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