The Quasicrystals Puzzle: An Introduction

Dan Shechtman; Linus Pauling.

[Ed Note: This is part 1 of a 4 part series discussing quasicrystals, which has been written in commemoration of Dan Schectman’s receipt of the 2011 Nobel Prize for Chemistry.  The science behind and debate over quasicrystals is a complicated one and we do not profess to be experts in the field.  What will follow today and over the next four weeks is our attempt to describe the science behind quasicrystals, including Linus Pauling’s role in its development.]

[All 3-D animations used in these posts were built by OSU student Geoff Bloom. Our thanks as well to Dr. Arthur Sleight, whose guidance was indispensable to the writing of these posts.]

In honor of his discovery of quasicrystals, Dan Shechtman, currently a distinguished professor at the Israel Institute of Technology (Technion), won the most recent Nobel Prize in Chemistry on October 5th, 2011.1  Quasicrystals, atomic structures whose order defies traditional crystallography, shocked the condensed matter science community when Shechtman and his team published their discovery in 1984.2  The strange qualities quasicrystals exhibited led to extensive research, debate, and even disbelief.

Among the scientists engaged in this debate was Linus Pauling, who studied quasicrystals intermittently between 1985 and 1993.  Pauling’s extensive background in crystallography – which began during his doctoral studies3, and was even part of the reason cited for his winning the Nobel Prize in 19544 – made him resistant to the discovery’s implications: that crystallography’s core principles about what was possible in solid structures were insufficient.  Pauling objected to the hypothesis that quasicrystals were an exception to the tenets of crystallography, and instead proposed a “multiple-twinning” theory that used, rather than contradicted, traditional crystallography’s assumptions to explain the strange qualities of quasicrystals.

Crystalline and Amorphous Solids

Before the discovery of quasicrystals, only two kinds of solid materials were thought to exist: crystalline and amorphous solids.  Crystalline solids are ordered structures that can be built from repeated substructures, called unit cells.5 These unit cells resemble bricks in a brick wall.  Let’s assume that a wall can be built from a whole number of bricks in each direction and the mortar between the bricks is empty space.  Each brick has the same shape, size, and arrangement of materials, and is oriented exactly the same way in space.  The entire brick wall, then, can be built from these bricks.  Similarly, an entire crystalline structure can be built out of a repeated unit cell.

A crystal can exhibit many kinds of symmetry, but it need only have translational symmetry in order to be defined as a crystal.6  Translational symmetry is the repetition of a single substructure by sliding it along an axis, such that the order of the atoms still matches exactly, and the entire structure along that axis can be recreated using only that repeated structure (this concept will be discussed again a bit later).  One familiar example of a crystalline structure is quartz.

Amorphous solids, on the other hand, have no symmetry and cannot be described using unit cells.  They are disordered.3  A good example of an amorphous solid is glass.

Notice that some compounds can have different structures altogether and still have the same chemical formula. Here, SiO2 has both an ordered crystalline structure (Quartz, left) and a disordered amorphous structure (glass, right).

Quasicrystals, too, are formed from substructures that can be compared to bricks, but such bricks are not of uniform size and shape, and are therefore not unit cells.  They do, however, fit together in such a way that the structure is more ordered than in amorphous solids, and even produce diffraction patterns that look similar to crystalline solids.  Yet, quasicrystals do not exhibit the translational symmetry necessary in crystalline structures.  Because of this, quasicrystals can exhibit other forms of symmetry previously thought impossible, such as 5-fold rotational symmetry.7

In 5-fold rotational symmetry, a pattern can be rotated around a point in a plane one-fifth of a complete turn, such that it looks identical to the original pattern.8  Certain kinds of rotational symmetry are typically associated with a lack of translational symmetry because the shapes whose numbers of sides coincide with the number of times a pattern can be rotated (like pentagons, in the case of 5-fold symmetry) cannot be used to completely fill a space.9  This concept, however, will be revisited later.

The fact that quasicrystals have some kinds of order that resemble crystalline structures, but also have “forbidden” rotational symmetry and lack translational symmetry is what makes them so controversial.

