Blake A.J.(ed.) Crystal Structure Analysis

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282 Introduction to twinning

domain. The 413 reﬂection in the grey domain would be unaffected

by twinning.

It is likely that the example given here would index readily on

the tetragonal supercell, but notice the bizarre systematic absences in

Fig. 18.4iv. Zones of unusual systematic absences are frequently a sign

that a crystal is non-merohedrally twinned. This pseudo-translational

symmetry should enable the true orthorhombic cell to be inferred, and it

can be characterized by a strongnon-origin peak in a Patterson synthesis

(see Section 18.10, Example 8).

In orthorhombic and higher systems potential non-merohedral twin

laws can often be derived from inspection of the unit cell dimensions.

In low-symmetry crystals the twin law is usually less obvious (general

procedures are given below), but it is possible to make a few general

observations that apply to monoclinic crystals. In these cases the twin

law is often found to be a 2-fold axis along the unit cell a-orc-axes. The

matrix for a two-fold rotation about the a-axis is:

⎛

⎜

⎝

100

0 −10

2c cos β

0 −1

⎞

⎟

⎠

The corresponding rotation about c is:

⎛

⎜

⎝

−10

2a cos β

0 −10

00 1

⎞

⎟

⎠

Likelytwinlawscanbe derived for monoclinic crystals by evaluating the

off-diagonal terms in these matrices; if near-rational values are obtained

the corresponding matrix should be investigated as a possible twin law.

18.8 The derivation of non-merohedral

twin laws

Diffraction patterns from non-merohedrally twinned crystals contain

many more spots than would be observed for an untwinned sam-

ple. Since individual spots may come from different domains of the

twin such diffraction patterns are frequently difﬁcult to index. Over-

lap between reﬂections may be imperfect in some or all zones of data

affected,and integration and data reductionneeds to be performed care-

fully. Software for integrating datasets from non-merohedral twins and

performing absorption corrections has recently become available [for

example, SAINT version 7 (Bruker-Nonius, 2002); TWINABS (Sheldrick,

2002)].

18.9 Common signs of twinning 283

Excellent programs such as DIRAX (Duisenberg, 1992) and

CELL_NOW (Sheldrick, 2005) have been developed to index diffrac-

tion patterns from non-merohedral twins. In many cases a pattern can

be completely indexed with two orientation matrices, and both these

programs offer procedures by which the relationship between these

alternative matrices is analyzed to suggest a twin law: if two domains

are indexed with orientation matrices A

and A

the twin law is given

by the product A

−1

It is usually the case that twinning can be described by a two-fold

rotation about a direct or reciprocal lattice direction. Indeed, it has been

shown by Le Page and Flack that, if two such directions are parallel, and

the vectors describing them have a dot product greater than two, then a

higher-symmetry supercell can be derived. The program CREDUC (Le

Page, 1982) is extremely useful for investigating this; it is available in

the Xtal suite of software (Hall et al., 1992), which can be downloaded

from http://www.ccp14.ac.uk. The same procedure is available in the

LePage routine in PLATON (Spek, 2003).

It is sometimes the case that the ﬁrst intimation the analyst has that

a crystal is twinned is during reﬁnement. Symptoms such as large,

inexplicable difference peaks and a high R factor may indicate that

twinning is a problem, while careful analysis of poorly ﬁtting data

reveals that they belong predominantly to certain distinct zones in

which |F

obs

is systematically larger than |F

calc

. If twinning is not

taken into account it is likely that these zones are being poorly mod-

elled, and that trends in their indices may provide a clue as to a

possible twin law. The computer program ROTAX (Cooper et al., 2002;

also available from http://www.ccp14.ac.uk) makes use of this idea to

identify possible twins laws. A set of data with the largest values of

[|F

obs

−|F

calc

]/σ (|F

obs

) is identiﬁed and the indices transformed by

two-fold rotations or other symmetry operations about possible direct

and reciprocal lattice directions. Matrices that transform the indices of

the poorly ﬁtting data to integers are identiﬁed as possible twin laws.

The analyst then has a set of potential matrices that might explain the

sourceof the reﬁnementproblems described above.Arelated procedure,

TwinRotMat, available in PLATON (Spek, 2003), works by identifying

reﬂections with very similar d -spacings.

