# Twins

*This web page was built from the slides used to the illustrate the session ‘Twinning. Don’t give up – yet’ presented at the BCA Autumn Meeting, Bristol, November 12th 1997. *

I am indebted to the CRYSTALS users who have passed on to me their experiences in working with twinned crystals. I am particularly grateful to Fred Einstein for constant nagging for changes, Bill Harrison for TLS data sets and Simon Parsons for several data sets including the TLQS data set shown later in this text. Marcus Neuburger rewrote the CRYSTALS manual on twinning, and Guy Badoux sent me a good list of useful references. We have, of course, added a few items based on George Sheldrick’s ideas.

The main theme is that twinning was once well understood by structure analysts working from photographic film, which greatly simplified indexing the data. The advent of the serial diffractometers made reliable indexing and data collection from twins more complicated. During the ’70s, the drift away from using collections of free standing programs towards using packages, some of which could not handle twins, made the treatment of twins seem even more difficult.

The emergence of area detector diffractometers means that data collection is becoming less complicated, and another of the major software systems now handles twinned refinements. The only difficulties which remain are for the analyst to recognise that they have twinning, and for them to solve the phase problem.

David Watkin 12/11/97

**Contents**

**Twinning References – Useful Starting Points**

**Difficulties with Twinned Structures**

**F or Fsq, -ve I or I.gt. 3 sigma(I)?**

**Twinning References – Useful Starting Points**

**1. General Introduction.**

- Crystal Structure Analysis, M.J.Buerger (1960), Wiley
- Fundamentals of Crystallography, Ed C.Giacovazzo, (1992), Oxford University Press
- X-ray Analysis and the Structure of Organic Molecules, J.D.Dunitz, (1960), Cornell University Press.

**2. Classification and Mathematical Treatment.**

- A tensor Classification of Twinning in Crystals, V.K.Wadhawan, (1997), Acta Cryst A53, 163-165.
- Characterisation of Twinning, A Santoro, (1974), Acta Cryst A30, 224-231.
- Classification of Triperiodic Twins, G.Donnay & J.D.H.Donnay, (1974), Canadian Mineralogist, 12, 422-425.
- The Derivation of the Twin Laws for (Pseudo)Merohedry by Coset Decomposition, H.D.Flack, (1987), Acta Cryst A43, 564-568.
- Twinning by Merohedry & X-Ray Crystal Structure Determination, M.Catti & G.Ferraris, (1976), Acta Cryst A32, 163-165.

**3. How its Computed.**

- On Structure Refinement Using Data from a Twinned Crystal. G.B.Jameson, (1982), Acta Cryst A38, 817-820
- Report No 63-RL-(3321G) (1963), P.R.Kennicott, General Electric Research Laboratory, Schenactady, New York.
- Structure and Stability of Carboxylate Complexes.C.K.Prout, J.R.Carruthers & F.J.C.Rossotti, (1971) J.Chem.Soc, (A), 3342-3349.

**4. Practical Aspects**

- Pseudo-Merohedral Twinning: The Treatment of Overlapped Data. (1969), C.T.Grainger, Acta Cryst A25, 427-434.
- The Interpretation of Pseudo-orthorhombic Diffraction Patterns, (1964) J.D.Dunitz, Acta Cryst 17, 1299-1304.

**5. Twins – a description**

- Two or more crystals of the same material intergrown so that the unit call (contents included) of the first is related to the unit cell of the second by a symmetry element.
- Such a geometrical relationship must exist in several specimens.
- Polysynthetic (lamellar) twinning is repeated twinning on a macro- or microscopic scale.
- If two structures that are different have grown together according to a law, the assembly is not a twin. Such a sample may show epitaxy, syntaxy or apotaxy.

