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What Are Quasicrystals?

Leonard Kelley holds a bachelor's in physics with a minor in mathematics. He loves the academic world and strives to constantly explore it.

The Strange World of Quasicrystals

Matter is usually understood to be some type of organization, whether it be loosely jointed like a gas or in a rigid structure like a solid. Sometimes those solid structures have repetition to them and form a crystal. But some solids exhibit a pattern which defies simple and yet remains valid, flying in the face of what should be allowable. This is the world of quasicrystals, and their theory brings together many fields of science into a beautiful form.

Crystal Origins

Before we can talk about how unique quasicrystals are, we should give a brief overview on crystals and their discovery. In the 18th century, mineralogy was a well-developed field. People recognized that they were pure and had a regular structure to them, but it was a mystery how the same matter could be arranged differently to form seemingly distinct objects, with angles and colors varying greatly. But in 1781, headway was made when Rene Just Hauy dropped a piece of calcareous spar and noticed the smooth sides the pieces had as well as the nice angles to them. He took a piece and cleaned off the rough leftover bits until solely flat faces remained. He noticed the shape was that of rhombohedrum. He took another sample and found he could do the same, in fact any sample demonstrated his ability to for the solid shape. He developed a stacking theory, where individual building blocks of that shape stack together, much like LEGOs do. This is essentially a proto-atomic theory over a century before it was developed! (Steinhardt 13-17, Sincell)

Over the next few decades he tests other minerals for this and uncovers a network of hidden geometries, each one giving the mineral its own identity. Altogether, he finds that minerals come in either tetrahedrons, triangular prisms, or parallelepipeds though is unable to explain why these are the only possible shapes. In 1801 he publishes Traite de Mineraolgie, documenting his findings and earning him the moniker “Father of Modern Crystallography” (Ibid et al.).

Eventually, crystallography is able to account for why these forms are possible and is able to predict their properties. It stems from the Three Principles of Crystallography: Pure substances form crystals provided sufficient time to bond, all crystals form one of the before mentioned shapes, and all crystals can be placed into symmetric categories known as tilings. These tilings, as we will see, are going to be crucial to quasicrystals (Steinhardt 18-21).

In crystals, our structure arises from combinations of 2D rectangles, squares, parallelograms, triangles, and hexagons arranged in such a way as to create a periodic structure in 3D. Of course, not all tiling combinations can match up well and it also depends on the mineral’s atomic make-up as well. This is where rotational symmetry can assist us. If I can rotate a tiling θ degrees and have the same configuration as before, then nθ rotations will work too. I can perform the rotation as many times as I want to, because it’s the same shape to me. The number of rotations we can perform in a 360 degree span gives us our n-fold symmetry. So, a parallelogram has 2-fold, a triangle 3-fild, a hexagon six-fold, a rectangle 2-fold, and an irregular tile 1-fold. To make patterns with tilings, these folds need to align nicely and with the fact that corners match up to make 360 helps us determine possible tilings (Ibid).

So, how many symmetries are there available to us as we build our 3D shapes? Turns out, its just 14, as uncovered by Bravais in 1848. These are known as the Bravais lattices, and based on how those are arranged with respect to rotational, reflectional, and translational symmetry gives us a total of 230 possible configurations! Crystallography is a great common ground for mathematics and physics, as we can see. To fully bring this from theory to fact, atomic structures needed to be examined. In 1912, Max von Laue developed X-ray diffraction to assist with this. Light, if moving with the right wavelength, can travel between atoms, and X-rays are perfect for this. After they pass by an atom and are influenced by its presence, the X-rays interfere with each other and create a diffraction pattern. If fired randomly, the pattern won’t present itself well, but if we look at the material through one of its rotational symmetry lines, we end up with a pattern of dots that defines the internal structure. And by altering the wavelength to change that dot pattern, we can develop an interior model of our substance (Steinhardt 22-4, Goldman).

