Mathematics of
quasicrystals:
point of view of a
crystallographer
D. Gratias
History
Mathematics of quasicrystals:
point of view of a crystallographer
Almost periodicity
Some basic questions
Experimental
diffraction patterns
Where are the
atoms ?
Using diffraction
experiments
Electron microscopy
Denis Gratias LEM (CNRS)/ONERA Châtillon
Conclusion
The early days
In 1982 Shechtman observes an apparent paradox between sharp
diffraction peaks and the 5-fold symmetry of the electron
diffraction patterns in rapidly solified (Al,Mn) alloys.
Mathematics of
quasicrystals:
point of view of a
crystallographer
D. Gratias
History
Almost periodicity
Some basic questions
Experimental
diffraction patterns
Where are the
atoms ?
Using diffraction
experiments
Electron microscopy
Conclusion
The early days
In 1982 Shechtman observes an apparent paradox between sharp
diffraction peaks and the 5-fold symmetry of the electron
diffraction patterns in rapidly solified (Al,Mn) alloys.
• Pionner works in crystallography in 1974 and first framework of
hyperspace description in 1977.
P. M. de Wolff, Acta Cryst. A30 (1974) 777,
A. Janner and T. Janssen Phys. Rev. B15 (1977) 643.
Mathematics of
quasicrystals:
point of view of a
crystallographer
D. Gratias
History
Almost periodicity
Some basic questions
Experimental
diffraction patterns
Where are the
atoms ?
Using diffraction
experiments
Electron microscopy
Conclusion
The early days
In 1982 Shechtman observes an apparent paradox between sharp
diffraction peaks and the 5-fold symmetry of the electron
diffraction patterns in rapidly solified (Al,Mn) alloys.
• Pionner works in crystallography in 1974 and first framework of
hyperspace description in 1977.
P. M. de Wolff, Acta Cryst. A30 (1974) 777,
A. Janner and T. Janssen Phys. Rev. B15 (1977) 643.
• Term ”quasicrystals” for ”quasiperiodic crystals”
H. Bohr, Ann. Math. 45 (1924) 29 ; ibid. 46 (1925) 1001 ; ibid.47 (1926) 237
A. S. Besicovic, Almost Periodic Functions, Cambridge University Press, 1932
Mathematics of
quasicrystals:
point of view of a
crystallographer
D. Gratias
History
Almost periodicity
Some basic questions
Experimental
diffraction patterns
Where are the
atoms ?
Using diffraction
experiments
Electron microscopy
Conclusion
The early days
In 1982 Shechtman observes an apparent paradox between sharp
diffraction peaks and the 5-fold symmetry of the electron
diffraction patterns in rapidly solified (Al,Mn) alloys.
• Pionner works in crystallography in 1974 and first framework of
hyperspace description in 1977.
P. M. de Wolff, Acta Cryst. A30 (1974) 777,
A. Janner and T. Janssen Phys. Rev. B15 (1977) 643.
• Term ”quasicrystals” for ”quasiperiodic crystals”
H. Bohr, Ann. Math. 45 (1924) 29 ; ibid. 46 (1925) 1001 ; ibid.47 (1926) 237
A. S. Besicovic, Almost Periodic Functions, Cambridge University Press, 1932
• First definitions of the cut and project technic and connection to
the simple cut method
M. Duneau and A. Katz, Phys. Rev. Lett. 54 (1985) 2688
V. Elser, Acta Cryst A 42, (1986)
P.A. Kalugin, A. Kitaeff and L. Levitov, JETP Lett. 41 (1985) 145
P. Bak, Phys. Rev. B 32 (1985) 5764
Mathematics of
quasicrystals:
point of view of a
crystallographer
D. Gratias
History
Almost periodicity
Some basic questions
Experimental
diffraction patterns
Where are the
atoms ?
Using diffraction
experiments
Electron microscopy
Conclusion
The early days
In 1982 Shechtman observes an apparent paradox between sharp
diffraction peaks and the 5-fold symmetry of the electron
diffraction patterns in rapidly solified (Al,Mn) alloys.
• Pionner works in crystallography in 1974 and first framework of
hyperspace description in 1977.