18.9 Common signs of twinning

The following list of common signs of twinning is based on that origi-

nally given by Herbst-Irmer and Sheldrick (1998). Use of these signs in

diagnosing twinning problems is illustrated in Section 18.10.

1. The metric symmetry of the lattice is higher than the Laue symmetry of

the diffraction pattern.

The reasons for this were discussed in Section 18.4. Three common

cases in small-molecule crystallography are as follows.

284 Introduction to twinning

•

Monoclinic P with β near 90

◦

(metrically orthorhombic); use

a two-fold axis along either a or c as the twin law.

• Triclinic, but transformable to monoclinic C; use a two-fold

rotation about the pseudo-monoclinic b-axis direction as the

twin law.

• Monoclinic P, but transformable to orthorhombic C; use a

two-fold rotation about one of the pseudo-orthorhombic cell

axes as the twin law. (The axis chosen should not correspond

to the monoclinic b-axis!)

If the twin scale factor is near 0.5, R

int

in the high-symmetry group

will be the same or only slightly higher than in the lower-symmetry

group. Even when the twin scale factor deviates signiﬁcantly

from 0.5 the higher symmetry R

int

may still be less than about

0.4; values of 0.60 or higher might be expected for untwinned

samples (although pseudo-symmetry in, for example, heavy-atom

positions can give rise to a similar effect).

2. The space group can not be determined, or, if it can, it is unusual.

Zones of systematic absences can be contaminated by overlap with

reﬂections from another domain in the twin.

What constitutes ‘unusual’ depends on the material being stud-

ied. For example, space group C 2/m is uncommon for molecular

compounds but not uncommon at all for ‘extended’ or ‘inorganic’

structures. In the author’s experience of molecular crystal struc-

tures, however, crystals appearing to be C-centred orthorhombic

are often (though not always) twinned monoclinic P, and those

appearing from systematic absences to be in C2, Cm or C2/m are

triclinic twins in P

1. Note that, even here, space group C 2 is quite

common for enantiopure compounds, and C2, Cm or C2/m are not

uncommon at all for ‘inorganic’ compounds such as metal oxides.

Finally, of course, some compounds really do crystallize in unusual

space groups. Unusual zones of absences may not be revealed by

a space group determination program, but can be identiﬁed by a

large peak in a Patterson map or by inspection of reciprocal lattice

plots.

3. High symmetry.

Low-symmetry tetragonal, trigonal, rhombohedral,hexagonal and

cubic crystals are always potentially twinned by merohedry; low-

symmetry trigonal crystals seem to be particularly prone. It is good

practice to test such structures for twinning as a matter of routine:

possible twin laws are given in Section 18.5. 95% of molecular crys-

tal structures are either triclinic, monoclinic or orthorhombic, and

so pseudo-merohedral twinning should always be kept in mind

when such a material appears to be tetragonal or higher symmetry.

High symmetry is common for ‘inorganic’ structures.

4. The value of |E

− 1| is low.

The reasons for this were discussed in Section 18.4.

5. The sample being studied has undergone a phase transition.

Examples 285

This was brieﬂy discussed in Section 18.3; and examples are

available in Gaudin et al. (2000) and Guelylah et al. (2001).

6. Indexing problems.

Perhaps the diffraction pattern did not index using default pro-

cedures. Alternatively, the unit cell volume may seem too high

(implying Z



> 3) or there is a very long cell axis; though both

of these features are possible for untwinned crystals, they are

unusual. Close inspection of peak proﬁles is a useful diagnos-

tic tool: twinning may be evidenced by a mixture of sharp and

split peaks in the diffraction pattern. Indexing problems are a

very common warning sign of non-merohedral twinning. Pseudo-

merohedral twins may be difﬁcult to index if peaks from different

domains overlap well at low resolution but not at high resolu-

tion: this may occur, for example, in a monoclinic crystal where β

deviates by more than ∼0.5

◦

from 90

◦

7. The structure does not appear to solve.

Most small-molecule structures solve readily with modern soft-

ware, and twinning should be considered in cases where automatic

solution fails (especially if the dataset appears to be of good qual-

ity). The possibility that the crystalbeing studiedis verydifferentin

composition from that intended should also be carefully explored.

