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. Goldman2,3. 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.

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.

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.

12. Widom Research Group. “Quasicrystals.” Carnegie Mellon University. N.d.

13. Marder, Jenny. “What are quasicrystals, and what makes them Nobel-worthy?” PBS Newshour Rundown News Blog. October 5, 2011.