Complete set
Abstract
Matrix methods for metric symmetry determination are fast, efficient, reliable, and, in contrast to reduction techniques, allow to establish simply all possible pseudo-symmetries in the vicinity of higher symmetry borders. It is shown that distances to borders may be characterized by one or a few monoaxial deformations measured by parameter ε, which corresponds to the relative change in the interplanar distance. The scope of this chapter is limited to a careful analysis of rhombohedral or monoclinic deformations occurring in hR lattices.
Keywords
- semi-reduced lattices
- lattice symmetry
- Bravais type border
- lattice deformation
1. Introduction
Chemical species are structurally classified by symmetry. The preliminary classification takes into account only translational properties, the
Classifications of unique lattice representatives obtained by the
2. Semi-reduced lattice descriptions
The concept of a semi-reduced lattice description (s.r.d.) has been given elsewhere [9]. The emphasis on the crystallographic features of lattices was obtained by shifting the focus (i) from the analysis of a lattice metric to the analysis of symmetry matrices [6], (ii) from the geometric interpretation of isometric transformation based on invariant subspaces to the orthogonality concept [7] extended to splitting indices [8], (iii) and from predefined cell transformations to transformations derivable via geometric information [6, 7]. It was shown that both corresponding arithmetic and geometric holohedries share the space distribution of symmetry elements and thus simplify the crystallographic description of structural phase transitions, especially those observed with the use of powder diffraction. Moreover, the completeness of s.r.d. types revealed a combinatorial structure of
The main result of introduced semi-reduced lattice representations consists in the extension of the famous characterization of Bravais lattices according to their metrical, algebraic, and geometric properties onto a wide class of primitive, less restrictive lattices (including Niggli-reduced, Buerger-reduced, nearly Buerger-reduced, and a substantial part of Delaunay-reduced). While the
and the subsequent geometric interpretation of the filtered matrices leads to mathematically stable and rich information on the individual transformation bringing the lattice into coincidence with itself (known as an
where Δ
It is obvious that symmetry operations fulfill the closure, associative, identity, and inverse axioms and form a group: an
Lattice | Metric | Lattice | Metric | Lattice | Metric | Lattice | Metric |
---|---|---|---|---|---|---|---|
2,2,1,0,0,−1 | 2,2,1,0,0,1 | 2,2,2,0,−1,−1 | 4,3,3,1,2,2 | ||||
2,1,2,0,−1,0 | 2,1,2,0,1,0 | 2,2,2,1,1,0 | 3,3,4,−2,−2,1 | ||||
1,2,2,−1,0,0 | 1,2,2,1,0,0 | 2,2,2,1,0,1 | 3,4,3,−2,1,−2 | ||||
1,1,1,0,0,0 | 2,2,2,0,1,1 | 4,3,3,1,−2,−2 | |||||
2,2,2,1,1,1 | 3,3,3,−1,−1,−1 | 2,2,2,1,−1,0 | 3,3,4,−2,2,−1 | ||||
2,2,2,−1,−1,1 | 3,3,3,1,1,−1 | 2,2,2,1,0,−1 | 3,4,3,−2,−1,2 | ||||
2,2,2,−1,1,−1 | 3,3,3,1,−1,1 | 2,2,2,0,1,−1 | 4,3,3,−1,−2,2 | ||||
2,2,2,1,−1,−1 | 3,3,3,−1,1,1 | 2,2,2,−1,1,0 | 3,3,4,2,−2,−1 | ||||
2,2,2,−1,−1,0 | 3,3,4,2,2,1 | 2,2,2,−1,0,1 | 3,4,3,2,−1,−2 | ||||
2,2,2,−1,0,−1 | 3,4,3,2,1,2 | 2,2,2,0,−1,1 | 4,3,3,−1,2,−2 |
In the s.r.d. approach, the primitive-to-Bravais transformations are not stored, but dynamically constructed, based on the geometric interpretation of symmetry matrices. Unfortunately, the classical symbol of a point or space symmetry operation bears information on an operation type and a 1D subspace (or 2D in the case of symmetry planes) of points invariant under this operation [10], but the information on the complement orthogonal subspace, invariant as a whole, is lost. In the developed
