The goal of Data Reduction

1
The goal of Data Reduction

• From a series of diffraction images (films),
  obtain a file containing the intensity (I) and
  standard deviation (s(I)) for each reflection,
  hkl.

Final intensities
Set of films
H K L I s 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

1. Index
2. Integrate
3. Merge

2
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
K
Plane L0
? -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
K,H,L
K,-H,L
H,K,L
H,-K,L
H
-H,K,L
-H,-K,L
-K,-H,L
-K,H,L
Average (merge together) symmetry related
reflections.
5
Three steps

• From a series of diffraction images, obtain a
  file containing the intensity (I) and standard
  deviation (s(I)) for each reflection, hkl.

H K L I s 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
2. integrate
Set of films
Final intensities
3. merge
1. index
6
Indexing
How do we find the correct h,k,l coordinate of
each reflection?
7
For a given spot on the film, we simply have to
trace the diffracted ray back to the reciprocal
latticepoint (h,k,l)
The answer is HKL3,2,2
3 lattice points in a direction
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
Unit cell parameters a,b,c, a,b,g
9
Which of the 15 parameters are set or known?
Which are unknown?
The wavelength of the incident radiation
Unit cell parameters a,b,c, a,b,g
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
valuecorresponding to the unit cell dimension.
16
Figure 3.
17
One dimensional Fourier transforms (7300
orientations)
18
Figure 4.
19
Lattice parameters determined
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.
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

22
Run autoindexing script
The autoindexing script is simply titled
a. Type denzo in the terminal window to
start the program Denzo. Then type _at_a
23
(No Transcript)
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?
1o
Film 2, exposed over 2 to 3 degrees
Film 1, exposed over 1 to 2 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
Unit cell parameters a,b,c, a,b,g
27
Integration
Adjust integration box size and background box
size. spot elliptical 0.6 0.6 background
elliptical 0.7 0.7
28
Insert refined unit cell and crystal orientation
parameters into integration script
(integ.dat).Type list to obtain refined
paramers..
Paste parameters into integration script
(integ.dat).
29
Integrated intensities are written to .x files
Film 1, exposed over 1 to 2 degrees
h k l flag I(profit)
I(prosum) c2 s(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
prok_001.img
prok_180.img
prok_002.img
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) c2 s(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
h k l flag I(profit)
I(prosum) c2 s(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) c2 s(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_180.x
prok_002.x
prok_001.x
31
h k l flag I(profit)
I(prosum) c2 s(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) c2 s(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
h k l flag I(profit)
I(prosum) c2 s(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) c2 s(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 -gt 360.x
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.

32
Is it Laue group 422 Or Laue group 4?
P422
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
Test existence of 4-fold Symmetry
and Perpendicular 2-fold symmetry
Test existence of 4-fold symmetry
33
j observations of the reflection 30 22 6
j H K L I
1 30 22 6 100
2 -30 -22 6 500
3 22 -30 6 300
ltIgt (100 500 300) / 3
300
SIj-ltIgt 100-300500-300300-300
200 200
0 400
SIj 100 500 300
900
Rsym 400/900 0.44 44
34
Statistics are analyzed as a function of
resolution (N shells).
Discrepancy between symmetry related reflections
(Rsym) increases with increasing resolution. Why?
Shell Rsym
100-5.0Å 0.04
5.0-3.0Å 0.06
3.0-2.0Å 0.08
2.0-1.7Å 0.15
35
SIGNAL TO NOISE RATIO (I/s)
Average I/s decreases with increasing
resolution High resolution shells with I/s lt2
should be discarded.
Shell I/sI
100-5.0 Å 20.0
5.0-3.0 Å 10.0
3.0-2.0 Å 7.0
2.0-1.7 Å 3.0
36
COMPLETENESS? What percentage of
reciprocal Lattice was measured for a
given Resolution limit? Better than 90 I hope.
Shell completeness
100-5.0Å 99.9
5.0-3.0Å 95.5
3.0-2.0Å 89.0
2.0-1.7Å 85.3
Overall 92.5
37
Assignment
38
Effect of mosaicity and wavelength spread Dl
39
Figure 5.
40
Figure 6.