What exactly are quasicrystals, often called the “third state of matter”? Professor Keiichi Edagawa of the Institute of Industrial Science, The University of Tokyo has been a pioneering figure in the field of quasicrystal research since its early days. This article explores his groundbreaking work—such as capturing the once-enigmatic formation process of quasicrystals on video—and delves into the unique nature of this mysterious structure.

An Impossible Substance in the Field of Crystallography

Metals, semiconductors, ceramics, and polymers — although these are completely different types of solid materials, they can broadly be classified at the structural level into two categories: crystalline, with a regular atomic arrangement, and amorphous (non-crystalline), with an irregular atomic arrangement. Glass and rubber are typical examples of amorphous materials.

Familiar examples of crystals include salt, snowflakes, and quartz. For atoms to be arranged in an orderly fashion, the structure must exhibit periodicity—a repeating pattern that extends indefinitely—as well as symmetry within the basic units that make up that pattern. A structure is said to have symmetry if, even when rotated around a certain axis or flipped at a certain point, it appears identical to before. A cube is a classic example. When cubes are packed tightly together without gaps, this arrangement creates periodicity, that is, a repeating structure.

A pattern that has symmetry and repeats periodically—that is the very definition of a crystal.

What happens, though, if we use a regular pentagon as the unit? A regular pentagon has five-fold symmetry, meaning it returns to its original appearance after five rotations of 72 degrees each. However, no matter how you shift or rotate them, regular pentagons cannot be arranged together without leaving gaps. In other words, they lack periodicity and thus fall outside of the definition of a crystal.

Five-fold symmetry. In nature, it is found everywhere, including flower petals and starfish.

However, in 1984, a solid with five-fold symmetry was reported. This quasicrystal became known as the third state of matter. The discovery of a substance previously deemed impossible in the field of crystallography was hailed as the most shocking event in solid-state physics in the latter half of the 20th century, and the man who discovered it, Dr. Daniel Shechtman of Israel, was awarded the Nobel Prize in Chemistry in 2011.

Professor Keiichi Edagawa of UTokyo-IIS has been studying quasicrystals, the oddballs of solid matter, for nearly 40 years.

Electron diffraction pattern of a quasicrystal (left) and Penrose lattice
The structure of a quasicrystal is an arrangement in three-dimensional space, and the corresponding two-dimensional arrangement is the Penrose lattice. The Penrose lattice consists of two types of rhombuses that cover the entire plane without gaps and have 5-fold symmetry (rotational symmetry of 72 degrees). Quasicrystals have a structure that is an extension of the Penrose lattice to three dimensions.

“It seems interesting. Why not look into this?”

He first encountered quasicrystals in the first year of his master’s program, when his academic supervisor suggested he explore the topic.

“At the time, it was a hot topic, and many people were looking into it in the hope finding new properties and functions. However, these efforts did not immediately lead to any groundbreaking applications. Gradually, the number of researchers declined, and the field as a whole as well as my own research went through a period of stagnation. I have continued to research quasicrystals alongside other topics because the ingenious and intricate order possessed by quasicrystals never ceases to amaze and inspire me. Most importantly, though, they remain a significant subject of study in the field of materials science,” said Edagawa.

Materials science is the academic field that studies the atomic arrangements (structures) of solid substances, investigates their physical properties, and elucidates the mechanisms that give rise to these properties. Once the mechanisms by which physical properties emerge from a substance’s structure are understood, it becomes possible to predict the properties of unknown materials from their structure and to design materials with enhanced properties.

How then can the structures of quasicrystals be verified?

“The primary methods for examining the structures of solid materials are X-ray diffraction and electron diffraction. It is possible to observe the directions in which X-rays or electron beams tend to scatter by placing a film behind a solid before targeting it (see the upper left portion of the diagram below). Imagine a screen along the surface of a body of water (see the upper right portion of the diagram below). If the screen has holes in it, waves passing through will create ripples that spread out from each opening. This phenomenon is called diffraction. When openings are arranged in a horizontal line, the diffracted waves passing through them will interfere with one another—either amplifying or canceling each other out—producing what’s known as interference. If many openings are arranged in a regular pattern, strong wave peaks will appear in specific directions. On the other hand, if the openings are arranged irregularly, waves will be generated in all directions with a certain intensity, resulting in a continuous waveform.”

