Transcript Document

Mario Bieringer__________________________________________________________________________
Neutron Powder Diffraction
Mario Bieringer
Department of Chemistry
University of Manitoba
presented at the
10th Canadian Neutron Summer School
June 15 – 18, 2009
Chalk River, Ontario
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10th Canadian Neutron Summer School
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June 16th, 2009
Mario Bieringer__________________________________________________________________________
Applications of Powder Diffraction
chemistry
physics
engineering
life sciences
biochemistry
materials science
geological sciences
archeology
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Applications of Powder Diffraction
diffraction = elastic scattering
momentum transfer at constant energy, i.e. incident l = diffracted l
Why do we care about diffraction experiments:
• identification of materials/phases
• determination of structural details, i.e. local
environments within an extended structure
Intensity (counts)
600
Si powder neutron diffraction pattern
C2 Chalk River
l = 1.329 Å
500
400
300
200
100
0
40
60
0
2 ( )
80
100
• evaluation of order and disorder in structures
• phase transitions (structural distortions
or reconstruction)
• chemical reactions
• relate structure and properties
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Routine Powder X-ray Diffraction:
M
D
A
S
Visible
Ref. Code
Counts
group-1_1
• Powder X-ray diffractogram
Chemical
Formula
red
01-073-2141 La2 O3
blue
00-005-0378 Ba C O3
green
01-089-5898 Cu O
10000
matched with the PDF database
• links to crystallographic
data if available (i.e structure
2500
details from ICSD)
• semi-quantitative analysis
0
20
30
40
50
60
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Position [°2Theta]
10th Canadian Neutron Summer School
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Routine Powder X-ray Diffraction:
M
D
A
S
XRA
Y
c
BE
A
M
Preferred Orientation
of flat sample mounts:
b
a
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Intensity (a.u.)
In-situ Powder X-ray Diffraction:
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What is a Powder?
powder = polycrystalline solid
large number of crystallites of mm length scale
ideal powders (for diffraction) show random orientation
of crystallites
 orientational average of single crystals
• large number of preparative methods available.
• powders can be prepared in large quantities (g, kg, etc.)
• fast synthesis
• real world materials are often polycrystalline
• in reality powders are often multiphasic
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Description of Crystallographic Structures
Unit Cell:
Smallest repeating unit capable of describing the entire crystal by
means of three non-parallel translation vectors.
 paralleliped
The 7 Crystal Systems
System
u.c. symmetry
unit cell parameters
cubic
m-3m
a=b=c
a = b = g = 90˚
hexagonal
6/mmm
a=b≠c
a = b = 90˚ g = 120˚
trigonal*
3/mmm
a=b≠c
a = b = 90˚ g = 120˚
tetragonal
4/mmm
a=b≠c
a = b = g = 90˚
orthorhombic
mmm
a≠b≠c
a = b = g = 90˚
monoclinic
2/m
a≠b≠c
a = g = 90˚ b ≠ 90˚
triclinic
-1
a≠b≠c
a ≠ b ≠ g ≠ 90˚
* also described as rhombohedral:
a=b=c
