Aperiodic Crystals

About 50 years ago, it was shown that there are solid state systems with perfect order but without lattice periodicity. These systems were called crystalline phases because of the order and incommensurate because of the lack of periodicity. They formed a challenge for crystallographers and physicists to understand the structure, the physical properties and the reason for their appearance. Later other classes of this type were found (occupation modulated crystals, incommensurate magnetic systems, incommensurate composites), the most important one being that of quasicrystals. The discovery of the latter class in 1982 caused a huge increase in interest. The first conferences on this new type of materials were called Modulated Crystals, later polytypes and quasicrystals were included in the title MOSPOQ. Nowadays these conferences continue under the name Aperiodic (Crystals). The field has become very active worldwide, and our insight into structure and properties has increased impressively. A brief sketch of the development of the field is given in this chapter.

The Thue–Morse system is a paradigm of singular continuous diffraction in one dimension. Here, we consider a planar generalisation, constructed by a bijective block substitution rule, which is locally equivalent to the squiral inflation rule. For balanced weights, its diffraction is purely singular continuous. The diffraction measure is a two-dimensional Riesz product that can be calculated explicitly.

The random local mixture of a family of primitive substitution rules with noble mean inflation multiplier is investigated. This extends the random Fibonacci example that was introduced by Godrèche and Luck in (J. Stat. Phys. 55:1–28,

1989

). We discuss the structure of the corresponding dynamical systems, and determine the entropy, an ergodic invariant measure and diffraction spectra.

We report recent progress in the problem of distinguishing convex subsets of cyclotomic model sets

Λ

by (discrete parallel) X-rays in prescribed

Λ

-directions. It turns out that for any of these model sets

Λ

there exists a ‘magic number’

m

Λ

such that any two convex subsets of

Λ

can be distinguished by their X-rays in any set of

m

Λ

prescribed

Λ

-directions. In particular, for pentagonal, octagonal, decagonal and dodecagonal model sets, the least possible numbers are in that very order 11, 9, 11 and 13.

In a recent paper

arXiv:1111.3617

, Lenz and Moody presented a method for constructing families of real solutions to the inverse problem for a given pure point diffraction measure. Applying their technique and discussing some possible extensions, we present, in a non-technical manner, some examples of homometric structures.

A lattice is called well-rounded, if its lattice vectors of minimal length span the ambient space. We show that there are interesting connections between the existence of well-rounded sublattices and coincidence site lattices (CSLs). Furthermore, we count the number of well-rounded sublattices for several planar lattices and give their asymptotic behaviour.

The unexpected discovery of ancient Islamic ornaments with quasicrystalline symmetries has triggered significant discussion and a number of debates on the mathematical sophistication of Islamic geometry and its generating principles. Astonishingly, eight centuries before its description in Modern Science, ancient artists had constructed patterns with perfect quasicrystalline formations. Recent studies have provided enough evidence to suggest that ancient designers, by using the most primitive tools (a compass and a straight edge), were able to resolve the complicated long-range principles of quasicrystalline formations. Derived from these principles, a global multi-level structural model is presented that is able to describe the global long-range order of octagon-based quasicrystalline formations in Islamic Architecture. This new method can be used as a general guiding principle for constructing infinite patches of octagon-based quasicrystalline formations, including Ammann–Beenker tiling, without the need for local strategies (matching, scaling, etc.) or complicated mathematics.

This paper introduces two tiles whose tilings form a one-parameter family of tilings which can all be seen as digitization of two-dimensional planes in the four-dimensional Euclidean space. This family contains the Ammann–Beenker tilings as the solution of a simple optimization problem.

A relatively simple substitution for the Robinson tilings is presented, which requires only 56 tiles up to translation. In this substitution, due to Joan M. Taylor, neighbouring tiles are substituted by partially overlapping patches of tiles. We show that this overlapping substitution gives rise to a normal primitive substitution as well, implying that the Robinson tilings form a model set and thus have pure point diffraction. This substitution is used to compute the Čech cohomology of the hull of the Robinson tilings via the Anderson–Putnam method, and also the dynamical zeta function of the substitution action on the hull. The dynamical zeta function is then used to obtain a detailed description of the structure of the hull, relating it to features of the cohomology groups.

