1. The goal of Data Reduction From a series of diffraction images (films), obtain a file containing the intensity ( I ) and standard deviation (  ( I )) for each reflection, hkl. H K L I  0 0 4 3295.4 174.0 0 0 8 482.1 28.7 0 0 12 9691.0 500.7 0 0 16 1743.9 67.4 0 0 20 5856.0 221.0 0 0 24 14066.5 436.2 0 0 28 9936.3 311.7 0 0 36 8409.8 273.4 0 0 40 790.5 32.8 0 0 44 103.4 18.4..... 37 7 0 28.5 16.2 37 7 1 110.1 10.9 37 7 2 337.4 13.3 37 7 3 98.5 10.6 37 7 4 25.9 10.7 Set of films Final intensities 1.Index 2.Integrate 3.Merge

2. 2,0,0 3,0,-1 4,0,-2 5,0,-3 5,0,-4 5,0,-5 4,0,-6 3,0,-7 2,0,-7 1,0,-8 0,0,-8 -1,0,-8 Indexing Assign an h,k,l coordinate to each reflection of the first image.

3. Integration Within a spot, sum up the intensity of each pixel. Repeat for each spot on each film.

4. Merge Average (merge together) symmetry related reflections. Plane L=0 K H ? -H,-K, L= -K, H, L= K,-H, L= H, K,-L= H,-K,-L= K, H,-L= -K,-H,-L= -H,-K,-L= K,-H,-L= -K, H,-L= -H, K, L= -K,-H, L= K, H, L Plane L=0 K H K,-H,L H,K,L -H,K,L -K,-H,L K,H,L -H,-K,L H,-K,L -K,H,L

5. Three steps From a series of diffraction images, obtain a file containing the intensity ( I ) and standard deviation (  ( I )) for each reflection, hkl. 2,0,0 3,0,-1 4,0,-2 5,0,-3 5,0,-4 5,0,-5 4,0,-6 3,0,-7 2,0,-7 1,0,-8 0,0,-8 -1,0,-8 Set of films 1. index 2. integrate 3. merge K,-H,L H,K,L -H,K,L -K,-H,L K,H,L -H,-K,L H,-K,L -K,H,L Final intensities H K L I  0 0 4 3295.4 174.0 0 0 8 482.1 28.7 0 0 12 9691.0 500.7 0 0 16 1743.9 67.4 0 0 20 5856.0 221.0 0 0 24 14066.5 436.2 0 0 28 9936.3 311.7 0 0 36 8409.8 273.4 0 0 40 790.5 32.8 0 0 44 103.4 18.4..... 37 7 0 28.5 16.2 37 7 1 110.1 10.9 37 7 2 337.4 13.3 37 7 3 98.5 10.6 37 7 4 25.9 10.7

6. 2,0,0 3,0,-1 4,0,-2 5,0,-3 5,0,-4 5,0,-5 4,0,-6 3,0,-7 2,0,-7 1,0,-8 0,0,-8 -1,0,-8 Indexing How do we find the correct h,k,l coordinate of each reflection?

7. What’s the h,k,l of this spot? 3 lattice points in a* direction 2 lattice points in b* direction For a given spot on the film, we simply have to trace the diffracted ray back to the reciprocal lattice point (h,k,l) The answer is HKL=3,2,2 What parameters must be defined to complete this construction?

8. 15 parameters must be determined to index a spot. The wavelength of the incident radiation Coordinates (X,Y) of the direct beam Coordinates (X,Y) for the spot position Unit cell parameters a,b,c,  The orientation of the unit cell axes with respect to the laboratory axes (  ).

9. Which of the 15 parameters are set or known? Which are unknown? The wavelength of the incident radiation Coordinates (X,Y) of the direct beam Coordinates (X,Y) for the spot position Unit cell parameters a,b,c,  The orientation of the unit cell axes with respect to the laboratory axes (  ).