a = b = g ≠ 90˚
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Description of Crystallographic Structures
Bravais Lattices:
Centering conditions can be described with translation vectors:
Lattice
Symbol
Primitive
lattice points
+(0, 0, 0)
1
Body centered I
+(½, ½, ½)
2
Base centered
A
+(0, ½, ½)
2
B
+(½,0, ½)
2
C
+(½, ½, 0)
2
F
+(0, ½, ½), +(½, ½, 0), +(½,0, ½)
4
Face centered
P
translations
A total of 14 Bravais Lattices exist (see next page for illustration)
cubic:
hexagonal:
trigonal:
tetragonal:
P, I, F
P
P
P, I
orthorhombic: P, I, F, (A,B,C)
monoclinic: P, C
triclinic: P
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figure taken from: www.chem.ox.ac.uk
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Diffraction Peaks due to Periodicity:
Due to the inherent crystal periodicity families of parallel virtual planes
can be constructed. The scattering probe diffracts off those planes and
generates a pattern that is characteristic of the spacing of the virtual
planes and the composition of the unit cell. Virtual planes are identified
by the Miller indices (h,k,l). The d-spacings, dhkl, (i.e. the perpendicular
distance, between planes) can be determined with Braggs law:
l = 2dhkl sin
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Diffraction Peaks due to Periodicity:
(001)
K2NiF4 structure: tetragonal phase
b
c
a
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Powder Diffraction and Symmetry:
Predict Bragg positions for primitive cells with V = 64 Å3 (l = 1.54 Å).
cubic
The higher the symmetry and
hexagonal
the smaller the cell volume the
tetragonal
smaller the number of diffraction
orthorhombic
peaks.
monoclinic
triclinic
18
21
24
27
30
0
2 ( )
33
36
peak multiplicities decrease from cubic to triclinic:
e.g. cubic
d(100) = d(-100) = d(010) = d(0-10) = d(001) = d(00-1)
orthorhombic
d(100) = d(-100) ≠ d(010) = d(0-10) ≠ d(001) = d(00-1)
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Description of Crystallographic Structures
Fractional coordinates:
Note: use the crystal system as the coordinate system
x = fraction along a axis
y = fraction along b axis
z = fraction along c axis
Symmetry Elements:
rotation axes:
inversion rotation axes:
mirror planes:
screw axes (helices):
glide planes:
1, 2, 3, 4, 6 fold
-1, -2, -3, -4, -6 fold
m
21, 31, 32, 41, 42, 43, 61, 62, 63, 64, 65
a, b, c, n, d
Absent reflections due to translations!
230 Space Groups: (International Tables for Crystallography: Vol. A)
e.g. I41amd
(number 141)
tetragonal system, body centered
41 screw axis, a glide, mirror, d glide
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Description of Crystallographic Structures
BaLaMnO4 (K2NiF4 structure type):
Space group I4/mmm
(# 139)
BaLaMnO_4
Lambda: 1.54178 Magnif: 1.0 FWHM: 0.200
Space grp: I 4/mmm Direct cell: 4.0000 4.0000 13.6568 90.00 90.00 90.00
radii =Z
013
Ba(56) = 0,0,0.35
La(57)) = 0,0,0.35
0
10
20
30
1 2 1 01 11 76
002048
123
020
022
0 10 50 6
002
112
011
004
114
110
Mn (25) = 0,0,0
40
50
O1(8)
= 0,½,0
O2(8)
= 0,0,1.4
60
Diffraction pattern:
No h+k+l = odd reflections
 Body centered e (½,½,½) translation)
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Diffraction Peak Intensities:
Structure Factor:
X-ray case:

