2 Origin, phases, and symmetry operators
Tải bản đầy đủ - 0trang
62
The origin problem, invariants, and seminvariants
Relation (3.3) suggests that the origin shift produces a phase shift that is proportional (modulo2π ) to the scalar product h · x0 . The phase shift vanishes if
c
h · x0 = n,
O
b
a
Fig. 3.2
Planes (232) constitute an equiphase surface for F 232 .
S
b
a
(3.4)
with n an integer value. Since (3.4) is the classical equation defining the set of
lattice planes with Miller indices h ≡ (h, k, l), the conclusion is: if the origin is
moved from O to any point O lying on the lattice planes h, then Fh does not
change its phase value. In a concise way, we say that the lattice planes h are
an equiphasic surface for the reflection h (see Fig. 3.2).
Figure 3.3 shows the equiphasic surface for the reciprocal vector h = (530).
This also illustrates that the phase variation (for shifts x0 normal to the lattice planes) must be faster for higher reflection indices; this result can also be
derived from relation (3.3). If the origin is moved along any vector S not lying
on the lattice planes (h, k, l), from one equiphasic plane to those adjacent, then
the phase of Fh will assume all possible values in the range (−π, π).
Let us now investigate how a change in origin modifies the matrix representation of the symmetry operators for a space group with point group order
equal to m. In their daily work, crystallographers use the origins tabulated in
the International Tables for Crystallography; e.g. in P1¯ the origin coincides
with an inversion centre, in P2 with a binary axis, etc. If, for some reason, they
are obliged to move the origin, they should know how the symmetry operators
change due to the origin shift. For a primitive unit cell with origin at O, the
symmetry operators Cs are defined by the relationship
Fig. 3.3
Equiphase surface for F 530 .
rjs = Cs rj = Rs rj + Ts , s = 1, . . . , m.
(3.5)
If x0 is the origin translation vector, in the new reference system, symmetry
equivalent points will be related by the relationship
rjs = C s rj = Rs rj + Ts , s = 1, . . . , m.
(3.6)
In order to derive the relationship between each Cs and its corresponding C s
we simply substitute into (3.6) the values
rjs = rjs − x0
and
rj = rj − x0 ,
obtaining
rjs − x0 = Rs rj − Rs x0 + Ts ,
or
rjs = Rs rj − (Rs − I)x0 + Ts , s = 1, . . . , m.
P¢
–r
O xo r
O¢
r¢
P
Since (3.7) and (3.5) must be identical whatever the value of rj , it follows that
Rs = Rs . Thus, a change of origin does not affect the rotation matrices but only
the translational components of the symmetry operators, and the translation
matrix changes according to
Ts = Ts + (Rs − I)x0 , s = 1, . . . , m.
Fig. 3.4
¯
Change of origin in P1.
(3.7)
(3.8)
¯ if we choose a new origin at a disFor instance, in the space group P1,
tance x0 from a centre of symmetry (see Fig. 3.4) then a point P, defined
The concept of structure invariant
by the positional vector r = r − x0 , will correspond to an equivalent point
P’ at −r − x0 ≡ −(r + 2x0 ). Since Rs = Rs , from (3.8) the new symmetry
operators arise:
⎡
⎤
⎡ ⎤
1 0 0
0
⎢
⎥
⎢ ⎥
R 1 = ⎣ 0 1 0 ⎦, T 1 = ⎣ 0 ⎦
0 0 1
0
⎡
⎤
⎡
⎤
1¯ 0 0
−2x0
⎢
⎥
⎢
⎥
R2 = ⎣ 0 1¯ 0 ⎦, T2 = ⎣ −2y0 ⎦,
0 0 1¯
−2z0
provided that x0 ≡ (xo , yo , zo ).
3.3 The concept of structure invariant
In Section 3.2 we showed that the phase of the reflection h changes if the origin
is shifted; consequently, φh cannot be directly determined from the experimental data. The opposite statement (i.e. φh may be determined from the data)
should be illogical; indeed the amplitudes are fixed by the structure, the phases
by our arbitrary choice of the origin.
How can we determine phases from experimental data? The only way
is to check if some products of structure factors can be identified which
remain invariant whatever the origin translation. In this case the values of
such combination are origin independent and therefore depend on the structure. Obviously we have to consider products of structure factors which contain
phase information. Let us consider the product
Fh1 Fh2 · . . . · Fhn = |Fh1 Fh2 · . . . · Fhn | exp i(φh1 + φh2 + · · · · · +φhn ) .
