1. What is the problem? How was the problem solved?
The Phase Problem or
The Problem of Crystallography
What is the problem?
How was the problem solved?

2. In the context of a single crystal x-ray study
A Structure Query (What did I make?)
Needed: A Single Crystal (0.2 mm)
Collection of X-Ray Diffraction Data
Solution of the Phase Problem
(To get a Preliminary Model)
Structure Model Parameter Refinement
Interpretation of the Result/Geometry Analysis
Validation of the Analysis
Report and Publication

3. Example of an Oil-mounted Crystal

5. Diffraction Pattern of a crystal rotated over 1 degree
H,K,L, I
Diffraction Pattern of a crystal rotated over 1 degree

6. Microscope versus X-ray diffraction
Glusker and Trueblood
Um 3D info zu erhalten, muss Objekt von verschiedenen Seiten betrachtet werden.
Falls Objekt opak: info ueber aeussere 3D Oberflaeche.
Falls Objekt teilweise transparent: vollstaendige Information (Computer-Tomographie).
same principle, no lenses

7. Bragg’s Law
nl = 2dsinq

8. Summation of waves A wave: wavelength, speed, amplitide (F), phase (j)
The result of a two waves
summation depends on
their amplitudes and (relative) phase
Um zu verstehen wie solche komplexe Muster entstehen können
und wodurch Lage und Intensität der Reflexe bestimmt sind, schauen wir uns erst
einmal die (hypothetische) Streuung an einem Einzelobjekt (z.b. Makromolekül) an.
Winkelbeziehungen bleiben erhalten.
Die Elektronen der einzelnen Streuzentren (Atome) werden zum Mitschwingen angeregt -> Emission von Kugelwellen (Huygensches Prinzip) in alle Richtungen.
Im folgenden behandeln wir nur elastische Streuung, d.h. kein Energieverlust (keine Wellenlängenänderung) durch den Streuprozess.
Betrachten wir nun eine dieser Richtungen.
Die von den einzelnen Streuzentren herrührenden Teilwellen interferieren miteinander.

10. Diffraction from any object
X-rays will scatter on each atom of the object:
scatter predominantly on electron shells, not nuclei
elastic (=same energy)
in all directions
The intensity of diffracted radiation in a particular direction will depend on the interference (=sum) of scattered waves from every atom of the object

11. The structure factor is a mathematical function describing the amplitude and phase of a wave diffracted from crystal lattice planes characterized by Miller indices h,k,l.
The structure factor may be expressed as
where the sum is over all atoms in the unit cell, xj,yj,zj are the positional coordinates of the jth atom, fj is the scattering factor of the jth atom, and αhkl is the phase of the diffracted beam.

13. Fourier Theory
Diffraction pattern related to object
Mathematical operation called Fourier Transform
Can be inverted to give pattern of electron density
Requires amplitude and phase of diffracted waves
The Phase Problem

14. Fourier Transform of a molecule
F T

15. Fourier Transform of a crystal

16. Diffraction as Fourier transform
Real space (x,y,z):
electron density r(x,y,z)
‘Reciprocal space’ (h,k,l):
diffracted waves F(h,k,l), j(h,k,l)
Physics tells us that the diffracted waves are Fourier transforms of
the electron density:
Moreover, a backward transform (synthesis) should bring us from waves
back to the electron density:
I.e. once we know the amplitudes and phases of diffracted waves
we can calculate the electron density!

17. Summary. Structure Factor F(hkl)
How to get to an electron density map from the intensities of reflections?
Intrinsic amplitude of scattered X-ray from each type of atom depends on its electron density r(xyz) (described by atomic scattering factor f)
f is used to describe scattering vector for each atom for one reflection (with Miller indices h,k,l)
Sum of atomic scattering vectors is molecular scattering factor F, also called structure factor (can be calculated for a given structure):
F(hkl) = ∫r(xyz) e-2pi(hx+ky+lz) Fourier Transformation

18. The Phase Problem
Continued: How to get to an electron density map from the intensities of reflections?
Fourier Transformation (other way around)
r(xyz) = ∫F(hkl) e2pi(hx+ky+lz)
Magnitude of F is proportional to square root of intensity:
|F(hkl)| ~ √ I(hkl)
Thus: r(xyz) = 1/NV ∑ ∑ ∑ √I(hkl) e-i(hkl) e2pi(hx+ky+lz)
h k l
Electron Density
Intensity
Phase
Missing Information to get electron density!

19. What is the Phase Problem?
From diffraction experiment; only measure the intensities (amplitude2)
Phase information is lost!
..hence the ‘phase problem’
Can we survive without the lost phase information?.....

20. Phases are more important than amplitudes!

21. Phase Problem
What we measure is not F directly, but the intensity I where
But F = (amplitude)ei(phase) and
so that we lose all the phase information and cannot do the inverse FT exactly

22. Phase Problem Phase has been lost What we need: What we have:
What we can get:
What we miss:
Phase and Amplitude of diffracted waves
Number of X-ray photons in each spot
Intensity = Amplitude2
Relative phase angles for different spots
Phase has been lost

23. What next? We can measure: We want to find out:
Diffraction spot intensities
Unit cell dimensions
Symmetry group
We want to find out:
Location of each atom in unit cell
Type of atom and expected F
Phase associated with each atom

24. Solution of the Phase Problem
Early Method: Trial & Error (Salts such as NaCl, Silicates etc.)
Patterson Methods (Heavy Atom)
Direct Methods (SHELXS, DIRDIF, SIR)
Isomorphous Replacement
New: Charge Flipping (Ab-initio)
Phase Problem Solved! Given reasonable data.