P. M. de Wolff, Acta Cryst. A30 (1974) 777,
A. Janner and T. Janssen Phys. Rev. B15 (1977) 643.
• Term ”quasicrystals” for ”quasiperiodic crystals”
H. Bohr, Ann. Math. 45 (1924) 29 ; ibid. 46 (1925) 1001 ; ibid.47 (1926) 237
A. S. Besicovic, Almost Periodic Functions, Cambridge University Press, 1932
• First definitions of the cut and project technic and connection to
the simple cut method
M. Duneau and A. Katz, Phys. Rev. Lett. 54 (1985) 2688
V. Elser, Acta Cryst A 42, (1986)
P.A. Kalugin, A. Kitaeff and L. Levitov, JETP Lett. 41 (1985) 145
P. Bak, Phys. Rev. B 32 (1985) 5764
• General lectures and reviews
M. Baake and U. Grimm ”a Guide to Quasicrystal Literature” in Directions in mathematical
quasicrystals, CRM monograph series Vol 13 AMS (2000) p 371
T. Janssen Crystallography of quasicrystals, Acta Cryst A 42 (1986) 261.
Mathematics of
quasicrystals:
point of view of a
crystallographer
D. Gratias
History
Almost periodicity
Some basic questions
Experimental
diffraction patterns
Where are the
atoms ?
Using diffraction
experiments
Electron microscopy
Conclusion
Which distributions of matter diffract ?
Mathematics of
quasicrystals:
point of view of a
crystallographer
D. Gratias
Important questions for crystallographers
History
I
Long range order (crystal order) and periodicity
I
Long range order and Bragg diffractions
Almost periodicity
Some basic questions
Experimental
diffraction patterns
Where are the
atoms ?
Using diffraction
experiments
Electron microscopy
Conclusion
Which distributions of matter diffract ?
Mathematics of
quasicrystals:
point of view of a
crystallographer
D. Gratias
Important questions for crystallographers
History
I
Long range order (crystal order) and periodicity
I
Long range order and Bragg diffractions
Almost periodicity
Some basic questions
Experimental
diffraction patterns
Where are the
atoms ?
From quasiperiodic tilings and algebraic number theory to
generalized cut-and-project sets
Y. Meyer Nombres de Pisot, nombres de Salem et analyse harmonique, Lecture Notes in Mathematics
117 (Springer) 1970
Y. Meyer, Algebraic Numbers and Harmonic Analysis, North-Holland 1972
P. A. B. Pleasant, Elementary and Analytic Theory of Numbers, Banach Center Publications 17, PWN
Polish Scientific Publishers, Warsaw (1984) 439
E. Bombieri and J.E. Taylor J de Physique Colloqiue 47 n7, C3—19, 1986
E. Bombieri and J.E. Taylor, Proceedings of Kovalevsky Symposium, Contemporary mathematics Series
Amer. Math. Soc.
(R. V. Moody this conference)
Using diffraction
experiments
Electron microscopy
Conclusion
Long range order and Bragg diffraction
Mathematics of
quasicrystals:
point of view of a
crystallographer
D. Gratias
There is no simple obvious correlation between Bragg peaks and
long range order.
History
Almost periodicity
Some basic questions
Experimental
diffraction patterns
Where are the
atoms ?
Using diffraction
experiments
Electron microscopy
Conclusion
Long range order and Bragg diffraction
Mathematics of
quasicrystals:
point of view of a
crystallographer
D. Gratias
There is no simple obvious correlation between Bragg peaks and
long range order.
History
Almost periodicity
Some basic questions
Experimental
diffraction patterns
Where are the
atoms ?
Using diffraction
experiments
Electron microscopy
Conclusion
Directions in Mathematical Quasicrystals, M. Baake and R. V. Moody Editors, CRM Monograph series,
Volume 13 American Mathematical Society (2000)
(S. Ben-Abraham this conference)
Experimental Diffraction sets
Bragg reflections distribute on a dense countable set of
points of reciprocal space.
Mathematics of
quasicrystals:
point of view of a
crystallographer
D. Gratias
History
Almost periodicity
Some basic questions
Experimental
diffraction patterns
Where are the
atoms ?
Using diffraction
experiments
Electron microscopy
Conclusion
Experimental Diffraction sets
Bragg reflections distribute on a dense countable set of
points of reciprocal space.
Mathematics of
quasicrystals:
point of view of a
crystallographer
D. Gratias
History
Almost periodicity
Some basic questions
Experimental
diffraction patterns
Where are the
atoms ?
