Crystallography in Canada
19th century basics spearhead 21st century progress:
Principal Research Officer, ICPET, National Research Council of Canada, Ottawa, Ontario
This is the "MISSYM story", as suggested by the Editor of this Newsletter issue. This very simple and small part of my work at NRC is now used to delegate so much quantum-modeling brainwork to computers that it may be worth consigning the sequence of concepts and events on paper.
In the fall of 1976, I was Post-doc with Gabrielle (Gai) Donnay (Martin, 1989), a diffractionist with great reputation and Crystallography Professor at the Geology Department of McGill University. Her husband, Joseph (José or J.D.H.) Désiré Hubert Donnay (1902-1994, see Donnay, 1962; Le Page, 1994) was Emeritus. J.D.H. was a legend of mineralogy, crystal morphology and twinning. They occupied facing desks in the same office. Together, they were a formidable team, essentially an encyclopedia of mineralogy, crystallography and diffraction. I had been assigned the task of teaching the yearly undergraduate Crystallography course to students from physics, chemistry and geology departments, while the Donnays were away on-and-off, a couple weeks at a time during a sabbatical year. Under any circumstances, teaching such a spectrum of nearly 40 undergraduate and graduate McGill students would have been a daunting task for any recent Ph.D. in Physics. Knowing that the lecture notes would be under the microscope of the Donnays each time they came back unannounced made things really daunting.
Maybe the very first theorem stated after definitions in any decent crystallography course is the Primitivity Theorem: A cell based on three shortest (meaning that none is shorter) non-coplanar lattice vectors is primitive. I tried to work out a simple proof for the lecture notes and failed. The theorem was printed in textbooks by Buerger, Friedel etc., but without proof. I knocked on J.D.H.'s door and asked what classroom proof he used for this theorem. Tall, straight, alert and friendly at 74, he was known to be the most knowledgeable living person for anything about geometrical crystallography. " Aaah, celui-là..." he said, hinting that something was ajar: even Bravais (1850), who had proved everything else, had not printed a proof for it, but was instead referring to a very complicated algebraic proof in a book printed twenty years earlier and that could not be found anymore. J.D.H. knew of other proofs as well, but all complicated and not suitable for teaching. I was extra careful to prove everything else, including the existence of seven crystal systems and no more, a fact that is usually taken for granted. Students were great and interested. Nobody dropped from the course or flunked the final exam on both geometrical crystallography and X-ray diffraction, to the amazement of the Donnays who actually checked that exam books at the bottom of the graded stack were up to their specs for graduation.
My second post-doc was at NRC's X-ray diffraction laboratory at the Chemistry Division in Ottawa, starting in 1977, working with Larry Calvert, Eric Gabe, Allen Larson and Yu Wang on the nuts and bolts of what would become the NRCCAD Fortran diffractometer program, the NRCVAX structure package and the CRYSTMET crystal structure database. The IUCr XIth Meeting in Warsaw in 1978 allowed two contributions provided that one would be in the session on " Crystallographic teaching". In addition to a poster on least-squares weights in accurate refinements, I then contributed the short 1976 proof for the existence of no more than seven crystal systems as a second poster (Le Page, 1978). Little did I know that, combined with the primitivity theorem (see full story in Le Page, 1992a), this superficially pointless classroom proof was the first quatrain of a lifelong saga that is still developing today at the rate of about one paper each three to five years. Applications are still bearing the most stunning fruits nearly each day.
The most pressing problems we had to tackle with diffractometer automation was recognition of the metric symmetry of the cell, derivation of crystal symmetry and its subsequent use in efficient data collection. The popular cell-reduction algorithm at that time was based on distinguishing 44 mutually exclusive cases for the matrix of dot products between edges of the Niggli cell (Niggli, 1928). It is summarized in International Tables for X-ray Crystallography (1969). Vol. I pp. 530-535. In addition to having generated an abundant literature of errata about conditions and of typos, the very basis of that approach was questionable to my eyes. As Niggli cell reduction requires knowledge of the lattice symmetry because it involves tests on equality, my feeling was that it is then logically circular to use the Niggli cell to determine lattice symmetry. Mathematically mutually exclusive cases may no longer be mutually exclusive when experimental error steps in, creating all sorts of problems having to do with pseudo symmetrical lattices or with the ordering of logical tests. Something more robust was needed.
