Review of Crystallography 1 Crystal basics m 1 Crvstal basic 1. Crystal ■2 Symmetry 2. Fundamental Characteristics of Crystals 3 Crystal Structure Analysis n 4 Crystal Chemistry E 5 Some Important Crystal Structures Why Solids? Early Ideas 6 ALL Compounds are solids under suitable nditions of temperature and pressure. Many exist o Crystals are solid -but solids are not necessarily crystalline o Crystals have symmetry(Kepler, 1611) and long o atoms in -fixed position range order I"simple"case- crystalline solid= Crystal Structure apes can be pa ked produce regular shapes (Hooke; Hauy, 1812) Why study crystal structures o description of solid n with other similar materials- 费 o correlation with physical properties Crystallinity Definition-Crvstal aA crystal may be defined as a collection of ator arranged in a pattern that is periodie in 3D. Crystals- A homogenous solid formed by a rily solids, but not all solids ar repeating, threedimensional pattern of crystalline(amorphous solids lack long range periodic atoms, Ions, or 1 molecules and having fixed order distances between constituent parts In a perfect single crystal, all atoms in the crystal are related either through translational symmetry or point symmetry POlycrystalline materials are made up of a great number of tiny ( m to nm) single crystals &Crystalline solids can be divided into two categorie extended and molecular
1 Review of Crystallography 1 Crystal Basics 2 Symmetry 3 Crystal Structure Analysis 4 Crystal Chemistry 5 Some Important Crystal Structures 1 Crystal Basics 1.Crystal 2.Fundamental Characteristics of Crystals Why Solids? µ ALL Compounds are solids under suitable conditions of temperature and pressure. Many exist only as solids. µ atoms in ~fixed position “simple”case ¾ crystalline solid Þ Crystal Structure Why study crystal structures? µ description of solid µ comparison with other similar materials ¾ classification µ correlation with physical properties Early Ideas Crystals are solid ¾ but solids are not necessarily crystalline Crystals have symmetry (Kepler, 1611) and long range order Spheres and small shapes can be packed to produce regular shapes (Hooke; Hauy,1812) ? Definition ¾ Crystal Crystals ¾ A homogenous solid formed by a repeating, three-dimensional pattern of atoms, ions, or molecules and having fixed distances between constituent parts. Crystallinity A crystal may be defined as a collection of atoms arranged in a pattern that is periodic in 3D. Crystals are necessarily solids, but not all solids are crystalline (amorphous solids lack long range periodic order). In a perfect single crystal, all atoms in the crystal are related either through translational symmetry or point symmetry. Polycrystalline materials are made up of a great number of tiny (mm to nm) single crystals Crystalline solids can be divided into two categories extended and molecular
Single Crystal and Single crystals Polycrystalline Materials array over the entire extent of the material ted arrangemen throughout the specim ment of atoms extends alline material: comprised of m ve different p all unit cells have the same orientation crystallographic orientation. There exist atomic n exist in nature mismatch within the regions where grains meet e can also be grown (eg. Si e without external constraints. will have flat gular faces Beautiful Crystals Polycrystalline Materials s Crystals of different 鲁s1Zes Grain Boundaries -mismatch between two ighboring crystals Polycrystalline Materials Basic Characteristic of Crystals OMost crystalline materials are composed of many a Homogeneity-Under macro small crystals called grains observation, the physics effect and chemical CRystallographic directions of adjacent grains are of a cry ire the usually random Anisot Physical properties of a CThere is usually atomic mismatch where two crystal differ according to the direction of grains meet -this is called a grain boundary measurement Most powdered materials have many randomly oriented grains
