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X-ray diffraction

X-ray diffraction. Meet in the LGRT lab Again, will hand in worksheet, not a formal lab report Revision exercise – hand in by April 17 th class. Diffraction summarized. The 6 lattice parameters ( a,b,c, a,b,g ) of a crystal determine the position of x-ray diffraction peaks.

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X-ray diffraction

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  1. X-ray diffraction • Meet in the LGRT lab • Again, will hand in worksheet, not a formal lab report • Revision exercise – hand in by April 17th class.

  2. Diffraction summarized • The 6 lattice parameters(a,b,c,a,b,g) of a crystal determine the position of x-ray diffraction peaks. • The contentsof the cell (atom types and positions) determine the relative intensity of the diffraction peaks. • If a diffraction peak can be identified with a Miller Index, the unit cell on the phase can usually be determined.

  3. Miller indices Miller index, hkl • Choose origin • Pick 1st plane away from origin in +a, +b direction • Find intercepts in fractional coordinates (“none = ∞”). • h = 1/(a-intercept) • k = 1/(b-intercept) • l = 1/(c-intercept) • If no intercept, index = 0 • Plane equation: ha + kb + lc = 1 (110) (100) (150) (2-10) • Same set of planes will be described if all 3 Miller indices are inverted (2-10) ≡ (-210) • Minus sign → bar

  4. Crystals facets correspond to Miller indices b • Haüy, 1784 • Crystals (like calcite) are made of miniscule identical subunits a (-120) (-100) (-1-10) (-110) (-2-10) (-120) • Facets can be described by low-order Miller indices (3-40) (340) Woolfson

  5. Miller indices For a lattice of known dimensions, the Miller indices can be used to calculate the d-spacing between hkl planes (a,b,c = lattice parameters). This d-spacing will determine where powder diffraction peaks are observed Origin at orange dot Miller index, hkl (-110) (100) (150) (2-10)

  6. Distance between planes (Miller indices)

  7. Bragg’s law derivation(angle of incidence = angle of diffraction) a b • x = d sinq • extra distance = 2d sinq = nl • nl= 2d sinq q d q q q q x d x x l = 2d sinq (for x-ray diffraction)

  8. Powder diffractometer

  9. KI (CsCl-type) x-ray pattern

  10. Crystals, diffraction, and Miller indices (001) (0-10) (010) (100)

  11. Coherent scattering from a row of atoms Will only happen when emission from all atoms is simultaneously stimulated.(Solid lines represent spatial regions where phase = 0 deg).

  12. Laue condition – vector description • Extra distance=a cos m - a cos n • Extra distance = -(a·S0) + (a·S) • Extra distance = l(a·s) s = (S – S0)/l S0 v1 cosq = v1·v2 v2 m a a a n S • Will have diffraction when: a cos m - a cos n = hl (h = integer) • Will have diffraction when: a·s = h (h = integer)

  13. Laue condition – 2D • Must have coherent scattering from ALL ATOMSin the lattice, no just from one row. • If color indicates phase of radiation scattered from each lattice point when observed at a distant site P, we see that scattering from rows is in phase while columns is out of phase, making net scattering from all 35 points incoherent and therefore NOT observable S0 b a S • Will have diffraction when: a·s = h (h = integer) • Will have diffraction when: b·s = k (k = integer) • Will have diffraction when: c·s = l (l = integer) Every lattice point related by translational symmetry will scattering in phase when conditions are met

  14. Single crystal diffraction • Used to solve molecular structure • Co(MIMT)2(NO3)2 example • Data in simple format – hkl labels + intensity + error 0 0 1 0.00 0.10 0 0 2 42.60 1.40 0 0 3 1.10 0.30 0 0 4 100.30 2.50 0 0 5 -0.30 0.50 0 0 6 822.30 16.70 0 0 7 -0.40 0.50 0 0 8 656.40 13.90 0 0 9 1.00 0.80 0 0 10 73.40 3.00 0 0 11 0.00 1.40 0 0 12 4.70 1.60 0 0 13 1.00 1.70 0 2 1 611.40 14.40 0 -2 -1 613.90 12.10 0 2 2 443.90 8.90 0 -2 -2 443.00 8.70 0 2 3 59.90 1.50 0 -2 -3 56.90 1.50 0 2 4 55.20 1.60 0 -2 -4 51.80 1.50

  15. h=2 h=1 h=0 h=-1

  16. h=2 l=10 l=9 l=8 l=7 l=6

  17. 3D view of reciprocal lattice (Lines connecting dots are unecessary) Massa

  18. Relation between direct and reciprocal cells Stout and Jensen

  19. Direct vs. reciprocal lattice direct or real Lattice vectors a, b, c Vector to a lattice point: d = ua + vb + wc Lattice planes (hkl) reciprocal Lattice vectors a*, b*, c* Vector to a reciprocal lattice point: d* = ha* + kb* + lc* Each such vector is normal to the real space plane (hkl) Length of each vector d* = 1/d-spacing (distance between hkl planes)

  20. Hexagonal lattice a* b* g changes from 120 to 60 deg a direct or real reciprocal b

  21. Distances between planes (d*) direct or real Vector to a reciprocal lattice point: d* = ha* + kb* + lc* Length of each vector d* = 1/d-spacing (distance between hkl planes) |d*| = (d*·d*)1/2 =(ha* + kb* + lc*) (ha* + kb* + lc*) =(ha*)2 + (kb*)2 + (lc*)2 + 2(ha*) (kb*) + 2(ha*) (lc*) + 2(kb*) (lc*) = h2a2* + k2b*2 + l2c*2 + 2klb*c*cosa* + 2lhc*a*cosb* + 2hka*cosg* reciprocal

  22. Density of CaRuO3 • Density = mass / volume • Mass of unit cell = (# Ca)(mCa) + (# Ru)(mRu) + (# O)(mO) • Volume of unit cell = abc = (3.7950 Å)3 • mamuNA = mgrams • M = molar mass (g/mol) • n = # f.u. per cell • NA = Avagadro’s # (1/mol) • V = cell volume

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