Ionic Conductors: Characterisation of Defect Structure Lecture 15 Total scattering analysis Dr. I. Abrahams Queen Mary University of London Lectures co-financed

Ionic Conductors: Characterisation of Defect

Structure

Lecture 15

Total scattering analysis

Dr. I. AbrahamsQueen Mary University of London

Lectures co-financed by the European Union in scope of the European Social Fund


What is total scattering analysis ?

A diffraction pattern of a fully ordered solid will contain peaks corresponding to reflections from particular sets of planes in the crystal lattice. In this type of solid, conventional analysis of the diffraction pattern gives an accurate picture of both long range and short range structure.

We have already seen that in disordered solids conventional diffraction analysis gives an average picture of the structure. From which carefull analysis of the average picture can yield information on the defect structure.

The long range order in a solid manifests itself in the sharp Bragg peaks observed in the diffraction pattern. However in addition to these Bragg peaks where is diffuse scattering that comes from short range correlations. Usually this information is not used.

In the total scattering approach both diffuse scattering and Bragg scattering are used to give a more complete picture of the structure.


Neutron diffraction patterns for Bi3Nb1-xYxO7-x

20.017.515.0

t. o. f [ ms ]

x = 0.2

x = 0.4

x = 0.6

x = 0.8

x = 1.0

x = 0.0

12.5


Diffuse Neutron Scattering

• Analysis of diffuse scattering gives information on local structure

• Analysis of Bragg scattering gives long range structure.

• Together the analysis gives a more complete picture of the defect structure


The basic theory behind total scattering analysis is essentially the same as for analysis of diffuse scattering for glasses, except that the Bragg data are additionally analysed.

As in analysis of glass diffraction data we transform the data to Q space where:

Theory of total scattering analysis

dQ

2

sin4

= Bragg angle, = wavelength and d = d-spacing.


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 total scattering cross section has components from Bragg and diffuse scattering. For a system containing N atoms:

DiffuseBragg

Qd

d

NQ

d

d

NQ

d

d

N

111

d

d

d

d into secondper scattered quanta ofnumber The

dQ

d

dQI

d

d

is the differential scattering cross section which is defined as:

Differential scattering cross section


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


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)


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


Reverse Monte Carlo modelling

RMC is a general simulation method based on experimental data, therefore the models can be simulated without bias. The procedure is a variation of the standard Metropolis Monte Carlo simulation, and is based on the random sampling of atom positions to drive structural models to be as consistent with the experimental data as possible.

The process involves an arrangement of N atoms (the configuration) that are generated within certain ranges in a three-dimensional box. Some of the atoms are selected randomly and moved a random amount, under periodic boundary conditions.

Each time, the difference (usually of the structure factors) between the new model and the data are recalculated, and only the difference minimizing movements are accepted otherwise the movements are rejected.


RMC methodology

1. Generation of the configuration of the system. N atoms are arranged in a three-dimensional box.

The box is based on the unit cell obtained from the Rietveld refinements and made by generating a 10 10 10 supercell of the crystallographic cell in P-1 symmetry.

2. Calculation of the correlation functions from the atom positions in theconfiguration usually:

S(Q), G(r), IBragg(Q) and GO-O(r)

where, IBragg(Q) is the intensity of Bragg scattering profile and GO-O(r) is the pairdistribution function for O-O pairs).


3. Calculation of the difference (2) between the measured correlation functions and the functions calculated from the configuration. e.g. for S(Q)

2

2

1obscalc

2)(

i

n

iii

QSQ

QSQS

Where the summation is over all n experimental data points, each with error σ (Qi). The total 2 is summed over all correlation functions.

4. Atom movements. One atom is selected at random. This atom is moved randomly in both direction and distance, which generates a new configuration. From the new configuration, a new set of correlation functions are calculated. If the value of 2 is lower than its previous value and the model satisfies the constraints, e.g. the minimum distance to other atoms in the system, the movement is accepted andsaved. Otherwise, the atom is returned to its previous position.


5. Cycling. The calculations and movements are repeated by returning to step 3. This procedure continues until the 2 value reaches equilibrium.

At this point, the model can be said to have converged.