Twinning reveals itself in the Patterson function, which becomes

a weighted superposition of the function derived from each

domain; this is discussed by Dauter (2003).

8. The reﬁnement is unsatisfactory.

The R factor may stick at a value much higher than R

int

; the

difference map may show inexplicable peaks; F

may be consis-

tently higher than F

for poorly ﬁtting data; or F

/F

 may be

systematically high for the weakest data.

18.10 Examples

Example 1. This example illustrates items 1, 2 and 7 in Section 18.9.

Crystals of the compound C

N (Fig. 18.2) diffracted rather weakly.

The unit cell appeared to be orthorhombic with dimensions a = 8.28,

b = 12.92, c = 41.67 Å. The volume ﬁts for Z = 8, the value of |E

− 1|

was 0.725. Z = 8 is not unusual for orthorhombic crystals; the c-axis

is long, but there were no other indexing solutions that were able to

account for all the reﬂections in the diffraction pattern. Although the

crystal was twinned the mean value of |E

− 1| is not abnormal for

a non-centrosymmetric structure. However, the space group assuming

orthorhombic symmetry appeared to be P22

2, which is very rare.

Mergingstatistics(R

int

)wereas follows: mmm,0.14;2/m with a unique,

0.13; 2/m with b unique 0.06, 2/m with c unique 0.09. The lowest R

int

assumed monoclinic symmetry with the b-axis of the orthorhombic

cell corresponding to the unique axis of the monoclinic cell. Notice,

286 Introduction to twinning

though, that merging in the higher-symmetry Laue class (mmm) yields

int

that is only moderately higher than in 2/m.

Taken with the space group information described above this seemed

to be a twin. The twin law used was

⎛

⎝

10 0

0 −10

00−1

⎞

⎠

and space group P2

was assumed. The symmetry of the lattice is mmm

(the order of this group is 8); the crystal structure belongs to point

group 2/m (order 4). Hence, we need to specify (8/4 ) − 1 = 1 twin

law (Eqn. 18.2).

The structure was difﬁcult to solve, and repeated attempts to ﬁnd a

solution in different direct methods packages were unsuccessful. The

molecule contains a rigid fragment, and a position and orientation for

one molecule (there are four in the asymmetric unit) was obtained by

Patterson search methods (DIRDIF, Beurskens et al., 1996) using the rigid

part of the molecule as a search fragment. The structure was completed

by iterative cycles of least-squares and Fourier syntheses (SHELXL97;

Sheldrick, 2008). A search for missed space-group symmetry did not

reveal any glide or mirror planes: the ﬁnal R factor was 0.1, and the twin

scale factor was 0.392(5).

Patterson methods are normally applied to the solution of heavy-

atom structures, but they are a valuable alternative to direct methods

when the latter fail for light-atom structures containing a rigid fragment.

Solution packages do not, as a rule, enable a twin law to be applied dur-

ing structure solution. The exception to this is the program SHELXD

(Sheldrick, 2008), which has proved to be very useful for solution of

twinned structures.

Example 2. This is an example of an apparently ‘impossible’ space

group (item 2 in Section 18.9), and also illustrates the comments made

about twinning in Sections 18.2 and 18.3. A nickel complex was appar-

ently orthorhombic P with cell dimensions a = 9.93, b = 10.95, c =

14.14 Å; all three cell angles were indistinguishable from 90

◦

, and R

int

was 0.079 for mmm symmetry. However, only the following systematic

absences were observed: h00 with h odd; 0k0 with k odd; 00l with l

odd, and hk0 with h + k odd – a pattern that is not consistent with any

orthorhombic space group. If the crystal system is taken to be mono-

clinic, with the original c-axis corresponding to the unique monoclinic

b direction, the space group is P2

/n. R

int

for 2/m symmetry is 0.039.

The structure solved for Ni and a few light-atom positions by direct

methods. The twin law

⎛

⎝

10 0

0 −10

00−1

⎞

⎠

was applied, and the remaining atoms were located in a difference map.

