(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.
 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 

 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 

 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 

 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 (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 spacegroup 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 phasedetermining process must also require a unique permissible origin congruent to the values assigned to the phases. More specifically, at the beginning of the structuredetermining 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 nset 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 originfixing 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 .