We introduce the mechanism of localized collective fluctuation of short-range ordered spin in a dodecahedral spin cluster in Zn–Mg–Tb icosahedral quasicrystals. In addition, we shall discuss the relation to the boson peak in topological glasses.

We discuss the slow dynamics mechanism of the excited carriers in the Al-based quasicrystal-like system. This glassy relaxation mechanism is closely related to the long recombination time of the excited carriers in Al–Pd–Re quasicrystals.

We study the electronic transport in quasiperiodic separable tight-binding models in one, two, and three dimensions. First, we investigate a one-dimensional quasiperiodic chain, in which the atoms are coupled by weak and strong bonds aligned according to the Fibonacci chain. The associated

d

-dimensional quasiperiodic tilings are constructed from the product of

d

such chains, which yields either the square/cubic Fibonacci tiling or the labyrinth tiling. We study the scaling behavior of the mean square displacement and the return probability of wave packets with respect to time. We also discuss results of renormalization group approaches and lower bounds for the scaling exponent of the width of the wave packet.

We performed first-principles FLAPW (Full potential Linearized Augmented Plane Wave) band calculations for Be

13

Mg and Be

13

Sb. Furthermore, we calculated the Hume–Rothery plot and

e

/

a

with the tetrahedron method from the case.output1 file generated from WIEN2k. These complex alloys belong to fcc structures with almost the same atom density as hcp Be. From the FLAPW-Fourier spectrum, we could point out that, in both alloys, the pseudogap is formed by Fs–Bz interactions with the spheres just coinciding to reciprocal lattice vectors, |

G

|=32,35,36 and 40.

We performed FLAPW electronic structure calculations with subsequent FLAPW-Fourier analysis for Tsai-type Cd

6

Ca containing 168 atoms per unit cell with space group

$Pm\bar{3}$

. The square of the Fermi diameter (2

k

F

)

2

,

e

/

a

and

critical

reciprocal lattice vector |

G

|

2

s were determined. The origin of the pseudogap across the Fermi level was interpreted in terms of the Hume–Rothery stabilization mechanism based on Fermi surface–Brillouin zone interactions (Fs–Bz) involved. The intuitively expected value of

e

/

a

=2.0 was confirmed. By extending our work to intermetallic compounds existing in the Cd–Ca binary alloy system, we determined the

e

/

a

values for Ca embedded in the polyvalent matrix Cd. The effective

e

/

a

for Ca was deduced to be two.

We have performed FLAPW electronic structure calculations with subsequent FLAPW-Fourier analysis for the low temperature phase Zn

6

Sc containing 336 atoms per unit cell with space group B2/b. The square of the Fermi diameter (2

k

F

)

2

, electrons per atom ratio

e

/

a

and

critical

reciprocal lattice vector |

G

|

2

s were determined. The origin of its pseudogap at the Fermi level was interpreted as arising from interference of electrons with (2

k

F

)

2

=79.0±0.2 with sets of lattice planes with |

G

|

2

ranging over 72 to 96. The work was extended to intermetallic compounds existing in M–Sc and M–Y (M=Zn, Cd and Al) binary alloy systems. The effective

e

/

a

values for Sc and Y were deduced to be 3.0 and 3.1, respectively.

We analyze transmission electron microscopy (TEM) images of self-assembled quasicrystals composed of binary systems of nanoparticles. We use an automated procedure that identifies the positions of dislocations and determines their topological character. To achieve this, we decompose the quasicrystal into its individual density modes, or Fourier components, and identify their topological winding numbers for every dislocation. This procedure associates a Burgers function with each dislocation, from which we extract the components of the Burgers vector after choosing a basis. The Burgers vectors that we see in the experimental images are all of lowest order, containing only 0s and 1s as their components. We argue that the density of the different types of Burgers vectors depends on their energetic cost.

