1. The four-circle single crystal diffractometer

5. (see Arndt & Willis, Single Crystal Diffractometry) The four-circle single crystal diffractometer (see Arndt & Willis, Single Crystal Diffractometry) Counter restricted to plane Counter rotated to appropriate Bragg angle Crystal rotated into reflecting position Counter restricted to plane Counter rotated to appropriate Bragg angle Crystal rotated into reflecting position

6. The four-circle single crystal diffractometer (see Arndt & Willis, Single Crystal Diffractometry) The four-circle single crystal diffractometer (see Arndt & Willis, Single Crystal Diffractometry) Counter restricted to plane Counter rotated to appropriate Bragg angle Crystal rotated into reflecting position Counter restricted to plane Counter rotated to appropriate Bragg angle Crystal rotated into reflecting position

7. The four-circle single crystal diffractometer (see Arndt & Willis, Single Crystal Diffractometry) The four-circle single crystal diffractometer (see Arndt & Willis, Single Crystal Diffractometry) Counter restricted to plane Counter rotated to appropriate Bragg angle Crystal rotated into reflecting position Counter restricted to plane Counter rotated to appropriate Bragg angle Crystal rotated into reflecting position

8. The four-circle single crystal diffractometer (see Arndt & Willis, Single Crystal Diffractometry) The four-circle single crystal diffractometer (see Arndt & Willis, Single Crystal Diffractometry) Counter restricted to plane Counter rotated to appropriate Bragg angle Crystal rotated into reflecting position Counter restricted to plane Counter rotated to appropriate Bragg angle Crystal rotated into reflecting position

9. The four-circle single crystal diffractometer Detect reflections one at a time (conventional counter) Thus, need to know where reflections are Detect reflections one at a time (conventional counter) Thus, need to know where reflections are

10. The four-circle single crystal diffractometer Detect reflections one at a time (conventional counter) Thus, need to know where reflections are To get reflection from a particular set of planes: locate reciprocal lattice pt wrt instrument coords rotate crystal so that reciprocal lattice vector coincident w/ diffraction vector Detect reflections one at a time (conventional counter) Thus, need to know where reflections are To get reflection from a particular set of planes: locate reciprocal lattice pt wrt instrument coords rotate crystal so that reciprocal lattice vector coincident w/ diffraction vector soso s S = s o – s

11. The four-circle single crystal diffractometer Counter restricted to plane Counter rotated to appropriate Bragg angle Crystal rotated into reflecting position Here, counter in position to receive reflection but crystal not in position to reflect Crystal is rotated around the axes to bring it to correct position for reflection Counter restricted to plane Counter rotated to appropriate Bragg angle Crystal rotated into reflecting position Here, counter in position to receive reflection but crystal not in position to reflect Crystal is rotated around the axes to bring it to correct position for reflection s soso S = s o – s

12. The four-circle single crystal diffractometer Counter restricted to plane Counter rotated to appropriate Bragg angle Crystal rotated into reflecting position Here, counter in position to receive reflection but crystal not in position to reflect Crystal is rotated around the axes to bring it to correct position for reflection Counter restricted to plane Counter rotated to appropriate Bragg angle Crystal rotated into reflecting position Here, counter in position to receive reflection but crystal not in position to reflect Crystal is rotated around the axes to bring it to correct position for reflection

13. The four-circle single crystal diffractometer Counter restricted to plane Counter rotated to appropriate Bragg angle Crystal rotated into reflecting position Here, counter in position to receive reflection but crystal not in position to reflect Crystal is rotated around the axes to bring it to correct position for reflection Counter restricted to plane Counter rotated to appropriate Bragg angle Crystal rotated into reflecting position Here, counter in position to receive reflection but crystal not in position to reflect Crystal is rotated around the axes to bring it to correct position for reflection

