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

International Tables for Crystallography (2010). Vol. B, ch. 2.2, pp. 215-216   | 1 | 2 |

Section 2.2.3. Origin specification

C. Giacovazzoa*

aDipartimento Geomineralogico, Campus Universitario, 70125 Bari, Italy, and Institute of Crystallography, Via G. Amendola, 122/O, 70125 Bari, Italy
Correspondence e-mail: carmelo.giacovazzo@ic.cnr.it

2.2.3. Origin specification

| top | pdf |

• (a) Once the origin has been chosen, the symmetry operators and, through them, the algebraic form of the s.f. remain fixed.

A shift of the origin through a vector with coordinates transforms into and the symmetry operators into , where

• (b) Allowed or permissible origins (Hauptman & Karle, 1953, 1959) for a given algebraic form of the s.f. are all those points in direct space which, when taken as origin, maintain the same symmetry operators . The allowed origins will therefore correspond to those points having the same symmetry environment in the sense that they are related to the symmetry elements in the same way. For instance, if for , then the allowed origins in Pmmm are the eight inversion centres.

To each functional form of the s.f. a set of permissible origins will correspond.

• (c) A translation between permissible origins will be called a permissible or allowed translation. Trivial allowed translations correspond to the lattice periods or to their multiples. A change of origin by an allowed translation does not change the algebraic form of the s.f. Thus, according to (2.2.3.2), all origins allowed by a fixed functional form of the s.f. will be connected by translational vectors such that where V is a vector with zero or integer components.

In centred space groups, an origin translation corresponding to a centring vector does not change the functional form of the s.f. Therefore all vectors represent permissible translations. will then be an allowed translation (Giacovazzo, 1974) not only when, as imposed by (2.2.3.3), the difference is equal to one or more lattice units, but also when, for any s, the condition is satisfied.

We will call any set of cs. or ncs. space groups having the same allowed origin translations a Hauptman–Karle group (H–K group). The 94 ncs. primitive space groups, the 62 primitive cs. groups, the 44 ncs. centred space groups and the 30 cs. centred space groups can be collected into 13, 4, 14 and 5 H–K groups, respectively (Hauptman & Karle, 1953, 1956; Karle & Hauptman, 1961; Lessinger & Wondratschek, 1975). In Tables 2.2.3.1–2.2.3.4 the H–K groups are given together with the allowed origin translations.

 Table 2.2.3.1| top | pdf | Allowed origin translations, seminvariant moduli and phases for centrosymmetric primitive space groups
H–K group
Space group Pmna
Pcca
Pbam
Pccn
Pbcm
Pmmm Pnnm
Pnnn Pmmn
Pccm Pbcn
Pban Pbca
Pmma Pnma
Pnna
Allowed origin translations (0, 0, 0); (0, 0, 0) (0, 0, 0) (0, 0, 0)
;
;
;
Vector seminvariantly associated with (l)
Seminvariant modulus (2, 2, 2) (2, 2) (2) (2)
Seminvariant phases

Number of semindependent phases to be specified 3 2 1 1
 Table 2.2.3.2| top | pdf | Allowed origin translations, seminvariant moduli and phases for noncentrosymmetric primitive space groups
H–K group
Space group P1 P2 Pm P222 Pmm2 P4 P3 P312 P31m P321 R3 R32
Pc P422 P31c R3m P23
Pcc2 P6 R3c
Pma2 P3m1 P6 P622   P432
P4mm P3c1
Pnc2 P4bm

Pba2
P4cc     P6mm
Pnn2 P4nc     P6cc

Allowed origin translations (x, y, z) (0, y, 0) (x, 0, z) (0, 0, 0) (0, 0, z) (0, 0, z) (0, 0, 0) (0, 0, z) (0, 0, 0) (0, 0, z) (0, 0, 0) (x, x, x) (0, 0, 0)