 

 

F( hkl )   f j exp 2ihxj  ky j  lz j  exp  B j sin 2  / l2
j
Neutron case:

F( hkl )   b j exp 2ihxj  ky j  lz j  exp  B j sin 2  / l2
j
Correction factors:
Powder Peak Intensities:
I ( hkl )  s p( hkl ) L A P( hkl ) F( hkl )
I ( hkl )  F( hkl )
F(hkl) = structure factor
fj
= X-ray form factor
b
= neutron scatt. length
h,k,l = Miller indices
xj, yj, zj = atomic
coordinates
of atom j
Bj = thermal parameter

= diffraction angle
l
= wavelength
S over entire unit cell
I(hkl) = intensity
2
2
s = scale factor
L = Lorentz-polarization
p = multiplicity
A = absorption correction
P = preferred orientation
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Diffraction Peak Intensities:
Scattering lengths
X-rays, f
scattering amplitude (10
-13
cm)
40
X-ray form factor, f:
sin/l=0Å
-1
sin/l=0.5Å
-1
35
30
25
20
neutrons, b
15
10
Temperature factor, Bj:
5
0
-5
0
10
20
30
40
50
60
70
80
90
atomic number Z
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Description of Crystallographic Structures
BaLaMnO4 (K2NiF4 structure type):
b
atom=x,y,z
Z
5.07fm Ba=0,0,0.35
56e
8.24fm La=0,0,0.35
57e
-3.73fm Mn=0,0,0
25e
5.80fm O1=0,½,0
8e
5.80fm O2=0,0,1.4
8e
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Peak Widths
coherence length  domain sizes  crystallite sizes  particle sizes
90
Scherrer equation:
l = wavelength
B = integral breadth
 = diffraction angle
80
60
Intensity (a.u.)
70
Intensity (a.u.)
0.9 l
D
B cos
B
80
60
50
area = 5 a.u.
40
20
40
0
30
4.80
20
4.85
4.90
4.95
5.00
5.05
5.10
5.15
5.20
0
2 ( )
10
The broader the peaks
the smaller the domains.
0
-10
4.8
4.9
5.0
5.1
5.2
5.3
5.4
5.5
5.6
0
2 ( )
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Diffraction from Randomly Oriented Crystals:
direct lattice
(crystal structure)
reciprocal lattice
(diffraction pattern)
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Diffraction from Randomly Oriented Crystals:
1 single crystal
few crystals
very large number
of microcrystals
Debye-Scherrer Camera
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Single Crystal versus Powder Diffraction:
Single crystal diffraction:
• Use 3-dimensional reciprocal space:
a*, b* and c* axes
• Use integrated intensities.
 unambiguous peak assignment
Powder diffraction:
• Use a large assembly of microcrystals
and record an average diffraction
pattern in 1-dimension.
• Use profile points.
 ambiguous peak assignment
Reminder: 1/d = 2 sin/l
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(100)
Intensity
(110)
(101)
(010)
1/d
Powder diffraction:
• Information gets buried in powder
average.
• Try to extract peak intensities or
fit the entire profile.
 Rietveld Method
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Powder X-ray vs. Powder Neutron Diffraction
simulations of Pr2Ta2O7Cl2 powder patterns
Powder X-ray Diffraction (l = 1.54059A)
Powder Neutron Diffraction (l = 1.54059A)
0
10
20
30
40
50
60
70
80
90
100
110
120
0
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2 ( )
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Simplified Powder Diffractogram
Intensity  * atomic/ionic positions
* temperature factors
* order/disorder
FWHM

* domain sizes
* habit
2 (˚)

