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## G6 Bravais General Analysis of Lattices (BGAOL)

by
Lawrence C. Andrews, Micro Encoder Inc.,
Herbert J. Bernstein, Bernstein+Sons, yaya@bernstein-plus-sons.com

A program to determine cells "close" to given cell to help find the Bravais lattice of highest symmetry consistent with the submitted cell.

Select the crystal
lattice centering:
Specify the cell edge lengths and angles:
_cell.length_a _cell.angle_alpha
_cell.length_b _cell.angle_beta
_cell.length_c _cell.angle_gamma
Specify the cell edge length esd's and angle esd's:
_cell.length_a_esd _cell.angle_alpha_esd
_cell.length_b_esd _cell.angle_beta_esd
_cell.length_c_esd _cell.angle_gamma_esd

## NOTICE

You may redistribute this program under the terms of the GPL.

Alternatively you may redistribute the functions and subroutines of this program as an API under the terms of the LGPL.

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## Background

BGAOL is an updated version of the program ITERATE that finds cells that are "close" to the cell given, in order to help find the Bravais lattice of highest symmetry consistent with the submitted cell. A central problem in the solution of every crystal structure is to determine the correct Bravais lattice of the crystal. Many methods have been described for finding the correct Bravais lattice. ITERATE is based on the G6 space approach of Andrews and Bernstein [1] An important alternative is Zimmermann and Burzlaff's DELOS [3] based on Delaunay reduction. DELOS has no explicit distance metric. BGAOL is a major revision to ITERATE informed by the analysis of the 15 5-dimensional boundary polytopes of the Niggli reduced cell cone and the associated transformation matrices and projectors [2].

## BGAOL

Bravais General Analysis of Lattices (BGAOL) is a program written in Fortran that starts with a given experimentally determined cell and finds cells of all possible symmetries that are close enough to the starting cell to be worth considering as alternative. For each of the alternative, the International Tables Niggli Lattice Character is given as well as the Bravais lattice type. BGAOL replaces the existing program ITERATE, making use of a better understanding of the geometry of the space of Niggli-reduced cells [1].

Correct identification of the Bravais lattice of a crystal is an important early step in structure solution. Niggli reduction is a commonly used technique. In [2] we investigated the boundary polytopes of the Niggli-reduced cone in the six-dimensional space G6 by organized random probing of regions near 1-, 2-, 3-, 4-, 5-, 6-, 7- and 8-fold boundary polytope intersections. We limited our consideration of valid boundary polytopes to those avoiding the mathematically interesting but crystallographically impossible cases of zero length cell edges. Combinations of boundary polytopes without a valid intersection or with an intersection that would force a cell edge to zero or without neighboring probe points are eliminated. 216 boundary polytopes are found. There are 15 5-D boundary polytopes of the full G6 Niggli cone, 53 4-D boundary polytopes resulting from intersections of pairs of the 15 5-D boundary polytopes, 79 3-D boundary polytopes resulting from 2-fold, 3-fold and 4-fold intersections of the 15 5-D boundary polytopes, 55 2-D boundary polytopes resulting from 2-fold, 3-fold, 4-fold and higher intersections of the 15 5-D boundary polytopes, 14 1-D boundary polytopes resulting from 3-fold and higher intersections of the 15 5-D boundary polytopes. The classification of the boundary polytopes into 5-, 4-, 3-, 2- and 1-dimensional boundary polytopes corresponds well to the Niggli classification and suggests other possible symmetries.

