- Crystals Manual

: 1. Introduction To The System; 2. Definitions And Conventions; 3. The Crystals Data-Base; 4. Initial Data Input; 5. Reflection Data Input; 6. Atomic And Structural Parameters; 7. Structure Factors And Least Squares; 8. Fourier Routines; 9. Analysis Of Results; 10. Twinned Crystals

Fri Jun 2 2000

Crystals Manual

Chapter 10: Twinned Crystals

: 10.1: Twinning - introduction; 10.2: Twinning Problems; 10.3: SORTING TWINNED STRUCTURE DATA - \REORDER

10.1: Twinning - introduction

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.

TLQS twins

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.

TLS twins

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.

10.2: Twinning Problems

The analysis of twinned structures is complicated by several issues.

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

a.

There is usually no difficulty in collecting data for TLS twins. For TLQS twins, each observation needs to be tagged to indicate which twin components (elements) contribute to the observation. This may be simply computed from the indices if the different lattices have a more-or-less exact relationship between them, of may need to be assigned more carefully if the twin obliquity causes only partial overlapping of some reflections. For doublet spots, it is important that either the whole doublet is integrated (tag '12'), or the principal component is separated out (tag '1').

b.

There may be serious difficulties in determining the space group. Trial and error may be the only procedure available.

c.

The space group used for data reduction and merging may not be that of the major component. A Space group showing the symmetry of the twinned diffraction data should be used initially. The correct space group should be used once data reduction is complete.

Twinning - Structure solution
In general, structure solution is the major difficulty in working with twinned crystals.

a.

For TLQS structures, if a substantial number of reflections are from the major component only, the structure may solve by traditional methods.

b.

For Class I TLS structures, structure solution is usually straight forward, the components of the twin differing only by the effects of anomalous scattering. Such twins (merohedral twins, or twinning by inversion) can be processed without futher reference to this part of the manual. All that needs to be refined is the Flack enantiopole parameter. See the main chapter on refinement.

c.

For Class II TLS structures, if the twin ratio is far from 50:50, the structure may solve by traditional methods.

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) ....

and similarly for F(calc). The v(i) are the volume fractions of the components contributing to the observation. A Fourier synthesis using /Fobs/ as coefficient is meaningless, since the phase alpha(calc) will belong to only one of the components. The terms needed for Fourier and other calculations are Fcalc(1), aplha(1) Fobs.v(1), i.e. only that contribution to Fo due to the principal element.

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

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

Overlap of reflections from both components will only occur when 'h' = 3n, giving the ELEMENT tag '12'. If the lattice is only sampled at r.l. points corresponding to the principal indexing, reflections with 'h'

3n will have the tag '1'.

Twinning - LISTS affected

LIST 5  - The number of twin elements and their values must be set
LIST 6  - The observed twinned data must be stored as /FOT/, and the 
          twin element tags be set
LIST 12 - The twin elements must be refined, and possibly constrained
LIST 13 - the key CRYSTAL TWIN=YES must be set
LIST 16 - The twin elements may be restrained
LIST 25 - This contains the twin laws

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
During initial data reduction the crystal must be given as untwinned in LIST 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. If there are more than two twin components, or the tag is not '12', the appropriate tag must be assigned to the affected reflections outside of CRYSTALS. The tag 12 can be inserted by CRYSTALS for any reflection without a tag. If the tags will all be '12', the normal LIST 6 input can be used, followed by the special call shown below.

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

After initial processing, LIST 13 should be changed to twinned, the correct space group entered, and the value of the observed structure factor stored as FOT, the Total or Twinned structure factor. This is done by a special call to the LIST 6 instructon, which also re-sets the TWIN flag in LIST 13.

\LIST 6
READ TYPE=TWIN
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 prohibited for twins. The special use on the LIST 6 instruction (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

10.3: SORTING TWINNED STRUCTURE DATA - \REORDER

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.

\REORDER

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 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.

The default output medium is usually to disk.

/REORDER
END

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, and 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
.......

The data can be processed in the true space group. LIST 6 input includes the 'element keys'. After data reduction, the data is stored as 'TWINNED' by the call to LIST 6 which saves the data in the .DSC file.

/LIST 1
REAL 7.2847 9.74 15.231 90 94.386 90
END

/SPACEGROUP
SYMBOL p 21/n
END

/list 13
crystal  friedel = no twinned=no
cond wave=1.5418
end

/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

/COMPOSITION
CONTENTS c  48  h  44  s  4  o  4  n  4
SCATT  CRSCP:SCATT
PROPER CRSCP:PROPERTIES
END

/LIST 23
MINIMISE  F-SQ=no
modify  anomalous=yes
END

/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

/CLOSE HKLI
/OPEN HKLI example.hkl
/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
/SYST
/SORT
/MERGE
END

/list 6
read type=twin
end

/SFLS
SCALE
END

/LIST 12
FULL FIRST(X'S, U[ISO]) UNTIL C(15)
equivalence element(1) element(2)
weight -1 element(2)
END

/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

Where It is the total observed intensity to which N components contribute, Ii is the intensity of component i , and vi is the amount of component i present in a given volume. The vi are known as the 'component scale factors', and are conventionally taken to be the amount of the given component present in a unit volume of the crystal, so that :

 SUM(vi) = 1      over all the components.

When a set of reflection data is handled for a twinned crystal, it is thus necessary to know which of the possible components contribute to the current reflection, and to be able to generate the indices of each of the components from a set of indices given in a standard reference system. If the indices of an component in its own reference system are given by the vector Hc and those in the standard system by H , the necessary interconversion is given by :

 Hc = R.H

R is a rotation matrix that describes the transformation of the indices. (The generation of the various sets of indices can be thought of as a rotation centred on the origin). The indices Hc are of necessity integers, but the components of H may in general take any value.

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.

Where X is the coordinate vector for any atom in the standard reference system, Xc is the coordinate vector for the same atom in the reference system for one of the components and H[T] indicates H transposed. The above expression may be rewritten as :

 H[T].R[T].Q.X = H[T].X

Where Q is the matrix that converts the atomic coordinates. Therefore :

 R[T].Q = I

Where I is the unit matrix. The matrix Q is thus given by:

 Q = R[TI]

Where R[TI] indicates R transposed and inverted. The coordinates therefore transform as :

 Xc = R[TI].X

Before any reflections can be processed, the matrices R must be provided. 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).