The Chemistry Division was a couple thousand feet from CISTI, the then recent scientific library for Canada. Although collections were created in 1974, they included excellent original editions of Bravais, Mallard, Voigt, Friedel, Wyckoff, Niggli etc. that are quite difficult to find elsewhere. A pristine copy of Seeber (1831) even existed in a locked room reserved to rare books. I could then see that J.D.H. had not exaggerated: the algebraic proof of the primitivity theorem sprawled there over about fifty pages.
After studying Bravais (1850) and dusting the 19th century vocabulary, I came up with the conclusion that, with u and h respectively designating a direct and a reciprocal vector from a primitive lattice, the pair of conditions {u·h=1 or 2; uÅ~h=0} was necessary and sufficient for lattice row [u] to be an even-order axis of metric symmetry. The only missing thing to transform this into a foolproof computer algorithm was limits on components of u and h. It can be proved that magnitudes of Miller indices of u and h cannot exceed 2 when referred to three shortest non-coplanar translations. Difficulties associated with experimental error were avoided: any set of three short translations could be used with no angular condition. Extension to detection of lattice pseudosymmetry was straightforward. Calling δ the angle between u and h, the case δ=0 corresponds to symmetry while the case δ<ε where ε is a small limiting value corresponds to pseudosymmetry because a small affine transformation can then create lattice symmetry from pseudosymmetry. The number of detected even-order axes is sufficient for crystal-system assignment, except for orthorhombic that has three mutually perpendicular twofold symmetry axes while rhombohedral has three coplanar ones, keeping complexity of branching to a very minimum. The lattice type is read off the values 1 or 2 of h·u for the metric symmetry axes. It became the CREDUC cell-reduction algorithm at the heart of NRCCAD, NRCVAX and the MDF checking software, and more recently CRYSTMET and Materials Toolkit. It has worked perfectly and never required any tweaking since the first day it implemented the necessary and sufficient pair of conditions above and identified one case in each crystal system. The work was presented at IUCr XII in Ottawa (Le Page, 1981) and published in Le Page (1982).
The next step was MISSYM that is meant to identify the symmetry of a structure model within a given distance tolerance. It is difficult to realize the significance of MISSYM for crystallography, now that the dust around R.E. (Dick) Marsh's campaign in the nineteen eighties has mostly settled. Dick has the incredible talent to "see" symmetry upon mere inspection of atom coordinates. He is very thorough in his re-analyses which involved re-typing deposited structure factors, as they were in printout form at the time. Some of his re-analyses are crystallographic gems demonstrating an extraordinary crystal-chemical sense. Dick's intentions were to raise the level of awareness and the crystallographic culture of Acta C contributors, or at least its editorial team. Results were not entirely up to expectations. Years after initially beating the drum about the item, awareness had risen to 100%, but Dick's batting average remained unchanged all along at about one marshed structure per Acta C issue. In spite of scientifically impeccable work and laudable intentions, Dick Marsh's campaign had the unintended adverse effect of creating waves across the whole body of science that were starting to bring the science of crystallography into disrepute. As crystal symmetry must re-establish the lattice within a fraction of a lattice translation, the rotation part R of a crystal-symmetry generator is therefore a lattice-symmetry operation. The CREDUC algorithm spells out Miller indices for the directions of the elements of metric symmetry in the reference system of the three shortest translations. All numerical expressions for R are then known in Cartesian axes. Extension of CREDUC to the determination of the symmetry of a model then only involves solving the equation x'=R·x+T for T on one pair of atoms with same nature and known Cartesian coordinates x and x'. Checking that those values of R and T transform each atom in the list into an atom with same nature, also in the list within the distance tolerance, then establishes the existence of symmetry in the structure model. This is the essence of the MISSYM algorithm presented at IUCr XIV in Perth (Le Page, 1987a) and published as (Le Page, 1987b) and of its MISSYM 1.1 version (Le Page, 1988) where defaults were tailored after the large number of cases exposed by Marsh until that date. Applied widely to past Acta C issues, MISSYM 1.1 flagged from defaults all structures that had been previously marshed as well as a few genuinely pseudosymmetrical structures, but nothing else as presented at IUCr XV in Bordeaux (Le Page, 1990). NRC gave the IUCr a license for the use of MISSYM for its own not-for-profit purposes. Brian McMahon inserted MISSYM in the IUCr checking package in the fall of 1989 (Dacombe & McMahon, 1990). Overnight, marshable structures became an essentially extinct species in IUCr journals. Dick Marsh refocused his campaign on chemistry journals with about the same success, while things went back to business as usual at Acta C and MISSYM went on with its own career as a filter in crystal-structure databases (Calvert, 1992; White, Rodgers & Le Page, 2002).