* crystal system
* unit cell dimensions
* space group
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X-ray vs Neutron Powder Diffraction?
XRD
NPD
phase I.D.
YES
NO
indexing
YES
YES
space group
YES
YES
structure refinement
YES
YES
light elements
NO
YES
neighbouring elements
NO
YES
structure solution
YES
YES
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When to Use Neutrons:
Ideally: always (most structure determinations and refinements benefit
from independent measurements (X-ray + neutron))
In particular:
• light and heavy elements present (e.g. oxides, fluoride, hydrides
(deuterides) of heavy metals)
• neighbouring elements: e.g. Al & Si, Fe & Co, Yb & Lu, Ti & V, etc.
• multiple site occupancies: e.g. disordered structures
• interest in true bulk properties (i.e averaging over large samples)
• complex experimental set ups: chemical reactions, high temperatures,
high pressures, low temperatures, magnetic fields, electrochemical
cells, chemical/structural processes etc.
• magnetic samples
• etc.
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Are Neutrons going to Answer Your Questions?
look up neutron scattering lengths and cross sections:
•
http://www.ncnr.nist.gov/resources/n-lengths/
Neutron News, Vol. 3, No. 3, 1992, pp. 29-37
Neutron scattering lengths and cross sections
Isotope
conc
Coh b
Inc b
Coh xs
Inc xs
Scatt xs
Abs xs
O
---
5.803
---
4.232
0.0008
4.232
0.00019
16O
99.762
5.803
0
4.232
0
4.232
0.0001
17O
0.038
5.78
0.18
4.2
0.004
4.2
0.236
18O
0.2
5.84
0
4.29
0
4.29
0.00016
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Are Neutrons going to Answer Your Questions?
LaFeAs(O1−xFx)
LiCoO2, LiNi1-xCoxO2, LiMn2O4
b(La=57) = 8.24 fm
b(Li=3) = -1.90 fm
b(Fe=26) = 6.58 fm
b(Co=27) = 2.49 fm
b(As=33) = 9.45 fm
b(Ni=28) = 10.3 fm
b(O=8) = 5.803 fm
b(Mn=25) = -3.73 fm
b(F=9) = 5.654 fm
b(O=8) = 5.803 fm
BiFeO3, BaTiO3, LnMnO3
La0.9Ba0.1Ga0.8Mg0.2O2.8
b(Bi=83) = 8.532 fm
b(La=57) = 8.24 fm
b(Fe 26) = 6.58 fm
b(Ba=56) = 5.07 fm
b(O=8) = 5.803 fm
b(Ga=31) = 7.288 fm
b(Mg=12) = 5.375 fm
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Two Neutron Powder Diffraction Methods
1. Continuous Wavelength (Reactor Sources):
• single (selectable) wavelength
detector wires
• continuous neutron flux
• position sensitive detectors,
record neutron rate as a
function of diffraction angle
• Bragg’s law: l = 2d sin
• Dd/d depends on:
- monochromator
- take-off angle,
- monochromator mosaicity
- and sample size
sample
2
M
collimator
m
on
o
sample environment
example: C2
http://neutron.nrc-cnrc.gc.ca/c2gen.html
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Two Neutron Powder Diffraction Methods
2. Time of Flight Method (Spallation Sources):
87-93deg
detector
28 - 32 deg
detector
example: HRPD (ISIS, U.K.)
2 choppers close to
the target (6m and 9m)
select neutron pulses
160-176 deg
backscattering
detector
sample
beam Stop
guide tube 100m
87-93 deg
detector
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http://www.isis.rl.ac.uk/crystallography/HRPD/index.htm
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Two Neutron Powder Diffraction Methods
2. Time of Flight Method (Spallation Sources):
• white radiation (including epithermal fraction)
• pulsed neutron source (50 to 60 Hz)
• very high neutron flux during neutron burst
• multitude of time of flight detectors, record neutron rate vs. time
m
m L
• Bragg’s law: l  2d sin 
de Broglie: l  n  n
h vn
ht
 t( hkl )  505.57 L d( hkl ) sin d
 t( hkl )  d( hkl )
mn = neutron mass, vn = neutron velocity, t = flight time,
