Lattices and Symmetry Scattering and Diffraction Physics Adapted

Lattices and Symmetry Scattering and Diffraction (Physics) (Adapted from: James A. Kaduk IIT Adjunct Faculty, Chemistry) 1

Bragg’s Law Fundamental statement of the way waves scatter off of planes of scatterers n = 2 dsin Rearranging, (1/2 d) = (1/n)sin / We can formulate this so that the n disappears, so that (1/2 d) = sin / 2

Bragg’s Law V. K. Pecharsky and P. Y. Zavalij, Fundamentals of Powder Diffraction and 3 Structural Characterization of Materials, p. 148 (2003)

Optical Diffraction PSSC Physics, Figure 18 -A, p. 202 -203 (1965) 4

Optical Diffraction PSSC Physics, Figure 18 -B, p. 202 -203 (1965) 5

Optical Diffraction D. Halliday and R. Resnick, Physics, p. 1124 (1962) 6

Scattering by One Electron Oscillation direction of the electron X-ray beam Electron Electric vector of The incident beam 7

Electromagnetic Waves E A A z E(t=0; z) = A cos 2 p(z/ ) 8

Consider a new wave displaced by a distance Z from the original wave: E Z new wave z original wave Z corresponds to a phase shift 2π(Z/λ) = α 9

Scattering by Two Electrons p s 0 2 r q 1 s 2 1 • Let the magnitudes of s 0 and s = 1/λ • Diffracted beams 1 and 2 have the same magnitude, but differ in phase because of the path difference p + q 10

Phase difference of wave 2 with respect to wave 1 2 pr • (s-s 0)/ = 2 pr • S/ where S = (s-s 0) 11

Scattering by an Atom 12

The Atomic Scattering Factor f(S) = ∫ r(r)exp(2 pir • S)d 3 r = 2 ∫ r(r)cos(2 pr • S)d 3 r 13

Atomic Scattering Factors V. K. Pecharsky and P. Y. Zavalij, Fundamentals of Powder Diffraction and Structural Characterization of Materials, p. 213 (2003) 14

Scattering from a Row of Atoms M. J. Buerger, X-ray Crystallography, Fig. 15, p. 33 (1942) 15

Scattering by a Plane of Atoms M. J. Buerger, X-ray Crystallography, Fig. 16, p. 34 (1942) 16

Scattering by a Unit Cell J. Drenth, Principles of Protein X-ray Crystallography, Fig. 4. 12 p. 80 (1999) 17

Scattering by a Unit Cell fj(S) = fjexp(2 pirj • S) for each atom j in the unit cell For the entire ensemble of atoms, F(S) = ∑j=1 n fjexp(2 pirj • S) 18

The crystallographer’s world view Reality can be more complex! 19

Molecules pack together in a regular pattern to form a crystal. There are two aspects to this pattern: Periodicity Symmetry First consider the periodicity… 20

To describe the periodicity, we superimpose (mentally) on the crystal structure a lattice. A lattice is a regular array of geometrical points, each of which has the same environment (they are all equivalent). 21

A Primitive Cubic Lattice (Cs. Cl) 22

A unit cell of a lattice (or crystal) is a volume which can describe the lattice using only translations. In 3 dimensions (for crystallographers), this volume is a parallelepiped. Such a volume can be defined by six numbers – the lengths of the three sides, and the angles between them – or three basis vectors. 23

A Unit Cell 24

Descriptions of the Unit Cell a, b, c, , , a, b, c x 1 a + x 2 b + x 3 c, 0 xn < 1 lattice points = ha + kb + lc, hkl integers domain of influence – Dirichlet domain, Voronoi domain, Wigner-Seitz cell, Brillouin zone 25

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The Reduced Cell • 3 shortest non-coplanar translations • Main Conditions (shortest vectors) • Special Conditions (unique) • May not exhibit the true symmetry 27

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Point Symmetry Elements • A point symmetry operation does not alter at least one point upon which it operates – Rotation axes – Mirror planes – Rotation-inversion axes (rotation-reflection) – Center Screw axes and glide planes are not point symmetry elements! 29

Symmetry Operations • A proper symmetry operation does not invert the handedness of a chiral object – Rotation – Screw axis – Translation • An improper symmetry operation inverts the handedness of a chiral object – – Reflection Inversion Glide plane Rotation-inversion 30

Not all combinations of symmetry elements are possible. In addition, some point symmetry elements are not possible if there is to be translational symmetry as well. There are only 32 crystallographic point groups consistent with periodicity in three dimensions. 31