The symmetry of the lattice is effectively mmm, and the crystal structure

Examples 287

belongs to point group 2/m, and as in example 1 we need to specify

(8/4) − 1 = 1 twin law. The ﬁnal R factor [based on F and data with

F > 4σ(F)] was 0.061, and the twin scale factor was 0.373(3).

Example 3. This example also illustrates a structure in which correct

space group determination was hindered by twinning. Twinning can

cause systematic absences from one domain to overlap with reﬂections

from a second domain, and this may yield a pattern of absences that is

inconsistent with any known space group (as we saw in Example 2), or

that leads to an incorrect space group assignment (as is illustrated here).

The diffraction pattern of a palladium complex was found to index

on a primitive monoclinic unit cell with dimensions a = 3.84, b = 9.73,

c = 21.20 Å, β = 95.4

◦

; R

int

= 0.037 for point group 2/m. This cell can

be transformed using the matrix

M =

⎛

⎝

100

−10−2

010

⎞

⎠

to a metrically orthorhombic C cell with dimensions a = 3.84, b = 42.20,

c = 9.73 Å, α = β = 90

◦

, γ = 89.8

◦

, but R

int

for mmm symmetry was

0.434. The merging statistics imply that the crystal is monoclinic. The

systematic absence data are summarized in Table 18.2.

The data in Table 18.2 appear to show a ‘clean’ set of absences for the

0k0 zone, but signiﬁcant intensity for the three h0l zones, indicating that

thespacegroupisP2

.Noticethe values of Iforthedifferentconditions

on h0l: that for h + k odd is over ten times smaller than either h odd or

l odd. In fact, the crystal is twinned, and the space group is P2

/n, but

the n-glide absences are contaminated by overlap of reﬂections from the

different domains.

As in the previous examples, we need to specify one twin law. A two-

fold rotation about either the a-orb-axis of the orthorhombic cell could

be used, but it is not necessary to use both (this can be proved using

coset decomposition). A two-fold rotation about the orthorhombic c-

axis should not be used as a twin law as this corresponds to the b-axis

Table 18.2. Systematic absence data for the palladium complex in Example

3. N is the number of data meeting the condition indicated in the ﬁrst row;

N (I > 3σ) is the number of these with signiﬁcant intensity; I and σ are the

intensity and uncertainty of the intensity, respectively. I indicates the mean

value of I. Data calculated using XPREP.

Condition

0k0,

k odd

h0l,

h odd

h0l,

l odd

h0l,

h + kodd

N 24 437 431 432

N (I > 3σ) 1 323 195 128

I 2.6 326.8 310.5 23.9

I/σ  0.5 7.6 5.4 2.8

288 Introduction to twinning

of the monoclinic cell, and a two-fold axis about this direction is part of

the monoclinic symmetry already.

With respect to the orthorhombic cell axes, a two-fold rotation about

the orthorhombic a-axis direction is given by the matrix

R =

⎛

⎝

10 0

0 −10

00−1

⎞

⎠

However, it is necessary to express this operation with respect to the

monoclinic axis system because this is being used to describe the

structure. The matrix M, which transforms the monoclinic cell to the

orthorhombiccell, was deﬁned above, and the requiredtwin law is given

by the triple matrix product M

−1

RM:

⎛

⎝

100

001

−0.5 −0.5 0

⎞

⎠

⎛

⎝

10 0

0 −10

00−1

⎞

⎠

⎛

⎝

100

−10−2

010

⎞

⎠

⎛

⎝

100

0 −10

−10−1

⎞

⎠

This procedure can be used whenever it is necessary to transform an

operation from one axis system to another.

Consider the effect of the twin law on the h0l reﬂections:

⎛

⎝

100

0 −10

−10−1

⎞

⎠

⎛

⎝

⎞

⎠

⎛

⎝

−h − l

⎞

⎠

For example, the systematicallyabsent 102 reﬂection will overlap with

the 10

3 reﬂection from the second domain: the 103 reﬂection is not sys-

tematically absent. This explains why the systematic absences for the

n-glide appear to have some intensity in Table 18.2.

Even though it was twinned this crystal structure solved easily, and

reﬁned to R = 0.042; the twin scale factor was only 0.07, which explains

the very different merging statistics in 2/m and mmm.

