From a series of diffraction images obtain the intensity I and standard deviation s I for each reflection hkl H K L I s 0 0 4 32954 1740 0 0 8 4821 287 ID: 586760
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Slide1
The goal of Data Processing
From a series of diffraction images, obtain the intensity (I) and standard deviation (s(I)) for each reflection, hkl.
H K L I s0 0 4 3295.4 174.00 0 8 482.1 28.70 0 12 9691.0 500.70 0 16 1743.9 67.40 0 20 5856.0 221.00 0 24 14066.5 436.20 0 28 9936.3 311.70 0 36 8409.8 273.40 0 40 790.5 32.80 0 44 103.4 18.4. . . . .. . . . .. . . . .37 7 0 28.5 16.237 7 1 110.1 10.937 7 2 337.4 13.337 7 3 98.5 10.637 7 4 25.9 10.7
Set of
360 images
Final intensities
IndexIntegrateMerge
3 x 10
9 bytes (3Gb)
8 x 105 bytes (800kB)Slide2
Indexing sounds like a trivial task
plane L=0
b*
a*
(6,2,0)Slide3
How many dimensions?
Proteinase K Crystal
Diffraction
Fourier transform
?
Detector
3
3
2
Record
a*
b*
c*Slide4
The Fourier transform of a 3D crystal
is a 3D reciprocal lattice
a
b
c
Unit cell lengths a, b, c
Atom coordinates x, y, z
Reciprocal cell lengths a*, b*, c*
Reflection coordinates h, k, l
a*
b*
c*Slide5
3D reciprocal lattice is projected on 2D detector
(projection is in direction of X-ray beam).
detectorIn an undistorted view of the reciprocal lattice, recorded reflections would reside on the surface of a sphere, not a plane.Slide6
A distortion-corrected representation of the reflectionsSlide7
R
estored depth of the diffraction pattern is evident from an orthogonal viewSlide8
Each circle corresponds to a different reciprocal lattice plane
Start indexing process using reflections within the same plane and lying near the origin.Slide9
Draw a
set of evenly spaced rowsSlide10
Draw the vector representing this repeat distance, a*, between
rowsSlide11
Draw a set of evenly spaced
columns
a*Slide12
Draw the vector representing this repeat distance,
b*,
between columns
a*Slide13
What is the angle between a* and b*?
a*
b*Slide14
Which of 14
Bravais Lattices has a=b and g=90°
CubicRhombohedralHexagonal/TrigonalTetragonalOrthorhombicMonoclinicTriclinicSlide15
Draw a set of evenly spaced
rows in orthogonal view
a*
b*Slide16
Draw the vector representing this repeat distance, a*, between rows
a*
b*Slide17
Draw a set of evenly spaced
columns in orthogonal view
a*
b*
a*Slide18
Draw the vector representing this repeat distance, c*, between columns
a*
b*
a*Slide19
Is the length of c* related to a* and b*?
What are the angles
a and b?
a*
b*
a*
c*Slide20
Which of 14
Bravais Lattices has a=b≠c and a=b=g=90°
CubicRhombohedralHexagonal/TrigonalTetragonalOrthorhombicMonoclinicTriclinicSlide21
What is the index of the lowest resolution reflection?
a*
b*Slide22
What is the index of the highest resolution reflection in the l=0 plane?
a*
b*
(-2,2,0)
(3,6,0)Slide23
FILM
X-ray beam
Determine unit cell length “a”
1,0,0 reflection
1,0,0
(0,0,0)
origin of reciprocal lattice,
Also known as
x-beam, y-beam
crystal
D
CF
=80
mm
Da*=
2.0 mm
a*
1/
l
a*
=
D
a
/
(l *
D
CF
)
a*
=
2.0mm
/
80mm*1.54
Å
a*
=
0.1623
Å
-1
a
=
61.6
Å
a*
/
D
a*
=
1/
l
/
D
CFSlide24
Review which experimental parameters were required to index a spot.
a*
b*
(-2,2,0)
Coordinates
of
the direct
beam,
(X,Y)
Coordinates (X,Y) for the spot position
Unit cell parameters
a,b,c
, a,b,g
The orientation of the unit cell axes with respect to the laboratory axes (
fyk
).
