The crystal gazer

Dan Shechtman, this year's Chemistry Nobel winner, faced strong criticism for his discovery of quasicrystals in 1982 before it was accepted.

THIS year's Nobel Prize in Chemistry was awarded to the 70-year-old Israeli scientist Dan Shechtman, who is a Distinguished Professor at Technion Israel Institute of Technology, Haifa, and holds the Philips Tobias Chair there. The coveted prize is for his discovery of quasicrystals, which caused a controversy at the time of its discovery nearly four decades ago. Shechtman became the object of all-round criticism, with Linus Pauling, one of the greatest chemists of the 20th century, famously saying, Danny Shechtman is talking nonsense. There is no such thing as quasicrystals, only quasi-scientists.

Shechtman made his discovery on April 8, 1982, while on sabbatical at the United States National Bureau of Standards (now the National Institute of Standards and Technology, NIST) and working on new, light alloys for aircraft. But it took nearly two and a half years for him to publish the results as he could convince no one of his findings; his boss at the NIST laboratory he was attached to asked him to leave the research group for bringing disgrace to the group. But Shechtman was not one to give up. He set about convincing others about the veracity of his finding and ultimately won the battle when scientists, including Pauling, finally conceded and were forced to reconsider the very nature of matter.

What are quasicrystals and why did the claim of its discovery become so controversial? To answer this, a little background on the knowledge of crystallography at the time is necessary. Shechtman was studying a somewhat unexplored alloy of aluminium (Al) with 10-14 per cent manganese (Mn). He had prepared a crystal of the material in the laboratory and put it under an electron microscope to observe it at the atomic level. He could not believe what he saw; it was contrary to all established laws about crystals and counter to what textbooks said and what he himself had been teaching his students. Eyn chaya kazo, he had said to himself in Hebrew, meaning There can be no such creature. Indeed, his discovery shook the very bedrock of solid-state science. So it was no wonder that he met with such criticism and hostility.

At a basic level, crystals are solids which have flat surfaces (facets) that intersect at characteristic angles. This picture was formalised by Ren Just Hy in 1784 into a theory of crystallography the science of crystals according to which crystals are solids that are ordered at a microscopic level. Solids rocks, minerals and ceramics have short-range and long-range orders to varying degrees. Short-range order is dictated by the chemical bonding requirements of the material's constituent atoms, and so all solids, including totally amorphous (non-crystalline) materials such as glass, possess short-range order. But the hallmark of crystallinity is long-range order, which is absent in amorphous materials.

Since the work of Hy, it has been believed that the only way that solids can have long-range order is when they have translational periodicity; that is, some basic structural unit repeating itself infinitely, much like the hexagon of a honeycomb, in all directions of the three-dimensional space. In other words, every crystal is a translational lattice of atoms (or molecules) characterised by the smallest structural unit, the unit cell, stacked together face-to-face to fill an infinite region of space without leaving any hole. Real crystals are, however, only finite but this idealised definition holds good in practice as unit cells are of the order of a few to tens of a nanometre (billionths of a metre) in size compared with physical crystals, which usually are hundreds of a micrometre (millionths of a metre) in size. The vast majority of unit cells form the bulk of the crystal; only a very small fraction forms part of a crystal's surface.

The theory of periodic ordering of crystals was enormously successful in predicting the characteristic angles of facets of crystals of a given type. In 1912, Max von Laue discovered X-ray diffraction in crystals, and William H. Bragg and William L. Bragg (father and son) subsequently developed the field of X-ray crystallography. When light passes through what is known as a diffraction grating, essentially a perforated plate, waves spread out in a semicircular fashion, reinforcing each other when their crests intersect and cancelling each other when the crests meet troughs. Correspondingly, a pattern of bright and dark areas a diffraction pattern appears on an imaging screen (Figure 1).