2 Single Crystal and Polycrystalline Materials ß Single crystal:atoms are in a repeating or periodic array over the entire extent of the material ß Polycrystalline material: comprised of many small crystals or grains. The grains have different crystallographic orientation.There exist atomic mismatch within the regions where grains meet. These regions are called grain boundaries. Single Crystals repeated arrangement of atoms extends throughout the specimen all unit cells have the same orientation exist in nature can also be grown (eg. Si) without external constraints, will have flat, regular faces Beautiful Crystals Polycrystalline Materials Crystals of different sizes orientations shapes Grain Boundaries -mismatch between two neighboring crystals Polycrystalline Materials Most crystalline materials are composed of many small crystals called grains Crystallographic directions of adjacent grains are usually random There is usually atomic mismatch where two grains meet ¾¾ this is called a grain boundary Most powdered materials have many randomly oriented grains Basic Characteristic of Crystals Homogeneity ¾¾ Under macroscopic observation,the physics effect and chemical composition of a crystal are the same. Anisotropy ¾¾ Physical properties of a crystal differ according to the direction of measurement
Anisotropy different packing. For instance, atoms along the edge of FCC unit cell are orie tame p l re rta lime mo berk a tenin properties of crysta tals. for instance. tI properties are isotropic. deformation depends on the directio iSome polycrystalline materials have grains n which a stress is applied. with preferred orientations(texture). so properties are dominated by those relevant to the texture orientation and the material exhibits anisotropic properties Law of Constancy of Interfacial Angle Crystal Shape sThe interfacial angles are constant for all IThe external shape of a crystal is referred to as its crystals if a given mineral with identical mposition at the same temperature eSince all crystals of the same substance will Typically see faces on crystals grown fron t E Not all crystals have well defined external faces have the same spacing between lattice points solution (they have the same crystal structure), the I Natural faces always have low indices(orientation sponding faces of the same can be described by Miller indices that are small mineral will be the same sThe symmetry of the lattice will determine the tThe faces that you see are the lowest energy faces angular relationships between crystal faces. Surface energy is minimized during growth This is a term that refers to the form that a crystal Crystal Habits takes as it grows pRismatic Crystal Habit Tabular rHombohedra dOdecahedral
3 Anisotropy Different directions in a crystal have different packing. For instance, atoms along the edge of FCC unit cell are more separated than along the face diagonal. This causes anisotropy in the properties of crystals, for instance, the deformation depends on the direction in which a stress is applied. In some polycrystalline materials, grain orientations are random, so bulk material properties are isotropic. Some polycrystalline materials have grains with preferred orientations (texture), so properties are dominated by those relevant to the texture orientation and the material exhibits anisotropic properties. Law of Constancy of Interfacial Angle The interfacial angles are constant for all crystals if a given mineral with identical composition at the same temperature. Since all crystals of the same substance will have the same spacing between lattice points (they have the same crystal structure), the angles between corresponding faces of the same mineral will be the same. The symmetry of the lattice will determine the angular relationships between crystal faces. Crystal Shape The external shape of a crystal is referred to as its Habit Not all crystals have well defined external faces Typically see faces on crystals grown from solution Natural faces always have low indices (orientation can be described by Miller indices that are small integers) The faces that you see are the lowest energy faces Surface energy is minimized during growth This is a term that refers to the form that a crystal takes as it grows. Prismatic Pyramidal Tabular Rhombohedra Dodecahedral Acicular Bladed Crystal Habits Crystal Habits
Law of Symmetry aLaw of Symmetry: Only 1, 2, 3, 4, 6 fold rotatio axis can exist in crystal Why snowflakes have 6 corners, never 5 or 7? 多3 By considering the packing of polygons in 2 dimensions, demonstrate why pentagons and Allowed Quasicrystal Structures(First in 1984) rotation a RoomMgoazno.s Ro Mgo. cde NOT 5.>6 Face-centred icosahedral R-Mg-Zn Primitiv Quasicrystal: AlFe Cu e P Dodecahedral morphology Rhombic triacontahedral Quasicrystalline Materials Amorphous solids bNon-periodic long-range ordered structures nIdeal solid crystals exhibits structural long range Rotational symmetry of diffraction patterns (e. order (LRO) FReal crystals contain imperfections, i.e., defects and impurities, which spoil the lRo AMorphous solids lack any LRO Ithough may exhibit short range order (SRO) Crystal ) Gas