L = flight distance, d = detector angle
• resolution: Dd/d depends on total flight path, L, and detector angle (d)
• use time focusing, large scattering volumes don’t degrade the resolution
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Data Comparison
1. Continuous Wavelength (Chalk River – C2 (Pr2Ta2O7Cl2) ): :
15000
Pr2Ta2O7Cl2
Low Temperature Powder Neutron Diffraction
C2 l = 2.37A
Intensity (a.u.)
12500
10000
7500
5000
2500
0
10
20
30
40
50
60
70
80
0
2 ( )
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Data Comparison
2. Time of Flight Method (IPNS – SEPD (Pr2Ta2O7Cl2) ):
d-range: 0.33-4.0Å
1000
d-range: 0.45-5.4Å
d-range: 0.85-10.2Å
400
800
800
600
500
300
500
600
300
6900
7000
7100
7200
7300
7400
7500
7600
7700
7800
T O F (m s )
IPNS - SEPD
0
BANK #1 (144.85 )
400
Intensity (a.u.)
400
Intensity (a.u.)
350
200
400
Intensity (a.u.)
In te n s ity ( a .u .)
600
700
300
IPNS - SEPD
0
BANK #2 (90 )
200
10000
15000
20000
25000
0
0
30000
IPNS - SEPD
0
BANK #3 (44 )
150
50
0
5000
200
100
100
0
250
5000
10000
15000
20000
25000
0
30000
5000
10000
TOF (ms)
15000
20000
25000
30000
TOF (ms)
TOF (ms)
450
500
0
500
0
IPNS - SEPD BANK #1 (144.85 )
IPNS - SEPD BANK #3 (44 )
400
0
IPNS - SEPD BANK #2 (90 )
400
300
200
350
Intensity (a.u.)
Intensity (a.u.)
Intensity (a.u.)
400
300
200
300
250
200
100
100
0
150
0
13000
14000
15000
16000
17000
TOF (ms)
10000
11000
12000
TOF (ms)
13000
100
5000
5500
6000
6500
7000
TOF (ms)
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Powder Data Analysis
1) Record Powder Diffractogram (neutron and/or X-ray)
2) Identify the phase(s) (X-ray database)
3) Index phase (X-ray or neutron), i.e. find unit cell (i.e. crystal system)
4) Identify space group (X-ray or neutron)
5) If a structural model is available (isostructural compound known)
proceed with Rietveld refinement
6) If no structural model is known proceed with ab-initio structure
solution
7) Consider X-ray and neutron combined analysis
8) Consider different wavelengths
9) Consider different temperatures
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Rietveld Method
Profile Fitting Method:
1.
use a crystallographic model and compute the corresponding
powder diffraction pattern
2.
compare simullation with experimental diffraction data
3.
minimize the difference between experimental and calculated
diffraction pattern in a least squares refinement by varying a
number of parameters:
S y   wi  yi  yci 
2
minimize:
wi = 1/yi
j
yi = observed intensity yci = calculated intensity
yci  s L( hkl ) F(2hkl ) fi 2i  2ci P( hkl ) A  ybi
j
s = scale factor, L(hkl) = Lorentz, polarization and multipicity factors
f(2i-2(hkl)) = reflection profile function, P(hkl) = preferred
orientation function, A = absorption factor, F(hkl) = structure factor,
ybi = background intensity
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Rietveld Method
Refinable Parameters:
GLOBAL PARAMETERS:
FOR EACH PHASE
zero point
atomic positions
instrumental profile
thermal parameters
profile asymmetry
site occupancies
background parameters
scale factor
sample displacement
lattice parameters
absorption
preferred orientation
crystallite size
microstrain
magnetic vectors
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Rietveld Method
Intensity (a.u.)
600
observed
calculated
difference
Bragg positions
300
0
14400
14500
14600
14700
14800
14900
15000
15100
TOF (ms)
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Rietveld Method
Evaluation of fitting quality (R-factors):
•
R-pattern, Rp:
Rp
y y


