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Chapter 10: Twinned Crystals
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The terminology in articles on twinning is complicated and sometimes contradictory, with the same term being used in different contexts by different authors. We shall use the following terms, based upon observations made from the whole reciprocal lattice.
It is assumed that sufficient reflections are measured to give a complete coverage of the asymmetric part of the r.l. for at least one (called the major) component of the twinned crystal.
Some, but possibly not all, of the reflections from the major component contain contributions from other twin components. Overlap is controlled by accidental relationships between cell parameters. If the relationship is very exact, so that all reflections are overlapped, the sample is a pseudo TLS twin.
Every reflection from the major component contains a constant fractional contribution from other components. The overlap is controlled by the crystal class rather than accidental relationships between cell parameters.
TLS twins - Class I Except for the effect of anomalous dispersion, the Laue symmetry of the diffracion pattern is the same as that of an un-twinned crystal.
TLS twins - Class II The Laue symmetry of the diffracion pattern is not the same as that of an un-twinned crystal.
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1. Identification that the crystal is indeed twinned.
Twinning - Initial clues
These may include may include:
a. Evident interpenetrating reciprocal lattices.
b. Split reflections, with a varying intensity ratio.
c. Systematic absences not conforming to any space group.
d. The ratio of intensties of equivalent reflections from different samples is not constant.
Other clues are:
a. Failure to solve the structure from apparently good data.
b. Irreducible R factor from seemingly good quality data.
c. Inexplicable strong residual peaks in the difference density map.
Twinning - Data collection and processing
Twinning - Structure solution
In general, structure solution is the major difficulty in working with twinned crystals.
Twinning - Structure Refinement
If the space group, trial structure, twin law and reflection components are known, this is straight forward. The sum of the twin fractions must be 1.0
Twin Data stored by CRYSTALS
For a twinned crystal the following equation holds.
Fsq(obs) = v1.Fsq(1) + v2.Fsq(2) ....
For a twin with two components, each observation may contain a contribution from each component, or from both. The reflections have to be 'tagged' to indicate which components are contributing, the ELEMENT coefficient in LIST 6 (section 5.3)
For a TLS twin, every observation contains a contribution from both components (though if it is a systematic absence for one component, the contribution will be zero). Since the tagging is the same for every reflection, it can be inserted automatically by CRYSTALS
For a TLQS twin, some observations will contain a contribution from the principal component, and some from both components, giving ELEMENT tags of '1' and '12' respectively. If additional observations have been made based on the reciprocal lattice of component 2, and are indexed with respect to lattice 2, they are given the tag '2'. If any of these also contain a contribution from component 1, the tag will be '21'.
Example 1. An orthorhombic space group with a~b, twinned by interchange of 'a' and 'b'. If 'a' is very similar to 'b', every observation 'hkl' will overlap with twin component 'khl', and the ELEMENT tag will be '12', the default. If a systematic absence from element 1 falls on element 2, the reflection should not be eliminated during data reduction, and will have the tag '12', even though the contrinbution from 1 is zero.
Example 2. A monoclinic crystal with 2cCos(beta)/a about 1/3. Twinning by a 2 fold rotation about 'a' gives a twin law
1 0 0 0 -1 0 -1/3 0 -1
Twinning - LISTS affected
LIST 5 - Parameters: the number of twin elements and their values must be set. LIST 6 - Reflections: the observed twinned data must be stored as /FOT/, and the twin element tags be set. LIST 12 - Constraint matrix: the twin elements must be refined, and possibly constrained. LIST 13 - Experimental info: the key CRYSTAL TWIN=YES must be set. LIST 16 - Restraints: the twin elements may be restrained. LIST 25 - This contains the twin laws themselves.
Twin List 5
The number of twin elements and their values must be given. Currently, the number of elements and their starting values cannot be input in \EDIT (though values can be changed later). Punch LIST 5, edit it, and re-input it, or use the SCRIPT EDLIST5.
\LIST 5 READ NATOM= NELEMENT= ELEMENT value(1) value(2) ... ATOM .......... ...... END
Twin List 6
For TLQS twins, the element tags (section 5.3) really depend upon exact experimental conditions, and should be computed by the data collection software. If a reflection is entered without a twin element tag (eg a SHELX HKL 4 file), CRYSTALS tries to compute the tag from the twin laws as follows:
h the index with respect to LIST 1 (cell) and LIST 2 (space group) (this is the index in LIST 6) T The twin law matrix. n the nominal index for the twinned reflection. n = T.h d the difference between an exact lattice point and the generated point. n-nint(n) s The length**2 of the difference vector, in A-2.
If 's' is less than the TWINTOLERANCE given on the LIST 6 MATRIX
directive, the twinned reflection is regarded as falling upon a primary
element reflection, and the element tag is updated to indicate this.
