diffraction experiments, datacloud.crm2.univ-lorraine.fr/pdf/bogota2018/suescun_diffraction.pdf ·...
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Diffraction Experiments, Data processing and reduction
Leopoldo SuescunLaboratorio de Cristalografía Química del Estado Sólido y Materiales,
Facultad de Química, Universidad de la República, Montevideo, Uruguay.
November 29th, 2018
![Page 3: Diffraction Experiments, Datacloud.crm2.univ-lorraine.fr/pdf/Bogota2018/Suescun_Diffraction.pdf · Representación esquemática de un cristal y su diagrama de difracción. Powder](https://reader034.vdocuments.site/reader034/viewer/2022042022/5e79dbeafa327909e23c67b5/html5/thumbnails/3.jpg)
Bragg’s Law
send2n hkl
2
d
2
2
2
2
2
2
1
c
l
b
k
a
hdhkl
For Orthogonal unit cells only
![Page 4: Diffraction Experiments, Datacloud.crm2.univ-lorraine.fr/pdf/Bogota2018/Suescun_Diffraction.pdf · Representación esquemática de un cristal y su diagrama de difracción. Powder](https://reader034.vdocuments.site/reader034/viewer/2022042022/5e79dbeafa327909e23c67b5/html5/thumbnails/4.jpg)
Bragg’s Law
send2n hkl
2
d
A
B
C D
2’
11’
d
2
2
2
2
2
2
1
c
l
b
k
a
hdhkl
For Orthogonal unit cells only
Phase shift:CB+BD
d
A
B
CD
CB=d*sin()
CB+BD=2d*sin()
For constructive interferenceCB+BD = n
![Page 5: Diffraction Experiments, Datacloud.crm2.univ-lorraine.fr/pdf/Bogota2018/Suescun_Diffraction.pdf · Representación esquemática de un cristal y su diagrama de difracción. Powder](https://reader034.vdocuments.site/reader034/viewer/2022042022/5e79dbeafa327909e23c67b5/html5/thumbnails/5.jpg)
Bragg’s Law & Laue Equations
• Another equation for Braggs law:
A
11’
2 2
0S
S
0ˆˆ SS
s
![Page 6: Diffraction Experiments, Datacloud.crm2.univ-lorraine.fr/pdf/Bogota2018/Suescun_Diffraction.pdf · Representación esquemática de un cristal y su diagrama de difracción. Powder](https://reader034.vdocuments.site/reader034/viewer/2022042022/5e79dbeafa327909e23c67b5/html5/thumbnails/6.jpg)
Bragg’s Law & Laue Equations
• Another equation for Braggs law:
2
0S
S
0ˆˆ SS
s
)(2)90cos(
ˆ
2
0 senss
Ss
s
![Page 7: Diffraction Experiments, Datacloud.crm2.univ-lorraine.fr/pdf/Bogota2018/Suescun_Diffraction.pdf · Representación esquemática de un cristal y su diagrama de difracción. Powder](https://reader034.vdocuments.site/reader034/viewer/2022042022/5e79dbeafa327909e23c67b5/html5/thumbnails/7.jpg)
2
)(2
)(2
dsen
sens
ds
sens
1
)(1
2
𝑠 𝑑
s
Bragg’s Law & Laue Equations
• Another equation for Braggs law:
![Page 8: Diffraction Experiments, Datacloud.crm2.univ-lorraine.fr/pdf/Bogota2018/Suescun_Diffraction.pdf · Representación esquemática de un cristal y su diagrama de difracción. Powder](https://reader034.vdocuments.site/reader034/viewer/2022042022/5e79dbeafa327909e23c67b5/html5/thumbnails/8.jpg)
Reciprocal lattice
a*
c*
b*
a
b
c
hkld
''' lkhd
Antitransform
FourierTransform
Crystal Structure Diffraction pattern
hkl
hklhkld
sr1*
'''
'''
*
'''
1
lkh
lkhlkhd
sr
0
V
bac
V
acb
V
cba
***
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Reciprocal lattice
Nomenclature:
a*
c*
b*
a
b
c
V
bac
V
acb
V
cba
***
𝑟ℎ𝑘𝑙∗ = ℎ 𝑎
∗+ 𝑘𝑏
∗+ 𝑙 𝑐
∗= 𝑠ℎ𝑘𝑙
𝑠ℎ𝑘𝑙 = ℎ 𝑎∗+ 𝑘𝑏
∗+ 𝑙 𝑐
∗
𝑠ℎ𝑘𝑙 = 𝑠ℎ𝑘𝑙
𝑠ℎ𝑘𝑙 = 𝑠ℎ𝑘𝑙 =1
𝑑ℎ𝑘𝑙
![Page 10: Diffraction Experiments, Datacloud.crm2.univ-lorraine.fr/pdf/Bogota2018/Suescun_Diffraction.pdf · Representación esquemática de un cristal y su diagrama de difracción. Powder](https://reader034.vdocuments.site/reader034/viewer/2022042022/5e79dbeafa327909e23c67b5/html5/thumbnails/10.jpg)
Reciprocal lattice and Ewald sphere
hkls
1
Rayos Xincidentes
hkl
hkld
s1
0Origen R.R.
