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. It broke into rhombohedra Intentional breakage of a rhombohedron produces smaller and smaller rhombohedra Calcite, CaCO 3

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

7. Unit cell, in yellow, gives directions and distances of translationally repeating unit 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. Molecules within the unit cell are related by symmetry The asymmetric unit here is one molecule, but may be several, or less than one. The number of molecules in the unit cell (Z) here is 4

11. Fractional Coordinates b c a x y z 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. 0 a b c 1 1/2 1/3 Miller Indices 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. 0 a b c 1 1/2 1/3 321 2 2/3 1 The hkl (321 in this case) actually refers to a set of parallel planes

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

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

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

18. Symmetry Review of Point Symmetry

19. chiral 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. proper rotation 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 foldIdentity 1. One fold,  = 360 o (Identity) Two fold 2. Two fold,  = 180 o Three fold 3. Three fold,  = 120 o Four fold 4. Four fold,  = 90 o Six fold 5. Six fold,  = 60 o n C n 12346C 1 C 2 C 3 C 4 C 6 The proper rotation axis is a line and is denoted by the symbol n (Hermann-Maugin) or C n (Schoenflies). Thus, the five proper crystallographic rotation axes are called 1, 2, 3, 4, 6, or C 1, C 2, C 3, C 4, C 6. Note: molecules have proper rotation axes of any value up to 

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 Rotate the motif through angle  Invert Invert the motif through a point on the rotational axis - this changes the chirality of the motif. improper rotation This “roto-inversion” is called an improper rotation

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

23. Example of a crystal with D 2h symmetry D 2h is the point group, containing C 2 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 C 2v symmetry, not C 2h

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

30. Net: a 2-D array of equispaced rows on a plane. a bbbb a b  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. 21212121 t In 3 Dimensions, also have combination of rotation and translation, called Screw Axes 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: P2 1 /c35.1% (monoclinic) P-123.0% (triclinic) C2/c8.1% (monoclinic) P2 1 2 1 2 1 7.9% (orthorhombic) P2 1 5.5% (monoclinic) Pbca3.5% (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. Nothing happens until...

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

48. In most orientations, there is no reflection

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

56. A Detector records the position and intensity of each reflected beam.

57. A computer records the position, intensity, and crystal orientation of each reflection

59. Intensity Data Collection (Single Crystal)

60. Kappa Apex II Diffractometer

62. Diffraction Pattern Recorded on Film Each spot has indices hkl Image from CCD Detector Intensity of each hkl measured Observed structure factor F o (hkl) proportional to √I hkl

63. Spacing of spots gives dimensions of unit cell Symmetry of pattern related to symmetry of crystal Intensities of spots related to electron density in unit cell Systematic absences yield clues about space group of crystal

64. Crystallographic Application of Fourier Summation In its application to crystallography, the Fourier coefficients are the structure factors F hkl (derived from measured intensities) and the continuous function is 3D electron density in the unit cell F hkl  (xyz) Structure Factors Electron density

65. real imaginary F hkl  hkl F hkl has an amplitude |F hkl | and a phase angle  hkl

67. Fourier Map From this, we can extract xyz values for atoms and generate a mathematical model of the crystal.

68. Structure Solution

69. Phase Problem So IF we know coordinates, phase can be calculated: Review of structure-Factor equation: F hkl = A hkl + iB hkl Where: And we can calculate electron density by Fourier series: Where:

70. But cannot calculate density unless we know phases So need phases to calculate density (and locate atoms) and need atom positions to calculate phases. A Classic Catch-22! What to do?

71. J. Karle, H. Hauptman (Nobel Prize in Chemistry, 1985) They developed statistical methods for estimating phases directly from measured |F o | values. Called “Direct Methods” and used in almost all small-molecule structure solutions today

72. Example Monoclinic space group P2 1 /n From Prof. Graca Vicente’s (LSU) laboratory 4508 total reflections, 340 reflections used in direct methods calculation

73. Incorrect solution (wrong phases)

74. Correct solution (approximately correct phases)

75. Final structure

76. A Fourier map, phased on all atoms. Note small peaks likely corresponding to H atoms Locate H atoms using a Difference Map, in which contribution from atoms already in model subtracted out.

77. Difference map showing H atoms

78. ORTEP drawing of model after least-squares fitting

79. Also get details of intermolecular contacts Example: Hydrogen bonding in D-mannitol

80. Structure Refinement

81. Linear Least Squares The "best" model is the one that minimizes the sum of squares of deviations with respect to the linear coefficients of the model function. As an example, a straight line is used as the model, 2 parameters: a & b The final result is obtained in one calculation y x y = ax +b

82. In crystallography, the observations are the structure-factor amplitudes |F o | The model is the set of adjustable parameters: xyz for each atom, displacement (thermal) parameters scale factor, etc., from which F c are calculated. So, we would like to find the best match between the F o and F c amplitudes by minimizing the sum of the squares of their differences :  ( |F o | - |F c | ) 2  w(F o 2 - F c 2 ) 2 In practice, most software refines on F 2 rather than F, and weights individual reflections according to their precision

83. But the equations are not linear So they must be approximated with linear equations and recycling is necessary until convergence is reached.

84. Anisotropic Displacement Parameters Each atom modeled with one isotropic displacement parameter U iso Each atom modeled with six anisotropic displacement parameters U 11 U 22 U 33 U 12 U 13 U 23

85. Is the result correct? This should be small for a correct refinement Typically 0.02 - 0.05, but may be higher for a weakly-scattering crystal. R =  |F obs – F calc | /  |F obs | Crystallographic R Factor Other indicators: Is the model chemically reasonable? Are the ellipsoids reasonably shaped? Are the H atoms visible in difference maps?

86. Absolute Structure Determination

87. Enantiomers give identical intensities And we usually get only relative configurations of chiral centers from X-ray data R S R S S R

88. But if a heavy atom is present and the wavelength is chosen carefully, enantiomeric crystals give a slightly different set of intensities, and one enantiomer fits the experimental data better than the other. In the last few years, new methods have been developed, and oxygen in CuK  radiation is “heavy” enough, with high-quality crystals.

89. (+) Nootkatone The major flavorant of grapefruit Insect repellent and termiticide towards Formosan termite Absolute configuration determined by X-ray methods Wikipedia.org A. M. Sauer, F. R. Fronczek, B. C. R. Zhu, W. E. Crowe, G. Henderson, R. A. Laine, Acta Cryst. C59, o254-o256 (2003).