*P van der Sluis, PhD Thesis ‘Single Crystals & X-ray Structure determination’, Utrecht, 1989.*

**Twins – Classification**

Early classifications were made on the basis of morphology, measured with contact or optical goniometers. The fine details of this scheme and the nomenclature, though useful for describing individual twins, are probably not useful for structure analysts. Classification based on the reciprocal lattice are simpler. Note that there is a rather inconsistent use of classification schemes in the literature. A fundamental division for the structure analyst is the distinction between twins in which every observable reciprocal lattice point contains contributions from both twin components, and those twins in which some points contain contributions from one component only.

**TLQS twins**have multiple diffraction patters. The ‘spots’ may be split, or give two distinct lattices.**TLS twins**show a single diffraction pattern.**Class I**(merohedral)

The twin operator belongs to the Laue symmetry of the untwinned crystal. The centre of symmetry can always be chosen as the operator. Except for the influence of Friedels Law, the intensities from a twinned sample are the same as those from an untwinned sample.

r.l. of twin has higher symmetry than r.l. of component

I_{t}= vI_{1}+ (1-v)I_{2}**Class II**(pseudo-merohedral)

The centre of symmetry is never the twin operator. Two reflections not equivalent by Laue symmetry contribute to a single observation.

r.l. of twin has same symmetry as r.l. of component.

**Detection of Twinning**

- External form – re-entrant angles.
- Partitioned extinction under polarising microscope.
- TLQS twins show split reflection – may vary with temperature. Indexing may be difficult.
- Twins by TLS with n=1 are difficult to recognise by X-ray diffraction. Different specimens may give different relative F values.
- Twins by TLS with n # 1 (n = 3 is common) often show peculiar systematic absences.
- Space groups with all reflection planes or all rotation axes (e.g. Pmmm) are suspicious, especially if there are other meaningless systematic absences.
- Structure wont solve from apparently good data.
- Irreducible R factor from apparently good diffraction data.

**Common Examples of Twinning**

- Twinning by a centre of inversion (TLS), e.g. P 21

Twin operator is -1 0 0 0 -1 0 0 0 -1

It = vI1 + (1-v)I2

In this case, the volume fraction v is the Flack Enantiopole, x. Most programs handle this as a special case. - Monoclinic with beta ~ 90. (TLQS)

If no substantial anomalous dispersion, the operator can be a mirror perpendicular to a or c, or a 2-fold rotation about a or c.

Twin operator is of the form 1 0 0 0 -1 0 0 0 -1 - Monoclinic with a ~ b.

This may also look like centred orthorhombic.

Rotation about 101 may generate curious systematic absences.

Twin law is of the form 0 0 1 0 -1 0 1 0 0 - Monoclinic with a ~ b, beta~ 120

May generate pseudo-hexagonal 3 fold twin (trilling)

Twin laws are of the form

-1 -1 0 1 0 0 0 0 1

0 1 0 -1 -1 0 0 0 1 - Orthorhombic with a ~ b

May simulate tetragonal.

Twin law is of the form 0 1 0 1 0 0 0 0 -1 - Introduction of a false mirror or 2-fold axis to a high symmetry space group, eg tetragonal, trigonal or hexagonal.

Twin Law is of the form 0 1 0 1 0 0 0 0 -1

**Example 1 – Inexplicable Absences (1) – **Black Nitrosylpentamminecobalt Dichloride

*(D. Dale & D. C. Hodgkin, J Chem Soc, (1965) 1364-1371; C. S. Pratt, B. A. Coyle & J. A. Ibers, J Chem Soc (1971) 2146-2151)*.

- Apparently tetragonal (
*b*unique), but with inexplicable absences - Actually orthorhombic,
*a*=10.45*b*=8.70*c*=10.45 C mcm - Twinned by rotation of 90 degrees about
*b*, twin law 0 0 -1 0 1 0 1 0 0 - Originally solved with the non-overlapping data and
*hkh*zone. - Eventually refined with diffractometer data using all reflections.

**Example 2 – Inexplicable Absences (2)** – The Interpretation of Pseudo-orthorhombic Diffraction Patterns,

(1964) J.D.Dunitz, Acta Cryst 17, 1299-1304.

This article gives several examples showing that apparently insoluble situations will yield to careful thought.

