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The list of the special k-vectors includes special points of symmetry and special lines of symmetry. The points A, H, HA, Γ, K, and KA are represented by red circles as they are special k-vector points of symmetry 32. The lines DT (indicated as Δ in its figure), P and PA are brown because they correspond to each of the three tertiary axes of the asymmetric unit and at the same time form its edges. The special k-vector lines T, TA and LE are coloured cyan as they do not form part of the edges of the representation domain. Together with the line LD (represented as Λ in the figure of P321) and the point M, they belong to the Wyckoff-position block 3j, i. e. all these different wave vectors belong to the same k-vector type. Its uni-arm description is achieved by the definition of two flagpoles, stretching out of the asymmetric unit along the Λ line. The flagpole LE1 is equivalent to LE, while the flagpole T1 U M0 U TA1 substitutes T U M U TA. The uni-arm description of the k-vector type 3j is shown in the last row of the Wyckoff-position block. The parameter description of the flagpoles and their parameter ranges with respect to the basis of the reciprocal-space group are given below the k-vector table.
The k-vector lines S, QA, and SA are dark blue as they are selected in CDML [2] to represent the special 2-fold symmetry lines along the edges of the representation domain. Together with the line Q and the point L, they belong to the special k-vector type of the Wyckoff-position 3k. As in the case of the 3j type, a uni-arm description can be achieved by defining two flagpoles stretching out of the representation domain along the k-vector-line Q.
It was already pointed out that special k-vector points and lines are brought out in colours only if they are chosen as orbit representatives of the corresponding k-vector type. For example, although SA1 is along a binary axis, it is not coloured as a special line (the pink colour indicates an asymmetric-unit edge) since it is not chosen as an orbit representative in any of the two descriptions: SA1 is substituted by SA in CDML description or by SA3 in the case of uni-arm description. Likewise, the points KA2 and KA0 (of symmetry 32) are not coloured as special points as they belong to the orbit of the special k-vector point KA, chosen as an orbit representative and shown in the diagram by a circle filled in red. (The fact that the three points belong to the same k-vector orbit is evident from their coefficients: KA(2/3,-1/3,0), KA2(2/3,2/3,0), KA0(-1/3,-1/3,0)).
The points L and M are examples of k-vector points whose little co-groups are not proper supergroups of the little co-groups of all points in their neighborhood. In fact, although L and M are explicitly listed by CDML as special k-vector points, they form part of the lines S and T and in the diagram they are represented by black circles filled in with white.
In the following, the Brillouin-zone diagrams of the space group R3 - the simplest of the rhombohedral space groups, are considered as an example. While the representation domains of the acute and obtuse unit cells are of a rather complicated form, the asymmetric units in both cases have topologically identical and relatively simple shape: it is a rhombus with an angle of 120° in the xy plane (an union of two equilateral triangles) with z extending from -1/6 to 1/6.
In the diagrams one can distinguish a single special k-vector type: it is a symmetry k-vector line along the three-fold axis. In the obtuse case, the correspondence between the CDML description of the special k-vector line and the the uni-arm description is straightforward as the necessary segment of the line 0, 0, z lies entirely inside the Brillouin zone: the uni-arm description of the Wyckoff position block 3a unifies the lines Γ and LE, and the two points Γ and T. In the hexagonal basis, it is described by the segment of the line 0, 0, z with z varying in the range (-1/2,1/2]. It is worth noting that although listed separately in CDML, the points Γ and T obviously belong to the same k-vector type as the lines Λ and LE, i.e. the points have the same symmetry as the points on the line and as such, they are represented by black circles filled in with white on the diagrams.
Due to the special shape of the Brillouin zone and the representation domain for the acute case (√3a > √2c), the special k-vector line corresponding to the Wyckoff position block 3a splits into several segments: the lines Γ and LE, located inside the Brillouin zone, and the lines P and PA (coloured dark blue) - at the border of the Brillouin zone. For the description of the end points of the segments it is necessary to introduce additional parameters as p0 and ld0 whose values depend on the specific relations between the lattice parameters. To enable uni-arm description, symmetry lines equivalent to P and PA, located outside the Brillouin zone and along (0, 0, z), are to be selected as orbit representatives. The uni-arm description of the special k-vector line is formed by the union of the lines Γ and LE, the flagpoles P1(∼ P) and PA1(∼ PA), and the points Γ and T2(∼ T). Its parameter description (0, 0, z) with z varying in the range (-1/2, 1/2], coincides with that of the obtuse case.