modifies only 1D subspace and in consequence retains the symmetry axis in [
3. Rhombohedral lattices in s.r.d.
It is difficult to classify or compare lattices that drastically change their class-dependent descriptions as a result of small deformations, structural phase transitions, or experimental errors. Such discontinuities in the Niggli-reduced space can be overcome by a deep mathematical treatment like in [11] or by applying a less restrictive method of Bravais cell assignment: Niggli reduction Delaunay reduction s.r.d. A wide class of lattices, including a trigonal and three cubic lattices, is considered here as ‘rhombohedral’ lattices. The actual form of a cell has no meaning, but a given lattice can be represented by a rhombohedron with equal sides
The angle
No. | cos(α) | α[°] | Description |
---|---|---|---|
1 | 1 | 0 | 1D |
2 | 1/2 | 60 | cF |
3 | 1/4 | 75.5225 | c/a = √(3) |
4 | 0 | 90 | cP |
5 | −1/8 | 97.1808 | c/a = √(3/3) |
6 | −1/4 | 104.4775 | c/a = √(3/5) |
7 | −1/3 | 109.4712 | cI |
8 | −1/2 | 120 | 2D |
Information contained in both Figure 1 and Table 2 explains discontinuities in descriptions of rhombohedral lattices. Descriptions of Niggli- or Buerger-reduced lattices must be changed during crossing characteristic angles 60° and 109.47°, since they are based on the shortest non-coplanar lattice vectors. Similarly, Bravais descriptions should reflect the increased symmetry for these angles (directions <001> reveal extra twofold and threefold symmetry). In sharp contrast to the above lattice representations, no drastic changes is necessary in semi-reduced descriptions of rhombohedral lattices, without losing relation with Bravais standardization.
4. Distance to the higher symmetry border: ε concept
In crystallography, it is crucial to standardize lattice descriptions and to assign one from the fourteen 3D Bravais types differentiated by symmetry. The process is straightforward for good quality data and faraway from the Bravais borders but in opposite cases, especially in the presence of unavoidable experimental errors, the solution cannot be unique. Usable distances should be defined to rank positive candidates. Most considerations about the calculation of such distances are devoted to the Niggli reduction, for example, see [11] and references contained therein; only some discuss the Buerger reduction [1, 7].
The geometric properties of matrices that transform an s.r.d. lattice into itself are utilized in the presented approach to the greatest degree, which form the
Δa/a% | Δb/b% | Δc/c% | Δα° | Δβ° | Δγ° | δ° | Operation |
---|---|---|---|---|---|---|---|
0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 1[]() |
0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 2[010](121) |
0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 2[1–10](1–10) |
0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 2[100](211) |
0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 3 + [−1–13](001) |
0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 3-[−1–13](001) |
0.00 | 2.47 | −2.41 | 0.00 | −0.79 | 0.79 | 1.95 | 2[01–1](01–1) |
2.47 | 2.47 | 0.00 | −2.38 | −2.38 | −1.59 | 1.95 | 2[001](112) |
2.47 | 0.00 | −2.41 | −0.79 | 0.00 | 0.79 | 1.95 | 2[−101](−1–1) |
0.00 | 2.47 | −2.41 | 0.00 | −0.79 | 0.79 | 1.95 | 2[−111](011) |
2.47 | 0.00 | −2.41 | −0.79 | 0.00 | 0.79 | 1.95 | 2[1–11](101) |
2.47 | 2.47 | 0.00 | −2.38 | −2.38 | −1.59 | 1.95 | 2[11–1](110) |
0.00 | 2.47 | −2.41 | 0.00 | −0.79 | 0.79 | 1.30 | 3 + [111](111) |
2.47 | 0.00 | −2.41 | −0.79 | 0.00 | 0.79 | 1.30 | 3-[111](111) |
0.00 | 2.47 | −2.41 | 0.00 | −0.79 | 0.79 | 1.30 | 3+ [3-1-1](100) |
2.47 | 2.47 | 0.00 | −2.38 | −2.38 | −1.59 | 1.30 | 3-[3-1-1](100) |
2.47 | 2.47 | 0.00 | −2.38 | −2.38 | −1.59 | 1.30 | 3 + [−13–1](010) |
2.47 | 0.00 | −2.41 | −0.79 | 0.00 | 0.79 | 1.30 | 3-[−13–1](010) |
2.47 | 0.00 | −2.41 | −0.79 | 0.00 | 0.79 | 1.95 | 4 + [−111](011) |
2.47 | 2.47 | 0.00 | −2.38 | −2.38 | −1.59 | 1.95 | 4-[−111](011) |
2.47 | 2.47 | 0.00 | −2.38 | −2.38 | −1.59 | 1.95 | 4 + (1–11](101) |
0.00 | 2.47 | −2.41 | 0.00 | −0.79 | 0.79 | 1.95 | 4-[1-11](101) |
2.47 | 0.00 | −2.41 | −0.79 | 0.00 | 0.79 | 1.95 | 4+ [11-1](110) |
0.00 | 2.47 | −2.41 | 0.00 | −0.79 | 0.79 | 1.95 | 4-[11-1](110) |
The filtering of symmetry matrices near cubic borders results in a rather big number (7 × 24) of quantitative data. As Table 3 shows, deviations are interrelated, not random. A maximal unsigned deviation well reflects this situation. Moreover, strict
It is clear from Table 1 that
(2, 2, 2, 1, 1, 1) = (2, 2, 2.1, 1, 1, 1) + ε(0, 0, 1, 0, 0, 0)
with the solution
The sign of
For rhombohedral lattices, two kinds of ε distances to the border (based on rhombohedral or monoclinic deformations) are generally analyzed. In more complicated cases, like cubic lattices modified by simultaneous rhombohedral and tetragonal distortions, few ε distances can be derived. Calculations are also possible in the presence of experimental errors, if they are smaller than distortions.
