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.


1 “Crystal Twinning.” University of Oklahoma Chemical Crystallography Lab, Department of Chemistry and Biochemistry. 11 April 2011. Web.

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.

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.


David and Clara Shoemaker

David and Clara Shoemaker working in an x-ray laboratory at Oregon State University, 1983.

Husband and wife crystallographers David and Clara Shoemaker were, in many respects, an unlikely couple.

David Shoemaker was born on May 12, 1920 in the tiny town of Kooskia, Idaho. Clara Brink was born on June 20, 1921 in Rolde, Holland. Both moved through their primary studies in orderly fashion and progressed to undergraduate work in chemistry – David at Reed College in Portland, Oregon, Clara at the University of Leiden.

In 1942 David received his bachelor’s degree from Reed and moved directly to the California Institute of Technology. Working under Linus Pauling, David quickly established himself as a promising doctoral candidate. His research was initially divided between Pauling’s expansive program of scientific war work and, later, a series of crystallographic investigations. While in Pasadena, David determined the structure of sixteen molecules, most notable among them threonine, an amino acid.

Upon receiving his Ph. D. in 1947, David – with the assistance of Pauling – was subsequently named a Guggenheim fellow, studying at both Oxford and the Institute for Theoretical Physics in Copenhagen. Aged 27, he was among the youngest of his era to receive a Guggenheim Fellowship.

Group photo of participants at the Conference on Current Problems of Physics. Copenhagen, Denmark. September 1947. Niels Bohr sits in the front row, far left. David Shoemaker is seated in the second row, fourth from right.

Group photo of participants at the Conference on Current Problems of Physics. Copenhagen, Denmark. September 1947. Niels Bohr sits in the front row, far left. David Shoemaker is seated in the second row, fourth from right.

Clara’s path through graduate studies was somewhat less smooth. She completed her undergraduate work at the University of Leiden in 1941, shortly before the Nazi occupation of the Netherlands and the subsequent closing of the university. Despite the turbulence of World War II, Clara was able to commence her graduate studies through the University of Utrecht, though much of her coursework was self-taught, conducted in her parents’ home. Despite these handicaps, Clara completed her doctoral examinations on time, in 1946, after which point she assumed an assistantship at the University of Utrecht and learned the techniques of x-ray crystallography, commuting one day per week to Amsterdam to study under the renowned crystallographer Caroline MacGillavry.

The years immediately following the close of hostilities were fruitful ones for both David and Clara. Having returned home from his Guggenheim trip, David was named a Senior Research Fellow at Caltech, where he solved the difficult structure of DL-serine and began the research program that came to define much of his (and Clara’s) career – a broad series of investigations into the structures of complex transition-metal phases. In the meantime, Clara became a full-time crystallographer, first studying crystal structures of monovalent ions at the University of Leiden and later working for one year at Oxford, where she conducted research on the crystal structure of vitamin B12 under Dorothy Hodgkin, the 1964 Nobel laureate in Chemistry.

In 1951 David was hired away from Caltech by the Massachusetts Institute of Technology, where he began investigating zeolite structures as an Assistant Professor. Two years later, dissatisfied with the working environment at the University of Leiden, Clara took a one-year leave of absence to work on transition metals at M.I.T. Her laboratory in Cambridge was run by David Shoemaker.

In 1954 David renewed Clara’s leave of absence for an additional year and by 1955 it was clear that Clara would not be returning to Europe – on August 5th, the couple was married. Shortly thereafter Clara transferred to Harvard Medical School to work under the biochemist Barbara W. Low. One year later, Clara gave birth to the couple’s only son. While caring for the newborn Robert, Clara worked from home on the International Tables of Crystallography.

The Shoemakers enjoyed a productive tenure at M. I. T. – David was promoted to full professor, began a lengthy service on the U. S. National Committee for Crystallography (including a three-year term as President) and published widely, including a textbook titled Experiments in Physical Chemistry, which would eventually run through six editions.

In 1970 David was elected President of the American Crystallography Association. That same year, the Shoemakers relocated to Oregon State University, where David had been hired to chair the Department of Chemistry. In reaction to the university’s nepotism guidelines, Clara arranged to work as Research Associate under Dr. Ken Hedberg – like David Shoemaker, a former graduate student of Linus Pauling. The arrangement lasted for several years until the university’s rules were relaxed.

Model of the crystal structure and superstructure of the K Phase, Mn77Fe4Si19. Model built by Clara B. Shoemaker, David P. Shoemaker and Ted E. Hopkins.

During his tenure as department chair, David oversaw two major building projects – the construction of a new chemistry laboratory facility and the renovation of the chemistry offices and research building. Over that same period of time, Clara trained several graduate students in techniques of x-ray crystallography, publishing papers with many of her protégés. The couple retired in 1984, though they continued to conduct important work on transition metal phases as well as the controversial topic of quasicrystals.

The Shoemakers remained close friends with Linus Pauling, though they did dispute certain of Pauling’s claims about the nature of quasicrystals. In 1995 David Shoemaker, himself in fading health, spoke of his long association with Pauling at a memorial conference organized at Oregon State University. David’s comments detailing his recollection of the discovery of the alpha-helix caused something of a stir in the audience, as the provenance of the alpha-helix work has long been a matter of some dispute.

David Shoemaker on the Discovery of the Alpha Helix

Afterward, Shoemaker offered this clarification:

My memory may have been faulty in claiming to have seen Pauling actually taping his cardboard amide linkages together to form a helix, but Professor William Lipscomb, in a talk that preceded mine, showed a drawing in Pauling’s own hand of an alpha-helix rolled out flat, showing what points the polypeptide chain joined together in the helix. The drawing was titled ‘alpha helix. First drawn March 1948. Linus Pauling.’ My visit to Oxford was from January to March 1948.

David Shoemaker died of kidney failure on August 24, 1995, some six months after the Pauling memorial conference. His wife Clara, a close friend of our department, passed away on September 30, 2009. Over the course of their professional association, David and Clara published thirty-six scientific papers together.

The David and Clara Shoemaker Papers are just one of the many collections held in the OSU Libraries Special Collections.