Quasicrystals

Fundamentals and Applications

Enrique Maciá-Barber

• Publisher: CRC Press
• Year: 2020
• Topic: Physical Sciences
• eBook ISBN:
  9781351209137
• Subtopic: Chemistry
• Index: Chemistry

This book provides an interdisciplinary guide to quasicrystals, the 2011 Nobel Prize in Chemistry winning topic, by presenting an up-to-date and detailed introduction to the many fundamental aspects and applications of quasicrystals science. It reviews the most characteristic features of the peculiar geometric order underlying their structure and their reported intrinsic physical properties, along with their potential for specific applications.

The role of quasiperiodic order in science and technology is also examined by focusing on the new design capabilities provided by this novel ordering of matter. This book is specifically devoted to promoting the very notion of quasiperiodic order, and to spur its physical implications and technological capabilities. It, therefore, explores the fundamental aspects of intermetallic, photonic, and phononic quasicrystals, as well as soft-matter quasicrystals, including their intrinsic physical and structural properties. In addition, it thoroughly discusses experimental data and related theoretical approaches to explain them, extending the standard treatment given in most current solid state physics literature. It also explores exciting applications in new technological devices of quasiperiodically ordered systems, including multilayered quasiperiodic systems, along with 2D and 3D designs, whilst outlining new frontiers in quasicrystals research.

This book can be used as a reader-friendly introductory text for graduate students, in addition to senior scientists and researchers coming from the fields of physics, chemistry, materials science, and engineering.

Key features:

• Provides an updated and detailed introduction to the interdisciplinary field of quasicrystals in a tutorial style, considering both fundamental aspects and additional freedom degrees provided by designs based on quasiperiodically ordered materials.

• Includes 50 fully worked out exercises with detailed solutions, motivating, and illustrating the different concepts and notions to provide readers with further learning opportunities.

• Presents a complete compendium of the current state of the art knowledge of quasicrystalline matter, and outlines future next generation materials based on quasiperiodically ordered designs for their potential use in useful technological devices.

Dr. Enrique Maciá-Barber is Professor of condensed matter physics at the Universidad Complutense de Madrid. His research interests include the thermoelectric properties of quasicrystals and DNA biophysics. In 2010 he received the RSEF- BBVA Foundation Excellence Physics Teaching Award. His book Aperiodic Structures in Condensed Matter: Fundamentals and Applications (CRC Press, Boca-Raton, 2009) is one of the Top Selling Physics Books according to YBP Library Services.

CHAPTER 1

Intermetallic Quasicrystals

1.1 WHAT IS A QUASICRYSTAL?

“A quasicrystal is the natural extension of the notion of a crystal to structures with quasiperiodic, rather than periodic, translational order”. (Dov Levine and Paul Joseph Steinhardt, 1984) [486]

1.1.1 The Al₈₆Mn₁₄ alloy electron diffraction picture

“It took the right instrument, an electron microscope, because while X-ray diffraction is great for studying many aspects of crystals, it cannot show you rotational symmetry” (Daniel Shechtman, 70th Birthday Celebration Symposium, 27 July 2011, Iowa State University, Ames).

“As I was studying rapidly solidified aluminium alloy which contained 25% manganese by transmission electron microscopy, something very strange and unexpected happened. There were 10 bright spots in the selected area diffraction pattern, equally spaced from the center and from one another. I counted them and repeat the count in the other direction and said myself: ‘There is no such animal’. In Hebrew: ‘Ein Chaya Kazo’”. (Daniel Shechtman commenting on his 8 April 1982 discovery [321], see Fig. 1.1)