This paper describes a new technique for solving the structure of quasicrystals. The technique is based on transformations between an average unit cell (AUC) and an envelope of diffraction peaks. For centrosymmetric structures like the Penrose tiling, the envelope makes it possible to determine the sign of the phase straight from the diffraction pattern. A Fourier transform of an envelope leads to a distribution of atomic positions within an AUC. Apart from theoretical and modeling aspects of the technique, the paper also presents the results of applying it to the well-known decagonal quasicrystal Al–Ni–Co.

Al

68

Pd

14.6

Co

17.4

, Al

69.8

Pd

13.8

Co

16.4

, Al

72

Pd

12.8

Co

15.2

, Al

73.8

Pd

11.9

Co

14.3

, and Al

76

Pd

11

Co

13

alloys annealed at 700

∘

C for 2000 h were studied. In the investigation, scanning electron microscopy including energy dispersive X-ray spectroscopy and electron backscatter diffraction, X-ray diffraction, and transmission electron microscopy were used. Altogether five near-equilibrium phases (

β

, U, Al

5

Co

2

,

ε

, Al

9

Co

2

) were identified. Transitions between

β

, U, and

ε

phases were also determined dependent on the alloy bulk metal composition. The experimental results were used to propose the partial isothermal section of the Al–Pd–Co phase diagram at 700

∘

C. The maximum solubilities at 700

∘

C of Pd in Al

9

Co

2

and Al

5

Co

2

were determined as 1.7 and 2.69 at.%, respectively.

This work is focused on the experimental investigation of intermetallic phases in Al

60

Co

29

Cu

11

, Al

63

Co

24

Cu

13

, and Al

67

Co

20

Cu

13

complex metallic alloys at near-equilibrium conditions. The alloys were long-term annealed at temperatures between 800 and 1150

∘

C and subsequently rapidly cooled to fix their high-temperature microstructures. Annealing temperatures were chosen with respect to the results of differential thermal analysis. Particular samples were studied by X-ray diffraction, scanning electron microscopy including energy dispersive X-ray spectroscopy and electron backscatter diffraction, and transmission electron microscopy. In the microstructures of particular samples, various combinations of D, B2, m, Al

5

Co

2

, and

Θ

-Al

2

Cu phases were identified depending on both bulk metal composition and thermal history.

The system Bi

2(

n

+2)

Mo

n

O

6(

n

+1)

is described with the superspace formalism. Considering the cationic distribution of the member with

n

=3, a superspace model is constructed beginning with a model previously proposed for the compound Bi

2

MoO

6

. The description of even members requires additional modifications. As a result, two superspace models are proposed for the different members of this system, depending on the parity of the parameter

n

. Both models have been checked through the Rietveld method combining synchrotron and neutron powder diffraction data.

An in-situ single-crystal X-ray diffraction study on tetragonal GdBaCo

2

O

5+

δ

with

δ

∼0.38 revealed that the crystal is pseudo-commensurate at room temperature with the magnitudes of the modulation vectors

q

1

and

q

2

parallel to the basal axes increasing gradually from the nearly commensurate value close to 1/3 upon heating. The basic structure of the compound is a double-layered perovskite type, having an alternating layer sequence [GdO

δ

]–[CoO

2

]–[BaO]–[CoO

2

] along the

c

axis. The oxygen deficiency of the crystal occurs only in the [GdO

δ

] layer, though it causes many positional modulations of constituent atoms in association with the valence fluctuation of Co cations between +2 and +3. Because of its pseudo-commensurate nature, the room temperature structure was also investigated by the commensurately-modulated approach as well as the conventional three-dimensional ones assuming a 3×3×2 supercell of the

P

4/

mmm

symmetry. These approaches successfully reproduced a prime structure of the compound, consisting of intersecting CoO

5

pyramidal arrays parallel to

a

and

b

axes. The incommensurate approach, on the other hand, also suggested a presence of a local disorder having a structural similarity with the high-temperature modification.

