Closing the Book on Quasicrystals

Linus Pauling at home, 1989.

[Part 4 of 4]

Linus Pauling was not the only scientist to offer an alternative theory for the nature of quasicrystals; one of the major competing theories, the “icosahedral glass” theory, was introduced and quickly abandoned by quasicrytals discoverers Dan Shechtman and Ilan Blech,¹ but further developed by physicists Peter W. Stephens and Alan I. Goldman^2,3. However, Shechtman was not the only scientist who held fast to and developed quasicrystal theory; a growing number of physicists and crystallographers began to support the idea that quasicrystals were legitimate exceptions that warranted redefining what qualified as a crystal.

In the November 1989 issue of Proceedings of the National Academy of Science, an article written by Drs. P.A. Bancel, P.A. Heiney, P.M. Horn, and P.H. Steinhardt, titled “Comment on a Paper by Linus Pauling” addressed Pauling’s continually-developing multiple-twinning hypothesis, responding in particular to his article, “So-called icosahedral and decagonal quasicrystals are twins of an 820-atom cubic crystal,” also published in PNAS.  Prior to the publication of Pauling’s “Icosahedral quasicrystals of intermetallic compounds are icosahedral twins of cubic crystals of three kinds,” the team sent their article to Pauling for his consideration. Pauling encouraged them not only to publish the article, but to publish it simultaneously with his own, so that they would appear in the same issue of PNAS.  Pauling himself communicated their finished manuscript to the journal.⁴

Just before submitting their “Comment on a Paper by Linus Pauling,” Bancel and his co-authors formed a sample of AlFeCu alloy that was considered “perfect” by refining the crystal creation process to produce extremely few anomalies and deformations.⁴

Shechtman’s MnAl6 diffraction photograph (left), versus Bancel, et. al’s “perfect” AlFeCu alloy (right). Note that while both exhibit five-fold rotational symmetry, Bancel, et. al’s image shows increased clarity and pattern regularity; thus its “perfection.”

They examined their new sample from the perspective of Pauling’s twinning hypothesis, and noted that Pauling would need to employ a unit cell containing nearly 100,000 atoms to describe an imperfect sample of AlFeCu alloy, and over 425,000 atoms to account for the team’s “perfect” samples. Such a structure, they argued, would be unfathomably complex, and an impractical model of the material’s structure.⁴

According to the team, the apparent success of Pauling’s hypothesis, in part, owed itself to the presence of structures called “phason strains.”⁴ Stresses applied to crystalline structures cause deformation. A variety of imaginary particles, called quasiparticles, exist for the sole purpose of explaining how physical reactions change the nature of certain subatomic particles.⁵ Instead of explaining, for example, how an electron’s behavior is modified by its interactions with electrons in surrounding atoms – a dauntingly complex task – one can simply substitute a particle that resembles an electron, but is more massive.⁵ Such an imaginary particle behaves quite similarly to the electron in its situation of interactions, but requires less complicated modeling, as it is essentially standing in for the behavior of a whole group of interacting particles.⁶

Two quasiparticles pertinent to crystallography are phonons and phasons. When external stress is applied to a crystalline structure, unit cells of that structure are distorted from their equilibrium shapes. This distortion is referred to as a “phonon strain.” When the stress is released, the return to equilibrium shape is modeled as the strain “relaxing” by transmitting phonon quasiparticles at the speed of sound. (Bear in mind that phonons do not really exist as particles, but are being employed for the sake of simplifying the model.) Effectively, the crystal structure returns to its original state immediately.⁴

An example of deformations resulting from external stress that distort the shapes of the structure’s parts. Note how the substructures in the diagram at the left have been reshaped considerably by the stress, producing a phonon strain (shown in the diagram to the right). Releasing the external stress should result in a nearly-immediate return to the original structure.

However, applied external stress can have another kind of effect on crystalline structure. Instead of distorting the overall unit cell shape, stress may rearrange unit cells without appreciable shape distortion. In “proper” crystals, described by uniform unit cells, such distortion would have no noticeable effect; the roughly identical parts would only be shuffled around, and the overall structure would look essentially identical. In contrast, the nonperiodic structure of quasicrystals – their lack of translational symmetry – means that rearranging parts of the overall pattern would change the structure noticeably. This rearrangement is referred to as a “phason strain.” Unlike phonon strains, they do not relax instantly once external stresses have been released. Instead, the process of returning to the ground state configuration may take hours, days, or even years. Thus, the shifted structure remains long after an explanation for the modification is visible.⁴

Here, external stress translates some of the substructures such that their arrangement has been markedly changed. Notice how the order of the Penrose tiles has changed (compared to the original arrangement on the right), disrupting matching rules. This changes how the substructures fit together, causing distortions in the boundaries between them. The result is a phason strain. When external stress is released, it will take a considerable amount of time before the phason strain dissipates.