Rayos Xdifractados
2
cristal
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hklshkl
hkld
s1
0
2
hkl
hkld
s1
1
1
hkl
hkl
d
d
2
1
2)sin(
1*
)sin(2 hkld
Reciprocal lattice and Ewald sphere
hklds
1)sin(
2
hkl
hkld
s1
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Reciprocal lattice and Ewald sphere
The Ewald sphere is the geometric site where the reciprocal lattice vectors fulfill the Laue equations or the family of planes fulfill Bragg’s Law therefore X-rays exit the crystal throught the reflection sphere at the point where the vector touches the sphere.
hkls
1
Rayos Xincidentes
hkl
hkld
s1
0Origen R.R.
Rayos Xdifractados
2
cristal
1
1
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1
Rayos Xincidentes 0
RetículoRecíproco
cristal
Rayos Xdifractados
Reciprocal lattice and Ewald sphere
***1
)sin(2
clbkahd
shkl
hkl
Shkl Shkl
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Esfera de ewald
hkls
1
Rayos X
incidentes
hkl
hkld
s1
Rayos X
difractados
2
Cristal
0Origen R.R.
1
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Esfera de ewald
Diffraction condition: when a RL point intersects the Reflection sphere X-ray will emerge from the cristal through that point.
1
Rayos X
incidentes 0
Retículo
Recíproco
cristal
Rayos X
difractados
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Ewald Sphere or Reflection Sphere
Diffraction condition: when a RL point intersects theReflection sphere X-ray will emerge from the cristal throughthat point
1
Rayos X
incidentes 0
Retículo
Recíproco
cristal
Rayos X
difractados
Rotate the crystal and the reciprocal lattice
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Limiting Sphere
Could all points of the Reciprocal Lattice be made toto intersect the reflection sphere for a given ?.
1
Rayos X
incidentes 0
Retículo
Recíproco
cristal
![Page 18: Diffraction Experiments, Datacloud.crm2.univ-lorraine.fr/pdf/Bogota2018/Suescun_Diffraction.pdf · Representación esquemática de un cristal y su diagrama de difracción. Powder](https://reader034.vdocuments.site/reader034/viewer/2022042022/5e79dbeafa327909e23c67b5/html5/thumbnails/18.jpg)
Limiting sphere
1
Rayos X
incidentes 0cristal
Retículo
Recíproco
![Page 19: Diffraction Experiments, Datacloud.crm2.univ-lorraine.fr/pdf/Bogota2018/Suescun_Diffraction.pdf · Representación esquemática de un cristal y su diagrama de difracción. Powder](https://reader034.vdocuments.site/reader034/viewer/2022042022/5e79dbeafa327909e23c67b5/html5/thumbnails/19.jpg)
Limiting sphere
1
Rayos X
incidentes 0cristal
Retículo
Recíproco
![Page 20: Diffraction Experiments, Datacloud.crm2.univ-lorraine.fr/pdf/Bogota2018/Suescun_Diffraction.pdf · Representación esquemática de un cristal y su diagrama de difracción. Powder](https://reader034.vdocuments.site/reader034/viewer/2022042022/5e79dbeafa327909e23c67b5/html5/thumbnails/20.jpg)
Limiting Sphere
1
Rayos X
incidentes 0cristal
Retículo
Recíproco
2
Those reciprocal lattice points that lay within 2/ of the origin (that have |s|< 2/ or d>/2)can be accessed in a diffraction experiment for a given .
![Page 21: Diffraction Experiments, Datacloud.crm2.univ-lorraine.fr/pdf/Bogota2018/Suescun_Diffraction.pdf · Representación esquemática de un cristal y su diagrama de difracción. Powder](https://reader034.vdocuments.site/reader034/viewer/2022042022/5e79dbeafa327909e23c67b5/html5/thumbnails/21.jpg)
• Using Bragg’s Law we can also calculate the maximum resolution (minimum d spacing) that we can measure in a diffraction experiment.
𝜆 = 2𝑑𝑠𝑖𝑛 𝜃 ⇒ 𝑑𝑚𝑖𝑛 =𝜆
2𝑠𝑖𝑛 𝜃𝑚𝑎𝑥=
𝜆
2𝑠𝑖𝑛 90=
𝜆
2
the minimum interplanar spacing accessible in the experiment depends on the wavelength. At 2=180 the minimum resolution is obtained, we cannot measure diffraction data beyond that angle.
Limiting Sphere
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0
1cristal
2
Limiting Sphere
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0
maxmin
2sd
Limiting Sphere
maxmin '''
2sd
’ > ,
d’min>dmin, s’max<smax
The shorter the better resolution for the same 2 angle.
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0
1cristal
2
Limiting SphereIf we assign each reciprocal lattice node the volume of the reciprocal unit cell, we can determine how many nodes are within the Limiting sphere, or the total number of reflections Nmax
accessible by the experiment.