A single crystal of Al

4

(Cr,Fe) grown by the Czochralski method has been investigated using X-ray and also neutron diffraction. The average structure of this crystal with a composition of Al

79.1

Cr

17.8

Fe

3.1

was found to be body-centered orthorhombic with the space group

Immm

. Using neutrons it was possible to distinguish the Fe and Cr positions within the structure. However, the diffraction patterns of both, X-ray as well as neutrons, showed additional reflections beyond the Bragg reflections violating the body-centering and reducing the space group symmetry to

Pmm

2. A renewed structure analysis taking also these additional reflections into account exhibits significant changes of about 30 % of the atomic positions. Features related to extra diffraction phenomena beyond the Bragg reflections are discussed in some detail.

Aperiodic alkane/urea inclusion compounds (UIC) are prototype composites which exhibit complex sequences of phases that can clearly be described in the (3+

d

) dimension crystallographic superspace. By simply changing the length of the guest alkane molecules (C

n

H

2

n

+2

) which pile up in the channels of the host urea honeycomb-like framework, it is, for instance, possible to have phase-ordering phase transition from 3 to (3+1) dimension in the case of

n

-heptane/urea (

n

=7), or as in the case of

n

-hexadecane/urea (

n

=16) or

n

-nonadecane/urea (

n

=19), a generalization to higher dimensions of the phase transitions found in modulated structures. Such results are successfully obtained with the help of high resolution diffraction methods.

The structure of Sr

3

TiNb

4

O

15

has been re-investigated using synchrotron X-ray powder diffraction data. Rietveld refinements of a structural model against these data were performed and confirmed a new unit cell and space group symmetry. Sr

3

TiNb

4

O

15

was found to possess

Pna

2

1

symmetry with a unit cell

a

=12.36081(2) Å,

b

=12.40288(2) Å,

c

=7.751270(10) Å.

The (1−

x

)Ta

2

O

5

⋅

x

Al

2

O

3

solid solution series was re-investigated using synchrotron X-ray powder diffraction and neutron powder diffraction data. Structures were refined using the Rietveld method and an incommensurately modulated composite structure approach in superspace group

Xmmm

(0

β

0)

s

00 with a composition dependent modulation wave vector. Variable temperature synchrotron X-ray powder diffraction data were collected between 27 and 990

∘

C. The magnitude of the modulation wave vector was found to be constant over most of the temperature range.

We discuss the crystal structure and phase transitions in the Cd–Ca system by comparing the phase diagram and pressure dependence of the potential energy curves for the rotation of the tetrahedral cores. It turns out that some phase boundaries especially between the cubic phases seem to be well described by the rotation of the tetrahedral cores. Based on this analysis, we predict expected differences in the phase diagram in the Cd–Y system. We also point out the connection between the orientational configurations of the tetrahedral cores and valency of the second element. Furthermore, we also mention the possibility of the binary Cd–Y QC.

For the construction of the average unit cell (AUC) within the statistical approach, the use of a so-called reference lattice is needed. The choice of the lattice constants is in general arbitrary. However, it is convenient to bind them with the reciprocal space vectors

k

and

q

(main and modulation vector,

q

=

k

/

τ

) which we use for indexing the diffraction pattern,

λ

k

=2

π

/

k

,

λ

q

=2

π

/

q

. AUC is a distribution of projections of atomic positions in real space on the reference lattice. With the choice of lattice as above, the shape of the AUC is related to the shape of the atomic surface (AS), used in the higher-dimensional approach. In this paper, the discussion on the choice of the set of wave vectors

k

and

q

is provided in terms of different geometrical bases used for a construction of Ammann–Kramer–Neri tiling (simply called Ammann tiling—AT, a model for icosahedral quasicrystal) and relation of the AUC and AS shapes. The dependence of the AUC shape on the choice of wave vectors is also demonstrated. Additionally, it is proved that the diffraction pattern does not depend on the basis chosen.