14. The four-circle single crystal diffractometer Counter restricted to plane Counter rotated to appropriate Bragg angle Crystal rotated into reflecting position Here, counter in position to receive reflection but crystal not in position to reflect Crystal is rotated around the axes to bring it to correct position for reflection Counter restricted to plane Counter rotated to appropriate Bragg angle Crystal rotated into reflecting position Here, counter in position to receive reflection but crystal not in position to reflect Crystal is rotated around the axes to bring it to correct position for reflection

15. The four-circle single crystal diffractometer Counter restricted to plane Counter rotated to appropriate Bragg angle Crystal rotated into reflecting position Here, counter in position to receive reflection but crystal not in position to reflect Crystal is rotated around the axes to bring it to correct position for reflection Counter restricted to plane Counter rotated to appropriate Bragg angle Crystal rotated into reflecting position Here, counter in position to receive reflection but crystal not in position to reflect Crystal is rotated around the axes to bring it to correct position for reflection

16. The four-circle single crystal diffractometer Counter restricted to plane Counter rotated to appropriate Bragg angle Crystal rotated into reflecting position Here, counter in position to receive reflection but crystal not in position to reflect Crystal is rotated around the axes to bring it to correct position for reflection Counter restricted to plane Counter rotated to appropriate Bragg angle Crystal rotated into reflecting position Here, counter in position to receive reflection but crystal not in position to reflect Crystal is rotated around the axes to bring it to correct position for reflection

17. The four-circle single crystal diffractometer Counter restricted to plane Counter rotated to appropriate Bragg angle Crystal rotated into reflecting position Counter restricted to plane Counter rotated to appropriate Bragg angle Crystal rotated into reflecting position   = 0 when  -axis along z

18. The four-circle single crystal diffractometer (see Arndt & Willis, Single Crystal Diffractometry) The four-circle single crystal diffractometer (see Arndt & Willis, Single Crystal Diffractometry) Counter restricted to plane Counter rotated to appropriate Bragg angle Crystal rotated into reflecting position Counter restricted to plane Counter rotated to appropriate Bragg angle Crystal rotated into reflecting position   = 0 when  -axis along z  = 0 when  -circle normal to x   = 0 when  -axis along z  = 0 when  -circle normal to x

19. The four-circle single crystal diffractometer (see Arndt & Willis, Single Crystal Diffractometry) The four-circle single crystal diffractometer (see Arndt & Willis, Single Crystal Diffractometry) Counter restricted to plane Counter rotated to appropriate Bragg angle Crystal rotated into reflecting position Counter restricted to plane Counter rotated to appropriate Bragg angle Crystal rotated into reflecting position   = 0 when  -axis along z   0 when   l   x 2  = 0 when counter at beam position   = 0 when  -axis along z   0 when   l   x 2  = 0 when counter at beam position

20. The four-circle single crystal diffractometer (see Arndt & Willis, Single Crystal Diffractometry) The four-circle single crystal diffractometer (see Arndt & Willis, Single Crystal Diffractometry) Counter restricted to plane Counter rotated to appropriate Bragg angle Crystal rotated into reflecting position Counter restricted to plane Counter rotated to appropriate Bragg angle Crystal rotated into reflecting position   = 0 when  -axis along z   0 when   l   x 2  = 0 when counter at beam position  arbitrary   = 0 when  -axis along z   0 when   l   x 2  = 0 when counter at beam position  arbitrary

21. The four-circle single crystal diffractometer Instrument alignment 1. Adjust tube to instrument - pinhole collimator sees tube focal spot Instrument alignment 1. Adjust tube to instrument - pinhole collimator sees tube focal spot

22. The four-circle single crystal diffractometer Instrument alignment 1. Adjust tube to instrument - pinhole collimator sees tube focal spot 2. Align goniometer using alignment crystal small sphere (< 0.3 mm) stable strong reflections low mosaicity minimum fluorescence ACA ruby crystals Instrument alignment 1. Adjust tube to instrument - pinhole collimator sees tube focal spot 2. Align goniometer using alignment crystal small sphere (< 0.3 mm) stable strong reflections low mosaicity minimum fluorescence ACA ruby crystals