This method is only an approximation, but may help to make otherwise
useless data useable. LIST 13 (section 4.13) will be automatically
updated to indicate that twinned data are being refined.
a)Analysis was started as untwinned, and the user wishes to convert to
a twinned refinement
\LIST 25 READ NELEMENT=2 MATRIX 1 0 0 0 1 0 0 0 1 MATRIX 0 1 0 1 0 0 0 0 1 END \LIST 6 READ TYPE=TWIN MATRIX TWINTOL=.001 END
b)Crystal identified as twinned, and data reduction, sorting and
merging done outside of CRYSTALS
If the reflection data has been preprocessed so that it is a full, unique, set for the corret space group, then the correct space group should be entered, and the reflections input as FOT directly. This tells CRYSTALS that the data is twinned.
\LIST 25 READ NELEMENT=2 MATRIX 1 0 0 0 1 0 0 0 1 MATRIX 0 1 0 1 0 0 0 0 1 END \OPEN HKLI TWINREF.HKL \LIST 6 READ F'S=FSQ NCOEF = 5 TYPE = FIXED CHECK = NO INPUT H K L /FOT/ SIGMA(/FO/) FORMAT (3F4.0,2F8.0) STORE NCOEF=9 OUTPUT INDICES /FO/ SIGMA(/FO/) /FOT/ /FC/ SQRTW ELEMENT CONTINUE RATIO/JCODE CORRECT MATRIX TWINTOL=.001 END
c)Data reduction, sorting and merging to be done in CRYSTALS
During initial data reduction (section 5.14) the crystal must be given as untwinned in LIST 13 (section 4.13), and the 'space group' should be that of the Laue Class of the intensity data, so that the symmetry of the data is preserved. In general, systematic absences should be preserved, unless centring of the cell matches for all twin components. Twin elelemt tags may be provided by an external program, or computed by CRYSTALS.
If there are special ELEMENT tags, use something like the following:
\OPEN HKLI twin.hkl \LIST 6 READ F'S=FSQ NCOEF = 6 TYPE = FIXED CHECK = NO INPUT H K L /FO/ SIGMA(/FO/) ELEMENTS FORMAT (3F4.0, 2F8.0, F3.0) STORE NCOEF=7 OUTPUT INDICES /FO/ SIGMA(/FO/) ELEMENTS RATIO/JCODE CORRECTIONS SERIAL END
\LIST 6 READ TYPE=TWIN MATRIX TWINTOL=.001 END
TWIN LIST 13
The keyword TWINNED must be set to YES for structure factor calculations. Because different components of a twin will probably have different extinction corrections, refinement of extinction is deprocated for twins. CRYSTALS prints a warning, then lets you continue at your own risk. The special use on the LIST 6 command (above) will update LIST 13 automatically.
\LIST 13 .... CRYSTALS FRIEDEL=NO TWIN=YES EXTINCTION=NO
Twin List 12
If all the element scale factors are refined simultaneously with the overall scale factor, the calculation will be singular. In general, the sum of the element scale factors is held at unity. For only two twin componenets, this can be done in LIST 12 as a constraint. For more, it can be done in LIST 16 as a restraint. The sum of the elements in input to LIST 5 should be unity.
\LIST 12 FULL ........ EQUIVALENCE ELEMENT(1) ELEMENT(2) WEIGHT -1 ELEMENT(2) END
Twin List 16
The sum of the element scale factors can be restrained to unity in LIST 16. In this case, they must all be freely refined in LIST 12.
\LIST12 FULL ........ CONTINUE ELEMENT SCALES END \LIST 16 SUM .0001 ELEMENT SCALES END
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For a twinned structure, after the data have been merged, it is advisable to re-sort the reflections, placing observations that contain contributions from elements with the same indices adjacent in the new LIST 6.
This directive initiates the re-sorting of reflections for a twinned structure. It is IMPERATIVE that the previous command has put the reflections on the disc. This is automatic if input is via a \LIST 6 command (section 5.3) or you can use the \LIST 6 READ TYPE=TWIN command.
STORE MEDIUM= This directive determines the output medium of the new LIST 6.
MEDIUM This parameter selects the output medium of the new LIST 6. The allowed values for this parameter are :
M/T DISC - DEFAULT VALUE.
Twins - backward compatability
Note that the key /FOT/ can be given in the initial data reduction if the crystal is also marked as twinned in LIST 13 (section
the observed intensity
input as /FOT/. This is preserved for backwards compatibility.
Twins - Worked Example
The data were provided by Simon Parsons, for a TLQS twin, where the bulk of the data is from only one component. For reciprocal lattice layers with h=3n, there is overlap from the second twin component. The 'elelent keys' are thus '12' for reflections with h=3n, otherwise '1'.
