Diffraction Experiments, Data processing and reduction

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2 Diffraction Experiments, Data processing and reduction Leopoldo Suescun Laboratorio de Cristalografía Química del Estado Sólido y Materiales, Facultad de Química, Universidad de la República, Montevideo, Uruguay. leopoldo@fq.edu.uy November 29th, 208

3 Bragg s Law 2 d n 2d sen d hkl h a hkl 2 2 k b 2 2 l c For Orthogonal unit cells only 2 2

4 Bragg s Law A 2 For constructive interference CB+BD = n n 2d sen hkl d C Phase shift: CB+BD CB=d*sin() B d C CB+BD=2d*sin() D 2 d B A D d hkl h a 2 2 k b 2 2 l c For Orthogonal unit cells only 2 2

5 Bragg s Law & Laue Equations Another equation for Braggs law: A 2 2 Ŝ s Sˆ Sˆ 0 Ŝ 0

6 Bragg s Law & Laue Equations Another equation for Braggs law: 2 Ŝ 0 Ŝ 0 ˆ ˆ S S s ) ( 2 ) cos(90 ˆ 2 0 sen s s S s s

7 Bragg s Law & Laue Equations Another equation for Braggs law: s 2sen( ) s 2dsen( ) s 2 sen( ) s d 2 s d

8 Reciprocal lattice a* c* b* a b c d hkl ' ' ' l d h k Antitransform Fourier Transform Crystal Structure Diffraction pattern hkl hkl hkl d s r * ' ' ' ' ' ' * ' ' ' l k h l k h l k h d s r 0 V b a c V a c b V c b a * * *

9 Reciprocal lattice c* b* b Nomenclature: a a* r hkl = h a + kb + l c = s hkl a* c b c c a b* c* V V a b V s hkl = h a + kb + l c s hkl = s hkl s hkl = s hkl = d hkl

10 Reciprocal lattice and Ewald sphere Rayos X difractados Rayos X incidentes s hkl cristal 2 s hkl d hkl 0 Origen R.R.

11 Reciprocal lattice and Ewald sphere shkl 2 sin( ) 2d hkl * dhkl 2 s hkl 0 d 2dhkl sin( ) hkl s 2 sin( ) d hkl s hkl d hkl s hkl d hkl

12 Reciprocal lattice and Ewald sphere Rayos X difractados Rayos X incidentes s hkl cristal 2 s hkl d hkl 0 Origen R.R. 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.

13 Reciprocal lattice and Ewald sphere Rayos X difractados S hkl S hkl Retículo Recíproco Rayos X incidentes cristal 0 2 s hkl sin( ) d hkl ha * kb * lc *

14 Esfera de ewald Rayos X difractados Rayos X incidentes s hkl Cristal 2 s hkl d hkl 0 Origen R.R.

15 Esfera de ewald Diffraction condition: when a RL point intersects the Reflection sphere X- ray will emerge from the cristal through that point. Rayos X difractados Rayos X incidentes 0 cristal Retículo Recíproco

16 Ewald Sphere or Reflection Sphere Diffraction condition: when a RL point intersects the Reflection sphere X-ray will emerge from the cristal through that point Rotate the crystal and the reciprocal lattice Rayos X incidentes 0 cristal Rayos X difractados Retículo Recíproco

17 Limiting Sphere Could all points of the Reciprocal Lattice be made to to intersect the reflection sphere for a given?. Rayos X incidentes 0 cristal Retículo Recíproco

18 Limiting sphere Rayos X incidentes 0 cristal Retículo Recíproco

19 Limiting sphere Rayos X incidentes 0 cristal Retículo Recíproco

20 Limiting Sphere Rayos X incidentes 0 cristal Those reciprocal lattice points that lay within 2/ of the origin (that have s < 2/ or d>/2) can be accessed in a diffraction Retículo Recíproco experiment for a given. 2

21 Limiting Sphere 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 d min = λ 2sin θ max = λ 2sin 90 = λ 2 the minimum interplanar spacing accessible in the experiment depends on the wavelength. At 2=80 the minimum resolution is obtained, we cannot measure diffraction data beyond that angle.

22 Limiting Sphere cristal 0 2

23 Limiting Sphere 2 d' min s' max ' >, d min >d min, s max <s max The shorter the better resolution for the same 2 angle. 0 2 dmin s max

24 Limiting Sphere If 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 N max accessible by the experiment. N max = V LS V = 4 3 π 2 λ V 3 = 32πV 3λ 3 cristal 0 2

25 Limiting Sphere N max = V LS V = 4 3 π 2 λ V BaTiO 3 at room temperature is tetragonal with space group P4mm, a=3.999 Å, c=4.07 Å and V= Å 3 Calculate the number of accessible reflections and the minimum d hkl available (maximum resolution) if: - (MoKa)= Å - (CuKa)=.548 Å Idem if you consider the structure of hen-egg lysozyme with space group P with a=60.0 Å, c=26.8 Å and V= Å 3 3 = 32πV 3λ 3

26 Experimental resolution When we perform an experiment, reaching the maximum available resolution requires to move the detector up to 2=80. Normally we would limit the resolution of the experiment to a diffractometer-limited angle (40 to 60 depending on the maker and limitations of the instrument). If we limit the scan to 2 max, s max =/[2sin( max )] and d min =2sin( max )/ the number of measurable reflections reduces to: N max = V LS V = π d min V = 4πV 3 3d min

27 The diffraction experiment Incident X-rays, crystal 0 beamstop

28 The diffraction experiment Incident X-rays, crystal 0 beamstop

29 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.

