X-ray diffraction

Laue’s and Bragg’s Treatments of X-ray diffraction

The path difference between the two rays reflected from successive planes isQA2 + 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

0.7-0.75 nm magneticfield of 11.7 T

SmC layers

53o

120 ºCO

O

H15C7O

H15C7O

OO

OC7H15

OC7H15

S

N N

Polymorphism of Discotic Mesophases

Discotic nematic (ND)

Discotic hexagonal columnarColh or Dh

Colho ordered;Colhd disordered

Discotic rectangular and tetragonal columnarColr or Dr; Colt or Dt

Discotic oblique columnarColob or Dob

uniaxial uniaxial

biaxial

biaxial

uniaxial

(Colh plastic crystal)

System Unit-cell symbol(s) Symmetry at lattice points

Unit-cell edges and angles

Oblique p (primitive unit cell) 2 a ≠ b, γ ≠ 90º, 120º

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

Diffraction Pattern of a Hexagonal Columnar Mesophase

7:4:3:1

hkkh

adhk

22

34

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).

(10)

(11)

a

Diffraction Pattern of a Hexagonal Columnar Mesophase

7:4:3:1

hkkh

adhk

22

34

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).

(10)

(01)

(11)

a

Diffraction Pattern of a Pseudo-hexagonal Columnar Mesophase

2

2

2

2

1

bk

ah

dhk 22 3khadhk

rectangular becomes pseudo-hexagonal if a = sq3 bSo that b2 = a2/3

2D X-ray Diffraction Pattern of a Mechanically Aligned Sample of the ‘Chemical Mixture’ (M = Cu)

sample

X-ray beamintercolumnar

21.0 Å

intracolumnar3.5 Å

Colh

X-ray diffraction pattern of Cu52 at 90 ºC

10 200

20000

40000

Inte

nsity

(co

unts

)

2 (°)

0 10 20 30

1000

1500

2000d100

d210

d200

d110

Inte

nsity

(co

unts

)

2 (°)

60 90 120 150

17.8

17.9

18.0

d 100

(Å)

T (°C)60 90 120 150

364

368

372

S (Å

2 )

T (°C)

Temperature dependence of X-ray diffraction pattern