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2 Origin, phases, and symmetry operators

# 2 Origin, phases, and symmetry operators

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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.

–r

O xo 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

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