Aperiodic crystalsIncommensurate modulated

crystals

b*

c*

From G. Pan, Thesis Orsay 1992

,

• Cuprate superconductor Bi2,2Sr1,8CuO2

• Incommensurate modulated phase• ‘‘Satellite’’ reflections around

main spots

b*c*

k

• 4 indices to index

Incommensurate? The NaNO2

case

P

Ferroelectric

Paraelectric

Phase diagram

Continuous variationof the

satellite position:

Incommensurate

Ferro Para

Inc.

From Dominique Durand, Thesis, LPS, Orsay

BCCDDevil straicaseUhrig (1989)

Incommensurate modulation

a

un

• Local property of the crystal has a periodicity

irrational with crystal periods

• Example: displacive modulation

• NaNO2 (electric polarisation), alliages (concentration wave), magnetism

• ADN, Coxeter helix

𝑹𝑢𝑣𝑤+𝒖0 sin(𝒌∙𝑹𝑢𝑣𝑤+𝜑 )¿

RS of modulated structures

• Calculation of the RS• Direct space given by:

• is non zero iff , 4 indices!

• Jacobi-Anger formula• Bessel function of order m

• and

𝑆 (𝒓 )=∑𝑢𝑣𝑤

𝛿(𝒓−(𝑹𝑢𝑣𝑤+𝒖0 sin (𝒌 ∙𝑹𝑢𝑣𝑤+𝜑)))

𝐹 (𝒒 )=𝑣∗ ∑h𝑘𝑙𝑚

𝐽𝑚(𝒒 ∙𝒖0)𝑒𝑖𝑚𝜑 𝛿(𝒒−𝑸h𝑘𝑙−𝑚𝒌)

𝑒𝑖𝑧 sin 𝜃=∑𝑚

𝐽𝑚 (𝑧)𝑒−𝑖𝑚 𝜃

𝐹 (𝒒 )=∫𝑆 (𝒓 )𝑒−𝑖𝒒 ∙𝒓𝑑3𝒓=∑𝑢𝑣𝑤

𝑒−𝑖𝒒 ∙𝑹𝑢𝑣𝑤𝑒− 𝑖𝒒 ∙𝒖0 sin (𝒌∙𝑹𝑢𝑣𝑤+𝜑 )

h=0

a*

h=1 h=2

k 2k 3k

m=

0 1 2 3-3 -2 -1

• Reciprocal space• Nodes of the RS flanked by satellites reflections at

• is non zero iff •

• Concept of space of higher dimension

RS of modulated structures

𝐹 (𝒒 )=𝑣∗ ∑h𝑘𝑙𝑚

𝐽𝑚(𝒒 ∙𝒖0)𝑒𝑖𝑚𝜑 𝛿(𝒒−𝑸h𝑘𝑙−𝑚𝒌)

Macroscopic consequence: calaverite

G0012

a*

c*

+q

-q

+2q

+3q

+4q

G2012

G2014

-

-

-

(201)-

(001)

𝑞=−0 .4095𝒂∗+0 .4492𝒄∗

• Calaverite : Au1-xAgxTe, gold ore• Facets does not satisfy Haüy’s Law

Composites crystals

• Entanglement of two crystals with mutually incommnsurate

lattice constants.a

a’

• Simple model: RS sum of the two RL

a*

b*=b’*

4 indices

b=b’

a’*

• In fact both lattices are intermodulated...

Structure of Ba

5.5 GPa 12.6 GPa

Phase IBody centred

Phase IIHexagonal

Phase IVTetragonal inc.

45 GPa

Phase VHexagonal

Phase IV : Self-hosting structure

Chains of Ba in a matrix of tetragonal Ba

0.341 nm

R.J. Nelmes, D.R Allan, M.I McMahon, et S.A. Belmonte, Phys. Rev. Lett., 83 (1999) 4081

Ch=0.4696 nm

(Centre terre 360=Gpa)

Quasi-cristaux• Electron diffraction by an Al-Mn alloy

D. Shechtman (Chemistry Nobel prize) et al. Phys. Rev. Lett. 53, 1951 (1984)• Quasicrystal discovered by accident (by serendipity) by Schechtman (1982)

Who was studying rapidly cooled alloys.

• Al alloys bad conductivity (I, T) • Fragile at 300 K, ductile at HT

• Diamagnetic• Tribologic properties, non stick surfaces

• AlMn rapidly cooled (imperfect)• 1986 : AlLiCu, stable quasicrystals

• 1988 : Perfect quasicrystals, AlCuFe, AlPdMn, AlPdRe• 2000 : Cd5,7Yb (Tsai, Nature)

Dodecahedralcrystal

of AlCuFe

Photo : Annick Quivy© CNRS - CECM, Vitry-Thiais

Are they twinned?X-ray photograph

Decagonal microcrystal Al0.63Cu0.175Co0.17Si0.02

From P. Launois et al., 1991

Linus Pauling (and others)suggested they were microtwins

Ex: 5-fold assembly of microcrystals

Microscopy and electron diffraction proved the QC existence

Quasicrystalline microscopic order: new phase of condensed matter

Electron beam (10 nm)

X-rays in the 90s (1-100 mm)now… also 50 nm

From M. Audier (1990)

72°

No!