4 Law of Symmetry Law of Symmetry: Only 1,2,3,4,6-fold rotation axis can exist in crystal. Why snowflakes have 6 corners, never 5 or 7? By considering the packing of polygons in 2 dimensions, demonstrate why pentagons and heptagons shouldn ’t occur. Empty space not allowed Quasicrystal: AlFeCu Allowed rotation axis: 1, 2, 3, 4, 6 NOT 5, > 6 Quasicrystal Structures (First in 1984) R0.09Mg0.34Zn0.57 Dodecahedral morphology R0.1Mg0.4Cd0.5 Rhombic triacontahedral morphology ED: 5 fold axis Face-centred icosahedral R-Mg-Zn Primitive icosahedral R-Mg-Cd Non-periodic long-range ordered structures Rotational symmetry of diffraction patterns (e.g. 5-fold, 10-fold) impossible for periodic crystals Quasicrystalline Materials Quasi-unit cells Amorphous Solids Ideal solid crystals exhibits structural long range order (LRO) Real crystals contain imperfections, i.e., defects and impurities, which spoil the LRO Amorphous solids lack any LRO [though may exhibit short range order (SRO)] Crystal Glass (amorphous) Gas
Quartz Crystal and Quartz Glass Transparent, amorphous solid Composition almost all silicon dioxide(SiO Quartz sand) Lead glass SiO.+ PbO, and K Quartz Crystal Quartz Glass Pyrex glass ? SiOz with BD3 so so% Green glass(cheap bottles)) FeO Blue glass Cobalt oxide+ Siog Fo mu ne Violet glass→ Yellow glass Uranium oxide SiO2 Red glass ]Gold and copper+ SiO 2 Liquid crystal From Crystal to Liquid Crystal Liquid crystals are a phase to Liquid tween that 捷图 LC typically rodshaped organic moieties about 25 湖鬻 Angstroms in length and liquid This is the structure change process of some molecule saiid tisd Orlat with long chains when increasing temperatures Principles of Liquid Crystal Displays +1 Entropy
5 Quartz Crystal and Quartz Glass Quartz Crystal Quartz Glass Glass ß Transparent, amorphous solid –Composition almost all silicon dioxide (SiO2 – Quartz sand) ß Ordinary glass ‡ 75% SiO2 ß Pyrex glass ‡ SiO2 with B2O3 ß Lead glass ‡ SiO2 + PbO, and K2O ß Green glass (cheap bottles) ‡ FeO + SiO2 ß Blue glass ‡ Cobalt oxide + SiO2 ß Violet glass ‡ Manganese + SiO2 ß Yellow glass ‡ Uranium oxide + SiO2 ß Red glass ‡ Gold and copper + SiO2 Liquid Crystal Liquid crystals are a phase of matter whose order is intermediate between that of a liquid and that of a crystal. The molecules are typically rod-shaped organic moieties about 25 Angstroms in length and their ordering is a function of temperature. From Crystal to Liquid Crystal to Liquid (a)crystal，(b)、(c) anisotropic liquids，(d) isotropic liquid This is the structure change process of some molecules with long chains when increasing temperatures Nematic Entropy driven formation of liquid crystals of rod-like colloids Isotropic Crystal Smectic = Direction of increasing density Principles of Liquid Crystal Displays No voltage voltage
Liquid Crystal Displays 2 Symmetry Smaller, lighter, with no radiation problems. Found on portables and notebooks, and starting to appear and more on desktops ASymmetry: Point Symmetry rather than emitted. Use of super-twisted crystals pace Symmetry have improved the viewing angle, and response rates 832 Point Groups of Crystals are improving all the time (necessary for tracking .Unit Cell, 7 Crystal Systems, Lattice Planes cursor accurately) Miller indices .Lattices and 14 Bravias Types of Lattices ●230 Space Group Crystal Symmetry Symmetry in Nature, Art and Math Symmetry is one idea by which man through the ages has tried to comprehend and create order Mathematics of Symmetry beauty and perfection. - Hermann Weyl Crystals Symmetry Physical Properties caused by Symmetry 米" Eiffel tower in Paris. France is a wonderful example of symmetry Macroscopic Symmetry Elements Point Symmetry Elements) Mirror Plane Symmetry mirror plane symmetry arise E Point symmetry elements operate to change the a°。o。°° when one half of an object is orientation of structural motifs A point symmetry operation does not alter at °。。o。 the mirror image of the other least one point that it operates on Symmetry Elements and Symmetry Operations 1. Mirror Planes -Reflection or Mirror 2. Center of Symmetry 3. Rotation axis -Rotate Can be folded in half 4. Rotoinversion axis -Rotate and inverse Seen externally with animals
