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.

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 (section 5.14) 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.
 

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 further 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.
 

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), alpha(1) Fobs.vol-fract(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 (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

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'.
 

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.

\LIST 5
READ NATOM=  NELEMENT=
ELEMENT value(1) value(2) ...
ATOM ..........
......
END

      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
 
The twin laws must be entered and CRYSTALS instructed to convert the reflection list to a twinned list.

      \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

After initial processing, LIST 13 (section 4.13) 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 instruction (which also sets the TWIN flag in LIST 13).

\LIST 6
READ TYPE=TWIN
MATRIX TWINTOL=.001
END

\LIST 13
....
CRYSTALS FRIEDEL=NO  TWIN=YES  EXTINCTION=NO

\LIST 12
FULL ........
EQUIVALENCE ELEMENT(1) ELEMENT(2)
WEIGHT -1 ELEMENT(2)
END

\LIST12
FULL ........
CONTINUE ELEMENT SCALES
END
\LIST 16
SUM .0001 ELEMENT SCALES
END

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.

      M/T
      DISC   -  DEFAULT VALUE.

/REORDER
END

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 

4.13), and the observed intensity input as /FOT/. This is preserved for backwards compatibility.

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 (reflection) 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.

\  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

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).