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Crystallography  RSC (ANU)
 Disordered Materials
 T
R Welberry ]
"One common thread is apparent in the writings
of all those who have published papers on the subject of optical transforms
 the great power of visual presentation in teaching, in stimulating thought
and in aiding the development of intuition, which still plays a major role
in solving the more complex diffraction problems."
C. A. Taylor 1975
Crystalline systems diffract Xrays which have a wavelength comparable
to the distances between atoms (~1 angstrom). The principles of Xray diffraction
can be demonstrated by light diffraction from optical diffraction masks.
These masks are 2D representations of physical systems "drawn" onto a transparency
(slide) using a photomation apparatus. The image is on a fine scale so
that a good sized diffraction pattern (Optical Transform) is formed when
the transparency is illuminated with a standard low power HeNe laser (or
similar).
Optical Transforms are a very useful educational tool for illustrating
a wide variety of diffraction phenomena from Bragg diffraction, through
diffuse scattering from disordered and amorphous systems to diffraction
from quasicrystals with noncrystallographic symmetry. The following figures
demonstrate the use of a selection of these optical transforms for modelling
the scattering patterns of amorphous and crystalline systems. (The full
list of available diffraction masks is given below).
Powders and Liquids
This
image shows diffraction from basic gratings, powder diffraction from
different symmetry lattices, and diffraction from a liquid.
Benzene
This
image shows diffraction from randomly distributed benzene molecules,
from a lattice of benzene molecules, from randomly distributed pairs and
quartets of molecules, and the effect of thermal diffuse scattering and
lattice vacancies on diffraction patterns.
Quasicrystals and Paracrystals
This
image shows the effect of space group symmetry on diffraction patterns,
quasicrystal diffraction patterns with two ordering schemes and a distorted
paracrystal diffraction pattern.
Simple Diffraction Apparatus
Some laser links
http://www.netinfo.com/~laser/index.html
http://www.optimaprec.com/index.htm
Some lens links
Edmund
Scientific http://www.edmundscientific.com/scientifics/scientifics.cfm
Optosigma
http://optics.org/optosigma/lenses/microobj.html
List of Optical diffraction masks available......
August 1998
The slides are designed to demonstrate various diffraction effects encountered
in crystallography, but some at least will have wider applicability.
The slides are in 35mm format and each contains a very finescale image
of a computergenerated model. The scale is chosen to be small so that
the optical diffraction pattern (Transform), which may be obtained using
a laser and a simple arrangement of lenses, is of a large enough size that
it may easily be viewed in a classroom with no further magnification required.
1af 1D gratings 3 different spacings, single or double slit 6
2a,b,c 2D powder patterns 3

Square lattice

Rectangular lattice

Hexagonal Lattice
3 2D liquid like structure 1

Shows first and second diffraction ring for comparison with 2c.
4a,b,c 2D Penrose tiling etc. 3

5fold symmetry  Penrose tiling

5fold symmetry  random tiling

8fold symmetry  random tiling.
5 Various Molecular crystals (hypothetical) showing different symmetries.
4

5a. p4 symmetry, 4 molecules per square cell arranged around a 4fold axis.

5b. p2gg symmetry, 2 molecules per rectangular cell, 2 glide planes

5c. pg symmetry, 2 molecules per rectangular cell, 1 glide plane

5d. p1 symmetry, 1 molecule per oblique cell, no symmetry
6 Disordered Molecular crystal showing diffuse scattering. 3

6a. short range order in one direction showing diffuse bands between Bragg
rows.

6b. short range order in one direction showing diffuse bands along Bragg
rows.

6c. short range order in two directions showing diffuse spots in centre
of cell.
7a,b,c Helical molecules, combinations of different pitches and radii.
3
8. Examples showing effect of molecular shape on transform of lattice.
10
c.f. Plates 11 & 12 of ' Atlas of Optical Transforms' by Harburn, Taylor
& Welberry

8a. single benzene molecules.