In quasicrystals, phason strains break icosahedral symmetry and change the ratio of distances between the structural parts, such that it is no longer a fixed irrational number. This has the effect of shifting diffraction peaks from their expected locations and distorting the regularity of the x-ray patterns. When such distortions are visible, the two logical conclusions from these shifts are that the crystal either has frozen-in phason strains or is formed from a very large, twinned unit cell.⁴

The reason why Pauling’s twinning model appeared to match experimental diffraction data, Bancel and his team argued, was because the unit cell it arrived at for each compound was comprised of the atoms between phason strains, which appeared to act as the boundaries to large, distinct unit cells. Twinning theory, they pointed out, also has the virtue of responding more directly to diffraction peak shifts, since it fits a unit cell specifically to the deformations a sample exhibits.⁴

Yet, the difficulty with Pauling’s model was that the data simply did not provide true evidence of twins; certain artifacts that result from twinning were not present as expected in the data.⁷ The immense unit cells the theory required made for impractical models that could not be generalized, especially given that they had to become larger and more complex as the quasicrystal samples they were modeling neared perfection. In fact, assuming every new sample of the same alloy has different diffraction peak shifts – a reasonable assumption, given that Bancel and his team demonstrated that quasicrystals can be refined to eliminate peak shifts almost entirely⁴ – the multiple-twinning hypothesis would technically require that a new unit cell be devised for each new sample, specifically tailored to its unique circumstances.

Peter Bancel.

Bancel concluded that Pauling’s multiple-twinning hypothesis was inadequate. Instead, Bancel proposed, acknowledging that quasicrystals are a legitimate exception to traditional crystallography tenets, though requiring restructuring of the definition of a crystal, made it possible to model the phenomenon more simply and accurately, and was therefore a better explanation than multiple-twinning.⁴

Other scientists studying the problem contacted Pauling, intending to persuade him of quasicrystal theory’s value. University of Pennsylvania physicist Paul Steinhardt wrote to Pauling in March 1989, reiterating the importance of his team’s “perfect” crystals and that their implications would “place very severe constraints on any multiple-twinning model.” He implied that Pauling had objected to Bancel’s stance and hypothesized that the diffraction patterns Bancel got from his sample only matched theoretical values because of a phenomenon called “multiple-scattering.” However, Steinhardt noted that Bancel had, as a follow-up to correspondence between Steinhardt and Pauling on the subject, done some “sample rotation experiments” to confirm that the diffraction data did, in fact, support Bancel’s claims.⁸

Simon C. Moss.

Devoted to his ideas, Pauling continued work on his multiple-twinning hypothesis, refining and applying the model to a variety of alloys. In 1993 Pauling corresponded with Dr. Simon C. Moss of the University of Houston Department of Physics. Moss addressed the fact that electron microscopy, diffraction evidence, and the twinning theory’s “absurdly large approximant cells” had all effectively ruled out Pauling’s model. He did concede that it could be possible that quasicrystalline forms may not be in their ground states, and that they may form “multi-domained complex crystals” (that is, twinned structures) at lower temperatures, offering a small concession to twinning theory’s potential. About quasicrystal theory itself, Moss wrote, “We will certainly keep you informed on our progress and perhaps, in time, bring you to our point of view. It is, I should say, rather widely held and well-supported by the data.”⁹

However, there is no indication that Moss, or the growing number of chemists and physicists supportive of quasicrystal theory, succeeded in swaying Pauling. Pauling’s written response to Moss was to point out a variety of small “horizontal and vertical layer lines,” visible in overexposed photographs, which he felt were inadequately described by quasicrystal theory, and to reiterate his belief that accrediting shifted diffraction peaks to the influence of phason strains was “unsatisfactory.” Though he acknowledged his theory would require very large unit cells – 52Å, 58Å, and perhaps even 66Å in width – he also pointed out that he had thought 70 years prior that a 30Å structure with 1000 atoms was overwhelmingly large, a structure later accepted as accurate and reasonable by the scientific community.¹⁰

Pauling seems to have defended the multiple-twinning theory until his death in 1994, despite the growing evidence and support for the theory that quasicrystals were, in fact, anomalies that required the field to rethink what forms of ordered solids were truly possible.

Today, in addition to being the motivation for the 2011 Nobel Prize for Chemistry, quasicrystals are finding potential use as insulation in engines,¹¹ materials for converting heat into electrical energy,¹¹ wear-resistant ball bearing coatings,¹² non-stick frying pan liners,¹² LED components,¹³ and parts in surgical instruments.¹³

Pauling’s dedication to his ideas, his profoundly complex solution to the perplexing nature of quasicrystals – one that attempted to reconcile the long-standing assumptions of the field of crystallography with an apparent exception – and his willingness to question the findings of colleagues, paying special attention to inconsistencies in their theories, highlight the intellectual drive and dynamic spirit that made Pauling a brilliant scientist. Further, it is perhaps Pauling’s genius that led him to so stubbornly pursue and defend his intricate multiple-twinning hypothesis, even after it seemed disproven. Despite the general agreement today that Pauling’s multiple-twinning theory was not an accurate explanation for quasicrystalline structure, his enthusiastic engagement in the quasicrystal debate demonstrates that in scientific discovery, even “wrong” ideas, when thoroughly investigated, are crucial to understanding the strange nature of the universe.

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1. Shechtman, D. and I Blech. “The Microstructure of Rapidly Solidified Al₆Mn.” Mettalurgical Transactions. 16A (1985): 1005-12.

2. Stephens, P.W. and A.I. Goldman. “Sharp Diffraction Maxima from an Icosahedral Glass.” Physical Review Letters 56 (1986): 1168-71.

