International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by Th. Hahn

International Tables for Crystallography (2006). Vol. A, ch. 4.3, pp. 71-73

Section 4.3.4. Tetragonal system

E. F. Bertauta

aLaboratoire de Cristallographie, CNRS, Grenoble, France

4.3.4. Tetragonal system

| top | pdf |

4.3.4.1. Historical note and arrangement of the tables

| top | pdf |

In the 1935 edition of International Tables[link], for each tetragonal P and I space group an additional C-cell and F-cell description was given. In the corresponding space-group symbols, secondary and tertiary symmetry elements were simply interchanged. Coordinate triplets for these larger cells were not printed, except for the space groups of class [\bar{4}m2]. In IT (1952)[link], the C and F cells were dropped from the space-group tables but kept in the comparative tables.

In the present edition, the C and F cells reappear in the sub- and supergroup tabulations of Part 7[link] , as well as in the synoptic Table 4.3.2.1,[link] where short and extended (two-line) symbols are given for P and C cells, as well as for I and F cells.

4.3.4.2. Relations between symmetry elements

| top | pdf |

In the crystal classes 42(2), 4m(m), [\bar{4}2\hbox{(}m\hbox{)}] or [\bar{4}m\hbox{(}2\hbox{)}], [4/m\;2/m\;\hbox{(}2/m\hbox{)}], where the tertiary symmetry elements are between parentheses, one finds [4 \times m = (m) = \bar{4} \times 2; \ 4 \times 2 = (2) = \bar{4} \times m.] Analogous relations hold for the space groups. In order to have the symmetry direction of the tertiary symmetry elements along [[1\bar{1}0]] (cf. Table 2.2.4.1[link] ), one has to choose the primary and secondary symmetry elements in the product rule along [001] and [010].

Example

In [P4_{1}2(2)\ (91)], one has [4_{1} \times 2 = (2)] so that [P4_{1}2] would be the short symbol. In fact, in IT (1935)[link], the tertiary symmetry element was suppressed for all groups of class 422, but re-established in IT (1952)[link], the main reason being the generation of the fourfold rotation as the product of the secondary and tertiary symmetry operations: [4 = (m) \times m] etc.

4.3.4.3. Additional symmetry elements

| top | pdf |

As a result of periodicity, in all space groups of classes 422, [\bar{4}m2] and [4/m\; 2/m\; 2/m], the two tertiary diagonal axes 2, along [[1\bar{1}0]] and [110], alternate with axes [2_{1}], the screw component being [{1 \over 2},\mp] [{1 \over 2}], 0 (cf. Table 4.1.2.2[link] ).

Likewise, tertiary diagonal mirrors m in x, x, z and [x,\bar{x},z] in space groups of classes 4mm, [\bar{4}2m] and [4/m\; 2/m\; 2/m] alternate with glide planes called g,1 the glide components being [{1 \over 2}], [\pm {1 \over 2}], 0. The same glide components produce also an alternation of diagonal glide planes c and n (cf. Table 4.1.2.2[link] ).

4.3.4.4. Multiple cells

| top | pdf |

The transformations from the P to the two C cells, or from the I to the two F cells, are [\eqalign{C_1 \hbox{ or } F_1\hbox{: }(\rm i)\quad {\bf a}' &= {\bf a} - {\bf b},\quad{\bf b}' = \phantom{-} {\bf a} + {\bf b},\quad{\bf c}' = {\bf c}\cr C_2 \hbox{ or } F_2\hbox{: }(\rm ii)\quad {\bf a}' &= {\bf a} + {\bf b},\quad{\bf b}' = - {\bf a} + {\bf b},\quad{\bf c}' = {\bf c}}] (cf. Fig. 5.1.3.5[link] ). The secondary and tertiary symmetry directions are interchanged in the double cells. It is important to know how primary, secondary and tertiary symmetry elements change in the new cells [{\bf a}',{\bf b}',{\bf c}'].