𝑁𝑚𝑎𝑥 = 𝑉𝐿𝑆𝑉∗ =
43𝜋
2𝜆
3
𝑉∗=32𝜋𝑉
3𝜆3
![Page 25: Diffraction Experiments, Datacloud.crm2.univ-lorraine.fr/pdf/Bogota2018/Suescun_Diffraction.pdf · Representación esquemática de un cristal y su diagrama de difracción. Powder](https://reader034.vdocuments.site/reader034/viewer/2022042022/5e79dbeafa327909e23c67b5/html5/thumbnails/25.jpg)
Limiting Sphere
BaTiO3 at room temperature is tetragonalwith space group P4mm, a=3.999 Å, c=4.017 Å and V=64.269 Å3
Calculate the number of accessible reflections and the minimum dhkl available (maximum resolution) if:- (MoKa)=0.71073 Å- (CuKa)=1.5418 Å
Idem if you consider the structure of hen-egg lysozyme with space group P43212 with a=60.01 Å, c=261.8 Å and V=942938 Å3
𝑁𝑚𝑎𝑥 = 𝑉𝐿𝑆𝑉∗ =
43𝜋
2𝜆
3
𝑉∗=32𝜋𝑉
3𝜆3
![Page 26: Diffraction Experiments, Datacloud.crm2.univ-lorraine.fr/pdf/Bogota2018/Suescun_Diffraction.pdf · Representación esquemática de un cristal y su diagrama de difracción. Powder](https://reader034.vdocuments.site/reader034/viewer/2022042022/5e79dbeafa327909e23c67b5/html5/thumbnails/26.jpg)
Experimental resolution
When we perform an experiment, reaching the maximum available resolution requires to move the detector up to 2=180.
Normally we would limit the resolution of the experiment to a diffractometer-limited angle (140 to 160 depending on the maker and limitations of the instrument).
If we limit the scan to 2max, smax=/[2sin(max)] and dmin=2sin(max)/
the number of measurable reflections reduces to:
𝑁𝑚𝑎𝑥 = 𝑉𝐿𝑆𝑉∗ =
43𝜋
1𝑑𝑚𝑖𝑛
3
𝑉∗=
4𝜋𝑉
3𝑑𝑚𝑖𝑛3
![Page 27: Diffraction Experiments, Datacloud.crm2.univ-lorraine.fr/pdf/Bogota2018/Suescun_Diffraction.pdf · Representación esquemática de un cristal y su diagrama de difracción. Powder](https://reader034.vdocuments.site/reader034/viewer/2022042022/5e79dbeafa327909e23c67b5/html5/thumbnails/27.jpg)
1crystal
The diffraction experiment
beam-stop
0
Incident
X-rays,
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1crystal
The diffraction experiment
beam-stop
0
Incident
X-rays,
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Single crystal data collection
• Modern CCD/CMOS area detectors allow for simultaneous collection of hundreds/thousands of diffraction intensities (reflections).
• Data is collected in frames produced by rotation of the crystal by a fraction of degree.
• Programs automatically integrate the images into individual indexed intensities.
![Page 30: Diffraction Experiments, Datacloud.crm2.univ-lorraine.fr/pdf/Bogota2018/Suescun_Diffraction.pdf · Representación esquemática de un cristal y su diagrama de difracción. Powder](https://reader034.vdocuments.site/reader034/viewer/2022042022/5e79dbeafa327909e23c67b5/html5/thumbnails/30.jpg)
X rays ()
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Single crystal intensities
• For each indexed reflection (dots inside the circles) an intensity value will be determined by a numerical integration procedure to provide the output in a h,k,l, Ihkl, sIhkllist that is used for structure determination and refinement.
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Single crystal intensities
• what should I do now?
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Reflections, all, unique, observed
• Part of the reflection list for the compound CH4N2S.
• During data collection, some reflections are measured more than once (red).
• Some are equivalent by symmetry (blue).
• Some are not observed (I<2sI) (grey).
• Etc…
![Page 34: Diffraction Experiments, Datacloud.crm2.univ-lorraine.fr/pdf/Bogota2018/Suescun_Diffraction.pdf · Representación esquemática de un cristal y su diagrama de difracción. Powder](https://reader034.vdocuments.site/reader034/viewer/2022042022/5e79dbeafa327909e23c67b5/html5/thumbnails/34.jpg)
Reflections, all, unique, observed
For the compound CH4N2S:a=7.6527 Å, b=8.5439 Å c=5.4858 Å, a=b=g=90°:=0.71073 Å, 2max=51.7°, smax=1.227 Å, dmin=0.815 Å
– 4119 intensities were integrated.
– 2227 were different– 2391 were observed (I>2sI)– 340 were unique
(unrelated by point group symmetry)
– 265 of the unique were observed.
![Page 35: Diffraction Experiments, Datacloud.crm2.univ-lorraine.fr/pdf/Bogota2018/Suescun_Diffraction.pdf · Representación esquemática de un cristal y su diagrama de difracción. Powder](https://reader034.vdocuments.site/reader034/viewer/2022042022/5e79dbeafa327909e23c67b5/html5/thumbnails/35.jpg)
Reflections, all, unique, observed
• Identification of useful intensities is required before structure determination since we need to know the degree of over-determination of the problem to know if the structure determination procedure will succeed.
• For the compound CH4N2S a=7.6527 Å, b=8.5439 Å c=5.4858 Å, a=b=g=90°: – 264 of the unique were observed.
– With this number of unique observed intensities only a small number of atomic parameters can be determined accurately. It is a requisite of the IUCr Journals to have 10 reflections per structural parameters so only around 25 structural parameters can be determined.