Difficulties with Twinned Structures:

- Knowing that the crystal is twinned. With serial diffractometers, the crystal may just be rejected as ‘no good’.
- Collecting the diffraction data. With serial diffractometers, the problem is to know exactly what has been recorded.
- Solving the Structure. Patterson or direct methods may not yield interpretable maps.
- Refining the Structure. With a suitable program, this is normally routine – if the twin law is known.

**A question…**

From ‘A CRYSTALS user’ Tue Oct 28 08:55:

To: David Watkin <david.watkin@chemistry.oxford.ac.uk>

Subject: Twin refinement

Hi David — apologies if this is a silly question, but is there any reason to prefer using /F/**2 over /F/ when refining twinned crystal structures? Accidentally refining on /F/ led to essentially the same answer as the /F/**2 refinement, with of course “the advantage” of Rw being approximately halved in magnitude. Thanks,

****

The following example is refined using F or Fsq as the ‘observation’, and shows that there is little evident difference.

Development of CRYSTALS depends heavily on users sending us interesting data sets to work on, even though they may have solved the difficulties with the analysis them selves.

Summary of available information

**Example 3 – C _{12} H_{11} N O S.**

This worked example is based on a good data set provided by Simon Parsons, at Edinburgh. We used it as an example of TLSQ twinning, and to demonstrate the effect of different refinement regimes. The following information was provided by Simon.

Summary of available information

AS19A2 – C_{12} H_{11} N O S, Space group – P 21/n

Data collected on a serial diffractometer.

Not all search reflections could be indexed

Solved by Dirdif Patterson.

Refined as twin

R1 (F>4sigmaF) = 6.1%

wR2 = 16.9%

Min and max rho-delta = -0.37, 0.28

Twin Law 2 fold rotation about a deduced from cell parameters.

2cCos(beta)/a is about 1/3.

1 0 0

0 -1 0

-1/3 0 -1

Data Reduction:

The following tables show some intensity statistics for the systematic absences, and Rint as a function of I/sigma(I)

Space group: P 21/n

Reflections measured: 3444

Systematic absences: Mean Fo=1.9, rms Fo/sigma = 23.3

The average value of the systematic absences is greater than zero (a systematic bias that may also under lie the weak observed reflections)

The merging R factor shows that the weak reflections don’t even agree between themselves.

Fo range | <0 | 0-1 | 1-2 | 2-4 | 4-8 | 8-16 | |

Mean_Fo | -.63 | .48 | 1.64 | 2.86 | 5.55 | 9.41 | |

Number | 103 | 119 | 8 | 8 | 5 | 9 | |

Fo/sigma range | <0 | 0-1 | 1-2 | 2-4 | 4-8 | 8-16 | >16 |

rms_Fo/sigma | 1.63 | .51 | 1.44 | 2.66 | 4.88 | 11.1 | 89.2 |

Number | 103 | 65 | 31 | 23 | 2 | 11 | 17 |

Range | I>10sigma | 10sigma>I>2sigma | I<2sigma |

Rint | 2.3% | 9.8% | 78.4% |

Structure solution/refinement:

Structure solved by SIR92 in default mode.

The Ui are the principal axes of the anisotropic temperature factors. For the un-twinned refinements, they are unacceptable, particularly for C15

Peak heights in range:

S 21 All data Riso =24.4 Rho delta -1.2 +2.5

O 7 Raniso=21.4 Rho delta -1.1 +2.4

N 7

C 7 I>3sigma Raniso=18.9 Rho delta -0.9 +1.9

. .

. . Principal axes of selected atoms

C15 2 Ui S .03 .04 .05

Q1 2 Ui N .02 .06 .07

Q2 2 Ui C15 .02 .03 .15

Why won’t it refine?

Check the available information. ‘E’ statistics may reveal something, as may analysis of residuals as a function of various parameters.

