I did not share their concerns, though, for one simple reason. I never thought Pauling’s alternative proposal was plausible. For one thing, Pauling’s multiple-twinning model of the icosahedral Al6Mn alloy was much more complicated than our quasicrystal explanation. And when it comes to science, the simplest explanation is usually the best.
Josh Socolar and I had established that the quasicrystal picture required four different building blocks (as shown in the color insert, image 2), each composed of tens of atoms, arranged in a quasiperiodic sequence. Pauling believed the materials were, instead, an intergrowth of many crystals oriented at different angles, a version of the multiple-twin idea that John Cahn had originally discussed with Dan Shechtman. According to Pauling’s theory, the repeating building blocks in each crystal had over eight hundred atoms per building block. To say it was more complicated than our theory would be a massive understatement.
I was actually becoming much more concerned about another competing idea that had begun to gain prominence around the same time Pauling was making his ideas known—the icosahedral glass model, a theory developed by Peter Stephens from Stony Brook University and Alan Goldman from Brookhaven National Laboratory. This was a much improved version of the Shechtman-Blech model.
The new icosahedral glass model proposed an atomic structure composed of icosahedron-shaped clusters arranged in a disordered pattern in space. That feature explained the word “glass” in the theory, because “glass” refers to materials with random atomic arrangements. In this model, the corners of each icosahedron-shaped cluster were aligned so that they pointed in the same directions in space. That feature was similar to the Shechtman-Blech model, with a notable improvement. Stephens and Goldman included an explanation of how to join the clusters together in a way that resulted in much smaller gaps and cracks.
The two models, icosahedral glass and our quasicrystal theory, could be distinguished, in principle, based on the sharpness and alignment of spots predicted for the diffraction pattern. A perfect quasicrystal produces a pattern of true pinpoints arranged in crisscrossing straight lines. The diffraction pattern for an icosahedral glass was predicted to be very similar, except the spots were fuzzy and did not align perfectly.
Unfortunately, the initial data was ambiguous because of the nature of the material being tested. To put it simply, Shechtman’s original aluminum-manganese sample was not very good quality. There was something inherently flawed about the alloy. Groups that had been trying to independently duplicate the sample were encountering the same problem.
The problem with the fuzziness and positions of the diffraction spots observed in the original sample had not been immediately apparent from the published photographs. Such images tend to be overexposed in ways that hide the flaws. But the issue was made glaringly obvious in the more precise X-ray diffraction patterns later made by Paul Heiney and Peter Bancel at Penn’s Laboratory for Research on the Structure of Matter.
Their X-ray diffraction lab was just across the street from my office at Penn, so I was able to study the test results as they were being completed. As someone who strongly believed in his own theory, I have to admit that I found the new diffraction images somewhat alarming. They clearly showed that the X-ray diffraction spots were fuzzy and imperfectly aligned, which did not match our predicted pattern. The results appeared to be a match to the competing glass model.
Things looked bad. But even so, I knew that the X-ray results would not necessarily sound the death knell for our theory. There might be a simple explanation for the fuzziness and small misalignments of the diffraction peaks. That was something that would naturally occur if the initial liquid mixture of elements being used to create a quasicrystal was cooled too quickly. The rapid cooling process tended to freeze in randomly placed defects and prevent the atoms from reaching the ideal arrangement.
As it turned out, all of the icosahedral Al6Mn samples that existed so far had been synthesized using a rapid cooling process. And with good reason. Whenever the material was cooled more slowly, it failed to form a quasicrystal. Instead, the aluminum and manganese atoms would totally rearrange themselves into a classic crystalline arrangement.
Joshua Socolar and I teamed up with the renowned condensed matter theorist Tom Lubensky to analyze the situation. The three of us developed a detailed theory to describe various distortions expected to occur in quasicrystal diffraction patterns because of defects introduced by the rapid cooling process. We found that the predictions could produce the same exact fuzziness and displacement of diffraction peaks that had been observed in X-ray experiments on Al6Mn. That meant our theory could be adapted to predict either sharp or fuzzy pinpoints, depending on the cooling process. So we were still in the running.
The icosahedral glass model was also still in the running, though, because it predicted fuzzy spots. To make matters worse, the data also allowed room for a version of Pauling’s idea of a multiple-twinned crystal, provided one allowed for at least eight hundred atoms in each repeating building block.
So in essence, all three models could explain Shechtman’s data.
In principle, there was yet another type of test that could settle the competition, one that involved heating instead of cooling. If one were to heat the sample gently over a long period of time, but not to such high temperatures as to make it melt, three different outcomes were possible. It would either form a more perfect quasicrystal with sharp peaks, as Dov and I had predicted, or it would form a more perfect multiple-twinned crystal in accordance with Pauling’s theory, or it would remain a disordered icosahedral glass with fuzzy peaks, in accordance with the Stephens-Goldman model.
But unfortunately, the heating test could never be performed with Shechtman’s alloy of Al6Mn because of its tendency to crystallize. Heating the alloy for even a short amount of time would destroy the icosahedral symmetry altogether, making it impossible to determine which theory was right.
