Non-crystalline materials and other things

1
Non-crystalline materials and other things

• By the end of this section you should
• know the difference between crystalline and
  amorphous solids and some applications for the
  latter
• understand how the different states affect the
  X-ray patterns
• be able to show the Ewald sphere construction for
  an amorphous solid
• be aware of different types of mesophases
• know the background to photonic crystals

2
Amorphous Solids

• So far we have discussed crystalline solids.
• Many solids are not crystalline - i.e. have no
  long range order.
• They can be thought of as solid liquids

3
Amorphous Solids

• The arrangement in an amorphous solid is not
  completely random
• 1) Coordination of atoms satisfied (?)
• 2) Bond lengths sensible
• 3) Each atom excludes others from the space it
  occupies.

? represented by radial distribution function,
g(r)
g(r) is probability of finding an atom at a
distance between r and r?r from centre of a
reference atom
Sometimes known as pair distribution function
4
Radial Distribution Function

• Take a reference atoms with radius a
• g(r) 0 for rlta
• g(r) ? 1 for large r
• At intermediate distances, g(r) oscillates around
  unity - short range order.

From any central atoms, the nearest neighbours
tend to have a certain pattern - though not so
rigidly as in a crystal
SiO4 - angles tend to 109.5º but are not exact
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Radial Distribution Function

• As we move out, the pattern becomes more and more
  varied until we reach complete disorder

X-ray diffraction can still give information on
the structure. X-rays scattered from atoms (not
planes) and interference effects will occur.
We use angle ?, though this does not relate to
any lattice plane as in Braggs law.
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Radial Distribution Function

• Scattered intensity depends on modulus - not
  direction - of K for an amorphous material.
• This means that diffraction patterns have
  circular symmetry rather than spots.

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Interference Function

• The interference function (i.e. scattering
  factor for amorphous materials) S(K) is given by

sinc Kr dr
where n is the no. of atoms per unit volume
and sinc ? sin ?/ ? S(K) is a Fourier transform
of g(r)-1 and
sinc Kr dK
8
Measurements

• We can measure the intensity, I(K), which (we
  assume) is directly related to S(K). Thus g(r)
  can be calculated from the interference effects
  in the (circular) diffraction pattern, and hence
  interatomic distances can be estimated.
• e.g. taking a radial cut from the centre of the
  pattern

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Measurements
Assignments made on expected distances between
atoms As we get further out, becomes less ideal
due to increased disorder
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Solid Liquids

• Diffraction patterns of an amorphous solid and a
  liquid of the same composition are very similar

The average structures are more or less the
same. Short range order less well developed in
liquid (peaks not so well defined)
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RDF in crystals

• We can also calculate this for a perfect crystal

Polonium, a 3.359 Å
This can allow analysis of not so perfect
crystals disorder Total diffraction
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Ewald Sphere for amorphous solids

• From previously

i.e. scattering depends only on modulus of K. So
we have a reciprocal sphere of radius K
intersecting with the Ewald sphere
This gives a circle where they intersect
diffraction pattern. (circle perp. to page)
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Back to EXAFS

• The Fourier transform of the EXAFS spectrum is
  also a radial distribution function

Intensity vs R (radius from central atom)
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Free volume

• Free volume (VF) defined as
• SV of glass/liquid - SV of corresponding crystal
• SV Volume per unit mass

15
Amorphous silicon

• Amorphous materials often not good conductors
  pathways blocked
• Crystalline silicon diamond structure, 4-fold
  coordination, regular (corner-sharing) tetrahedra
• Amorphous silicon mostly 4-fold coordination,
  fairly regular tetrahedra BUT

• not all atoms 4-fold coordinated
• dangling bonds
• Can be terminated by H atoms

kypros.physics.uoc.gr/resproj.htm
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Uses

• Method of production means it can be deposited
  over large areas thin films, flexible
  substrates
• Photovoltaics e.g. solar cells

Energy conversion not so efficient as crystalline
Si, but more energy efficient to produce
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Photovoltaics

• Instead of heat, light causes electron/hole pairs
• Cell made of pn junction - photons absorbed in
  p-layer.
• p-layer is tuned to the type of light - absorbs
  as many photons as possible
• move to n-layer and out to circuit.

http//solarcellstringer.com/
http//www.nrel.gov/data/pix/Jpegs/07786.jpg
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Mesophases
Normally a solid melts to give a liquid. In
some cases, an intermediate state exists called
the mesophase (middle). Substances with a
mesophase are called liquid crystals
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Liquid Crystals and Mesophases
Cholesteryl benzoate
Thanks to Toby Donaldson
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What types of molecules show liquid crystalline
behaviour?