A diffraction experiment directly probes the degree of order in a solid. If there is order in the crystal, the diffraction pattern shows a set of spots called Bragg peaks against a dark background. The longer the range of order in the material, the sharper the peaks. In the following 70 years of literally exploding X-ray crystallography research, all observed diffraction patterns were in complete agreement with the predictions of the theory and the basic notion that order in crystals was due to the periodic arrangement of unit cells to fill all space.

In his study in 1982, Shechtman used electrons instead of light or X-rays, and his grating was a three-dimensional array of atoms in the rapidly quenched Al-Mn alloy. The diffraction pattern was similar to observed patterns until then, but in some very crucial aspect it was different. It had 10 bright spots arranged in a circle, something he had never seen before in all his years of research (Figure 2). According to the basic tenets of crystallography, a diffraction pattern with 10 dots in a circle was impossible. So Shechtman counted and recounted the dots in the pattern; there were 10. Four or six would have been understandable, but not 10.

To understand this one needs to understand another geometrical property of crystals. The periodicity, or infinite repetitiveness of a basic pattern, also dictates another important property of crystals: they remain unchanged only under certain operations of rotation. In other words, they have only certain rotational symmetries. A chessboard, for example, has a fourfold symmetry; it can be rotated four times, one-fourth of a rotation each time, and it will look the same. Similarly, the hexagon of a honeycomb can be rotated six times. Now a crystal can be imagined to be similar to a tessellation in two dimensions, say of a bathroom floor with a single type of tile. It can be easily seen that a bathroom floor cannot be covered with pentagonal tiles (Figure 3).

Analogously, to be compatible with translational periodicity, there is a restriction on the allowed rotational symmetries in a crystal. It can be rigorously proved mathematically that only two-, three-, four- and sixfold rotations are allowed. Fivefold rotations (or any n-fold rotation with n > 6) are disallowed. In three dimensions, there can be different rotational symmetries along different axes, but they are restricted to the same allowed set. This is the crystallographic restriction theorem.

But what Shechtman saw seemed to have a 10-fold symmetry; rotation by 36 degrees left the pattern unchanged. In the notebook where he recorded his observations of April 1982, he wrote 10 fold??? against the observation numbered 1725 (Figure 4). By reorienting the sample in different ways, he found that the crystal itself did not have a 10-fold symmetry but actually had the symmetries of an icosahedron, a geometrical object with 20 identical equilateral triangular faces: six axes of fivefold symmetry, along with 10 axes of threefold symmetry and 15 axes of twofold symmetry (Figures 5a and 5b). What he found was similar to a sphere, such as a football, with patches of six-cornered polygons or hexagons alone. To make the sphere, one needs both pentagons and hexagons (Figure 6).

When Shechtman tried to share his discovery with his colleagues, he faced extreme scepticism. The chief point of contention was that what he had seen was a twin crystal. It was a well-known fact that many disallowed diffraction patterns were artefacts of superposition of multiple allowed, but rotated, crystal symmetries, a phenomenon known as twinning, which is caused by sudden changes in direction during crystal growth. However, Shechtman had tested the sample repeatedly to rule out twinning. Electron microscopy helped him in this by allowing him to study smaller and smaller regions of the sample.

After he returned to the Technion in 1983-end, he found in one colleague, Ilan Blech, a sympathetic listener. Together they worked out a model based on pentagonal symmetry to explain the peculiar diffraction pattern. In the summer of 1984, they submitted the article to Journal of Applied Physics, but it was rejected almost immediately. Shechtman then approached John Cahn, a well-known figure in the field who had brought him to the NIST in the first place, to look at his data. Cahn's first reaction, too, was that this was a case of twinning. But on taking a closer look, he got interested and consulted a French crystallographer, Denis Gratias, who found Shechtman's data reliable. In November 1984, Shechtman, Cahn, Blech and Gratias finally got the article published the presence of Cahn was the key in Physical Review Letters.