Penrose tiling

• Two types of tiles• Matching rules

Some quasicrystals can be modelledby penrose tilings

Exemple: Al-Fe-Cu

36° 72°

Indexation of quasicrystals diffraction diagram

• FT of the Penrose tiling Indexed by 4 vectors

arithmetically independent

a1*

a4*

a3*

• 4 indices• Z-module of rank 4

• How to index a non periodical lattice?

a2*

∑𝑖=1

𝑛

𝑛𝑖𝑎𝑖∗=0⟺ {𝑛𝑖}𝑖=0

Indexation of QC

• Diagramm of icosahedral QCare indexed by 6 indices

• All form a Z-module of rank 6

XY

a5*a4*

a1*

a3*

a2*

a6*

Z

: golden ratio

𝜏=1+√52

=2cos36 ° ≅ 1.618

𝑸h𝑘𝑙h′𝑘′ 𝑙′={ h+𝜏 h′

𝑘+𝜏𝑘 ′𝑙+𝜏 𝑙 ′

With etc.

Definition of a crystal

IUCr 1991‘‘By ‘crystal’

we mean any solid having an essentially discrete diffraction diagram,

and by ‘aperiodic crystal’ we mean any crystal

in which three-dimensional lattice periodicity can be considered to be absent.’’

« Par cristal on désigne un solide

dont le diagramme de diffraction estessentiellement discret

et par cristal apériodiqueon désigne un cristal

dans lequel la périodicité tridimensionnellepeut être considérée absente »

Z-module

Let’s consider an ‘’object’’ the FT of which is a

Z-module of finite rank:

vectors of Z-module of rank n; indices

• 3D lattice : Miller indices; • Incommensurate ; • Icosahedral quasicrystal

𝑭 (𝒒 )=∑{𝑛 𝑖}𝒄 ({𝑛𝑖 })𝛿(𝒒−∑

𝑖=1

𝑛

𝑛𝑖𝒂𝑖∗)

Let’s calculate the inverse FT of

Superspace

H is a periodical functionin a superspace of dimension n

: cut of an periodic ‘‘crystal’’ in the n-superspace

by an ‘’hyper-line’ of equations

𝐻 (𝛼1∗𝑥 ,𝛼2

∗𝑥 ,…𝛼𝑛∗𝑥 )=𝑆 (𝑥)

𝐻 ( 𝑦1, 𝑦 2,… 𝑦𝑛)=∑{𝑛𝑖 }𝑐 ( {𝑛𝑖})𝑒

𝑖∑𝑖=1

𝑛

𝑛 𝑖𝑦 𝑖

Inverse FT of

In 1D:

mutually irrationnals, such as

Example in 2D Cut of 2D latticeby a band with

irrationnal tangent

+

Projection of pointson the line

=

Penrose tilings:2D cuts of 4D superspace

Fibonacci sequence

Exemples

2D lattice + cut

1D crystal

Composite cristal

Incommensurate modulation

Quasicrystal

Quasicrystal: cut and projection

• Basis are called ‘‘atomic surfaces’’

Quasicristal

• Discontinuous atomic surfaces

Physical spaceSlope : t Fibonacci sequence

Perp. space

• Where are the atoms• Refinement of electron density in the superespace

• Decoration of Penrose tilings• Approximants

Rational slope:approximant

Phason: displacement in the perpendicular space

• Crystal translation…• Relative sliding of the two crystals in composites

• Sliding of incommensurate modulation• Atomic rearrangements in quasicrystals

Perp. space

Edagawa PRL 2000

Phasons in quasicrystals: atomic motions

Aperiodic order

Indexation of diffraction diagramof a body in dimension D

by a finite number N of indices(Case of all known ‘‘crystals’’)

This body is aperiodic if .It is possible to obtained this crystal by a D-cut

of a N-superspace

Is there something beyond quasicrystals?

Yes, almost periodicity If is a function continuous of

is a -pseudoperiod (p.p)if

F is almost periodic iffThe set of its ε-pseudoperiods is relatively dense (it exists a max distance between neighbouring

p.p.)

Every periodic function in a.p.!

T=76T=151

Esentially discrete…

Bohr theorem (Harald, Niels’ brother) :

is almost periodic

is the limit of a series .

The tiling ‘‘chair’’is limit-periodic

Z-module of infinite rank

In nature nothing has been found to be limit-periodic

Peaks in

∑𝒏𝑐𝑛𝑒

−𝑖 𝜆𝑛𝑥