Electron Density in Crystals

1. Electron Density in Crystals • Since the crystal is made up of repeating unit cells, the electron density in the cell must be periodic! • This is called the Charge Density Wave. • In 1-dimension, the length of the charge density wave is the cell length.

2. One-Dimensional Crystal

3. Charge Density Wave

4. Fourier Series

5. Wave Components One wavelength/cell h=1 ten wavelengths/cell h=10

6. Correct Phasing

7. Incorrect Phasing

8. Diffraction in 1-D • is the electron density at any point x F(h) is called the structure factor and is indexed by the value h F is related to the square root of the intensity of the diffracted beam. Fo is the observed structure factorFc calculated

9. A Comparison • The Charge Density Wave (CDW) can be decomposed into Fourier components which when summed give the CDW. • The CDW can also be calculated from a Fourier sum of the diffracted waves. • The structure factors F represent the Fourier components of the CDW • The amplitude F is the Fourier coeffcient!

10. Shift of Origin

11. Shift of Origin • Clearly this shift does nothing to change the Fourier Coefficients since the shape of the wave is unchanged. • All that is needed is to change the phase of each wave by ½ wavelength. • The location of the atoms is in the phases and not the coefficients!! • Cannot tell positions from diffraction data

12. Fourier Domains • In the crystal the electron density makes up real space. • Real space has the symmetry of the space group • The Fourier Transfer generates reciprocal space • Reciprocal Space is the diffraction space • Reciprocal space has the same symmetry as real space.

13. Views of Reciprocal Space

14. NMR

16. Real Reciprocal Relations • Reciprocal lengths and angles are indicated by a * (a* b* c* a* b* g*) • a*.a=1 a*.b=0 a*.c=0 etc. • Therefore a* is perpendicular to the bc plane, b* to the ac plane and c* the ab plane.

17. Reciprocal cell

19. 1-D Centered Cell

20. Expand to 3-D • The only waves possible must be components of the 3 1-d systems along a, b, and c. • Use h along a k along b and l along c • F(hkl) is the 3-d structure factor

21. Diffraction Data 1 1 7 537.22 13.60 1 1 8 517.09 12.86 -1 -1 -9 712.87 17.72 1 1 9 716.10 17.99 -1 -1 -10 341.13 9.31 1 1 10 348.02 9.23 -1 -1 -11 8.12 0.46 1 1 11 8.14 0.42 -1 -1 -12 0.26 0.21 1 1 12 0.16 0.22 -1 -1 -13 177.19 5.10 1 1 13 176.55 4.88 -1 -1 -14 363.25 9.10 1 1 14 372.53 10.10 • H K L F2 s(F2)

22. 2-fold Screw Axes • Along b have –x,1/2+y,-z • For points on the axis (i.e. 0,y,0) this looks like a translation only. • Just like 1-d centered cell • Result for 0k0 h=2n systematic presence • If the screw does not run through origin the same presences exit because cannot detect the origin shift!

23. 31 Along C axis • In this case only waves with l=3 are allowed • Thus the condition is 00l, l=3n

24. Glide perpendicular to b • As before, points in the ac plane see only translation. • Translate ½ along c • Allowed h0l l=2n • For n glide translation is ½ along the ac diagonal resulting in h0l h+l=2n

25. Centering • C centering adds (1/2,1/2,0) to all points. • Therefore for all hkl h+k=2n • I centering adds (1/2,1/2,1/2) to all points • Therefore for all hkl h+k+l=2n

27. C2/c

28. Pnma