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

Non-crystalline materials and other things

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”

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

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

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

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.

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

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:

Measurements

Assignments made on expected distances between atoms

As we get further out, becomes less “ideal” due to increased disorder

“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)

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”

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)

Intensity vs R (radius from central atom)

Back to EXAFS

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

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

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

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

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

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

Liquid Crystals and Mesophases

Friedrich Reinitzer (1857-1927)

Thanks to Toby Donaldson

Crystal 145.5 °C LC 178.5°C I

Cholesteryl benzoate

• 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

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.

Polarised light microscopy

Thanks to Toby Donaldson

IsotropicLiquid crystallinecholesteric

Crystalline

Mesophases

If we increase temperature, we can see how the disordering occurs:

Mesophases - more detail

(a) smectic phase - from the Greek for soap,

A C

Layers are preserved, but order between and within layers is lost

Smectic

Thanks to Toby Donaldson

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

Nematic Phase, N

Thanks to Toby Donaldson

Isotropic Liquid

Mesophases - XRD

Example - mix of powder (circles) and ordering (arcs)

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

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

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

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

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

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

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

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/

Artificial Photonics

Massive research area (esp. in Scotland!)

Control areas of differing refractive index, e.g.

d

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

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

“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

Artificial Opal

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

Inverse Opal

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

Templating to produce…

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.

Uses

From: K Inoue & K. Ohtaka: “Photonic crystals” ( Springer, NewYork,2003).

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