In fact, more than three decades since its discovery, experiments still cannot definitively determine whether Shechtman’s Al6Mn material is a genuine quasicrystal, an icosahedral glass, or one of Linus Pauling’s multiple-twinned crystals.
That dilemma partially explains why it took so long for the scientific community to accept quasicrystals as a new form of matter.
The second reason for the delay in acceptance was more theoretical in nature, based on a thorough study of Penrose tiling. Critics who preferred the icosahedral glass picture argued that a true quasicrystal was an unattainable state because there was no plausible way to “grow” one.
To crystallographers, the word “grow” means forming crystals slowly from a liquid mixture of atoms. One can make sugar crystals, commonly known as rock candy, by dissolving lots of sugar in water and then waiting a few days for the sugar crystals to form. Similar processes occur in nature and in the laboratory. What occurs on a microscopic scale is that, beginning from some small cluster of atoms in the liquid, more and more atoms attach until the cluster “grows” to a visible size. For this to occur, it is important that the atoms maintain a regular periodic order whenever they attach. Since atoms in the liquid randomly approaching a cluster only interact with the nearest atoms in that cluster, there must be simple forces or, equivalently, simple rules that determine where atoms attach and where they do not.
Common experience in constructing Penrose tilings suggested that simple “growth rules” like this do not exist for quasicrystals. Imagine you decided to cover a large surface with a Penrose pattern using a pile of fat and skinny rhombus tiles. You know about matching rules, so you would make sure that any tile you added would join in accordance with the matching interlocks Penrose prescribed. Your goal is to completely cover the surface without leaving any gaps.
You might guess that this could be easily accomplished. After all, Penrose showed that it is possible to completely cover a surface, even one of infinite extent, using his interlocking tiles.
But you would be dead wrong. The Penrose tiling is
like a challenging jigsaw puzzle composed of only two shapes. There is a valid solution to the problem, a way in which all of the jigsaw puzzle pieces can interconnect. But finding that precise solution requires patience and a lot of trial and error.
If you started putting the tiles together one by one, chances are that you would run into difficulty after only a dozen or so pieces even if you meticulously followed all of the interlock rules each time you added a tile. You would eventually run into a spot where neither a fat nor a skinny tile could fit. You could start over and try again making different choices. Odds are, though, that you would not get much further.
The problem is that Penrose interlock rules only ensure that an added tile is properly aligned with its immediate neighbors. The rules do not ensure that the added tile is properly aligned with respect to the rest of the tiles in the pattern. So unless you are lucky, some of the tiles already added to distant parts of the pattern will be in conflict. And that conflict only becomes apparent when you suddenly reach a point where no tiles can fit. Scientists call that type of dead end a defect.
If you continued adding tiles, you would soon find yourself creating another defect. And then another and another and another. By the time you put together hundreds of tiles, you would have so many defects that you would hardly recognize the result as a Penrose tiling.
Of course, Penrose proved that it was possible for the tiles to be arranged in a perfect, gapless pattern. But he never claimed that one could construct a pattern by putting his tiles together in an arbitrary order. In fact, he was well aware of the fact that the proper arrangement was nearly impossible to find.
If that problem occurred for Penrose tiles with matching rules, critics argued, the same must occur for atoms attaching one by one to a cluster to form a quasicrystal: So many defects would form during growth that it would be nearly impossible to form anything resembling a true quasicrystal. Skeptics concluded that for all practical purposes, a perfect quasicrystal was an unattainable state of matter.
This was a true low point in the quasicrystal story. The two problems were seemingly insurmountable. The best experiments on Al6Mn were being conducted with a rapid cooling process that always created fuzzy spots in the X-ray diffraction pattern, instead of the sharp pinpoints we had predicted. And now, there appeared to be a strong conceptual argument that quasicrystals were effectively an impossible state of matter.
The debate was resolved by two breakthroughs. One theoretical and the other experimental.
* * *
YORKTOWN HEIGHTS, JULY 1987: The theoretical breakthrough came with the discovery of an alternative to Penrose’s interlocking rules, which we called “growth rules.” They made it possible to add tiles one by one to a pattern without making any mistakes or creating any defects. The growth rules were inspired by yet another visit to the IBM Thomas J. Watson Research Center in Yorktown Heights, New York. This time I had been invited to spend the summer continuing my research on quasicrystals.
One day while working at the Center, a researcher named George Onoda invited me and his colleague David DiVincenzo to lunch. He wanted to discuss a new idea about how to avoid defects in Penrose tilings. I had known George for several years. We met during my first sabbatical at IBM in 1984, around the same time that Dov and I published our first papers on quasicrystals. I had known David since he had been a graduate student at Penn.
When we sat down to lunch, George explained that he was familiar with the problem of frequent defects created by following the Penrose interlocking rules. He had struggled with the issue and found that he could construct additional rules that could ensure defects were made less frequently. That idea sounded intriguing. So we quickly finished our lunch and moved to a nearby meeting area where we could sit around a big circular coffee table. George took out a box full of paper Penrose tiles and began demonstrating his new rules.