10. How is the unit cell and crystal orientation determined? Acta Cryst. (1999), D55, 1690-1695

11. Figure 1.

12. Choose principle axes by inspection

13. One dimensional Fourier transforms (7300 orientations) Find all pairs of spots that can be connected by a vector of given orientation, but any length. e.g.

14. One dimensional Fourier transforms (7300 orientations) Find all pairs of spots that can be connected by a vector of given orientation, but any length. e.g.

15. Figure 3. You will find some vector lengths are represented in the diffraction pattern much more frequently than others. These vector lengths differ by integral multiples of one particular value…corresponding to the unit cell dimension.

16. Figure 3.

17. One dimensional Fourier transforms (7300 orientations)

18. Figure 4.

19. A group of 30 possible non-linear vectors are calculated. 3 vectors at a time are combined to give a basis set of direct-space primitive unit cells. For each combination of 3 vectors, a distortion index is evaluated which describes how the observed fitted reciprocal lattice deviates from the 14 Bravais lattices. A chi squared statistics describes the deviations of the observed reflections from the theoretical lattice. Lattice parameters determined

20. Xdisplay and Peak Search 1) Display first image in your data set with xdisplay. xdispccd images/my.img 2) Press “Peak Search”. Red circles indicate position of prominent peaks (spots). 3) Evaluate whether you need more or fewer peaks. 4) Pres “OK” 5) Spot positions (x,y) are written to a file “peaks.file.”

21. Peaks.file 7777 0.0 0.0 1 1 height X Y frame 13 2695.7 1350.5 1 1 27 2669.5 1062.4 1 1 16 2570.6 1143.5 1 1 26 2569.4 1302.4 1 1 30 2562.5 1592.5 1 1 32 2554.5 1902.4 1 1 32 2524.5 1103.4 1 1 22 2514.5 1523.8 1 1 12 2503.4 1316.6 1 1 21 2494.5 1949.5 1 1 15 2492.5 1923.4 1 1 35 2488.5 1721.5 1 1 17 2483.5 1870.6 1 1 12 2479.4 1212.5 1 1 32 2465.5 1452.5 1 1 15 2456.4 638.4 1 1 13 2444.7 900.7 1 1 14 2437.6 1183.4 1 1 23 2436.4 1969.4 1 1 Etc…………………………………………..

22. Run autoindexing script The autoindexing script is simply titled “a.” Type “denzo: in the terminal window to start the program Denzo. Then type @a

24. Select a space group with desired Bravais Lattice (e.g. new space group P4) Predicted pattern should match observed diffraction pattern. “go” to refine

25. Necessary to index film 2 from scratch? film3? etc? 1o1o Film 1, exposed over 1 to 2 degrees Film 2, exposed over 2 to 3 degrees

26. The first film provides all the parameters need to predict the location of every spot on every film. The wavelength of the incident radiation Coordinates (X,Y) of the direct beam Coordinates (X,Y) for the spot position Unit cell parameters a,b,c,  The orientation of the unit cell axes with respect to the laboratory axes (  ).

27. Integration Adjust integration box size and background box size. spot elliptical 0.6 0.6 background elliptical 0.7 0.7

28. Paste parameters into integration script (integ.dat). Insert refined unit cell and crystal orientation parameters into integration script (integ.dat).Type “list” to obtain refined paramers..

29. Integrated intensities are written to.x files Film 1, exposed over 1 to 2 degrees h k l flag I(profit) I(prosum)  2  (I) cos incid. X pix Y pix 29 20 -33 1 202.3 200.8 1.36 17.4 0.556 6.4 1353.0 29 21 -31 1 102.1 105.0 1.08 7.7 0.560 16.8 1421.5 30 26 -19 1 1291.2 1323.2 1.19 50.0 0.554 23.9 1808.7 31 28 -11 1 1554.0 1618.7 1.26 95.1 0.536 24.2 2061.6 29 22 -29 1 24.0 25.2 1.29 1.2 0.564 28.0 1489.1 ……………………………………………………….