![Page 36: Diffraction Experiments, Datacloud.crm2.univ-lorraine.fr/pdf/Bogota2018/Suescun_Diffraction.pdf · Representación esquemática de un cristal y su diagrama de difracción. Powder](https://reader034.vdocuments.site/reader034/viewer/2022042022/5e79dbeafa327909e23c67b5/html5/thumbnails/36.jpg)
Reflections, all, unique, observed
• Even though we don’t need 3855 of the 4119 collected intensities for the structure determination and refinement, the redundancy allows for symmetry determination and data consistency checks, for the determination of other effects such as absorption or radiation damage of the crystal and to reduce the total uncertainty of the refined parameters due to a reduction in the uncertainty of the observed intensities.
• Rint (internal consistency of symmetry-related reflections) and Rsigma (average I/sI for all reflections) are good measures of the consistency between equivalent reflections (symmetry-related reflections that should have equal intensities) and the overall uncertainty of the recorded intensities respectively.
![Page 37: Diffraction Experiments, Datacloud.crm2.univ-lorraine.fr/pdf/Bogota2018/Suescun_Diffraction.pdf · Representación esquemática de un cristal y su diagrama de difracción. Powder](https://reader034.vdocuments.site/reader034/viewer/2022042022/5e79dbeafa327909e23c67b5/html5/thumbnails/37.jpg)
Single crystal intensities
• Since the reciprocal space is the Fourier Transform of the crystal and the crystal is (assumed) periodic, all the information in the reciprocal space is condensed in the reciprocal lattice points .
• Therefore the intensities of the RL points contain all the structural information and should allow to determine the structure.
• We will start by determining the type of space group symmetry of the crystal.
• Intensities in the reciprocal lattice are related to the structure factors (Fhkl) through a series of corrections that should be accounted for correctly, so one can determine the structure from the intensities after proper data Processing and Reduction.
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Powder Intensities
• If one set of reciprocal lattice points can be associated to each crystal in the sample, consider a polycrystalline (powder) sample:– A very large number (ideally > 107 crystallites)– Of (ideal) homogeneous size between 1 and 10 mm– Ideally randomly oriented in space and packed homogeneously
• The Reciprocal Space becomes populated with a large number of copies of the same RL but randomly rotated around the origin of the reciprocal space.
• RL vectors shkl with length |shkl|=1/dhkl are distributed randomly around the origin defining the radii of a sphere.
• Intersection of the spheres with the Ewald sphere produce circles through where diffracted rays come out of the sample forming diffraction cones.
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1
Rayos X
incidentes 0
Retículo
Recíproco
cristal
Rayos X
difractados
Powder Intensities
• For one crystal we found:
![Page 40: Diffraction Experiments, Datacloud.crm2.univ-lorraine.fr/pdf/Bogota2018/Suescun_Diffraction.pdf · Representación esquemática de un cristal y su diagrama de difracción. Powder](https://reader034.vdocuments.site/reader034/viewer/2022042022/5e79dbeafa327909e23c67b5/html5/thumbnails/40.jpg)
One nanocrystal illuminated by an
electron beam will give a diffraction
pattern corresponding to a single
crystal (the difference between ED
and XRD are a matter of other
school).
Powder Intensities
![Page 41: Diffraction Experiments, Datacloud.crm2.univ-lorraine.fr/pdf/Bogota2018/Suescun_Diffraction.pdf · Representación esquemática de un cristal y su diagrama de difracción. Powder](https://reader034.vdocuments.site/reader034/viewer/2022042022/5e79dbeafa327909e23c67b5/html5/thumbnails/41.jpg)
Representación esquemática de un cristal y su diagrama
de difracción.
Powder Intensities
One small crystals will define a Reciprocal lattice with an orientatioconsisten with the relative orientation of the crystal with the X-ray beam.
![Page 42: Diffraction Experiments, Datacloud.crm2.univ-lorraine.fr/pdf/Bogota2018/Suescun_Diffraction.pdf · Representación esquemática de un cristal y su diagrama de difracción. Powder](https://reader034.vdocuments.site/reader034/viewer/2022042022/5e79dbeafa327909e23c67b5/html5/thumbnails/42.jpg)
Four crystals in different orientations will diffract together but theirReciprocal lattices points are rotated by the same angle of the crystals.