30 X rays ()

31 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, I hkl, si hkl list that is used for structure determination and refinement.

32 Single crystal intensities what should I do now?

33 Reflections, all, unique, observed Part of the reflection list for the compound CH 4 N 2 S. 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

34 Reflections, all, unique, observed For the compound CH 4 N 2 S: a= Å, b= Å c= Å, a=b=g=90 : = Å, 2 max =5.7, s max =.227 Å, d min =0.85 Å 49 intensities were integrated were different 239 were observed (I>2s I ) 340 were unique (unrelated by point group symmetry) 265 of the unique were observed.

35 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 CH 4 N 2 S a= Å, b= Å c= 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 0 reflections per structural parameters so only around 25 structural parameters can be determined.

36 Reflections, all, unique, observed Even though we don t need 3855 of the 49 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. R int (internal consistency of symmetry-related reflections) and R sigma (average I/sI for all reflections) are good measures of the consistency between equivalent reflections (symmetryrelated reflections that should have equal intensities) and the overall uncertainty of the recorded intensities respectively.

37 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 (F hkl ) 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.

38 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 > 0 7 crystallites) Of (ideal) homogeneous size between and 0 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 s hkl with length s hkl =/d hkl 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.

39 Powder Intensities For one crystal we found: Rayos X difractados Rayos X incidentes 0 cristal Retículo Recíproco

40 Powder Intensities 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).

41 Powder Intensities One small crystals will define a Reciprocal lattice with an orientatio consisten with the relative orientation of the crystal with the X-ray beam. Representación esquemática de un cristal y su diagrama de difracción.

42 Powder Intensities Four crystals in different orientations will diffract together but their Reciprocal lattices points are rotated by the same angle of the crystals.

43 Powder Intensities 4 crystallites 40 crystallites diffracting together

44 Powder Intensities 40 crystallites 200 crystallites

45 Difracción de rayos X de polvo The Fourier Transform of a polycrystallines sample is the sum of the FTs for each of the crystallites in the sample (ideally >0 7 crystals of equal size in the -0 mm range with random orientations) s hkl =/d X rays () Fourier Transform d hkl 45

46 Powder diffraction data collection A policrystalline sample (powder) is composed of an infinite (0 0 ) number of micrometric (-0 mm) crystal with a perfectly random order and its Fourier Transform d* hkl =/d=s hkl Rayos X () Fourier Transform d hkl 46

47 Powder diffraction data collection The powder pattern is obtained from the intersection of the reciprocal lattice of the powder (spheres) and the Ewald sphere. Rayos X () / 2 4 Intersections are circles and X-rays come from the sample exiting the Ewald sphere in cones of 4 angle. 47

48 Powder diffraction data collection Depending on the detector system geometry (and the sample shape and orientation respect to the beam different kinds of diffraction patterns are obtained. / / Rayos X () Sample Transmission Geometry (frequently found in Synchrotron labs) Reflection Geometry (frequently found in conventional labs) 48

49 Powder diffraction data collection Depending on the detector system geometry (and the sample shape and orientation respect to the beam different kinds of diffraction patterns are obtained. Rayos X () Transmission Geometry (frequently found in Synchrotron labs) 49

50 Powder diffraction data collection Position-sensitive detector with/without energy discrimination OFF ON Rayos X () Transmission Geometry (frequently found in Synchrotron labs) 50

51 2 Powder diffraction data collection Debye-Scherrer Camera with cylindrical film Rayos X () Transmission Geometry (Debye Scherrer Camera) 5

52 Powder diffraction data collection 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) Rayos X () Transmission Geometry (frequently found in Synchrotron labs) 52

53 Powder diffraction data collection We can integrate the detector counts into a diffraction pattern 53

54 Powder diffraction data collection We can numerically integrate the detector counts into a diffraction pattern 54

55 Powder diffraction data collection Depending on the geometry of the diffractometer different procedures for data collection are performed. Source Sample 2 55 Modern laboratory powder diffractometer

56 Powder intensities

57 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 s hkl or d hkl ) Metric overlap due to equal d-spacing of reflections with different hkl but similar d hkl Convolution overlap for independent reflections with D2<instrumental resolution Sometimes this makes it impossible to determine individual intensities

58 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.

59 Powder Intensity Extraction Both Pawley and LeBail method are based on the Rietveld equation: s j y y S LPAI 2 2 obsi bi hkl i j hkl hkl Pawley: I hkl is refined as a variable intensity I hkl to fit the corresponding peak/s. Le Bail: I hkl is calculated from refined F obs 2. F hkl = is the starting value and then replaced by F obs in successive LS cycles to achieve the best fit and best I hkl.