## The Fifteen 5-D Niggli-cone Boundaries

All of the primitive lattice types can be represented as combinations of the 15 5-D boundary polytopes. All of the non-primitive lattice types can be represented as combinations of the 15 5-D boundary polytopes and of the 7 special-position subspaces of the 5-D boundary polytopes. This study provided a new, simpler and arguably more intuitive basis set for the classification of lattice characters and helped to illuminate some of the complexities in Bravais lattice identification, allowing for a new embedding-based distance calculation in BGAOL and helping to prune the tree of alternate cells to be considered. The same embedding-based distance calculation is a promising tool for database searches. Below are the fifteen 5-D boundary polytopes of Niggli-reduced cells in G6. Boundary polytopes 1, 2, 3, 4, 5, 7, A, D and F each have special position subspaces containing cells that are mapped onto themselves by the Niggli-reduction transform of the specified boundary polytope. The special-position subspaces are identified by the conditions to be added to the conditions that define the boundary polytope itself. For a given boundary polytope Γ, the column "Condition" gives the G6 constraints (prior to closure) of the boundary polytope. When taken with the "Special-Position Subspace" constraint in the last column, the result is the entirety of the special-position subspace $\html'Γ'↖\html'^'$. The Special-Position Subspace'' constraint by itself is Γ'. Boundary polytopes 1 and 2 apply in both the all acute ($+ + +$) and all obtuse ($- - -$) branches of the Niggli-reduced cone. Boundary polytopes 8, B, E and F are restricted to the all obtuse ($- - -$) branch of the Niggli-reduced cone, N. Boundary polytopes 6, 7, 9, A, C and D are restricted to the all acute ($+ + +$) branch of N. While the boundary polytopes 3, 4 and 5 are boundaries of both the all acute ($+ + +$) and all obtuse ($- - -$) branches, the common special position subspace of those polytopes is just $g_4 = g_5 = g_6 = 0$ which is part of the ($- - -$) branch.
ClassBoundaryConditionSpecial-Position Subspace
Equal cell edges1all$g_1 = g_2$$g_4 = g_5 2allg_2 = g_3$$g_5 = g_6$
Ninety degrees3all$g_4 = 0$$g_5 = g_6 = 0 4allg_5 = 0$$g_4 = g_6 = 0$
5all$g_6 = 0$$g_4 = g_5 = 0 Face diagonal6+ + +g_2 = g_4 and g_5 \html'≥' g_6(none) 7+ + +g_2 = g_4 and g_5 < g_6$$g_5 = g_6/2$
8- - -$g_2 = -g_4$(none)
9+ + +$g_1 = g_5$ and $g_4 \html'≥' g_6$(none)
A+ + +$g_1 = g_5$ and $g_4 < g_6$$g_4 = g_6/2 B- - -g_1 = -g_5(none) C+ + +g_1 = g_6 and g_4 \html'≥' g_5(none) D+ + +g_1 = g_6 and g_4 < g_5$$g_4 = g_5/2$
E - - -$g_1 = - g_6$(none)
Body diagonalF- - -$g_1+g_2+g_3+g_4+g_5+ g_6 = g_3$ $g_1-g_2-g_4+g_5=0$

## Niggli Lattice Characters in Terms of the Niggli Boundaries

Roof/NiggliSymbol ITLatticeChar BravaisLatticeType G6Subspace G6BoundaryPolytope Roof/NiggliSymbol ITLatticeChar BravaisLatticeType G6Subspace G6BoundaryPolytope 44A 3 $cP$ $(r,r,r,0,0,0)$ $12345 = 12{3↖\text'^'} = 12{4↖\text'^'} = 12{5↖\text'^'}$ 51A 16 $oF$ $(r,r,s,-t,-t,-2r+2t)$ $1F1' = {1↖\text'^'}F$ 44C 1 $cF$ $(r,r,r,r,r,r)$ 12679ACD 51B 26 $oF$ $(r,s,t,r/2,r,r)$ $ADA' = {A↖\text'^'}D$ 44B 5 $cI$ $(r,r,r,-2r/3,-2r/3,-2r/3)$ $12F2'F' = 1{2↖\text'^'}{F↖\text'^'}$ 52A 8 $oI$ $(r,r,r,-s,-t,-2r+s+t)$ 12F 45A 11 $tP$ $(r,r,s,0,0,0)$ $1345 = 1{3↖\text'^'} = 1{4↖\text'^'} = 1{5↖\text'^'}$ 52B 19 $oI$ $(r,s,s,t,r,r)$ 29C = 2AD 45B 21 $tP$ $(r,s,s,0,0,0)$ $2345 = 2{3↖\text'^'} = 2{4↖\text'^'} = 2{5↖\text'^'}$ 52C 42 $oI$ $(r,s,t,-s,-r,0)$ 58BF 45D 6 $tI$ $(r,r,r,-r+s,-r+s,-2s)$ $12FF' = 12{F↖\text'^'}$ 53A 33 $mP$ $(r,s,t,0,-u,0)$ 35 45D 7 $tI$ $(r,r,r,-2s,-r+s,-r+s)$ $12F2' = 1{2↖\text'^'}F$ 53B 35 $mP$ $(r,s,t,-u,0,0)$ 45 45C 15 $tI$ $(r,r,s,-r,-r,0)$ 158BF 53C 34 $mP$ $(r,s,t,0,0,-u)$ 34 45E 18 $tI$ $(r,s,s,r/2,r,r)$ $2ADA' = 2{{A↖\text'^'}}D$ 57B 17 $mI$ $(r,r,s,-t,-u,-2r+t+u)$ 1F 48A 12 $hP$ $(r,r,s,0,0,-r)$ 134E 57C 27 $mI$ $(r,s,t,u,r,r)$ 9C = AD 48B 22 $hP$ $(r,s,s,-s,0,0)$ 2458 57A 43 $mI$ $(r,s,t,-s+u,-r+u,-2u)$ $FF' = {F↖\text'^'}$ 49C 2 $hR$ $(r,r,r,s,s,s)$ $121'2' = {1↖\text'^'}{2↖\text'^'}$ 55A 10 $mC$ $(r,r,s,t,t,u)$ $11' = {1↖\text'^'}$ 49D 4 $hR$ $(r,r,r,-s,-s,-s)$ $121'2' = {1↖\text'^'}{2↖\text'^'}$ 55A 14 $mC$ $(r,r,s,-t,-t,-u)$ $11' = {1↖\text'^'}$ 49B 9 $hR$ $(r,r,s,r,r,r)$ 1679ACD 55B 20 $mC$ $(r,s,s,t,u,u)$ $22' = {2↖\text'^'}$ 49E 24 $hR$ $(r,s,s,-s+r/3,-2r/3,-2r/3)$ $2F2'F' = {2↖\text'^'}{F↖\text'^'}$ 55B 25 $mC$ $(r,s,s,-t,-u,-u)$ $22' = {2↖\text'^'}$ 50C 32 $oP$ $(r,s,t,0,0,0)$ $345 = {3↖\text'^'} = {4↖\text'^'} = {5↖\text'^'}$ 56A 28 $mC$ $(r,s,t,u,r,2u)$ $AA' = {A↖\text'^'}$ 50D 13 $oC$ $(r,r,s,0,0,-t)$ 134 56C 29 $mC$ $(r,s,t,u,2u,r)$ $DD' = {D↖\text'^'}$ 50E 23 $oC$ $(r,s,s,-t,0,0)$ 245 56B 30 $mC$ $(r,s,t,s,u,2u)$ $77' = {7↖\text'^'}$ 50A 36 $oC$ $(r,s,t,0,-r,0)$ 35B 54C 37 $mC$ $(r,s,t,-u,-r,0)$ 5B 50B 38 $oC$ $(r,s,t,0,0,-r)$ 34E 54A 39 $mC$ $(r,s,t,-u,0,-r)$ 4E 50F 40 $oC$ $(r,s,t,-s,0,0)$ 458 54B 41 $mC$ $(r,s,t,-s,-u,0)$ 58