The concept of a quantitative measure between the probe cell and cells with higher symmetry based on monoaxial deformations is thus outlined, but for practical applications this idea should be thoroughly investigated in s.r.d. This study provides analyses and two real-life examples limited to rhombohedral lattices.
5. Distances between hR and cubic lattices
In the case being considered, the semi-reduced
5.1. hR -cF border
Let us have
deformation | deformation | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2.1 | 2.1 | 2.1 | 1.1 | 1.1 | 1.1 | −0.1·(111) | 2.1 | 2.1 | 2 | 1 | 0 | 1.1 | −0.1·(110) |
2 | 2 | 2.1 | 1 | 1 | 1 | −0.1·(001) | 2 | 2.1 | 2.1 | 1.1 | 0 | 1 | −0.1·(011) |
2 | 2.1 | 2 | 1 | 1 | 1 | −0.1·(010) | 2 | 2 | 2.1 | 1 | 0 | 1 | −0.1·(001) |
2.1 | 2 | 2 | 1 | 1 | 1 | −0.1·(100) | 2.1 | 2 | 2 | 1 | 0 | 1 | −0.1·(100) |
2.1 | 2.1 | 2.1 | −1 | −1 | 1.1 | −0.1·(−1–11) | 2.1 | 2.1 | 2 | 0 | 1 | 1.1 | −0.1·(110) |
2 | 2 | 2.1 | −1 | −1 | 1 | −0.1·(001) | 2.1 | 2 | 2.1 | 0 | 1.1 | 1 | −0.1·(101) |
2 | 2.1 | 2 | −1 | −1 | 1 | −0.1·(010) | 2 | 2.1 | 2 | 0 | 1 | 1 | −0.1·(010) |
2.1 | 2 | 2 | −1 | −1 | 1 | −0.1·(100) | 2 | 2 | 2.1 | 0 | 1 | 1 | −0.1·(001) |
2.1 | 2.1 | 2.1 | −1 | 1.1 | −1 | −0.1·(1–1 − 1) | 2 | 2.1 | 2.1 | 1.1 | -1 | 0 | −0.1·(011) |
2 | 2 | 2.1 | −1 | 1 | −1 | −0.1·(001) | 2.1 | 2 | 2.1 | 1 | −1.1 | 0 | −0.1·(−101) |
2 | 2.1 | 2 | −1 | 1 | −1 | −0.1·(010) | 2 | 2.1 | 2 | 1 | −1 | 0 | −0.1·(010) |
2.1 | 2 | 2 | −1 | 1 | −1 | −0.1·(100) | 2.1 | 2 | 2 | 1 | −1 | 0 | −0.1·(100) |
2.1 | 2.1 | 2.1 | 1.1 | −1 | −1 | −0.1·(−11–1) | 2 | 2.1 | 2.1 | 1.1 | 0 | −1 | −0.1·(011) |
2 | 2 | 2.1 | 1 | −1 | −1 | −0.1·(001) | 2.1 | 2.1 | 2 | 1 | 0 | −1.1 | −0.1·(1–10) |
2 | 2.1 | 2 | 1 | −1 | −1 | −0.1·(100) | 2.1 | 2 | 2 | 1 | 0 | −1 | −0.1·(100) |
2.1 | 2 | 2 | 1 | −1 | −1 | −0.1·(010) | 2 | 2 | 2.1 | 1 | 0 | −1 | −0.1·(001) |
2 | 2.1 | 2.1 | −1 | −1 | 0 | −0.1·(01–1) | 2.1 | 2 | 2.1 | 0 | 1.1 | −1 | −0.1·(101) |
2.1 | 2 | 2.1 | −1 | −1 | 0 | −0.1·(−101) | 2.1 | 2.1 | 2 | 0 | 1 | −1.1 | −0.1·(1–10) |
2 | 2.1 | 2 | −1 | −1 | 0 | −0.1·(010) | 2 | 2.1 | 2 | 0 | 1 | −1 | −0.1·(010) |
2.1 | 2 | 2 | −1 | −1 | 0 | −0.1·(100) | 2 | 2 | 2.1 | 0 | 1 | −1 | −0.1·(001) |
2 | 2.1 | 2.1 | −1 | 0 | −1 | −0.1·(01–1) | 2.1 | 2 | 2.1 | −1 | 1.1 | 0 | −0.1·(101) |
2.1 | 2.1 | 2 | −1 | 0 | −1 | −0.1·(1–10) | 2 | 2.1 | 2.1 | −1.1 | 1 | 0 | −0.1·(01–1) |
2.1 | 2 | 2 | −1 | 0 | −1 | −0.1·(001) | 2 | 2.1 | 2 | −1 | 1 | 0 | −0.1·(010) |
2 | 2 | 2.1 | −1 | 0 | −1 | −0.1·(100) | 2.1 | 2 | 2 | −1 | 1 | 0 | −0.1·(100) |