Using Diffraction

Condensed matter scientists determine the structures of atomic solids using methods generally called “diffraction”.  Samples are either isolated to a single crystal, or crushed into a fine powder so that the faces of it are randomly oriented.10  Then, a narrow beam of electrons, x-rays, or neutrons is shot at the material.6  The beam interacts with the sample’s atoms, such that some of it passes through, and other parts of it are scattered.  These scatterings produce a diffraction pattern, which provides information about the spacing between atoms.10  The places where these scatterings intersect most intensely form pronounced spikes on the detector readout called “Bragg peaks”,10 named after William L. and William H. Bragg, a father/son team who pioneered the use of x-ray diffraction in the early 20th century.11  The pattern and its corresponding Bragg peaks are used to model the structure of the sample material.

Diffraction using single crystals provides perhaps the clearest information about symmetry, because it produces distinct spots.  However, single-crystal diffraction only works when the material, in fact, exists as singular crystals.12
In some cases, crystals exist in a “twinned” state.  Here, two (or more, in the case of “multiple twinning”) structures have identical structural arrangements, but are oriented in different directions in space (more technically, they are said to have different domains).  These structures are then embedded in one another to effectively become a new crystalline structure.  Because of this, twinning tends to add “false” symmetry to diffraction patterns.13

This electron micrograph shows multiple-twinning of five crystals around an icosahedral seed. Notice that the crystals appear to have identical structures, but are oriented in different directions and embedded within one another. This particular alloy is Al5Mn.

Pauling’s View

Pauling’s reaction to the news of the discovery of quasicrystals was that they were not a new form of material existing somewhere between crystalline and amorphous solids, but that they were really twinned crystals, made from the same unit cells seen in crystalline materials.  The substructures from which the unit cells were made, he claimed, were oriented differently in space and embedded in one another as twins.  The five-fold symmetry that appeared, Pauling argued, was the result of multiple twins – the impact of the many domains of the twinned substructures – and not found in the single substructures themselves.  This way, the “forbidden” five-fold symmetry would not actually be “true” symmetry, because it would not all happen in one domain.  Instead, it would be a “false” symmetry made by the multiple twins.

Because Pauling believed quasicrystals were formed from multiply-twinned crystals, and not single crystals, he decided to rely on finely-crushed powder samples and x-ray beams.  That way, the patterns that resulted would not be influenced by twinning, since the domains of the twins would be randomly oriented.  However, the trade-off was that the powder patterns were harder to work with, since they produced rings, rather than spots, making the symmetry unclear.14 This meant that, to form a model for his twinning theory, Pauling had to rely on ring measurements that were often difficult to determine precisely.

Two kinds of electron diffraction patterns. On the left is a single-crystal electron diffraction pattern of an Al-Mn composite. On the right is a powdered-sample electron diffraction pattern of an Al-Mn-Ce alloy, Al93Mn6Ce3. Note the distinct spots in the single-crystal pattern, and the difficult-to-measure rings in the powder pattern.

Despite this limitation, the model that Pauling developed was a work of genius – intricate, complex, and impressive in its spatial reasoning.  It initially seemed like a promising hypothesis.  However, it was unsupported by evidence, and gradually abandoned by the condensed matter community.

Understanding Pauling’s complex alternative hypothesis requires some basic tools from crystallography.

One of the primary ways to discuss atomic structures is through symmetry, particularly the previously-mentioned reflective, translational, and rotational symmetry functions.  If part of a sample’s structure can be reflected over an axis – without modifying the arrangement of atoms – and its structure is identical, it is said to have reflective symmetry.  Samples that have translational symmetry, as previously mentioned, can be modeled by sliding a fixed atomic arrangement along an axis at specific intervals, such that at those intervals, the atomic arrangement does not change.  A sample has translational periodicity for each dimensional axis along which this is possible.