Penrose Tilings

If you cool a liquid slowly into a solid, the atoms can arrange themselves into a crystal provided the atoms are all the same with “simple interatomic forces” allowing for a stackable structure. But what if the liquid is rapidly cooled? There isn’t enough time for the atoms to organize themselves, so we should anticipate the solid structure would be random. Paul Steinhardt and his team simulated several scenarios where “the cooling and solidification process” was altered and in 1980 found that surprisingly an icosahedron pattern was found! No matter where you looked, a pentagonal pattern arose (Steinhardt 29-35, Goldman).

But that is a 5-sided shape and a regular pentagon would have 5-fold symmetry. That doesn’t stack up nicely, hence why it was never considered for atomic structures. This finding was just a simulation, but the probability charts indicated it very likely that it matched a real phenomenon. Does it violate crystallography up to this point? No, because the laws as established prior refer to macroscopic structures and not to the atomic components themselves. On a small atom count we can certainly get weird stackings, but as the number of atoms grows, the chances of the shape taking hold decrease because of its inherent instability. The lack of neat stacking in addition to interatomic forces are enough to disrupt the overall structure from taking hold. And yet the simulation showed the stacking sticking for 1000s of atoms of the same kind, something that shouldn’t happen (Ibid).

The central idea behind the contradiction lie in Penrose tilings, or special types of polygons that combine in non-periodic ways. That is, I can fit them together but I will never have an overarching pattern to predict the next step. Roger Penrose found the first ones in 1974, with a kite containing three 72 degree angles and a 144 degree angle and a dart, or convex polygon, with a 72 degree and two 36 interior angles and a 144 degree exterior angle. It is essentially a specific way to break down a rhombus with interior angles of 108 and 72. By placing the edges of the tiles together in certain ways, we can create interlocking shapes and thus a new pattern whose only catch is a lack of periodic repetition to it (Steinhardt 40-9).

You essentially have to trial-and-error your way in finding more combinations. Is there a non-random way to achieve this? Steinhardt and team did find a 5-fold symmetry in Penrose tilings where we make something from a 5-pointed star and radiate outward, generating less of these stars and in no discernable pattern. It was not a reassuring finding. Some loophole should exist, something that theory could describe (Ibid).

The kites and darts.

The kites and darts.

Ammann Bars

Enter Robert Ammann. He was able to find a big matching tool that has become known as Ammann bars which offered some insight into what was going on. When presented with a Penrose tiling set, we can draw lines though each of my tiles in such a way that when the shape is put together, the lines all connect and are straight and parallel to each other! And there can me multiple sets of these lines at different angles but parallel to lines in that configuration. Steinhardt looked at these lines with the 5-fold symmetry of those Penrose tiles and found that the angle between different configurations was 108 degrees, which is the same as the exterior angle of a pentagon! Again, the beauty of mathematics is strongly spotlighted here (56-61).

But it certainly doesn’t end there. The distances between our parallel lines is always one of two values, with one being smaller than the other. Sometimes you can have a sequence of several smaller distances then a few larger ones. If you look at the ratio of the first 3 lines its 2:1. For the first 5 its 3:2 and the first 8 its 5:3. Those are Fibonacci numbers! If you take the ratio of these lengths towards infinity, it turns out to be the Golden Ratio, a hallmark of mysterious mathematics. We have just related a major geometrical concept to two of mathematics awesome topics. And we can construct it too! It’s a physical relationship and not just pure mathematics! This quasiperiodic nature of the tilings couldn’t have a matter counterpart…could it? (Ibid).

The sets of parallel lines, one wider (W) than the other narrow (N) set.

The sets of parallel lines, one wider (W) than the other narrow (N) set.