2.1 | 2.1 | 2 | 0 | −1 | −1 | −0.1·(1–10) | 2.1 | 2.1 | 2 | −1 | 0 | 1.1 | −0.1·(110) |
2.1 | 2 | 2.1 | 0 | −1 | −1 | −0.1·(−101) | 2 | 2.1 | 2.1 | −1.1 | 0 | 1 | −0.1·(01–1) |
2 | 2.1 | 2 | 0 | −1 | −1 | −0.1·(010) | 2.1 | 2 | 2 | −1 | 0 | 1 | −0.1·(100) |
2 | 2 | 2.1 | 0 | −1 | −1 | −0.1·(001) | 2 | 2 | 2.1 | −1 | 0 | 1 | −0.1·(001) |
2 | 2.1 | 2.1 | 1.1 | 1 | 0 | −0.1·(011) | 2.1 | 2.1 | 2 | 0 | −1 | 1.1 | −0.1·(110) |
2.1 | 2 | 2.1 | 1 | 1.1 | 0 | −0.1·(101) | 2.1 | 2 | 2.1 | 0 | −1.1 | 1 | −0.1·(−101) |
2.1 | 2 | 2 | 1 | 1 | 0 | −0.1·(100) | 2 | 2.1 | 2 | 0 | −1 | 1 | −0.1·(010) |
2 | 2.1 | 2 | 1 | 1 | 0 | −0.1·(010) | 2 | 2 | 2.1 | 0 | −1 | 1 | −0.1·(001) |
The interpretation of 4 × 16 items in Table 4 is very easy due to the fact that Miller indices of planes perpendicular to the unique threefold axis are given explicitly in the deformation symbols. In the considered situation, the operation on
(2.1, 2, 2.1, 0, −1.1, 1) - 0.1·(−1·-1, 0·0, 1·1, 1·0, 1·-1, −1·0) = (2, 2, 2, 0, −1, 1)
(2, 2.1, 2, 0. -1, 1) - 0.1·(0·0, 1·1, 0·0, 0·1, 0·0, 0·1) = (2, 2, 2, 0, −1, 1)
(2, 2, 2.1, 0, −1, 1) - 0.1·(0·0, 0·0, 1·1, 1·0, 1·0, 0·1) = (2, 2, 2, 0, −1, 1)
Assigning the symmetry group to the final
5.2. hR -cI border
The
hR metric | deformation | hR metric | deformation | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
3.1 | 3.1 | 3.1 | −1 | −1 | −1 | −0.1·(111) | 3.1 | 4.4 | 3.9 | −2.6 | 1.3 | −2.2 | −0.1·(1–23) |
3.1 | 3.9 | 3.1 | −1 | −1 | −1 | −0.1·(−13–1) | 3.1 | 4.4 | 3.1 | −1.8 | 0.9 | −2.2 | −0.1·(−121) |
3.1 | 3.1 | 3.9 | −1 | −1 | −1 | −0.1·(−1–13) | 3.1 | 4.4 | 3.1 | −2.2 | 0.9 | −1.8 | −0.1·(12–1) |
3.9 | 3.1 | 3.1 | −1 | −1 | −1 | −0.1·(3–1-1) | 3.9 | 4.4 | 3.1 | −2.2 | 1.3 | −2.6 | −0.1·(3–21) |
3.1 | 3.1 | 3.1 | 0.9 | 0.9 | −1 | −0.1·(−1–11) | 4.4 | 3.9 | 3.1 | 1.3 | −2.2 | −2.6 | −0.1·(−231) |
3.1 | 3.1 | 3.9 | 1.3 | 1.3 | −1 | −0.1·(113) | 4.4 | 3.1 | 3.9 | 1.3 | −2.6 | −2.2 | −0.1·(−213) |
3.1 | 3.9 | 3.1 | 1.3 | 0.9 | −1 | −0.1·(−131) | 4.4 | 3.1 | 3.1 | 0.9 | −2.2 | −1.8 | −0.1·(21–1) |
3.9 | 3.1 | 3.1 | 0.9 | 1.3 | −1 | −0.1·(3–11) | 4.4 | 3.1 | 3.1 | 0.9 | −1.8 | −2.2 | −0.1·(2–11) |
3.1 | 3.1 | 3.1 | 0.9 | −1 | 0.9 | −0.1·(−11–1) | 3.1 | 3.1 | 4.4 | −1.8 | 2.2 | −0.9 | −0.1·(112) |
3.1 | 3.1 | 3.9 | 1.3 | −1 | 0.9 | −0.1·(−113) | 3.1 | 3.1 | 4.4 | −2.2 | 1.8 | −0.9 | −0.1·(−1–12) |
3.1 | 3.9 | 3.1 | 1.3 | −1 | 1.3 | −0.1·(131) | 3.1 | 3.9 | 4.4 | −2.6 | 2.2 | −1.3 | −0.1·(−13–2) |
3.9 | 3.1 | 3.1 | 0.9 | −1 | 1.3 | −0.1·(31–1) | 3.9 | 3.1 | 4.4 | −2.2 | 2.6 | −1.3 | −0.1·(3–12) |
3.1 | 3.1 | 3.1 | −1 | 0.9 | 0.9 | −0.1·(1–1-1) | 3.1 | 4.4 | 3.9 | −2.6 | −1.3 | 2.2 | −0.1·(−1–23) |
