Crystallography and Diffraction. Theory and Modern Methods of Analysis

Lecture 15Amorphous diffraction

Dr. I. AbrahamsQueen Mary University of London

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Unlike crystalline solids, amorphous solids show no regular repeating structure that can be defined by a lattice.

In these solids atoms show a distribution of environments, that typically manifest themselves as a broadening of peaks in spectroscopic techniques such as NMR and IR.

In diffraction experiments amorphous materials show no Bragg peaks, but they do exhibit scattering that can be analysed.

Diffraction from Amorphous solids

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One approach to understanding the diffraction patterns of glasses is to consider what happens to a crystalline solid and progressively introduce disorder.

Consider a simple crystalline solid with a primitive unit cell and only one atom per cell.

The a-axis of the unit cell is equal to the diameter of the atom.

We can stack the unit cells in directions X,Y,Z.

If we stack N1 atoms in direction X, N2 atoms in direction Y and N3 atoms in direction Z to give a cubic crystal containing N1 N2 N3 atoms, then we can calculate the intensity I at angle as:

Progressive disorder approach

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23

22

21

222

1

0321

1

0

1

0

32

sin8exp

sin4

sin4sin3

3

2

2

1

1

NNN

aR

a

RR

where

R

RRNNNNNNMNPfI

NN

N

NN

N

NN

N

M is the multiplicity = 8, 4 or 2 (if all Nj > 0, two Nj > 0, one Nj > 0 respectively) Is the standard deviation in Å in the distribution of inter-atomic distances P is the polarization factorf 2 is the square of the atomic scattering factor.

The sum is carried out over all values of N1 N2 and N3 for which 0 < (N1+N2+N3) < N

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As we increase i.e. increase the level of disorder the pattern broadens.

This method does not lend itself easily to more complex systems and so other methods of analysis are used.

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k0 is the incident wave vector of magnitude 2/

kf = final wave vector of magnitude magnitude of k0

Q = scattering vector = k0 – kf = 4/

Typical diffraction experiment

The scattering vector

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In a diffraction experiment the Intensity I(Q) measured at the detector of angle d is given by:

Where is the scattering cross section, is the flux and

The differential scattering cross section has components from distinct and self scattering. For a system containing N atoms:

selfdistinct

Qd

d

NQ

d

d

NQ

d

d

N

111

d

d

d

d into secondper scattered quanta ofnumber The

dQd

dQI

d

d

is the differential scattering cross section which is defined as:

Differential scattering cross section

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Distinct diffraction is the diffraction from different atomic sites and self diffraction is the diffraction from individual atomic sites. For a system with N atoms of n chemical species:

n

bcQFQd

d

N

21

F(Q) is the total interference function, c is the fraction of chemical species and b is the scattering length of species .

As we are mostly interested in the distribution of one species () around another () we can define F(Q) as:

1,

QSbbccQFn

Where S is known as the partial structure factor.

Structure factors and correlation functions

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NN

iji ij

ij

Qr

Qr

NccSS

,

,

sin11

The partial structure factor is given by:

Where rij is the radial distance between scatterers i and j and the symbol denotes thermal average. The partial pair distribution functions g are obtained by Fourier transformation of S

dQQrQSQr

rg

drQrrgrq

QS

sin12

11

sin14

1

002

0

0

Where 0 is the total number density of atoms = N/V (V = volume)

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The number of atoms around atoms in a spherical shell i.e. the partial coordination number is given by integration of the partial radial distribution function.

2

1

204

r

r

drrrgcn

The total pair correlation function G(r) is derived by Fourier transform of the total interference function F(Q).

00

2sin

2

1dQQrQqF

rrG

The total correlation function T(r) is given by:

2

04 brGrrT where n

bcb

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rGrD 04

Another correlation function that is often used is the differential correlation fund D(r)

2

1

1

21

n

iii

n

iii

X

Qfc

Qfcdd

NQF

GX(r) is obtained by Fourier transformation of FX(Q) as before. GX(r) can also be written as:

11,

2

1

rg

Kc

KKccrG ij

n

jin

iii

jijiX

Where Ki is the effective number of electrons for species i.

For X-ray scattering we need to use the X-ray scattering factor rather than the scattering length. The total interference function for X-rays is given by:

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Neutron diffraction correlation functions for lithium borate glasses

Swenson et al. Phys Rev. B. 52 (1995) 9310

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Q-ranges

In order to get good radial distribution function data the range of Q should be large. Typically neutron data allows Q ranges up to ca. 50 Å-1 while in X-ray data the maximum useable Q value is close to 20 Å-1

For laboratory X-ray data, Cu tubes have maximum Q value around 8 Å-1. Ag tubes increase the Q-max to ca. 20 Å-1

Synchrotron radiation is commonly used for X-ray experiments.

As we have seen before it is not just the Q-range that is important but the different sensitivities of X-rays and neutrons to different elements and their isotopes that make the choice of radiation important.

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Comparison of X-ray and neutron contrast in cobalt, lead and magnesium phosphate glasses.Hoppe et al. J. Non-Crystalline Solids 293-295 (2001) 158

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Total correlation functions for phosphate glasses of composition NaM(P3O9) (M = Ca, Sr, Ba)

T. Di Cristina PhD Thesis Queen Mary Univ. of London 2004

Fit to correlation functions for phosphate glasses of compositionNaSr(P3O9)

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Sample M = Ba M = Sr M = Ca 35Ca 40Ca

rP-NBO / Å 1.4796(5) 1.4801(9) 1.4800(6) 1.4893(-) 1.4891(-)

½2NBOP u / Å 0.0343(5) 0.0346(9) 0.0362(5) 0.0364(5) 0.0374(6)

nP-NBO 1.83(3) 1.85(5) 1.87(3) 2.10(3) 2.11(3)

rP-BO / Å 1.6018(9) 1.6001(16) 1.5977(10) 1.6028(11) 1.6006(12)

½2BOP u / Å 0.0497(9) 0.0497(14) 0.0490(9) 0.0490(-) 0.0490(-)

nP-BO 1.99(3) 2.01(5) 1.96(4) 1.75(2) 1.72(3)

Total nP-O 3.82(6) 3.85(10) 3.83(7) 3.85(5) 3.83(6)

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RMC Modelling of diffraction data

ND

XRD

Reverse Monte Carlo modelling of diffraction data is a very powerful way of structure elucidation allowing for individual pair correlations to modelled.

e.g. Calcium metaphosphate glass. Wetherall et al. J. Phys. C. Condens Mater. 21 (2009) 035109

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Bibliography

For more detailed discussion of the theory of diffraction in amorphous solids see

1. Neutron and x-ray diffraction studies of liquids andGlasses, Henry E Fischer, Adrian C Barnes and Philip S Salmon, Rep. Prog. Phys. 69 (2006) 233–299

2. X-Ray Diffraction Procedures For Polycrystalline and Amorphous Materials 2nd Edition Harold P Klug and Leroy E Alexander, Wiley 1974.