The CREDUC algorithm was again key to interpretation of convergent-beam electron-diffraction (CBED) patterns for cell volumes (Le Page and Downham, 1991) or least-squares cell data (Le Page, 1992b). Those concepts and developments were applied to rapid phase analysis in electron microscopy, and presented at IUCr XVI in Beijing (Le Page, Y., Chenite, A. and Rodgers, J. R., 1993).
It is well known since Mallard (1885) that twinning is due to metric symmetry or pseudosymmetry of a multiple cell, and characterized by its twin law with small obliquity and small twin index. Although Cesaro (1886) dabbled with the problem, more than a century later, no software existed that would be capable of spelling out all possible twin laws within maximum index and maximum obliquity for a given material. The main change required to adapt the CREDUC algorithm to the prediction of binary twins was then to relax the limit on the dot product condition and the limit on Miller indices to a same integer 2n value. If the dot product for a solution is p, the twin index is either p or p/2, depending on twin-lattice centering. The value of δ is the twin obliquity while that of ε is the maximum obliquity. The corresponding program called OBLIQUE was presented at IUCr XVIII in Glasgow (Le Page, 1999) and published as Le Page (2002). It has since been spelling out possible binary twin laws within maximum obliquity and maximum index without missing one or proposing a wrong one, again because it is based on theorems. This is a big help for the identification of twin laws from single-crystal diffraction patterns for all of organic, inorganic and recently macromolecular materials.
Le Page, Klug & Tse (1996 a), presented at IUCr XVII in Seattle and published as Le Page, Klug & Tse (1996 b), describes a step-by-step manual procedure to use MISSYM to derive conventional crystallographic descriptions from the P1 models in primitive axes used by ab-initio DFT quantum modeling. This was again a timely contribution as quantum modeling was becoming feasible with expensive supercomputers that had recently sprouted around the planet. By that time, the performance of inexpensive off-the-shelf PCs was already within one to two orders of magnitude of what was needed to perform quantum computations on real systems. As an order of magnitude corresponds to five years in Moore's law terms, it became clear that a spectacular rampup in the use of quantum software was to be anticipated for shortly after Y2K. Anybody who dabbled with quantum software knows how tedious and time-consuming the creation of the model and its input files, as well as the interpretation of output files can be. It could then be foreseen that by around 2005, the power of even a modest computing setup would exceed the data creation and interpretation capability of an individual, thus creating a bottleneck. Exactly like Eric Gabe, Allen Larson, Yu Wang, Peter White and myself had created in the 1970s the NRCCAD and NRCVAX software packages that automate to a great degree the work of the inorganic structural crystallographer, I similarly set up in 1998 with Paul Saxe from Materials Design Inc. to automate to a very significant extent the job of the quantum modeler in a first package called MedeA. We first created and automated the least-squares symmetry-general calculation of the elastic tensor from total energy (Le Page & Saxe, 2001) and from stress calculations (Le Page & Saxe, 2002). At the heart of this automation is the programming of the step-by-step procedure described in Le Page, Klug and Tse (1996b). The work was submitted to the XIXth IUCr Congress in Geneva (Le Page, Saxe & Rodgers, 2002a) and printed as (Le Page, Saxe & Rodgers, 2002b). The next implementation, called Materials Toolkit with Innovative Materials Technologies Inc. (Le Page and Rodgers, 2005) is much more versatile and includes powerful extensions over the capabilities described in Le Page, Saxe & Rodgers (2002b). Its achievements include exploratory computation of thermomechanical properties of existing or prospective materials by the thousand, contributions to seismology, permanent storage of toxic or radioactive elements, correction of experimental errors in published elastic tensors for reference materials, design of a coating for turbine blades in jet engines that increases their resistance to erosion by nearly an order of magnitude, now certified for commercial flights and commercialized etc.
What am-I doing now? Mostly three things. It is quite common nowadays that experimental data alone is too scant to allow deductive scientific conclusions. I actively seek such bogged-down studies within ICPET. Complementing the study with quantum modeling often allows definite conclusions to be drawn by giving unambiguous support to a model, even in the absence of a competing one. This is the first thing. The second thing that I do is introduce young crystallographers around me to the automated application of quantum methods as in Mercier & Le Page (2008). The combined result of the first two items is enthralling joint publications with my ICPET and IRC crystallographer colleagues that get noticed internationally (see Acta B highlights in IUCr Newsletter, fall 2005 and spring 2007). They also get noticed at NRC as shown on Fig. 1 where the team of ICPET and IRC crystallographers receives a much coveted NRC award for Research Excellence. This also establishes factually that Materials Toolkit is an extremely powerful and versatile package, at the cutting edge for both Structural Science and for Materials Science. The third thing I do is expand the range of automated applications in Materials Toolkit (see Le Page, 2006) while getting methods and results published. The latest such study has been the precise quantum calculation of surface tension on crystal faces for metals, currently being refereed.