3.1 | 3.1 | 3.9 | −1 | 1.3 | 0.9 | −0.1·(1–13) | 3.1 | 4.4 | 3.1 | −1.8 | −0.9 | 2.2 | −0.1·(121) |
3.1 | 3.9 | 3.1 | −1 | 0.9 | 1.3 | −0.1·(13–1) | 3.1 | 4.4 | 3.1 | −2.2 | −0.9 | 1.8 | −0.1·(1–21) |
3.9 | 3.1 | 3.1 | −1 | 1.3 | 1.3 | −0.1·(311) | 3.9 | 4.4 | 3.1 | −2.2 | −1.3 | 2.6 | −0.1·(32–1) |
3.1 | 3.1 | 4.4 | 2.2 | 1.8 | 0.9 | −0.1·(−112) | 4.4 | 3.1 | 3.9 | −1.3 | −2.6 | 2.2 | −0.1·(−2–13) |
3.1 | 3.9 | 4.4 | 2.6 | 2.2 | 1.3 | −0.1·(132) | 4.4 | 3.9 | 3.1 | −1.3 | −2.2 | 2.6 | −0.1·(23–1) |
3.1 | 3.1 | 4.4 | 1.8 | 2.2 | 0.9 | −0.1·(−11–2) | 4.4 | 3.1 | 3.1 | −0.9 | −1.8 | 2.2 | −0.1·(211) |
3.9 | 3.1 | 4.4 | 2.2 | 2.6 | 1.3 | −0.1·(312) | 4.4 | 3.1 | 3.1 | −0.9 | −2.2 | 1.8 | −0.1·(2–1-1) |
3.1 | 4.4 | 3.9 | 2.6 | 1.3 | 2.2 | −0.1·(123) | 3.1 | 3.9 | 4.4 | 2.6 | −2.2 | −1.3 | −0.1·(−132) |
3.1 | 4.4 | 3.1 | 2.2 | 0.9 | 1.8 | −0.1·(1–2-1) | 3.1 | 3.1 | 4.4 | 1.8 | −2.2 | −0.9 | −0.1·(11–2) |
3.1 | 4.4 | 3.1 | 1.8 | 0.9 | 2.2 | −0.1·(12–1) | 3.1 | 3.1 | 4.4 | 2.2 | −1.8 | −0.9 | −0.1·(112) |
3.9 | 4.4 | 3.1 | 2.2 | 1.3 | 2.6 | −0.1·(321) | 3.9 | 3.1 | 4.4 | 2.2 | −2.6 | −1.3 | −0.1·(3–1-2) |
4.4 | 3.1 | 3.9 | 1.3 | 2.6 | 2.2 | −0.1·(213) | 3.1 | 4.4 | 3.1 | 1.8 | −0.9 | −2.2 | −0.1·(−12–1) |
4.4 | 3.1 | 3.1 | 0.9 | 2.2 | 1.8 | −0.1·(2–11) | 3.1 | 4.4 | 3.9 | 2.6 | −1.3 | −2.2 | −0.1·(−123) |
4.4 | 3.1 | 3.1 | 0.9 | 1.8 | 2.2 | −0.1·(−2–11) | 3.1 | 4.4 | 3.1 | 2.2 | −0.9 | −1.8 | −0.1·(121) |
4.4 | 3.9 | 3.1 | 1.3 | 2.2 | 2.6 | −0.1·(231) | 3.9 | 4.4 | 3.1 | 2.2 | −1.3 | −2.6 | −0.1·(3–2-1) |
3.1 | 3.1 | 4.4 | −2 | −2 | 0.9 | −0.1·(−112) | 4.4 | 3.1 | 3.9 | −1.3 | 2.6 | −2.2 | −0.1·(2–13) |
3.1 | 3.9 | 4.4 | −3 | −2 | 1.3 | −0.1·(13–2) | 4.4 | 3.9 | 3.1 | −1.3 | 2.2 | −2.6 | −0.1·(−23–1) |
3.1 | 3.1 | 4.4 | −2 | −2 | 0.9 | −0.1·(−112) | 4.4 | 3.1 | 3.1 | −0.9 | 1.8 | −2.2 | −0.1·(−211) |
3.9 | 3.1 | 4.4 | −2 | −3 | 1.3 | −0.1·(31–2) | 4.4 | 3.1 | 3.1 | −0.9 | 2.2 | −1.8 | −0.1·(211) |
The last three lines give:
(4.4, 3.9, 3.1, −1.3, 2.2, −2.6) − 0.1·(−2·−2, 3·3, −1·−1, −1·3, −1·−2, −2·3) = (4, 3, 3, −1, 2, −2)
(4.4, 3.1, 3.1, −0.9, 1.8, −2.2) − 0.1·(−2·−2, 1·1, 1·1, 1·1, 1·−2, −2·1) = (4, 3, 3, −1, 2, −2)
(4.4, 3.1, 3.1, −0.9, 2.2, −1.8) − 0.1·(2·2, 1·1, 1·1, 1·1, 1·2, 2·1) = (4, 3, 3, −1, 2, −2)
The assigning of a symmetry group to a modified metric or comparing it with Table 1 reveals
The presence of random errors complicates the derivation of ε and Δ
(4.41, 3.08, 3.12, −0,98, 2,23, −1.9) + ε·(2·2, 1·1, 1·1, 1·1, 1·2, 2·1) = (4, 3, 3, −1, 2, −2)
gives ε = −0,093, which can be considered as a rather interesting result.
