X-ray Crystallography, an Overview

1. X-ray Crystallography, an Overview Frank R. Fronczek Department of Chemistry Louisiana State University Baton Rouge, LA Feb. 5, 2014

2. “Long before there were people on the earth, crystals were already growing in the earth’s crust. On one day or another, a human being first came across such a sparkling morsel of regularity lying on the ground or hit one with his stone tool, and it broke off and fell at his feet, and he picked it up and regarded it in his open hand, and he was amazed.” M. C. Escher From “Approaches to Infinity”

3. Topics 1. Crystals 2. Point Symmetry (Brief Review) 3. Space Group Symmetry 4. Diffraction and Fourier Analysis 5. Intensity Data Collection 6. Structure Solution and Refinement 7. Absolute Structure

4. René Just Haüy 1743-1822

5. Calcite, CaCO3 It broke into rhombohedra Intentional breakage of a rhombohedron produces smaller and smaller rhombohedra

6. For Calcite This led to concept of the “unit cell”

7. Unit cell, in yellow, gives directions and distances of translationally repeating unit The three axes are labelled a, b, and c, and may have different lengths

8. If we want to indicate only the translational regularity and not the structure itself, we can do so with an array of points called a lattice

9. Important to distinguish between the structure and the lattice, which is just an array of points which indicates the regularity of the structure.

10. The number of molecules in the unit cell (Z) here is 4 Molecules within the unit cell are related by symmetry The asymmetric unit here is one molecule, but may be several, or less than one.

11. Fractional Coordinates c z b x y a x is the fractional coordinate in the a direction y in the b direction z in the c direction

12. To Completely Describe the Structure, Must Determine: Dimensions of the Unit Cell Symmetry of the Unit Cell Coordinates of all the atoms in the Asymmetric Unit

13. Miller Indices c 1 1/2 b 0 1/3 a Orientations of planes in space are given by indices hkl which are the reciprocals of the fractional intercepts For example, this is the 321 plane

14. c 2 1 321 b 0 1/2 1 1/3 2/3 a The hkl (321 in this case) actually refers to a set of parallel planes

15. c dhkl hkl b 0 a The perpendicular distance between the planes is called the d spacing for the set of planes

16. c b a Indices can be positive or negative Note the meaning of a zero index

17. c b a Miller Indices of the Cubic, Octahedral, and Dodecahedral Faces of a Crystal in the Cubic System Natural crystal faces tend to have low-numbered Miller indices

18. Symmetry Review of Point Symmetry

19. The motif is the smallest part of a symmetric pattern, and can be any asymmetric chiral* object. To produce a rotationally symmetric pattern, place the same motif on each spoke. The pattern produced is called a proper rotation because it is a real rotation which produces similarity in the pattern. *not superimposable on its mirror image, like a right hand. The pattern is produce by a four-fold proper rotation.

20. Normal crystals contain only five kinds of proper rotational symmetry: One fold, f = 360o (Identity) Two fold, f = 180o Three fold, f = 120o Four fold, f = 90o Six fold, f = 60o Note: molecules have proper rotation axes of any value up to  The proper rotation axis is a line and is denoted by the symbol n (Hermann-Maugin) or Cn (Schoenflies). Thus, the five proper crystallographic rotation axes are called1, 2, 3, 4, 6, or C1, C2, C3, C4, C6.

21. There is another, quite different way to produce a rotationally symmetric pattern: put motifs of the opposite hand on every other spoke. The imaginary operation required to do this is: Rotate the motif through angle f Invert the motif through a point on the rotational axis - this changes the chirality of the motif. This “roto-inversion” is called an improper rotation

22. – i 1 or Inversion Center The simplest is a one-fold improper rotation, which is just inversion through a center, with symbol Another common type of improper rotation is the mirror

23. Example of a crystal with D2h symmetry D2h is the point group, containing C2 axes, mirrors, and inversion center.

24. There is an infinite number of point groups Crystals can fall into only 32 of them, because crystals can have only 1,2,3,4 and 6-order rotation axes The 32 point groups can be further categorized into 7 Crystal Systems Triclinic Monoclinic Orthorhombic Trigonal Tetragonal Hexagonal Cubic

25. Importance of Getting the Symmetry Right A canoe should have C2v symmetry, not C2h

26. Space Group Symmetry

27. Space groups Extend 32 crystallographic point groups by adding translational symmetry elements to form new groups. How many 3-D space groups are there?

28. How many orderly ways are there to pack identical objects of arbitrary shape to fill 3-dimensional space? To answer this, need to extend point symmetry to include periodic structures: Crystals!

29. a a a a a' a' a' Translational Symmetry Can describe the repetition by the direction and distance The set of lattice points describing the 1-D translation is a row

30. a g a b b b b b Net: a 2-D array of equispaced rows on a plane. Unit Cell is “building block” of this 2-D lattice, and is described by a, b, and , the angle between them.

34. Centered Lattice

35. There are five 2-D lattices. Now stack up these nets to form 3-D lattices. Get unit cell with 3 axes and 3 angles Get several new types of centered lattices: End-centered, body-centered, face-centered 14 3-D lattices in all, called Bravais lattices

36. The 14 Bravais Lattices

39. New kind of translational symmetry element: Glide plane Combination of mirror and translation by 1/2 cell length Note: 2 directions associated with glide plane: Direction of mirror and direction of translation.

40. t In 3 Dimensions, also have combination of rotation and translation, called Screw Axes 21 Can also have 3fold, 4fold and 6fold screw axes

41. 3-Dimensional Space Groups • Combine the 14 Bravais lattices with: • 32 crystallographic point groups • Screw Axes • Glide Planes Get 230 3-D space groups So exactly (only?) 230 orderly ways to pack identical objects of arbitrary shape to fill 3-dimensional space.

42. Are the 230 space groups equally represented by actual crystal structures? NO! In Cambridge Structural Database (~700,000 structures containing “organic” carbon) 83.1% of all structures are In just 6 space groups: P21/c 35.1% (monoclinic) P-1 23.0% (triclinic) C2/c 8.1% (monoclinic) P212121 7.9% (orthorhombic) P21 5.5% (monoclinic) Pbca 3.5% (orthorhombic) No other space group with >2% All 230 space groups represented with at least one. P4mm only one structure. 25 space groups <10 structures.

43. X-Ray Diffraction

44. Monochromatic X-Rays from a fixed source The crystal remains in the incident beam during rotation

45. Start rotating the crystal in the xray beam.

46. Start rotating the crystal in the xray beam. Nothing happens until ...

47. dhkl    2 “reflection” θ = Bragg angle, 2θ = scattering angle λ = 2dhkl sin θ (Bragg Equation) λ = wavelength of X-rays dhkl = “interplanar spacings” in the crystal ("reflections" are produced by the diffraction of X-rays)

48. In most orientations, there is no reflection

49. In most orientations, there is no reflection

50. In most orientations, there is no reflection, but different reflections occur at selected orientations.