23. The four-circle single crystal diffractometer Instrument alignment 1. Adjust tube to instrument - pinhole collimator sees tube focal spot 2. Align goniometer using alignment crystal small sphere (< 0.3 mm) stable strong reflections low mosaicity minimum fluorescence ACA ruby crystals Instrument alignment 1. Adjust tube to instrument - pinhole collimator sees tube focal spot 2. Align goniometer using alignment crystal small sphere (< 0.3 mm) stable strong reflections low mosaicity minimum fluorescence ACA ruby crystals

24. The four-circle single crystal diffractometer Instrument alignment 2. Align goniometer using alignment crystal Centre crystal in goniometer using telescope Approx. 2  zero align Locate strong reflection from crystal Centre diffracted beam from crystal Shift all 4 axes until reflection centred Instrument alignment 2. Align goniometer using alignment crystal Centre crystal in goniometer using telescope Approx. 2  zero align Locate strong reflection from crystal Centre diffracted beam from crystal Shift all 4 axes until reflection centred

25. The four-circle single crystal diffractometer Instrument alignment 2. Align goniometer using alignment crystal Centre crystal in goniometer using telescope Approx. 2  zero align Locate strong reflection from crystal Centre diffracted beam from crystal Shift all 4 axes until reflection centred Repeat at –2  to find 2  = 0  and  zeroes set during this process Instrument alignment 2. Align goniometer using alignment crystal Centre crystal in goniometer using telescope Approx. 2  zero align Locate strong reflection from crystal Centre diffracted beam from crystal Shift all 4 axes until reflection centred Repeat at –2  to find 2  = 0  and  zeroes set during this process

26. The four-circle single crystal diffractometer Crystal alignment Use nearly same procedure - adjust goniometer head arcs Crystal alignment Use nearly same procedure - adjust goniometer head arcs

27. The four-circle single crystal diffractometer Crystal alignment Use nearly same procedure - adjust goniometer head arcs Films & other flat specimens Can use laser for initial alignment Adjust specimen height Rotate , adjusting goniometer head arcs until laser spot stationary Crystal alignment Use nearly same procedure - adjust goniometer head arcs Films & other flat specimens Can use laser for initial alignment Adjust specimen height Rotate , adjusting goniometer head arcs until laser spot stationary

28. The four-circle single crystal diffractometer Crystal alignment Use nearly same procedure - adjust goniometer head arcs Films & other flat specimens Can use laser for initial alignment Adjust specimen height Rotate , adjusting goniometer head arcs until laser spot stationary – repeat w/ x-ray reflection Crystal alignment Use nearly same procedure - adjust goniometer head arcs Films & other flat specimens Can use laser for initial alignment Adjust specimen height Rotate , adjusting goniometer head arcs until laser spot stationary – repeat w/ x-ray reflection

29. The four-circle single crystal diffractometer Flat specimen application - texture analysis What is texture (preferred orientation)? Flat specimen application - texture analysis What is texture (preferred orientation)?

30. The four-circle single crystal diffractometer Flat specimen application - texture analysis What is texture (preferred orientation)? 1st – the stereographic projection Flat specimen application - texture analysis What is texture (preferred orientation)? 1st – the stereographic projection

31. The four-circle single crystal diffractometer Flat specimen application - texture analysis What is texture (preferred orientation)? Now consider {100} pole distribution in polycrystalline sheet materials (cubic) – use stereographic projection representation crystals randomlycrystals all aligned oriented Flat specimen application - texture analysis What is texture (preferred orientation)? Now consider {100} pole distribution in polycrystalline sheet materials (cubic) – use stereographic projection representation crystals randomlycrystals all aligned oriented

32. The four-circle single crystal diffractometer Flat specimen application - texture analysis What is texture (preferred orientation)? Now consider {100} pole distribution in polycrystalline sheet materials (cubic) – use stereographic projection representation crystals randomlycrystals all aligned oriented For real textured matl, pole figure is somewhere betwn these Flat specimen application - texture analysis What is texture (preferred orientation)? Now consider {100} pole distribution in polycrystalline sheet materials (cubic) – use stereographic projection representation crystals randomlycrystals all aligned oriented For real textured matl, pole figure is somewhere betwn these