Hunting for Quasicrystals

To find out, a 3D model of the quasiperiodic behavior would have to be constructed. The icosahedrons with their 5-fold symmetry only display that in 6 directions, making combinations challenging. Rhombohedra’s, similar to our rhombi from before, have other folds that can be potentially exploited for patterning. With wider and narrower options being used, we can build new patterns with two rhombohedra’s in a similar fashion to the Penrose tilings of 2D. This led to the first quasiperiodic solid being found, known as the rhombic triacontahedron. It is a 30-sided shape with all rhombi as faces, with no gaps anywhere in the interior but no overall tiling pattern to it either! (64-66)

What would such a material yield under X-ray diffraction? We know what the pattern is, so it’s just a matter of performing the scattering and seeing our dot pattern. It ends up being a more intricate pattern, with different configurations depending on how I am examining the object. With the pattern in tow, it was just a matter of someone testing materials and finding the match (Steinhart 70-8, Sincell).

And indeed, on April 8, 1982 it happened as Dan Shechtman was examining a sample of crystal Al­6Mn. Unaware of the aforementioned results, he recorded the diffraction pattern using electron scattering instead and found a 10-fold pattern. At the time, it was thought that the result was a consequence of multiple twining effects, or when 2 crystals are looked at the same time and overlap to create different patterns. Such an incident is rather common and so no one paid much heed to the finding. In fact, a frequent check of the behavior is using a narrower beam, which should eliminate the 10-fild symmetry. When this did not happen, naturally scientists knew they had something interesting. Eventually, word got around and the first quasicrystal was formally recognized (Ibid et al).

Examples of Tiling Patterns and Their X-Ray Diffraction Results

what-are-quasicrystals
what-are-quasicrystals
what-are-quasicrystals

Defending Quasicrystals

As it should be the case in science, the results were put under fire, with alternate theories as to what was really found being offered. One of these people was Linus Pauling, who went back to the multiple twinning model for inspiration. He offered a different atomic configuration which while more complicated than the quasicrystal model, could still be explained by the overlapping layer idea. Peter Stephens and Alan Goldman developed an icosahedral glass-model, where the basic components are randomly scattered (hence the glass in the name) but the icosahedrons corners all point in the same direction. The diffraction pattern should, under the right conditions, provide a different pattern in terms of arrangement and dot size than the quasicrystal model did. Stechtman’s sample wasn’t adequate for this due to a lack of replication by other scientists (Steinhardt 101-4, Goldman).

Once new samples were found and analyzed, they seemed to promote the glass model over the quasicrystals, but these were generated via rapid cooling. Was the process fast enough or did it allow something that wasn’t truly a quasicrystal to form and therefore render the comparison moot? Steinhardt, along with Joshua Socolar and Tom Lubensky, decided to find out by seeing if the quasicrystal model could generate a defect pattern too. Sure enough, it’s possible to generate that model too if the crystal is rapidly cooled. Another good test would be to heat the material. If it’s a quasicrystal, the diffraction pattern will be shaper but if a glass it will be fuzzier, for the structure will partially collapse (Ibid).

Building Quasicrystals

However, a more pressing concern than all these comparisons lay in the actual construction of a quasicrystal. After all, they have no overall pattern from which we can algorithmically build one up. How could nature possibly create one? After all, it has to be simple if a general construction is to be achieved. Man built his model off of trial and error, so does nature follow suit? We should therefore expect many, many defects in samples that we find, and yet most are pure and free of error (Steinhardt 105-6).

The breakthrough came in July of 1987 when a set of growth rules was established that allowed Penrose tiles to be added without error. The discovery was spearheaded by the IBM Thomas J. Watson Research Center in Yorktown Heights, and it involves an alternative to Penrose interlocking rules. The tiles run through a process of updating, where a dead end is found and so a rule to fill the void is established, and so the process repeats. By finding a tile that will fill in a corner meeting area, otherwise known as a vertex, with the Penrose tiles at our disposable, it’s just the probability of the right fit which completes the corners. Penrose tiles had been all about edges before, but now it was a vertex game too (107-9).

