non-crystalline materials and other things
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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 - PowerPoint PPT PresentationTRANSCRIPT
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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
Non-crystalline materials and other things
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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”
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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
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Radial Distribution Function
Take a reference atoms with radius a
g(r) = 0 for r<a
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 Bragg’s law.
sin2
K
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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:
}1)r(g{r4n1)K(S 02
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
0
}1)K(S{K4n
11)r(g 2 sinc Kr dK
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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
a2a3
a 2a 3aPolonium, 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:
sin2
K
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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Intensity vs R (radius from central atom)
Back to EXAFS
• The Fourier transform of the EXAFS spectrum is also a radial distribution function
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Free volume
Free volume (VF) defined as:
SV of glass/liquid - SV of corresponding crystal
SV = Volume per unit mass
crystal
liquid
glass
Temperature
Volume
Tg
Glass transition
Tm
melting
VF
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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…
kypros.physics.uoc.gr/resproj.htm
• …not all atoms 4-fold coordinated• …“dangling bonds”
Can be terminated by H atoms
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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
Friedrich Reinitzer (1857-1927)
Thanks to Toby Donaldson
Crystal 145.5 °C LC 178.5°C I
Cholesteryl benzoate
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• Anisometric molecular shape
NC OCnH2n+1
Calamitic liquid crystals
OR
RO
RO
OR
OR
OR
Discotic liquid crystals
What types of molecules show liquid crystalline behaviour?
Thanks to Toby Donaldson
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Polarised light microscopy
Birefringent Lysozyme crystals viewed by polarised light microscopyhttp://www.ph.ed.ac.uk/~pbeales/research.html
Otto Lehmann (1855-1922)
Mostly now used in geology
Gases, liquids, unstressed glasses and cubic crystals are all isotropic
One refractive index – same optical properties in all directions
Most (90%) solids are anisotropic and their optical properties vary depending on direction.
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Polarised light microscopy
Thanks to Toby Donaldson
IsotropicLiquid crystallinecholesteric
Crystalline
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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,
A C
Layers are preserved, but order between and within layers is lost
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Smectic
Thanks to Toby Donaldson
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Mesophases - more detail
(b) nematic phase - from the Greek for thread,
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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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
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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 fermionshttp://ab-initio.mit.edu/photons/tutorial/ - S.G Johnston
1887 19872-D
periodic intwo directions
3-D
periodic inthree directions
1-D
periodic inone direction
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Bragg’s Law – wider applications
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
n = 2d sin
Opal (SiO2.nH2O)
A fossilised bone!
Silica spheres 150-300 nm in diameter – ccp/hcp
http://www.mindat.org/gallery.php?min=3004
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Bragg’s Law – wider applications
We replace the d-spacing, from Bragg’s 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!!
2r
r
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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:
dn2
hcE
r
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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/
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Artificial Photonics
Massive research area (esp. in Scotland!)
Control areas of differing refractive index, e.g.
d
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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)
T. Baba et al, Yokohama National University
From amorphous silicon – 3D, 1.3 – 1.5 m
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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
Yurii A. Vlasov, Xiang-Zheng Bo, James C. Sturm & David J. Norris., Nature 414, 289-293 (2001)
Templating to produce…
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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 Bragg’s law to a different scale length leads us to consider photonic crystals