33. The four-circle single crystal diffractometer Flat specimen application - texture analysis {100} pole distribution in polycrystalline sheet materials (cubic) – use stereographic projection representation Use four-circle system to get pole density distribution Flat specimen application - texture analysis {100} pole distribution in polycrystalline sheet materials (cubic) – use stereographic projection representation Use four-circle system to get pole density distribution

34. The four-circle single crystal diffractometer Flat specimen application - texture analysis {100} pole distribution in polycrystalline sheet materials (cubic) – use stereographic projection representation Use four-circle system to get pole density distribution How?? Flat specimen application - texture analysis {100} pole distribution in polycrystalline sheet materials (cubic) – use stereographic projection representation Use four-circle system to get pole density distribution How??

35. The four-circle single crystal diffractometer Another preferred orientation application – drawn polymers (fiber texture) Pole figure Another preferred orientation application – drawn polymers (fiber texture) Pole figure FA

36. The four-circle single crystal diffractometer Another preferred orientation application – drawn polymers (fiber texture) Pole figure Use four-circle system to get pole density distribution How?? Another preferred orientation application – drawn polymers (fiber texture) Pole figure Use four-circle system to get pole density distribution How?? FA

37. The four-circle single crystal diffractometer Unknown crystal orientation – initial reflection search  scans at various  s for a sequence of 2  s Centre reflections w/ half shutters of detector aperture Get , , , 2  for each reflection Unknown crystal orientation – initial reflection search  scans at various  s for a sequence of 2  s Centre reflections w/ half shutters of detector aperture Get , , , 2  for each reflection

38. The four-circle single crystal diffractometer Unknown crystal orientation – initial reflection search  scans at various  s for a sequence of 2  s Centre reflections w/ half shutters of detector aperture Get , , , 2  for each reflection Need ~25 reflections to index (get (hkl)s and lattice params) Then get orientation matrix Unknown crystal orientation – initial reflection search  scans at various  s for a sequence of 2  s Centre reflections w/ half shutters of detector aperture Get , , , 2  for each reflection Need ~25 reflections to index (get (hkl)s and lattice params) Then get orientation matrix

39. The four-circle single crystal diffractometer Unknown crystal orientation – initial reflection search Get orientation matrix Two coord. Systems: diffractometer – xyz (orthogonal) recip. lattice – a*b*c*(may be oblique) Unknown crystal orientation – initial reflection search Get orientation matrix Two coord. Systems: diffractometer – xyz (orthogonal) recip. lattice – a*b*c*(may be oblique)

40. The four-circle single crystal diffractometer Unknown crystal orientation – initial reflection search Get orientation matrix Two coord. Systems: diffractometer – xyz (orthogonal) recip. lattice – a*b*c*(may be oblique) Use orthogonal recip lattice coord. System H orthog = BH B is matrix that transforms from oblique to orthog. System and H xyz = UH orthog Unknown crystal orientation – initial reflection search Get orientation matrix Two coord. Systems: diffractometer – xyz (orthogonal) recip. lattice – a*b*c*(may be oblique) Use orthogonal recip lattice coord. System H orthog = BH B is matrix that transforms from oblique to orthog. System and H xyz = UH orthog

41. The four-circle single crystal diffractometer To get reflection from a particular reflection: locate recip. Lattice pt. wrt instrument coords. rotate crystal so that recip. lattice vector is coincident w/ diffraction vector For the latter: To get reflection from a particular reflection: locate recip. Lattice pt. wrt instrument coords. rotate crystal so that recip. lattice vector is coincident w/ diffraction vector For the latter:

42. The four-circle single crystal diffractometer 1 2 3

43. To get reflection from a particular reflection:

44. The four-circle single crystal diffractometer