Both X-rays and electron beams possess wave-like properties, so when they strike a solid material, the interference of the resulting diffracted waves creates a diffraction pattern. When this happens, each atom acts like an opening in a screen (see the upper center portion of the diagram below). If X-rays or electron beams pass through materials in which the atoms are arranged in a regular pattern as in a crystal, a dot pattern will appear. In contrast, disordered materials (amorphous solids) produce a blurred diffraction pattern.

“Dot patterns also appear in the diffraction patterns of quasicrystals. This indicates that they possess order over a far longer range than the typical atomic spacing, which sets them apart from amorphous solids. Moreover, quasicrystal diffraction patterns can exhibit five-fold, ten-fold, and even twelve-fold rotational symmetries. We can mathematically prove that such rotational symmetries cannot exist in periodic structures, and therefore, we can conclude that quasicrystals are not crystals, which are defined by their periodic atomic arrangements. The discovery of quasicrystals—a third state of matter—came more than 70 years after the phenomenon of X-ray diffraction was first observed in 1912, making it all the more astonishing to learn that such a substance actually existed in nature. Quasicrystals overturned a foundational concept in the field of crystallography,” said Edagawa.

The Moment They Were Captured for the First Time Ever!

So, then, how exactly are quasicrystals formed? The mechanism by which atoms come together to create the unique atomic arrangement of quasicrystals had remained a mystery for many years. Although various theoretical models had been proposed, no one had been able to experimentally verify them.

In 2015, Edagawa, together with his student Keisuke Nagao, succeeded in observing the growth of quasicrystals in real time through a high-resolution observation technique that makes use of a transmission electron microscope. They extracted still images from the recorded video and conducted analyses. Previously proposed models suggested that quasicrystals grew by consistently maintaining order through sophisticated local rules. However, their observations revealed that atoms initially assemble without preserving order, and when disorder occurs, they rearrange themselves to correct it. Quasicrystals form as this process continues to repeat.

“We also discovered that this correction process is linked to a degree of structural freedom unique to quasicrystals, known as phason. This is the fundamental mechanism behind the formation of order in quasicrystals,” said Edagawa.

The paper reporting their findings was published in the physics journal Physical Review Letters, in which it was selected as a Featured Article. (K. Nagao et al., Phys. Rev. Lett. 115, 075501 (2015)．)

Edagawa revealed the secret to succeeding in such a challenging experiment as follows.

“We heated the sample to around 900°C inside a transmission electron microscope and conducted high-resolution observations of the quasicrystal grain growth process, and anyone who has used a transmission electron microscope will understand just how challenging this is. It actually took us over three years from the start of the experiment before we were finally able to obtain clear observations. We were convinced that succeeding at an observation would lead to a major breakthrough, and I’m glad that we were persistent.”

Quasicrystal Research Continues to Unfold

Recently, Edagawa, together with Associate Professor Yuki Tokumoto from the same research group, discovered the world’s second quasicrystal exhibiting superconductivity. Their discovery marked another significant leap forward in the field.

In Japan, research into quasicrystals is once again gaining momentum, driven by Grants-in-Aid for Scientific Research on Innovative Areas, JST-CREST, and other large-scale projects. This renewed momentum is undoubtedly thanks to researchers like Edagawa, who have continued to illuminate the field and lay its foundation over the years. “I don’t want to just follow in someone else’s footsteps. I want to keep venturing into uncharted territory.” This approach to research embodies the very uniqueness of quasicrystals themselves.

Keisuke Nagao, Tomoaki Inuzuka, Kazue Nishimoto, and Keiichi Edagawa*, “Experimental Observation of Quasicrystal Growth”, Phys. Rev. Lett (2015),
DOI: 10.1103/PhysRevLett.115.075501

Yuki Tokumoto*, Kotaro Hamano, Sunao Nakagawa, Yasushi Kamimura, Shintaro Suzuki, Ryuji Tamura and Keiichi Edagawa*, “Superconductivity in a van der Waals layered quasicrystal”, Nature Communications (2024),
DOI: 10.1038/S41467-024-45952-2

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