Vector seminvariantly associated with (h, k, l) (h, k, l) (h, k, l) (h, k, l) (h, k, l) (l) (l)
Seminvariant modulus (0, 0, 0) (2, 0, 2) (0, 2, 0) (2, 2, 2) (2, 2, 0) (2, 0) (2, 2) (3, 0) (6) (0) (2) (0) (2)
Seminvariant phases if if ;
(mod 3) (mod 6)       ;
Allowed variations for the semindependent phases , if , if , if , if , if if (mod 3)
if (mod 2)
Number of semindependent phases to be specified 3 3 3 3 3 2 2 2 1 1 1 1 1
 Table 2.2.3.3| top | pdf | Allowed origin translations, seminvariant moduli and phases for centrosymmetric non-primitive space groups
H–K group
Space groups Immm Fmmm
Ibam Fddd
Cmcm Ibca
Cmca Imma
Cmmm
Cccm
Cmma
Ccca
Allowed origin translations (0, 0, 0) (0, 0, 0) (0, 0, 0) (0, 0, 0) (0, 0, 0)

Vector seminvariantly associated with (l)
Seminvariant modulus (2, 2) (2, 2) (2) (2) (1, 1, 1)
Seminvariant phases ; ; All
Number of semindependent phases to be specified 2 2 1 1 0
 Table 2.2.3.4| top | pdf | Allowed origin translations, seminvariant moduli and phases for noncentrosymmetric non-primitive space groups
H–K group
Space group C2 Cm Cmm2 C222 Amm2 Imm2 I222 F432 F222 I4 I422 Fmm2 I23
Cc Abm2 Iba2 F23 Fdd2
Ccc2   Ama2 Ima2     I4mm   I432
Aba2       I4cm

Allowed origin translations (0, y, 0) (x, 0, z) (0, 0, z) (0, 0, 0) (0, 0, z) (0, 0, z) (0, 0, 0) (0, 0, 0) (0, 0, 0) (0, 0, z) (0, 0, 0) (0, 0, 0) (0, 0, z) (0, 0, 0)

Vector seminvariantly associated with (k, l) (h, l) (h, l) (h, l) (h, l) (h, l) (h, l) (l) (l) (l)
Seminvariant modulus (0, 2) (0, 0) (2, 0) (2, 2) (2, 0) (2, 0) (2, 2) (2) (4) (0) (2) (4) (0) (1, 1, 1)
Seminvariant phases with (mod 4) with (mod 4) All
Allowed variations for the semindependent phases , if , if , if , if if (mod 2)
if (mod 2)
if (mod 2)
if 1 (mod 2)
All
Number of semindependent phases to be specified 2 2 2 2 2 2 2 1 1 1 1 1 1 0
• (d) Let us consider a product of structure factors being integer numbers.

The factor is the phase of the product (2.2.3.5). A structure invariant (s.i.) is a product (2.2.3.5) such that Since are usually known from experiment, it is often said that s.i.'s are combinations of phases for which (2.2.3.6) holds.

, , , , are examples of s.i.'s for .

The value of any s.i. does not change with an arbitrary shift of the space-group origin and thus it will depend on the crystal structure only.

• (e) A structure seminvariant (s.s.) is a product of structure factors [or a combination of phases (2.2.3.7)] whose value is unchanged when the origin is moved by an allowed translation.

Let 's be the permissible origin translations of the space group. Then the product (2.2.3.5) [or the sum (2.2.3.7)] is an s.s., if, in accordance with (2.2.3.1), where r is a positive integer, null or a negative integer.

Conditions (2.2.3.8) can be written in the following more useful form (Hauptman & Karle, 1953): where is the vector seminvariantly associated with the vector and is the seminvariant modulus. In Tables 2.2.3.1–2.2.3.4, the reflection seminvariantly associated with , the seminvariant modulus and seminvariant phases are given for every H–K group.

The symbol of any group (cf. Giacovazzo, 1974) has the structure , where L stands for the lattice symbol. This symbol is underlined if the space group is cs.

By definition, if the class of permissible origin has been chosen, that is to say, if the algebraic form of the symmetry operators has been fixed, then the value of an s.s. does not depend on the origin but on the crystal structure only.