Using diffraction
experiments
Electron microscopy
Conclusion
For experimentally observing ”Bragg peaks” it is necessary
that the peaks of intensity larger than a given positive
threshold distribute on a discrete set of wavevectors.
Experimental Diffraction patterns
The diffraction data are used to compute the self correlation
function by Fourier transforming the experimentally observed
intensities.
Mathematics of
quasicrystals:
point of view of a
crystallographer
D. Gratias
History
Almost periodicity
Some basic questions
Experimental
diffraction patterns
Where are the
atoms ?
Using diffraction
experiments
Electron microscopy
Conclusion
Experimental Diffraction patterns
The diffraction data are used to compute the self correlation
function by Fourier transforming the experimentally observed
intensities.
Going further requires modelling the structure at some stage. In
the framework of cut and project fo example, it consists, among
other adjustements, in adjusting the shapes and locations of the
acceptance windows. Simplest models are based on windows
having polyhedral shapes.
Mathematics of
quasicrystals:
point of view of a
crystallographer
D. Gratias
History
Almost periodicity
Some basic questions
Experimental
diffraction patterns
Where are the
atoms ?
Using diffraction
experiments
Electron microscopy
Conclusion
Experimental Diffraction patterns
The diffraction data are used to compute the self correlation
function by Fourier transforming the experimentally observed
intensities.
Going further requires modelling the structure at some stage. In
the framework of cut and project fo example, it consists, among
other adjustements, in adjusting the shapes and locations of the
acceptance windows. Simplest models are based on windows
having polyhedral shapes.
Constraints in the refinement process :
I
Invariant positions of the Bragg reflections
I
Invariant symmetry
I
Invariant overall stoichiometry
I
(Atoms should be kept at distances large enough for being
physically acceptable)
Mathematics of
quasicrystals:
point of view of a
crystallographer
D. Gratias
History
Almost periodicity
Some basic questions
Experimental
diffraction patterns
Where are the
atoms ?
Using diffraction
experiments
Electron microscopy
Conclusion
Experimental Diffraction patterns
The diffraction data are used to compute the self correlation
function by Fourier transforming the experimentally observed
intensities.
Going further requires modelling the structure at some stage. In
the framework of cut and project fo example, it consists, among
other adjustements, in adjusting the shapes and locations of the
acceptance windows. Simplest models are based on windows
having polyhedral shapes.
Constraints in the refinement process :
I
Invariant positions of the Bragg reflections
I
Invariant symmetry
I
Invariant overall stoichiometry
I
(Atoms should be kept at distances large enough for being
physically acceptable)
G. Bernuau and M. Duneau, Fourier analysis of Deformed Model Sets in Directions in Mathematical
Quasicrystals, M. Baake and R. V. Moody edts, CRM monograph series Vol 13
Mathematics of
quasicrystals:
point of view of a
crystallographer
D. Gratias
History
Almost periodicity
Some basic questions
Experimental
diffraction patterns
Where are the
atoms ?
Using diffraction
experiments
Electron microscopy
Conclusion
Example of i-AlPdMn
The initial atomic model is made on 3 windows and generates B
and M type atomic clusters in real space.
Mathematics of
quasicrystals:
point of view of a
crystallographer
D. Gratias
History
Almost periodicity
Some basic questions
Experimental
diffraction patterns
Where are the
atoms ?
Using diffraction
experiments
Electron microscopy
The M are made of an occupied center, an internal dodecahedron
partially occupied (7/20), an icosidodecahedron and a large
icosahedron.
Conclusion
Example of i-AlPdMn
The initial atomic model is made on 3 windows and generates B
and M type atomic clusters in real space.
Mathematics of
quasicrystals:
point of view of a
crystallographer
D. Gratias
History
Almost periodicity
Some basic questions
Experimental
diffraction patterns
Where are the
atoms ?
Using diffraction
experiments
Electron microscopy
The M are made of an occupied center, an internal dodecahedron
partially occupied (7/20), an icosidodecahedron and a large
icosahedron.
But actual Mackay atomic clusters are built with an inner
icosahedron with an empty center instead of of the partially
occupied dodecahedron.
Conclusion
Mathematics of
quasicrystals:
point of view of a
crystallographer
D. Gratias
To transform M clusters into Mackay clusters, we first
remove the inner dodecahedra and the central atoms. . .
History
Almost periodicity
Some basic questions
Experimental
diffraction patterns
Where are the
atoms ?