60 Powder Intensity Extraction After a Powder Intensity Extraction an h,k,l,i hkl 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.

61 Powder vs Single Crystal intensities Conventional X-ray powder diffraction data for Thiourea (CH 4 N 2 S) down to a resolution of 0.84 Å (2=33 for CuK a ) provides ~342 unique reflections of which ~249 do not overlap significantly (D2<FWHM/2) of which ~224 are observed (I>2s I criterion). Conventional X-ray single crystal diffraction data for Thiourea (CH 4 N 2 S) down to a resolution of 0.84 Å (2=50 for MoK a ) provides 340 unique reflections (none overlaping) of which 265 are observed (I>2s I criterion). Peak overlap produces a reduction of at least 40% of the candidate observed intensities, these calculations are made for a small unit cell wilth relatively high symmetry so things get worse for low-symmetry/large-cell problems.

62 Powder vs Single Crystal intensities In either case for Thiourea (CH 4 N 2 S) we ended up with files containing h, k, l, I hkl and si hkl lists that have to be correctly processed before submitting the problem for structural determination.

63 Intensity-Structure Factor relation The intensities can be related to the structure factors of the crystal by the equation: I hkl =kalp F hkl 2 k= scale factor A= absorption correction L= Lorentz correction p= Polarization correction

64 Intensity-Structure Factor relation I hkl =kalp F hkl 2 k= scale factor Depends 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)

65 Intensity-Structure Factor relation I hkl =kalp F hkl 2 A= absorption correction m I I0e Calculated from the Lambert-Beer Law Cylindric powder sample: X-rays () Absorption profile of a plate-shaped single crystal of a Pb and I-containing sample sorted by run number. k k 0... A e k ( * sen 2 3 mr k mr k m 0mRk 2 3 R / 2 ( )e *sen ( ) * sen( ) m es el coeficiente de absorción lineal de la muestra y R el radio del cilindro (capilar). 4

66 Intensity-Structure Factor relation I hkl =kalp F hkl 2 L= Lorentz correction L sen( 2 ) 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.

67 Intensity-Structure Factor relation I hkl =kalp F hkl 2 p= Polarization correction P ( cos 2 2 ( 2 )) Sealed tube with filter: p=/2+cos 2 (2)/2 Synchrotron radiation: p= Sealed tube with IBM: p=/2+cos 2 (2 m )cos 2 (2)/2

68 Intensity-Structure Factor relation For a spherical (or low absorbing m<) crystal: F rel (hkl) =(I(hkl)/Lp) /2 =(I Lp (hkl)) /2 k F rel (hkl) 2 =ki Lp (hkl)= F(hkl) 2 F rel (hkl) is the relative structure factor modulus (determined assuming k includes all unknown corections to the intensity that are independent of hkl or.

69 Wilson Plot With a little help of mathematics we can estimate k to get the structure factors to its correct magnitude, and as a bonus we can estimate the average B for the crystal: F( hkl ) gaj f F( hkl ) aj e j B j j aj aj 2i( sin( ) 2 g e e 2i( hx hx Atomic scattering factor f j j ky j ky j lz lz j j ) ) e B j sin( ) 2 Atomic displacement (thermal, Debye-Waller) parameter 69

70 Wilson Plot With a little help of mathematics we can estimate k to get the structure factors to its correct magnitude, and as a bonus we can estimate the average B for the crystal: f a g a f a e B a sin( ) 2 B=5 B=2

71 Wilson Plot With a little help of mathematics we can estimate k to get the structure factors to its correct magnitude, and as a bonus we can estimate the average B for the crystal: ln(k) ln I j Lp f hkl aj 2 2B ave sin 2 lnk

72 72 Intensity scaling and statistics Normalized Structure Factor E(hkl): F( hkl ) E( hkl ) j 2 g aj e 2i( hx F( hkl ) j f j ky aj j lz j 2 ( ) ) 2 < E(hkl) 2 > =< F 0 (hkl) 2 > /S j f 2 aj=s j f 2 aj/s j f 2 aj=

73 73 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 by atoms and thermal motion. With the calculation of E(hkl) we eliminate the effect of scattering amplitude decay with the scattering angle. g a f a e B a sin( ) 2 g a f a B=5 B=2 0

74 74 Intensity scaling and statistics E(hkl) are fundamental for structural determination both from single crystal or powder diffraction data by Direct Methods or the Patterson Method. The distribution of E(hkl) values with 2 is also useful since it follows 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 the systematic absences in many cases) or to analyze the symmetry of special projections of the structure that may be useful to solve disorder problems or other rarities of crystal structures.

75 75 Intensity scaling and statistics We have shown that < E(hkl) 2 > = It is possible to demonstrate that if a structure is: Centrosymmetric: < E(hkl) 2 - > =0.968 Non-centrosymmetric: < E(hkl) 2 - > =0.736 This is not only valid for every hkl but for special groups of them such as 0kl, h0l and hk0 that are the projections of crystal structure along [00], [00] and [00] respectively. In the afternoon we will check how symmetry affects the diffraction pattern and extract the space group from the analysis of diffraction intensities.