## How BGAOL Works

BGAOL starts with a probe cell $g$ in G6 and projects it onto each of the 15 boundaries, keeping the projected images that lie within the error bounding box around the probe and within the Niggli cone. In this case, 2 boundaries are shown, which we call $\text'Ω'$ and $\text'Θ'$. The higher symmetry boundary $\text'ΩΘ'$ formed by the intersection of $\text'Ω'$ and $\text'Θ'$ happens to lie outside of the error bounding box. However for each of the cell projections it finds that are within the error bounding box, BGAOL applies the transformation associated with the boundary, in this case $M_{\text'Ω'}$, and keeps the resulting cell $M_{\text'Ω'}(P_{\text'Ω'}(g))$ if it is nearly reduced. Non-duplicate cells are added to the list until no more are found, and then each cell is tested by projection for its distance from each Niggli lattice character. The distance is computed as if working within the Niggli cone embedded in a higher dimensional space, so that the distance from, say, $P_{\text'Ω'}(g)$ to $M_{\text'Ω'}(P_{\text'Ω'}(g))$ is treated as zero. Thus, in the example shown, even though $\text'ΩΘ'$ is outside the error bounding box, using the embedding distance, it is sufficiently close to $M_{\text'Ω'}(P_{\text'Ω'}(g))$ for it to be accepted as a candidate.

## Note on the Matches Reported

The program on this Web page implements a search in G6 for the various Bravais lattices that the user's cell may fit. For each lattice type, the best metric match is reported. If the higher symmetry type is actually correct, then that is likely to be the best cell from which to start further refinement. However, the possibility exists that one of the rejected cells (which did not match as well) was actually the correct one to use. The reason for this ambiguity is experimental error and its propagation in the transformations of the lattices in the program. Fortunately, the rejected cells are usually quite similar to the accepted one.

A note on standard deviations: First, even in the best of circumstances, standard deviations of unit cell dimensions from 4-circle diffractometer data are always underestimated (by at least a factor of 2). In addition, the points chosen for the determination are often not well distributed (for example all in the first octant of orthorhombic lattices). These less than optimal choices cause substantial systematic error. The experimental errors are amplified in the mathematical conversions between various lattices that any lattice search program must perform. It is not a rare occurrence for angles to be incorrect by 0.5 degrees in initial unit cell determinations.

Note: Even in most well determined unit cells, the actual errors in the edge lengths is 0.2 to 0.5 parts per thousand. (Note that reproducibility of the measurements is substantially better, leading to the illusion that diffractometers produce excellent unit cell parameters). Use of standard deviations that are too small is a common reason for failure of Bravais lattice searches. For small molecules, 0.1 Angstroms is a reasonable error for the edge lengths, for proteins, 0.4 to 0.5 (or even more for preliminary measurements). Accurate unit cell parameters must by determined by a number of more complex methods and must include extrapolation to remove systematic effects. For an excellent summary, see "Xray Structure Determination", G.H.Stout and L.H.Jensen, Wiley, 1989.

## References

[1] L. C. Andrews and H. J. Bernstein. Lattices and reduced cells as points in 6-space and selection of Bravais lattice type by projections. Acta Crystallogr., A44:10091018, 1988.
[2] L. C. Andrews and H. J. Bernstein. The Geometry of Niggli Reduction. arXiv, 1203.5146v1 [math-ph], 2012. arxiv.org/abs/1203.5146.
[3] H. Zimmermann and H. Burzlaff. DELOS A computer program for the determination of a unique conventional cell. Zeitschrift fu r Kristallographie, 170:241 246, 1985.

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Updated 21 July 2012.