- Analysis of variance:
- Agreement analysis on
**h**indexThe following table shows the R and weighted R factor as a function of the**h**index. Note the relatively high R factors for layers with**h**=3n -
Residual as a function of index **‘h’**(all data) (I >3 sigma) LAYER No. >/Fo/< >/Fc/< R(%) Rw(%) R(%) Rw(%) -8 9 1.03 1.2 29.45 42.65 19.45 27.82 -7 40 1.47 1.5 20.04 23.36 14.50 16.94 **-6****72****1.65****1.6****39.80****44.35****33.76****38.03**-5 94 1.62 1.7 17.50 20.33 13.94 15.31 -4 113 1.76 1.9 17.00 18.06 12.72 13.78 **-3****117****2.86****2.0****38.48****49.45****37.10****46.01**-2 135 2.32 2.5 12.02 13.50 11.44 12.06 -1 139 2.47 2.7 10.59 10.89 10.03 10.06 0 141 3.13 2.6 20.15 19.23 18.05 17.92 1 141 2.65 2.9 14.01 13.14 11.27 11.89 2 141 2.18 2.4 14.79 15.73 11.56 11.77 **3****127****2.98****2.1****34.65****44.08****31.80****40.89**4 111 1.73 1.9 17.64 19.80 12.59 13.06 5 89 1.51 1.6 17.98 22.79 14.81 16.49 **6****62****2.03****1.5****36.08****43.10****32.33****37.64**7 35 0.87 1.2 30.74 43.96 20.92 23.05 8 5 1.25 1.7 18.48 44.87 .01 .01 TOTALS 1571 2.24 2.16 21.42 26.02 18.91 23.24

Wilson Plot from SIR92

Note that U(iso) is normal. The plot itself also looked okay.

**********************************

* y = s**2 *

* x = ln <i> / sigfsq *

* ( w ) = wilson *

* ( * ) = calc *

**********************************

* intercept = -3.20153 *

* slope = -6.56201 *

* b(iso) = 3.28101 *

* u(iso) = 0.04155 *

* scale = 24.57010 *

* scale*f(obs.)**2 = f(abs.)**2 *

**********************************

Pseudo-translation information from SIR92

SIR92 revealed something curious about the reflections with h=3n.

The mean value of E**2 was about twice normal.

+++++++++++++++++++++++++++++++++

*** pseudotranslation section ***

*** program searched for pseudo-translational symmetry ***

class(es) of reflections probably affected by pseudotranslational effects:

condition number of >E**2< figure mean fract. scatt. power

reflections of merit in pseudotranslation (m.f.s.p.)h = 3n 1000 1.358 1.65 17 %

** pseudotranslational symmetry will be neglected in subsequent steps ***

+++++++++++++++++++++++++++++++++++

Getting an Answer:

- Some of the initial search reflections would not index
- SIR92 showed that there was something curious with reflections for which h = 3n
- Refinement has hung at R = 19%
- Agreement analysis shows something curious for reflections with h = 3n.

Try masking out of the refinement reflections with h = 3n

All sigma, h#3n Ui S .03 .05 .06 R=9.9% rho delta -.5 .4

Ui N .03 .04 .05

Ui C15 .02 .04 .07

I>3sigma, h#3n Ui S .03 .04 .05 R=4.3% rho delta -.2 .2

Ui N .03 .04 .06

Ui C15 .03 .04 .07

If we mask out the h=3n reflections, the R factor looks O, and the temperature factors have improved. Can AS19A2 be twinned?

The bottom line:

Treat as twinned by 2 fold rotation about ‘a’

Twin Law

1 0 0

0 -1 0

-1/3 0 -1

All sigma, Fo Ui S .03 .05 .06 R=5.5% rho delta -.4 .3

Ui N .03 .04 .05

Ui C15 .03 .04 .08

All sigma, Fsq Ui S .03 .05 .05 R=5.6% rho delta -.4 .3

Ui N .03 .04 .05

Ui C15 .03 .04 .07

Twin ratio refines to 0.52 : 0.48

Comparison of results – Refinement of a twin under different regimes:

Refinement to convergence of AS19 under different regimes. In each case, a weighting scheme was chosen to give S (goodness of fit) of about unity, and a uniform distribution of w(delta)sq against Fo (delta is Fo-Fc or Fosq-Fcsq).

