chem cryst Xray Crystallography University of Oxford

Twinning. Don't give up  yet.
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 Sheldricks ideas.
Some of the 'slides' still
need a bit more text adding.
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
1 General Introduction.
 Crystal Structure Analysis, M.J.Buerger (1960), Wiley
 Fundamentals of Crystallography, Ed C.Giacovazzo, (1992), Oxford
University Press
 Xray 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, 163165.
 Characterisation of Twinning, A Santoro, (1974), Acta Cryst A30,
224231.
 Classification of Triperiodic Twins, G.Donnay & J.D.H.Donnay,
(1974), Canadian Mineralogist, 12, 422425.
 The Derivation of the Twin Laws for (Pseudo)Merohedry by Coset
Decomposition, H.D.Flack, (1987), Acta Cryst A43, 564568.
 Twinning by Merohedry & XRay Crystal Structure Determination,
M.Catti & G.Ferraris, (1976), Acta Cryst A32, 163165.
3 How its Computed.
 On Structure Refinement Using Data from a Twinned Crystal.
G.B.Jameson, (1982), Acta Cryst A38, 817820
 Report No 63RL(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),
33423349.
4 Practical Aspects
 PseudoMerohedral Twinning: The Treatment of Overlapped Data.
(1969), C.T.Grainger, Acta Cryst A25, 427434.
 The Interpretation of Pseudoorthorhombic Diffraction Patterns,
(1964) J.D.Dunitz, Acta Cryst 17, 12991304.
 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 & Xray Structure
determination', Utrecht, 1989.
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
It = vI1 + (1v)I2

Class II (pseudomerohedral)
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.
 External form  reentrant 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 Xray
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.
 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 + (1v)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 2fold 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 pseudohexagonal 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 2fold 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
Black Nitrosylpentamminecobalt Dichloride
D.Dale & D.C.Hodgkin, J Chem Soc, (1965) 13641371
C.S.Pratt, B.A.Coyle & J.A.Ibers, J Chem Soc (1971) 21462151
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 nonoverlapping data and hkh zone.
Eventually refined with diffractometer data using all reflections
The Interpretation of Pseudoorthorhombic Diffraction Patterns,
(1964) J.D.Dunitz, Acta Cryst 17, 12991304.
This article give several examples showing that apparently insoluble
situations will yield to careful thought.
 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.
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.
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.
AS19A2  C12 H11 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 rhodelta = 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
Systematic absences and Rint
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.
Systematic absences
Fo range  <0  01  12 
24  48  816 

Mean_Fo  .63  .48  1.64 
2.86  5.55  9.41 
Number  103  119  8 
8  5  9 




Fo/sigma range  <0  01  12 
24  48  816  >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 
Rint as function of I/sigma
Range  I>10sigma  10sigma>I>2sigma  I<2sigma 
Rint  2.3%  9.8%  78.4% 
Structure solved by SIR92 in default mode.
The Ui are the principal axes of the anisotropic temperature factors.
For the untwinned 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
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 index
The 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 *
**********************************
 Pseudotranslation 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 pseudotranslational 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 ***
+++++++++++++++++++++++++++++++++++
 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?
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
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 FoFc or
FosqFcsq).
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.
Refinement of AS19
 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.
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 pseudocentric, pseudocentred, 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 nonroutine, 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 silverradiation 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 oneparameter
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), SpringerVerlag, 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, 994996.

'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)

F.L.Hirshfeld & D.Rabinovich, (1973), Acta Cryst A29, 510513.

(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, Xray Analysis and the Structure of Organic Molecules, (1979),
Cornell University Press, pp 208209.

'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 515516.

'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'
[David Watkin 
Keith Prout 
Xray diffraction suite 
Z prime 
Twins  Mogul
]