  • (i) Primary symmetry elements

    In P groups, only two kinds of planes, m and n, occur perpendicular to the fourfold axis: a and b planes are forbidden. A plane m in the P cell corresponds to a plane in the C cell which has the character of both a mirror plane m and a glide plane n. This is due to the centring translation [{1 \over 2},{1 \over 2},0] (cf. Chapter 4.1[link] ). Thus, the C-cell description shows2 that [P4/m..] (cell a, b, c) has two maximal k subgroups of index [2], [P4/m..] and [P4/n..] (cells [{\bf a}',{\bf b}',{\bf c}']), originating from the decentring of the C cell. The same reasoning is valid for [P4_{2}/m..] .

    A glide plane n in the P cell is associated with glide planes a and b in the C cell. Since such planes do not exist in tetragonal P groups, the C cell cannot be decentred, i.e. [P4/n..] and [P4_{2}/n..] have no k subgroups of index [2] and cells [{\bf a}',{\bf b}',{\bf c}'].

    Glide planes a perpendicular to c only occur in [I4_{1}/a\;(88)] and groups containing [I4_{1}/a] [[I4_{1}/amd\ (141)] and [I4_{1}/acd\ (142)]]; they are associated with d planes in the F cell. These groups cannot be decentred, i.e. they have no P subgroups at all.

  • (ii) Secondary symmetry elements

    In the tetragonal space-group symbols, one finds two kinds of secondary symmetry elements:

    • (1) 2, m, c without glide components in the ab plane occur in P and I groups. They transform to tertiary symmetry elements 2, m, c in the C or F cells, from which k subgroups can be obtained by decentring.

    • (2) [2_{1},b,n] with glide components [{1 \over 2},0,0{\hbox{;}}\; 0,{1 \over 2},0{\hbox{;}}\; {1 \over 2},{1 \over 2},0] in the ab plane occur only in P groups. In the C cell, they become tertiary symmetry elements with glide components [{1 \over 4},\! - {1 \over 4}, 0]; [{1 \over 4},{1 \over 4},0]; [{1 \over 4}], [{1 \over 4}], [{1 \over 2}]. One has the following correspondence between P- and C-cell symbols: [\eqalign{&P . 2_{1} . = C .. 2_{1}\cr &P . b . = C .. g_{1} \hbox{ with } g({\textstyle{1 \over 4}}, {\textstyle{1 \over 4}}, 0)\hbox{ in } \quad x, x - {\textstyle{1 \over 4}}, z\cr &P . n . = C .. g_{2} \hbox{ with } g({\textstyle{1 \over 4}}, {\textstyle{1 \over 4}}, {\textstyle{1 \over 2}})\hbox{ in } \quad x, x - {\textstyle{1 \over 4}}, z,}] where [(g_{1})^{2}] and [(g_{2})^{2}] are the centring translations [{1 \over 2},{1 \over 2},0] and [{1 \over 2},{1 \over 2},1]. Thus, the C cell cannot be decentred, i.e. tetragonal P groups having secondary symmetry elements [2_{1}], b or n cannot have klassengleiche P subgroups of index [2] and cells [{\bf a}',{\bf b}',{\bf c}'].

  • (iii) Tertiary symmetry elements

    Tertiary symmetry elements 2, m, c in P groups transform to secondary symmetry elements in the C cell, from which k subgroups can easily be deduced [(\rightarrow)]: [\matrix{P..m\ =&C.m.\longrightarrow& P.m.{\hbox to 3pt{}}\cr g&b{\hbox to 12pt{}}& P.b.{\hbox to 3pt{}}\cr P..c\ =&C.c.\longrightarrow &P.c.{\hbox to 4pt{}}\cr n&n\qquad&P.n.{\hbox to 4pt{}}\cr P..2\ =&C.2.\longrightarrow & P.2.{\hbox to 4pt{}}\cr {\hbox to 6pt{}}2_{1}&2_{1}{\hbox to 3pt{}}&P.2_{1}.}] Decentring leads in each case to two P subgroups (cell [{\bf a}',{\bf b}',{\bf c}']), when allowed by (i) and (ii).

