Crystallography Fundamentals: Structure Analysis and Factors

1. Characterization: Classification: Long range ordering periodicity  unit cell Symmetry  7 crystal systems  230 space groups Structural information: Unit cell  Miller indices (h, k, l)  d spacing Relative intensity  atomic positions

2. X-ray Crystallography Introduction Crystal Diffraction Diffraction  Structure • Structure Amplitudes, Fhkl • Atomic scattering factors • Fourier transfer Fhkl  (x, y, z) • Least squares refinement • Structure properties --- Distance / angles ; packing etc. • Structure data base

3. Structural Analysis: Data measurement:h, k, l, dhkl , nhkl , Ihkl( F2hkl ) Phase determination  heavy atom method; direct method; multiple scattering Fourier transformation: F2  (r); reciprocal  real space Least squares refinement: (r)  xi, yi, zi, ui Structural model: bond distances; bond angles; atomic thermal vibration etc.

4.  l  d Bragg Diffraction

5. Direction (h, k, l) Amplitude Fhkl Phase  Direction  (nhkl) Amplitude fi Phase i P1 P2 f1 f2   Resultant Xtotal X1 X2  0 0 0   90 90 90 180 180 180 f1 f2 270 270 270 360 360 450 450 450 540 540 540 Xtotal (f1f2)cos X1 f1cos X2 f2cos P1 1 P2 f1 2   f2 Resultant Xtotal X2 X1  0 0 0   B’ f2 90 90 90 F  f2sin2 f2 f1 Xtotal f1cos(1)  f2cos(2) 180 180 180 F 2 • cos (f1cos1  f2cos2)  sin (f1sin1  f2sin2) 270 f1 f1sin1  1 360 360 A’ 450 450 450 f2sin2 540 540 f1sin1 540 X1 f1cos(1) X2 f2cos(2) Structure Factor Calculation (A)2  (B’)2  F2cos2  F2sin2  F2 Combination of Wave F  A’  iB’ F ei  Let Xtotal F cos(  )  F cos cos  F sin sin

6. Structure Amplitude (Factor) X1 X2 rn S0 S R Detector

7. Pathlength difference bet. atom n at rn and the origin at 0 Electric field at rn : E0 : electric field amplitude of the incident beam at rn Electric field at P (defection)：

8. 2θ

9. mth order Zeroth order ’ mth order  E ’  E’ a  G D F Incident beam

10. Bh Fh  Ah If centrosymmetric and no anomalous scatter: Bh=0; α=0 orπ

11. h

12. Centric case with non-anomalous scatterers Z pt. charge B = 0 atomic sphere (fixed atom at ri) T=0K B > 0 with thermal vibration T as function of B : Thermal parameter

13. when at (000) Centric case with anomalous scatterers NA : No. of atom types in an asymmetric unit NE : No. of symm. elements of the space group Ri : sym. operator h : (h, k, l) xj : (xj, yj, zj) if then

14. 2 Å Considering nuclear thermal vibration As a point scatterer thermal vibration  electron density smearing if isotropic  spherically symmetric

15. Tanisotropic  thermal vibration as an ellipsoid 3×3 matrix  6 elements uij symmetric: u12= u21; u23 = u32; u13 = u31 based on a, b, c,-axis

16. Thermal ellipsoid diagonize eigen function (three principle axes of the ellipsoid) eigen value u1 u2 u3

17. Constraint in Thermal vibration In case of ab-plane mirror plane sym: U11 , U22 , U33 ， U12 U13=U23=0 x → x y → y z → -z U11 U22 U33 U12 U13 U23 U12=U12 U13 →-U13；U23 →-U23 ∴U13=U23=0

18. Systematic Absences P21 for any atom i at x y z R1 x -x y+1/2 -z R2x

19. 2(cos2ky i sin2ky) when k  2n  0 when k  2n1 Let   hx + lz When   0 i.e. h  0 and I  0 i.e. for 0k0 reflections G0k0  cos2ky  cos2[ky  (k/2)]  i {sin2ky  sin2[ky  (k/2)]}  cos2ky  cos2kycosk  sin2kysink  i sin2ky  i sin2kycosk  i cos2kysink

20. Systematic Absences(space group extinction from translational sym. elements)

24. Difference in phase 2 X-ray Beam Atom Atomic Scattering

25.  r d Atomic Scattering If the atom is a point charge (compared w.r.t the wavelength), it scatters as Z (atomic number) Scattering amplitude amplitude scattered by atom Eatom amplitude scattered by free e- Ee (factor) ρ(r) ：electron density around nucleus dq = ρ(r) dV q: charge V : volume

27. For example fZ Int’l Table of X-ray CrystallographyVol C Ψ mostly from HF p.477 sinθ/λ Analytical form : Vol C.p500.

28. f0 for atoms from Z = 1 to 90

29. coefficients of analytical form in atomic scattering factors

31. Gd Sm 30 30 20 f” f” 20 f” Gd 10 10 30 0 0 20 10 10 f’ f’ 10 20 20 30 30 0 f’ 0 30 20 10 185 1705 1715 184 1710 (Å) (Å) f” Sm 30 20 10 0 f’ 0 30 20 10 Anomalous Scattering (b) (a) • Fig. : • Anomalous scattering terms f’ and f” for： • gadolinium near the L3 edge； • samarium near the L3 edge (d) (c) • Fig. : • Plot in the complex plane of f’  i f”： • gadolinium near the L3 edge； • samarium near the L3 edge

32. Atomic f’ and f”

33. Relationship between Fhkl & ρuvw Atomic scattering Crystal scattering general form with continuous ρ(u,v,w) Corresponds to the intensity of h,k,l refln. reciprocal space corresponds to the electron density at u,v,w position direction space