Settling the Debate

The same year as the vertex breakthrough was made, an experimental one was also achieved. An-Pang Tsai and the team from the Tohoku University in Sendai, Japan. They were able to make a new quasicrystal made of aluminum, copper, and iron. It was not made via rapid cooling, which involves rapidly rotating the material to achieve a cooling rate of 1 million Kelvin a second, but over a slow multiday formation. It was practically defect free and displayed 15-fold symmetry. The diffraction pattern of the material revealed sharp points along perfectly straight lines, agreeing with the quasicrystal model. Over the years opponents of the idea have tried to modify their idea but the consensus nowadays is definitely in the quasicrystal camp (Steinhart 110-114, Goldman)

The quasicrystal made at Tohoku.

The quasicrystal made at Tohoku.

A Stellar Origin

It would be remiss of me if I did not mention the khatyrkite quasicrystal samples Steinhardt recovered from Kamchatka. No, they were not made there but found there, indicating some local source that was later determined to be from a meteorite based on the metal content found with the sample as well as the presence of chondrules, which were building blocks from the early solar system. He knew to look for fragments at that site after uncovering a 5-fold symmetry in an archival diffraction image, with the sample being identified as being from there. Many grains were recovered, so let’s dive into what they can inform us (Steinhardt 302-5).

Grain 121 had chondrules with it that help confirm the astronomical origin known from the previous sample. Around the crystal was also a matrix containing the chemical markers in it. Both the matrix and the chondrules hint at the exact nature of the source, that being a carbonaceous chondrite meteorite (311-6).

Meanwhile, Grain 122 had an icosahedrite pattern as did Grain 123, but the latter was in direct contact with meteoric material, showing a common lineage. Once the samples were sent to Caltech, an oxygen isotope test was conducted to show unequivocally that the crystals and chondrules were the same age. Somehow, a natural quasicrystal formed in space a long time ago and made its way to Earth (Ibid).

One of the recovered samples.

One of the recovered samples.

But the big question is how could such an object form? Other grains provided further clues. Take Grain 125, the best case of “contact between oxygen-bearing silicate and khatyrkite, the crystal AL-Cu alloy most abundant in the samples.” By examining the edges of contact between them, one can maybe get a feel for the forces experienced in the formation. Under an electron microscope, scientist spotted filaments of nearly pure aluminum in the khatyrkite, something never before seen in a mineral. To create such a feature, the material had to be molten and then rapidly cooled, trapping those strands. Figuring out the silicate portion, however, is trickier because of averaging out effects potentially blurring the potential explanation. Scientist’s needed a refined sample, so they brought in an ion beam to carve out a thin slice and then examine it under an electron microscope (325-333).

Turns out, a complex structure was present, also indicating a rapid cooling event for meteoric material. And truly telling was the presence of ahrensite, whose formation can only arise from pressures greater than 150,000 atmospheres, and at temperatures greater than 2000 degrees Fahrenheit! All the signs point to a space collision of some sort. The shock wave helped compress the material and stretch it in a rapid fashion. With heat and pressure provided, our quasicrystal was able to form. But, was this collision during the tumultuous beginnings of our Solar System or was it sometime afterward? (Ibid).

Timing It Just Right

To help identify the age of the meteorite, we need to get a feel for how stable our icosahedrite is at high pressures. This is because a post-collision could either form the material or alter was already there, so by identifying the response of the quasicrystal to high pressure we can find which of our options is more viable. Using synthetically derived icosahedrites, the test objects were subjected to x-ray diffraction and had their dot pattern tracked as temperature and pressure increased. No discernable changes were spotted, hinting at the possibility of the quasicrystal originating from the formation of the solar system and not being impacted by space collisions. A synthesis test mimicking the conditions of a space collision showed quasicrystal formation closely in line with the samples found (but not completely the same), providing even more evidence (Steinhardt 337-342, Asimow).