When looking backward and extrapolating forward, I am happy to see that crystallography has constantly remained at the cutting edge of knowledge and technology since 1850. Its activities are surfing from one wave to another wave, gaining momentum each time through innovation and greater automation of complex repetitive tasks. From what I can see, this is not going to slow down soon as quantum modeling of materials, a branch of modern crystallography, is currently undergoing a phase of explosive expansion in its capabilities and volume of output. This comes from greater automation harnessing the exponential increase of available computing power. This expansion will ultimately be a success only if the software that automates it rests on theorems and not on ad-hoc considerations suffering from particular cases and exceptions.
References Bravais, A. (1850). J. Ec. Polytech. 19 (33) 1-128. Calvert, L.D. (1992). Acta Crystallogr. B48, 113-114. Cesaro, G. (1886). Bull. Soc. Fr. Min. 9, 222-242. Dacombe, M.H. and McMahon, B. (1990). Acta Crystallogr. A46, p. C-454, Abstract MS-15.02.03. Donnay, J.D.H. (1962). pp 564-569 in Fifty years of X-ray diffraction. Edited by P.P. Ewald. Also available as: http://www.iucr.org/iucr-top/publ/50YearsOfXrayDiffraction/index.html Le Page, Y. (1978). Acta Crystallogr. A34, p. S-389, Abstract 18-2-8. Le Page, Y. (1981). Acta Crystallogr. A37, p. C-338, Abstract 18-1-02. Le Page, Y. (1982). J. Appl. Cryst. 15, 255-259. Le Page, Y. (1987a). Acta Crystallogr. A43, p. C-293, Abstract 18-1-2. Le Page, Y. (1987b). J. Appl. Cryst. 20, 264-269. Le Page, Y. (1988). J. Appl. Cryst. 21, 983-984. Le Page, Y. (1990). Acta Crystallogr. A46, p. C-454, Abstract 15-02-03. Le Page, Y. (1992a). J. Appl. Cryst. 25, 661-662. Le Page, Y. (1992b). Microscopy research. 21, 158-165. Le Page, Y. (1994). ACA Newsletter, Winter 1994, 14-15. Le Page, Y. (1999). XVIIIth Meeting of the IUCr in Glasgow. Abstract book, p.186, Abstract M12-CC.001. Le Page, Y. (2002). J. Appl. Cryst. 35, 175-181. Le Page, Y. (2006). MRS bulletin 31, 981-984 (2006). Le Page, Y., Chenite, A. and Rodgers, J. R. (1993). Acta Crystallogr. A49, p. C59, Abstract PS-02-08-26. Le Page, Y. and Downham, D.A. (1991). J. Electr. Microsc. Tech. 18, 437-439. Le Page, Y., Klug, D.D. and Tse, J.S. (1996 a). Acta Crystallogr. A54, p. C-91, Abstract MS03-06-02. Le Page, Y., Klug, D.D. and Tse, J.S. (1996 b). J. Appl. Cryst. 29, 503-508. Le Page, Y. and Rodgers, J.R. (2005). J. Appl. Cryst. 38, 697-705. Le Page, Y. and Saxe, P.W. (2001). Phys. Rev. B63, 174103. Le Page, Y. and Saxe, P.W. (2002). Phys. Rev. B65, 104104. Le Page, Y., Saxe, P.W. and Rodgers, J.R. (2002a). Acta Crystallogr. A58, p.C215. Le Page, Y., Saxe, P.W. and Rodgers, J.R. (2002b). Acta Crystallogr. B58, 349-357. Mallard, F.E. (1885). Bull. Soc. Fr. Min. 8, 452-469. Martin, R.F. (1989). Amer. Mineral. 74, 491-493. Mercier, P.H.J. and Le Page, Y. (2008). Acta Crystallogr. B64, 131-143. Niggli, P. (1928). Krystallographische und strukturtheoretische Grundbegriffe. Handbuch der Experimentalphysik, Vol. 7, part 1, pp. 108-176. Leipzig: Akademische Verlagsgesellschaft. Seeber, L.A. (1831). Untersuchungen über die Eingenschaften der positiven ternären quadratischen Formen. Freiburg. White, P.S., Rodgers, J.R. and Le Page, Y. (2002). Acta Cryst. B58, 343-348. |
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