In the case of multiple data sets (e.g. X-ray and neutron diffraction data), the overall agreement parameter 2

all includes a summation over all data types.

m

m22

all


Worked example - Bi3YO6

Conventional analysis of Bi3YO6

1.0 1.5 2.0 2.5 3.0

0

20

40

60

80

100

(a)Backscattering

d - spacing / Angstrom

Ab

s n

eu

tro

n c

ou

nts

Fit to neutron diffraction data for Bi3YO6

Detail of Y coordination in Bi3YO6


Bi3YO6 thermal variation of oxide ion distribution

There is a small change in the oxide ion distribution with temperature

Oxide ions per cell

Site 25oC 800oC % change

8c 3.128 3.104 -0.8

32f 2.368 2.336 -1.35

48i 0.528 0.816 +54.5

Vac 2.504 2.556 +5.2

8 c

8c

3 2 f

32f

4 8 i

48i

V a c

Vac

0

0.5

1

1.5

2

2.5

3

3.5

Nu

mb

ers

per

cel

l

8c 32f 48i Vac


Problems with Conventional Approach to Structure Elucidation

The conventional approach to analysis of defect structure relies on detailed analysis of the average crystal structure and assumes ions/atoms behave according to their known crystal chemistry to derive models of local structure.

There are several problems associated with this approach.

• Atoms/ions do not always behave according to their established crystal chemistry.

• In “fully disordered” systems where, the conventional approach ignores the diffuse scattering which contains information on local ordering.

• All crystallographic approaches effectively use a static model to model a dynamic system.


2 4 6 8 10 12

0

10

11

12

Obs. RMC fitting Diff.

S(Q

) / A

rb. U

nit

Q / Å-1

4 8 12 16 20 24-0.20.0

10.0

10.2

10.4

10.6

10.8

11.0 Obs. RMC fitting Diff.

G(r

) / A

rb. U

nit

r / Å

2 4 6 8 10 12

0

10

11

12

Obs. RMC fitting Diff.

S(Q

) / A

rb. U

nit

Q / Å-1

5 10 15 20 25-0.20.0

10.0

10.2

10.4

10.6

10.8

11.0 Obs. RMC fitting Diff.

G(r

) / A

rb. U

nit

r / Å

RT

800C

S(Q) G(r)

Total scattering for Bi3YO6


Pair Correlations – g(r)

-0.1

0.9

1.9

2.9

3.9

4.9

0 5 10 15 20 25r / Å

g(r

)

Bi-O

Y-O

O-O

-0.1

0.9

1.9

2.9

3.9

4.9

0 5 10 15 20 25

r / Å

g(r

)

Bi-O

Y-O

O-O

25oC

800oC


Coordination in Bi3YO6

1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0-5

0

5

10

15

20

25

30

35

40

Bi-O Y-O

Fre

qu

ency

/ %

Distance / Å

1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0-5

0

5

10

15

20

25

30

35

40

Bi-O Y-O

Fre

qu

ency

/ %

Distance / Å

25oC: Bi CNav = 4.0

800oC Bi CNav = 3.9

25oC: Y CNav = 5.0

800oC Y CNav = 4.9


Vacancy ordering in Bi3YO6

<111> ordering

O Angle ratio 6:6:3 <110> ordering

Angle ratio 6:7:2 <100> ordering

Angle ratio 7:6:2

There are three characteristic O-Bi-O angles for the ideal fluorite structure: 70, 109 , 180 . The ratio of the number of these angles per Bi atom gives information on the vacancy ordering.

Ideal

Angle ratio 12:12:4


Angular Distribution Function AO-M-O() in Bi3YO6

109180 180 7070 109

Angles 71.4o 108.0o 180o

Ratio 6 : 8 : 2

Angles 72.8o 106.3o 180o

Ratio 5 : 9 : 2

800oC25oC


<111> <110> <100>

Vacancy ordering

Angle ratio 4:3:2 <110><110>

Vacancy ordering

Angle ratio 6:12:2<100> <110>

Vacancy ordering

Angle ratio 10:8:2

If one considers the ions on the 48i site as a Frenkel defect, leaving a vacancy on the fluorite site, then in Bi3YO6 there are 3 vacancies per metal atom

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Ionic Conductors: Characterisation of Defect Structure Lecture 15 Total scattering analysis Dr. I. Abrahams Queen Mary University of London Lectures co-financed