1. O AK R IDGE N ATIONAL L ABORATORY U. S. D EPARTMENT OF E NERGY 1 Handling Single Crystals in Inelastic Neutron Scattering Experiments Mark Lumsden Center for Neutron Scattering ORNL To be published, Journal of Applied Crystallography With Lee Robertson and Mona Yethiraj

2. O AK R IDGE N ATIONAL L ABORATORY U. S. D EPARTMENT OF E NERGY 2 Outline:  History and current state  Better way – what it enables in experiments – how to use it in practice  Basic concept  Application to triple-axis  Application to time-of-flight spectrometers

3. O AK R IDGE N ATIONAL L ABORATORY U. S. D EPARTMENT OF E NERGY 3 How single crystals were handled in the past (and present)  Triple-axis: usually have to work with a set of orthogonal vectors. If symmetry is less than orthorhombic, this is very difficult. Measurements are restricted to a plane.  TOF spectrometers: very little capability for handling single crystals during an experiment. Most things need to be calculated in a manual manner. Some limited abilities in data analysis/visualization.  Must be a better way!! UB matrix

4. O AK R IDGE N ATIONAL L ABORATORY U. S. D EPARTMENT OF E NERGY 4 What it enables in experiments:  Ability to handle any crystal symmetry (becoming increasingly important)  For triple-axis, allows movements (in h,k,l,E space) out of scattering plane  For TOF spectrometers, allows full mapping of measured intensity into h,k,l,E space  Simplifies data analysis/visualization  Requires less intervention from user  Provides considerable ease-of-use

5. O AK R IDGE N ATIONAL L ABORATORY U. S. D EPARTMENT OF E NERGY 5 How do you do an experiment on MAPS Complicated sample alignment using GENIE. From this alignment, you can manually calculate what is needed to align some direction with incident beam Manually insert a bunch of information into mslice to allow you to look at data (extremely difficult with lower than orthorhombic sym.) Possible experiment with UB matrix Measure 2 or more reflections and tell the computer what their indices are – UB matrix calculation completed!! Tell the computer where (in reciprocal space) you’d like to go (e.g. put c-axis along k i with a-axis vertical ) Convert full measured data set into h,k,l,E Run ‘mslice’ with very little data input – only what you want to look at (e.g. I want a slice in the (hk0) plane with a specific E and l range). NOTE: this would now work for ANY crystal symmetry!!

6. O AK R IDGE N ATIONAL L ABORATORY U. S. D EPARTMENT OF E NERGY 6

7. O AK R IDGE N ATIONAL L ABORATORY U. S. D EPARTMENT OF E NERGY 7 UB Matrix Myths Only works for 4-circle diffractometer Only works for diffraction NO – can easily be adapted to any goniometer NO – we have extended the formalism to handle inelastic case

8. O AK R IDGE N ATIONAL L ABORATORY U. S. D EPARTMENT OF E NERGY 8 UB Matrix – basic concept William R. Busing and Henry A. Levy, Acta. Cryst. 22, 457 (1967)  Based on rotation matrices  2 important matrices involved:  B transforms lattice constants into orthonormal space (crystal coord. system)  U transforms from crystal coord. system into instrument coord. system  What does this mean?? Coordinate mapping from angle space into reciprocal space!!  Basic Equation: Q L =  NUBQ  What do you need to do to implement it:  Scheme to calculate U from measured reflections (well tested schemes already exist – need Q =UBQ )  Convert from angles to h,k,l,E (simple Q=(UB) -1 Q )  Covert from h,k,l,E to angles - problem is usually overdetermined – need some appropriate constraints

9. O AK R IDGE N ATIONAL L ABORATORY U. S. D EPARTMENT OF E NERGY 9 Triple-axis implementation Busing and Levy’s formalism dealt with 4-circle diffractometer Needed to derive expressions for different goniometer Needed to generalize implementation to handle inelastic case

10. O AK R IDGE N ATIONAL L ABORATORY U. S. D EPARTMENT OF E NERGY 10 Triple-axis implementation (cont.) All of Busing and Levy’s calculations use this vector!

11. O AK R IDGE N ATIONAL L ABORATORY U. S. D EPARTMENT OF E NERGY 11 Triple-axis implementation (cont.) Calculate U: Measure N reflections and specify their h,k,l Ways to calculate U: 1.2 non-collinear reflections measured and lattice constants given 2. 3 or more non-coplanar reflections given. This will calculate U and fully refine the lattice constants. h,k,l,E from given angles: 1. From k i, k f, , calculate Q and  2. Measured s1 gives  s1  3.  together with , gives u 4. Q=(UB) -1 Q gives h,k,l

12. O AK R IDGE N ATIONAL L ABORATORY U. S. D EPARTMENT OF E NERGY 12 TOF Spectrometer Implementation Need to handle k f out of plane

13. O AK R IDGE N ATIONAL L ABORATORY U. S. D EPARTMENT OF E NERGY 13 TOF Spectrometer Implementation As we can now calculate u from measured angles, we can use identical procedures to calculate U. Q=(UB) -1 Q provides a mapping from measurement to h,k,l,E space!! (Given k i,k f, ,  ’,s1, ,  observation)

14. O AK R IDGE N ATIONAL L ABORATORY U. S. D EPARTMENT OF E NERGY 14 TOF Angle Calculations (case 1) Place a plane horizontal Specify plane with either plane normal or a pair of vectors within the plane – either can be converted to a plane normal unit vector in the -coordinate system, u  (using Q =UBQ): To get plane normal vertical, we need:

15. O AK R IDGE N ATIONAL L ABORATORY U. S. D EPARTMENT OF E NERGY 15 TOF Angle Calculations (case 2) Case of no goniometer arcs,  = =0 Start by converting Q=(h,k,l) into -coordinate system All angles defined!!

16. O AK R IDGE N ATIONAL L ABORATORY U. S. D EPARTMENT OF E NERGY 16 Conclusions  We have extended the UB matrix formalism of Busing and Levy to handle inelastic experiments on TOF spectrometers.  Implementation of this formalism at SNS should greatly simplify and enhance single crystal experiments on TOF spectrometers