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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