In ITA the monoclinic space group C2/c (No. 15) is described in six settings: depending on the cell choices, there are three descriptions for each of the unique axis b and unique axis c settings. The Brillouin-zone database contains k-vector tables and figures of two settings of C2/c, namely, the settings A112/a (unique axis c, cell choice 1) and C12/c1 (unique axis b, cell choice 1). The space group C2/c belongs to the arithmetic crystal class 2/mC which includes also the space group C2/m (No. 12). In the following, we discuss shortly the k-vector table and figure of A112/a, and then proceed with the derivation of the data of C12/c1 from those of A112/a.
The reciprocal-space group of A112/a is isomorphic to the symmorphic space group A112/m, i.e. the list of special Wyckoff positions of A112/m indicates the special k-vector types of A112/a. In fact, the Wyckoff-position data, including multiplicities, Wyckoff letters, site-symmetry groups and coordinate triplets of the k-vector tables are taken directly from ITA. For the determination of the parameter ranges one starts by defining the parameter region (or space) of a Wyckoff position (line, plane, or space) which is inside the unit cell. The ratio of order of the site-symmetry group (representing those operations which leave the parameter space fixed pointwise) and the order of the stabilizer (which is the set of all symmetry operations modulo integer translations which leave the parameter space invariant as a whole) give the independent fraction of the parameter space (i.e. of the volume of the unit cell, or of the area of the plane, or of the length of the line). For example, the parameter space of the line 0, 0, z in the unit cell is determined by the variation of z in the range (-1/2 < z < 1/2). The order of the site symmetry group is 2 while its stabilizer is of order 4 (the group 2/m), so the independent segment is exactly 1/2 of the parameter space in the unit cell, e.g., (0 < z < 1/2). In a similar way, the independent area of the plane x, y, 0 is exactly 1/2 of the area of the plane in the unit cell. For its uni-arm description, it is necessary to introduce a wing, stretching outside of the asymmetric unit x, y, 0 : 0 < x < 1/2, -1/2 < y < 0 (coloured in pink in the figure of the space group A112/a).
The labels of the special k-vector points, lines and planes and their coordinates listed in the first two columns are taken directly from Table 3.9 (c) of CDML. The correspondence between the special k-vectors listed by CDML and the Wyckoff positions of ITA follows from the relation between the primitive basis {g1, g2, g3} used by CDML and the conventional ITA basis {kx, ky, kz} (cf. Table 3.4 of CDML):
The k-vector data of C12/c1 (i. e. of the arithmetic crystal class 12/m1C) can be derived from those of A112/a (i. e. of 112/mA) utilising the relationship between the two settings descriptions of ITA. The transformation matrix P, specifying the relation between the basis {ab, bb, cb} of the setting unique axis b (cell choice 1) and the basis {ac, bc, cc} of the setting unique axis c (cell choice 1) reads:
| (ab,bb,cb) = (ac,bc,cc) P = (ac,bc,cc) |  |
0 | 0 | 1 |  |
(cf. Table 5.1.3.1 of ITA) |
| 1 | 0 | 0 | ||||
| 0 | 1 | 0 |
The coordinate triplets
|
xb | |
| yb | ||
| zb |
|
xb | |
= P-1 | |
xc | |
= | |
0 | 1 | 0 | |
|
xc | |
= | |
xc | |
| yb | yc | 0 | 0 | 1 | yc | yc | |||||||||||||
| zb | zc | 1 | 0 | 0 | zc | zc |
|
1/2 | |
⇒ | |
1/4 | |
| 1/4 | 1/4 | |||||
| 1/4 | 1/2 |
Under a coordinate transformation of the bases in direct space (ab, bb, cb) = (ac, bc, cc) P, the corresponding k-vector coefficients transform according to
| P = | |
0 | 0 | 1 | |
| 1 | 0 | 0 | |||
| 0 | 1 | 0 |
| (kp1,b kp2,b kp3,b) = (kp1,c kp2,c kp3,c) | |
0 | 0 | 1 | |
= | (kp1,c kp2,c kp3,c) | ⇒ kp1,b = kp2,c kp2,b = kp3,c kp3,b = kp1,c |
| 1 | 0 | 0 | ||||||
| 0 | 1 | 0 |
The special k-vectors of C12/m1 keep the CDML labels of the k-vectors of A112/m from which they are derived.
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