• Anisometric molecular shape

Thanks to Toby Donaldson
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Polarised light microscopy
Mostly now used in geology Gases, liquids,
unstressed glasses and cubic crystals are all
isotropic One refractive index same optical
properties in all directions
Otto Lehmann (1855-1922)
Most (90) solids are anisotropic and their
optical properties vary depending on direction.
Birefringent Lysozyme crystals viewed by
polarised light microscopy http//www.ph.ed.ac.uk/
pbeales/research.html
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Polarised light microscopy
Thanks to Toby Donaldson
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Mesophases
If we increase temperature, we can see how the
disordering occurs
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Mesophases - more detail
(a) smectic phase - from the Greek for soap,
smegma
A C
Layers are preserved, but order between and
within layers is lost
25
Smectic
Thanks to Toby Donaldson
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Mesophases - more detail
(b) nematic phase - from the Greek for thread,
nemos
Layers are lost, but the molecules remain
aligned If we looked at this end on, it would
look like a liquid
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Nematic Phase, N
Thanks to Toby Donaldson
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Isotropic Liquid
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Liquid Crystals
Novel phase structures
SmA phase
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Mesophases - XRD
Example - mix of powder (circles) and ordering
(arcs)
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LCDs

• LCs sandwiched between two cross polarisers
• twist in LC allows light to pass through
• Applied voltage removes twist and light no longer
  passes through

http//www.geocities.com/Omegaman_UK/lcd.html
http//www.edinformatics.com/inventions_inventors/
lcd.htm
32
Photonic Crystals

• 1887 Lord Rayleigh noted Bragg Diffraction in
  1-D Photonic Crystals

1987 Eli Yablonovitch Inhibited spontaneous
emission in solid state physics and electronics
Physical Review Letters, 58, 2059, 1987 Sajeev
John Strong localization of photons in certain
disordered dielectric super lattices Physical
Review Letters, 58, 2486, 1987
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Basics of photonics

• Periodic structures with alternating refractive
  index

Photonic band gap analogous to electronic band
gap Weakly interacting bosons vs strongly
interacting fermions
http//ab-initio.mit.edu/photons/tutorial/ - S.G
Johnston
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Braggs Law wider applications
n? 2d sin ?

• This is a general truth for any 3-d array.
• If we imagine the atoms as larger spheres,
  then
• d becomes larger
• ? becomes larger visible light
• This is the basis for photonic crystals

Opal (SiO2.nH2O) A fossilised bone! Silica
spheres 150-300 nm in diameter ccp/hcp
http//www.mindat.org/gallery.php?min3004
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Braggs Law wider applications

• We replace the d-spacing, from Braggs law,
• with the optical thickness nrd
• where nr is the refractive index (e.g. of the
  silica in opal)

n? 2nrd sin ?
nr is 1.45 in opal so
n? 2.9 d sin ?
This gives ?max 2.9 d for normal incidence
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Geometry of packed spheres

• If we assume the spheres close pack, then we
  can calculate d

sin 60 d/2r d 1.73 r
?max 2.9 d So ?max 5r (approx.) for normal
incidence We now need to manipulate d!!
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Photonic band gap

• From above ?max 2nrd at this ?, no light
  propagates
• And from de Broglie E hc/ ?
• So in photonic crystals, we define the photonic
  band gap

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Photonics in nature
J. Zi et al, Proc. Nat. Acad. Sci. USA, 100,
12576 (2003) Blau, Physics Today 57, 18 (2004)
http//newton.ex.ac.uk/research/
39
Artificial Photonics

• Massive research area (esp. in Scotland!)
• Control areas of differing refractive index, e.g.

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The first experiment

• An array of small holes 1mm apart were drilled
  into a piece of material which had refractive
  index 3.6.
• Calculate the wavelength of light trapped by
  this material

?max 2nrd 2 x 3.6 x 0.001 7.2 x 10-3
m Microwaves
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Woodpile crystal

• Logs of Si 1.2 ?m wide

K. Ho et al., Solid State Comm. 89, 413 (1994)
H. S. Sözüer et al., J. Mod. Opt. 41, 231 (1994)
http//www.sandia.gov/media/photonic.htm
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Artificial photonic crystals

• S. G. Johnson et al., Nature. 429, 538 (2004)

From amorphous silicon 3D, 1.3 1.5 ?m
T. Baba et al, Yokohama National University
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Artificial Opal

• D. Norris, University of Minnesota
  http//www.cems.umn.edu/research/norris/index.html
  

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Inverse Opal
Templating to produce

• Yurii A. Vlasov, Xiang-Zheng Bo, James C. Sturm
  David J. Norris., Nature 414, 289-293 (2001)

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Inverse Opal

• Silica spheres with a refractive index of 1.45
• 1.3 ?m

Q Calculate d (and hence the radius of the
spheres) from this information.
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Uses

• From K Inoue K. Ohtaka Photonic crystals (
  Springer, NewYork,2003).

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Summary

• Amorphous materials show short range order and
  have have various applications e.g. in
  photovoltaics
• X-ray interference effects still occur, leading
  to circular diffraction patterns which relate to
  g(r), the radial distribution function and the
  scattered X-ray intensity depends on the modulus
  of the scattering vector, K
• States intermediate between crystalline and
  liquid exist - mesophases - such as nematic and
  smectic
• These have wide applications, an example being
  LCDs
• Extension of Braggs law to a different scale
  length leads us to consider photonic crystals