3. Letter to Linus Pauling from Paul Steinhardt, David Rittenhouse Laboratory, University of Pennsylvania Department of Physics. Mar. 15, 1989.

4. Bancel, Peter A., Paul A. Heiney, Paul M. Horn, and Paul J. Steinhardt. “Comment on a Paper by Linus Pauling.” Proceedings of the National Academy of Sciences in the United States of America: 86.22 (1989): 8600-1.

5. Ford, Chris. “Physics of Nanoelectronic Systems: Lecture Notes, Chapter 7.” Semiconductor Physics Group, Cavendish Laboratory, University of Cambridge Department of Physics. Jan. 2011. http://www.sp.phy.cam.ac.uk/teaching/np/chapter7.pdf

6. Mattuck, Richard D. “The Many-Body Problem for Everybody.” A Guide to Feynman Diagrams in the Many-Body Problem. Second Edition. 1976. McGraw-Hill.

7. Steinhardt, Paul J and Stellan Ostlund. The Physics of Quasicrystals. Singapore: World Scientific Publishing, 1987. Online. 310-12.

8. Steinhardt, Paul. Letter to Linus Pauling. 14 March 1989.

9. Moss, Simon C. Letter to Linus Pauling. 3 February 1993.

10. Pauling, Linus. Letter to Simon C. Moss. 26 March 1993.

11. Lannin, Patrick and Veronica Ek. “Ridiculed crystal work wins Nobel for Israeli.” Reuters. October 6, 2011. http://ca.reuters.com/article/topNews/idCATRE7941EP20111006?pageNumber=2&virtualBrandChannel=0

12. Widom Research Group. “Quasicrystals.” Carnegie Mellon University. N.d. http://euler.phys.cmu.edu/widom/research/qc/quasi.html

13. Marder, Jenny. “What are quasicrystals, and what makes them Nobel-worthy?” PBS Newshour Rundown News Blog. October 5, 2011. http://www.pbs.org/newshour/rundown/2011/10/quasicrystals-win-chemistry-nobel.html

Shechtman and Pauling Debate Quasicrystal Theory

Linus Pauling, 1986.

[Part 3 of 4]

David and Clara Shoemaker were not the only scientists who felt that Linus Pauling’s quasicrystals hypothesis, while admirable, was unsubstantiated by experimental data.

In fact, in “Metallic Phase with Long-Range Orientational Order and No Translational Symmetry,” the article that introduced quasicrystals to the scientific community, Dan Shechtman and co-author Ilan Blech noted that twins were initially suspected as being the reason for the unusual structure, but that after subjecting the crystals to a broad range of experiments and even using data from the x-ray diffraction patterns themselves, they determined that their sample was not composed of twins.

Their argument centered on the fact that twins should have been visible when they employed a method called “dark-field microscopy,” which illuminates whole grains, and that the twins should have changed, in some fashion, the patterns resulting from their electron diffraction experiments at various resolutions. Furthermore, they argued, twinning would not interfere with matching a Bravais lattice to a crystal in an x-ray diffraction pattern. Therefore, the fact that scientists were having difficulty assigning lattices – and, by extension, unit cells – to quasicrystals would not, in itself, be an indication that their structure was based on twins. Because of the overwhelming evidence against twins, the research team concluded that the sample “[did] not consist of multiply twinned regular crystal structures.”¹

Pauling, however, cautioned Shechtman and Blech against discounting the influence of twins. In a letter dated April 24, 1985, he mentioned that certain previously analyzed crystalline structures shared some of the characteristics of the newly-discovered quasicrystals, and that there had been structures in the past that initially surprised researchers, but were found to accord with existing crystallography paradigms. To develop his theory, Pauling requested copies of Shechtman and Blech’s MnAl₆ x-ray diffraction data.²

Shechtman, who was working at Technion, the Israel Institute of Technology in Haifa, responded promptly. In a letter received by the Linus Pauling Institute on May 15th, he emphasized that a number of different experiments had yielded no evidence of twins, and that other research teams had backed his findings. Nonetheless, Shechtman included prints of the x-ray diffraction patterns and copies of data on the sample’s Bragg peaks.³

Pauling’s handwritten notes on a copy of MnAl6 diffraction patterns in Shechtman and Blech, et. al. “Observation par microscopie électronique de la quasi-periodicite dans un solide a symetrie icosaedrique” (roughly, “An Observation by Electron Microscopy of the Quasiperiodicity in a Solid Having Icosahedral Symmetry”) from the French journal, Comptes Rendus de l’Académie des Sciences, 1985. The notes are concerned with calculating interplanar distances between the diffraction peaks shown, as well as extrapolating this to a 23.36Å unit cell.

In a response dated June 6, 1985, Pauling thanked Shechtman for the diffraction images, and requested permission to use them in an article introducing his multiple-twinning theory. He explained, very briefly, his hypothetical twinned, twenty-icosahedral structure as the basis for the icosahedral symmetry seen in quasicrystals, and mentioned certain pieces of evidence that he felt Shechtman and his team had overlooked. In particular, Pauling noticed three to five weak lines in the x-ray diffraction patterns that, to Pauling, supported a twinned structure.⁴

Then Pauling made a strange offer: he asked Shechtman if he would like to co-author an article on twinning with him. The difficulty, Pauling pointed out, would be that Shechtman would have to contradict himself and, to some extent, negate his own findings. In essence, Pauling offered Shechtman an opportunity to admit that he was wrong.⁴

It is unclear what, exactly, Shechtman’s response was; no letter from him about the offer is preserved in the Pauling Papers, and the tone of Pauling’s next letter implies that the two did not discuss it.