Using diffraction
experiments
Electron microscopy
Conclusion
Mathematics of
quasicrystals:
point of view of a
crystallographer
. . . and replace them by small icosahedra.
This is achieved by adding small windows as satellites around
the bc (bc 0 ) and increase a little the interatomic distances :
D. Gratias
History
Almost periodicity
Some basic questions
Experimental
diffraction patterns
Where are the
atoms ?
Using diffraction
experiments
Electron microscopy
Conclusion
New set of Windows
Mathematics of
quasicrystals:
point of view of a
crystallographer
D. Gratias
History
Almost periodicity
Some basic questions
Experimental
diffraction patterns
Where are the
atoms ?
Using diffraction
experiments
Electron microscopy
Conclusion
A typical slice of the modified atomic struture
Mathematics of
quasicrystals:
point of view of a
crystallographer
D. Gratias
History
Almost periodicity
Some basic questions
Experimental
diffraction patterns
Where are the
atoms ?
Using diffraction
experiments
Electron microscopy
Conclusion
Optimizing the refinement process
Mathematics of
quasicrystals:
point of view of a
crystallographer
D. Gratias
History
Optimizing the process of atomic structure determination in
the framework of ideal quasiperiodic structures is a vey
difficult task.
Almost periodicity
Some basic questions
Experimental
diffraction patterns
Where are the
atoms ?
Using diffraction
experiments
Electron microscopy
Conclusion
Optimizing the refinement process
Mathematics of
quasicrystals:
point of view of a
crystallographer
D. Gratias
History
Optimizing the process of atomic structure determination in
the framework of ideal quasiperiodic structures is a vey
difficult task.
I
great difficulties in preparing good homogeneous
samples (non-congruent solidification, possible phase
transformations during the cooling of the single grains),
I
limitations in the process of collecting the weakest
diffraction data (important statistical errors),
I
need for additional technics and physical measurements
for deciphering the possible atomic configurations
Almost periodicity
Some basic questions
Experimental
diffraction patterns
Where are the
atoms ?
Using diffraction
experiments
Electron microscopy
Conclusion
Dynamical diffraction of fast electrons
An important experimental tool is the electron microscope (tool at
the origin of the discovery of quasicrystals). But electron
diffraction is a strongly dynamical process.
Mathematics of
quasicrystals:
point of view of a
crystallographer
D. Gratias
History
Almost periodicity
Some basic questions
Experimental
diffraction patterns
Where are the
atoms ?
Using diffraction
experiments
Electron microscopy
Conclusion
Dynamical diffraction of fast electrons
An important experimental tool is the electron microscope (tool at
the origin of the discovery of quasicrystals). But electron
diffraction is a strongly dynamical process.
Mathematics of
quasicrystals:
point of view of a
crystallographer
D. Gratias
History
Almost periodicity
Some basic questions
Experimental
diffraction patterns
Where are the
atoms ?
Using diffraction
experiments
Electron microscopy
Conclusion
where Ĥ(z) is the scattering Hamiltonian, χ2 = k02 − kz2 and V (z)
is the (quasiperiodic) diffraction potential :
X
1 X
b
|qi(q 2 − χ2 )hq| +
|qihq|V (z)|q 0 ihq 0 |)
H(z)
=
(
2kz q
0
q,q
The initial state of the diffracting electrons being |χi, the state at
the exit of the thin foil at thickness z,is given :
X
|Ψ(z)i =
|jie −iEj z hj|χi
j
Mathematics of
quasicrystals:
point of view of a
crystallographer
D. Gratias
History
Almost periodicity
Some basic questions
Experimental
diffraction patterns
Where are the
atoms ?
Using diffraction
experiments
Electron microscopy
Conclusion
The initial state of the diffracting electrons being |χi, the state at
the exit of the thin foil at thickness z,is given :
X
|Ψ(z)i =
|jie −iEj z hj|χi
Mathematics of
quasicrystals:
point of view of a
crystallographer
D. Gratias
History
Almost periodicity
Some basic questions
Experimental
diffraction patterns
j
The diffracted intensity associated to the Bragge peak q is given
by :
X
Iq (z) = |hq|Ψ(z)i|2 = |
hq|jie −iEj z hj|χi|2 .
j
Where are the
atoms ?
