⬇ Download Markdown  ·  ⬇ Download LaTeX  ·  ⬇ original  ·  view original  ·  CSV: table 1

1 March 10, 2023

arXiv:2302.13833v2 [cond-mat.mtrl-sci]

9

Mar 2023

Delone lattice studies in C 3 , the space of three complex variables

Lawrence C. Andrews a * and Herbert J. Bernstein b a Ronin Institute, 9515 NE 137th St, Kirkland, WA, 98034-1820 USA, and b Ronin

Institute, c/o NSLS-II, Brookhaven National Laboratory, Upton, NY, 11973 USA.

E-mail: [email protected] lattice ; reduction ; Delone ; Selling ; C 3 Abstract

The Delone (Selling) scalars, which are used in unit cell reduction and in lattice type determination, are studied in C3 , the space of three complex variables. The three complex coordinate planes are composed of the six Delone scalars. Note: In his later publications, Boris Delaunay used the Russian version of his surname, Delone.

1. Introduction

The scalars used by Delaunay (1932) in his formulation of Selling reduction (Selling, 1874) are (in the conventional order) b·c, a·c, a·b, a·d, b·d, c·d, where d = −a− b− c.

PREPRINT: Acta Crystallographica Section A A Journal of the International Union of Crystallography

2 (As a mnemonic device, observe that the first three terms use α, β, and γ, in that order, and the following terms use a, b, c, in that order.) Andrews et al. (2019b) chose to represent the Selling scalars in the space S6 , {s1 , s2 , s3 , s4 , s5 , s6 } (defined in the order above), as a way to create a metric space for the measurement of the distance between lattices. Andrews et al. (2019b) also considered the representation of this space as the space of three complex dimensions, C3 or {c1 , c2 , c3 } . In C3 , in terms of the Selling scalars, a vector is defined as {(s1 , s4 ) , (s2 , s5 ) ,(s3 , s6 ) }, where the real and imaginary parts of each are the “opposite” scalars according to the definition of Delaunay (1932) (see Andrews et al. (2019a)). As a mnemonic device, note that the complex components involve (α, a), (β, b), and (γ, c). Additionally, each complex term uses all four 3-space vectors; for example, c1 is (b · c, a · d). Andrews et al. (2019b) considered the matrix representations of the reflections in C3 (and in S6 ). This paper describes the boundary transformations at the edges of the fundamental unit of C3 . In S6 , the fundamental unit is the all negative orthant, which contains only and all of the reduced cells. In S6 and C3 , the boundaries located where any S6 scalar (or correspondingly inC3 , the real or imaginary part) equals to zero. The rationale for this work was that it might lend insights into the topology of the space of lattices.

2. Notation

Complex numbers will be represented in Cartesian format (x, y), where x is the real part and y is the imaginary part. We will represent a vector in C3 by {(x1 , y1 ) , (x2 , y2 ) , (x3 , y3 ) } as an alternative to {(s1 , s4 ) , (s2 , s5 ) ,(s2 , s6 ) }. Next we define the operators in C3 that will be used in the matrix descriptions of IUCr macros version 2.1.15: 2021/03/05

3
the transformations at the boundaries of the fundamental unit.
See Table 1.

Table 1. The Result column values give the products of each operator as applied to (x j , y j )

. Operator Usage Result Name Mr Mr (cj ) (−xj , −xj + yj ) Minus real Mi Mi ( cj ) (xj − yj , −yj ) Minus imag Pr Pr (cj ) (xj , xj ) Plus real Pi Pi ( cj ) (yj , yj ) Plus imag

RI

Mr (cj ) (−xj , −xj + yj ) Minus real Mi ( cj ) (xj − yj , −yj ) Minus imag Pr (cj ) (xj , xj ) Plus real Pi ( cj ) (yj , yj ) Plus imag R (cj ) xj Real I (cj ) yj Imaginary