Crystal-to-detector distance
The wavelength of the incident radiationSlide25
What are some reasons why indexing might be inaccurate or unreliable?
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,
a,b,g
The orientation of the unit cell axes with respect to the laboratory axes (
fyk
).Slide26
Need a program that can index spots from multiple lattice planes without manually aligning crystalSlide27
Automatic indexing algorithm explained
Acta Cryst. (1999), D55, 1690-1695Slide28
Locate reflections positions
(peaks of high intensity)
1) Display first image in your data set with2) Press “Peak Search”. Red circles indicate position of prominent peaks (spots).3) Evaluate whether you need more or fewer peaks.4) Press “OK”5) Spot positions (x,y) are written to a file “peaks.file.”
Peak Search
177 peaks foundSlide29
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 1Etc…………………………………………..Slide30
Project vectors onto a line.
Measure the length of each projection.Slide31
Look for incremental differences in lengths.
1
2
3
4
Distribution of lengths is not incremental, it is continuousSlide32
Rotate, 7300 orientations tested.Slide33
Projected vectors for rotated image.Slide34
Sort the vector projections by length
. Count the number of observations of each length.
2 vectors of length 28.5 mm
5 vectors of length 26.5 mm
5 vectors of length 24.5 mm
10 vectors of length 22.5 mm
et cetera
Note: Projected vectors have a quantized values (distribution looks like steps).
The incremental difference
D
is proportional to the reciprocal cell length
1
2
3
4
D
D
D
D
D
D
D
D
D
D
D
D
D
D Slide35
Fourier analysis of length histogram reveals cell dimension.
Unit cell length = 62.5
Å
1-D Fourier Transform
A cosine wave with periodicity of 62.5
Å
is a major contributor to the 1-D FT.Slide36
Run autoindexing script
The
autoindexing script is simply named “a.”Type “denzo” to start the program.Then type @a to pass instructions to Denzo.Slide37Slide38
Select a space group with desired Bravais Lattice
(
e.g. new space group P4)Predicted pattern should match observed diffraction pattern.“go” to refineSlide39
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 parameters.
.Slide40
It is not necessary to index following images from scratch.
1
o
Film 1, exposed over 1 to 2 degrees
Film 2, exposed over 2 to 3 degreesSlide41
Integration
1)Draw boundaries of each reflection
2) Sum up the intensities recorded on each pixel within boundary.3) Repeat for each spot on each film.Slide42
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……………………………………………………….Slide43
One .x file for each film
Film 1, exposed over
1 to 2 degrees
Film 2, exposed over
2 to 3 degrees
Film 360, exposed over 360 to 361 degrees
h k l flag I(profit) I(prosum) c2 s(I) cos incid. X pix Y pix29 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 2061.629 22 -29 1 24.0 25.2 1.29 1.2 0.564 28.0 1489.129 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 pix29 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 2061.629 22 -29 1 24.0 25.2 1.29 1.2 0.564 28.0 1489.129 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 2061.629 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)
c
2
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
prok_001.img
prok_001.x
prok_002.img
prok_360.img
prok_002.x
prok_360.xSlide44
h k l flag I(profit) I(prosum)
c2 s(I) cos incid. X pix Y pix29 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 2061.629 22 -29 1 24.0 25.2 1.29 1.2 0.564 28.0 1489.129 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 2061.629 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
Accuracy will improve if we Merge multiple observations of the same reciprocal lattice pointBut, we must test if rotational symmetry exists between lattice points.
h k l flag I(profit) I(prosum) c2
s(I) cos incid. X pix Y pix29 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 2061.629 22 -29 1 24.0 25.2 1.29 1.2 0.564 28.0 1489.129 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 2061.629 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)
c
2 s(I) cos incid. X pix Y pix29 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)
c
2
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
->
360.xSlide45
Choose point group symmetry
(4 or 422)Slide46
Is it
Point group 422
Or Point 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-foldsymmetry
Test existence of
4-fold Symmetry
andPerpendicular2-fold symmetrySlide47
j observations of the reflection 30 22 6
<
I>= (550 + 500 + 543) / 3 = 531
j
HK
LI
130
22
6
5502
-30
-22
6
500
3
30
-22
-6
543
R
merge
=
S
|
I
j
-
<
I
>
|
S
I
j
Discrepancy
between symmetry related reflections
S
|
I
j
-
<
I
>
|
=
|550-
531
|+|500-
531
|+|543-
531
|
=
19
+
31
+
12 =
62
S
Ij = 550
+ 500 + 543 = 1593
Rsym = 62/1593 = 0.126 = 12.6%
?
