Quasicrystals

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Quasicrystals
AlCuLi QC Rhombic triacontrahedral grain
Typical decagonal QC diffraction pattern (TEM)
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Quasicrystals
Diffraction pattern for 8-fold QC
Diffraction pattern for 12-fold QC
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Quasicrystals
Principal types of QCs icosahedral
decagonal
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Quasicrystals
Principal types of QCs icosahedral
decagonal metastable (rapid
solidifcation) stable (conventional
solidification)
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Quasicrystals
Principal types of QCs icosahedral
decagonal metastable (rapid
solidifcation) stable (conventional
solidification) QCs usually have compositions
close to crystalline phases - the "crystalline
approximants"
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Quasicrystals
While pentagons (108 angles) cannot tile to fill
2D 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
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Quasicrystals
While pentagons (108 angles) cannot tile to fill
2D space, two rhombs w/ 72 36 angles can -
if matching rules are followed
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Quasicrystals
Fourier transform of this Penrose tiling gives a
pattern which exhibits 5 (10) - fold symmetry
very similar to diffraction patterns for
icosahedral QCs
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Quasicrystals

10
Quasicrystals

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Quasicrystals
t
1

Diffraction pattern (in reciprocal space) of
icosahedral QC can be indexed w/ 6 six integers -
axes along 6 icosahedron directions qi (referred
to Cartesian qx, qy, qz)
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Quasicrystals
t
1

Diffraction pattern (in reciprocal space) of
icosahedral QC can be indexed w/ 6 six integers -
axes along 6 icosahedron directions qi (referred
to Cartesian qx, qy, qz) q1 (1 t 0)
q2 (t 0 1) q3 (t 0 1) q4 (0 1
t) q5 (1 t 0) q6 (0 t 1)
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Quasicrystals
t
1

• t (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 qi (referred
to Cartesian qx, qy, qz) q1 (1 t 0) q2
(t 0 1) q3 (t 0 1) q4 (0 1
t) q5 (1 t 0) q6 (0 t 1)
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Quasicrystals
Diffraction pattern (in reciprocal space) of
icosahedral QC can be indexed w/ 6 six integers -
axes along 6 icosahedron directions qi (referred
to Cartesian qx, qy, qz) q1 (1 t 0) q2
(t 0 1) q3 (t 0 1) q4 (0 1
t) q5 (1 t 0) q6 (0 t 1) Thus,
icosahedral QC is periodic in 6D
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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
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Quasicrystals
Diffraction pattern (in reciprocal space) of
icosahedral QC can be indexed w/ 6 six integers -
axes along 6 icosahedron directions qi (referred
to Cartesian qx, qy, qz) q1 (1 t 0)
q2 (t 0 1) q3 (0 1 t) q4 (1 t
0) q5 (t 0 1) q6 (0 1 t) Thus,
icosahedral QC is periodic in 6D But not in
3D To understand this, consider periodic 2D
crystal
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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/ t - does not pass through
lattice nodes except origin
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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 2nd dimension) - shown here
as lines through the 2D lattice nodes
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Quasicrystals
To understand this, consider periodic 2D crystal
Must be some kind of structure in the extended
space (the 2nd 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
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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 t
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Quasicrystals
To understand this, consider periodic 2D crystal
Think of 2 spaces - "parallel" (real) "perp"
(extended)
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Quasicrystals
Consider incommensurate crystals
Need additional dimension to completely describe
structure
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Quasicrystals
Consider incommensurate crystals
Similar to quasiperiodic case
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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
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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
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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
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Quasicrystals
More on indexing Note strangeness of axial
directions 63.43 from q1
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Quasicrystals
More on indexing Use Cartesian system basis
vectors down 3 2-folds Then indices
are (hh't, kk't, ll't) Usually given
as (h/h' k/k' l/l') Ex (210010) gt
(2/2 0/2 0/0) (111111) gt (0/2 2/2
0/0)