If part of a sample’s structure can be rotated – without changing atomic order or interatomic distances – within a plane to get a result identical to the original structure, the sample has rotational symmetry.  Oftentimes, structures with rotational symmetry are said to have “n-fold” symmetry, where n is the number of times per 360 degrees a structure can be rotated in a plane to get an identical atomic arrangement.  For example, if a structure can be rotated 90 degrees to achieve an identical-looking structure, it is said to have 4-fold rotational symmetry, since it can be rotated four times before simply assuming its original orientation.  Traditional crystallography holds that rotational symmetry can be found only in 2-, 3-, 4-, and 6-fold varieties.8

The reason for this lies in the fact that, within a given plane, a structure generally resembles a polygon with the number of sides equal to or an integer multiple of its n-fold symmetry.  Many polygons with numbers of sides other than the accepted n-fold symmetries (like pentagons, which have five sides) cannot be translated to tile an entire surface.  Since the primary criterion for a material qualifying as a crystal is translational symmetry, materials formed from shapes that cannot be translated to fill the entire structure are not crystalline.  Therefore, rotational symmetry and translational symmetry correlate, and the fact that quasicrystals had abnormal n-fold rotational symmetry implied that they also lacked translational periodicity.9

It is important to note that, while it is sometimes possible to find reflective, translational, or rotational symmetry for a small part of a sample’s structure, it must be possible to model the entire structure of the sample in a given plane using the symmetry function in order for the sample to have that form of symmetry in that plane.

If the above image, by MC Escher, is taken to exist in non-hyperbolic space, then there is no part of it that can be translated to fill the entire space. The image does, however, have reflective symmetry across the vertical axis, and three-fold rotational symmetry.

For example, notice that the pattern below has reflective symmetry across the perfectly vertical axis.  Its entire structure can be modeled by reflecting the left half of the structure over the vertical axis and onto the right half, without changing the arrangement of shapes.  It also has rotational symmetry, since the entire structure can be rotated 120 degrees to get an identical-looking arrangement.

However, it does not have translational symmetry.  Notice that there are small substructures that are repeated, and that could be isolated and slid along a line to model part of the structure.  But, repeating these small substructures would not describe the entire structure; there are still shapes in-between the substructures that are left out when the smaller piece is translated.  Since there is no smaller substructure in the pattern above that can be translated in order to model the entire pattern, this structure lacks translational periodicity along the two dimensional axes of this plane.

In three-dimensional space, unit cells are used to model an atomic structure in its entirety.  A unit cell is the most concise representation of a crystalline structure; it is the minimum amount of information required to build the entire structure in all three dimensions.15  Recall the previous discussion about modeling spaces with bricks and their similarity to unit cells.  A closely related concept, called a “Bravais lattice”, helps describe the structure of a unit cell,15 and can be repeated in two- or three-dimensional space to model a crystal’s structure; there are five unique Bravais lattices in two-dimensions, and fourteen in three-dimensions.16  Crystalline solids can be matched to Bravais lattices, but amorphous and quasicrystalline solids cannot, because they cannot be modeled with unit cells.

Penrose Tiles

The most convenient approximation used in modeling quasicrystals involves a mathematical relationship developed by British theoretical physicist Sir Roger Penrose, called Penrose tiles.  Penrose tiles are small non-identical shapes that can be arranged so as to fill a space completely, resembling the interaction between substructures in quasicrystals.  The simplest Penrose tilings can be made with two tiles – typically, a “fat” and a “thin” rhombus.  Most arrangements of Penrose tiles are either periodic – with definite translational symmetry, resembling a crystalline structure – or random – arranged in a disordered way, resembling amorphous structures.   The randomness of Penrose tilings varies, and the number of potential arrangements is infinite.17

An important aspect of Penrose tiles is that they can have additional rules as to how they can be arranged, called “matching rules”.  One simple form of matching rules is adding directional arrows to each tile edge and requiring that tiles can only be placed next to one another if the edges they share have the same directional arrows.  This has the effect of limiting possible arrangements, since, as the crystal gets larger, the shapes of spaces and the directional arrows limit considerably which tiles can go next to one another.  The result often leads to a pattern with substructures that repeat themselves at times throughout the structure, but cannot be translated to describe the entire space.  They also frequently have unusual n-fold rotational symmetry.  In short, these special tilings make patterns that resemble the translational aperiodicity and somewhat-developed order of quasicrystals.17

A Penrose tiling made from “fat” and “thin” rhombuses. Note the reflective symmetry across the vertical axis, and 5-fold rotational symmetry, but that there is no subsection of this pattern that can be translated to reconstruct the entire image faithfully.

