1. 1 Quasicrystals AlCuLi QC Rhombic triacontrahedral grain Typical decagonal QC diffraction pattern (TEM)

2. 2 Quasicrystals Diffraction pattern for 8-fold QC Diffraction pattern for 12-fold QC

3. 3 Quasicrystals Principal types of QCs: icosahedral decagonal

4. 4 Quasicrystals Principal types of QCs: icosahedral decagonal metastable (rapid solidifcation) stable (conventional solidification)

5. 5 Quasicrystals Principal types of QCs: icosahedral decagonal metastable (rapid solidifcation) stable (conventional solidification) QCs usually have compositions close to crystalline phases - the "crystalline approximants"

6. 6 Quasicrystals While pentagons (108° angles) cannot tile to fill 2=D space, two rhombs w/ 72° & 36° angles can - if matching rules are followed N.B. - see definitive & comprehensive book on tiling by Grünbaum and Shepherd

7. 7 Quasicrystals While pentagons (108° angles) cannot tile to fill 2=D space, two rhombs w/ 72° & 36° angles can - if matching rules are followed

8. 8 Quasicrystals Fourier transform of this Penrose tiling gives a pattern which exhibits 5 (10) - fold symmetry – very similar to diffraction patterns for icosahedral QCs

9. 9 Quasicrystals

10. 10 Quasicrystals

11. 11 Quasicrystals Diffraction pattern (in reciprocal space) of icosahedral QC can be indexed w/ 6 six integers - axes along 6 icosahedron directions q i (referred to Cartesian q x, q y, q z ) 1 

12. 12 Quasicrystals Diffraction pattern (in reciprocal space) of icosahedral QC can be indexed w/ 6 six integers - axes along 6 icosahedron directions q i (referred to Cartesian q x, q y, q z ) q 1 = (1  0) q 2 = (  0 1) q 3 = (  0 1) q 4 = (0 1  ) q 5 = (1  0) q 6 = (0  1) 1 

13. 13 Quasicrystals  = (1 + 5)2  = 1.618… Diffraction pattern (in reciprocal space) of icosahedral QC can be indexed w/ 6 six integers - axes along 6 icosahedron directions q i (referred to Cartesian q x, q y, q z ) q 1 = (1  0) q 2 = (  0 1) q 3 = (  0 1) q 4 = (0 1  ) q 5 = (1  0) q 6 = (0  1) 1 

14. 14 Quasicrystals Diffraction pattern (in reciprocal space) of icosahedral QC can be indexed w/ 6 six integers - axes along 6 icosahedron directions q i (referred to Cartesian q x, q y, q z ) q 1 = (1  0) q 2 = (  0 1) q 3 = (  0 1) q 4 = (0 1  ) q 5 = (1  0) q 6 = (0  1) Thus, icosahedral QC is periodic in 6D

15. 15 Quasicrystals Also consider: to periodically tile in 2-D – need three translation vectors if 5-fold, reasonable cell is pentagon – need additional dimension to fill space (tile) – more translation vectors

16. 16 Quasicrystals Diffraction pattern (in reciprocal space) of icosahedral QC can be indexed w/ 6 six integers - axes along 6 icosahedron directions q i (referred to Cartesian q x, q y, q z ) q 1 = (1  0) q 2 = (  0 1) q 3 = (0 1  ) q 4 = (1  0) q 5 = (  0 1) q 6 = (0 1  ) Thus, icosahedral QC is periodic in 6D But not in 3D To understand this, consider periodic 2D crystal:

17. 17 Quasicrystals To understand this, consider periodic 2D crystal: The 2D crystal is not in our observable world - what IS seen is the cut along E But cut along E may or may not pass through lattice nodes Cut shown has slope 1/  - does not pass through lattice nodes except origin

18. 18 Quasicrystals To understand this, consider periodic 2D crystal: But can observe both real structure and diffraction pattern for this 1D quasiperiodic crystal Must be some kind of structure in the extended space (the 2 nd dimension) - shown here as lines through the 2D lattice nodes

19. 19 Quasicrystals To understand this, consider periodic 2D crystal: Must be some kind of structure in the extended space (the 2 nd dimension) - shown here as lines through the 2D lattice nodes Some of the lines intersect "the real world" cut E, thereby allowing observation of the real quasiperiodic structure

20. 20 Quasicrystals To understand this, consider periodic 2D crystal: Note short & long segments in real real world cut - form "Fibonacci sequence": s l sl lsl sllsl lslsllsl sllsllslsllsl …….. if s = 1, l = 

21. 21 Quasicrystals To understand this, consider periodic 2D crystal: Think of 2 spaces - "parallel" (real) & "perp" (extended)

22. 22 Quasicrystals Consider incommensurate crystals: Need additional dimension to completely describe structure

23. 23 Quasicrystals Consider incommensurate crystals: Similar to quasiperiodic case

24. 24 Quasicrystals There are 16 space groups for the 6-D point group 532 w/ P, I, F 6-d cubic lattices The 6-D structure & the parallel & perpendicular subspaces are all invariant under the operations of 532

25. 25 Quasicrystals There are 16 space groups for the 6-D point group 532 w/ P, I, F 6-d cubic lattices The 6-D structure & the parallel & perpendicular subspaces are all invariant under the operations of 532 To visualize 6-D structure, must make 2-D cuts which necessarily must show both parallel & perp spaces

26. 26 Quasicrystals There are 16 space groups for the 6-D point group 532 w/ P, I, F 6-d cubic lattices The 6-D structure & the parallel & perpendicular subspaces are all invariant under the operations of 532 To visualize 6-D structure, must make 2-D cuts which necessarily must show both parallel & perp spaces This cut has 2-folds along parallel & perp directions

27. 27 Quasicrystals More on indexing – Note strangeness of axial directions – 63.43° from q 1

28. 28 Quasicrystals More on indexing – Use Cartesian system – basis vectors down 3 2-folds Then indices are: (h+h' , k+k' , l+l'  ) Usually given as: (h/h' k/k' l/l') Ex: (210010) ––> (2/2 0/2 0/0) (111111) ––> (0/2 2/2 0/0)