5.3. hR -cP border
To all cells contained in Tables 4,5 exact
Symbol | Symbol | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | 1 | 2 | −1 | 0 | 0 | 1 | 1 | 2 | −1 | 0 | 0 | ||
1 | 1 | 2 | 0 | −1 | 0 | 1 | 1 | 2 | 0 | −1 | 0 | ||
1 | 1 | 2 | 0 | 1 | 0 | 1 | 1 | 2 | 0 | 1 | 0 | ||
1 | 1 | 2 | 1 | 0 | 0 | 1 | 1 | 2 | 1 | 0 | 0 | ||
1 | 2 | 1 | −1 | 0 | 0 | 1 | 2 | 1 | −1 | 0 | 0 | ||
1 | 2 | 1 | 1 | 0 | 0 | 1 | 2 | 1 | 0 | 0 | −1 | ||
1 | 2 | 1 | 0 | 0 | −1 | 1 | 2 | 1 | 0 | 0 | 1 | ||
1 | 2 | 1 | 0 | 0 | 1 | 1 | 2 | 1 | 1 | 0 | 0 | ||
2 | 1 | 1 | 0 | 1 | 0 | 2 | 1 | 1 | 0 | −1 | 0 | ||
2 | 1 | 1 | 0 | 0 | 1 | 2 | 1 | 1 | 0 | 0 | −1 | ||
2 | 1 | 1 | 0 | 0 | −1 | 2 | 1 | 1 | 0 | 0 | 1 | ||
2 | 1 | 1 | 0 | −1 | 0 | 2 | 1 | 1 | 0 | 1 | 0 | ||
1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 3 | −1 | −1 | 0 | ||
1 | 2 | 2 | −1 | −1 | 1 | 1 | 1 | 3 | −1 | 1 | 0 | ||
1 | 2 | 2 | −1 | 1 | −1 | 1 | 1 | 3 | 1 | −1 | 0 | ||
1 | 2 | 2 | 1 | −1 | −1 | 1 | 1 | 3 | 1 | 1 | 0 | ||
1 | 2 | 2 | −1 | −1 | 0 | 1 | 3 | 1 | −1 | 0 | −1 | ||
1 | 2 | 2 | −1 | 0 | −1 | 1 | 3 | 1 | −1 | 0 | 1 | ||
1 | 2 | 2 | 1 | 1 | 0 | 1 | 3 | 1 | 1 | 0 | −1 | ||
1 | 2 | 2 | 1 | 0 | 1 | 1 | 3 | 1 | 1 | 0 | 1 | ||
1 | 2 | 2 | 1 | −1 | 0 | 3 | 1 | 1 | 0 | −1 | −1 | ||
1 | 2 | 2 | 1 | 0 | −1 | 3 | 1 | 1 | 0 | −1 | 1 | ||
1 | 2 | 2 | −1 | 1 | 0 | 3 | 1 | 1 | 0 | 1 | −1 | ||
1 | 2 | 2 | −1 | 0 | 1 | 3 | 1 | 1 | 0 | 1 | 1 | ||
2 | 1 | 2 | 1 | 1 | 1 | 2 | 1 | 3 | −1 | −2 | 1 | ||
2 | 1 | 2 | −1 | −1 | 1 | 2 | 1 | 3 | −1 | 2 | −1 | ||
2 | 1 | 2 | −1 | 1 | −1 | 2 | 1 | 3 | 1 | −2 | −1 | ||
2 | 1 | 2 | 1 | −1 | −1 | 2 | 1 | 3 | 1 | 2 | 1 | ||
2 | 1 | 2 | −1 | −1 | 0 | 2 | 3 | 1 | −1 | −1 | 2 | ||
2 | 1 | 2 | 0 | −1 | −1 | 2 | 3 | 1 | −1 | 1 | −2 | ||
2 | 1 | 2 | 1 | 1 | 0 | 2 | 3 | 1 | 1 | −1 | −2 | ||
2 | 1 | 2 | 0 | 1 | 1 | cP80 | 2 | 3 | 1 | 1 | 1 | 2 | |
2 | 1 | 2 | 1 | −1 | 0 | cP81 | 3 | 1 | 2 | −1 | −2 | 1 | |
2 | 1 | 2 | 0 | 1 | −1 | cP82 | 3 | 1 | 2 | −1 | 2 | −1 | |
2 | 1 | 2 | −1 | 1 | 0 | 3 | 1 | 2 | 1 | −2 | −1 | ||
2 | 1 | 2 | 0 | −1 | 1 | 3 | 1 | 2 | 1 | 2 | 1 | ||
2 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 3 | −2 | −1 | 1 | ||
2 | 2 | 1 | −1 | −1 | 1 | 1 | 2 | 3 | −2 | 1 | −1 | ||
2 | 2 | 1 | −1 | 1 | −1 | 1 | 2 | 3 | 2 | −1 | −1 | ||
2 | 2 | 1 | 1 | −1 | −1 | 1 | 2 | 3 | 2 | 1 | 1 | ||
2 | 2 | 1 | −1 | 0 | −1 | 1 | 3 | 2 | −2 | −1 | 1 | ||
2 | 2 | 1 | 0 | −1 | −1 | 1 | 3 | 2 | 2 | −1 | −1 | ||
2 | 2 | 1 | 1 | 0 | 1 | 1 | 3 | 2 | 2 | 1 | 1 | ||
2 | 2 | 1 | 0 | 1 | 1 | 1 | 3 | 2 | −2 | 1 | −1 | ||
2 | 2 | 1 | 1 | 0 | −1 | 3 | 2 | 1 | −1 | −1 | 2 | ||
2 | 2 | 1 | 0 | 1 | −1 | 3 | 2 | 1 | −1 | 1 | −2 | ||
2 | 2 | 1 | −1 | 0 | 1 | 3 | 2 | 1 | 1 | −1 | −2 | ||
2 | 2 | 1 | 0 | 0 | 1 | 3 | 2 | 1 | 1 | 1 | 2 |
For all
In the neighborhood of cubic symmetry, the semi-reduced
6. Distances between hR and monoclinic lattices: composed deformations
As mentioned earlier, the symmetry axis splits orthogonally 3D lattice into union of 1D lattice and 2D lattice and is stable during uniaxial deformation in 1D direction. But a twofold axis is less restrictive in comparison with higher order axes, and in this case 2D lattice can also be modified. This complicates the modeling of