A further test of age lies in comparing neon to helium isotope levels. In space, radiation impacts the amounts of both of these because they form as a result of the bombardment. If some huge impact occurs, the levels are effectively reset at zero, so by seeing what amounts we have and compare the percentage of isotopes present we can estimate the age of intense conditions. The levels present point to a collision anytime from 100 million years ago to as a far back as 1 billion years ago. Via carbon dating, we know the Khatyrka meteorite fell to Earth about 7000 years ago, so we know the impact with the planet didn’t contribute to the reset isotopes. That quasicrystal definitely formed in space (Ibid et al).

The Future

Other grains seemed to paint this picture too, so clearly the next step is to look at the parent object. And surely, how daunting could that possibly be, trying o finds a small unknown object traveling through our solar system? Turns out, surprisingly, not as tough as it should have been. Julia 89, a Mars/Jupiter main-belt asteroid, has chemical links to the Khatyrka meteorite. This object is 150 kilometers across and is from a known collision group that occurred about a few 100 million years ago. Hmmm, how convenient! And Julia 89 is a CV3 chrondrite, which is also like our sample! (Steinhardt 361-3)

Meanwhile, advancements in the development of quasicrystals have improved, with the actual formation being witnessed by Uli Wiesner (Cornell University) and team. While examining silica particles guide themselves into binding via the micelle process, a structure appeared that was non-repeating but clearly present as a bunch of tilings! Under x-ray diffraction, the mesoporous silica nanoparticle displaced 12-fold symmetry, also confirming the quasicrystal nature of the material. Filming the formation process was tricky because of the electron microscope being used, which isn’t conducive to filming motion. So, the team just took the formation process and stopped it at many points, taking images as they went along. Stitching these together created a visual for the formation of the quasicrystal. They found that the formation process was impacted by motion (the more stirring occurred, the greater the formation was) and also and by chemicals (with mesitylene being used to encourage bonding). Further insights are still to come, but what a fantastic step this is (Lovell)!

A transmission electron microscope image of the nanoparticle and the tiling pattern of it.

A transmission electron microscope image of the nanoparticle and the tiling pattern of it.

And if that wasn’t cool enough, potential insights into quasicrystal internal mechanics have been found, based on research by Keiichi Edagawa (University of Tokyo) and team. They created an aluminum-copper-cobalt quasicrystal by melting the components together at 1173 degrees Celsius, then cooling it to room temperature. Differentiating rhombi and pentagon tiles formed, according to electron microscope scans. By causing a single atom to become excited, a phonon (or sound wave acting like a particle) will travel though the material. This in turn causes phasons to develop, which literally cause the structure to change momentarily. Hexagonal rhombi become pentagons and vice versa, depending on the location of the action. But what is crazy is that once the phason is complete the configuration returns to normal. This perhaps can give more clues to quasicrystal formation (Sincell).

So when you need a prime example of the beauty of math with the wild frontiers of physics, look no further than quasicrystals.

Works Cited

Asimow, Paul D. et al. “Shock synthesis of quasicrystals with implications for their origin in asteroid collisions.” PNAS Vol. 113, No. 26. 7077-7081.

Goldman, Alan I. and Peter W. Stephens. “The Structure of Quasicrystals.” Scientific American Apr. 1991. Print. 44-53.

Lovell, Daryl. “Engineering team images tiny quasicrystals as they form.” Innovations-report.com. innovations report, 18 Aug. 2017. Web. 11 Feb. 2020.

Sincell, Mark. “Physicists Glimpse How Quasicrystals Boogie.” Sciencemag.org. Science, Vol. 289, 1 Sept. 2000. Web. 13 Feb. 2020.

Steinhardt, Paul. The Second Kind of Impossible. Simon & Schuster, New York. 2019. Print. 13-24, 29-33, 40-49, 56-61, 64-6, 70-78, 101-114, 311-6, 325-333, 337-342, 361-3.

This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualized advice from a qualified professional.

© 2021 Leonard Kelley

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