    In I groups, 2, m and d occur as tertiary symmetry elements. They are transformed to secondary symmetry elements in the F cells. I groups with tertiary d glides cannot be decentred to P groups, whereas I groups with diagonal symmetry elements 2 and m have maximal P subgroups, due to decentring.

4.3.4.5. Group–subgroup relations

| top | pdf |

Examples are given for maximal k subgroups of P groups (i), of I groups (ii), and for maximal tetragonal, orthorhombic and monoclinic t subgroups.

4.3.4.5.1. Maximal k subgroups

| top | pdf |

  • (i) Subgroups of P groups

    The discussion is limited to maximal P subgroups, obtained by decentring the larger C cell (cf. Section 4.3.4.4[link] Multiple cells).

    • Classes [\bar{4}], 4 and 422

      Examples

      • (1) Space groups [P\bar{4}] (81) and [P4_{p}\ (p = 0, 1, 2, 3)] (75–78) have isomorphic k subgroups of index [2], cell [{\bf a}',{\bf b}',{\bf c}'].

      • (2) Space groups [P4_{p}22\ (p = 0, 1, 2, 3)] (89, 91, 93, 95) have the extended C-cell symbol [\matrix{\noalign{\vskip 13pt}C4_{p} 2\ 2\hfill\cr }2_{1}\hfill\cr}], from which one deduces two k subgroups, [P4_{p}22] (isomorphic, type [{\bf IIc}]) and [P4_{p}2_{1}2] (non-isomorphic, type IIb), cell a′, b′, c′.

      • (3) Space groups [P4_{p}2_{1}2] (90, 92, 94, 96) have no k subgroups of index [2], cell [{\bf a}',{\bf b}',{\bf c}'].

    • Classes [\bar{4}m2], 4mm, 4/m, and 4/mmm

      Examples

      • (1) [P\bar{4}c2\ (116)] has the C-cell symbol [\matrix{\noalign{\vskip 13pt}C\bar{4}2\ c\hfill\cr }2_{1}\hfill\cr}], wherefrom one deduces two k subgroups, [P\bar{4}2c] and [P\bar{4}2_{1}c], cell a′, b′, c′.

      • (2) [P4_{2}mc\ (105)] has the C-cell symbol [\matrix{\noalign{\vskip 13pt}C4_{2}cm\hfill\cr }n\hfill\cr}], from which the k subgroups [P4_{2}cm] (101) and [P4_{2}nm\ (102)], cell [{\bf a}', {\bf b}', {\bf c}'], are obtained.

      • (3) [P4_{2}/mcm\ (132)] has the extended C symbol [\matrix{\noalign{\vskip 13pt}C4_{2}/mmc\hfill\cr / }n\; b\hfill\cr}], wherefrom one reads the following k subgroups of index [2], cell [{\bf a}', {\bf b}', {\bf c}']: [P4_{2}/mmc], [P4_{2}/mbc], [P4_{2}/nmc], [P4_{2}/nbc].

      • (4) [P4/nbm\ (125)] has the extended C symbol [\matrix{\noalign{\vskip 13pt}C4/amg_{1}\hfill\cr  bb\hfill\cr}] and has no k subgroups of index [2], as explained above in Section 4.3.4.4[link] .

  • (ii) Subgroups of I groups

    Note that I groups with a glides perpendicular to [001] or with diagonal d planes cannot be decentred (cf. above). The discussion is limited to P subgroups of index [2], obtained by decentring the I cell. These subgroups are easily read from the two-line symbols of the I groups in Table 4.3.2.1[link].

    Examples

    • (1) I4cm (108) has the extended symbol [\matrix{\noalign{\vskip 13pt}I 4 c e\hfill\cr4_{2} b m\hfill\cr}]. The multiplication rules [4 \times b = m = 4_{2} \times c] give rise to the maximal k subgroups: P4cc, [P4_{2}bc], P4bm, [P4_{2}cm].