In mid-July 1985, Pauling again wrote to Shechtman, revealing a proposed structure composed of “icosatwins” and 1000-atom unit cells. He acknowledged that the unit cell size was unusually large, but stood by his hypothesis, noting that he himself had discovered a few complex crystal structures during the 1920s, when 1200-atom cells were inconceivable. He also mentioned that, since he had not received any correspondence from Shechtman indicating interest in co-authorship, he had sent the article he offered Shechtman off to the journal Nature for review.⁵ (This article would be accepted and published later that year, and was the first article in which Pauling discussed his twinning hypothesis.⁶)

Shortly after this exchange, correspondence indicates that Shechtman and Pauling met in person at the Linus Pauling Institute of Science and Medicine in Palo Alto. Mentioning that he would be in America on business, and available during the week of August 19, 1985, Shechtman wrote that he hoped they could meet and discuss quasicrystals in detail.⁷ Pauling obliged and extended an invitation. Though the details of the meeting are unclear, it does not seem to have gone especially well.

Dan Shechtman, 1983. (Credit: Technion – Israel Institute of Technology)

In a letter dated September 3, 1985, Shechtman wrote to Dr. Sten Samson of the California Institute of Technology (and a former graduate student under Pauling), carbon-copying Pauling, and enclosing a sample of a mostly – but not entirely – icosahedral, rapidly cooled, powdered ribbon of material. He wrote that he was fulfilling a request made by Pauling, and that Samson had, apparently, been forewarned that he would be receiving the sample and was aware of the analyses that Pauling wanted performed. The tone of Shechtman’s letter seems somewhat begrudging and reticent.⁸

The results of Samson’s analysis are not preserved among the Pauling Papers, but correspondence between Pauling and Shechtman appears to have dissipated for nearly a year. In mid-April 1986, Shechtman sent a brief letter requesting that Pauling keep him apprised of his activity with regard to quasicrystals.⁹ No reply on Pauling’s part is preserved amongst his papers.

It seems it was not until August 12, 1986 that Pauling again wrote to Shechtman. Addressing both Shechtman and his original quasicrystal article co-author Ilan Blech, Pauling claimed that he had found an error of 14.4% in their scale for the electron diffraction patterns of MnAl₆ – the sample used in their first article – and that, adjusted against this error, the data strongly supported Pauling’s twinning hypothesis. He went on to call the error “easily avoidable” and (in a somewhat patronizing tone) detailed a fairly basic process by which the team could have – but, Pauling presumed, did not – verify their scale’s accuracy. Pauling’s tone was curt. “This error caused me several months of unnecessary effort,” he wrote. He went on to say that, after correcting this alleged error, “there no longer remains any doubt about the nature of the ‘icosahedral quasicrystals.’ They are twins of a cubic crystal with edge 26.73Ǻ.” Pauling closed his letter by noting: “Verification of my statement that your scale is in error by 14.4% would, of course, provide additional evidence for the foregoing conclusion.”¹⁰

Though Shechtman’s response is not among Pauling’s correspondence, it seems to have humbled Pauling. A letter dated September 8, 1986 – written less than a month after Pauling’s terse note – reveals Pauling to be in a more deferential mood. He thanked Shechtman for his letter and photographs, and acknowledged that they clearly showed that Shechtman and Blech’s original quasicrystals scale was correct. The error, Pauling wrote, occurred from misinterpreting what he called a “statement about scale made to me by another investigator.” Despite acknowledging his error, Pauling did not directly apologize, but instead went on to state concerns about a different set of photographs showing decagonal relationships, estimating calculations based on them to be off by 8%.¹¹ The gap in correspondence in the Pauling Papers implies that Pauling and Shechtman did not correspond again for an entire year.

However sparse the exchange between Shechtman and Pauling during that time, it did not mirror a reduction in Pauling’s work on his twinning hypothesis. The news that Shechtman and Blech’s scale was indeed accurate caused a small crisis for Pauling, who was certain that the “correction” he proposed would justify his structure. For most of the month of October 1986, Pauling contemplated the multiple-twinning structure at either his ranch in Big Sur or on airplanes and in hotel rooms between peace talks and chemistry lectures. Dozens of pages of calculations, diagrams, and hypotheses on legal pads reveal a constant refinement of the twinning theory. Pauling’s meticulousness in noting the date and time reveals that the “quasicrystal problem,” as he called it, occupied his mind even into the early morning hours.¹²

On October 16, 1986, Pauling arrived at a 920-atom unit cell composed of eight 117-atom clusters arranged snugly at 90-degree angles of rotation, repeating along all three axes. Triumphantly, Pauling wrote “Hurray! The Quasi Problem is Solved.¹³”

After weeks of calculations and creative spatial configurations, Pauling believed he finally explained quasicrystals in terms of Bravais lattices, announcing, “Hurray! The Quasi Problem is Solved.”. However, further thought led him to reconsider, and he later added the phrase, “But see page 11!!!” on which he recalculated and abandoned his findings.