To properly understand Daniel Shechtman’s astonishment, we must recall that 20th century crystallography started in 1912 with the introduction of X-ray diffraction methods by Max von Laue (1879–1950), William Henry Bragg (1862–1942), and William Lawrence Bragg (1890–1971), who assumed that atoms were arranged in ordered crystals according to regular periodic arrays on a lattice. During the seventy years elapsed until Shechtman’s discovery almost all reported crystalline structures were perfectly compatible with that assumption, and accordingly the main attribute of crystalline solid state was progressively identified with the presence of a periodic distribution of matter density, ρ(r), inside the considered samples. Ideally assuming that atoms can be treated as material points, such a distribution can be envisioned as a lattice of points in Euclidean three-dimensional space, so that the vectors describing the lattice points have the general form r = n₁e₁ + n₂e₂ + n₃e₃, where {e_i} is a suitable vector basis and the coordinates are integer numbers (i.e., n_i ∈ ℤ). Within this framework, lattice periodicity is given by the invariance condition ρ(r + R₀) = ρ(r), where R₀ = m₁ e₁ + m₂e₂ + m₃e₃ (m_i ∈ ℤ) belongs to the lattice-related vectorial space. Physically, lattice periodicity guarantees that the local atomic environment we have at a given volume determined by the vector basis {e_i} inside the crystal exactly repeats itself throughout the space.

Figure 1.1 Electron diffraction pattern corresponding to the Al₈₆Mn₁₄ alloy quasicrystal discovered by Shechtman. A 10-fold symmetry axis around the origin can be clearly appreciated. (Reprinted figure with permission from Shechtman D, Blech I, Gratias D, and Cahn J W, Physical Review Letters 53, 1951, 1984. Copyright (1984) by the American Physical Society).

Periodic arrangements in Euclidean space involve rotations along with translations. We say that a given structure possesses a m-fold rotational symmetry if it is left unchanged when rotated by an angle 2π/m, with respect to a certain axis, fixed a given point. The integer m ∈ ℕ is called the order of the rotational symmetry (or of its related symmetry axis). Therefore, the lattice remains the same under rotations relating lattice points to each other, which are represented by a set of matrices of the form

$R_{\alpha }=\left(\begin{array}{ccc}{n}_{11}& n_{12}& n_{13}\\ n_{21}& n_{22}& n_{23}\\ n_{31}& n_{32}& n_{33}\end{array}\right),$

whose entries are integer numbers. On the other hand, on a properly chosen basis, pure rotations can be expressed in terms of the orthogonal matrices¹

${\text{M}}_X=\left(\begin{array}{ccc}1& 0& 0\\ 0& c& -s\\ 0& s& c\end{array}\right),{\text{M}}_Y=\left(\begin{array}{ccc}c& 0& s\\ 0& 1& 0\\ -s& 0& c\end{array}\right),{\text{M}}_Z=\left(\begin{array}{ccc}c& -s& 0\\ s& c& 0\\ 0& 0& 1\end{array}\right),$

(1.1)

with c ≡ cos φ and s ≡ sin φ, describing counterclock rotations by an angle φ around the X, Y, and Z Cartesian axes, respectively. Accordingly, R_α = BM_αB⁻¹, where B denotes the basis matrix. Now, since the trace of a matrix is invariant under a basis transformation, the periodicity condition ρ(R_αr) = ρ(r) relating lattice points, implies that tr(R_α) = tr(M_α), so that we obtain the so-called crystallographical restriction [381, 840],

$1+2\cos \phi =n_{11}+n_{22}+n_{33}\equiv n\in \mathbb{Z},$

(1.2)

which can be rewritten in the convenient form cos φ = (n − 1)/2. The bo...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Table of Contents
• Preface
• Chapter 1 ■ Intermetallic Quasicrystals
• Chapter 2 ■ Hamiltonian Quasicrystals
• Chapter 3 ■ Physical Properties of Intermetallic Quasicrystals
• Chapter 4 ■ Photonic and Phononic Quasicrystals
• Chapter 5 ■ Actual and Prospective Applications of Quasicrystals
• Chapter 6 ■ New Frontiers in Quasicrystals Science
• Bibliography
• Index