The structure that Shechtman had discovered was termed quasicrystals, short for quasiperiodic crystals, by Paul Steinhardt and Dov Levine in a paper that appeared a mere five weeks later from the University of Pennsylvania. This was the first of a series of papers in which the two physicists set up much of the initial theoretical foundations of the field. These developments did not really stop criticism of Shechtman's work, and many (notably Pauling) continued to interpret his results as caused by twinning. But, many crystallographers had seen similar patterns earlier in fact, as far back as in 1956, in an aluminium-lithium-copper system but had ignored them thinking that they were due to twinning.

Another key reason why most crystallographers did not consider Shechtman's data reliable was his use of the electron microscope, which was considered imprecise. In fact, as Shechtman has said, most of them did not know how to use the electron microscope, whereas he considered it an ideal tool to study small crystals despite its relative inaccuracy. It was also not easy to repeat the experiment with X-rays because X-ray diffraction studies require much larger single crystals.

It was only in 1987 that scientists in Japan and France were able to grow quasicrystals large enough to be studied with X-rays and verify that Shechtman had indeed seen a crystal with a fivefold symmetry. The turning point for Shechtman came only after he showed these pictures at a conference in Perth the same year. His achievement began to gain recognition slowly and even Pauling was perhaps veering towards accepting quasicrystals when he invited Shechtman to write a joint article with him on the subject. Sadly, Pauling died before this could materialise.

[L]ike many discoveries, it was difficult to convince many people, especially the old established generation of X-ray crystallographers, Shechtman said in the customary interview to the Nobel Foundation after the announcement. Crystallographers believed in X-ray results, which are of course very accurate. But the X-rays are limited and electron microscopy filled the gap, and so the discovery of quasicrystals could have been discovered only by electron microscopy, and the community of crystallographers, for several years, was not willing to listen. But then we had the results from X-rays on quasicrystals and then the community joined. And that process took a few years.

The fivefold way

But still Shechtman did not have any idea what the strange creature looked like on the inside. Symmetry may be fivefold, but the basic question that could not be answered immediately was how the atoms were packed. The answer was actually contained in the explorations of mathematicians some years earlier involving games with mosaics. In the 1960s, mathematicians were trying to solve the problem of laying a mosaic with a limited number of tiles and such that the pattern never repeated itself. That is, it would be an aperiodic mosaic. In 1966, an American mathematician showed that it was possible to do so with 20,000 different tiles. But over the years, with more and more solutions for aperiodic mosaic appearing, the number of tiles required got smaller and smaller. The most elegant solution came in 1976 from the British mathematician Roger Penrose, who produced aperioid mosaics with just two tiles, a fat and a thin rhombus, and the tiling had a fivefold symmetry (Figure 7).

Incidentally, it was discovered later that certain medieval architecture, called Girih patterns, dating to the 13th century, had aperiodic mosaics that used only five tiles. Such mosaics can be seen at the Darb-I Imam Shrine in Iran and the Alhambra Palace in Spain. Penrose tiling has been used in the analyses of such architectural aperiodic mosaics.

Even before Shechtman made his discovery, Alan Mackay, a crystallographer, used the Penrose tiling in a curious way. He wanted to find out whether atoms could form aperiodic patterns. He used the Penrose mosaic to make holes at the intersections of the mosaic to represent atoms and used this pattern as a diffraction grating. He got a 10-fold symmetric pattern, 10 bright dots in a circle, just as Shechtman observed later. The actual theoretical connection between Mackay's model and Shechtman's data was made by Steinhardt and Levine in their 1984 paper. As one of the referees to the paper of Shechtman and others, Steinhardt had got a chance to read the paper even before its publication and, being aware of Mackay's model experiment, could quickly give an explanation for Shechtman's observations. Thus, just as squares or hexagons provide the geometric analogues for periodic crystals, Penrose tiling provides the geometric analogue for aperiodic quasicrystals.