(3.9)
According to (3.2) an origin translation will modify (3.9) into
F h1 F h2 · . . . · F hn = Fh1 Fh2 · . . . · Fhn exp[−2π i(h1 + h2 + · · · + hn ) · x0 ].
(3.10)
Relation (3.10) suggests that the product of structure factors (3.9) is invariant
under origin translation if
h1 + h2 + · · · + hn = 0.
(3.11)
Products of structure factors which satisfy (3.11) are called structure invariants
(s.i.), since their values do not depend on the origin, and therefore depend only
on the structure (Hauptman and Karle, 1953).
The simplest examples of s.i. are:
1. For n = 1, relation (3.11) confirms F 000 as the simplest structure invariant
(it is equal to the number of electrons in the unit cell).
2. For n = 2, relation (3.11) reduces to h1 + h2 = 0 or, in other notation,
h2 = −h1 . Accordingly, the product Fh F−h = |Fh |2 is a structure invariant
(which agrees well with the obvious expectation that an observation does
not depend on the origin we choose).
63
64
The origin problem, invariants, and seminvariants
3. For n = 3, relation (3.11) reduces to h1 + h2 + h3 = 0. Accordingly,
Fh1 Fh2 F−(h1 +h2 ) = |Fh1 Fh2 F−(h1 +h2 ) | exp i(φh1 + φh2 − φh1 +h2 )
(3.12)
is a s.i., specifically called triplet invariant.
4. For n = 4, relation (3.11) defines the quartet invariant,
Fh1 Fh2 Fh3 F−(h1 +h2 +h3 ) = |Fh1 Fh2 Fh3 F−(h1 +h2 +h3 ) |
× exp i(φh1 + φh2 + φh3 − φh1 +h2 +h3 ) .
Quintet, sextet, etc. s.i.s are defined by analogy.
Frequently the terms triplet, quartet, quintet invariant are referred to as:
(a) a product of normalized structure factors like Eh1 Eh2 E−(h1 +h2 ) ,
Eh1 Eh2 Eh3 E−(h1 +h2 +h3 ) , etc.;
(b) the sum of phases rather than to the product of structure factors. For
example, we will refer to(φh1 + φh2 − φh1 +h2 ) as a triplet invariant, to
(φh1 + φh2 + φh3 − φh1 +h2 +h3 ) as a quartet invariant, and so on. In equivalent notation, we can also write triplet invariants as (φh + φk − φh+k )
or (φh − φk − φh−k ), and quartet invariants as (φh + φk + φl − φh+k+l ) or
(φh − φk − φl − φh−k−l ).
Let us now suppose that, at a certain step of the phasing process, a model
structure is available and that Fp is the corresponding structure factor. Then
a new type of s.i. may be devised which simultaneously contains F and Fp
structure factors (see Sections 7.2 and 7.5); we will see that such invariants are
very useful for facilitating the passage from the model to the target structure.
Examples of this second type of invariant (the reader will easily see below that
origin translations do not modify the value of the invariants) are:
n = 2 : Fh F−ph , or in terms of phase cosine cos(φh − φph );
n = 3 : Fh Fk F−h−k
or
(φh + φk + φ−h−k ),
Fh Fk Fp−h−k
or
(φh + φk + φp−h−k ),
Fph Fk F−h−k
or
(φph + φk + φ−h−k ),
Fh Fpk F−h−k
or
(φh + φpk + φ−h−k ),
Fh Fpk Fp−h−k
or
(φh + φpk + φp−h−k ),
Fph Fk Fp−h−k
or
(φph + φk + φp−h−k ),
Fph Fpk F−h−k ,
or
(φph + φpk + φ−h−k ),
Fph Fpk Fp−h−k
or
(φph + φpk + φp−h−k ).
Allowed or permissible origins in primitive space groups
Any of the above invariants may be estimated from the amplitudes of the
corresponding observed and calculated structure factors.
Similar expressions may be obtained for quartets, quintets, etc.
