X-ray diffraction Laue’s and Bragg’s Treatments of X-ray diffraction The path difference between the two rays reflected from successive planes is QA2 + A2R = 2d sin θ Constructive interference is obtained if the path difference is equal to an intergral number of wavelengths (Bragg’s Law) 2d sin θ = nλ X-ray diffraction of N Phases X-ray diffraction of Sm Phases 2D WAXS Pattern of a Tetracatenar SmC Phase H15C7O 120 ºC OC7H15 H15C7O OC7H15 O O O O S N N 53o SmC layers 0.7-0.75 nm magnetic field of 11.7 T Polymorphism of Discotic Mesophases uniaxial uniaxial biaxial Discotic nematic (ND) uniaxial Discotic hexagonal columnar Colh or Dh Colho ordered; Colhd disordered biaxial Discotic oblique columnar Colob or Dob (Colh plastic crystal) Symmetry at lattice points Discotic rectangular and tetragonal columnar Colr or Dr; Colt or Dt System Unit-cell symbol(s) Unit-cell edges and angles Oblique p (primitive unit cell) 2 Rectangular p, c (centred unit cell) 2 mm a ≠ b, γ = 90º Square p 4 mm a = b, γ = 90º Hexagonal p 6 mm a = b, γ = 120º a ≠ b, γ ≠ 90º, 120º Diffraction Pattern of a Hexagonal Columnar Mesophase d hk a (11) a 4 2 h k 2 hk 3 (10) 1: 3 : 4 : 7 with hk = kh = 10 is set to be 1 An additional peak can appear at 1.323 which was explained by a periodic electron density distortion (DeGennes, J. Phys. Letters, 1983 (44), L657). Diffraction Pattern of a Hexagonal Columnar Mesophase (01) d hk a 4 2 h k 2 hk 3 a (10) (11) 1: 3 : 4 : 7 with hk = kh = 10 is set to be 1 An additional peak can appear at 1.323 which was explained by a periodic electron density distortion (DeGennes, J. Phys. Letters, 1983 (44), L657). Diffraction Pattern of a Pseudo-hexagonal Columnar Mesophase d hk 1 h k 2 2 b a 2 2 rectangular becomes pseudohexagonal if a = sq3 b So that b2 = a2/3 d hk h a 2 3k 2 2D X-ray Diffraction Pattern of a Mechanically Aligned Sample of the ‘Chemical Mixture’ (M = Cu) sample Colh X-ray beam intercolumnar 21.0 Å intracolumnar 3.5 Å X-ray diffraction pattern of Cu52 at 90 ºC 2000 Intensity (counts) 40000 Intensity (counts) d100 d110 1500 d200 d210 1000 20000 0 10 20 2 (°) 0 10 20 2 (°) 30 Temperature dependence of X-ray diffraction pattern 18.0 17.9 S (Å2) d100 (Å) 372 368 17.8 364 60 90 T (°C) 120 150 60 90 T (°C) 120 150