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Applications ofApplications of thethe RietveldRietveldmethodmethod
Thierry Roisnel
Laboratoire de Chimie du Solide et Inorganique Molculaire
UMR6511 CNRS Universit de Rennes 1 (France)
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Crystal structuresMagnetic structures (neutron data)
1.1 Isomorphic or partially known structure
structure refinement + difference Fourier maps examination
1.2 Unknown structure
1.1. Unit cell determination
1.2 Space group determination1.3 Crystal structure determination:1.4 Structure refinement (Rietveld method)
1. Structure refinement
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1.1 Unit cell determination
indexation of diffraction pattern:. Determination of cell parameters. Crystal symmetry. (hkl) Miller indices for every Bragg line
*2 2 *2 2 *2 2 *2 * * * * * *
2 . 2 . 2 .= = + + + + +hkl hklQ d h a k b l c klb c lhc a hka b
Simple problem:
Needs only a list of angular positions of Bragg lines:
. Accuracy and precision of Bragg positions !
!! . Single phase ?. Diffractometer zero-shift ?
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1.2 Space group determination
. Whole profile refinement of the diffraction pattern (FullProf):. No structural model. Cell parameters previously determined (1.1). Space group: Lau group
. Detailed examination of systematic absences:. Extinctions due to Bravais lattice. Extinctions to symmetry elements
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1.3 Crystal structure determination
. Extraction of integrated intensities + Use of traditionnalmethods used in single crystal crystallography:
. Direct methods (EXPO, SHELXS )
. Patterson map
. Direct space methods:. Genetic algorithm, Monte Carlo, Simulating annealing
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. X-rays (conventionnal):
. Instrumental resolution" weak cell distortions
. Laboratory device
. Form factor:. decrease with sin/:
" Low diffracted intensity at high angles
. Increase with Z" diffracted intensities dominated by heavy atoms
1.4 Crystal structure refinement: X-rays versusneutron data
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. Neutrons : interaction neutron with atom nuclei
. Scattering power is independent of atomic number:
" localization of light atoms (H/D, Li):" important contrast: cationic distribution (ex: Fe/Mn)
. Low absorption:" experiments versus external parameter (temperature,
pressure)
. Scattering powder is independent of sin/:" accurate thermal displacement parameters
. Gaussian profile function
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. Neutron diffraction
. Instrumental resolution
. Need a particular installation (nuclear reactor, spallation source)
. Large volume of powdered sample (2-3 cm3)
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FullProf exercices
(http://www-llb.cea.fr/fullweb/fp2k/fp2k_exercices.htm)
Exercice #1: Analysis of the spinel MgAl2O4
(ref. V. Montouillot, PhD Thesis, Univ. Orlans 1998)
Spinel structure : cubic unit cell (space group: F d 3 m)
AB2X4: A2+ (Mg, Ca, FeII, Zn) tetrahedral (Td) &B3+ (Al, FeIII, MnIII, ) octaedral (Oh) sitesX: O2-
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AB2X4 spinel structrure
Different ways to fill the Td (nTd=8) and Oh sites (nOh=4):
( ) ( )Td Oh
2 3 2 31 x x x 2 x 4A B A B O+ + + +
x=0
x=1
( ) ( )Td Oh2 3
1 2 4A B O+ +
( ) )Td Oh
3 2 31 1 1 4B A B O+ + +
" direct spinel
" inverse spinel
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Exercice #1: Analysis of the spinel MgAl2O4
Space Group: F d -3 m
Setting 1:Tetrahedral A-positions (Mg,Al) 8a (1/4,1/4,1/4)
Octahedral B-positions (Mg,Al) 16d (5/8,5/8,5/8)Oxygen positions (O) 32e (x,x,x) x=0.386
Setting 2:Tetrahedral A-positions (Mg,Al) 8a (1/8,1/8,1/8)Octahedral B-positions (Mg,Al) 16d (1/2,1/2,1/2)
Oxygen positions (O) 32e (x,x,x) x=0.261
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Exercice #1: Analysis of the spinel MgAl2O4
Two neutron powder diffraction data files:
MgAl2O4s.dat -> Sample obtained by conventional hightemperature solid state reaction
neutron powder diffractometer: 3T2 (LLB, Saclay)
. = 1.227
. Resolution parameters: U = 0.276, V=0.340, W=0.147
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MgAl2O4 exercice
Refine the structure allowing the distribution of Mn and
Al between the two available cationic sites. Calculate the value of the spinel structure:
3
3B in Td siteB total
+
+ =
Winplotr (mgal2o4s)
Introduction of a structural model: Rietveld refinement
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Introduction of a structural model: Rietveld refinement
JBT=0 cristal. Structure factor
IRF=0 automatic generation of reflections from the space group symbol!Phase 1: MgAl2O4-sharp
!Nat Dis Ang Pr1 Pr2 Pr3 Jbt Irf Isy Str Furth ATZ Nvk Npr More
5 0 0 0.0 0.0 1.0 0 0 0 0 0 0.00 0 7 0
F d 3 m
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Introduction of a structural model: Rietveld refinement
!Phase 1: MgAl2O4-sharp
!Nat Dis Ang Pr1 Pr2 Pr3 Jbt Irf Isy Str Furth ATZ Nvk Npr More
5 0 0 0.0 0.0 1.0 0 0 0 0 0 0.00 0 7 0F d 3 m
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Micro-structural features:
* apparent size of coherent domains in the direction perpendicularto the diffraction planes
* apparent strains in the direction perpendicular to the diffractionplanes:
Origin of micro-strains: . Defaults: dislocations, vacancies
. local fluctuations of composition (solid solutions )
2.Micro-structure refinement
( ) ( ) ( )x f x g x =
Observed profile Intrinsic profileInstrumental profile
h l d
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Diffracted intensity for a phase is proportional to the quantity of irradiatedmatter (absorption effects are not taken into account).