Yet, a small note in the margin underneath redirects to work done two days later, in which Pauling recalculated his cell size based on Shechtman and Blech’s electron diffraction photos. The new cell was still composed of eight 117-atom clusters, but as many as 72 of those atoms were shared, making the unit cell 840 atoms instead of 920. Pauling also concluded that the cell would be essentially body-centered cubic.¹⁴ Eventually that cell also succumbed to scrutiny, and on October 24, Pauling considered an 804-atom unit cell.¹⁵

Finally, Pauling concluded that the unit cell likely contained 820 atoms, formed from 104-atom clusters sharing outer electron shells. It was this structure that formed the core of Pauling’s article, “Evidence from x-ray and neutron powder diffraction patterns that the so-called icosahedral and decagonal quasicrystals of MnAl₆ and other alloys are twinned cubic crystals,” published in June 1987 in the Proceedings of the National Academy of Science (PNAS).

Pauling’s clusters were arranged such that each was at the corner of a unit cube and surrounded by twelve more clusters in the shape of a nearly regular icosahedron. He noted, “All of the clusters have the same orientation, and any one cluster could serve as a seed for twinning.” To defend this structure, Pauling pointed to its considerable correlation with diffraction patterns, and argued that any mismatching that occurred around cube edges was due to slight variations in alloy compositions. He also noted that another alloy, Mg₃₂(Al₉Zn)₄₉, is known to have a tightly-packed cluster-based structure, making such a structure in MnAl₆ not unprecedented. In fact, Pauling argued, the intense heating and the rapid cooling process used to form MnAl₆ crystals (which are typically referred to as “rapidly quenched”) likely led to closely packed alloy clusters.¹⁶

Papers arranged on Pauling’s desk, 1987.

By the time of the article’s publication, over a one-hundred other alloys with icosahedral quasicrystalline structures had been found.¹⁷ Pauling, like many scientists, began to expand his theory to account for even more alloy structures, beyond the first anomalous discovery, MnAl₆.

Breaking the apparent year of silence, Pauling wrote once more to Shechtman on October 6, 1987. By now his tone had shifted from cordially tense scientific competition to that of camaraderie. In addition to thanking Shechtman for the many glossy x-ray photographs and diffraction data calculations he had provided him over the years, Pauling also thanked Shechtman for his very discovery of quasicrystals. He wrote,

This discovery has resulted in a great contribution to crystallography and metallurgy, in that it has stimulated hundreds of investigators to study alloys and has led to much additional knowledge about intermetallic compounds….Your discovery has also made me happier…. For over two years I have worked on this problem, and have enjoyed myself while doing it. I estimate that I have spent nearly 1,000 hours just thinking about this whole question, and more than 1,000 hours making calculations, and writing papers.

The fruit of this labor, for Pauling, was the discovery of “five new complicated structures,” the details of which Pauling shared with Shechtman. He even – in what looks like a conciliatory acknowledgement of Shechtman’s expertise – asked Shechtman to check on the diffraction patterns of a slowly cooling structure to see if the intensity spots shifted in position or intensity.¹⁸

Furthermore, Pauling reiterated his previous desire to write a paper with Shechtman about two- and six-fold symmetry, saying, “I should be very pleased if a paper could be published with the authors Shechtman and Pauling.” He even acknowledged, to some extent, the tension that had existed between them, writing, “I hope that we can cooperate in the attack on this problem. I have the impression from referees’ reports on papers that I have submitted to Physical Review Letters that at least one of the referees considers me to be an antagonist.” Though Pauling did not agree with this assessment, or apologize for any of his behavior, he did write, “It is something like the situation between the United States and the Soviet Union. It would be much better if they were to cooperate in attacking world problems, rather than to function as antagonists.”¹⁸

Responding quickly to this letter, Shechtman cabled Pauling, thanking him and saying that his communication “made me very happy.”¹⁹ In a follow-up letter sent November 10th, Shechtman expressed interest in collaborating with Pauling on a joint quasiperiodic structures article, and offered to host Pauling at Technion in Haifa, covering all of his expenses.²⁰ In response, Pauling wrote that he was cutting down on his amount of travel, and so would not likely travel to Israel. Rather, Pauling wrote, the nature of their collaboration could be such that Pauling would send Shechtman manuscripts for his consideration and input, implying that, while Shechtman was welcome to visit the Institute in Palo Alto, their collaboration would be a long-distance one.²¹

That letter, dated December 8, 1987, is the last archived bit of correspondence between Pauling and Shechtman. The two never co-authored an article.

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1  Shechtman, D., I. Blech, D. Gratias, and J.W. Cahn. “Metallic Phase with Long-Range Orientational Order and No Translational Symmetry.” Physical Review Letters 53.20: 1951-3 (1984).

2  Pauling, Linus. Letter to Dan Shechtman and Ilan Blech. 24 April 1985. Ava Helen and Linus Pauling Papers, Sci 4.005.2.

3  Shechtman, Dan. Letter to Linus Pauling. 15 May 1985.

4  Pauling, Linus. Letter to Dan Shechtman. 6 June 1985.

5  Pauling, Linus. Letter to Dan Shechtman. 10 July 1985.

6  Pauling, Linus. “Apparent icosahedral symmetry is due to directed multiple twinning of cubic crystals.” Nature 317 (October 1985): 512-14.