30. One.x file for each film Film 1, exposed over 1 to 2 degrees Film 2, exposed over 2 to 3 degrees Film 180, exposed over 180 to 181 degrees h k l flag I(profit) I(prosum)  2  (I) cos incid. X pix Y pix 29 20 -33 1 202.3 200.8 1.36 17.4 0.556 6.4 1353.0 29 21 -31 1 102.1 105.0 1.08 7.7 0.560 16.8 1421.5 30 26 -19 1 1291.2 1323.2 1.19 50.0 0.554 23.9 1808.7 31 28 -11 1 1554.0 1618.7 1.26 95.1 0.536 24.2 2061.6 29 22 -29 1 24.0 25.2 1.29 1.2 0.564 28.0 1489.1 29 20 -33 1 202.3 200.8 1.36 17.4 0.556 6.4 1353.0 29 21 -31 1 102.1 105.0 1.08 7.7 0.560 16.8 1421.5 30 26 -19 1 1291.2 1323.2 1.19 50.0 0.554 23.9 1808.7 31 28 -11 1 1554.0 1618.7 1.26 95.1 0.536 24.2 2061.6 29 20 -33 1 202.3 200.8 1.36 17.4 0.556 6.4 1353.0 29 21 -31 1 102.1 105.0 1.08 7.7 0.560 16.8 1421.5 30 26 -19 1 1291.2 1323.2 1.19 50.0 0.554 23.9 1808.7 31 28 -11 1 1554.0 1618.7 1.26 95.1 0.536 24.2 2061 h k l flag I(profit) I(prosum)  2  (I) cos incid. X pix Y pix 29 20 -33 1 52.3 50.8 1.36 17.4 0.556 6.4 1353.0 29 21 -31 1 102.1 105.0 1.08 7.7 0.560 16.8 1421.5 30 26 -19 1 1291.2 1323.2 1.19 50.0 0.554 23.9 1808.7 31 28 -11 1 1554.0 1618.7 1.26 95.1 0.536 24.2 2061.6 29 22 -29 1 24.0 25.2 1.29 1.2 0.564 28.0 1489.1 29 20 -33 1 202.3 200.8 1.36 17.4 0.556 6.4 1353.0 29 21 -31 1 102.1 105.0 1.08 7.7 0.560 16.8 1421.5 30 26 -19 1 1291.2 1323.2 1.19 50.0 0.554 23.9 1808.7 31 28 -11 1 1554.0 1618.7 1.26 95.1 0.536 24.2 2061.6 29 20 -33 1 202.3 200.8 1.36 17.4 0.556 6.4 1353.0 29 21 -31 1 102.1 105.0 1.08 7.7 0.560 16.8 1421.5 30 26 -19 1 1291.2 1323.2 1.19 50.0 0.554 23.9 1808.7 31 28 -11 1 1554.0 1618.7 1.26 95.1 0.536 24.2 2061 h k l flag I(profit) I(prosum)  2  (I) cos incid. X pix Y pix-29 -20 33 1 212.3 220.8 1.36 17.4 0.556 6.4 1353.0 29 21 -31 1 102.1 105.0 1.08 7.7 0.560 16.8 1421.5 30 26 -19 1 1291.2 1323.2 1.19 50.0 0.554 23.9 1808.7 31 28 -11 1 1554.0 1618.7 1.26 95.1 0.536 24.2 2061.6 29 22 -29 1 24.0 25.2 1.29 1.2 0.564 28.0 1489.1 29 20 -33 1 202.3 200.8 1.36 17.4 0.556 6.4 1353.0 29 21 -31 1 102.1 105.0 1.08 7.7 0.560 16.8 1421.5 30 26 -19 1 1291.2 1323.2 1.19 50.0 0.554 23.9 1808.7 31 28 -11 1 1554.0 1618.7 1.26 95.1 0.536 24.2 2061.6 29 20 -33 1 202.3 200.8 1.36 17.4 0.556 6.4 1353.0 29 21 -31 1 102.1 105.0 1.08 7.7 0.560 16.8 1421.5 30 26 -19 1 1291.2 1323.2 1.19 50.0 0.554 23.9 1808.7 31 28 -11 1 1554.0 1618.7 1.26 95.1 0.536 24.2 2061 prok_001.img prok_001.x prok_002.img prok_180.img prok_002.x prok_180.x