Powder Intensities
![Page 43: Diffraction Experiments, Datacloud.crm2.univ-lorraine.fr/pdf/Bogota2018/Suescun_Diffraction.pdf · Representación esquemática de un cristal y su diagrama de difracción. Powder](https://reader034.vdocuments.site/reader034/viewer/2022042022/5e79dbeafa327909e23c67b5/html5/thumbnails/43.jpg)
40 crystallites diffracting together4 crystallites
Powder Intensities
![Page 44: Diffraction Experiments, Datacloud.crm2.univ-lorraine.fr/pdf/Bogota2018/Suescun_Diffraction.pdf · Representación esquemática de un cristal y su diagrama de difracción. Powder](https://reader034.vdocuments.site/reader034/viewer/2022042022/5e79dbeafa327909e23c67b5/html5/thumbnails/44.jpg)
40 crystallites 200 crystallites
Powder Intensities
![Page 45: Diffraction Experiments, Datacloud.crm2.univ-lorraine.fr/pdf/Bogota2018/Suescun_Diffraction.pdf · Representación esquemática de un cristal y su diagrama de difracción. Powder](https://reader034.vdocuments.site/reader034/viewer/2022042022/5e79dbeafa327909e23c67b5/html5/thumbnails/45.jpg)
• The Fourier Transform of a polycrystallines sample is the sum of the FTsfor each of the crystallites in the sample (ideally >107 crystals of equalsize in the 1-10 mm range with random orientations)
X rays ()
Difracción de rayos X de polvo
45
Fourier Transform
dhkl
shkl =1/d
![Page 46: Diffraction Experiments, Datacloud.crm2.univ-lorraine.fr/pdf/Bogota2018/Suescun_Diffraction.pdf · Representación esquemática de un cristal y su diagrama de difracción. Powder](https://reader034.vdocuments.site/reader034/viewer/2022042022/5e79dbeafa327909e23c67b5/html5/thumbnails/46.jpg)
• A policrystalline sample (powder) is composed of an “infinite” (1010) number of micrometric (1-10 mm) crystal with a perfectly randomorder and its Fourier Transform
Rayos X ()
46
Fourier Transform
dhkl
d*hkl =1/d=shkl
Powder diffraction data collection
![Page 47: Diffraction Experiments, Datacloud.crm2.univ-lorraine.fr/pdf/Bogota2018/Suescun_Diffraction.pdf · Representación esquemática de un cristal y su diagrama de difracción. Powder](https://reader034.vdocuments.site/reader034/viewer/2022042022/5e79dbeafa327909e23c67b5/html5/thumbnails/47.jpg)
• The powder pattern is obtained from the intersection of thereciprocal lattice of the powder (spheres) and the Ewaldsphere.
Rayos X ()
47
1/
Powder diffraction data collection
Intersections are circles and X-rays come from the sample exiting the Ewaldsphere in cones of 4 angle.
2
4
![Page 48: Diffraction Experiments, Datacloud.crm2.univ-lorraine.fr/pdf/Bogota2018/Suescun_Diffraction.pdf · Representación esquemática de un cristal y su diagrama de difracción. Powder](https://reader034.vdocuments.site/reader034/viewer/2022042022/5e79dbeafa327909e23c67b5/html5/thumbnails/48.jpg)
• Depending on the detector system geometry (and the sample shapeand orientation respect to the beam different kinds of diffractionpatterns are obtained.
Rayos X ()
48
1/
Powder diffraction data collection
Transmission Geometry (frequently found in Synchrotron labs)
Sample
1/
Reflection Geometry (frequently found in conventional labs)
![Page 49: Diffraction Experiments, Datacloud.crm2.univ-lorraine.fr/pdf/Bogota2018/Suescun_Diffraction.pdf · Representación esquemática de un cristal y su diagrama de difracción. Powder](https://reader034.vdocuments.site/reader034/viewer/2022042022/5e79dbeafa327909e23c67b5/html5/thumbnails/49.jpg)
49
• Depending on the detector system geometry (and the sample shapeand orientation respect to the beam different kinds of diffractionpatterns are obtained.
Powder diffraction data collection
Rayos X ()
Transmission Geometry (frequently found in Synchrotron labs)
![Page 50: Diffraction Experiments, Datacloud.crm2.univ-lorraine.fr/pdf/Bogota2018/Suescun_Diffraction.pdf · Representación esquemática de un cristal y su diagrama de difracción. Powder](https://reader034.vdocuments.site/reader034/viewer/2022042022/5e79dbeafa327909e23c67b5/html5/thumbnails/50.jpg)
OFFON
50
Rayos X ()
Powder diffraction data collection
Transmission Geometry (frequently found in Synchrotron labs)
Position-sensitive detector with/without energy discrimination
![Page 51: Diffraction Experiments, Datacloud.crm2.univ-lorraine.fr/pdf/Bogota2018/Suescun_Diffraction.pdf · Representación esquemática de un cristal y su diagrama de difracción. Powder](https://reader034.vdocuments.site/reader034/viewer/2022042022/5e79dbeafa327909e23c67b5/html5/thumbnails/51.jpg)
• Debye-Scherrer Camera with cylindrical film
51
Rayos X ()
Powder diffraction data collection
Transmission Geometry (Debye Scherrer Camera)
2
![Page 52: Diffraction Experiments, Datacloud.crm2.univ-lorraine.fr/pdf/Bogota2018/Suescun_Diffraction.pdf · Representación esquemática de un cristal y su diagrama de difracción. Powder](https://reader034.vdocuments.site/reader034/viewer/2022042022/5e79dbeafa327909e23c67b5/html5/thumbnails/52.jpg)
• En función de la geometría del detector de rayos X se obtienen los distintos tipos de diagramas de polvo. Con una pantalla plana (Image Plate)
52
Rayos X ()
Powder diffraction data collection
Transmission Geometry (frequently found in Synchrotron labs)
![Page 53: Diffraction Experiments, Datacloud.crm2.univ-lorraine.fr/pdf/Bogota2018/Suescun_Diffraction.pdf · Representación esquemática de un cristal y su diagrama de difracción. Powder](https://reader034.vdocuments.site/reader034/viewer/2022042022/5e79dbeafa327909e23c67b5/html5/thumbnails/53.jpg)
• We can integrate the detector counts into a diffraction pattern
53
Powder diffraction data collection
![Page 54: Diffraction Experiments, Datacloud.crm2.univ-lorraine.fr/pdf/Bogota2018/Suescun_Diffraction.pdf · Representación esquemática de un cristal y su diagrama de difracción. Powder](https://reader034.vdocuments.site/reader034/viewer/2022042022/5e79dbeafa327909e23c67b5/html5/thumbnails/54.jpg)
• We can numerically integrate the detector counts into a diffraction pattern
54
Powder diffraction data collection
![Page 55: Diffraction Experiments, Datacloud.crm2.univ-lorraine.fr/pdf/Bogota2018/Suescun_Diffraction.pdf · Representación esquemática de un cristal y su diagrama de difracción. Powder](https://reader034.vdocuments.site/reader034/viewer/2022042022/5e79dbeafa327909e23c67b5/html5/thumbnails/55.jpg)
• Depending on the geometry of the diffractometer different procedures for data collectionare performed.
