International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 1.4, pp. 118-119   | 1 | 2 |

Section 1.4.3.3. Symbolic representation of A and B

U. Shmuelia

1.4.3.3. Symbolic representation of A and B

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We shall first discuss the symbols for the space groups that are not associated with a unique axis. These comprise the triclinic, orthorhombic and cubic space groups. The symbols are also used for the seven rhombohedral space groups which are referred to rhombohedral axes (IT I, 1952[link]; IT A, 1983[link]).

The abbreviation of triple products of trigonometric functions such as, e.g., denoting [\cos (2 \pi hx) \sin (2 \pi ky) \cos (2 \pi lz)] by csc, is well known (IT I, 1952[link]), and can be conveniently used in representing A and B for triclinic and orthorhombic space groups. However, the simplified expressions for A and B in space groups of higher symmetry also possess a high degree of regularity, as is apparent from an examination of the structure-factor tables in Volume I (IT I, 1952[link]), and as confirmed by the preparation of the present tables. An example, illustrating this for the cubic system, is given below.

The trigonometric structure factor for the space group [Pm\bar{3}] (No. 200) is given by[\eqalignno{A &= 8[\cos (2\pi hx) \cos (2\pi ky) \cos (2\pi lz)\cr &\quad + \cos (2\pi hy) \cos (2\pi kz) \cos (2\pi lx)\cr &\quad + \cos (2\pi hz) \cos (2\pi kx) \cos (2\pi ly)], &(1.4.3.2)}]and the sum of the above nine-function block and the following one:[\eqalignno{&8[\cos (2\pi hx) \cos (2\pi kz) \cos (2\pi ly)\cr &\quad + \cos (2\pi hz) \cos (2\pi ky) \cos (2\pi lx)\cr &\quad + \cos (2\pi hy) \cos (2\pi kx) \cos (2\pi lz)] &(1.4.3.3)}]is the trigonometric structure factor for the space group [Pm\bar{3}m] (No. 221, IT I, 1952[link], IT A, 1983[link]). It is obvious that the only difference between the nine-function blocks in (1.4.3.2)[link] and (1.4.3.3)[link] is that the permutation of the coordinates xyz is cyclic or even in (1.4.3.2)[link], while it is noncyclic or odd in (1.4.3.3)[link].

It was observed during the generation of the present tables that the expressions for A and B for all the cubic space groups, and all the relevant hkl subsets, can be represented in terms of such `even' and `odd' nine-function blocks. Moreover, it was found that the order of the trigonometric functions in each such block remains the same in each of its three terms (triple products). This is not surprising since each of the above space groups contains threefold axes of rotation along [111] and related directions, and such permutations of xyz for fixed hkl (or vice versa) are expected. It was therefore possible to introduce two permutation operators and represent A and B in terms of the following two basic blocks:[\eqalignno{\hbox{Epqr} &= \hbox{p}(2\pi hx) \hbox{q}(2\pi ky) \hbox{r}(2\pi lz) + \hbox{p}(2\pi hy) \hbox{q}(2\pi kz) \hbox{r}(2\pi lx)\cr &\quad + \hbox{p}(2\pi hz) \hbox{q}(2\pi kx) \hbox{r}(2\pi ly) &(1.4.3.4)}]and[\eqalignno{\hbox{Opqr} &= \hbox{p}(2\pi hx) \hbox{q}(2\pi kz) \hbox{r}(2\pi ly) + \hbox{p}(2\pi hz) \hbox{q}(2\pi ky) \hbox{r}(2\pi lx)\cr &\quad + \hbox{p}(2\pi hy) \hbox{q}(2\pi kx) \hbox{r}(2\pi lz), &(1.4.3.5)}]where each of p, q and r can be a sine or a cosine, and appears at the same position in each of the three terms of a block. The capital prefixes E and O were chosen to represent even and odd permutations of the coordinates xyz, respectively.

For example, the trigonometric structure factor for the space group [Pa\bar{3}] (No. 205, IT I, 1952[link], IT A, 1983[link]) can now be tabulated as follows:[\let\normalbaselines\relax\openup2pt\matrix{\phantom{-}A \hfill & B \hfill & h + k \hfill & k + l \hfill & h + l\hfill\cr \phantom{-}\hbox{8Eccc} \hfill & 0 \hfill & \hbox{even} \hfill & \hbox{even}\hfill &\hbox{even}\hfill\cr -8 \hbox{Ecss}\hfill & 0\hfill & \hbox{even}\hfill & \hbox{odd}\hfill & \hbox{odd}\hfill\cr -8 \hbox{Escs}\hfill & 0\hfill & \hbox{odd}\hfill & \hbox{even} \hfill &\hbox{odd}\hfill\cr -8 \hbox{Essc}\hfill & 0\hfill & \hbox{odd}\hfill & \hbox{odd}\hfill & \hbox{even}\hfill\cr}](cf. Table A1.4.3.7[link]), where the sines and cosines are abbreviated by s and c, respectively. It is interesting to note that the only maximal non-isomorphic subgroup of [Pa\bar{3}], not containing a threefold axis, is the orthorhombic Pbca (see IT A, 1983[link], p. 621), and this group–subgroup relationship is reflected in the functional forms of the trigonometric structure factors; the representation of A and B for Pbca is in fact analogous to that of [Pa\bar{3}], including the parities of hkl and the corresponding forms of the triple products, except that the prefix E – associated with the threefold rotation – is absent from Pbca. The expression for A for the space group [Pm\bar{3}m] [the sum of (1.4.3.2)[link] and (1.4.3.3)[link]] now simply reads: [A = 8 (\hbox{Eccc} + \hbox{Occc})].