y
i
ci
i
  wi  yi  yci 2 

R-weighted pattern, Rwp: Rwp  
  w  y 2 
i
i


1/ 2
•
•
R-Bragg factor, RB:
RB
I


( hkl )
(' obs' )  I ( hkl ) (calc)
 I ('obs' )
i
•
R-expected, Re:
 N  P  
Re  
2
w
y
  i i 
1/ 2
Goodness of fit, S:
S
Rwp
Re
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10th Canadian Neutron Summer School
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Rietveld Method
A large number of Rietveld programs are available:
To a large extent these programs have similar capabilities with
individual strengths:
Rietveld Programs:
GSAS
FullProf
Rietica
Rietan
BGMN
TOPAS
etc.
http://www.ccp14.ac.uk/
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10th Canadian Neutron Summer School
slide 40
June 16th, 2009
Mario Bieringer__________________________________________________________________________
Rietveld Method
1000
Pr2Ta2O7Cl2 (PTOC449)
Powder Neutron Diffraction
0
IPNS - SEPD BANK #1 (144.85 )
800
600
Intensity (a.u.)
800
Intensity (a.u.)
600
400
200
0
400
5000
6000
7000
8000
9000
10000
TOF (ms)
200
observed
calculated
difference
Bragg
0
5000
10000
15000
20000
25000
30000
TOF (ms)
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10th Canadian Neutron Summer School
slide 41
June 16th, 2009
Mario Bieringer__________________________________________________________________________
Rietveld Method
Refinement of Pr2Ta2O7Cl2:
Parameters:
Data:
varied: all atomic positions
SEPD (IPNS)
anisotropic thermal parameters
3 scattering banks
unit cell parameters
11005 data points
background parameters
60 parameters varied
scale factor
peak shape parameters
Results:
Rwp:
0.0448
Rp:
0.0280
Re:
0.0342
S:
1.17
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10th Canadian Neutron Summer School
slide 42
June 16th, 2009
Mario Bieringer__________________________________________________________________________
Compare Single Crystal and Powder Results
Pr2Ta2O7Cl2:
space group: I2/m
X-ray single X-tal
atom
x
y
1
powder neutron (TOF)
z_________
x
y
z
:
Pr
0.6925(1)
0
0.5270(1)
0.69114(24)
0
0.5273(4)
Ta
0.0343(0)
0
0.2361(1)
0.03451(13)
0
0.23741(24)
Cl
0.8219(3)
½
0.3891(6)
0.82304(13)
½
0.39002(25)
O1
0.0448(8)
½
0.2060(18)
0.04401(18)
½
0.21012(37)
O2
0.9190(8)
0
0.0679(16)
0.91882(16)
0
0.07140(34)
O3
0.8244(9)
0
0.7519(17)
0.82666(16)
0
0.75746(32)
O4
½
½
0
½
½
0
a = 14.109(1) Å
a = 14.10763(22) Å
b = 3.9247(3) Å
b = 3.92548(5) Å
c = 6.9051(6) Å
c = 6.90610(10) Å
b = 92.953(8)0
b = 92.9626(14)0
1
2
2
U. Schaffrath and R. Gruhn Naturwissenschaften, 75 140 (1988)
P. Baudry and M. Bieringer (2003)
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10th Canadian Neutron Summer School
slide 43
June 16th, 2009
Mario Bieringer__________________________________________________________________________
Powder X-ray vs. Neutron Diffraction
Li4MgReO6:
an order-disorder study
4000
Fhkl 
powder X-RAY diffraction pattern for
Li4MgReO6
l = 1.54059Å

n 1
bn expi 2 hxn  kyn  lzn 
powder NEUTRON diffraction pattern for
Li4MgReO6
l = 1.54059Å
40000
3000
30000
Intensity (a.u.)
Intensity (a.u.)
N
2000
1000
0
20000
10000
0
20
40
60
80
100
20
2 ( )
0
Fhkl 
N