H, K, L=
-H,-K, L=
H,-K,-L Slide48
Merge
Average (merge together) symmetry related reflections.
Plane L=0
b*
a*
?
-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
b*
a*
K,-H,-L
H,K,L
-H,K,-L
-K,-H,L
K,H,L
-H,-K,L
H,-K,-L
-K,H,-LSlide49
Discrepancy between symmetry related reflections (R
sym
) increases with increasing resolution. Why?Shell
R
sym100-5.0Å
0.04 5.0-3.0Å
0.063.0-2.0Å
0.08
2.0-1.7Å
0.15Statistics are analyzed as a function of resolution (N shells).Slide50
Average I/
s
decreases with increasing resolutionHigh resolution shells with I/s <2 should be discarded.
Shell
I/sI100-5.0
Å20.05.0-3.0
Å10.03.0-2.0
Å
7.02.0-1.7 Å
3.0
SIGNAL TO NOISE RATIO (I/
s
)Slide51
COMPLETENESS?
What percentage of reciprocal
Lattice was measured for a givenResolution limit?Better than 90% I hope.Shell
completeness
100-5.0Å99.9%
5.0-3.0Å95.5%
3.0-2.0Å89.0%
2.0-1.7
Å85.3%
Overall 92.5%Slide52
AssignmentSlide53
FILM
X-ray beam
Layer line screen blocks diffracted rays from upper layers ( that is, l
≠
0)
1,0,0 reflection
1,0,0
1/
l
(0,0,0)
origin of reciprocal lattice,
Also known as
x-beam, y-beam
crystal
Da*
a*
1,0,1Slide54
Knowing the orientation of the
reciprocal lattice allows prediction of the position of each reflection on the detector
a
b
a
b
. . . . .
. . . . .
. . . . .
. . . . .
. .
.
. .
. . . . .
. . . . .
. . . . .
. . . . .
b*
a*
1/
l
(0,0)
(0,-1)
(0,-2)
(0,-3)
(-1,3)
X-ray
beam
crystal
detectorSlide55
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 just 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?Slide56
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.
Indices h,k,l are coordinates of the reflections, analogous to how atom positions are described by coordinates x,y,z.Slide57
X-ray scattering from a 2D crystal
a
b
Crystal
Xrays
DetectorSlide58
Diffracted intensities arise as if reflecting from families of planes --William Bragg.
(0,1) planes
a
b
Crystal
(1,1) planes
(2,1) planes
(3,1) planes
(0,2) planes
(1,2) planes
(2,2) planes
(3,2) planes
Xrays
DetectorSlide59
(0,1) planes
Crystal
(1,1) planes
(2,1) planes
(3,1) planes
(0,2) planes
(1,2) planes
(2,2) planes
(3,2) planes
a
b
Xrays
Detector
What are the chances of observing reflections from all these planes in a single orientation?
Excellent
ZeroSlide60
Crystal
(1,1) planes
a
b
Xrays
Detector
The planes must be oriented w.r.t the X-ray beam such that the difference in path length of each scattered ray is n
l
.
Difference in path lengths is not n
l
.
Scattered waves are out of phase.
Total destructive interference.
(1,1) reflection not observed in this crystal orientation.