7  Shechtman, Dan. Letter to Linus Pauling. 3 July 1985.

8  Shechtman, Dan. Letter to Sten Samson, cc Linus Pauling. 3 September 1985.

9  Shechtman, Dan. Letter to Linus Pauling. 20 April 1986.

10  Pauling, Linus. Letter to Dan Shechtman and Ilan Blech. 12 August 1986.

11  Pauling, Linus. Letter to Dan Shechtman. 8 September 1986.

12  Pauling, Linus. Hand-numbered series of LP Quasicrystal Notes and Calculations, October 7-10, 1986; October 12-27, 1986.

13  Pauling, Linus. Hand-numbered series of LP Quasicrystal Notes and Calculations, October 16, 1986.

14  Pauling, Linus. Hand-numbered series of LP Quasicrystal Notes and Calculations, October 18, 1986.

15  Pauling, Linus. Hand-numbered series of LP Quasicrystal Notes and Calculations. October 24, 1986.

16  Pauling, Linus. “Evidence from x-ray and neutron powder diffraction patterns that the so-called icosahedral and decagonal quasicrystals of MnAl₆ and other alloys are twinned cubic crystals.” Proceedings of the National Academy of Sciences 84:12 (1987) 3951-3.

17  The Scientist, “Quasicrystal Research: Where The Action Was In 1988”. May 29, 1987.

18  Pauling, Linus. Letter to Dan Shechtman. 6 October 1987.

19  Shechtman, Dan. Telegram to Linus Pauling. 26 October 1987.

20  Shechtman, Dan. Letter to Linus Pauling. 10 November 1987.

21  Pauling, Linus. Letter to Dan Shechtman. 8 December 1987.

The Pauling Theory of Quasicrystals

Clara Shoemaker, Linus Pauling and David Shoemaker, Oregon State University, 1983.

[Part 2 of 4]

The introduction of a new discovery, quasicrystals, challenged the underlying assumptions of crystallography itself. Some researchers theorized that quasicrystals were a new material existing as an intermediate state between amorphous and crystalline solids, and others proposed that quasicrystals were a new subset of crystalline structures; these hypotheses are generally referred to as “quasicrystal theory.” A number of scientists resisted the theoretical changes quasicrystals posed, preferring instead to explain the phenomenon with the existing rules of crystallography. Among them was Linus Pauling, who proposed a remarkably complex alternative to quasicrystal theory known as the “multiple twinning” hypothesis.

Prior to the discovery of quasicrystals, crystallography held that some structures exhibited a phenomenon called “twinning.” In twinning, crystals with the same structure exist in different domains – that is, they are oriented so they are essentially facing in different directions – but are embedded within each other, effectively making a new structure altogether.¹

One way to visualize twinning is to imagine crystals as being formed from “clusters” of small sets of atoms. However, some of the clusters share their “end atoms,” such that two clusters stem from a shared set. These clusters are thus “twinned.”²

Pauling felt certain that quasicrystalline structure could be explained by multiple twinning between atomic clusters in the crystal. Analyzing Dan Shechtman’s article, he asserted that a large, roughly cubic unit cell with twinning clusters was responsible for the apparent icosahedral symmetry.³

To get help in developing the multiple twinning hypothesis and testing some initial predictions, Pauling approached Oregon State University crystallographer David Shoemaker, and his wife, Clara, also a crystallographer in her own right. David had previously worked with Pauling on x-ray diffraction while studying under him as a graduate student. In a speech given in 1995 at the Oregon State University symposium, “Life and Work of Linus Pauling: A Discourse on the Art of Biography,” he recalled Pauling insisting that, contrary to Shechtman’s claim, the MnAl₆ structures he had found could be indexed to a Bravais lattice – albeit through a complex interchange of twins. Above all, Pauling was certain that the rules of crystallography did not need to be modified to accommodate quasicrystals.⁴

Pauling’s theoretical structure, which was, according to Pauling himself, devised over “a couple days of work” in early 1985, is complex, but forms an explanation for quasicrystalline structure that does not require modifying the definition of a crystal. Instead of directly analyzing a MnAl₆ alloy, Pauling focused on a MnAl₁₂ alloy with icosahedral symmetry and twinning. Using the icosahedral structure as a framework, he imagined each of the vertices of the shape (essentially, the centers of the atomic spheres packed to make the shape) as representing an Aluminum (Al) atom, and the point at the center, between the packed spheres, as representing a Manganese (Mn) atom. Each of the twelve Al atoms is therefore linked to a single central Mn atom.²

A regular icosahedron made from spheres representing atoms. The twelve vertices of the icosahedron (blue) are Aluminum atoms, and the interior atom (red) is Manganese. Notice how equilateral triangular “faces” are formed between three of the vertices. Also note that the “bonds” linking the atoms are only approximations to show the relationships between atoms, and that in the actual MnAl6 alloy, the atoms are linked in “metallic bonds,” which have different properties from “true” bonds. [Animation by Geoff Bloom]

Pauling also assumed that each icosahedral MnAl₁₂ structure is adjacent to exactly four other MnAl₁₂ icosahedra, and shares a face with each one. Such an arrangement would allow for each of the twelve Al vertex atoms in the original MnAl₁₂ icosahedron to be at the vertex of a shared triangular face. Effectively, this would make each Al atom linked with two Mn atoms – the Mn atom at the center of its original icosahedron, and the Mn atom at the center of the new icosahedron with which it shares a face.²It is this “link” that implies the “twinning” integral to Pauling’s theory.

