- if comp. surface --- irregular, interpenetratingContact Contact & & Penetration Penetration twins twins Both are Both are simple twins simple twins only two parts only two partspolysynthetic polysynthetic & & cyclic cyclic twins twins Both are Both are multiple multiple twins twins only two parts only two parts