crystal lattices
Boris Gruber's contributions to mathematical crystallography
^{a}Research with Neutrons and Muons, Paul Scherrer Institut, Forschungsstrasse 111, Villigen PSI, 5232, Switzerland
^{*}Correspondence email: hans.grimmer@bluewin.ch
Boris Gruber made fundamental contributions to the study of crystal lattices, leading to a finer classification of lattice types than those of Paul Niggli and Boris Delaunay before him.
Keywords: Boris Gruber; crystal lattices; lattice characters; Buerger cells; Niggli cell; Bravais types; genera.
1. Introduction
Paul Niggli (Niggli, 1928) used the results of Gotthold Eisenstein (Eisenstein, 1851) on the reduction of ternary positive quadratic forms to define for any given a with a unique In this way he obtained 44 lattice characters, which constitute a finer classification of lattices than the 14 Bravais types. Martin J. Buerger considered all primitive cells characterized by the shortest three noncoplanar translations (Buerger, 1957, 1960), known as `Buerger cells'. Many lattices have more than one Buerger cell, one of which corresponds to Niggli's choice.
Boris Gruber, a Czech mathematician working in Prague at Charles University, became interested in crystallographic lattices and obtained a number of important results, some of which I will mention here. He showed that a given lattice can have up to five Buerger cells (Gruber, 1973), a value reached only by certain lattices of anorthic Bravais type (aP). Consider for a Buerger cell the absolute values by which its angles α, β and γ deviate from 90°. Gruber showed that Niggli chose the Buerger cell for which the sum of these three absolute values is maximal (Gruber, 1989). As the number of free parameters characterizing a Buerger cell is largest for aP lattices (a, b, c, α, β, γ), it is most important to classify such lattices according to their crystallographic properties. Whereas only two among the 44 lattice characters concern this Bravais type, 43 among Grubers 127 genera correspond to aP lattices (Gruber, 1997a). This paper was an important step towards the present efforts of dividing crystallographic lattices into a large number of equivalence classes, to which the MACSMIN 2022 meeting in Liverpool, UK, was devoted. Gruber also gave necessary and sufficient conditions satisfied by conventional cells for the 14 Bravais types (see International Tables for Crystallography, Volume A, 2002).
2. Boris Gruber and his contributions to mathematical crystallography
Boris Gruber was born in 1921 in the early years of the Czechoslovak Republic. He spent his childhood first in Pilsen then in Brno. He went to a classical grammar school where Latin and Greek were the main subjects. He graduated in 1940, when all Czech universities were closed. To avoid his deportation to national socialist Germany, his parents enrolled him in a technical vocational school. After his second diploma, he joined a factory in Adamov, where his father and younger brother were already employed. During the liberation of Brno, the family went to Bílovice out of fear that their house could be the target of air raids, which indeed happened.
As soon as the war ended, Boris enrolled at Charles University in Prague to study mathematics and physics. His future wife, Zdeňka Kamarýtová, was in the same group as him while he worked on his dissertation. They married in 1951 and had two children, a son Jiří and a daughter Dana.
Boris began his academic career as an assistant to his former professor and later received an offer to teach introductory mathematics courses at the Faculty of Electrical Engineering of the Czech Technical University in Prague. The faculty was located in Poděbrady, a spa resort east of Prague on the river Elbe (Labe in Czech). As he refused to join the communist party, there was no future for him in Poděbrady. He therefore returned in 1958 to the Faculty of Mathematics and Physics in Prague, where he devoted himself to scientific work but was not allowed to teach. His first two publications were in Czech, a paper on the van der Waals equation of state (Gruber, 1950) and a study of the foundations of geometry (Gruber, 1957). Early on, he started using computers and creating programs for them, which led to the papers Numerical determination of the relative minimum of a function of several variables by quadratic interpolation (Gruber, 1967) and On the possibility of applying a computer when solving the fourcolour problem (Gruber, 1971). These publications show that he was able to contribute valuable results to many different fields: physics, pure mathematics and applied mathematics.
However, most by far of Boris Gruber's scientific work concerned crystal lattices. In his early work he developed an algorithm for determining the symmetry and stacking properties of the planes (hkl) in the 14 Bravais types of lattices. In 1966 he presented a preliminary account of the algorithm at the 7th Congress of the International Union of Crystallography (IUCr), which was held in Moscow. This research led to the paper Algorithm for determining the symmetry and stacking properties of the planes (hkl) in a Bravais lattice (Gruber, 1970a). The paper On the shortest lattice vectors in a threedimensional translational (Bravais) lattice appeared in the same year (Gruber, 1970b).
