Twinning
Twinning
• Like disorder but of unit cell orientation…
– In a perfect single crystal, unit cells are all
found in the same orientation.
• We can consider a group of the same orientation
to belong to a domain. Here only one domain
exists.
– In a twinned crystal, some unit cells may not
be in the same orientation as others
• Multiple domains exist
Twinning
• Why is that a problem?
– Orientation of the unit cell relates to the
orientation of the diffraction pattern
– Overlap of reflections can occur
– Overlap obscures correct intensities from us
– We need accurately measured intensities to
solve and refine a good structure
Twinning
• Non-merohedral twinning
– The data collected can usually be seen to contain two
different diffraction patterns.
– Imagine two crystals stuck together in an arbitrary
fashion. They will each produce a discrete diffraction
pattern which will overlay.
– For a true twin the crystals must grow through each
other but cracked crystals, small passengers, etc
produce the same effect in the diffraction pattern and
therefore may also be dealt with in the same way.
Twinning
Contains intensities
from
(3domain
(-1
4)
5)
1 (3 4) reflection
domain 2 (-1 5) reflection
Twinning
• Merohedral twinning
– One diffraction pattern observed
– Diffraction pattern fully overlapped.
– The intensity of all reflections is due to the
addition of two (or more) different reflections.
– Can be difficult to detect (virtually impossible
at the diffractometer)
Twinning
• Merohedral twinning
– Is a consequence of unit cell shape.
– The unit cell shape may have different
symmetry from its contents.
– Imagine the following scenario…
Twinning
Twinning
Contains intensities
from
domain 1 (3 4) reflection
domain 2 (-4 -3) reflection
Twinning
• More common in higher symmetry crystal
systems
– Orthorhombic
– Tetragonal
– Etc
• N.B. Pseudo-merohedral twinning can occur
when a low symmetry crystal system is close in
shape to higher symmetry crystal system.
– e.g. monoclinic with close to 90°
– e.g. orthorhombic with a close to b in length
Twinning
• Treatment
– Two ways
• Index both domains prior to integration
– reflection files (HKLF 5 + HKLF4)
» each measured, overlapped intensity is assigned
multiple Miller indices linked to each domain
» ratio of domains is determined during refinement
• Detect twinning after integration (during
refinement)
– reflection file (HKLF 4)
» only one set of indices for each measured intensity
» need a twin law to relate domains to one another
Describing a twin
• Need a means of relating the two
components mathematically
– They are related by symmetry a operation
• e.g. rotation or mirror
– Symmetry operations transform coordinates
– Matrices transform coordinates and therefore
describe symmetry operations
• Use a Matrix!
– A matrix relating two domains is called a twin
law.
Matrix algebra
• Matrix is a 2D array of numbers
– E.g. a 3x3 matrix contains three rows and 3
columns (always state rows then columns)
( )
0 1 2
3 4 5
6 7 8
Matrix algebra
• Condition to multiply matrixes.
–
–
–
–
–
–
(m x n) x (p x r)
Letters describe dimensions of matrix.
n must be equal to p, result will be a (m x r) matrix.
E.g. 3x3 matrix multiplied by a 3x1 matrix gives a 3x1
e.g. 4x3 matrix multiplied by a 3x2 matrix gives a 4x2
You cannot multiply a 3x3 by a 2x2 for example
Matrix algebra
( )( ) = ( )
1 2 3
4 5 6
7 8 9
1
2
3
14
32
50
( (7x1)
(1x1) + (8x2)
(4x1)
(2x2) + (9x3)
(5x2)
(3x3) ) = 50
(6x3)
14
32
How does this relate?
• Since a twin is essentially a rotation of
some unit cells with respect to others we
can relate one to the other by describing
the rotation
– matrices can describe rotations
• a rotation transforms one set of coordinates into
another
• we call the rotation matrix a twin law
Simple rotations
z’
z
2
y
x
x’
y’
New axes in terms of old ones
x’ = -x
y’ = -y
z’ = z
So to convert…
( )( ) = ( )
-1 0 0
0 -1 0
0 0 1
x
y
z
this describes 2-fold about z
-x
-y
z
Data Integrated as a Twin
• Twinning detected prior to integration
– Files given, HKLF 4, HKLF 5
– HKLF 4 file is to allow initial structure solution
– HKLF 5 file is for refinement
• Ins file needs…
– a BASF instruction
• This is a scale factor which determines relative quantity of
each domain
• Put in top (header) section
• Requires an initial value, e.g. 50:50 is
– BASF 0.5
Data Integrated as a twin
• Ins file needs…
– HKLF instruction near end of file needs to
match reflection file format or shelx will
complain
• Use HKLF 4 to solve
• Use HKLF 5 to refine
• N.B.
– HKLF 5 data is ‘twin law aware’ so you do not
need to specify the twin law
Handling twinned data
• Second case, not detected at integration
– Files given, HKLF 4 (as normal)
– Warning signs
•
•
•
•
•
Can’t solve
Odd Q-peaks
Unusual ADP’s
Q-peaks look like a ‘ghost’ molecule superimposed on your model
Unexpectedly high r-factor
– Determining the twin law
• Try an additional symmetry element if pseudo-merohedral
– e.g. for monoclinic with beta close to 90°
» try 2-fold about a or c
• Use Rotax or TwinRotMat in Platon to determine the law
– Requires an fcf file.
» Use ACTA or LIST 4 in the ins file.
Ins File
• Ins file needs…
– A TWIN instruction which specifies the twin law on one line
• e.g.
( -1 0 0 )
( 0 -1 0 )
( 0 0 1)
• becomes TWIN -1 0 0 0 -1 0 0 0 1
– A BASF instruction as before
• N.B.
– No need to change the HKLF instruction
OR
• Can try to generate an HKLF 5 file from the HKLF 4 file and Twin
Law, using WinGX.
Did it work?
• Check the BASF value after refinement
• Check the change in R-Factor
• Hopefully any problems you had will go
away.
Using Rotax
• Using rotax
– Can run through WinGX or standalone.
– Output will give potential twin laws with a figure of
merit (f.o.m.).
• In general f.o.m. should be less than ~5 (which is default cutoff.
• Smaller is better
• Note number of rejected reflections
– 30 reflections are used in total, a max of 15 may be discarded
» If 15 are discarded, and f.o.m. changes significantly twin
law is less credible but still worth trying.
Example output
On command line
Direct direction
1.
Direct direction
0.
Reciprocal direction
0.
Reciprocal direction
0.
Reciprocal direction -5.
Reciprocal direction -12.
Direct direction
2.
Direct direction
5.
0. 0.
1. 0.
1. 0.
0. 1.
0. 2.
0. 5.
0. 5.
0. 12.
f.o.m.
f.o.m.
f.o.m.
f.o.m.
f.o.m.
f.o.m.
f.o.m.
f.o.m.
=
=
=
=
=
=
=
=
In List file (rotax.out or rotax.lst)
[
1.000
0.000
0.000]
[
0.000
-1.000
0.000]
[ -0.822
0.000
-1.000]
Figure of merit =
7.06
15 reflections omitted
Figure of merit with no omissions =
9.67
7.06
0.00
0.00
7.06
5.98
8.39
5.98
8.39