This paper describes the average unit cell (AUC) approach and its application to a derivation of the structure factor for quasicrystals. Scaling plays a special role in this approach. The TAU2-scaling in AUC simplifies the formulae for the structure factor. The TAU-scaling of peak positions distinguishes quasicrystals from twins, indicating a phase transition at some critical concentration of atoms.

HAADF (high-angle annular detector dark-field) images with

Cs

-corrected scanning transmission electron microscopy (STEM) have been observed for a W-(AlCoNi) crystalline phase and two-types of decagonal quasicrystals in Al

71.5

Co

25.5

Ni

3

and Al

72.5

Co

17.5

Ni

10

alloys. We have secured positive evidence that three-dimensional arrangements of transition-metal (TM) atoms of decagonal quasicrystals can be directly derived from the arrangements of bright dots in HAADF-STEM images, which correspond to individual TM atoms, by reference to results on HAADF-STEM observation and the structure of W-(AlNiCo) determined by X-ray diffraction analysis. We could conclude that pure TM atomic sites and mixed TM sites (with Al atoms) on A and B planes stacking along the periodic axis are located at the lattice points of a Penrose lattice with a bond length of 0.25 nm. In both planes atomic sites form pentagonal tilings with bond lengths of 0.47 nm and 0.76 (=0.47⋅

τ

) nm, respectively, in both the Al

71.5

Co

25.5

Ni

3

and Al

72.5

Co

17.5

Ni

10

decagonal quasicrystals, whose structures were formally characterized as rhombic and pentagonal tilings of atom columnar clusters with a bond length of 2 nm.

A crystalline approximant, which is related to Al–Cu–Co decagonal quasicrystals with two aperiodic planes stacking along the periodic axis, in an Al

66

Cu

15

Co

19

alloy annealed at 900

∘

C for 36 h has been studied by high-angle annular detector dark-field (HAADF) observations with

C

s-corrected scanning electron microscopy (STEM). Observed HAADF-STEM images represent individual transition-metal (TM) atoms as bright dots, and so a three-dimensional arrangement of TM atoms in the approximant can be derived from the arrangement of bright dots. The structure has an orthorhombic unit cell with

a

0

=10.1 nm,

b

0

=0.4 nm and

c

0

=6.7 nm, formed by an ordered arrangement of two types of atom columnar clusters in a

τ

3

-inflated monoclinic Al

13

Co

4

structure formed by a network of pentagons with an edge-length of 2 nm. The TM atoms in the two planes stacking along the

b

-axis are located at lattice points of a Penrose lattice with a bond length of 0.25 nm and pentagonal tilings with bond lengths of 0.47 and 0.76 nm.

A variety of orthorhombic approximant

ϵ

n

(

n

=6, 16, 22, and 28)-phases exist in Al–Pd–(Mn, Fe, Co, Rh) systems. HREM images and the corresponding electron diffraction patterns show that the

ϵ

16

phase in an annealed Al

80

Pd

11

Co

9

alloy exhibits a locally disordered structure consisting of pentagonal and banana-shaped tiles with an edge length of 0.76 nm. This paper demonstrates a feasible structural model for the

ϵ

16

phase in the Al–Pd–Co system by single-crystal X-ray diffraction coupled with HREM.

The structure of the approximant

τ

2

-Al

3

Co (

P

2/

m

:

a

=3.9831(3) nm,

b

=0.8127(1) nm,

c

=3.2182(3) nm, and

β

=108.03(1)

∘

), associated with the decagonal quasicrystals with a period of 0.8 nm, was analyzed using a high-angle annular detector dark-field (HAADF) observation with

C

s-corrected scanning transmission electron microscopy (STEM). The HAADF-STEM image clearly showed the arrangement of individual Co atoms as bright dots. The contrast among the atoms in the lattice led to an image of the fundamental structure of the

τ

2

-Al

3

Co phase, composed of an ordered arrangement of pentagonal columnar units with edge lengths of 0.47 nm. The arrangements of atoms in the columnar units were quantitatively determined by single crystal X-ray diffraction (XRD). The results demonstrate that the pentagonal columnar units form common tiles in the shape of a squashed hexagon, a pentagonal star, and a crown. Among the tiles, the pentagonal star composed of 10 pentagonal units was similar to that found in the W-(AlNiCo) approximant for the Al–Ni–Co decagonal quasicrystal (DQC).