Two icosahedra sharing a face. Note how there are fewer than 24 Aluminum atoms. This is because the atoms along the shared face are part of both icosahedra. Each of these shared Aluminum atoms is therefore “linked” to two Manganese atoms – the central atoms of each icosahedron. [Animation by Geoff Bloom]

Pauling noted that imaginary lines between the Mn atom within the original icosahedron and the Mn atoms at the centers of the four adjacent icosahedra would point toward the corners of a structure similar to a regular tetrahedron (a three-dimensional structure with four equilateral-triangle faces, resembling a pyramid). That is, one can imagine that the Mn atoms in the centers of the four outer icosahedra could be connected with lines to form a regular tetrahedron.² The interior angle of the tetrahedron (with two of the Mn atoms at the corners of the tetrahedron at each end, and the central Mn atom at the “middle” of the angle) would be 109.5 degrees² – the ideal tetrahedral bond angle, which Pauling himself proved to be the most efficient in 1930.⁵

Four icosahedra surrounding and sharing faces with a central icosahedron to form a tetrahedron. Note again that there are fewer Aluminum atoms than there would be were the icosahedra separate from one another. This again shows that the atoms along the shared faces are part of both icosahedra containing that face. [Animation by Geoff Bloom]

Pauling also noted that 109.5 degrees is very close to the 108.0 degrees found between lines connecting three adjacent vertices in a pentagon. Thus, he predicted that icosahedra would arrange themselves at approximate 108.0-degree angles relative to one another to form a pentagonal ring, the first three of which would be from the tetrahedral shape (two at the vertices and the icosahedron at the center of the tetrahedron), and the other two supplied by a nearby tetrahedron. This would “bend” the internal tetrahedral angle slightly.²

Icosahedra sharing faces and forming a pentagonal ring. Each of these icosahedra would be part of a tetrahedron (not shown, for simplicity), which would bend slightly to make the smaller 108° interior angle of the pentagon. [Animation by Geoff Bloom]

This complex pentagonal ring, in turn, acts as a face of a larger three-dimensional shape, a regular dodecahedron. A dodecahedron is formed from twelve regular-pentagon faces, and is a common structure for intermetallic compounds. It also has twenty vertices. At each vertex would be an icosahedron, and each face would be a pentagonal ring of icosahedra. Therefore, each dodecahedron would be made from twenty twinned icosahedra.²

A simplified representation of the dodecahedron formed through linked pentagonal rings. Note the pentagonal faces. Each of the spheres at the vertices now represents an icosahedron; there are a total of twenty in the dodecahedron. [Animation by Geoff Bloom]

An alternate way to look at the structure is to imagine that the tetrahedra (formed from five multiply-twinned icosahedra) come together to form dodecahedra, such that the center of each tetrahedron sits at the corner where three pentagonal faces meet, and the lines connecting the three icosahedra on the “base” of the tetrahedron to its center icosahedron would form the edges of the pentagonal faces of the dodecahedron. These tetrahedra would then share end-points, such that there would only be a total of twenty icosahedra in the dodecahedral structure.

A dodecahedron formed from pentagonal rings (outlined in green). Note, too, the tetrahedron that exists at the corner of three pentagonal faces (outlined in yellow), demonstrating how slight modifications of tetrahedra formed from icosahedra eventually lead to the dodecahedral shape. The extra icosahedra (represented here by spheres, for simplicity’s sake) attached to the dodecahedron’s vertices demonstrate the presence of complete tetrahedra, and allude to ways for the dodecahedra to fit together in a larger structure. [Animation by Geoff Bloom]

By arranging these dodecahedra, Pauling initially arrived at an intricate structure containing 136 Mn atoms and 816 Al atoms (though this number changed many times throughout Pauling’s development of his theory), a structure he felt represented the unit cell of the alleged MnAl₆ quasicrystal.²

The structure of clathrate hydrate, above, is not identical to Pauling’s proposed twinning model, but is similar. Pauling used this known structure as a foundation for his proposed quasicrystal unit cell, which also uses staggered dodecahedra – except a much larger number of them.

Pauling felt experimental data substantiated his twinning model for a variety of reasons. First, his initial calculations for the unit cell size – approximately 26.73Å – matched x-ray powder images given to him by Shechtman.⁴ Second, Pauling had found what he called faint “layer” lines in the powder images that he felt were not adequately explained by quasicrystal theory, but instead matched structures with multiple twins.³ Third, Pauling noted that the Bragg peaks were shifted from their expected locations in ways that could be accounted for by his twinning model, but could not be addressed with the model for quasicrystal growth; that is, some atoms were in unexpected positions that could not yet be explained by any other theory of how quasicrystals arranged themselves.³ Most of all, Pauling’s repeated insistence on his experience with and integral role in shaping crystallography shows that he resisted changing what were considered foundational concepts, and strongly believed that the tenets of crystallography were sound enough for explaining what others were quick to call an exception.