55
Powder diffraction data collection
2Sample
Source
Modern
laboratory powder diffractometer
![Page 56: Diffraction Experiments, Datacloud.crm2.univ-lorraine.fr/pdf/Bogota2018/Suescun_Diffraction.pdf · Representación esquemática de un cristal y su diagrama de difracción. Powder](https://reader034.vdocuments.site/reader034/viewer/2022042022/5e79dbeafa327909e23c67b5/html5/thumbnails/56.jpg)
Powder intensities
![Page 57: Diffraction Experiments, Datacloud.crm2.univ-lorraine.fr/pdf/Bogota2018/Suescun_Diffraction.pdf · Representación esquemática de un cristal y su diagrama de difracción. Powder](https://reader034.vdocuments.site/reader034/viewer/2022042022/5e79dbeafa327909e23c67b5/html5/thumbnails/57.jpg)
Powder Intensities
• Due to either systematic properties or experimental conditions in a powder pattern a large number of reflections overlap. – Symmetry overlap of equivalent reflections (all showing
identical |shkl| or dhkl)
– Metric overlap due to equal d-spacing of reflections with different hkl but similar dhkl
– Convolution overlap for independent reflections with D2<instrumental resolution
Sometimes this makes it impossible to determine individual intensities
![Page 58: Diffraction Experiments, Datacloud.crm2.univ-lorraine.fr/pdf/Bogota2018/Suescun_Diffraction.pdf · Representación esquemática de un cristal y su diagrama de difracción. Powder](https://reader034.vdocuments.site/reader034/viewer/2022042022/5e79dbeafa327909e23c67b5/html5/thumbnails/58.jpg)
Powder Intensity Extraction
• In modern times two methods are systematically used for intensity extraction from Powder Data:– Pawley fit
– Le Bail fit
• In both cases the procedure provides a set intensities corresponding to hkl indices of the corresponding contributions to each peak.
• In the case of severe overlap (no true multiplet is visible) the intensity extraction gives equal intensity to all the components of the fitted peak.
![Page 59: Diffraction Experiments, Datacloud.crm2.univ-lorraine.fr/pdf/Bogota2018/Suescun_Diffraction.pdf · Representación esquemática de un cristal y su diagrama de difracción. Powder](https://reader034.vdocuments.site/reader034/viewer/2022042022/5e79dbeafa327909e23c67b5/html5/thumbnails/59.jpg)
Powder Intensity Extraction
• Both Pawley and LeBail method are based on the Rietveld equation:
j hkl
hklihkljbiobsi 22LPAIsSyy
Pawley:Ihkl is refined as a variable
intensity Ihkl to fit the corresponding peak/s.
Le Bail:Ihkl is calculated from refined
|Fobs|2 . |Fhkl|=1 is the starting
value and then replaced by |Fobs| in successive LS cycles to achieve the best fit and best Ihkl.
![Page 60: Diffraction Experiments, Datacloud.crm2.univ-lorraine.fr/pdf/Bogota2018/Suescun_Diffraction.pdf · Representación esquemática de un cristal y su diagrama de difracción. Powder](https://reader034.vdocuments.site/reader034/viewer/2022042022/5e79dbeafa327909e23c67b5/html5/thumbnails/60.jpg)
Powder Intensity Extraction
• After a Powder Intensity Extraction an h,k,l,Ihkl list is obtained but all available hkl values are those non-overlaping so they are all “unique” reflections.
• Accidental peak overlap reduces the number of observations further by a percentage up to 25% for large-cell/low-symmetry materials.
![Page 61: Diffraction Experiments, Datacloud.crm2.univ-lorraine.fr/pdf/Bogota2018/Suescun_Diffraction.pdf · Representación esquemática de un cristal y su diagrama de difracción. Powder](https://reader034.vdocuments.site/reader034/viewer/2022042022/5e79dbeafa327909e23c67b5/html5/thumbnails/61.jpg)
Powder vs Single Crystal intensities
• Conventional X-ray powder diffraction data for Thiourea(CH4N2S) down to a resolution of 0.84 Å (2=133° for CuKa) provides ~342 unique reflections of which ~249 do not overlap significantly (D2<FWHM/2) of which ~224 are observed (I>2sI criterion).
• Conventional X-ray single crystal diffraction data for Thiourea (CH4N2S) down to a resolution of 0.84 Å (2=50° for MoKa) provides 340 unique reflections (none overlaping) of which 265 are observed (I>2sI
criterion).