Penrose tilings are not used exclusively for quasicrystal modeling, but, because they tend to closely match electron diffraction data,17 are fairly easy to make, and are less abstract than many other visual depictions of quasicrystals, they are the most common depictions of quasicrystals.

Icosahedral Symmetry

Some quasicrystals exhibit a quality called icosahedral symmetry, meaning their atoms appear to be arranged roughly in three-dimensional shapes with twelve corners and twenty equilateral-triangular faces.  Icosahedral symmetry correlates with 5-fold rotational symmetry in part because the lines that connect opposite corners of icosahedra operate as axes about which 5-fold rotational symmetry occurs; each of these connecting lines ends in the very center of a pentagonal shape, formed from five equilateral-triangle faces, and rotations 1/5 of a full circle about that center point will produce an identical-looking structure.9

A regular icosahedron with twelve vertices and twenty faces. Note that, when opposite corners of the icosahedron are connected with a line (here, the green line), the edges around one of those end points form a pentagon (outlined in red). This region implies the 5-fold rotational symmetry associated with icosahedra.

Icosahedra cannot be arranged with their faces touching in such a way that a given three-dimensional space can be completely filled by them; other shapes must be inserted between the icosahedra to fill those gaps.9  Because an icosahedron cannot, alone, be translated along axes in three-dimensional space to arrive at a complete model of a crystal, an icosahedron cannot be a unit cell.

However, it makes sense that solids would tend to have icosahedral structures.  Traditionally, atoms are modeled as spheres, since they exist in roughly spherical spaces, with a nucleus of protons and neutrons at the center, orbited by electrons in “clouds”.  To maximize space, a structure would have to pack as many of these roughly-spherical atoms together as possible.

Sphere Packing

Here’s where the mathematical concept of sphere-packing comes in handy.  After a great deal of consideration, mathematicians agreed that the highest possible number of non-deformed spheres that can be arranged so that they are touching one another in a space is twelve.18  Therefore, if each of these spheres represents an atom, and the goal is to maximize the amount of matter in a given space, the best possible structure will have twelve atoms closely arranged around one another.  If these atoms are of the same element (and therefore the same size), and are arranged around another element of smaller size (which would exist at the center of the structure), they would form a structure so that each of the larger atoms is at one of twelve corners, and the spaces between these corners (equidistant) become the “edges” of equilateral triangle faces – in short, they would form an icosahedron.9

A regular icosahedron made from spheres representing atoms – twelve at the vertices, and one smaller atom at the center. Notice how equilateral triangular “faces” are formed between three of the vertices. (Animation courtesy of Geoff Bloom.)

This is an idealized model, of course, but such a structure would minimize “empty” space between atoms.   Certain alloys, particularly those involving transition elements (like manganese, one of the elements in the first observed instances of quasicrystals), are best described by an icosahedral symmetry model.9  However, since, reiterating the above, icosahedra cannot be translated to fill a space completely, they are not suitable as unit cells.  As a result, solids exhibiting icosahedral symmetry – including many kinds of quasicrystals – generally lack translational periodicity.

Still a Puzzle

Because of the aperiodicity associated with icosahedral symmetry and the presumed impossibility of 5-fold rotational symmetry, Shechtman was shocked when he had found an alloy (MnAl6, containing Aluminum and Manganese) that exhibited these traits – so much so that, in a 1985 letter to Pauling, he admitted his initial disbelief and detailed at least four other kinds of experiments to which he subjected his findings, in addition to asking other researchers to review and duplicate his results.

The reviewers agreed: the alloy Shechtman studied had icosahedral and five-fold rotational symmetry, exhibiting some kinds of order similar to crystals, but not translational symmetry.19  Shechtman’s study, “Metallic Phase with Long-Range Orientational Order and No Translational Symmetry”, was successfully published in 1984 in the major peer-reviewed journal, Physical Review Letters,2 and the name “quasicrystals” developed soon after.