Gruber determined the number of Buerger cells for each of the 14 Bravais types (Gruber, 1973): In seven types there is only one Buerger cell, cubic facecentred lattices have 2, in three types there is either 1 or 2, in two types there are 1, 2 or 3 and in the triclinic (anorthic) type there are from 1 to 5. The main results of this paper were obtained during the author's stay at the University of Surrey in the UK, made possible by a grant from the UK Science Research Council.
Křivý and Gruber proposed an algorithm for determining the Niggli cell, starting from an arbitrary ). This algorithm is often used in modern datareduction software: five among the 100 citations of the paper (up to 22 December 2022) appeared in 2022.
of a threedimensional (Křivý & Gruber, 1976In August 1984, Boris Gruber took part in the Paul Niggli Symposium organized by the Swiss Society for Crystallography. He gave one of the principal lectures, entitled `Cell reduction' (Gruber, 1984). In this lecture, he stated that the fact that the Niggli cell has no obvious geometrical meaning may look disturbing to physicists and crystallographers. He discussed the Buerger cell with smallest surface, which led in 1989 to his paper Reduced cells based on extremal principles. He also mentioned the 24 Delaunay–Voronoi types (also called `Symmetrische Sorten') (Delaunay, 1933) which, like Niggli's 44 `Gitterarten', are a finer classification of lattices than the 14 Bravais types. The abstract of his lecture is shown in Fig. 1. This line of his thoughts led in 1997 to his paper Classification of lattices: a new step (Gruber, 1997a).
I had been asked to preside over the session on cell reduction, to which Peter Engel (Bern) and Boris Gruber contributed. We got on well, as shown by the card that he sent me after the symposium (Fig. 2).
In 1989, Gruber succeeded in giving a geometric interpretation of the Niggli choice (Gruber, 1989). He considered the surface S of the cell defined by the vectors , and ,
and the deviation W, defined as
Gruber considered the four kinds of Buerger cells for which S or W are minimal or maximal, respectively. He showed that the Niggli cell is the Buerger cell with the largest value of W. Boris Gruber thanked his wife for carefully checking his calculations.
In 1991, de Wolff and Gruber gave an exact definition of Niggli's 44 `Gitterarten', which they called `lattice characters'. In order to represent the characters graphically, they used the projection of the Nigglireduced basis vector c on the a, b plane (de Wolff & Gruber, 1991).
In his paper Topological approach to the Niggli lattice characters (Gruber, 1992) Gruber represented any lattice by a point in , the fivedimensional Euclidean space. He showed that the image of a lattice character is a maximal connected subset of the image of the Bravais type that contains the character. If instead of the Buerger cell with W maximal, another of the four possibilities mentioned above is used to obtain a unique cell, another splitting of Bravais types into lattice characters results. This shows that lattice characters are not as fundamental a concept as the Bravais types.
In 1997 Gruber found a classification of lattices that was finer than Bravais types and of major significance to crystallography. In his paper Classification of lattices: a new step (Gruber, 1997a), he constructed a classification into 127 classes, called `genera', which is finer than classifications into Bravais types, Delaunay–Voronoi types (Delaunay, 1933) or lattice characters (see Table 1). Lattices of the same genus agree in the number of Buerger cells, the densest directions and planes, and the symmetry of these planes. Even the formulae for the conventional cells are the same.

Anorthic lattices have six free parameters. Subdividing this Bravais type into a large number of cases according to their crystallographic properties was very welcome. Monoclinic lattices, with four free parameters, were also subdivided into a large number of cases with different crystallographic properties.
The same year, Gruber published another paper, Alternative formulae for the number of sublattices (Gruber, 1997b). In this paper he considered lattices in ndimensional Euclidean space and presented formulae for the number of sublattices of a given index k. They are based on the decomposition of the index k into a product of prime numbers and have the form of a rational function of these primes. Compared with other methods known at the time, they gave the result in a quicker and easier way.