Peak overlap produces a reduction of at least 40% of the candidate observed intensities, these calculations are made for a small unit cell wilthrelatively high symmetry so things get worse for low-symmetry/large-cell problems.
![Page 62: Diffraction Experiments, Datacloud.crm2.univ-lorraine.fr/pdf/Bogota2018/Suescun_Diffraction.pdf · Representación esquemática de un cristal y su diagrama de difracción. Powder](https://reader034.vdocuments.site/reader034/viewer/2022042022/5e79dbeafa327909e23c67b5/html5/thumbnails/62.jpg)
Powder vs Single Crystal intensities
• In either case for Thiourea (CH4N2S) we ended up with files containing h, k, l, Ihkl and sIhkl lists that have to be correctly processed before submitting the problem for structural determination.
![Page 63: Diffraction Experiments, Datacloud.crm2.univ-lorraine.fr/pdf/Bogota2018/Suescun_Diffraction.pdf · Representación esquemática de un cristal y su diagrama de difracción. Powder](https://reader034.vdocuments.site/reader034/viewer/2022042022/5e79dbeafa327909e23c67b5/html5/thumbnails/63.jpg)
Intensity-Structure Factor relation
• The intensities can be related to the structure factors of the crystal by the equation:
Ihkl=kALp|Fhkl|2
k= scale factorA= absorption correctionL= Lorentz correctionp= Polarization correction
![Page 64: Diffraction Experiments, Datacloud.crm2.univ-lorraine.fr/pdf/Bogota2018/Suescun_Diffraction.pdf · Representación esquemática de un cristal y su diagrama de difracción. Powder](https://reader034.vdocuments.site/reader034/viewer/2022042022/5e79dbeafa327909e23c67b5/html5/thumbnails/64.jpg)
Intensity-Structure Factor relation
Ihkl=kALp|Fhkl|2
k= scale factorDepends on:
– Incident beam intensity– Data collection time per point/frame– Absorption of x-rays in the diffraction path– Detector efficiency– Diffractometer geometry (slits, filters,
monochromators, etc)– …
![Page 65: Diffraction Experiments, Datacloud.crm2.univ-lorraine.fr/pdf/Bogota2018/Suescun_Diffraction.pdf · Representación esquemática de un cristal y su diagrama de difracción. Powder](https://reader034.vdocuments.site/reader034/viewer/2022042022/5e79dbeafa327909e23c67b5/html5/thumbnails/65.jpg)
Intensity-Structure Factor relation
Ihkl=kALp|Fhkl|2
A= absorption correction m
eII 0
Cylindric powder sample:
m es el coeficiente de absorción lineal de la muestra y R el radio del cilindro (capilar).
X-rays ()
43
32
210 RkRkRkRk
eAmmmm
...
0456.26)(sen*109561.0e)(sen*01911.099978.25(k
697653.1k
)(sen*024514.02/1
1
0
2
Absorption profile of a plate-shaped single crystal of a Pb and I-containing sample sorted by run number.
Calculated from the Lambert-Beer Law
![Page 66: Diffraction Experiments, Datacloud.crm2.univ-lorraine.fr/pdf/Bogota2018/Suescun_Diffraction.pdf · Representación esquemática de un cristal y su diagrama de difracción. Powder](https://reader034.vdocuments.site/reader034/viewer/2022042022/5e79dbeafa327909e23c67b5/html5/thumbnails/66.jpg)
Intensity-Structure Factor relation
Ihkl=kALp|Fhkl|2
L= Lorentz correction
)2(sen
1L
L accounts for the area (volume) difference of the intersection of the Ewald Sphere (a thick line due to wavelength dispersion) and each diffraction spot or sphere with 2 angle.
![Page 67: Diffraction Experiments, Datacloud.crm2.univ-lorraine.fr/pdf/Bogota2018/Suescun_Diffraction.pdf · Representación esquemática de un cristal y su diagrama de difracción. Powder](https://reader034.vdocuments.site/reader034/viewer/2022042022/5e79dbeafa327909e23c67b5/html5/thumbnails/67.jpg)
Intensity-Structure Factor relation
Ihkl=kALp|Fhkl|2
p= Polarization correction
Sealed tube with filter: p=1/2+cos2(2)/2
Synchrotron radiation: p=1
Sealed tube with IBM: p=1/2+cos2(2m)cos2 (2)/2
2
))2(cos1(P
2
![Page 68: Diffraction Experiments, Datacloud.crm2.univ-lorraine.fr/pdf/Bogota2018/Suescun_Diffraction.pdf · Representación esquemática de un cristal y su diagrama de difracción. Powder](https://reader034.vdocuments.site/reader034/viewer/2022042022/5e79dbeafa327909e23c67b5/html5/thumbnails/68.jpg)
Intensity-Structure Factor relation
• For a spherical (or low absorbing m<1) crystal:
|Frel(hkl)|=(I(hkl)/Lp)1/2=(ILp(hkl))1/2
k|Frel(hkl)|2=kILp(hkl)=|F(hkl) |2
|Frel(hkl)| is the relative structure factor modulus
(determined assuming k includes all unknown corections to the intensity that are independent of hklor .
