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Guide to the k-vector tables

The k-vector tables consists of two parts: (i) ’ Litvin & Wike’ description and (ii) ’Plane-group description’. The first three columns under the heading ’Litvin & Wike’ refer to the description of k-vectors found in Tables 24 and 25 of Litvin and Wike, 1991 (abbreviated as L&W) [3]. It consists of labels of k-vectors (column 1), their parameter descriptions (column 2) and their layer little co-group (column 3 for primitive lattices and column 4 for c-centred). Note that L&W substitute the Greek-character labels for the symmetry points and lines inside the Brillouin zone by a symbol consisting of two Roman characters, e. g. GM for Γ, LD for Λ, etc. In order to enable the uni-arm description new k-vector types, equivalent to those of L&W are added to the k-vector lists. Equivalent k-vector (related by the sign ∼) are designated by the same labels; additional indices distinguish the new k-vectors.

Different k-vectors with the same L&W label always belong to the same k-vector type, i. e. they correspond to the same Wyckoff position. k-vectors with different L&W labels may either belong to the same or to different types of k-vectors. When k-vectors with different L&W labels belong to the same k-type, the corresponding parameter descriptions are followed by the letters ’ex’. Symmetry points or lines of symmetry of L&W, related to the same Wyckoff position, are grouped together in a block. In the k-vector tables, neighbouring Wyckoff-position blocks are distinguished by a slight difference in the background colour. The parameter description of the uni-arm region (for a discussion of uni-arm description the reader is referred to [4]) of a k-vector type is shown in the last row of the correponding Wyckoff position block.

The wave-vector coefficients of L&W (column 2 of the k-vector tables) refer always to a primitive basis irrespective of whether the conventional description of the group in ITE is with respect to a centred or primitive basis. For that reason, for layer groups with centred lattices, the wave-vector coefficients with respect to the usual conventional reciprocal basis, i. e. dual to the conventional centred basis, are listed in the column under the heading ’Conventional’ of the k-vector tables.

The layer little co-group data of each k-vector is listed under the heading ’Layer little co-group’ of the k-vector tables. The layer little co-groups are subgroups of the point group of the layer group and are described by oriented point-group symbols (as is customary for site-symmetry groups of Wyckoff positions).

The data for the crystallographic classification scheme of the wave vectors are listed under the heading ’Plane-group description’ in the k-vector tables. The columns ’Wyckoff positions’ show the data of ’multiplicity’, ’Wyckoff letter’ and ’site symmetry’ of the Wyckoff positions of the corresponding symmorphic plane group P₀ of ITA [2] which is isomorphic to the reciprocal-plane group (P)^*. The multiplicity of a Wyckoff position divided by the number of lattice points in the conventional unit cell of ITA is equal to the number of arms of the star of the corresponding k-vector. The alphabetical sequence of the Wyckoff positions determines the sequence of the L&W labels. Unlike in ITA, the tables start with the Wyckoff letter ’a’ for the Wyckoff position of the highest site symmetry. Site symmetries are described by means of oriented point-group symbols which are also links to more detailed information on the symmetry operations of the site-symmetry group. Besides the shorthand description, the matrix-column representation and the geometric representation of the symmetry operations of the site-symmetry group, the program also provides a table with the relationship between the symmetry operations of the site-symmetry group and the layer little co-group.

The parameter description of the Wyckoff positions are shown in the last column of the wave-vector tables under the heading ’Coordinates’. It consists of a representative coordinate doublet of the Wyckoff position and algebraic statements for the description of the independent parameter range. In some cases, the algebraic expressions are substituted by the designation of the parameter region in order to avoid clumsy notation. Because of the dependence of the shape of the Brillouin zone on the lattice parameter relations there may be vertices of the Brillouin zone with a variable coordinate. If such a point is displayed and designated in the tables and figures by an upper-case letter, then the label of its variable coefficient used in the parameter-range descriptions is the same letter but typed in lower case.

Because of the isomorphism between P₀ and (P)^* the coordinate doublets of the Wyckoff positions of P₀ can be related to as k-vector coefficients (k_a1, k_a2) determined with respect to the conventional ITA basis of P₀.

At the bottom of the web page with the k-vector table one finds an auxiliary tool which allows the complete characterization of a wavevector of the reciprocal space (not restricted to the first Brillouin zone): given the k-vector coefficients referred either to a primitive (L&W) or to a conventional basis, the program assigns the k vector to the corresponding wavevector symmetry type, specifies its L&W label, and calculates the layer little co-group and the arms of the k-vector star.