3. Matrices of boundary transformations

For the boundary at s1 : (the real component of c1 ) . M r 0 0 Pr iℜ ℜ Pr iℑ ℑ M r 0 0 Pr iℑ ℑ Pr iℜ ℜ M r 0 0 Pr ℜ iℜ Pr ℑ iℑ M r 0 0 Pr ℑ iℑ Pr ℜ iℜ For the boundary at s4 : (the imaginary component of c1 ) . M i 0 0 Pi iℜ ℜ Pi iℑ ℑ M i 0 0 Pi iℑ ℑ Pi iℜ ℜ M i 0 0 Pi ℜ iℜ Pi ℑ iℑ M i 0 0 Pi ℑ iℑ Pi ℜ iℜ For the boundary at s2 (the real component of c2 ) : i ℜ Pr ℜ 0 Mr 0 i ℑ Pr ℑ i ℑ Pr ℑ 0 Mr 0 i ℜ Pr ℜ ℜ Pr iℜ 0 Mr 0 ℑ Pr iℑ ℑ Pr iℑ 0 Mr 0 ℜ Pr iℜ For the boundary at s5 : (the imaginary component of c2 ) . i ℜ Pi ℜ 0 Mi 0 i ℑ Pi ℑ i ℑ Pi ℑ 0 Mi 0 i ℜ Pr i ℜ ℜ Pi iℜ 0 Mi 0 ℑ Pi iℑ ℑ Pi iℑ 0 Mi 0 ℜ Pi iℜ For the boundary at s3 (the real component of c3 ) : i ℜ ℜ Pr i ℑ ℑ Pr 0 0 Mr i ℑ ℑ Pr i ℜ ℜ Pr 0 0 Mr ℜ iℜ Pr ℑ iℑ Pr 0 0 Mr ℑ iℑ Pr ℜ iℜ Pr 0 0 Mr IUCr macros version 2.1.15: 2021/03/05

4 For the boundary at s6 (the imaginary component of c3 ) : i ℜ ℜ Pi i ℑ ℑ Pi 0 0 Mi i ℑ ℑ Pi i ℜ ℜ Pi 0 0 Mi ℜ iℜ Pi ℑ iℑ Pi 0 0 Mi ℑ iℑ Pi ℜ iℜ Pi 0 0 Mi

4. Basics

The standard representation of the identity operation is c′ = 1 0 0 0 1 0 0 0 1 c. The identity in C3 can also be written: c′ = 1 0 0 0 ℜ + iℑ 0 0 0 ℜ + iℑ c. Delaunay (1932) does not consider the boundary transformations in detail. How- ever, he uses them to define the process of Selling reduction. For example in S6 , he lists the following as one of the possible results for a transformation on s1 : {-s1 , -s1 + s2 , s1 + s3 , s1 + s5 , s1 + s4 , s1 + s6 } . The third boundary transform for s1 implements this operation and interchanges the real part of c3 and the imaginary part of c2 : M r 0 0 Pr ℜ iℜ Pr ℑ iℑ Considered in C3 , Delone’s alternate transformation for the s1 boundary would exchange the real of c2 with the imaginary part of c3 . That is the fourth matrix in the list for s1 . The other two transformations for s1 can be generated from the two we have just displayed by the ”exchange operation” (Andrews et al., 2019b) applied to the second and third C3 coordinates. Delone did not describe the latter two trans- formations, perhaps because even a single transformation was adequate to implement reduction. He had already listed two. IUCr macros version 2.1.15: 2021/03/05

5

5. Graphical display of projections

The two-dimensional nature of the three coordinates of C3 suggests their use for graphical display. As an example, we use Phospholipase A2 (retrieved from the Protein Data Bank (Bernstein et al., 1977)), which has had several similar or identical structures deter- mined (Le Trong & Stenkamp, 2007). Andrews et al. (2019b) found additional cases (see Table 2) PDB id Centering a b c α β γ 1DPY R 57.98 57.98 57.98 92.02 92.02 92.02 1FE5 R 57.98 57.98 57.98 92.02 92.02 92.02 1G0Z H 80.36 80.36 99.44 90 90 120 1G2X C 80.95 80.57 57.1 90 90.35 90 1U4J H 80.36 80.36 99.44 90 90 120 2OSN R 57.10 57.10 57.10 89.75 89.75 89.75

PDB id Centering a b c α β γ 1DPY R 57.98 57.98 57.98 92.02 92.02 92.02 1FE5 R 57.98 57.98 57.98 92.02 92.02 92.02 1G0Z H 80.36 80.36 99.44 90 90 120 1G2X C 80.95 80.57 57.1 90 90.35 90 1U4J H 80.36 80.36 99.44 90 90 120 2OSN R 57.10 57.10 57.10 89.75 89.75 89.75 Table 2. Phospholipase A2 unit cells Below, Figure 1 shows the unit cells as reported (the centering of lattices has not been removed). The following figures show various transformations and embellish- ments of the reported cells.

Coordinate c 1 Coordinate c 2 -s

4 -s 1 -s 2

Fig. 1. Phospholipase A2 unit cells as reported.
IUCr macros version 2.1.15: 2021/03/05

Coordinate c 3 -s

5 -s 3 -s 6 6 Coordinate c 1 Coordinate c 2 -s

4 -s 1 Coordinate c 3 -s

5 -s 2 -s 6 -s 3

Fig. 2. The unit cells Niggli-reduced. The similarity of the 3 projections is indicative
of the exact or nearly exact rhombohedral symmetry.