Using diffraction
experiments
Electron microscopy
Conclusion
The initial state of the diffracting electrons being |χi, the state at
the exit of the thin foil at thickness z,is given :
X
|Ψ(z)i =
|jie −iEj z hj|χi
The diffracted intensity associated to the Bragge peak q is given
by :
X
Iq (z) = |hq|Ψ(z)i|2 = |
hq|jie −iEj z hj|χi|2 .
j
Images are computed by back Fourier transforming the
q-components of the state |Ψ(t)i at the exit face of the specimen
b
after multiplicating by the transfert function T(q)
that contains
the optical aberrations of the microscope :
XX
−iEj z
b
hρ|Ψ(z)i =
hρ|qiT(q)hq|jie
hj|χi
j
D. Gratias
History
Almost periodicity
Some basic questions
Experimental
diffraction patterns
j
q
Mathematics of
quasicrystals:
point of view of a
crystallographer
Where are the
atoms ?
Using diffraction
experiments
Electron microscopy
Conclusion
Mathematics of
quasicrystals:
point of view of a
crystallographer
Variation of the lowest eigenvalues as function of the
number of diffracting beams and variation of the image as
function of thickness and defocus.
10
8
●
●
●
●
●
●
6
●
Ej
Conclusion
●
●
●
●
●
●
●
-2
●
-4
0,0
Almost periodicity
Some basic questions
Experimental
diffraction patterns
Using diffraction
experiments
Electron microscopy
●
0
History
Where are the
atoms ?
●
●
●
4
2
D. Gratias
●
●
0,2
0,4
<j|x>
0,6
0,8
Intensity of the central beam as a function of the
thickness z
Mathematics of
quasicrystals:
point of view of a
crystallographer
D. Gratias
History
The different curves show the result of the calculations for
different sizes of the q-basis
Almost periodicity
Some basic questions
Experimental
diffraction patterns
Where are the
atoms ?
Using diffraction
experiments
Electron microscopy
Conclusion
on the left : increasing qk size at constant q⊥ ; on the right
increasing q⊥ at constant qk : numerical ”stabilization” is
achieved for relatively small thicknesses z
Simulation of i-AlPdMn
Mathematics of
quasicrystals:
point of view of a
crystallographer
D. Gratias
History
Almost periodicity
Some basic questions
Experimental
diffraction patterns
Where are the
atoms ?
Using diffraction
experiments
Electron microscopy
Conclusion
Simulated HREM image and diffraction f iAlPdMn along a
5f direction in exact Laue symmetric orientation with 1231
beams (74 orbits).
Mathematics of
quasicrystals:
point of view of a
crystallographer
D. Gratias
History
The image changes drastically with
I
the number of beams used for diagonalizing the
scattering Hamiltonian
I
the thicknes and the optical aberrations
Stable images are obtained for small thicknesses (several
nanometers)
Almost periodicity
Some basic questions
Experimental
diffraction patterns
Where are the
atoms ?
Using diffraction
experiments
Electron microscopy
Conclusion
CBED simulations
CBED images can be used to better differentiate between
models but the sample thickness must be important (250
nanometers)
Mathematics of
quasicrystals:
point of view of a
crystallographer
D. Gratias
History
Almost periodicity
Some basic questions
Experimental
diffraction patterns
Where are the
atoms ?
Using diffraction
experiments
Electron microscopy
Conclusion
CBED simulations
CBED images can be used to better differentiate between
models but the sample thickness must be important (250
nanometers)
Mathematics of
quasicrystals:
point of view of a
crystallographer
D. Gratias
History
Almost periodicity
Some basic questions
Experimental
diffraction patterns
Where are the
atoms ?
Using diffraction
experiments
Electron microscopy
Conclusion
A few questions as conclusion
I
Methodology
Mathematics of
quasicrystals:
point of view of a
crystallographer
D. Gratias
History
Almost periodicity
Some basic questions
Experimental
diffraction patterns
Where are the
atoms ?
Using diffraction
experiments
Electron microscopy
Conclusion
A few questions as conclusion
I
Methodology
I
quasiperiodic pattern experimental recognition
Mathematics of
quasicrystals:
point of view of a
crystallographer
D. Gratias
History
Almost periodicity
Some basic questions
Experimental
diffraction patterns
Where are the
atoms ?
Using diffraction
experiments
Electron microscopy
Conclusion
A few questions as conclusion
I
Methodology
I
I
quasiperiodic pattern experimental recognition
refinement of structures : deformability of model sets
( ?)