3.4 Allowed or permissible origins in primitive
space groups
In Section 3.2 it has been shown that fixing the symmetry operators Cs
(and through this the algebraic form of the structure factor) is equivalent to
selecting the class of allowed origin. In order to simplify the calculations
during structural analysis and in order to handle the symmetry more easily it is convenient, in practice, to choose the origin on one or more of the
symmetry elements. Thus, it is usual to choose the origin on high-order symmetry elements when they are present: in cs. (centrosymmetric) space groups
it may be convenient to locate the origin on an inversion centre. This frequently corresponds with the choices given in the International Tables for
Crystallography.
Moving the origin from one site to another usually modifies the algebraic
representation of the symmetry operators. We define an allowed or permissible origin as all those points in direct space which, when taken as the origin,
maintain the same symmetry operators Cs . The allowed origins will therefore
correspond to points having the same ‘symmetry environment’, in the sense
that they are related to the symmetry elements in the same way. For instance,
if the origin is located on an inversion centre, all the inversion centres in P-1
that are compatible with symmetry operators Cs , given by
⎡
⎤
⎡ ⎤
⎡
⎤
⎡ ⎤
1¯ 0 0
1 0 0
0
0
⎢
⎥
⎢ ⎥
⎢
⎥
⎢ ⎥
⎥
⎢ ⎥
⎢ ¯ ⎥
⎢ ⎥
R1 = ⎢
⎣ 0 1 0 ⎦, T1 = ⎣ 0 ⎦, R2 = ⎣ 0 1 0 ⎦, T2 = ⎣ 0 ⎦,
0 0 1
0
0
0 0 1¯
will be permissible origins. To each functional form of the structure factor
there will be a class of permissible origins which, since they are all related to
the symmetry elements in the same way, will be said to be equivalent. These
constitute a class of equivalent origins or equivalence class.
Recognizing permissible origins is in general quite simple, through visual
inspection of the space group diagram in the International Tables for
Crystallography. We shall now see how to define permissible origins using
an algebraic procedure.
Let O be an origin compatible with a fixed algebraic form of the structure
factor; all other origins belonging to the same equivalence class can be defined
in a very simple way using relation (3.7). Since a shift of origin must leave
R s = Rs , it will be sufficient, in order to keep the symmetry operators
Cs and thus the algebraic form of the structure factor unchanged, to have
Ts = Ts for all values of s. More generally, because of the periodicity of crystal
lattices, it will be sufficient to have Ts − Ts = V, where V is a vector with zero
65
66
The origin problem, invariants, and seminvariants
or integer components. All origins allowed by a fixed functional form of the
structure factor will be connected by translational vectors x0 such that
(Rs − I) x0 = V,
s = 1, 2, . . . , m.
(3.13)
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.
Let us now consider some examples of the above concepts. In the space
group P2/m (compare Fig. 3.5) the origin is chosen on an inversion centre with
b as a twofold axis; the general equivalent positions in the unit cell are
(x, y, z),
Fig. 3.5
Space group P2/m.
(¯x, y¯ , z¯),
(¯x, y, z¯),
The symmetry operators are then
⎡
⎤
1 0 0
⎢
⎥
⎥
R1 = I = ⎢
⎣ 0 1 0 ⎦,
0 0 1
⎡
1¯ 0 0
(x, y¯ , z).
⎡
⎤
⎢
⎥
¯ ⎥
R2 = ⎢
⎣ 0 1 0 ⎦,
0 0 1¯
⎤
⎡
⎢
⎥
⎥
R3 = ⎢
⎣ 0 1 0 ⎦,
0 0 1¯
1¯ 0 0
1 0 0
⎤
⎢
⎥
¯ ⎥
R4 = ⎢
⎣ 0 1 0 ⎦,
0 0 1
T1 = T2 = T3 = T4 = 0.
The Rs − I matrices are
⎡
0 0 0
⎤
⎡
⎢
⎥
⎥
R1 − I = ⎢
⎣ 0 0 0 ⎦,
0 0 0
⎡
⎤
2¯ 0 0
⎢
⎥
⎥
R3 − I = ⎢
⎣ 0 0 0 ⎦,
0 0 2¯
⎤
⎢
⎥
¯ ⎥
R2 − I = ⎢
⎣ 0 2 0 ⎦,
0 0 2¯
⎡
⎤
0 0 0
⎢
⎥
¯ ⎥
R4 − I = ⎢
⎣ 0 2 0 ⎦.