25. Recovering phases…the Patterson function
Invented by W. L. Patterson for small molecules
Patterson map is calculated with the square of structure factor amplitude and a phase of zero
This is an interatomic vector map
Each peak corresponds to a vector between atoms in the crystal
Peak intensity is the product of electron densities of each atom

26. Patterson Function
Number of spots = N2 where N = # atoms - gets huge!

28. From model building to developing a structure Fourier synthesis
Once phases have been determined as accurately as possible, an e-density map (the Fourier transform of the structure factors with phases) is prepared and interpreted.
Use observed structure amplitudes |Fobs(hkl)| with best phases, best(hkl) to give experimentally derived structure factors, Fexp(hkl):
Fexp(hkl) = |Fobs(hkl)| exp[ i best(hkl)]
Fourier transform creates e-density exp(xyz):
Iterative Process: Model is subtly changed and structure factors are calculated (Fcalc) and compared with measured Fobs to see how well a model fits the data.

29. Direct Methods
Derive phases directly from the magnitudes
Positivity and atomicity
Equal atom structure

30. Structure Factors

31. Sign relationship. An important outcome of the Sayre Equation
Sh  Sh’ Sh - h’ or S-hSh’ Sh - h’  +1
Sign relationship. An important outcome of the Sayre Equation

33. D. Harker and J. Kasper
D. Sayre
1950’s J. Karle and H. Hauptman
I. L. Karle and J. Karle
1970’s M. M. Woolfson
Nobel Prize awarded to H. Hauptman and J. Karle

34. Isomorphous replacement
Used early 1900’s for small molecules by Groth (1908); Beevers and Lipson (1934) ; Robertson (1935)
Perutz (1956) and Kendrew (1958) used it on proteins
Use of heavy atom substitution in a crystal
“Isomorphous” – same shape
“Replacement” – heavy atom might be replacing light salts or solvent molecules
Why heavy atoms? large atomic numbers; Contribute disproportionately to the intensities
Principle: change the crystal (with heavy atoms); perturb the structure factors; conclude phases from how the structure factors are perturbed
Collect two (or more) datasets: ‘native’ and ‘derivative’ data

35. The centrosymmetric space group was confirmed by the absence of a piezoelectric and pyroelectric effect, which is seen in resorcinol
The unit cell contains two centrosymmetric molecules. The metal atom and the four surrounding isoindole nitrogen atoms must all lie strictly in one plane

36. If a reflection from the free compound corresponds in phase to a peak at the centre of symmetry, then the same reflection from the metal compound will be of greater intensity owing to the addition of the peak due to the metal atom alone, and the phase constants of both these reflections will be 0 (or positive sign). If, however, the reflection from the free compound corresponds in phase to a trough at the centre of symmetry, then the addition of the peak due to the metal atom alone will tend to fill up this trough, and the resultant reflection from the metal compound will usually be of smaller intensity, the phase constants of both reflections in this case being x (or negative sign). If the reflection from the free compound corresponds to a very small trough (weak intensity), the phase may be reversed by the addition of the metal atom. When the structure amplitudes are expressed in absolute measure there is practically no ambiguity in the principal (h01) z one, and the phase constants of these reflections, numbering about 300, from phthalocyanine and its nickel derivative have been determined with certainty.

37. The usual difficulty of the unknown phase constant, has been overcome by comparing absolute measurements of corresponding measurements from nickel phthalocyanine and the metal-free compound which leads to a direct determination of all the significant phase constants in the (h0l) zones of the two compounds, numbering about 300. A Fourier analysis of these results determines two co-ordinates of each carbon and nitrogen atom in the structure and the regularity of the projection shows beyond doubt that the molecule is planar.

38. Isomorphous replacement contd…
Calculate the Isomorphous difference Fiso
Use Fiso in direct or Patterson method to deduce position of heavy atoms
Then deduce possible values for (protein) phase angles

39. Completing the structure
Given the Diffraction Data and (Approximate) Phases a 3D Electron Density Map can be Calculated.
rx,y,z = 1/V Shkl Fhkl exp{-2pi(hx + ky + lz)}
Fhkl = |Fhkl|exp(fhkl)
Following is a section through such a map

42. Interpretation in Terms of Atoms
Position of highest density => Position x,y,z
Deviation of the density shape from the ideal atomic electron density => Thermal motion parameters:
Isotropic: U(iso) or
Anisotropic: U11,U12,U13,U22,U23,U33
(Displacement Parameters) => ORTEP
Note: ORTEP does NOT represent the electron distribution

43. Structural Refinement

44. Interpretation in Terms of Bonds
Bonds between atoms of type A and B are assigned on the basis of atomic covalent radii with: d < R(A) + R(B) + 0.4
‘Crystallographic Bonds’ are not necessarily Chemical Bonds.
Van der Waals Radii are used to detect isolated molecular species or short contacts.

45. Display Options
Ball-and-Stick
Simple but may hide problems with a structure.
ORTEP
Often preferred because it visualizes most model parameters and possible problems.
CPK
Spacefilling PLOT illustrating the shape etc

46. A pretty picture, but what about the numbers …

49. Summary. Central Formulae
Diffraction Spots: 2dhkl sinqhkl = nl
Solve the Phase problem. Get the model
Electron Density Map (3D Fourier Map)
rx,y,z = 1/V Shkl Fhkl exp{-2pi(hx + ky + lz)}
Structure Factor (Improve the model)
Fhkl(calc) = Sj fj Tj,hkl exp{2pi(hxj+kyj+lzj)}
Least Squares Model Refinement
Minimize: Shkl [whkl(|Fhkl(obs)|2-|Fhkl(calc)|2)]2
Convergence Criteria: R1, wR2, S

50. End of First Lecture