The true nature of quasicrystals is still not completely understood.  Whether quasicrystals are a new subset of crystalline structures, requiring a redefinition of what qualifies as a crystal (changing a definition that has existed for a considerably long time), or are some kind of exception existing as an intermediate state between crystalline and amorphous solids remains a controversial matter even today.  However, after a great deal of debate, and impassioned dedication on Pauling’s part, the multiple-twinning hypothesis was effectively ruled out as an explanation for the quasicrystal phenomenon – the developments of which will be discussed next week.


References

1“The Nobel Prize in Chemistry 2011: Dan Shechtman.” Nobelprize.org. 28 Nov 2011. http://www.nobelprize.org/nobel_prizes/chemistry/laureates/2011/shechtman.html
2Shechtman, D., I. Blech, D. Gratias, and J.W. Cahn. “Metallic Phase with Long-Range Orientational Order and No Translational Symmetry.” 53.20: 1951-3 (1984).
3Paradowski, Robert. “Pauling Chronology: Linus Pauling as a Graduate and Postdoctoral Student at the California Institute of Technology.” The Ava and Linus Pauling Papers. Oregon State University Special Collections. 2006. http://scarc.library.oregonstate.edu/coll/pauling/chronology/page8.html
4“The Nobel Prize in Chemistry 1954: Linus Pauling.” Nobelprize.org. 28 Nov 2011. http://www.nobelprize.org/nobel_prizes/chemistry/laureates/1954/
5Genack, Azriel Z. “Solids.” Physics 204 lecture presentation. Department of Physics, Queens College, City University of New York. http://www.physics.qc.edu/pages/genack/Physics204/7%20Crystal%20Structure%20and%20Classification%20of%20Solids.ppt
6Janot, C. Quasicrystals: A Primer. 2nd eed. Oxford: Clarendon Press, 1994. 1.
7Weber, Steffen. “Quasicrystals.” JCrystalSoft. 2011. http://www.jcrystal.com/steffenweber/qc.html
8Rice University. “Quasicrystals: Somewhere Between Order and Disorder.” 29 May 2007. http://www.quasicrystals.org/
9Linus Pauling Institute of Science and Medicine Newsletter. “Icosahedral Symmetry.” Vol.2 , Issue 9, Fall 1986. p. 4-5.
10Li, Youli. “Introduction to X-ray Diffraction.” Materials Research Lab at UCSB. University of California Santa Barbara. N.d. http://www.mrl.ucsb.edu/mrl/centralfacilities/xray/xray-basics/index.html
11Schields, Paul J. “Bragg’s Law and Diffraction.” Center for High Pressure Research. State University of New York at Stony Brook Department of Earth & Space Sciences. 29 Jan 2010. http://www.eserc.stonybrook.edu/projectjava/bragg/
12Clark, Christine M. and Barbara L. Dutrow. “Single-crystal X-ray Diffraction.” Integration Research and Education: Geochemical Instrumentation and Analysis. Carleton College Science Education Resource Center. 10 Mar 2012. Web. .
13“Crystal Twinning.” University of Oklahoma Chemical Crystallography Lab, Department of Chemistry and Biochemistry. 11 Apr 2011. Web. .
14“Powder Diffraction Methods.” Purdue University Department of Chemistry. N.d. Web.
15The Bodner Research Group. “Unit Cells.” Purdue University Division of Chemistry Education. N.d. http://chemed.chem.purdue.edu/genchem/topicreview/bp/ch13/unitcell.php
16Van Zeghbroeck, Bart J. “Bravais Lattices.” Principles of Semiconductor Devices. 1997. Online text. University of Colorado Department of Electrical, Computer, and Energy Engineering. http://ecee.colorado.edu/~bart/book/bravais.htm
17Janot, C. Quasicrystals: A Primer. 2nd ed. Oxford: Clarendon Press, 1994. 30-35.
18The Bodner Research Group. “The Structure of Metals.” Purdue University Division of Chemistry Education. N.d. http://chemed.chem.purdue.edu/genchem/topicreview/bp/ch13/structure.php
19Shechtman, Dan. Letter to Linus Pauling. 15 May 1985.

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