2dsin
qSlide61
Reciprocal Lattice
a*
b*
a
b
Crystal
Xrays
Detector
1/
l
Sphere of reflection shows the relationship between Bragg plane (HKL) and location of reflection (HKL) on detectorSlide62
RECIPROCAL LATTICE
a*
b*
(0,1) planes
length=1/d
0,1
(0,1)
a
b
Crystal
Detector
Xrays
(0,1) Bragg planes produce (0,1) reciprocal lattice pointSlide63
RECIPROCAL LATTICE
a*
b*
(1,1) planes
length=1/d
1,1
(1,1)
a
b
(0,1)
Crystal
Detector
Xrays
(1,1) Bragg planes produce (1,1) reciprocal lattice pointSlide64
RECIPROCAL LATTICE
a*
b*
(2,1) planes
length=1/d
2,1
(1,1)
(0,1)
a
b
(2,1)
Crystal
Detector
Xrays
(2,1) Bragg planes produce (2,1) reciprocal lattice pointSlide65
RECIPROCAL LATTICE
a*
b*
(3,1) planes
length=1/d
3,1
(1,1)
(0,1)
a
b
(2,1)
(3,1)
Crystal
Detector
Xrays
(3,1) Bragg planes produce (3,1) reciprocal lattice pointSlide66
RECIPROCAL LATTICE
a*
b*
(0,2) planes
length=1/d
0,2
(1,1)
(0,1)
(2,1)
(3,1)
(0,2)
a
b
Crystal
Detector
Xrays
(0,2) Bragg planes produce (0,2) reciprocal lattice pointSlide67
RECIPROCAL LATTICE
a*
b*
(1,2) planes
length=1/d
1,2
(1,1)
(0,1)
(2,1)
(3,1)
(0,2)
(1,2)
a
b
Crystal
Detector
Xrays
(1,2) Bragg planes produce (1,2) reciprocal lattice pointSlide68
RECIPROCAL LATTICE
a*
b*
(2,2) planes
length=1/d
2,2
(1,1)
(0,1)
(2,1)
(3,1)
(2,2)
(0,2)
(1,2)
a
b
Crystal
Detector
Xrays
(2,2) Bragg planes produce (2,2) reciprocal lattice pointSlide69
RECIPROCAL LATTICE
a*
b*
(3,2) planes
length=1/d
3,2
(1,1)
(0,1)
(2,1)
(3,1)
(2,2)
(0,2)
(1,2)
(3,2)
a
b
Crystal
Detector
Xrays
(3,2) Bragg planes produce (3,2) reciprocal lattice pointSlide70
RECIPROCAL LATTICE
a*
b*
(1,1)
(0,1)
(2,1)
(3,1)
(2,2)
(0,2)
(1,2)
(3,2)
a
b
Crystal
Detector
Xrays
And so on... to fill out reciprocal lattice.
(1,0)
(2,0)
(3,0)
(1,-2)
(2,-2)
(3,-2)
(2,-1)
(1,-1)
(3,-1)
(0,-1)
(0,-2)
(-1,1)
(-1,2)
(-1,0)
(-1,-2)
(-1,-1)
(-2,1)
(-2,2)
(-2,0)
(-2,-2)
(-2,-1)
(-3,1)
(-3,2)
(-3,0)
(-3,-2)
(-3,-1)
In this crystal orientation
Which reflections will appear on the detector?Slide71
RECIPROCAL LATTICE
a*
b*
(1,1) planes
(1,1)
a
b
Crystal
Detector
Xrays
What operation must we perform to the crystal in order to observe the (1,1) reflection?
33
°Slide72
RECIPROCAL LATTICE
(1,1) planes
a*
b*
(1,1)
Crystal
Detector
Xrays
Rotate crystal to observe (1,1) reflection.
a
b
(1,1) reflection is located here!Slide73
a
b
a
b
(1,1) planes
(1,1)
Crystal
Detector
Xrays
A crystal rotation that brings the reciprocal lattice point (HKL) in contact with sphere of reflection, satisfies Bragg’s Law for reflection (HKL).
q
1/
l
d*
d*/2 =
1/
l
•sin
q
2d
•
sin
q
=
l
}
(1,1) reflection is located here!
RECIPROCAL LATTICESlide74
a
b
a
b
(1,1) planes
(1,1)
Crystal
Detector
Xrays
Indexing - We observe the location of a reflection on the detector. Which set of Bragg planes produced it?
There is a reflection located here!
What is it’s index?
RECIPROCAL LATTICE
a*
b*
How many increments of a* and b* is this point from the origin?
d*=ha*+ kb*+ lc*Slide75
1/
l
RECIPROCAL PLANE
Crystal
Detector
Xrays
We collected a series of images, each covering 1
° rotation
. Here we rotate 15° per image.
5
°
D
CF
X,YSlide76
RECIPROCAL PLANE
Crystal
Detector
Xrays
We collected a series of exposure while crystal rotates.