6 Smaller, lighter, with no radiation problems. Found on portables and notebooks, and starting to appear more and more on desktops. Less tiring than c.r.t. (Cathode -ray tube) displays, and reduce eye-strain, due to reflected nature of light rather than emitted. Use of super -twisted crystals have improved the viewing angle, and response rates are improving all the time (necessary for tracking cursor accurately). Liquid Crystal Displays 2 Symmetry Symmetry: Point Symmetry Space Symmetry 32 Point Groups of Crystals Unit Cell, 7 Crystal Systems, Lattice Planes, Miller indices Lattices and 14 Bravias Types of Lattices 230 Space Groups Crystal Symmetry ß Mathematics of Symmetry ß Crystal’s Symmetry ß Physical Properties caused by Symmetry Symmetry in Nature, Art and Math Symmetry is one idea by which man through the ages has tried to comprehend and create order, beauty and perfection. ¾ Hermann Weyl Eiffel tower in Paris, France is a wonderful example of symmetry Macroscopic Symmetry Elements (Point Symmetry Elements) Point symmetry elements operate to change the orientation of structural motifs A point symmetry operation does not alter at least one point that it operates on Symmetry Elements and Symmetry Operations: 1. Mirror Planes — — Reflection or Mirror 2. Center of Symmetry — — Inverse 3. Rotation Axis — — Rotate 4. Rotoinversion Axis — — Rotate and inverse Mirror Plane Symmetry Mirror plane symmetry arises when one half of an object is the mirror image of the other half •Can be folded in half •Seen externally with animals s s
Mirror Plane Symmetry Symmetry Operation Reflect sThis molecule has two mirror planes flips all points in the ci .One is horizontal, in the plane of the asymmetric unit over a paper and bisects the H-C-H bond line which is called the MOther is vertical, perpendicular to mirror and thereby the plane of the paper and bisects the changes the handedness of CkC· Cl bonds WY any figures in the asymmetric unit. The points along the mirror has reflectional symmetry if an line are all invariant lane can divide the crvstal into halves points under a reflection. hich is the mirror image of the other Rotational Symmetry Symmetry Operation Rotation turns all the points in the o Rotated about a point symmetric unit around one o Allows chirality o In crystals limited to 1,2,3,4,and6 he handedness of figures in rotations nly invariant point Symmetry Axis of Rotation Rotational Symmetry We say a crystal has a symmetry coincidence upon rotation axis of rotation when we can turn it the pattern looks exactly the same. nfold axis of rotational Think of the center of a pizza. If it the same size and have the same then the pizza could be turned and you couldn't tell the difference element for which the operation is a rotation of 3607n retry. The below has rotational symmetry of graphite 60 degrees
7 This molecule has two mirror planes: One is horizontal, in the plane of the paper and bisects the H-C-H bonds Other is vertical, perpendicular to the plane of the paper and bisects the Cl-C-Cl bonds A crystal has reflectional symmetry if an imaginary plane can divide the crystal into halves, each of which is the mirror image of the other. Mirror Plane Symmetry Symmetry Operation Reflection qflips all points in the asymmetric unit over a line, which is called the mirror and thereby changes the handedness of any figures in the asymmetric unit. The points along the mirror line are all invariant points under a reflection. Rotational Symmetry Rotated about a point Allows chirality In crystals limited to 1,2,3,4, and 6 rotations Symmetry Operation Rotation turns all the points in the asymmetric unit around one point, the center of rotation. A rotation does not change the handedness of figures in the plane. The center of rotation is the only invariant point. Symmetry Axis of Rotation We say a crystal has a symmetry axis of rotation when we can turn it by some degree about a point and the pattern looks exactly the same. Think of the center of a pizza. If it is made so that all the pieces are the same size and have the same ingredients in the same places, then the pizza could be turned and you couldn't tell the difference. This means the pizza has rotational symmetry. The pizza below has rotational symmetry of 60 degrees. Rotational Symmetry coincidence upon rotation about the axis of 360°/n Þ n-fold axis of rotational symmetry graphite O H Symbol for a symmetry element for which the operation is a rotation of 360°/n C2 = 180°, C3=120°, C4 = 90°, C5 = 72°, C6 = 60°, etc