0 0 0
It is easy to verify that the translations
⎡
⎤
⎡
⎤
n1 /2
0
⎢
⎥
⎢
⎥
⎣ 0 ⎦
⎣ n2 /2 ⎦
0
2¯ 0 0
0
⎡
⎤
0
⎢
⎥
⎣ 0 ⎦,
n3 /2
with n1 , n2 , n3 integer numbers, satisfy (3.13) and therefore connect origins
belonging to the same equivalence class. Any combination of the above three
Allowed or permissible origins in primitive space groups
67
translations will also connect origins allowed by the given functional form of
the structure factor. From the three basic translations
⎡1⎤
⎡ ⎤
⎡ ⎤
0
0
2
⎢ ⎥
⎢1⎥
⎢ ⎥
⎢ 0 ⎥,
⎢ ⎥,
⎢ 0 ⎥,
⎣ ⎦
⎣2⎦
⎣ ⎦
1
0
0
2
we can derive a sort of lattice of permissible translations. Within a single unit
cell the permissible origins will be defined by the translation vectors
⎡ ⎤ ⎡1⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡1⎤ ⎡1⎤ ⎡1⎤
0
0
0
0
2
2
2
2
⎢ ⎥ ⎢ ⎥ ⎢1⎥ ⎢ ⎥ ⎢1⎥ ⎢ ⎥ ⎢1⎥ ⎢1⎥
⎢ 0 ⎥, ⎢ 0 ⎥, ⎢ ⎥, ⎢ 0 ⎥, ⎢ ⎥, ⎢ 0 ⎥, ⎢ ⎥, ⎢ ⎥.
⎣ ⎦ ⎣ ⎦ ⎣2⎦ ⎣ ⎦ ⎣2⎦ ⎣ ⎦ ⎣2⎦ ⎣2⎦
1
1
1
1
0
0
0
0
2
2
2
2
(3.14)
In the space group P4/n, when the origin is chosen on 4¯ (compare Fig. 3.6)
¯
at 14 , 14 , 0 from an inversion centre, the general equivalent positions are
(x, y, z),
(y, x¯ , z¯),
1
1
1
1
− x, − y, z¯ , (¯x, y¯ , z),
+ x, + y, z¯ ,
2
2
2
2
1
1
1
1
− y, + x, z ,
+ y, − x, z ,
2
2
2
2
The Rs − I matrices are
⎡
0 0 0
⎤
⎡
2¯ 0 0
⎤
⎢
⎥
⎥,
0
0
0
R1 − I = ⎢
⎣
⎦
0 0 0
⎢
⎥
¯ 0 ⎥,
R2 − I = ⎢
0
2
⎣
⎦
0 0 2¯
⎡
⎡
2¯ 0 0
⎤
0 0 0
⎤
⎢
⎥
¯ ⎥
R3 − I = ⎢
⎣ 0 2 0 ⎦,
0 0 0
⎢
⎥
⎥
R4 − I = ⎢
⎣ 0 0 0 ⎦.
0 0 2¯
⎡
⎡
1¯ 1¯ 0
⎤
1¯ 1 0
⎤
⎢
⎥
¯ ⎥
R5 − I = ⎢
⎣ 1 1 0 ⎦,
0 0 2¯
⎢
⎥
¯ ¯ ⎥
R6 − I = ⎢
⎣ 1 1 0 ⎦,
0 0 2¯
⎡
⎡
1¯ 1¯ 0
⎤
⎢
⎥
¯ ⎥
R7 − I = ⎢
⎣ 1 1 0 ⎦,
0 0 0
1¯ 1 0
⎤
⎢
⎥
¯ ¯ ⎥
R8 − I = ⎢
⎣ 1 1 0 ⎦.
0 0 0
(¯y, x, z¯),
Fig. 3.6
Space group P4/n.
68
The origin problem, invariants, and seminvariants
A lattice of permissible origins with basic translations,
⎡1⎤ ⎡ ⎤ ⎡ ⎤
0
0
2
⎢ ⎥ ⎢1⎥ ⎢ ⎥
⎢ 0 ⎥, ⎢ ⎥, ⎢ 0 ⎥,
⎣ ⎦ ⎣2⎦ ⎣ ⎦
1
0
0
2
satisfies relations (3.13) for s = 1, 2, 3, 4; but in order to also satisfy (3.13) for
s ≥ 5, the sum and the difference of the components of the translation vector x0
in the (a, b) plane must be integer numbers. The allowed translations defining
the permissible origins will be
⎡ ⎤ ⎡ ⎤ ⎡1⎤ ⎡1⎤
0
0
2
2
⎢ ⎥ ⎢ ⎥ ⎢1⎥ ⎢1⎥
⎢ 0 ⎥, ⎢ 0 ⎥, ⎢ ⎥, ⎢ ⎥.