Quasicrystals and aperiodic mosaics also display another intriguing symmetry property. The golden ratio of mathematics and art (5 + 1)2, a mathematical constant denoted by the Greek letter (tau) is linked intimately to them. For instance, the acute and obtuse rhombic tiles in Penrose tiling repeat along each symmetry direction with frequencies whose ratio is . Similarly, the ratio of various distances in between atoms in quasicrystals has been found to be related to .

Since its discovery in 1982, hundreds of quasicrystals, with different compositions and symmetries, have been synthesised. The first quasicrystals were thermodynamically unstable. The first stable icosahedral quasicrystals were synthesised in 1987 in a ternary iron-copper-aluminium (Fe-Cu-Al) system. Stable quasicrystals can be grown to large sizes and exhibit certain typical features of well-ordered crystalline systems, and some have been found to have a very high degree of order. These stable, high-quality samples were important to the present understanding of quasicrystalline structure. The discovery of a binary, stable icosahedral quasicrystal in calcium-cadmium and ytterbium-cadmium systems in 2000 was another key development, which provided the basis for the detailed structural elucidation of icosahedral quasicrystals.

However, only in 2009, as a result of the persistent efforts of Steinhardt and co-workers, was the first naturally occurring quasicrystal discovered. A new kind of mineral, an Al-Cu-Fe icosahedrite with a 10-fold symmetry, was found in the Khatyrka river in Chukotka in eastern Russia. Quasicrystals have also been found by a Swedish company in one of the most durable kinds of steel with extremely good characteristics. The steel has been found to consist of two phases: hard steel quasicrystals embedded in a softer kind of steel. Quasicrystals seem to function as a kind of armour. This steel is now used in products such as razor blades and special needles for eye surgery.

Quasicrystals are typically hard but, despite being so, are brittle like glass. Periodic structures greatly enhance the thermal and electronic transport properties of solids. In the absence of such periodicity, quasicrystals behave more like glass, with unusual transport properties rendering them bad conductors of heat and electricity. This property makes them suitable as thermoelectric materials that convert heat into electricity. Quasicrystals are being investigated at present as components for light-emitting diodes (LEDs) and as heat-insulating materials in engines. Because they also have non-stick surface properties, they are being used as surface coatings in frying pans.

Although translational periodicity was never proved to be a necessary condition for order, it had formed the basis for the definition of a crystal as adopted by the International Union of Crystallography (IUCr): A crystal is a substance in which constituent atoms, molecules, or ions, are packed in a regularly ordered, repeating three-dimensional pattern. But Shechtman's discovery of quasicrystals that had long-range order but lacked translational periodicity led to the dismantling of this definition in 1992 and the giving way to a broader definition that includes aperiodic structures such as quasicrystals as well. The IUCr has now defined a crystal to be any solid having an essentially discrete diffraction diagram. In the special case that 3-dimensional lattice periodicity can be considered to be absent, the crystal is aperiodic, it said.

This definition is also open-ended in the sense that nature may be hiding more secrets from us: there may be another class of non-periodic structures, as yet unknown, with discrete diffraction patterns, meaning long-range order. An overly restrictive definition could again result in a mistake some time in the future. The discovery, as Shechtman said in his post-Nobel announcement interview, was also a humbling one; nature sprang a surprise in a field that was regarded closed by the scientific community. Even though 40 years have passed since Shechtman's revolutionary discovery, many questions remain. As the mathematician Marjorie Senechal writing on quasicrystals said, The long answer to What is a quasicrystal?' is no one is sure. But the short answer is it is a crystal with forbidden symmetry.

Discoveries and research on properties of quasicrystals since Shechtman's discovery show that scientists have only scratched the surface of the science of crystals. But his discovery certainly opened up a vast new field of materials that is continuously being explored. But the most important lesson from Shechtman's work is, as he said, a good scientist is somebody who is willing to listen to news in science which are not expected, is open-minded and willing to listen to other scientists when they find something. Daring to question established knowledge is as important a trait in a scientist as domain expertise and skill.