1. A Brief Description of the Crystallographic Experiment
Much of what is done in obtaining a crystal structure appears strange at first.
The point of this lecture is to give an overview of what is happening and why.

2. What we want.
The idea of the analysis is to determine the location of all the atoms in a solid.
Since the the solid is made up of many atoms this is daunting. For example in 40g of NaCl there are 6.02x1023 formula units
However, a crystalline solid is made up of repeating units called unit cells. Thus the problem is reduced to determining the contents of one unit cell which when translated builds up the crystal. For NaCl this reduces to finding 4 Na and 4 Cl ions
The unit cells can contain from 1 to 192 repeating sub units

3. The identical sub units are related to each other within the unit cell by the crystallographic symmetry of the unit cell which is called the space group. The space group name describes the symmetry of the unit cell the same way D4h describes the symmetry of a molecule.
In most cases the sub group called the asymmetric unit contains one formula unit including any solvents or host molecules.
Therefore the problem of finding all the atoms in the crystal becomes one of determining the atoms in the asymmetric unit; for NaCl one Na and one Cl

4. What do we mean by atoms?
We treat the idea of an individual atom as if it were something that could be picked up by small tweezers.
In reality the atom is mostly empty space with a very small nucleus.
Presently there is no way to magnify the atoms so the can be directly observed. Even if this could be done what would it look like?

5. Real Space
Since in a crystal every atom has a regular location, a coordinate system can be built which gives the location of any atom in the crystal.
This is called real space and is what we want to determine.
The locations are given as coordinates (x,y,z) within the unit cell. Since these vaules are in one unit cell x,y,z are called fractional coordinates. The fractional part indicated where in the unit cell the atom is located while the integer part labels which unit cell.

6. Electron Density
Since atoms cannot be seen we need some other property to measure them.
Electron density is easily determined.
Thus in locations in the crystal where there is electron density an atom is located. Note this is an assumption!
The greater the density the greater the atomic number of the atom.

7. 1-D Real Space

8. 2-Dimensions In this case both axes are needed to represent location.
The intensity can be given in two ways.
It can be provided as numbers at a regular basis. This is not particularly useful
Lines of equal values can be used to form contour maps which are more instructive.

9. A numerical pressure map

10. A pressure contour map

11. Crystallographic Electron Density

12. 3-d Maps
The old way of building 3-d maps was to draw 2-d contour maps on clear plastic sheets and then stack the sheets so you could look down in the third dimension
Fortunately today maps are rarely output. Instead software scans the map and calculates where the peak maxima are.

14. Obtaining the Electron Density
The unit cell is too small for any experiment to directly measure the electron density inside it.
First a single unit cell would be required.
Secondly a probe small enough to fit at various locations inside the cell and measure the values would be required.

15. Fourier Transform
While it may not make sense yet one way to solve the measurement problem is to Fourier transform real space into a new space.
Assume for the moment this will get us somewhere.
This new transformed space will be called reciprocal space.

16. The Properties of Reciprocal Space
Reciprocal space is quite weird.
The dimensions of the axes are in units of 1/d or in the case of crystals Å-1. Å is the symbol for Angstrom units or 10-8 cm.
The coordinates of reciprocal space are given by (h,k,l)
h,k, and l must be integers!

17. Observing Reciprocal Space
Obviously, reciprocal space can be calculated by performing a Fourier transform on real space. Since real space is what we are trying to determine this is not helpful. It is also possible to transform reciprocal space to real space.
Reciprocal space can also be observed by using any interacting radiation that has a wavelength about the length of the unit cell. This would be a couple of Angstrom or less.

18. Crystal Diffraction
High energy neutrons interact with the atomic nuclei and if the wavelength is right then neutron diffraction can be observed.
High energy electrons or x-rays (electromagnetic radiation) can interact with the electron density. X-rays are perfect as they have a wavelength in the range of Å.
This observed diffraction is the Fourier transform of real space or is reciprocal space and can be observed.

19. Reciprocal and Real Space
Obviously, there must be a relationship between real and reciprocal space.
If the edges of the unit cell are a, b and c then the edges of the reciprocal cell are a*, b*, and c*
Furthermore the symmetry of the real cell must be reflected in the reciprocal cell
Obviously there is a way of relating the real and reciprocal cells. Note in general a* is not equal to 1/a.

20. Reciprocal Space

21. So it seems easy
Measure the intensity of the diffraction spots and determine their h,k,l and then do a Fourier transform to see the electron density.
The first problem is that amplitude of the diffraction is needed (called F) and the intensity (I) is measured.
I α F2
Since F has a sign (called the phase) this information is lost when F is calculated. For every 1000 data collected there are 2(1000-1) (5.36x10300) possible assignment of signs.

22. Maybe Phases are not improtant.
It should be pointed out that the phase equals sign equivalence only applies to crystals with a center of inversion.
For crystals lacking this symmetry the phases are more complex.
Lets see if we really need the phases.

25. Duck Intensity Cat Phases

26. Cat Intensity Duck Phases

27. The Phase Problem Obviously the phases are important.
The problem cannot proceed without them and because of the nature of X-rays they can not be determined as part of the data collection.
So the direct Fourier approach fails.
However, it is important we can calculate the data and hence the phases if we know what real space looks like.

28. How to proceed
1. The location of at least 15% of the electron density must be determined. This density could be in one or a few heavy atoms or many light atoms. Obtaining this initial information is called solving the structure.
2. Once we have an initial model, the coordinates and adps of the atom(s) can be adjusted to give the best fit between the observed data (Fc) and the calculated data (Fo). The data can then be calculated by a Fourier transform of real space to reciprocal space.

29. 3. The calculated Fc's contain the phase information though they may not all be correct as the model is not complete. These phases can then be applied to Fo and now a Fourier transform can be run on Fo to get a better electron density map and atoms added.
4. This cycle is repeated until all the atoms are found, all the parameters are refined and their values converged.

30. Homework
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