8b. pairs of benzene molecules separated by cell translation vector a.

8c. quartets of benzene molecules at corners of oblique cell axb.

8d. lattice of benzene molecules with oblique cell axb.

8e. lattice of benzene molecules with oblique cell axb. with
random (thermal) displacements (showing TDS)

8f. lattice of benzene molecules with oblique cell axb. with
random 50% occupancy of sites (showing DDS, disorder diffuse scattering)

8g. single 5atom asymmetric molecules.

8h. pairs of 5atom asymmetric molecules separated by cell translation
vector a.

8i. quartets of 5atom asymmetric molecules at corners of oblique cell
axb.

8j. lattice of 5atom asymmetric molecules with oblique cell axb.
9. Lattice modulations. 2
c.f. Plate 20 of ' Atlas of Optical Transforms' by Harburn, Taylor &
Welberry

9a. in preparation

9b. in preparation

9c. Transverse wave perturbing lattice, wavevector in general direction.

9d. Longitudinal wave perturbing lattice, wavevector in general direction.
10.
Highly distorted lattices (Hosemann paracrystals). 6
Defined in terms of transverse correlation r_{T},
longitudinal correlation r_{L},
and variance s^{2}.

14a. r_{T}=0.95;
r_{L}=0.95;
s^{2}=0.5.

14b. r_{T}=0.95;
r_{L}=0.95;
s^{2}=1.0

14c. r_{T}=0.99;
r_{L}=0.95;
s^{2}=0.5

14d. r_{T}=0.99;
r_{L}=0.95;
s^{2}=1.0

14e. r_{T}=0.95;
r_{L}=0.99;
s^{2}=0.5

14f. r_{T}=0.95;
r_{L}=0.99;
s^{2}=1.0
11a,b Novelty transforms 2

a. Slide whose transform is picture of bird & fish.

b. Slide whose transform is picture of building.
12. Various Lattice functions  demonstration multiplication/convolution.
6
c.f. Plates 10 &13 of ' Atlas of Optical Transforms' by Harburn, Taylor
& Welberry

12a. Lattice Function 8x8

12b. Lattice Function 3x4

12c. Lattice Function 5x4

12d. Lattice masked by small circular aperture

12e. Lattice masked by larger circular aperture

12f. Lattice masked by rectangular aperture inclined at 26 degrees.
13. Development of 1D chain  multiplication/convolution  SINC function.
5
c.f. Plates 1 & 10 of ' Atlas of Optical Transforms' by Harburn, Taylor
& Welberry

13a. Airy Disc pattern (Single atoms distributed randomly)

13b. Airy Disc modulated by fringes (Pairs of atoms with constant spacing)

13c. Airy Disc modulated by fringes (Pairs of atoms with constant larger
spacing)

13d. Airy Disc modulated by narrower fringes (Chains of 4 atoms with constant
spacing)

13e. Diffraction pattern of row of dots (Long 1D Chains of atoms with
constant spacing)
14.
Atomic Size Effect  4
see Welberry, T.R. (1986) J. Appl. Cryst. 19,
382389

14a. Random 50:50 distribution of two atom types, scattering factors f_{big}
and f_{small}, on perfect lattice

14b. Same Distribution as 14a with sizeeffect relaxation f_{big}
>f_{small}

14c. Same Distribution as 14a with sizeeffect relaxation f_{big}
<f_{small}

14d. Same Distribution as 14a with sizeeffect relaxation f_{big}
=f_{small}
Total number of slides.................. 58
Optical Diffraction Slides may be ordered by writing to:
Dr. T.R.Welberry,
Research School of Chemistry,
Australian National University,
Canberra City, ACT 0200,
Australia.
Email:welberry@rsc.anu.edu.au
Cost of slides is A$20 per slide.
Overhead transparencies showing enlarged portions of each slide may
be obtained for an extra A$5 per slide.
Web page maintained by T.R.Welberry  ANU  last update August
2000
welberry@rsc.anu.edu.au