31. h k l flag I(profit) I(prosum)  2  (I) cos incid. X pix Y pix 29 20 -33 1 202.3 200.8 1.36 17.4 0.556 6.4 1353.0 29 21 -31 1 102.1 105.0 1.08 7.7 0.560 16.8 1421.5 30 26 -19 1 1291.2 1323.2 1.19 50.0 0.554 23.9 1808.7 31 28 -11 1 1554.0 1618.7 1.26 95.1 0.536 24.2 2061.6 29 22 -29 1 24.0 25.2 1.29 1.2 0.564 28.0 1489.1 29 20 -33 1 202.3 200.8 1.36 17.4 0.556 6.4 1353.0 29 21 -31 1 102.1 105.0 1.08 7.7 0.560 16.8 1421.5 30 26 -19 1 1291.2 1323.2 1.19 50.0 0.554 23.9 1808.7 31 28 -11 1 1554.0 1618.7 1.26 95.1 0.536 24.2 2061.6 29 20 -33 1 202.3 200.8 1.36 17.4 0.556 6.4 1353.0 29 21 -31 1 102.1 105.0 1.08 7.7 0.560 16.8 1421.5 30 26 -19 1 1291.2 1323.2 1.19 50.0 0.554 23.9 1808.7 31 28 -11 1 1554.0 1618.7 1.26 95.1 0.536 24.2 2061 360 frames, 1 degree rotation each With.x files, we can map intensities onto a reciprocal lattice 1)Accuracy will improve if we Merge multiple observations of the same reciprocal lattice point 2)But, we must test if rotational symmetry exists between lattice points. h k l flag I(profit) I(prosum)  2  (I) cos incid. X pix Y pix 29 20 -33 1 202.3 200.8 1.36 17.4 0.556 6.4 1353.0 29 21 -31 1 102.1 105.0 1.08 7.7 0.560 16.8 1421.5 30 26 -19 1 1291.2 1323.2 1.19 50.0 0.554 23.9 1808.7 31 28 -11 1 1554.0 1618.7 1.26 95.1 0.536 24.2 2061.6 29 22 -29 1 24.0 25.2 1.29 1.2 0.564 28.0 1489.1 29 20 -33 1 202.3 200.8 1.36 17.4 0.556 6.4 1353.0 29 21 -31 1 102.1 105.0 1.08 7.7 0.560 16.8 1421.5 30 26 -19 1 1291.2 1323.2 1.19 50.0 0.554 23.9 1808.7 31 28 -11 1 1554.0 1618.7 1.26 95.1 0.536 24.2 2061.6 29 20 -33 1 202.3 200.8 1.36 17.4 0.556 6.4 1353.0 29 21 -31 1 102.1 105.0 1.08 7.7 0.560 16.8 1421.5 30 26 -19 1 1291.2 1323.2 1.19 50.0 0.554 23.9 1808.7 31 28 -11 1 1554.0 1618.7 1.26 95.1 0.536 24.2 2061 h k l flag I(profit) I(prosum)  2  (I) cos incid. X pix Y pix 29 20 -33 1 202.3 200.8 1.36 17.4 0.556 6.4 1353.0 29 21 -31 1 102.1 105.0 1.08 7.7 0.560 16.8 1421.5 30 26 -19 1 1291.2 1323.2 1.19 50.0 0.554 23.9 1808.7 31 28 -11 1 1554.0 1618.7 1.26 95.1 0.536 24.2 2061.6 29 22 -29 1 24.0 25.2 1.29 1.2 0.564 28.0 1489.1 29 20 -33 1 202.3 200.8 1.36 17.4 0.556 6.4 1353.0 29 21 -31 1 102.1 105.0 1.08 7.7 0.560 16.8 1421.5 30 26 -19 1 1291.2 1323.2 1.19 50.0 0.554 23.9 1808.7 31 28 -11 1 1554.0 1618.7 1.26 95.1 0.536 24.2 2061.6 29 20 -33 1 202.3 200.8 1.36 17.4 0.556 6.4 1353.0 29 21 -31 1 102.1 105.0 1.08 7.7 0.560 16.8 1421.5 30 26 -19 1 1291.2 1323.2 1.19 50.0 0.554 23.9 1808.7 31 28 -11 1 1554.0 1618.7 1.26 95.1 0.536 24.2 2061 h k l flag I(profit) I(prosum)  2  (I) cos incid. X pix Y pix 29 20 -33 1 202.3 200.8 1.36 17.4 0.556 6.4 1353.0 29 21 -31 1 102.1 105.0 1.08 7.7 0.560 16.8 1421.5 30 26 -19 1 1291.2 1323.2 1.19 50.0 0.554 23.9 1808.7 31 28 -11 1 1554.0 1618.7 1.26 95.1 0.536 24.2 2061.6 29 22 -29 1 24.0 25.2 1.29 1.2 0.564 28.0 1489.1 29 20 -33 1 202.3 200.8 1.36 17.4 0.556 6.4 1353.0 29 21 -31 1 102.1 105.0 1.08 7.7 0.560 16.8 1421.5 30 26 -19 1 1291.2 1323.2 1.19 50.0 0.554 23.9 1808.7 31 28 -11 1 1554.0 1618.7 1.26 95.1 0.536 24.2 2061.6 29 20 -33 1 202.3 200.8 1.36 17.4 0.556 6.4 1353.0 29 21 -31 1 102.1 105.0 1.08 7.7 0.560 16.8 1421.5 30 26 -19 1 1291.2 1323.2 1.19 50.0 0.554 23.9 1808.7 31 28 -11 1 1554.0 1618.7 1.26 95.1 0.536 24.2 2061 prok_001 -> 360.x