13
°Slide77
RECIPROCAL PLANE
Crystal
Detector
Xrays
We collected a series of exposure while crystal rotates.
15
°Slide78
RECIPROCAL LATTICE
Crystal
Xrays
Map onto reciprocal lattice
Image rotation
0-15
°Slide79
RECIPROCAL LATTICE
Crystal
Xrays
Map onto reciprocal lattice
Image rotation
0-15
°
15-30°Slide80
RECIPROCAL LATTICE
Crystal
Xrays
Map onto reciprocal lattice
Image rotation
0-15
°
15-30°
30-45°Slide81
RECIPROCAL LATTICE
Crystal
Xrays
Map onto reciprocal lattice
Image rotation
0-15
°
15-30°
30-45°
45-60°Slide82
RECIPROCAL LATTICE
Crystal
Xrays
Map onto reciprocal lattice
Image rotation
0-15
°
15-30°
30-45°
45-60°
60-75°Slide83
RECIPROCAL LATTICE
Crystal
Xrays
Map onto reciprocal lattice
Image rotation
0-15
°
15-30°
30-45°
45-60°
60-75°
75-90°Slide84
RECIPROCAL LATTICE
Crystal
Xrays
Map onto reciprocal lattice
Image rotation
0-15
°
15-30°
30-45°
45-60°
60-75°
75-90°
90-105°Slide85
a*
b*
RECIPROCAL LATTICE
Crystal
Xrays
Map onto reciprocal lattice
Image rotation
0-15
°
15-30°
30-45°
45-60°
60-75°
75-90°
a
bSlide86
a*
b*
RECIPROCAL LATTICE
Crystal
Xrays
Map onto reciprocal lattice
Image rotation
0-15
°
15-30°
30-45°
45-60°
60-75°
75-90°
(1,-2)
(2,-2)
(3,-2)
(0,-2)
(-1,-2)
(-2,-2)
(-3,-2)
(2,-1)
(1,-1)
(3,-1)
(0,-1)
(-1,-1)
(-2,-1)
(-3,-1)
(1,0)
(2,0)
(3,0)
(-1,0)
(-2,0)
(-3,0)
(1,1)
(0,1)
(2,1)
(3,1)
(-1,1)
(-2,1)
(-3,1)
(2,2)
(0,2)
(1,2)
(3,2)
(-1,2)
(-2,2)
(-3,2)
a
bSlide87
a*
b*
RECIPROCAL LATTICE
Crystal
Xrays
Map onto reciprocal lattice
a
bSlide88
3D reciprocal lattice is projected on 2D detector.
detectorSlide89
3 steps of data reduction
IndexingAssign H,K,L values to each reflectionWe learn unit cell parametersa,b,c, a,
b,gIdentify which of 14 Bravais LatticesIntegrationSum up the number of X-ray photons that intercepted the detector for each reflection, H,K,L.We learn the intensity values for each recorded reflection, H,K,LScale and MergeAverage together reflections related by symmetryCalculate scale factor for each image to minimize discrepancies between measurements of symmetry related reflections.We obtain the final symmetry averaged data set.We learn the space group symmetry
H K L I
s
37 7 1 28.5 9.2-37 -7 1 30.1 10.9 7 37 1 37.4 13.3 37 7 -1 28.7 10.6-37 -7 -1 25.9 9.7 37 7 2 337.4 13.3 37 7 3 98.5 10.6 37 7 4 25.9 10.7
H K L I s 37 7 1 30.1 10.7 37 7 2 337.4 13.3 37 7 3 98.5 10.6 37 7 4 25.9 10.7Slide90
Outline 1
Overview –We will process diffraction data in 3 steps. Briefly:1) indexing- assign coordinates (h,k,l) to each reflection in the data set2) integration- extract intensity values from the diffraction images for each reflection, h,k,l3) scaling and merging -average together the multiple intensity measurements related by symmetry.Indexing is the most challenging step of the three.The concept of indexing sounds trivial –locate spots on the image and assign them coordinates on the reciprocal lattice. But, complexity arises from the fact that the diffraction pattern has 3 dimensions and the detector used for data collection has only 2 dimensions. The 3D reciprocal lattice is projected on a 2D surface in our diffraction experiment.
