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# Symmetry

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### Symmetry

1. 1. Lecture 2: Crystal Symmetry
2. 2. Crystals are made of infinite number of unit cells Unit cell is the smallest unit of a crystal, which, if repeated, could generate the whole crystal. A crystal’s unit cell dimensions are defined by six numbers, the lengths of the 3 axes, a, b, and c, and the three interaxial angles, α, β and γ.
3. 3. A crystal lattice is a 3-D stack of unit cellsCrystal lattice is an imaginative grid system in three dimensions inwhich every point (or node) has an environment that is identical to thatof any other point or node.
4. 4. SymmetryA state in which parts on opposite sides of a plane,line, or point display arrangements that are related toone another via a symmetry operation such astranslation, rotation, reflection or inversion.Application of the symmetry operators leaves theentire crystal unchanged.
5. 5. Symmetry ElementsRotation turns all the points in the asymmetric unit around one axis, the center of rotation. A rotation does not change the handedness of figures. The center of rotation is the only invariant point (point that maps onto itself).
6. 6. Symmetry elements: rotation
7. 7. Symmetry elements: rotation
8. 8. Symmetry ElementsTranslation moves all the points in the asymmetric unit the same distance in the same direction. This has no effect on the handedness of figures in the plane. There are no invariant points (points that map onto themselves) under a translation.
9. 9. Symmetry ElementsScrew axes (rotation + translation) rotation about the axis of symmetry by 360°/n, followed by a translation parallel to the axis by r/n of the unit cell length in that direction. (r < n)
10. 10. 120° rotation1/3 unit cell translation
11. 11. Symmetry ElementsInversion, or center of symmetry 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.
12. 12. Symmetry ElementsMirror plane or Reflection flips 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 (points that map onto themselves) under a reflection.
13. 13. Symmetry elements:mirror plane and inversion center The handedness is changed.
14. 14. Symmetry ElementsGlide reflection (mirror plane + translation) reflects the asymmetric unit across a mirror and then translates parallel to the mirror. A glide plane changes the handedness of figures in the asymmetric unit. There are no invariant points (points that map onto themselves) under a glide reflection.
15. 15. Symmetries in crystallography• Crystal systems• Lattice systems• Space group symmetry• Point group symmetry• Laue symmetry, Patterson symmetry
16. 16. Crystal system• Crystals are grouped into seven crystal systems, according to characteristic symmetry of their unit cell.• The characteristic symmetry of a crystal is a combination of one or more rotations and inversions.
17. 17. 7 Crystal Systems orthorhombic hexagonal monoclinic trigonal cubic tetragonal triclinicCrystal System External Minimum Symmetry Unit Cell PropertiesTriclinic None a, b, c, al, be, ga,Monoclinic One 2-fold axis, || to b (b unique) a, b, c, 90, be, 90Orthorhombic Three perpendicular 2-folds a, b, c, 90, 90, 90Tetragonal One 4-fold axis, parallel c a, a, c, 90, 90, 90Trigonal One 3-fold axis a, a, c, 90, 90, 120Hexagonal One 6-fold axis a, a, c, 90, 90, 120Cubic Four 3-folds along space diagonal a, a, ,a, 90, 90, 90
18. 18. Auguste Bravais Lattices(1811-1863) • In 1848, Auguste Bravais demonstrated that in a 3-dimensional system there are fourteen possible lattices • A Bravais lattice is an infinite array of discrete points with identical environment • seven crystal systems + four lattice centering types = 14 Bravais lattices • Lattices are characterized by translation symmetry
19. 19. Four lattice centering typesNo. Type Description1 Primitive Lattice points on corners only. Symbol: P.2 Face Centered Lattice points on corners as well as centered on faces. Symbols: A (bc faces); B (ac faces); C (ab faces).3 All-Face Centered Lattice points on corners as well as in the centers of all faces. Symbol: F.4 Body-Centered Lattice points on corners as well as in the center of the unit cell body. Symbol: I.
20. 20. Tetragonal lattices are either primitive (P) or body-centered (I) C centered lattice = Primitive lattice
21. 21. Monoclinic lattices are either primitive or C centered
22. 22. Point group symmetry• Inorganic crystals usually have perfect shape which reflects their internal symmetry• Point groups are originally used to describe the symmetry of crystal.• Point group symmetry does not consider translation.• Included symmetry elements are rotation, mirror plane, center of symmetry, rotary inversion.
23. 23. Point group symmetry diagrams
24. 24. There are a totalof 32 point groups
25. 25. N-fold axes with n=5 or n>6 does not occur in crystalsAdjacent spaces must be completely filled (no gaps, nooverlaps).
26. 26. Laue class, Patterson symmetry• Laue class corresponds to symmetry of reciprocal space (diffraction pattern)• Patterson symmetry is Laue class plus allowed Bravais centering (Patterson map)
27. 27. Space groupsThe combination of all available symmetry operations (32point groups), together with translation symmetry,within the all available lattices (14 Bravais lattices) leadto 230 Space Groups that describe the only ways in whichidentical objects can be arranged in an infinite lattice.The International Tables list those by symbol andnumber, together with symmetry operators, origins,reflection conditions, and space group projectiondiagrams.
28. 28. A diagram from International Table of Crystallography
29. 29. Identification of the Space Group is called indexing the crystal.The International Tables for X-ray Crystallography tell us a hugeamount of information about any given space group. For instance,If we look up space group P2, we find it has a 2-fold rotation axisand the following symmetry equivalent positions: X , Y , Z -X , Y , -Zand an asymmetric unit defined by: 0≤x≤ 1 0≤y≤ 1 0 ≤ z ≤ 1/2 An interactive tutorial on Space Groups can be found on-line in Bernhard Rupp’sCrystallography 101 Course: http://www-structure.llnl.gov/Xray/tutorial/spcgrps.htm
30. 30. Space group P1Point group 1 + Bravais lattice P1
31. 31. Space group P1barPoint group 1bar + Bravais lattice P1
32. 32. Space group P2Point group 2 + Bravais lattice “primitive monoclinic”
33. 33. Space group P21Point group 2 + Bravais lattice “primitive monoclinic”,but consider screw axis
34. 34. Coordinate triplets, equivalent positionsr = ax + by + cz,Therefore, each point can be described by its fractionalcoordinates, that is, by its coordinate triplet (x, y, z)
35. 35. Space group determination• Symmetry in diffraction pattern• Systematic absences• Space groups with mirror planes and inversion centers do not apply to protein crystals, leaving only 65 possible space groups.
36. 36. A lesson in symmetry from M. C. Escher
37. 37. Another one:
38. 38. Asymmetric unitRecall that the unit cell of a crystal is the smallest 3-D geometricfigure that can be stacked without rotation to form the lattice. Theasymmetric unit is the smallest part of a crystal structure fromwhich the complete structure can be built using space groupsymmetry. The asymmetric unit may consist of only a part of amolecule, or it can contain more than one molecule, if the moleculesnot related by symmetry.
39. 39. Matthew Coefficient• Matthews found that for many protein crystals the ratio of the unit cell volume and the molecular weight is between 1.7 and 3.5Å3/Da with most values around 2.15Å3/Da• Vm is often used to determine the number of molecules in each asymmetric unit.• Non-crystallographic symmetry related molecules within the asymmetric unit