The first two columns show bond lengths for refinement using all data (including negative observations). The 3rd and 4th column contain results after eliminating refections with I < 3 sigma(I). The final column contains results after eliminating weak reflections *and also eliminating all reflections with h = 3n. About one third of the independent observations have been eliminated.*

The lower part of the table shows means and esds for bonds in the phenyl group.

ALL DATA | I>3 SIGMA(I) | I>3 sigma,h#3n | ||||

Fsq | Fo | Fsq | Fo | Fo | ||

S(1)-C(2) | 1.688(6) | 1.684(7) | 1.683(5) | 1.684(5) | 1.698(8) | |

C(2)-N(3) | 1.358(8) | 1.357(9) | 1.364(7) | 1.367(6) | 1.347(10) | |

C(2)-C(7) | 1.433(9) | 1.438(11) | 1.436(8) | 1.431(8) | 1.435(11) | |

N(3)-C(4) | 1.375(9) | 1.378(11) | 1.375(9) | 1.369(8) | 1.388(11) | |

N(3)-C(9) | 1.456(8) | 1.453(10) | 1.467(7) | 1.469(7) | 1.475(11) | |

C(4)-C(5) | 1.333(12) | 1.345(14) | 1.348(10) | 1.349(9) | 1.363(12) | |

C(5)-C(6) | 1.388(11) | 1.379(13) | 1.392(9) | 1.393(9) | 1.378(14) | |

C(6)-C(7) | 1.368(10) | 1.369(12) | 1.385(9) | 1.380(8) | 1.366(11) | |

C(7)-O(8) | 1.352(8) | 1.350(10) | 1.339(7) | 1.346(7) | 1.353(10) | |

C(9)-C(10) | 1.517(9) | 1.510(10) | 1.492(8) | 1.492(8) | 1.494(12) | |

(mean) | ||||||

C(10)-C(11) | 1.400(8) | 1.401(10) | 1.400(7) | 1.395(7) | 1.399(3) | 1.366(11) |

C(10)-C(15) | 1.396(8) | 1.393(11) | 1.391(8) | 1.391(8) | 1.393(2) | 1.392(11) |

C(11)-C(12) | 1.369(9) | 1.376(11) | 1.370(8) | 1.370(7) | 1.371(3) | 1.391(11) |

C(14)-C(15) | 1.371(11) | 1.379(13) | 1.383(10) | 1.385(10) | 1.380(6) | 1.393(12) |

C(12)-C(13) | 1.387(11) | 1.395(13) | 1.390(9) | 1.385(9) | 1.389(4) | 1.368(13) |

C(13)-C(14) | 1.383(12) | 1.380(14) | 1.368(10) | 1.366(10) | 1.374(9) | 1.384(14) |

(mean) | 1.384(12) | 1.387(10) | 1.384(13) | 1.382(13) | 1.382(12) |

Notice that all regimes give substantially the same results, providing a partial answer to the CRYSTALS users question.

F or Fsq, -ve I or I>n sigma(I)?

The Hirshfeld & Rabinovich paper (Acta Cryst A29, (1973), 510) arguing for the use of Fsq and retention of -ve intensities is well known, and may serve as a starting point for answering this question. In practice, problems fall into two broad classes.

- The structure is ‘routine’
- There are ample data at 3sigma(I), (where sigma(I) for area detectors may need to be multiplied by a factor of 3 to 5 to bring it into the same sale as serial diffractometers – see Kirschbaum et al below).
- The structure is not pseudo-centric, pseudo-centred, does not have a marginal Flack enantiopole parameter, etc.
- In the case of routine structures, the final atomic parameters will be substantially the same whether the refinement is with F or Fsq, and whether the weak reflections are included or rejected. Older crystallographers may recall that lowering the sigma threshold to get more reflections into a refinement usually had no impact on the parameters values, though the sus (esds) usually rose a little. Fsq refinement may have a wider radius of convergence (though modern Direct Methods usually give models well within convergence for F or Fsq), but is sensitive to bad observations (though modern diffractometers rarely yield totally spurious observations). In principle the standard uncertainty of Fsq is readily calculated, though in practice weights derived from it need to be well massaged, as do the weights for F refinement.