2 Rotation Axis (ZINJAH) 32

3 Rotation Axis (ZIRNAP) 33

4 Rotation Axis (FOYTAO) 34

6 Rotation Axis (GIKDOT) 35

-1 Inversion Center (ABMQZD) 36

m Mirror Plane (CACVUY) 37

-3 Rotary Inversion Axis (DOXBOH) 38

-4 Rotary Inversion Axis (MEDBUS) 39

-6 Rotary Inversion Axis (NOKDEW) 40

21 Screw Axis (ABEBIS) 41

31 Screw Axis (AMBZPH) 42

32 Screw Axis (CEBYUD) 43

41 Screw Axis (ATYRMA 10) 44

42 Screw Axis (HYDTML) 45

43 Screw Axis (PIHCAK) 46

61 Screw Axis (DOTREJ) 47

62 Screw Axis (BHPETS 10) 48

63 Screw Axis (NAIACE) 49

64 Screw Axis (TOXQUS) 50

65 Screw Axis (BEHPEJ) 51

c Glide (ABOPOW) 52

n Glide (BOLZIL) 53

d (diamond) Glide (FURHUV) 54

Not all space groups are possible for protein crystals, because the chirality of the building blocks requires that the assembled molecule be chiral. 55

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Scattering by a Crystal 57

Total scattering by a crystal For a unit cell with its own origin at ta + ub + vc the scattering is F(S) e 2 pita • Se 2 piub • Se 2 pivc • S and the total scattering by the crystal is K(S) = F(S) ∑t=0 n 1 e 2 pita • S ∑u=0 n 2 e 2 piub • S ∑v=0 n 3 e 2 pivc • S 58

Fourier Transform The Fourier transform is an equation to calculate the frequency, amplitude and phase of each sine wave needed to make up any given signal x(t): F(f) = ∫+x(t)ei 2 pftdt 59

Brightness image / Fourier transform http: //cns-alumni. bu. edu/~slehar/fourier. html 60

Kevin Cowtan’s Fourier Duck http: //www. ysbl. york. ac. uk/~cowtan/fourier/magic. html 61

Kevin Cowtan’s Fourier Cat http: //www. ysbl. york. ac. uk/~cowtan/fourier/magic. html 62

Magnitudes from duck and phases from cat 63

Magnitudes from cat and phases from duck 64

A tail of two cats http: //www. ysbl. york. ac. uk/~cowtan/fourier/coeff. html Magnitudes only 65

A similar structure – tail-less Manx cat 66

Known magnitudes and Manx phases “an |Fo| map” Despite the fact that the phases contain more structural information about the image than the magnitudes, the missing tail is restored at about half of its original weight. This occurs only when the phases are almost correct. The factor of one half arises because we are making the right correction parallel to the estimated phase, but no correction perpendicular to the phase (and <cos 2>=1/2). There is also some noise in the image. 67

“A 2|Fo|-|Fc| map” 68

The “reflecting planes” are lattice planes 69

The Ewald Construction 70

Mosaicity (Mosaic Spread) B. D. Cullity and S. R. Stock, Elements of X-ray Diffraction, p. 175 (2001). 71

The Patterson Function Let P(u, v, w) = P(u) = (1/V)∑h |F(h)|2 cos(2 ph • u) Or (if there is no anomalous dispersion) P(u) = (1/V)∑h |F(h)|2 e 2 pih • u 72

What does the Patterson function mean? Consider an alternate expression P(u) = ∫r r(r)r(r+u)dvreal This shows that the Patterson function is actually a type of correlation function of the electron density with itself, or, to put it another way, it’s a convolution of the electron density with its inverse. 73

More about Structure Factors F(S) = ∑j=1 n fje 2 pirj • S For noncentrosymmetric structures, the following reformulation may be useful: F(S) = ∑j=1 n fjcos(2 pirj • S) + i ∑j=1 n fjsin(2 pirj • S) = A(S) + i. B(S) 74

Falloff in intensity with resolution The falloff makes statistical analysis awkward. Intensities are normally measured on an arbitrary scale. For large numbers of welldistributed S <|F(S)|2> = <I(abs, S)> = ∑ifi 2 But we estimate fi 2 as 2 / 2 2 0 2 2 B sin fi = (fi ) e i Where Bi is a measure of the mobility of atom i. 75

The Wilson Plot 2 / 2 2 Bsin Ke <I(S)> = K<I(abs, S)> = ∑i(fi 0)2 ln {<I(S)> / ∑i(fi 0)2} = ln K + 2 Bsin 2 / 2 http: //www. ysbl. york. ac. uk/~mgwt/CCP 4/EJD/bms 10. html 76

Normalized Structure Factors 77

Least Squares Refinement • Assume observations are uncorrelated • If Error distributions are Gaussian Observations are weighted by 1/ 2 LS gives maximum likelihood estimates of xj • Restraints count as observations (weights? ) E. Prince, Mathematical Techniques in Crystallography and Materials Science, Springer-Verlag (1994) 78

Accuracy vs. Precision • The standard uncertainty (esd) is a measure of precision • LS yields minimum variances • Correlations • Systematic errors • Accuracy? International Tables for Crystallography, Volume F, Section 18. 5 79

The Macromolecular CIF Dictionary (mm. CIF) International Tables for Crystallography, Volume G Section 4. 5 pages 295 -443 (!) 80