(3.15)
⎣ ⎦ ⎣ ⎦ ⎣2⎦ ⎣2⎦
1
1
0
0
2
2
Let us now apply equation (3.13) to some n.cs. (non-centrosymmetric) space
groups. In the space group P3, with the origin on a threefold axis (compare Fig.
3.7), the general equivalent positions are
(x, y, z),
Fig. 3.7
Space group P3.
The Rs − I matrices are
⎡
⎤
0 0 0
⎢
⎥
⎥
R1 − I = ⎢
⎣ 0 0 0 ⎦,
0 0 0
(¯y, x − y, z),
⎡
(y − x, x¯ , z).
1¯ 1¯ 0
⎤
⎢
⎥
¯ ⎥
R2 − I = ⎢
⎣ 1 2 0 ⎦,
0 0 0
⎡
2¯ 1 0
⎤
⎢
⎥
¯ ¯ ⎥
R3 − I = ⎢
⎣ 1 1 0 ⎦.
0 0 0
We can easily verify that origin translations with components 12 , 13 in the
(a, b) plane satisfy (3.13) for all values of s. Note, however, that relations
(3.13) do not imply any restriction to shifts in the z direction. The allowed
translations in the unit cell are then given by
⎡ ⎤ ⎡2⎤ ⎡1⎤
0
3
3
⎢ ⎥ ⎢1⎥ ⎢2⎥
⎢ 0 ⎥, ⎢ ⎥, ⎢ ⎥.
(3.16)
⎣ ⎦ ⎣3⎦ ⎣3⎦
z
z
z
Let us now consider, as a further example, space group R3 (compare
Fig. 3.8). With rhombohedral axes and the origin on a threefold axis, the
general equivalent positions are
(x, y, z),
Fig. 3.8
Space group R3.
The Rs − I matrices are
⎡
⎤
0 0 0
⎢
⎥
⎥
R1 − I = ⎢
⎣ 0 0 0 ⎦,
0 0 0
(z, x, y),
⎡
(y, z, x).
1¯ 0 1
⎤
⎢
⎥
¯ ⎥
R2 − I = ⎢
⎣ 1 1 0 ⎦,
0 1 1¯
⎡
1¯ 1 0
⎤
⎢
⎥
¯ ⎥
R3 − I = ⎢
⎣ 0 1 1 ⎦.
1 0 1¯
The concept of structure seminvariant
In order to satisfy relationships (3.13), one has to choose translation vectors
with equal components, i.e.
⎡ ⎤
x
x0 = ⎣ x ⎦.
(3.17)
x
The reader will easily be able to find the permissible translations for any other
primitive space group using the algebraic procedure described in this section.
There are two important points that should also be noted. The first is related
to the absence in equation 3.13 of the translational components of the symmetry operators. The distribution in direct space of the allowed origins is only
dependent on the rotation matrices; for a given space group the allowed translations are therefore independent of the chosen form of the structure factor.
More explicitly, let us consider the space group P4/n, mentioned above. When
¯ the general equivalent positions
the origin is chosen at 1¯ at 14 , 14 , 0 from 4,
become
1
1
1
1
1
(x, y, z), (¯x, y¯ , z¯),
− x, − y, z ,
+ x, + y, z¯ ,
y¯ , + x, z¯ ,
2
2
2
2
2
1
1
1
+ y, x¯ , z¯ ,
− y, x, z ,
y, − x, z ;
2
2
2
The reader can immediately verify that the rotation matrices are unchanged
in the new reference system, even if the algebraic form of the structure factor
is modified; (3.15) are therefore the only allowed translations in this case, as
well. Note that the four inversion centres in Fig. 3.6 do not all belong to the
same equivalence class, because of the presence of the n glide.
The second point to stress is that the permissible translations, depending
on the rotation matrices only, will remain the same in those space groups in
which simple axes or planes are substituted by screw axes or glide planes (for
instance, P2/m, P21 /m, P2/c, and P21 /c).