- The structure is non-routine, and suffers from one or more of the above problems.
- In this case no single recipe will resolve the problem. Simply refining on Fsq is unlikely to provide a clear answer. Adding in large numbers of essentially ‘unobserved’ reflections will contribute nothing to finding a solution, particularly if the weak reflections are systematically over estimated (check the mean value of the systematic absences). Imagine collecting the full silver-radiation sphere for a crystal of a soft organic small molecule, in P 21. The fact that the 0 1 0, 0 3 0, 0 5 0, 0 7 0 are very weak is significant. Nothing can be gleaned from the observation that the 0 39 0, 0 40 0 and 0 41 0 are also unobserved. Dumping masses of weak reflections into a refinement (something now very easy to do with the advent of area detectors) not only serves no useful purpose, it actually degrades the usefulness of one-parameter estimators (such as the Hamilton weighted R factor), unless the weight of these refections is insignificant. Prince (1994) has suggested how the important reflections (strong or weak) may be identified, but few programs systems include this calculation.

**If the question of F or Fsq, 3 sigma or -ve reflections becomes significant to a structure analysis, there are certainly more fundamental questions to be answered.**

Other interesting reading is:

- J.S.Rollett, Crystallographic Computing Techniques, (1976), ed F.R.Ahmed et al, Munksgaard, pp 414.
- ‘We have considered the relationship between M and N The minima of these functions can be made to coincide ……..and both functions, properly weighted, give the same true minimum.’

- J.S.Rollett, Crystallographic Computing Techniques, (1976), ed F.R.Ahmed et al, Munksgaard, pp 414.
- ‘This led us to predict that N might have false minima not present in M, and we have found a case in which this was so.’

- E.Prince, Mathematical Techniques in Crystallography and Material Science, (1994), Springer-Verlag, pp 125
- ‘Therefore the results (of refinement) will be substantially identical whether Fsq or F is used as the data value, and which is used is largely a matter of local fashion.’

- A.J.Wilson, (1976), Acta Cryst A32, 994-996.
- ‘Refinement in R1 (against Fo) can be regarded as a special case of refinement in R2 with weights

w2 = w1 (sqrt(Io)+sqrt(Ic))**-2

where the w1’s are the weights that would have been used in R1.’

(put otherwise, w1 = w2 (sqrt(Io)+sqrt(Ic))**2)

- ‘Refinement in R1 (against Fo) can be regarded as a special case of refinement in R2 with weights
- F.L.Hirshfeld & D.Rabinovich, (1973), Acta Cryst A29, 510-513.
- (referring to neglect of negative observations) ‘our limited experience indicates that in real situations the effect of biased data (distributions) on the structurally interesting parameters is rarely large enough to matter’

- J.D.Dunitz, X-ray Analysis and the Structure of Organic Molecules, (1979), Cornell University Press, pp 208-209.
- ‘The best way to decide whether the structure is really or only nearly centrosymmetric is to scrutinize the few reflections that are most sensitive…. The sensitive reflections are the ones, generally weak, for which the calculated value of A (real part of the structure factor) is close to zero’

- E.Prince & W.L.Nicholson, in Structure & Statistics in Crystallography, (1984), ed A.J.C.Wilson, Adenine Press, pp 191
- ‘Actually, omitting a weak reflection, or any reflection, cannot bias the parameter estimates, as may be shown ….’

- K.Kirschbaum, A.Martin & A.A.Pinkerton, þ/2 Contamination in CCD area- detector data, (1997), J.Appl.Cryst. 30, pp 515-516.
- ‘Some reduction in the number of ‘observed’ systematic absences is noted, …. The improvement in the data is shown to be insignificant for routine data collection’