• (f) Suppose that we have chosen the symmetry operators and thus fixed the functional form of the s.f.'s and the set of allowed origins. In order to describe the structure in direct space a unique reference origin must be fixed. Thus the phase-determining process must also require a unique permissible origin congruent to the values assigned to the phases. More specifically, at the beginning of the structure-determining process by direct methods we shall assign as many phases as necessary to define a unique origin among those allowed (and, as we shall see, possibly to fix the enantiomorph). From the theory developed so far it is obvious that arbitrary phases can be assigned to one or more s.f.'s if there is at least one allowed origin which, fixed as the origin of the unit cell, will give those phase values to the chosen reflections. The concept of linear dependence will help us to fix the origin.

• (g) n phases are linearly semidependent (Hauptman & Karle, 1956) when the n vectors seminvariantly associated with the are linearly dependent modulo , being the sem­invariant modulus of the space group. In other words, when is satisfied. The second condition means that at least one exists that is not congruent to zero modulo each of the components of . If (2.2.3.10) is not satisfied for any n-set of integers , the phases are linearly semindependent. If (2.2.3.10) is valid for and , then is said to be linearly semidependent and is an s.s. It may be concluded that a seminvariant phase is linearly semidependent, and, vice versa, that a phase linearly semidependent is an s.s. In Tables 2.2.3.1–2.2.3.4 the allowed variations (which are those due to the allowed origin translations) for the semindependent phases are given for every H–K group. If is linearly semindependent its value can be fixed arbitrarily because at least one origin compatible with the given value exists. Once is assigned, the necessary condition to be able to fix a second phase is that it should be linearly sem­independent of .

Similarly, the necessary condition to be able arbitrarily to assign a third phase is that it should be linearly sem­independent from and .

In general, the number of linearly semindependent phases is equal to the dimension of the seminvariant vector (see Tables 2.2.3.1–2.2.3.4). The reader will easily verify in (h, k, l) P (2, 2, 2) that the three phases , , define the origin (o indicates odd, e even).

• (h) From the theory summarized so far it is clear that a number of semindependent phases , equal to the dimension of the seminvariant vector , may be arbitrarily assigned in order to fix the origin. However, it is not always true that only one allowed origin compatible with the given phases exists. An additional condition is required such that only one permissible origin should lie at the intersection of the lattice planes corresponding to the origin-fixing reflections (or on the lattice plane h if one reflection is sufficient to define the origin). It may be shown that the condition is verified if the determinant formed with the vectors seminvariantly associated with the origin reflections, reduced modulo , has the value ±1. In other words, such a determinant should be primitive modulo .

For example, in the three reflections define the origin uniquely because Furthermore, in define the origin uniquely since

• (i) If an s.s. or an s.i. has a general value ϕ for a given structure, it will have a value −ϕ for the enantiomorph structure. If , π the s.s. has the same value for both enantiomorphs. Once the origin has been assigned, in ncs. space groups the sign of a given s.s. , π can be assigned to fix the enantiomorph. In practice it is often advisable to use an s.s. or an s.i. whose value is as near as possible to .

References

Giacovazzo, C. (1974). A new scheme for seminvariant tables in all space groups. Acta Cryst. A30, 390–395.
Hauptman, H. & Karle, J. (1953). Solution of the Phase Problem. I. The Centrosymmetric Crystal. Am. Crystallogr. Assoc. Monograph No. 3. Dayton, Ohio: Polycrystal Book Service.
Hauptman, H. & Karle, J. (1956). Structure invariants and seminvariants for non-centrosymmetric space groups. Acta Cryst. 9, 45–55.
Hauptman, H. & Karle, J. (1959). Table 2. Equivalence classes, seminvariant vectors and seminvariant moduli for the centered centrosymmetric space groups, referred to a primitive unit cell. Acta Cryst. 12, 93–97.
Karle, J. & Hauptman, H. (1961). Seminvariants for non-centrosymmetric space groups with conventional centered cells. Acta Cryst. 14, 217–223.
Lessinger, L. & Wondratschek, H. (1975). Seminvariants for space groups and . Acta Cryst. A31, 521.