Coordinate c 1 Coordinate c 2 -s

4 -s 1 -s 5 -s 2 Coordinate c 3 -s

6 Coordinate c 3 -s

6 -s 3

Fig. 3. The unit cells Delone reduced.

Coordinate c 1 Coordinate c 2 -s

4 -s 1 -s 2 -s 5 -s 3

Fig. 4. The unit cells, Niggli and five copies were perturbed 2% orthogonally to S6 vector. IUCr macros version 2.1.15: 2021/03/05

7 Coordinate c 1 Coordinate c 2 -s

4 -s 1 Coordinate c 3 -s

5 -s 2 -s 6 -s 3

Fig. 5. The unit cells Niggli, and five copies were perturbed 10% orthogonally to S6 vector.

6. Summary

The transformation matrices shown above demonstrate the considerable regularity of Selling reduction, as used by Delaunay (1932), in comparison to Niggli reduction (Niggli, 1928). While all the boundaries of the non-positive orthant of S6 are essentially the same (and related by reflections), the boundaries formed in Niggli reduction are of multiple types, and the fundamental unit of the space representing Niggli-reduced cells (G6 , see Andrews & Bernstein (2014)) is non-convex. The matrices also display one of the unique properties of C3 , which makes it a useful conceptual representation. For the scalar being transformed, it and its opposite scalar create a unique row in each transformation matrix. These rows contain only the minus operator and zeros, highlighting the unique relationship of that pair of scalars. Several aspects of C3 are evident from inspecting the matrices. First, each boundary has four possible transformations that can be applied. Since each of the transforma- tions at boundaries are self-inverse, they are the same transformations that would be used in the process of cell (lattice) reduction. Delaunay (1932) and Delone et al. (1975) give only two choices, presumably for simplicity, omitting transformations that IUCr macros version 2.1.15: 2021/03/05

8 use the ”exchange” operator (see Andrews et al. (2019b)).

7. Availability of code

The C ++ code for C3 is available in github.com, in https://github.com/duck10/

LatticeRepLib.git.

Acknowledgements

Careful copy-editing and corrections by Frances C. Bernstein are gratefully acknowl-
edged. Our thanks to Jean Jakoncic and Alexei Soares for helpful conversations and
access to data and facilities at Brookhaven National Laboratory.

Funding information

Funding for this research was provided in part by: US Department of Energy Offices
of Biological and Environmental Research and of Basic Energy Sciences (grant No.
DE-AC02-98CH10886; grant No. E-SC0012704); U.S. National Institutes of Health
(grant No. P41RR012408; grant No. P41GM103473; grant No. P41GM111244; grant
No. R01GM117126, grant No. 1R21GM129570); Dectris, Ltd.

References

Andrews, L. C. & Bernstein, H. J. (2014). J. Appl. Cryst. 47(1), 346 – 359. Andrews, L. C., Bernstein, H. J. & Sauter, N. K. (2019a). Acta Crys. A75(1), 115–120. Andrews, L. C., Bernstein, H. J. & Sauter, N. K. (2019b). Acta Cryst. A75(3), 593–599. Bernstein, F. C., Koetzle, T. F., Williams, G. J. B., Meyer, Jr., E. F., Brice, M. D., Rodgers, J. R., Kennard, O., Shimanouchi, T. & Tasumi, M. (1977). J. Mol. Biol. 112, 535 – 542. Delaunay, B. N. (1932). Z. Krist. 84, 109 – 149. Delone, B. N., Galiulin, R. V. & Shtogrin, M. I. (1975). J. Sov. Math. 4(1), 79 – 156. Le Trong, I. & Stenkamp, R. E. (2007). Acta Cryst. D63, 548 – 549. Niggli, P., (1928). Krystallographische und Strukturtheoretische Grundbegriffe, Handbuch der Experimentalphysik, Vol. 7, part 1. Akademische Verlagsgesellschaft, Leipzig. Selling, E. (1874). Journal f¨ur die reine und angewandte Mathematik (Crelle’s Journal), 1874(77), 143 – 229.

Synopsis

The space C3 is explained in more detail than in the original description. Boundary transfor- mations of the fundamental unit are described in detail. A graphical presentation of the basic coordinates is described and illustrated. IUCr macros version 2.1.15: 2021/03/05

Coordinate c 1 -s 1 Coordinate c 2 -s

4

-s 2 -s 5 -s 3