This structure could have been solved in P2

, but reﬁnement would

have been unstable; the extra symmetry could have been located by a

program such as PLATON/ADDSYM or MISSYM. Symmetry checking

should be carried out as a matter of routine for all crystal structures, but

it is particularly important to do this for twinned structures because of

the extra pitfalls attendant on space group determination.

Example 4. The diffraction pattern measured from a crystal of a nickel

complex of composition C

NiO

indexed on the monoclinic C-

centred unit cell a = 15.20, b = 54.49, c = 10.14 Å, β = 90.73

◦

. R

int

for

2/m symmetry was 0.076. The space group appeared to be one of C2, Cm

or C2/m; solution in each of these was attempted, but no recognizable

structure solution was obtained. Merging in Laue class

1 yielded R

int

0.038, which is somewhat better than in 2/m, and this indicated that the

structure was really triclinic.

Examples 289

The conventional triclinic settingof the unit cell is a = 10.14,b = 15.20,

c = 28.29 Å, α = 74.42, β = 89.80 and γ = 89.27

◦

, and transformation is

accomplished with the matrix:

⎛

⎝

00−1

100

0.5 −0.5 0

⎞

⎠

(a cell reduction program, such as XPREP, will providethis information).

The twin law is a two-fold rotation about the pseudo-monoclinic b

direction, but this needs to be expressed with respect to the triclinic

axes. As in Example 3, this requires the formation of a triple matrix

product:

⎛

⎝

00−1

100

0.5 −0.5 0

⎞

⎠

⎛

⎝

−10 0

010

00−1

⎞

⎠

⎛

⎝

010

01−2

−10 0

⎞

⎠

⎛

⎝

−100

0 −10

0 −11

⎞

⎠

where the three matrices to be multiplied are (from right to left) the

triclinic to monoclinic transformation, a two-fold axis about b in the

monoclinic cell and the monoclinic to triclinic transformation. Note that

the ﬁrst and third matrices are the inverses of each other.

The crystal structure solved readily by direct methods in P

1, and

reﬁned to R = 0.037, with a twin scale factor of 0.2668(7). Z



for this

structure was 4, which is unusually high, though symmetry checking

using PLATON/ADDSYM did not indicate any missed translational or

other symmetry.

An alternative, but equivalent strategy in this example would have

been to work in the non-standard setting C

1, using the pseudo-

monoclinic axis system and the twin law

⎛

⎝

−10 0

010

00−1

⎞

⎠

This might have been preferred on the grounds that use of the non-

standard space groupsetting made the choice of twin law more obvious.

Example 5. The crystal structure of B

is tetragonal. R

int

was 0.020

in point group 4/m, but 0.060 in 4/mmm. The absences were consistent

with space group I4

/a; even though this space group is centrosym-

metric the mean value of |E

− 1| was only 0.686. The ambiguous Laue

symmetry, the high metric symmetry and low mean value of |E

−1|were

taken as signs that the structure could be twinned (signs 1, 3 and 4 in

Section 18.9).

The structure was solved easily by direct methods (SIR92) in default

mode in I4

/a but, on reﬁning the structure, R appeared to stick at

0.23, even with anisotropic displacement parameters for all atoms. Low-

symmetry tetragonal structures are always susceptible to twinning via

290 Introduction to twinning

Table 18.3. Coset decomposition of 4/mmm with respect to 4/m (calculated using

TWINLAWS). The notation 2

[100]

indicates a two-fold axis along [100], m

[100]

is a

mirror plane perpendicular to [100] and 4

[001]

is a rotation of +90

◦

about [001].

[001]

−

[001]

−1 m

[001]

−4

[001]

−4

−

[001]

[100]

[010]

[−110]

[110]

[100]

[010]

[−110]

[110]

one of the symmetry elements of point group 4/mmm that is not present

in 4/m. One such operator is a two-fold axis about [110], expressed by

the matrix

⎛

⎝

01 0

10 0

00−1

⎞

⎠

Application of this matrix as a twin law, together with reﬁnement of

the twin scale factor, caused R to drop immediately to 0.023; the twin

scale factor was 0.416(2).