Δa/a [%] | Δb/b [%] | Δc/c [%] | Δα [°] | Δβ [°] | Δγ [°] | δ [°] | Operation |
---|---|---|---|---|---|---|---|
Deformation 0.001·(01–1) | |||||||
0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 2[01–1](01–1) |
0.0500 | −0.0500 | 0.0000 | −0.0800 | 0.0800 | 0.0000 | 0.0934 | 2[1–10](1–10) |
0.0500 | 0.0000 | −0.0500 | −0.0800 | 0.0000 | 0.0800 | 0.0934 | 2[−101](−101) |
0.0500 | 0.0000 | −0.0500 | −0.0800 | 0.0000 | 0.0800 | 0.0000 | 3 + [111](111) |
0.0500 | −0.0500 | 0.0000 | −0.0800 | 0.0800 | 0.0000 | 0.0000 | 3-[111](111) |
Deformation 0.001·(011) | |||||||
0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 2[01–1](01–1) |
0.0500 | −0.0500 | 0.0000 | 0.0496 | −0.0496 | 0.0000 | 0.0556 | 2[1–10](1–10) |
0.0500 | 0.0000 | −0.0500 | 0.0496 | 0.0000 | −0.0496 | 0.0556 | 2[−101](−101) |
0.0500 | 0.0000 | −0.0500 | 0.0496 | 0.0000 | −0.0496 | 0.0532 | 3 + [111](111) |
0.0500 | −0.0500 | 0.0000 | 0.0496 | −0.0496 | 0.0000 | 0.0532 | 3-[111](111) |
Deformation 0.001·(01–1) + 0.001·(011) | |||||||
0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 2[01–1](01–1) |
0.1000 | −0.0999 | 0.0000 | −0.0304 | 0.0304 | 0.0000 | 0.0774 | 2[1–10](1–10) |
0.1000 | 0.0000 | −0.0999 | −0.0304 | 0.0000 | 0.0304 | 0.0774 | 2[−101](−101) |
0.1000 | 0.0000 | −0.0999 | −0.0304 | 0.0000 | 0.0304 | 0.0532 | 3 + [111](111) |
0.1000 | −0.0999 | 0.0000 | −0.0304 | 0.0304 | 0.0000 | 0.0532 | 3-[111](111) |
The ε-deformations are additive by the definition, but this feature is also valid for geometric images (excluding
7. The distances for phospolipase A2
For a comparative study of different distances between a probe cell and the items in protein database (PDB), McGill and others [2] used unit cells of phospolipase A2 discussed in [12], which concluded that items 1g2x, 1u4j, and 1fe5 describe the same structure. Study, among other interesting conclusions, showed a similarity only between 1g2x and 1u4j cells for all applied distances. This result is also confirmed by analysis based on
1g2x | 80.949 | 80.572 | 57.098 | 90° | 90.35° | 90° | C |
1u4j | 80.36 | 80.36 | 99.44 | 90° | 90° | 120° | R |
1fe5 | 57.98 | 57.98 | 57.98 | 92.02° | 92.02° | 92,02° | P |
3260.18 | 3261.15 | 3261.15 | 15.22 | 14.12 | 14.12 | original | |
ε = −1.04·(011) +0.07·(01–1) | deformation: monoclinic | ||||||
3260.18 | 3260.18 | 3260.18 | 14.12 | 14.12 | 14.12 | hR | |
ε = −14.12·(111) | deformation: rhombohedral | ||||||
3246.06 | 3246.06 | 3246.06 | 0.00 | 0.00 | 0.00 | cP | |
3251.28 | 3251.28 | 3251.28 | 22.41 | 22.41 | 22.41 | original | |
ε = −22,41·(111) | deformation: rhombohedral | ||||||
3228.87 | 3228.87 | 3228.87 | 0.00 | 0.00 | 0.00 | cP | |
3361.68 | 3361.68 | 3361.68 | −118.49 | −118.49 | −118.49 | original | |
ε = 118,49·(111) | deformation: rhombohedral | ||||||
3480.17 | 3480.17 | 3480.17 | 0.00 | 0.00 | 0.00 | cP |
The monoclinic deformation of 1g2x cell is very small. Rhombohedral distances ε to the cubic border are similar for 1g2x and 1u4j, but drastically different in comparison with that in 1fe5. Moreover, the different sign suggests that if one agrees that all three items describe the same structure it must also allow the possibility that the true symmetry is cubic. It is also visible that this method is sensitive for much smaller (then analyzed) deviations from the symmetry borders.