n 1
40
60
80
100
2 ( )
0
f n (2 ) expi 2 hxn  kyn  lzn 
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10th Canadian Neutron Summer School
slide 44
June 16th, 2009
Mario Bieringer__________________________________________________________________________
Powder X-ray vs. Neutron Diffraction
Li4MgReO6:
Li+
Mg2+
Re6+
O2-
neutrons
b (10-12m)
-1.90
5.37
9.2
5.803
X-rays
Z
3 (2)
12 (10)
75 (69)
8 (10)
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10th Canadian Neutron Summer School
slide 45
June 16th, 2009
Mario Bieringer__________________________________________________________________________
High Temperature Studies
Neutron diffraction and IN-SITU reactions.
Preparation of microwave dielectric oxides of composition Ba3ZnTa2O9 (BZT).
3 BaCO3 + ZnO
+
Ta2O5
----
Ba3ZnTa2O9 + 3 CO2
c
a
a
a
a
a
a
a
a
a
a
a
a
a
(a)
a
(b)
(c)
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10th Canadian Neutron Summer School
slide 46
June 16th, 2009
Mario Bieringer__________________________________________________________________________
High Temperature Studies
Only a high flux neutron diffractometer
will permit collection of diffraction
patterns within one 1 minute.
GEM (ISIS, U.K.)
TOF instrument with
more than 6000 detectors
and medium resolution
(18 – 20 m flight path).
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10th Canadian Neutron Summer School
slide 47
June 16th, 2009
Mario Bieringer__________________________________________________________________________
High Temperature Studies
3 BaCO3 + ZnO
*
+
Ta2O5
*
----
*
Ba3ZnTa2O9 + CO2
*
*
BZT
T (0C)
Rhombohedral
BaCO3and intermediate phases
Starting
materials
* =BZT
=ZnO
=Ta2O5
=BaCO3
d-spacing (Å)
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10th Canadian Neutron Summer School
slide 48
June 16th, 2009
Mario Bieringer__________________________________________________________________________
Low Temperature Studies
Crystal structure and magnetic long range order in Ba0.05Sr0.95LaMnO4
1 phase Rietveld
refinement T=125K
phase 1:
Ba0.05Sr0.95LaMnO4
crystal structure
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10th Canadian Neutron Summer School
slide 49
June 16th, 2009
Mario Bieringer__________________________________________________________________________
Low Temperature Studies
Crystal structure refinement and unit cell evolution in Ba0.05Sr0.95LaMnO4
tetragonal:
13.18
3.790
I4/mmm
a=b≠c
3.788
13.17
a-axis (A)
O2-
13.16
3.784
c-axis (A)
3.786
Mn3+
3.782
13.15
3.780
0
(Ba2+,
La3+)
Sr2+,
20
40
60
80
100
120
T (K)
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10th Canadian Neutron Summer School
slide 50
June 16th, 2009
Mario Bieringer__________________________________________________________________________
Resources
R.A. Young,‘The Rietveld Method’, IUCr Monographs on Crystallography 5, Oxford University Press, New
York (1993)
W.I.F. David, K. Shankland, L.B. McCusker, Ch. Baerlocher,‘Structure Determination from Powder
Diffraction Data’, IUCr Monographs on Crystallography 13, Oxford University Press, New York (1993)
J. Baruchel, J.L. Hodeau, M.S. Lehmann, J.R. Regnard, C. Schlenker,‘Neutron and Synchrotron Radiation
for Condensed Matter Studies’, Volume 1, 2 and 3, Springer Verlag, Berlin (1993)
G.E. Bacon, ‘Neutron Diffraction’, Claredon Press, Oxford, 3rd edition (1975)
Allen C. Larson, Robert von Dreele, ‘GSAS - General Structure Analysis System’
Juan Rodriguz-Carvajal, ‘An Introduction to the Program FULLPROF 2000’, Saclay, 2001
(ftp://ftp.cea.fr/pub/llb/divers/fullprof.2k/ )
Mark Ladd, Rex Palmer, ‘Structure Determination by X-Ray Crystallography’, Kluwer Academic, New
York, 2003, 4th Edition
Vitalij K. Pecharsky, Peter Y. Zavalij, ‘Fundamentals of Powder Diffraction and Structural Characterization
of Materials’, Springer, 2005
G.L. Squires, ‘Introduction to the theory of thermal neutron scattering’, Dover Publications, New York,
1978
Georg Will, ‘Powder Diffraction: The Rietveld Method and the Two-Stage Method’, Springer, Heidelberg,
2006
Abraham Clearfield, Joseph Reibenspies, Nattamai Bhuvanesh, ‘Principles and Applications of Powder
Diffraction’ Wiley,2008
Rietveld program downloads, manuals and tutorials can be found at: http://www.ccp14.ac.uk
More: Structures: http://www.chem.ox.ac.uk/icl/heyes/structure_of_solids/lecture1/Lec1.html
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10th Canadian Neutron Summer School
slide 51
June 16th, 2009