The orders of 4/mmm and 4/m are 16 and 8, respectively. Therefore

we need to consider (16/8) − 1 = 1 twin law (Eqn. 18.2). Coset decom-

position of 4/mmm with respect to 4/m yields the data in Table 18.3. The

elements in the ﬁrst line of the table are those of point group 4/m; any

of the elements in the second line of the table could have been used as

a twin law: the two-fold axis along [110] was used above, but use of a

two-fold axis along [100] or [010], or a mirror perpendicular to [−110]

would have modelled the data equally well.

Example 6. The compound Et

−

crystallizes with a metrically

hexagonal unit cell, of dimensions a = 8.254 and c = 6.996 Å (Churakov

and Howard, 2004). The systematic absences were consistent with space

groups P6

mc and P31c, and merging in 6mm and 31m yielded similar

statistics. The mean value of |E

− 1| was 0.678, slightly lower than

expected for a non-centrosymmetric space group. The data could be

modelled in P6

mc, though the structure was disordered; R was 0.054,

though the Flack parameter was rather imprecise [0.0(4)], and the high-

est difference map peak was +0.82 eÅ

−3

, which is high for a compound

of this composition.

The high symmetry, reﬁnement statistics, the low mean value of

− 1| and the similar merging in 6mm and 31c point to twinning

(1, 3, 4, and 8 in Section 18.9), and so the structure was also solved and

reﬁned in P31c. This yielded an ordered model. The R factor was 0.072

before twinning was modelled, but application of a twin law (see below)

caused R to drop to 0.019, with difference map extremes of +0.18 and

−0.09 eÅ

−3

. These statistics are clearly superior to those obtained in

mc, and illustrate the comment made by Herbst-Irmer and Sheldrick

(1998) that it is worth investigating the possibility of twinning before

investing time and effort in disorder modelling.

Examples 291

Table 18.4. Coset decomposition of 6/mmm with respect to 31m (calculated

using TWINLAWS). The notation used is similar to that in Table 18.3.

[001]

−

[001]

[210]

[120]

[−110]

[001]

−

[001]

[110]

[100]

[010]

[210]

[120]

[−110]

−1 −3

[001]

−3

−

[001]

[100]

[010]

[110]

−6

[001]

−6

−

[001]

The orders of 6/mmm (the lattice holohedry) and 31m are 24 and 6,

respectively, and so to investigate twinning completely we need to con-

sider (24/6) − 1 = 3 different twin laws (i.e. the crystal could consist of

up to four domains). Table 18.4 shows coset decomposition of 6/mmm

and 31m. The ﬁrst line in the table shows the elements of 31m, and the

second line a set of possible merohedral twin laws that could model a

second domain; Churakov and Howard used the mirror perpendicular

to [100], expressed by the matrix

⎛

⎝

−100

110

001

⎞

⎠

but any of the other elements in row two of Table 18.4 would have

worked equally well.

31m is a non-centrosymmetric point group, and so the ‘absolute struc-

ture’ should be determined: two Flack parameters are needed, one for

each of the domains so far identiﬁed. It is easy to forget to do this, but use

of coset decomposition ensures the absolute structure will be correctly

treated! The third line of Table 18.4 contains the element

1: inclusion of

the inversion operator (or any other other elements in row 3) as a second

twin law would model twinning by inversion (i.e. the Flack parameter)

in the ﬁrst domain of the crystal. The elements in the fourth row would

enable the Flack parameter in the second domain to be reﬁned. In the

widely used program SHELXL the instructions

TWIN −100110001−4

BASF 0.25 0.25 0.25

would ensure that all twin laws were included in the model; other

programs may need each twin law matrix to be input explicitly.

After reﬁnement, the scale factors for the four domains of the crystal

were 0.46(4), 0.48(4), 0.05(5) and 0.01(4). The last two scale factors are the

Flack parameters for the ﬁrst two domains; the fact that they are very

near zero with small standard uncertainties shows that the absolute

structures of the ﬁrst and second domains are correct.

Example 7. Afurther example of the importance of coset decomposition

in the analysis of twinned crystals is found in the crystal structure of the