32. Is it Laue group 422 Or Laue group 4? P4 H, K,L -H,-K,L -K, H,L K,-H,L H, K,-L H,-K,-L K, H,-L -K,-H,-L P422 Test existence of 4-fold symmetry Test existence of 4-fold Symmetry and Perpendicular 2-fold symmetry

33. j observations of the reflection 30 22 6 = (100 + 500 + 300) / 3 = 300 jHKL I 130226100 2-30-226500 322-306300 R sym =  | I j - |  I j Discrepancy between symmetry related reflections  | I j - | = |100-300|+|500-300|+|300-300| = 200 + 200 + 0 = 400  I j = 100 + 500 + 300 = 900 R sym = 400/900 = 0.44 = 44%

34. Discrepancy between symmetry related reflections (R sym ) increases with increasing resolution. Why? ShellR sym 100-5.0Å0.04 5.0-3.0Å0.06 3.0-2.0Å0.08 2.0-1.7Å0.15 Statistics are analyzed as a function of resolution (N shells).

35. Average I/  decreases with increasing resolution High resolution shells with I/  <2 should be discarded. Shell I/   100-5.0 Å20.0 5.0-3.0 Å10.0 3.0-2.0 Å7.0 2.0-1.7 Å3.0 SIGNAL TO NOISE RATIO (I/  )

36. COMPLETENESS? What percentage of reciprocal Lattice was measured for a given Resolution limit? Better than 90% I hope. Shell completeness 100-5.0Å99.9% 5.0-3.0Å95.5% 3.0-2.0Å89.0% 2.0-1.7Å85.3% Overall 92.5%

37. Assignment

38. Effect of mosaicity and wavelength spread 

39. Figure 5.

40. Figure 6.