Volume A of International Tables for Crystallography (2002) contains a chapter Further properties of lattices (Chapter 9.3) by Gruber, where he presents short versions of his results in less technical language, illustrated with three figures. Most important is Section 9.3.4, which deals with conventional cells for the 14 Bravais types. He starts with a remark on Section 9.1.7 written by H. Burzlaff and H. Zimmermann. Gruber shows that the conditions given in Table 9.1.7.2 for a cell to be the of a given nonanorthic Bravais type are necessary but not always sufficient. In Table 9.3.4.1 he gives conditions that are necessary and sufficient (Fig. 3). Note that Gruber's notion of a differs in several ways from the notion used by Burzlaff and Zimmermann: The orthorhombic cells are chosen such that a < b and b < c. A primitive rhombohedral cell is chosen for hR, not a rhombohedrally centred hexagonal cell. Gruber takes the centred monoclinic lattice as Icentred, not Ccentred, which simplifies the results. Bravais type aP is not mentioned in the table. All lattices that do not satisfy any of the conditions given for the other 13 Bravais types are of type aP.
The column `Conditions' of the table contains footnotes, some of which I give here in abbreviated form. Row tI: For the lattice is cF. Row oC: For the lattice is hP. Row hR: For α = 60° the lattice is cF, for α = arccos(−1/3) the lattice is cI. In row mI Gruber gives the conditions for which the lattice is hR. Therefore, one may say that cF is a limiting case of tI as well as of hR. In his footnotes Gruber addressed nontrivial limiting cases but did not mention the trivial ones, for instance that cP is a limiting case of tP. I am convinced that it would have been easy for him to collect all limiting cases and wonder why he did not find it worthwhile.
In Section 9.3.5 Gruber defined `conventional characters' as follows: Two lattices of the same Bravais type belong to the same conventional character if and only if one lattice can be deformed into the other in such a way that the conventional parameters of the deformed lattice change continuously from the initial to the final position without change of the Bravais type. The 22 conventional characters are a classification of lattices intermediate between the 14 Bravais types and the 44 lattice characters.
Gruber published his last paper in 2009, many years after retirement. Its title is On a minimum tetrahedron in a threedimensional lattice. Part I. Lattices with a shortest basis fulfilling (Gruber, 2009). At the end of the paper, which he devoted to the memory of his wife Zdeňka, he wrote `The author hopes to find a successor who will complete the whole problem by solving it also in the negative region a + b + c = min, '.
In 2015 I published a paper entitled Partial order among the 14 Bravais types of lattices: basics and applications (Grimmer, 2015). In this paper I determined the limiting cases of Bravais types and illustrated the results with figures.
I asked Mois Aroyo whether he was interested in including my results in the second online edition of Volume A of International Tables for Crystallography, which he was preparing. He suggested extending Table 9.3.4.1 in Volume A of International Tables for Crystallography (2002) to show which Bravais types are limiting cases of more general ones. I suggested that Boris Gruber could do this. When Gruber informed Aroyo that his health did not allow him to do so, I took over. In the second online edition of Volume A of International Tables for Crystallography (2016) the result is shown in the chapter Further properties of lattices (which in this edition is Chapter 3.1.4, by B. Gruber and H. Grimmer). Table 3.1.4.1 shows that the limiting Bravais types are obtained simply by replacing in the column `Conditions' of the old Table 9.3.4.1 (Fig. 3) any < or ≠ symbols by =. The result is illustrated in Fig. 4 (which is Fig. 3.1.4.3 in the chapter).
A photograph of Boris Gruber aged 90 is shown in Fig. 5. He died in 2016.
3. Summary
In his talk at the Paul Niggli Symposium in 1984, Gruber sketched his ideas for a finer classification of lattices that would subdivide the Bravais types, particularly those of low symmetry, in a way significant to crystallography. This idea culminated in his paper Classification of lattices: a new step (Gruber, 1997a). This paper is close to the efforts to which the MACSMIN 2022 meeting was devoted, the main difference being that experimental errors in determining the lattice parameters were not considered in the papers of Gruber, as was pointed out by Andrews & Bernstein (2014).
I admire the foresight and perseverance with which Boris Gruber contributed very valuable results on crystal lattices, even though he was not allowed to teach and his international contacts were restricted for a long time.
Acknowledgements
I am very grateful to Jiří Gruber for the photograph shown in Fig. 5 and for information on the life of his father.
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