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.

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

(3 4)(-1 5)

Contains intensitiesfromdomain 1 (3 4) reflectiondomain 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

Contains intensitiesfromdomain 1 (3 4) reflection

Twinning

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 23 4 56 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

( )( ) = ( )1 2 34 5 67 8 9

123

143250

Matrix algebra

( (1x1) + (2x2) + (3x3) ) = 14( (4x1) + (5x2) + (6x3) ) = 32( (7x1) + (8x2) + (9x3) ) = 50

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 rotationsz

y

x x’

y’

z’

New axes in terms of old ones

x’ = -xy’ = -yz’ = z

So to convert…

( )( ) = ( )-1 0 0 0 -1 0 0 0 1

xyz

-x-y z

this describes 2-fold about z

2

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 cut-

off.• 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 outputOn command line

Direct direction 1. 0. 0. f.o.m. = 7.06Direct direction 0. 1. 0. f.o.m. = 0.00Reciprocal direction 0. 1. 0. f.o.m. = 0.00Reciprocal direction 0. 0. 1. f.o.m. = 7.06Reciprocal direction -5. 0. 2. f.o.m. = 5.98Reciprocal direction -12. 0. 5. f.o.m. = 8.39Direct direction 2. 0. 5. f.o.m. = 5.98Direct direction 5. 0. 12. f.o.m. = 8.39

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 = 7.0615 reflections omittedFigure of merit with no omissions = 9.67