We will call any set of cs. or n.cs. space groups having the same allowed origin translation a Hauptman–Karle (H–K) family. The 94 n.cs. primitive space
groups can be collected in 13 H–K families and the 62 primitive cs. groups
in 4 H–K families. The second and the third rows of Tables 3.1 and 3.2 show
these results explicitly.
It is worth emphasizing, for the benefit of the reader who wishes to refer
to the literature for further details, that this concept of permissible origins is
formally different from that given in the classical papers of Hauptman and
Karle (1953b, 1956). We have followed the treatment described by Giacovazzo
(1974a) because this provides a more general treatment.
3.5 The concept of structure seminvariant
In Section 3.3 we showed that products of structure factors (or the sum of
phases) exist, the values of which do not depend on the origin, but only on the
structure; we called them structure invariant (s.i.). Their importance is related
to the following specific property: their phase values may be estimated from
measured amplitudes.
69
70
The origin problem, invariants, and seminvariants
Table 3.1 Allowed origin translations, seminvariant moduli, and phases for centrosymmetric primitive space groups
H–K family
Space group
(h, k, l)P(2, 2, 2)
(h + k, l)P(2, 2)
(l)P(2)
(h + k + l)P(2)
P1¯
Pmna
P m4
P 4n mm
P3¯
R3¯
P m2
Pcca
P 4m2
P 4n cc
¯
P31m
¯
R3m
P 2m1
Pbam
P 4n
P 4m2 mc
¯
P31c
¯
R3c
P 2c
Pccn
P 4n2
P 4m2 cm
¯
P3m1
Pm3¯
P 2c1
Pbcm
P m4 mm
P 4n2 bc
¯
P3c1
Pn3¯
Pmmm
Pnnm
P m4 cc
P 4n2 nm
P m6
Pa3¯
Pnnn
Pmmn
P 4n bm
P 4m2 bc
P 6m3
¯
Pm3m
Pccm
Pbcn
P 4n nc
P 4m2 nm
P m6 mm
¯
Pn3n
Pban
Pbca
P m4 bm
P 4n2 mc
P m6 cc
¯
Pm3n
Pmma
Pnma
P m4 nc
P 4n2 cm
P 6m3 cm
¯
Pn3m
P 6m3 mc
Pnna
Allowed origin translations
0, 12 ,
(0, 0, 0);
1
2 , 0, 0
1
2
(0, 0, 0)
;
1
1
2 , 0, 2
0, 0,
0, 12 , 0 ;
1 1
2, 2,0
1 1
2, 2,0
1 1 1
2, 2, 2
1 1 1
2, 2, 2
0, 0,
1
2
;
1
2
(0, 0, 0)
0, 0,
1
2
(0, 0, 0)
1 1 1
2, 2, 2
Vector hs seminvariantly
associated with h = (h, k, l)
Seminvariant modulus ωs
(h, k, l)
(h + k, l)
(l)
(h + k + l)
(2, 2, 2)
(2, 2)
Seminvariant
φeee
φeee ; φooe
Number of semi-independent
phases to be specified
3
2
(2)
φeee ; φeoe
φoee ; φooe
1
(2)
φeee ; φooe
φoeo ; φeoo
1
We now introduce a related concept, structure seminvariant (s.s.). Let us
suppose that, for a given space group, the symmetry operators Cs , and therefore
the allowed origins, have been fixed. If x0 is a generic translation, equation (3.3) suggests that the phase of a structure factor will change, after an
origin shift, by a finite quantity. The central question for this section is: do
particular phases or combinations of phases exist, the values of which do not
change when the origin is moved within the set of allowed origins? If yes, we
will say that the phases or the combination of phases are s.s. (Hauptman and
Karle, 1956).