Rotation Axis(Cn) Center of Symmetry In general present if you can draw a straight line from any n fold rotation axis s rotation by(360/n) point, through the center, to an equal distance the other side, and arrive at an identical point Can rotate by120° about the o·Cl bond and the molecule looks identical This is called a rotation ax tation axis, as rotate by 120(= 888 3603)to reach an identical o。。0 Center of No center of I symmetry ats symmetry Symmetry Operation n Axis - Symmetry Inversion Axis of rotary Inversion Rotoinversion Axis(Sn or n):nfold rotation combined with an inversion side of a center of symmetry has a similar point at an equal distance the er of 1=i 2 3=3-fold rotation Mac Symmetry Elements Point groups Electrical resistance Thermal expansion Magnetic susceptibility= Macroscopic symmetry 8 =3-fold rotation Macroscopically measured Translation symmetry mirror plane Combination of mirror, center of symmetry rotational symmetry, center of inversion point groups
8 Can rotate by 120° about the C-Cl bond and the molecule looks identical Þ the H atoms are indistinguishable. This is called a rotation axis Þ in particular, a three fold rotation axis, as rotate by 120° (= 360°/3) to reach an identical configuration Rotation Axis (Cn) In general: n-fold rotation axis = rotation by (360°/n) “present if you can draw a straight line from any point, through the center, to an equal distance the other side, and arrive at an identical point”. Center of symmetry at S No center of symmetry (x,y,z) (-x,-y,-z) Center of Symmetry (Inversion symmetry) i Symmetry Operation Inversion every point on one side of a center of symmetry has a similar point at an equal distance on the opposite side of the center of symmetry. Rotoinversion Axis ¾¾ Symmetry Axis of Rotary Inversion Rotoinversion Axis (Sn or ) : n-fold rotation combined with an inversion. n 1 = i 2 = m 3 = 3-fold rotation + inversion 4 6 =3-fold rotation with perpendicular mirror plane Macroscopic Symmetry Elements: Point Groups Electrical resistance Thermal expansion Magnetic susceptibility Elastic constants Macroscopically measured properties Þ Macroscopic symmetry XÛ Translation symmetry Combination of mirror, center of symmetry, rotational symmetry, center of inversion Þ point groups
Point group F Point groups have symmetry about a single point at Point group (point symmetryAll molecules characterized by 32 different combinations of t Symmetry elements are geometric entities about hich a symmetry operation can be performed. In symmetry elements int group, all symmetry elements must pass hrough the center of mass (the point). A symmetry There are two naming systems commonly used opentical to the initial object. object n describing symmetry elements tical tool 1. The Schoenflies notation used extensively by lize and simplify many spectroscopists s in chemistry. A group consists of a set of 2. The Hermann-Mauguin or international ns) that completely describe the notation preferred by crystallographers All Combinations of point Symmetry Elements Are Not Possible Deduction of 32 Point gro The allowed combinations of point symmetry rOtation: 1, 2-,- 4- 6-fold= 5 point group Point Symmetry Elements gRotation-inversion: 2,3,4,6 Compatible With 3D Translations →5 point group among32 COmbinations of rotation: 222. 223 224. 226. 23 nat ry Elemont symbol 432=6 point group among 32 cOmbinations of rotation and an inversion or a Rotoinversion Axis n-1, 2, 3,4,E mirror:15 point group among 32 Mirror I point group among 32 Deduction of 32 Point Grou Schoenflies Symbols 1、2、3、4、6 fold rotation axis exist in crystal, which re marked as Cr Ca Ca C and ce tThe combination of rotation axis can deduce ich only 、23432 which 4 Cn: cyclic, the po the order of the rotation ax g D,: dihedral, the group point which generated onsists of only rotation axis from the combination of 2-fold axis. n is the have in order of the main rotation axis 9T: tetrahedral he combination of rotation axi *0: octahedral with higher order L6L2 3L4L, 3L4L26L2 L 2L