8. hR -mC dilemma in α-Cr2O3, α-Fe2O3, CaCO3
The crystal structures of BiFeO3, as well as of α-Cr2O3, α-Fe2O3, CaCO3 are usually described as trigonal, but there are motivations that come from systematic (
Let us look at the published data obtained for the monoclinic model [3, 4]. Cell parameters were recalculated to the primitive form, which was not Niggli. The strict symmetry had geometric description 2 [1, 2, 3, 4, 5, 6, 7, 8, 9, 10](1–10). Therefore, it was assumed that composite deformation ε1·(1–10) + ε2·(110) brings these monoclinic cells to the rhombohedral ones. The BiFeO3 cell data were not available but all the data for
9. Summary
Generally, border problems cannot be overlooked in s.r.d. Small, but not negligible, values of discrepancy parameters indicate the border problem and give some measure to the higher symmetry border. Deviations in isometric actions on the investigated cell can be explained by monoaxial deformations measured by parameter
Moreover,
The concept is outlined and tested for
References
- 1.
Maciček J, Yordanov A. BLAF – A robust program for tracking out admittable Bravais lattice(s) from the experimental unit-cell data. Journal of Applied Crystallography. 1992; 25 :73-80 - 2.
McGill KJ, Asadi M, Karkasheva MT, Andrews LC, Bernstein HJ. The geometry of Niggli reduction: SAUC – Search of alternative unit cells. Journal of Applied Crystallography. 2014; 47 :360-364 - 3.
Stękiel M, Przeniosło R, Sosnowska I, Fitch A, Jasiński JB, Lussier JA, Bieringer M. Lack of a threefold rotation axis in α -Fe2O3 and α -Cr2O3 crystals. Acta Crystallographica. 2015; B71 :203-208 - 4.
Przeniosło R, Fabrykiewicz P, Sosnowska I. Monoclinic deformation of calcite crystals at ambient conditions. Physica B. 2016; 496 :49-56 - 5.
Himes VL, Mighell AD. A matrix method for lattice symmetry determination. Acta Crystallographica. 1982; A38 :748-749 - 6.
Himes VL, Mighell AD. A matrix approach to symmetry. Acta Crystallographica. 1987; A43 :375-384 - 7.
Le Page Y. The derivation of the axes of the conventional unit cell from the dimensions of the Buerger-reduced cell. Journal of Applied Crystallography. 1982; 15 :255-259 - 8.
Stróż K. Space of symmetry matrices with elements 0, ±1 and complete geometric description; its properties and application. Acta Crystallographica. 2011; A67 :421-429 - 9.
Stróż K. Symmetry of semi-reduced lattices. Acta Crystallographica. 2015; A71 :268-278 - 10.
Stróż K. A systematic approach to the derivation of standard orientation-location parts of symmetry-operation symbols. Acta Crystallographica. 2007; A63 :447-454 - 11.
Andrews CL, Bernstein HJ. The geometry of Niggli reduction: BGAOL – Embedding Niggli reduction and analysic of boundaries. Journal of Applied Crystallography. 2014; 47 :346-359 - 12.
Le Trong I, Stenkamp RE. An alternate description of two crystal structures of phospholipase A2 from Bungarus caeruleus . Acta Crystallographica. 2007;D :548-54963