The s.s. are important because they can be estimated from measurements,
provided that the symmetry operators have been fixed (and they usually remain
fixed during the full phasing process). Indeed, the fixed nature of the symmetry
operators restricts the allowed origins to a specific subset of points, within
Table 3.2 Allowed origin translations, seminvariant moduli, and phases for non-centrosymmetric primitive space groups
H–K family
(h, k, l)
(h, k, l)
(h, k, l)
(h, k, l)
(h, k, l)
(h + k, l)
P(0, 0, 0) P(2, 0, 2) P(0, 2, 0) P(2, 2, 2) P(2, 2, 0) P(2, 0)
(h + k, l)
P(2, 2)
(h – k, l)
P(3, 0)
(2h + 4k + 3l)
P(6)
(l)P(0)
Space group
P1
P2
P21
Pm
Pc
P222
P2221
P21 21 2
P21 21 21
Pmm2
Pmc21
Pcc2
Pma2
Pca21
Pnc2
Pmn21
Pba2
Pna21
Pnn2
P4
P41
P42
P43
P4mm
P4bm
P42 cm
P42 nm
P4cc
P4nc
P42 mc
P42 bc
P3
P31
P32
P3m1
P3c1
P312
P31 12
P32 12
P6
¯
P6m2
¯
P6c2
P31m
P31c
P6
P61
P65
P64
P63
P62
P6mm
P6cc
P63 cm
P63 mc
Allowed origin
translations
(x, y, z)
(0, y, 0)
0, y, 12
1
2 , y, 0
1
1
2 , y, 2
(x, 0, z)
x, 12 , z
(0, 0, 0)
1
2 , 0, 0
0, 12 , 0
0, 0, 12
0, 12 , 12
1
1
2 , 0, 2
1 1
,
,
2 2 0
1 1 1
2, 2, 2
(0, 0, z)
0, 12 , z
1
2 , 0, z
1 1
2, 2,z
(0, 0, z)
1 1
2, 2,z
P4¯
P422
P421 2
P41 22
P41 21 2
P42 22
P42 21 2
P43 22
P43 21 2
¯
P42m
¯
P42c
¯ 1m
P42
¯ 1c
P42
¯
P4m2
¯
P4c2
¯
P4b2
¯
P4n2
(0, 0, 0)
0, 0, 12
1 1
2, 2,0
1 1 1
2, 2, 2
(0, 0, z)
1 2
3, 3,z
2 1
3, 3,z
(0, 0, 0)
0, 0, 12
1 2
3, 3,0
1 2 1
3, 3, 2
2 1
3, 3,0
2 1 1
3, 3, 2
(0, 0, z)
(h + k + l)
P(0)
(h + k + l)
P(2)
P321
P31 21
P32 21
P622
P61 22
P65 22
P62 22
P64 22
P63 22
¯
P62m
¯
P62c
R3
R3m
R3c
R32
P23
P21 3
P432
P42 32
P43 32
P41 32
¯
P43m
¯
P43n
(0, 0, 0)
0, 0, 12
(x, x, x)
(0, 0, 0)
1 1 1
2, 2, 2
(l)P(2)
Table 3.2 (Continued)
H–K family
Vector hs
seminvariantly
associated with
h = (h, k, l)
Seminvariant
modulus ωs
Seminvariant
phases
Allowed
variations for
the semiindependent
phases
Number of semiindependent
phases to be
specified
(h, k, l)
(h, k, l)
(h, k, l)
(h, k, l)
(h, k, l)
(h + k, l) (h + k, l) (h – k, l)
P(0, 0, 0) P(2, 0, 2) P(0, 2, 0) P(2, 2, 2) P(2, 2, 0) P(2, 0)
P(2, 2)
P(3, 0)
(2h + 4k + 3l)
P(6)
(l)P(0)
(l)P(2)
(h + k + l)
P(0)
(h, k, l)
(h, k, l)
(h, k, l)
(h, k, l)
(h, k, l)
(h + k, l) (h + k, l) (h – k, l)
(2h + 4k + 3l)
(l)
(l)
(h + k + l) (h + k + l)
(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)
φ 000
φ e0e
φ 0e0
φ eee
φ ee0
φ ee0
φ oo0
φ eee
φ ooe
φ hke
φh,k,h+
¯ k¯
φ eee ; φ ooe
φ oeo ; φ ooe
∞ ,
2
2 if
h=l=0
∞ ,
2 if
l=0
∞ ,
2 if
l=0
2
φ hkl if
2h + 4k + 3l ≡ 0
(mod 6)
2 if
h≡k
(mod 3) 3 if
l ≡ 0 (mod 2)
φ hk0
∞ ,
2 if
k=0
φ hk0 if
h–k≡0
(mod 3)
∞ ,
3 if
l=0
∞
2
3
3
3
2
2
1
l
∞
3
3
2
1
(h + k + l)
P(2)
∞
1
2
1