      Similarly, I4mm (107) has the P subgroups P4mm, [P4_{2}nm], P4nc, [P4_{2}mc], i.e. I4mm and I4cm have all P groups of class 4mm as maximal k subgroups.

    • (2) [I4/mcm\ (140)] has the extended symbol [\matrix{\noalign{\vskip 13pt}I 4/m \;ce\hfill\cr4_{2}/n b m\hfill\cr}]. One obtains the subgroups of example (1) with an additional m or n plane perpendicular to c.

      As in example (1), [I4/mcm] (140) and [I4/mmm] (139) have all P groups of class [4/mmm] as maximal k subgroups.

4.3.4.5.2. Maximal t subgroups

| top | pdf |

  • (i) Tetragonal subgroups

    The class [4/mmm] contains the classes [4/m, 422, 4mm] and [\bar{4}2m]. Maximal t subgroups belonging to these classes are read directly from the standard full symbol.

    Examples

    • (1) [P4_{2}/m\;bc\ (135)] has the full symbol [P4_{2}/m\;2_{1}/b\;2/c] and the tetragonal maximal t subgroups: [P4_{2}/m], [P4_{2}2_{1}2], [P4_{2}bc], [P\bar{4}2_{1}c], [P\bar{4}b2].

    • (2) [I4/m\;cm\ (140)] has the extended full symbol [{\matrix{\noalign{\vskip 13pt}I 4/m{\hbox to 1pt{}}2/c{\hbox to 5pt{}}2/e\hfill\cr4_{2}/n2_{1}/b 2_{1}/m\hfill\cr}}] and the tetragonal maximal t subgroups [I4/m], I422, I4cm, [I\bar{4}2m], [I\bar{4}c2]. Note that the t subgroups of class [\bar{4}m2] always exist in pairs.

  • (ii) Orthorhombic subgroups

    In the orthorhombic subgroups, the symmetry elements belonging to directions [100] and [010] are the same, except that a glide plane b perpendicular to [100] is accompanied by a glide plane a perpendicular to [010].

    Examples

    • (1) [P4_{2}/m bc\ (135)]. From the full symbol, the first maximal t subgroup is found to be [P2_{1}/b\;2_{1}/a \;2/m] (Pbam). The C-cell symbol is [C4_{2}/m\;cg_{1}] and gives rise to the second maximal orthorhombic t subgroup Cccm, cell [{\bf a}',{\bf b}',{\bf c}'].

    • (2) [I4/m\;cm\ (140)]. Similarly, the first orthorhombic maximal t subgroup is [\matrix{\noalign{\vskip 13pt}I c c m\hfill\crb a n\hfill\cr}] (Ibam); the second maximal orthorhombic t subgroup is obtained from the F-cell symbol as [\matrix{\noalign{\vskip 13pt}F c{\hbox to 3pt{}} c{\hbox to 3pt{}} m\hfill\crm m n\hfill\cr}] (Fmmm), cell a′, b′, c′.

    These examples show that P- and C-cell, as well as I- and F-cell descriptions of tetragonal groups have to be considered together.

  • (iii) Monoclinic subgroups

    Only space groups of classes 4, [\bar{4}] and [4/m] have maximal monoclinic t subgroups.

    Examples

    • (1) [P4_{1}\;(76)] has the subgroup [P112_{1}\ (P2_{1})]. The C-cell description does not add new features: [C112_{1}] is reducible to [P2_{1}].

    • (2) [I4_{1}/a\ (88)] has the subgroup [I112_{1}/a], equivalent to [I11 2/a\;(C2/c)]. The F-cell description yields the same subgroup [F11\;2/d], again reducible to [C2/c].

References

Internationale Tabellen zur Bestimmung von Kristallstrukturen (1935). 1. Band, edited by C. Hermann. Berlin: Borntraeger. [Revised edition: Ann Arbor: Edwards (1944). Abbreviated as IT (1935).]
International Tables for X-ray Crystallography (1952). Vol. I, edited by N. F. M. Henry & K. Lonsdale. Birmingham: Kynoch Press. [Abbreviated as IT (1952).]








































to end of page
to top of page