However, Pauling’s twinning model had significant problems. David Shoemaker recalled having initial success with the x-ray diffraction patterns, finding that they matched Pauling’s calculations for the unit cell side length, at 26.73Ǻ. Then, when Pauling revisited his calculations to confirm their accuracy, the work hit a snag. Instead of a 26.73Ǻ unit cell side, Pauling realized his calculations called for a 23.36Ǻ side – a difference of about 15%. From Shoemaker’s perspective, this made the theory implausible. “I don’t think he was successful,” Shoemaker stated with respect to Pauling’s argument. “We [David and Clara] examined the figures ourselves and were unable to find any justification for the twinning theory there. So we, perhaps understandably, lost interest in it, but he continued on.”⁴

Pauling began arguing for multiple-twinning in late 1985. In an interview with John Maddox, writing for the journal Nature, he first publicly introduced his ideas, showcasing a 1120-atom unit cell for describing the MnAl₁₂ structure. His conclusion: “Crystallographers can now cease to worry that the validity of one of the accepted bases of their science has been questioned.”⁶ Shortly afterward, Pauling submitted a letter to the magazine Science News, which the periodical titled “The nonsense about quasicrystals.” In it, Pauling writes:

There is no doubt in my mind that my explanation of the quasicrystal phenomenon is correct. I have now accounted for the atomic arrangement seen on the electron micrographs. I trust that my paper containing these additional arguments will be published in Physical Review Letters. I think that it is interesting that an inter-metallic compound that I investigated in 1922, and whose structure was determined 40 years later, has the same structure as these ‘quasicrystals’, but without the twinning that they show. This is the compound sodium dicadmide, which is mentioned in my Nature article. It is also interesting that the scientific journals are printing scores of papers about exotic explanations of the observation but that I have had difficulty getting my papers on the subject published. I think that I am almost the only, perhaps really the only, x-ray crystallographer who has become interested in this subject. The explanation probably is that the other x-ray crystallographers felt that the nonsense about quasicrystals would soon fade away. That is how I felt for about five months, and then I finally decided that I would look into the matter.⁷

The letter was published January 4th, 1986, only three months after the publication of Pauling’s first article on the subject (the “Nature article” he mentions).⁸ His claim that no other x-ray crystallographers were interested in quasicrystals was an exaggeration, but the bulk of the scientists concerned with the subject were, in fact, physicists and not analytical chemists.

The dismissive tone that Pauling took toward quasicrystal theory would maintain itself throughout the rest of his career. That some referees at Physical Review Letters allegedly felt Pauling was behaving as “an antagonist” toward quasicrystal theorists⁹ – and perhaps Shechtman in particular – is not surprising, given the tone of Pauling’s debut letter. Describing the discovery and related research as “nonsense,” saying that “real” x-ray crystallographers avoided the matter and hoped it would “fade away,” and referring to initial explanations by other scientists as being “exotic” are actions imbued with condescending overtones. Further, Pauling’s mention of his own extensive expertise in the crystallography field, coupled with his seemingly patronizing line, “I finally decided I would look into the matter” makes it tempting to conclude that Pauling believed, perhaps a bit too strongly, in his own superiority.

Pauling’s hypothesis was a true masterpiece in its complexity, but it had major faults. Perhaps most damaging was the fact that no evidence of twins, a vital part of Pauling’s theory, had been found at all in the quasicrystals themselves.¹⁰ Though Pauling’s structure was certainly complex, and seemed to fit some of the evidence, his overconfidence, and the objections of other scientists, meant that conflict was looming on the horizon.

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References

1 “Crystal Twinning.” University of Oklahoma Chemical Crystallography Lab, Department of Chemistry and Biochemistry. 11 April 2011. Web. http://xrayweb.chem.ou.edu/notes/twin.html.

2 Linus Pauling Institute of Science and Medicine Newsletter. “Icosahedral Symmetry.” Vol.2 , Issue 9, Fall 1986. p. 4-5.

3 Pauling, Linus. Letter to Dan Shechtman. 6 June 1985.

4 Shoemaker, David. “My Memories and Impressions of Linus Pauling.” The Life and Work of Linus Pauling (1901-1994): A Discourse on the Art of Biography. Oregon State University. LaSells Stewart Center, Corvallis, OR. 1 March 1995. Symposium Presentation.

5 Paradowski, Robert. “Pauling Chronology: Early Career at the California Institute of Technology.” The Ava and Linus Pauling Papers.Oregon State University Special Collections & Archives Research Center. 2006. http://scarc.library.oregonstate.edu/coll/pauling/chronology/page8.html

6 Quoted in Peterson, I. “Probing Deeper Into Quasicrystals.” Science News 128.18 (1985): 278-9.

7 Pauling, Linus. “‘The nonsense about quasicrystals.'” Science News 129.1 (1986): 3.

8 Pauling, Linus. “Apparent icosahedral symmetry is due to directed multiple twinning of cubic crystals.” Nature 317 (October 1985): 512-14.

9 Pauling, Linus. Letter to Dan Shechtman. 6 October 1987.

10 Steinhardt, Paul J and Stellan Ostlund. The Physics of Quasicrystals. Singapore: World Scientific Publishing, 1987. Online. 310-12.

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.

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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.