As pointed out above, the permutation operators also apply to rhombohedral space groups that are referred to rhombohedral axes (Table A1.4.3.6)[link], and the corresponding expressions for R3 and [R\bar{3}] bear the same relationship to those for P1 and [P\bar{1}] (Table A1.4.3.2)[link], respectively, as that shown above for the related [Pa\bar{3}] and Pbca.

When in any given standard space-group setting one of the coordinate axes is parallel to a unique axis, the point-group rotation matrices can be partitioned into [2\times 2] and [1\times 1] diagonal blocks, the former corresponding to an operation of the plane group resulting from the projection of the space group down the unique axis. If, for example, the unique axis is parallel to c, we can decompose the scalar product in (1.4.2.19)[link] and (1.4.2.20)[link] as follows:[\eqalignno{{\bf h}^{T} ({\bi P}_{s} {\bf r} + {\bf t}_{s}) &=\pmatrix{h &k} \left[\pmatrix{P_{11} &P_{12}\cr P_{21} &P_{22}\cr} \pmatrix{x\cr y\cr} + \pmatrix{t_{1}\cr t_{2}\cr}\right]&\cr&\quad + l(P_{33}z + t_{3}), &(1.4.3.6)\cr}]where the first scalar product on the right-hand side of (1.4.3.6)[link] contains the contribution of a plane group and the second product is the contribution of the unique axis itself. The above decomposition often leads to a convenient factorization of A and B, and is applicable to monoclinic, tetragonal and hexagonal families, the latter including rhombohedral space groups that are referred to hexagonal axes.

The symbols used in Tables A1.4.3.3[link], A1.4.3.5[link] and A1.4.3.6[link] are based on such decompositions. In those few cases where explicit expressions must be given we make use of the convention of replacing [\cos (2 \pi u)] by [\hbox{c}(u)] and [\sin (2 \pi u)] by [\hbox{s}(u)]. For example, [\cos [2 \pi (hy + kx)]] etc. is given as [{\rm c}(hy + kx)] etc. The symbols are defined below.

  • Monoclinic space groups (Table A1.4.3.3)[link]

    The following symbols are used in this system:[\eqalign{{\rm c}(hl) &= \cos [2\pi (hx + lz)], \quad {\rm c}(hk) = \cos [2\pi (hx + ky)]\cr {\rm s}(hl) &= \sin [2\pi (hx + lz)], \quad \, {\rm s}(hk) = \sin [2\pi (hx + ky)]\cr} \eqno(1.4.3.7)]so that any expression for A or B in the monoclinic system has the form [K \hbox {p}(hl) \hbox {q}(ky)] or [K \hbox {p}(hk) \hbox {q}(lz)] for the second or first setting, respectively, where p and q can each be a sine or a cosine and K is a numerical constant.

  • Tetragonal space groups (Table A1.4.3.5)[link]

    The most frequently occurring expressions in the summations for A and B in this system are of the form[\hbox{P(pq)} = \hbox{p}(2\pi hx) \hbox{q}(2\pi ky) + \hbox{p}(2\pi kx) \hbox{q}(2\pi hy) \eqno(1.4.3.8)]and[\hbox{M(pq)} = \hbox{p}(2\pi hx) \hbox{q}(2\pi ky) - \hbox{p}(2\pi kx) \hbox{q}(2\pi hy), \eqno(1.4.3.9)]where p and q can each be a sine or a cosine. These are typical contributions related to square plane groups.

  • Trigonal and hexagonal space groups (Table A1.4.3.6)[link]

    The contributions of plane hexagonal space groups to the first term in (1.4.3.6)[link] are[\eqalign{p_{1} &= hx + ky, \quad p_{2} = kx + iy, \quad p_{3} = ix + hy,\cr q_{1} &= kx + hy, \quad q_{2} = hx + iy, \quad q_{3} = ix + ky,\cr} \eqno(1.4.3.10)]where [i = -h -k] (IT I, 1952[link]). The symbols which represent the frequently occurring expressions in this family, and given in terms of (1.4.3.10)[link], are[\eqalign{\hbox{C}(hki) &= \cos (2\pi p_{1}) + \cos (2\pi p_{2}) + \cos (2\pi p_{3})\cr \hbox{C}(khi) &= \cos (2\pi q_{1}) + \cos (2\pi q_{2}) + \cos (2\pi q_{3})\cr \hbox{S}(hki) &= \sin (2\pi p_{1}) + \sin(2\pi p_{2}) + \sin (2\pi p_{3})\cr \hbox{S}(khi) &= \sin (2\pi q_{1}) + \sin (2\pi q_{2}) + \sin (2\pi q_{3})\cr} \eqno(1.4.3.11)]and these quite often appear as the following sums and differences:[\eqalign{\hbox{PH(cc)} &= \hbox{C}(hki) + \hbox{C}(khi), \quad \hbox{PH(ss)} = \hbox{S}(hki) + \hbox{S}(khi)\cr \hbox{MH(cc)} &= \hbox{C}(hki) - \hbox{C}(khi), \quad \hbox{MH(ss)} = \hbox{S}(hki) - \hbox{S}(khi).\cr} \eqno(1.4.3.12)]The symbols defined in this section are briefly redefined in the appropriate tables, which also contain the conditions for vanishing symbols.

References

International Tables for Crystallography (1983). Vol. A, Space-Group Symmetry, edited by Th. Hahn. Dordrecht: Reidel.
International Tables for X-ray Crystallography (1952). Vol. I, Symmetry Groups, edited by N. F. M. Henry & K. Lonsdale. Birmingham: Kynoch Press.








































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