Recall that the Fourier transform of a 3D crystal is a 3D reciprocal lattice. What are the names of the three coordinates used to index atoms in crystal space? (x,y,z) What are the names of the three coordinates used to index reflections in diffraction space? (h,k,l) What are the repeating unit dimensions along x,y,z? along h,k,l?Indexing would be trivial if our diffraction images captured an undistorted view of the 3D reciprocal lattice like these. Ideally, reflections would be divided neatly into sections, aligned with reciprocal cell axes and without breaks in the pattern, as shown. Does our data come like this? No. It looks like these images: misoriented, distorted from projection, and discontinuous. It is possible to collect undistorted images of the 3D lattice using a precession camera, but unfortunately, it is inefficient and time consuming to collect data with a precession camera.Our task of indexing is to assign h,k,l values to spots in our diffraction images. In so doing, we can map them onto an undistorted 3D reciprocal lattice, computationally—as shown here.Indexing conceptsWe are going to use a program, Denzo, to help us index the thousands of reflections that we recorded. However, I would also like to show you how to index by inspection so you gain an intuitive feeling for indexing.1) Take one of the many diffraction images that we recorded and eliminate from it the distortion due to projection of the 3D pattern onto a 2D detector. We can do this by taking into consideration the curvature of Ewald’s sphere of reflection. Recall that a reflection is recordable only when the corresponding planes in the crystal are oriented in the beam in such a way that satisfies Bragg’s law. That is, the photons reflected from the planes differ in path length precisely by integer multiples of the wavelength. This condition is satisfied when the reciprocal lattice point crosses the sphere of reflection. So, all the reflections recorded on the film originate from a curved surface. Eliminating the distortion from projecting this curved surface onto a 2D detector involves re-introducing the 3rd dimension and restoring the curvature to the diffraction pattern. Like this. We can now see undistorted, but sparsely populated, 3D reciprocal lattice. Show orthogonal views.2) Identify the sets of evenly spaced rows and columns in the reciprocal lattice. Draw a set of evenly spaced lines through columns of spots. Draw a set of evenly spaced lines through rows of spots. 3) Find the reciprocal cell lattice parameters. Draw the vector representing this repeat distance, a*, between columns. Draw the vector representing the repeat distance, b*, between rows.Slide91
Outline 2
4) Note the angle between a* and b*? This is gamma. 5) Note the relationship between the lengths of a* and b*, if any.6) Narrow down the possibilities of choice of Bravais lattice. Note, current info suggests either primitive tetragonal or cubic.7) Identify the third unit cell length, c*. Identify the sets of evenly spaced columns in the reciprocal lattice.
Draw the vector representing this repeat distance, c*, between columns. What angle does c* make with a* and b*? These are beta and gamma. 8) Is the crystal primitive tetragonal or cubic?9) What is the index for this reflection?Review. In our intuitive indexing process, the following parameters were either measured or derived from measurements. Which of the following parameters were measured directly from the diffraction experiment? Which parameters were derived from the measurements. Why might indexing fail?Indexing in practiceAutoindexing using Denzo. It uses a computer algorithm to perform the same analysis as we just did here.Identify spots. Uses an algorithm, not very good. You use your eyes and pattern recognition.Identify rows and columns of spots. It performs a systematic search of all orientations of the image for periodicities among spot locations. It does this by projecting each spot on a line. In certain orientations the lengths of these projections will differ by integer multiples of a constant, corresponding to the reciprocal cell length.A one-dimensional Fourier transform of the vector lengths identifies this increment, a*.Our task of indexing is to assign h,k,l values to spots in our diffraction images so we can map them onto an undistorted 3D reciprocal lattice, computationally—as shown here.Indexing conceptsWe are going to use a program, Denzo, to help us index the thousands of reflections that we recorded. However, I would also like to show you indexing by inspection so you become familiar with the concepts of indexing.Distortions due to projection of the 3D pattern onto a2) integration- for each reflection we sum the intensity values for all pixels within the reflection boundary3) scaling and merging –for each unique reflection (h,k,l) average all intensity measurements of (h,k,l) and its symmetry mates. Determine scale factors to obtain the best agreement between I(h,k,l) values measured from different images.Slide92Slide93Slide94Slide95