Sure enough, George’s rules were an improvement. We still encountered a dead end, a gap that could not be filled, but we were able to put together two dozen tiles before that first occurrence. Once we understood how George’s new rules worked, we noticed that we could add yet another rule that would make the process even better. And after we tried that rule, we discovered yet another rule that would lead to even more improvement. Each of us took turns adding new rules over the next two hours until, suddenly, we found that we could cover the entire table with George’s tiles without making any mistakes or adding any more rules.
It must have been a strange sight to see three scientists hunched over a table intent on constructing a homemade paper puzzle. But if anyone happened to take exception, we would have never noticed. The more time we spent on the project, the more absorbing it became. None of us had ever expected to find rules that made it possible to put together so many Penrose tiles without making a defect.
Unfortunately, we only managed to achieve this success after drawing up a long laundry list of seemingly arcane rules, things like “if a configuration like such-and-such occurs, add a fat tile to this particular edge of it.” But then, as I began to study the list more closely, I noticed that the entire list of rules could be restated compactly if expressed in terms of adding tiles to something called an “open vertex.”
A vertex of a tiling is any point where the corners of several tiles meet. An open vertex is where there remains a wedge of space to add more tiles.
The long list of new rules we devised were able to be reduced to one sentence: Only add a tile to a vertex if there is a unique choice that produces a legal vertex, one that is allowed in a perfect Penrose tiling; otherwise, randomly choose another vertex and try again.
Could such a simple rule really work? Proving it mathematically was a challenge that took several months. Once again, I contacted Josh Socolar, who was by now considered a world expert on tilings. It had been a few years since the two of us had first theorized why quasicrystals might form, using matching interlock rules, Ammann lines, and deflation-inflation rules. Now, using a brilliant combination of computer programming and mathematical reasoning that Josh had devised, we were able to demonstrate that all three properties were essential to proving that the new vertex rule worked without fail, with the minor technical twist that the initial seed cluster of tiles included a configuration known to Penrose tilers as a “decapod.”
Our new growth rules were fundamentally different from Penrose’s original matching rules. Those rules constrain the way two tiles can join along an edge. The growth rules constrain the way a group of tiles can join around a vertex. Just like the matching rules, though, the growth rules could result from realistic atomic interactions in which the forces between atoms only extend across a few atomic bond lengths.
The growth rules surprised the scientific community. Among those most astounded was Roger Penrose. I first met Roger in 1985 when I invited him to Penn to meet with both my theory group and my experimental colleagues working on quasicrystals. I eagerly showed him all the research that his ingenious invention had inspired. Roger was the epitome of modesty and graciousness. With his crisp British accent, he politely asked hundreds of questions and generously shared his own ideas. We quickly established a great rapport which continues to this day, as we share interests in both quasicrystals and cosmology.
In 1987, though, Roger was still convinced that the skeptics were right. Based on the problems encountered while constructing Penrose tilings, he believed it was impossible for atoms with ordinary interatomic forces to form highly perfect quasicrystals. A few years later, though, he changed his mind. In 1996, I was invited to a 65th birthday fest at Oxford University to honor Roger and his many historic contributions. The event gave me the opportunity to show Roger the mathematical proof of our growth rules. As a memento, I gave him a rare set of our three-dimensional building blocks (color insert, 2), which he gratefully accepted.
It would take us nearly three more decades before we could complete the proof for growth rules in three dimensions. Although the same principles apply as for t
wo-dimensional Penrose tiles, arriving at a proof is much more difficult. It is much harder to visualize three-dimensional building blocks and there are many more configurations to consider. Josh and I put the problem aside until 2016, when we decided to revisit it using improved visualization techniques. By then, Josh was a professor at Duke University and was joined by his talented undergraduate student, Connor Hann. Together, the three of us finally completed the proof.
Finding growth rules for Penrose tilings in two dimensions had been enough to decimate the skeptics’ conceptual argument that a perfect quasicrystal was an unattainable state. But would it ever be possible to find a combination of elements that would form a perfect quasicrystal in the laboratory?
* * *
SENDAI, JAPAN, 1987: Even before our paper on growth rules was published, a scientist a quarter of the way around the world from us solved that problem.
An-Pang Tsai and his collaborators at Japan’s Tohoku University announced the discovery of a beautiful new icosahedral quasicrystal composed of aluminum, copper, and iron. Unlike quasicrystals synthesized earlier, Tsai’s sample did not require rapid cooling. As a result, it could be annealed, meaning it could be heated gently for days without transforming into a crystal. The annealed quasicrystal was nearly defect-free and had a solid, beautifully faceted shape that clearly showed its inherent five-fold symmetries.
The image, which appears below, may appear ordinary at first glance, like a faceted diamond or quartz crystal. But this is far from ordinary. These were the first-ever absolutely perfect pentagonal facets ever seen, and a major scientific advance compared to the disordered, feathery structures formed by Shechtman’s Al6Mn alloy.
The Second Kind of Impossible Page 9