Sections of reflection file 'example.hkl'
-6 0 0 2.16 1.08 12 -6 0 -1 -0.47 0.93 12 -6 0 -2 24.98 1.63 12 ...... -6 -2 0 1.64 0.95 12 -6 -2 -1 8.40 1.06 12 -6 -2 -2 3.33 1.18 12 -5 5 1 10.61 1.22 1 -5 5 2 0.75 0.96 1 ........ -4 0 3 -0.45 0.63 1 -4 0 4 4.73 0.82 1 -4 0 5 -0.78 0.71 1 -4 0 6 48.40 1.69 1 -4 0 7 0.12 0.68 1 -4 0 8 -0.35 0.83 1 -3 -7 0 7.68 1.24 12 -3 -7 -1 13.11 1.45 12 -3 -7 -2 13.89 1.36 12 .......
\ Input the cell parameters \LIST 1 REAL 7.2847 9.74 15.231 90 94.386 90 END \ Input the space group \SPACEGROUP SYMBOL p 21/n END \ Input the experimental data \list 13 crystal friedel = no twinned=no cond wave=1.5418 end \ Input the twin laws, including the identity matrix \ which corresponds to the first component of the \ twin, i.e. the one it was indexed on. \list 25 read nele=2 matrix 1 0 0 0 1 0 0 0 1 matrix 1 0 0 0 -1 0 -.33333 0 -1 end \ Input scattering factors (list 3) and cell contents \ (list 29) using the composition command: \COMPOSITION CONTENTS c 48 h 44 s 4 o 4 n 4 SCATT CRSCP:SCATT PROPER CRSCP:PROPERTIES END \ Specify how the SFLS calculations should be done: \LIST 23 MINIMISE F-SQ=no modify anomalous=yes END \ Input a whole model: scale parameter, twin element scales and \ the atom parameters. \list 5 read natom = 5 nelem=2 overal scale=.2 elem .5 .5 atom s 1 1.0000 0.0398 0.9390 0.3740 0.3888 atom n 2 1.0000 0.0617 0.6708 0.1939 0.3428 atom o 3 1.0000 0.0460 0.6967 0.4265 0.5265 atom c 4 1.0000 0.0416 0.9097 0.0426 0.2936 atom c 5 1.0000 0.0317 0.7467 0.2938 0.3989 end \ Open a file on the device called 'HKLI' \CLOSE HKLI \OPEN HKLI example.hkl \ Read data from that device into LIST 6 in the \ specified format and leave space for the specified \ keys. \list 6 READ F'S=FSQ NCOEF = 6 TYPE = FIXED CHECK = NO INPUT H K L /FO/ SIGMA(/FO/) ELEMENT FORMAT (3F4.0, 2F8.0,f4.0) STORE NCOEF=7 OUTPUT INDICES /FO/ SIGMA(/FO/) RATIO/JCODE CORRECTIONS SERIAL ELEMENT END \ Remove systematic absences and move hkl indices by symmetry so that \ they fall into a unique volume of reciprocal space: \SYST \ Sort the reflections: \SORT \ Merge adjacent reflections with the same indices: \MERGE END \ Store the reflections and at the same time, guess the element \ key using the twin laws in L25 to predict if overlap is likely. \List 6 read type=twin end \ Compute the scale factor \SFLS SCALE END \ Set up the matrix of constraint (aka the refinement \ directives): \LIST 12 FULL FIRST(X'S, U[ISO]) UNTIL C(15) equivalence element(1) element(2) weight -1 element(2) END \ Carry out one cycle of least squares refinement: \SFLS REF END
Twinning - Mathematical aspects
In a twinned crystal, two or more separate components or ELEMENTS contribute to the diffraction pattern, and the observed intensities may contain contributions from any one of the possible twin component In addition, the amount of each twin component present in a specified unit of volume is not restricted, and in general will vary between different samples of the same material.
The expression for an observed intensity in such a case is given by :
It = v1*I1 + v2*I2 + . . + vn*In
SUM(vi) = 1 over all the components.
Hc = R.H
The interconversion of atomic coordinates between the various reference systems in a twinned crystal can also be expressed in terms of R :
Hc[T].Xc = H[T].X for any component.
H[T].R[T].Q.X = H[T].X
R[T].Q = I
Q = R[TI]
Xc = R[TI].X
Before any reflections can be processed, the matrices R must
These are given in LIST 25, which must contain one matrix for
each possible component.
(If the standard system is chosen as that of component 1, for example,
the first R matrix will be the unit matrix, which must be given
as it is not assumed).
© Copyright Chemical Crystallography Laboratory, Oxford, 2011. Comments or queries to Richard Cooper - firstname.lastname@example.org Telephone +44 1865 285019. This page last changed on Wednesday 27 April 2011.