Today in Inorganic…. Symmetry elements and operations Properties of Groups Symmetry Groups, i.e.,...

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Transcript of Today in Inorganic…. Symmetry elements and operations Properties of Groups Symmetry Groups, i.e.,...

  • Slide 1
  • Today in Inorganic. Symmetry elements and operations Properties of Groups Symmetry Groups, i.e., Point Groups Classes of Point Groups How to Assign Point Groups Previously: Welcome to 2011!
  • Slide 2
  • x Symmetry may be defined as a feature of an object which is invariant to transformation There are 5 types of symmetry elements. 1. Mirror plane of reflection, z y
  • Slide 3
  • Symmetry may be defined as a feature of an object which is invariant to transformation There are 5 types of symmetry elements. 2. Inversion center, i z y x
  • Slide 4
  • Symmetry may be defined as a feature of an object which is invariant to transformation There are 5 types of symmetry elements. 3. Proper Rotation axis, C n where n = order of rotation z y x
  • Slide 5
  • Symmetry may be defined as a feature of an object which is invariant to transformation There are 5 types of symmetry elements. y 4. Improper Rotation axis, S n where n = order of rotation Something NEW!!! C n followed by z
  • Slide 6
  • Symmetry may be defined as a feature of an object which is invariant to transformation There are 5 types of symmetry elements. 5. Identity, E, same as a C 1 axis z y x
  • Slide 7
  • When all the Symmetry of an item are taken together, magical things happen. The set of symmetry operations NOT elements) in an object can form a Group A group is a mathematical construct that has four criteria (properties) A Group is a set of things that: 1) has closure property 2) demonstrates associativity 3) possesses an identity 4) possesses an inversion for each operation
  • Slide 8
  • Lets see how this works with symmetry operations. Start with an object that has a C 3 axis. 1 1 2 2 3 3 NOTE: that only symmetry operations form groups, not symmetry elements.
  • Slide 9
  • Now, observe what the C 3 operation does: 1 1 2 2 3 3 3 3 1 1 2 2 2 2 3 3 1 1 C3C3 C32C32
  • Slide 10
  • A useful way to check the 4 group properties is to make a multiplication table: 1 1 2 2 3 3 3 3 1 1 2 2 2 2 3 3 1 1 C3C3 C32C32
  • Slide 11
  • Now, observe what happens when two symmetry elements exist together: Start with an object that has only a C 3 axis. 1 1 2 2 3 3
  • Slide 12
  • Now, observe what happens when two symmetry elements exist together: Now add one mirror plane, 1. 1 1 3 3 11 2 2
  • Slide 13
  • Now, observe what happens when two symmetry elements exist together: 1 1 2 2 3 3 3 3 2 2 C3C3 11 11 1 1
  • Slide 14
  • Heres the thing: Do the set of operations, {C 3 C 3 2 1 } still form a group? 1 1 2 2 3 3 3 3 1 1 2 2 3 3 2 2 1 1 How can you make that decision? C3C3 11 11
  • Slide 15
  • This is the problem, right? How to get from A to C in ONE step! 1 1 2 2 3 3 3 3 1 1 2 2 3 3 2 2 1 1 What is needed? C3C3 11 11 ACB
  • Slide 16
  • 1 1 2 2 3 3 3 3 1 1 2 2 3 3 2 2 1 1 What is needed? Another mirror plane! C3C3 11 11 1 1 2 2 3 3 22
  • Slide 17
  • 3 3 1 1 2 2 3 3 2 2 1 1 And if theres a 2 nd mirror, there must be . 33 11 1 1 2 2 3 3 22
  • Slide 18
  • 3 3 1 1 2 2 3 3 2 2 1 1 Does the set of operations {E, C 3 C 3 2 1 2 3 } form a group? 33 11 1 1 2 2 3 3 22 1 1 2 2 3 3 3 3 1 1 2 2 2 2 3 3 1 1 C3C3 C32C32
  • Slide 19
  • The set of symmetry operations that forms a Group is call a Point Groupit describers completely the symmetry of an object around a point. Point Group symmetry assignments for any object can most easily be assigned by following a flowchart. The set {E, C 3 C 3 2 1 2 3 } is the operations of the C 3v point group.
  • Slide 20
  • The Types of point groups If an object has no symmetry (only the identity E) it belongs to group C 1 Axial Point groups or C n class C n = E + n C n ( n operations) C nh = E + n C n + h (2n operations) C nv = E + n C n + n v ( 2n operations) Dihedral Point Groups or Dn class D n = C n + nC2 (^) D nd = C n + nC2 (^) + n d D nh = C n + nC2 (^) + h Sn groups: S 1 = C s S 2 = C i S 3 = C 3h S 4, S 6 forms a group S 5 = C 5h
  • Slide 21
  • Linear Groups or cylindrical class Cv and Dh = C + infinite sv = D + infinite sh Cubic groups or the Platonic solids.. T: 4C3 and 3C2, mutually perpendicular Td (tetrahedral group): T + 3S4 axes + 6 s O: 4C3, mutually perpendicular, and 3C2 + 6C2 Oh (octahedral group): O + i + 3 sh + 6 sd Icosahedral group: Ih : 6C5, 10C3, 15C2, i, 15 s
  • Slide 22
  • Whats the difference between: v and h 1 1 2 2 3 3 3 3 1 1 2 2 3 3 2 2 1 1 h is perpendicular to major rotation axis, C n vv v is parallel to major rotation axis, C n hh
  • Slide 23
  • 5 types of symmetry operations. Which one(s) can you do?? Rotation Reflection Inversion Improper rotation Identity
  • Slide 24
  • 1 1 2 2 3 3 1 1 2 2 3 3
  • Slide 25
  • 1 1 2 2 3 3 C3C3 11 11
  • Slide 26
  • 1 1 2 2 3 3 C3C3 1 1 2 2 3 3 22