![Page 69: Diffraction Experiments, Datacloud.crm2.univ-lorraine.fr/pdf/Bogota2018/Suescun_Diffraction.pdf · Representación esquemática de un cristal y su diagrama de difracción. Powder](https://reader034.vdocuments.site/reader034/viewer/2022042022/5e79dbeafa327909e23c67b5/html5/thumbnails/69.jpg)
Wilson Plot
• With a little help of mathematics we can estimate k to get thestructure factors to its correct magnitude, and as a bonus wecan estimate the average B for the crystal:
j
)lzkyhx(i2
aj
B
ajaj
j
B)lzkyhx(i2
aj
jjj
2)sin(j
2)sin(jjjj
eg)hkl(F
efg
eef)hkl(F
Atomic scattering factor
Atomic displacement (thermal, Debye-Waller)parameter
69
![Page 70: Diffraction Experiments, Datacloud.crm2.univ-lorraine.fr/pdf/Bogota2018/Suescun_Diffraction.pdf · Representación esquemática de un cristal y su diagrama de difracción. Powder](https://reader034.vdocuments.site/reader034/viewer/2022042022/5e79dbeafa327909e23c67b5/html5/thumbnails/70.jpg)
Wilson Plot
• With a little help of mathematics we can estimate k to get thestructure factors to its correct magnitude, and as a bonus wecan estimate the average B for the crystal:
2)sin(aB
aa efg
B=2B=5
af
![Page 71: Diffraction Experiments, Datacloud.crm2.univ-lorraine.fr/pdf/Bogota2018/Suescun_Diffraction.pdf · Representación esquemática de un cristal y su diagrama de difracción. Powder](https://reader034.vdocuments.site/reader034/viewer/2022042022/5e79dbeafa327909e23c67b5/html5/thumbnails/71.jpg)
Wilson Plot
• With a little help of mathematics we can estimatek to get the structure factors toits correct magnitude, and as a bonus we can estimate theaverage B for the crystal:
klnsin
B2f
hklIln
2
ave
j
2
aj
Lp
ln(k)
![Page 72: Diffraction Experiments, Datacloud.crm2.univ-lorraine.fr/pdf/Bogota2018/Suescun_Diffraction.pdf · Representación esquemática de un cristal y su diagrama de difracción. Powder](https://reader034.vdocuments.site/reader034/viewer/2022042022/5e79dbeafa327909e23c67b5/html5/thumbnails/72.jpg)
Intensity scaling and statistics
• Normalized Structure Factor E(hkl):
< |E(hkl)|2>=< |F0(hkl)|2>/Sjf2
aj=Sjf2
aj/Sjf2
aj=1
j
2
aj
2
2
)(f
)hkl(F)hkl(E
72
j
)lzkyhx(i2
ajjjjeg)hkl(F
![Page 73: Diffraction Experiments, Datacloud.crm2.univ-lorraine.fr/pdf/Bogota2018/Suescun_Diffraction.pdf · Representación esquemática de un cristal y su diagrama de difracción. Powder](https://reader034.vdocuments.site/reader034/viewer/2022042022/5e79dbeafa327909e23c67b5/html5/thumbnails/73.jpg)
Intensity scaling and statistics
• The Normalized Structure Factors E(hkl) are a way of expressing the structure factors considering only the effect of amplitude decay by interference among radiation scattered byatoms and thermal motion.
• With the calculation of E(hkl) we eliminate the effect of scattering amplitude decay with the scattering angle.
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2)sin(aB
aa efg
B=2B=5
a
a
fg
1
0
![Page 74: Diffraction Experiments, Datacloud.crm2.univ-lorraine.fr/pdf/Bogota2018/Suescun_Diffraction.pdf · Representación esquemática de un cristal y su diagrama de difracción. Powder](https://reader034.vdocuments.site/reader034/viewer/2022042022/5e79dbeafa327909e23c67b5/html5/thumbnails/74.jpg)
Intensity scaling and statistics
• E(hkl) are fundamental for structural determination both fromsingle crystal or powder diffraction data by Direct Methods orthe Patterson Method.
• The distribution of E(hkl) values with 2 is also useful since itfollows the symmetry of the structure.
• A number of statistic indicators based on E(hkl) are in general used to distinguish among centric and acentric space groups(task that cannot be performed only by inspecting thesystematic absences in many cases) or to analyze thesymmetry of special projections of the structure that may beuseful to solve disorder problems or other rarities of crystalstructures.
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![Page 75: Diffraction Experiments, Datacloud.crm2.univ-lorraine.fr/pdf/Bogota2018/Suescun_Diffraction.pdf · Representación esquemática de un cristal y su diagrama de difracción. Powder](https://reader034.vdocuments.site/reader034/viewer/2022042022/5e79dbeafa327909e23c67b5/html5/thumbnails/75.jpg)
Intensity scaling and statistics
• We have shown that <|E(hkl)|2> =1
• It is possible to demonstrate that if a structure is:
– Centrosymmetric: <|E(hkl)2-1|> =0.968
– Non-centrosymmetric: <|E(hkl)2-1|> =0.736
• This is not only valid for every hkl but for special groups of them such as 0kl, h0l and hk0 thatare the projections of crystalstructure along [100], [010] and [001] respectively.
75
In the afternoon we will check how symmetry affects the diffraction pattern and extract the space group from the analysis of diffraction intensities.