9 Point Groups Point groups have symmetry about a single point at the center of mass of the system. Symmetry elements are geometric entities about which a symmetry operation can be performed. In a point group, all symmetry elements must pass through the center of mass (the point). A symmetry operation is the action that produces an object identical to the initial object. Group theory is a very powerful mathematical tool that allows us to rationalize and simplify many problems in chemistry. A group consists of a set of symmetry elements (and associated symmetry operations) that completely describe the symmetry of an object. Point Group Point group (point symmetry)¾¾All molecules characterized by 32 different combinations of symmetry elements There are two naming systems commonly used in describing symmetry elements 1. The Schoenflies notation used extensively by spectroscopists 2. The Hermann-Mauguinor international notation preferred by crystallographers All Combinations of Point Symmetry Elements Are Not Possible The allowed combinations of point symmetry elements are called point groups Point Symmetry Elements Compatible With 3D Translations Deduction of 32 Point Groups Rotation: 1-, 2-, 3-, 4-, 6-foldÞ 5 point group among 32 Rotation-inversion: Þ 5 point group among 32 Combinations of rotation: 222, 223, 224, 226, 23, 432 Þ 6 point group among 32 Combinations of rotation and an inversion or a mirror:15 point group among 32 mirror: 1 point group among 32 Deduction of 32 Point Groups 3L 3L44L36L2 24L3 L66L2 L22L2 1、2、3、4、6 fold rotation axis exist in crystal, which are marked as C1、C2、C3、C4 and C6 . The combination of rotation axis can deduce：222、223、 224、226、23、432，which are marked as D2、D3、D4、 D6、T and O. The above 11 point groups consists of only rotation axis and do not have inversion axis. Schoenflies Symbols vCn : cyclic, the point group which only one rotation axis,n is the order of the rotation axis. vDn : dihedral, the group point which generated from the combination of 2-fold axis, n is the order of the main rotation axis). vT: tetrahedral vO: octahedral The combination of rotation axis with higher order
The 32 Point Groups 32 Crystallographic Point Groups 圃 cHc【cA dd mirror plane to the above 1l basic point groups, the dding mirror plane intersect ation, no new symmetry 6,6, Tetragonal S4. C4h D 2)Mirror plane is vertical to the main rotation axis, Py axis, and is Cubie agonal to the neighbo 令 Attention: adding F、toDa、 T and o is equal to dk、Tb and O respectively, Oa is equal to Oh. Identifying Point Groupss= Identifying Point Groups(1)The sandry is acedia Ye Pefecfy Ootheca a We can use a flow chart such mey(事 in this process are. mine the symmetry is Isthere at leof ene Iso rotting 2. Determine if there is a 卤面 incipal rotation axis. 3. Determine if there otation axes perpendicular to he principal axis. 4. Determine if there 5. Assign point group Identifying Point Groups(2, mris n Identifying Point Groups(3) ardr gee nis of rotation t axis of roation C Are theren Cax Is there a pane of symmetry o perpendicular to De perpend calar to the ipd us C parallel to the principe
10 The 32 Point Groups C1 C2 C3 C4 C6 D2 D3 D4 D6 T O 1 1 +Ph Cs C2 h C3 h C4 h C6h D2h D3 h D4h D6h Th Oh 2 2 +Pv - - C2 v C3 v C4 v C6v - - - - - - - - - - - - 2 6 +Pd - - - - - - - - - - D2d D3 d - - - - Td - - 2 9 +C Ci - - C3 i - - - - - - - - - - - - - - - - 3 1 n S4 3 2 Add mirror plane to the above 11 basic point groups, the adding mirror plane intersect at one point with other symmetry elements, and in addition, no new symmetry types are formed, thus there are three ways: 1)Mirror plane is horizontal with the main rotation axis, Ph 2)Mirror plane is vertical to the main rotation axis, Pv 3)Mirror plane is vertical to the main rotation axis,and is diagonal to the neighboring 2-fold axis), Pd vAttention: adding Pv to Dn、T and O is equal to Dnh、Th and Oh respectively, Od is equal to Oh . 32 Crystallographic Point Groups Crystal System Number of Point Groups Herman-Mauguin Point Group Schoenflies Point Group Triclinic 2 1,`1 C1, Ci Monoclinic 3 2, m, 2/m C2, Cs , C2h Orthorhombic 3 222, mm2, mmm D2, C2v, D2h Trigonal 5 3,`3, 32, 3m, `3m C3, S6, D3, C3v, D3d Hexagonal 7 6,`6, 6/m, 622, 6mm, `62m, 6mm C6, C3h, C4h, D6, C6v, D3h , D6h Tetragonal 7 4,`4, 4/m, 422, 4mm,`42m, 4/mmm C4, S4, C4h, D4, C4v, D2d, D4h Cubic 5 23, m3, 432, `432, m`3m T, Th, O, Td, Oh We can use a flow chart such as this one to determine the point group of any object. The steps in this process are: 1. Determine the symmetry is special. 2. Determine if there is a principal rotation axis. 3. Determine if there are rotation axes perpendicular to the principal axis. 4. Determine if there are mirror planes. 5. Assign point group. Identifying Point Groups Identifying Point Groups (1) Identifying Point Groups (2) Identifying Point Groups (3)
