Basics of XRD part III

Transcription

1 Basics of XRD part III Dr. Peter G. Weidler Institute of Functional Interfaces IFG 1 10/31/17 KIT The Research University of the Helmholtz Association Name of Institute, Faculty, Department

2 Overview why XRD? Information inside XRD data Basics of crystallography Basics of X-ray scattering Sample preparation Basics of instrumentation Data evaluation positions, phase identification lattice parameters quantitative XRD November 2017

3 Why XRD? Identification of material Quantification Solid state properties substitution, stress strain... changes upon heating, sorption,... So, what do we want November 2017

4 Intensity of diffraction peaks I hkl = I o K M P L G A E T F hkl ² I o = primary intensity K = scale-factor (cps, counts,...) M = multiplicity (planes of the same form {hkl} have the same scattering angle θ and therefore contribute to the same peak P = polarization-factor (angular dependency of dipole radiation) L = Lorentz-factor (single X-tal: time plane remains in diffraction position) G = geometry-factor ( e.g. powder: intensity on the cone circumference) A = absorption-factor (x-ray will be absorbed by the material) E = extinction-factor (decrease of intensity by secondary beams) T = temperature-factor (thermal movement of the atoms in the lattice) F hkl ² = structure-factor November 2017

5 F hkl ² = Structure-factor position r n of atom n in unit cell m 1 m 2 m 3 plane wave approximation ε p is value of electric field at observation point P November 2017

6 F hkl ² = Structure-factor Structure-Factor November 2017

7 F hkl ² = Structure-factor Laue-equations --> Bragg-Law for plane h'k'l' expressing cell vectors in terms of unit vectors a i and fractional coordinates x n, y n, z n, r n = x n a 1 + y n a 2 + z n a 3 : (s - s o ) r n lattice plane hkl fractional coordinates November 2017

8 h k l --> lattice plane b 1 b 2 b 3 --> reciprocal space unit cell is scattering vaector H hkl on a reciprocal lattice point?? fractional coordinates: where is which atom placed in the unit cell with parameters a 1 a 2 a November 2017

9 Atomic scattering factor November 2017

10 Atomic scattering factor f o ~ Z November 2017

11 example face-centered lattice fcc or F for every atom (total number n) in position Xn Yn Zn ---> three identical atoms in (Xn+1/2, Yn+1/2, Zn) ; (Xn+1/2, Yn, Zn+1/2) ; (Xn, Yn+1/2, Zn+1/2) November 2017

12 example face-centered lattice fcc or F -->rock salt NaCl --> NaCl as example Na and Cl both form a fcc lattice!! Cl: (0,0,0); (1/2,1/2,0); (1/2,0,1/2); (0,1/2,1/2) Na: (1/2,1/2,1/2); (0,0,1/2); (0,1/2,0); (1/2,0,0) Kleber et al. Einführung in die Kristallographie, 17ed November 2017

13 Rock Salt NaCl Using (0,0,0) for Cl and (1/2,1/2,1/2) for Na: finding atomic scattering factor f Cl and f Na

14 example face-centered lattice fcc or F using values of ~ 18 for f Cl and ~ 10 for f Na for the atomic scattering factor (for small values of sinθ/λ ~ number of e - in atom) then for h,k,l, even F hkl = 4(f Cl + f Na ) ~ 112 for h,k,l, odd F hkl = 4(f Cl - f Na ) ~ 32 for h,k,l mixed F hkl = 0 ICDD # hkl rel. Int no mixed indices!! November 2017

15 Halite NaCl Cu K α1, November 2017

16 crystallite size effect on line width x-tal size ~ 1 µm x-tal size ~ 10 nm... intensity not influenced!! November 2017

17 occupancy effect on intensity octahedral site tetrahedral site Fe site density reduced on octa- and tetrahedral position for about 14% November 2017

18 intermediate summary different factors --> intensity instrumental parameters sample parameters Structure Factor Atomic Scattering Factor November 2017

19 Sample Preparation... How important is it? Most important part of whole XRD activity!!!! What are the main feature to watch: sample height/position preferred orientation grain size of powder stress/strain induction by milling!... and some more November 2017

20 sample preparation -- sample position shift of peak positions change in intensity --> errors in qualitative and quantitative phase analysis indexing of lattice parameters corrective: internal standard November 2017

21 sample preparation -- sample position Δ 2θ = -2 S cos θ / R R= goniometer radius S downward, away from goniometer center (S a positive number!), results in a negative Δ 2θ, that is the peak position will be displaced to lower 2theta If measured angle is below correct angle, then sample too low. Sample Displacement 2θ Δ 2θ S 2theta 38,6 38,4 38,2 38,0 37,8-1,5-1 -0,5 0 0,5 1 1,5 position / / / mm November 2017

22 sample preparation -- preferred orientation (001) sample holder intensity of (001) peak stronger than at random orientation --> in mixture --> overestimation of this phase!! (e.g. clay) Attention by plates, flakes, needles... --> errors in qualitative and quantitative phase analysis Rietveld-refinement corrective: --> milling --> dispersion in (XRD-amorphous) matrix --> more tricks (depending on sample) November 2017

23 sample preparation -- milling effects grains too large: absorption preferred orientation homogeneity of sample --> errors in qualitative and quantitative phase analysis Rietveld refinement --> milling... BUT too long milling destroys crystallinity or induces stress/strain ---> peak width broadening --> XRD before and after milling!!! rule of thumb: approx. 5 µm ok for most materials November 2017

24 sample preparation...and some more... particle statistics: random powder orientation enough particles each (hkl) present in each orientation to diffract example: quartz α-sio 2 10x10x0.2 mm³ = 20 mm³...some calculations yielding the number of particles in diffraction conditions: To achieve PO-error better than 1% it needs more the 53'000 particles with 1µm --> almost impossible --> Rotation of sample November 2017

25 sample preparation...and some more... intensity variation vs. sample position during rotation at fixed angle grain size /µm Zevin & Kimmel Quantitative X-ray Diffractometry, Springer November 2017

26 Measurement - what to observe what do I want to know: - just quick check for phase presence/absence - lattice parameters determination --> site substitution unknown sample - quantitative work --> length of 2-θ range: short or long (sections?) --> step width -- resolution of single peaks, peak-overlap rule of thumb: 1/5 and 1/3 of FWHM --> counting time per step rule of thumb: 10'000 counts at maximum absolute --> availability of XRD-facility example: θ, step θ, 5 sec --> 11h 6' θ, step θ, 5 sec --> 16h 40' 5 ½ hrs November 2017

27 End of part 1 --> questions!!!!!!!!!!! Basics of instrumentation Data evaluation positions, phase identification lattice parameters quantitative XRD November 2017

28 Acknowledgment Bruker AXS do Brasil and Bruker AXS Germany, Knielingen CEFET UMFG INCT-Acqua November 2017