Algebraic relationships for structure seminvariants

Giacovazzo, C.

doi:10.1107/97809553602060000555

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

pdf | chapter contents | chapter index | related articles

International Tables for Crystallography (2006). Vol. B. ch. 2.2, p. 224 | 1 | 2 |

Section 2.2.5.8. Algebraic relationships for structure seminvariants

C. Giacovazzo^a ^*

^aDipartimento Geomineralogico, Campus Universitario, I-70125 Bari, Italy
Correspondence e-mail: c.giacovazzo@area.ba.cnr.it

2.2.5.8. Algebraic relationships for structure seminvariants

| top | pdf |

According to the representations method (Giacovazzo, 1977a, 1980a,b):

(i) any s.s. Φ may be estimated via one or more s.i.'s $[\{\psi\}]$ , whose values differ from Φ by a constant arising because of symmetry;
(ii) two types of s.s.'s exist, first-rank and second-rank s.s.'s, with different algebraic properties:
(iii) conditions characterizing s.s.'s of first rank for any space group may be expressed in terms of seminvariant moduli and seminvariantly associated vectors. For example, for all the space groups with point group 422 [Hauptman–Karle group P(2, 2)] the one-phase s.s.'s of first rank are characterized by $[\eqalign{(h, k, l) &\equiv 0 \hbox{ mod } (2,2,0) \hbox{ or } (2,0,2) \hbox{ or } (0,2,2)\cr (h \pm k, l) &\equiv 0 \hbox{ mod } (0, 2) \hbox{ or } (2, 0).}]$

The more general expressions for the s.s.'s of first rank are

(a) $[\Phi = \varphi_{\bf u} = \varphi_{{\bf h} ({\bi I} - {\bi R}_{\alpha})}]$ for one-phase s.s.'s;
(b) $[\Phi = \varphi_{{\bf u}_{1}} + \varphi_{{\bf u}_{2}} = \varphi_{{\bf h}_{1} - {\bf h}_{2} {\bi R}_{\beta}} + \varphi_{{\bf h}_{2} - {\bf h}_{1} {\bi R}_{\alpha}}]$ for two-phase s.s.'s;
(c) $[\Phi = \varphi_{{\bf u}_{1}} + \varphi_{{\bf u}_{2}} + \varphi_{{\bf u}_{3}} = \varphi_{{\bf h}_{1} - {\bf h}_{2} {\bi R}_{\beta}} + \varphi_{{\bf h}_{2} - {\bf h}_{3}{\bi R}_{\gamma}} + \varphi_{{\bf h}_{3} - {\bf h}_{1} {\bi R}_{\alpha}}]$ for three-phase s.s.'s;
$[\eqalign{\quad(d)\; \Phi &= \varphi_{{\bf u}_{1}} + \varphi_{{\bf u}_{2}} + \varphi_{{\bf u}_{3}} + \varphi_{{\bf u}_{4}} \cr &= \varphi_{{\bf h}_{1} - {\bf h}_{2} {\bi R}_{\beta}} + \varphi_{{\bf h}_{2} - {\bf h}_{3} {\bi R}_{\gamma}} + \varphi_{{\bf h}_{3} - {\bf h}_{4} {\bi R}_{\delta}} + \varphi_{{\bf h}_{4} - {\bf h}_{1} {\bi R}_{\alpha}}}\hfill]$ for four-phase s.s.'s; etc.

In other words:

(a) $[\varphi_{\bf u}]$ is an s.s. of first rank if at least one h and at least one rotation matrix $[{\bi R}_{\alpha}]$ exist such that $[{\bf u} = {\bf h}({\bi I} - {\bi R}_{\alpha})]$ . $[\varphi_{\bf u}]$ may be estimated via the special triplet invariants $[\{\psi\} = \varphi_{\bf u} - \varphi_{\bf h} + \varphi_{{\bf h} {\bi R}_{\alpha}}. \eqno(2.2.5.33)]$ The set $[\{\psi\}]$ is called the first representation of $[\varphi_{\bf u}]$ .
(b) $[\Phi = \varphi_{{\bf u}_{1}} + \varphi_{{\bf u}_{2}}]$ is an s.s. of first rank if at least two vectors $[{\bf h}_{1}]$ and $[{\bf h}_{2}]$ and two rotation matrices $[{\bi R}_{\alpha}]$ and $[{\bi R}_{\beta}]$ exist such that $[\cases{{\bf u}_{1} = {\bf h}_{1} - {\bf h}_{2} {\bi R}_{\beta}\cr {\bf u}_{2} = {\bf h}_{2} - {\bf h}_{1} {\bi R}_{\alpha}.\cr} \eqno(2.2.5.34)]$ Φ may then be estimated via the special quartet invariants $[\{\psi\} = \varphi_{{\bf u}_{1} {\bi R}_{\alpha}} + \varphi_{{\bf u}_{2}} - \varphi_{{\bf h}_{2}} + \varphi_{{\bf h}_{2} {\bi R}_{\beta} {\bi R}_{\alpha}} \eqno(2.2.5.35a)]$ and $[\{\psi\} = \{\varphi_{{\bf u}_{1}} + \varphi_{{\bf u}_{2} {\bi R}_{\beta}} - \varphi_{{\bf h}_{1}} + \varphi_{{\bf h}_{1} {\bi R}_{\alpha} {\bi R}_{\beta}}\}. \eqno(2.2.5.35b)]$ For example, $[\Phi = \varphi_{123} + \varphi_{\bar{7}\bar{2}\bar{5}}]$ in $[P2_{1}]$ may be estimated via $[\{\psi\} = \varphi_{123} + \varphi_{\bar{7}\bar{2}\bar{5}} - \varphi_{\bar{3}K\bar{1}} + \varphi_{3K1}]$ and $[\{\psi\} = \varphi_{123} + \varphi_{7\bar{2}5} - \varphi_{4K4} + \varphi_{\bar{4}K\bar{4}},]$ where K is a free index.

The set of special quartets (2.2.5.35a) and (2.2.5.35b) constitutes the first representations of Φ.

Structure seminvariants of the second rank can be characterized as follows: suppose that, for a given seminvariant Φ, it is not possible to find a vectorial index h and a rotation matrix $[{\bi R}_{\alpha}]$ such that $[\Phi - \varphi_{\bf h} + \varphi_{{\bf h} {\bi R}_{\alpha}}]$ is a structure invariant. Then Φ is a structure seminvariant of the second rank and a set of structure invariants ψ can certainly be formed, of type $[\{\psi\} = \Phi + \varphi_{{\bf h} {\bi R}_{p}} - \varphi_{{\bf h} {\bi R}_{q}} + \varphi_{{\bf l} {\bi R}_{i}} - \varphi_{{\bf l} {\bi R}_{j}},]$ by means of suitable indices h and l and rotation matrices $[{\bi R}_{p}, {\bi R}_{q}, {\bi R}_{i}]$ and $[{\bi R}_{j}]$ . As an example, for symmetry class 222, $[\varphi_{240}]$ or $[\varphi_{024}]$ or $[\varphi_{204}]$ are s.s.'s of the first rank while $[\varphi_{246}]$ is an s.s. of the second rank.

The procedure may easily be generalized to s.s.'s of any order of the first and of the second rank. So far only the role of one-phase and two-phase s.s.'s of the first rank in direct procedures is well documented (see references quoted in Sections 2.2.5.9 and 2.2.5.10).

References

Giacovazzo, C. (1977a). A general approach to phase relationships: the method of representations. Acta Cryst. A33, 933–944.Google Scholar

Giacovazzo, C. (1980a). Direct methods in crystallography. London: Academic Press.Google Scholar

Giacovazzo, C. (1980b). The method of representations of structure seminvariants. II. New theoretical and practical aspects. Acta Cryst. A36, 362–372.Google Scholar

International Tables for Crystallography (2006). Vol. B. ch. 2.2, p. 224