Distinctive diffuse rings around Bragg positions have been observed in the diffraction patterns of a crystal of the N-terminal fragment of the Gag protein from Feline Foamy Virus. It is shown that these are caused by geometric frustration as molecules try to pack on the triangular

a

–

b

mesh of the space group P6

1

22. The disorder prohibits conventional structure solution. The possibility of using the diffuse scattering to aid solution is explored using Reverse Monte Carlo modelling.

Using quasielastic neutron scattering (QENS) and molecular dynamics (MD) simulations, dynamical disorder was shown to be present in the Zn

6

Sc cubic 1/1-approximant to Tsai type quasicrystals. This dynamical disorder originates from reorientations of the innermost tetrahedron shell inside the Tsai type clusters’ building blocks. To enable such a rotational motion inside a close-packed alloy, a unique dynamical flexibility is necessary. We present a study of the tetrahedron dynamics with respect to this structural flexibility.

Quasicrystals are structures with long range order but no translational symmetry. Besides phonons, quasicrystals posses additional hydrodynamic modes called phasons. In a recent article (Kromer et al., Phys. Rev. Lett. 108:218301,

2012

), the trajectories of colloidal particles in a laser field with decagonal symmetry were studied when the phasonic displacement was changed. Here we generalize the results to laser fields with eight- and twelve-fold symmetry. In principle, the method can also be used to predict collective rearrangements of atoms due to phasons in intrinsic quasicrystalline systems.

Recently it has been shown that some low order approximants to decagonal or icosahedral quasicrystals provide excellent activity and selectivity for hydrogenation of alkynes. Our recent works on Al

13

Co

4

and AlPd compounds demonstrated that the catalytically active surfaces in both cases are surfaces with (pseudo-)five-fold symmetry. Ab-initio DFT calculations have been used to identify the reaction centers and to construct a detailed atomistic scenario for the acetylene to ethylene hydrogenation. It was found that the activity of the catalysts is not promoted by the transition metal (TM) atoms alone but by a cluster of Al atoms centered at a slightly protruding TM atom. In the present contribution, we demonstrate that local configurations of Al and TM atoms favorable for selective catalysis of the hydrogenation reactions naturally appear at Al–TM surfaces with pentagonal symmetry. We discuss the possibility to use surfaces of the Al–TM quasicrystals and their approximants as catalysts for hydrogenation reactions.

We have studied the effect of leaching treatments on the surface microstructure and chemical composition of Al-based quasicrystals. The high symmetry surfaces of single grain icosahedral (

i

-) Al–Cu–Fe and decagonal (

d

-) Al–Ni–Co quasicrystals and a polygrain

i

-Al–Pd–Re quasicrystal with random surface orientation were leached with NaOH solution at varying times and the resulting surfaces were characterized by scanning electron microscopy, energy dispersive X-ray analysis and X-ray photoelectron spectroscopy. The leaching treatments preferentially remove Al producing nanoparticles of the transition metals and their oxides. The leached fivefold surface of

i

-Al–Cu–Fe exhibits micron sized dodecahedral cavities on which the nanoparticles are precipitated. However, no specific microstructure has been observed on the tenfold surface of

d

-Al–Ni–Co and the polygrain

i

-Al–Pd–Re. The quasicrystalline surface can be regained after polishing the leached layer, indicating that leaching occurs only in a limited depth from the surface. This was revealed by low energy electron diffraction after the surface was prepared under ultra high vacuum conditions. These results provide important information for preparation of model catalysts of nanoparticles of catalytically active metals on quasicrystal surfaces.