Mathematics of
quasicrystals:
point of view of a
crystallographer
D. Gratias
History
Almost periodicity
Some basic questions
Experimental
diffraction patterns
Where are the
atoms ?
Using diffraction
experiments
Electron microscopy
Conclusion
A few questions as conclusion
I
Methodology
I
I
I
quasiperiodic pattern experimental recognition
refinement of structures : deformability of model sets
( ?)
(2D) quasiperiodic Hamiltonians (approaching the
lowest eigenstates)
Mathematics of
quasicrystals:
point of view of a
crystallographer
D. Gratias
History
Almost periodicity
Some basic questions
Experimental
diffraction patterns
Where are the
atoms ?
Using diffraction
experiments
Electron microscopy
Conclusion
A few questions as conclusion
I
Methodology
I
I
I
I
quasiperiodic pattern experimental recognition
refinement of structures : deformability of model sets
( ?)
(2D) quasiperiodic Hamiltonians (approaching the
lowest eigenstates)
Diffraction and Bragg peaks model sets : needs for a
general definition of crystals
Mathematics of
quasicrystals:
point of view of a
crystallographer
D. Gratias
History
Almost periodicity
Some basic questions
Experimental
diffraction patterns
Where are the
atoms ?
Using diffraction
experiments
Electron microscopy
Conclusion
A few questions as conclusion
I
Methodology
I
I
I
I
I
quasiperiodic pattern experimental recognition
refinement of structures : deformability of model sets
( ?)
(2D) quasiperiodic Hamiltonians (approaching the
lowest eigenstates)
Diffraction and Bragg peaks model sets : needs for a
general definition of crystals
Space Symmetry from geometric superimposition to
undistinguishability ;
Mathematics of
quasicrystals:
point of view of a
crystallographer
D. Gratias
History
Almost periodicity
Some basic questions
Experimental
diffraction patterns
Where are the
atoms ?
Using diffraction
experiments
Electron microscopy
Conclusion
A few questions as conclusion
I
Methodology
I
I
I
I
I
I
quasiperiodic pattern experimental recognition
refinement of structures : deformability of model sets
( ?)
(2D) quasiperiodic Hamiltonians (approaching the
lowest eigenstates)
Diffraction and Bragg peaks model sets : needs for a
general definition of crystals
Space Symmetry from geometric superimposition to
undistinguishability ;
What technic, if it exists, is the one-to-one signature of
long range order ?
Mathematics of
quasicrystals:
point of view of a
crystallographer
D. Gratias
History
Almost periodicity
Some basic questions
Experimental
diffraction patterns
Where are the
atoms ?
Using diffraction
experiments
Electron microscopy
Conclusion
A few questions as conclusion
I
Methodology
I
I
I
I
I
I
I
quasiperiodic pattern experimental recognition
refinement of structures : deformability of model sets
( ?)
(2D) quasiperiodic Hamiltonians (approaching the
lowest eigenstates)
Diffraction and Bragg peaks model sets : needs for a
general definition of crystals
Space Symmetry from geometric superimposition to
undistinguishability ;
What technic, if it exists, is the one-to-one signature of
long range order ?
Beyond quasiperiodic order : what physical properties
are the bases of long range order (matching rules /
growth rules)
Mathematics of
quasicrystals:
point of view of a
crystallographer
D. Gratias
History
Almost periodicity
Some basic questions
Experimental
diffraction patterns
Where are the
atoms ?
Using diffraction
experiments
Electron microscopy
Conclusion
A few questions as conclusion
I
Methodology
I
I
I
I
I
I
I
I
quasiperiodic pattern experimental recognition
refinement of structures : deformability of model sets
( ?)
(2D) quasiperiodic Hamiltonians (approaching the
lowest eigenstates)
Diffraction and Bragg peaks model sets : needs for a
general definition of crystals
Space Symmetry from geometric superimposition to
undistinguishability ;
What technic, if it exists, is the one-to-one signature of
long range order ?
Beyond quasiperiodic order : what physical properties
are the bases of long range order (matching rules /
growth rules)
What mathematical properties are pertinent for
physicists (for the experimental point of view :
structural stability with respect to fluctuations).
Mathematics of
quasicrystals:
point of view of a
crystallographer
D. Gratias
History
Almost periodicity
Some basic questions
Experimental
diffraction patterns
Where are the
atoms ?
Using diffraction
experiments
Electron microscopy
Conclusion