3.Quantitative phase analysis and
Rietveld method
For a multiple phase pattern:
1
( )=
= +
h, h,
hci i i
N
b I T T S
With:
SScale factor for phase
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3.Quantitative phase analysis
. .
mS
Z V
. . .m S M Z V
with m
ZV
Mass of the phase in the sample
Molecular weight
Number of molecules per unit cell
Volume of the unit cell
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3.Quantitative phase analysis
1 1
( )
.( )= =
= = N N
i i i
i i
m S ZMV
Wm S ZMV
determination of weight fraction w of every phase present in the sample
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3.Quantitative phase analysis in FullProf
Take care of:
. Sample preparation: homogeneity, number of particleswith random orientation (beware of preferential orientation)
. Correct calculation of structure factors for every phase
3 Q tit ti h l i i F llP f
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3.Quantitative phase analysis in FullProf
2
2
1 1
( ) . .
.( ) . .= =
= =
i
N N
i i i
i
ii i
i
t AS ZMV f V S
S
TZ
t
W
S ZMV f V ATZ
with
Scale factor in FullProf (refinable variable)FullProf parameter
Brindley factor (particle absorption contrastfactor).t is tabulated as a function of (i-).RFullProf parameter
S2= i ii iiZ MATZ f t
it
3 Q tit ti h l i i F llP f
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3.Quantitative phase analysis in FullProf
Used to transform the site multiplicities in PCR FullProf input file, totheir real values. For a stoichimetric phase, f= 1 if thesemultiplicities are calculated by dividing the Wyckoff multiplicity mof the site by the general multiplicity Mof the space group.Otherwise, f=occ.M/m, where occ. is the occupation number in thePCR file
if
In order to GET PROPER VALUES OF WEIGHT FRACTIONS LETTHE PROGRAM RE-CALCULATE ATZ by putting them to ZERO.The correct ATZ value to be rewritten in the PCR input file.
(First atom in the PCR file has to be fully occupied)
3 Quantitative phase analysis in FullProf
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3.Quantitative phase analysis in FullProf
1. Crystal structure has to be refined:
JBT=0 (IRF=0)
refine the structural parameters as usually
2. Crystal structure is well known:
2.1 create hkl file containing hkl list with corresponding F2(JLKH=5)
2.2 refine the pattern without entering atomic positionsJBT=-3, IRF=2 (Profile Matching mode with constantrelative intensities for the current phase, butrefinable scale factor)
Winplotr (Si3N4)
3 Rietveld Q P A
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3.Rietveld Q.P.A.
easy to operate (automatic analysis in FullProf) no internal standard
non destructive method
up to 16 phases in FullProf
polymorphism
neutron case: large amounts of powder analysis (real samples) industrial applications (cements, clays )
structure model dependent: {Fhkl} have to be known
beware of preferential orientation
3.Quantitative phase analysis:
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.Q p ydetermination of an amorphous phase
content ?
If an amorphous phase is present in the sample:S Cm m m=
Sample weight
Amorphous phase weight Crystalline partweight
Amorphous phase weight fraction:
1CA
S S
mm
m m=
3.Quantitative phase analysis:
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Q p ydetermination of an amorphous phase
content ?
. . . .m k S M Z V =
1 1
. . . .N N
C i i i i i
i i
m m k S M Z V = =
= =
For each crystalline phase:
For N crystalline phase sample:
Introduction of an internal standard in the powder:(mst
is known)
. . . .st st st st stm k S M Z V =
. . .
st
st st st st
mk
S M Z V =
3.Quantitative phase analysis:
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Q p ydetermination of an amorphous phase
content ?
Remarks:
same strategy can be applied for unknownstructure phase
difficult to manage for bulk sample
if the structure of the amorphous phase is known,a QPA can be realized through broadening linetreatment for the amorphous phase (small coherentdiffraction domains)
3 Rietveld Q P A : -Si N4 and Si N4 mixture
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3.Rietveld Q.P.A.: -Si3N4 and _Si3N4 mixture
Neutron data: Si3N4: 93% wgtSi3N4: 7% wgt
3 Rietveld Q P A : -Si3N4 and Si3N4 mixture
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3.Rietveld Q.P.A.: Si3N4 and _Si3N4 mixture
X-rays data: Si3N4: 91.6% wgtSi3N4: 8.4% wgt
Some references on Q P A by Rietveld method
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Some references on Q.P.A.by Rietveld method
R.J. Hill & C.J. Howard, J. Appl. Cryst. 20, 467-476 (1987)Quantitative phase analysis from neutron powder diffraction data using the
Rietveld method
G.W. Brindley, Phil. Mag. 36, 347-369 (1945)The effect of grain or particle size on X-ray reflections from mixedpowders and alloys considered in relation to the quantitative determination ofcrystalline substances by X-ray methods
D.L. Bish & S.A. Howard, J. Appl. Cryst. 21, 86-91 (1988)Quantitative phase analysis using the Rietveld method
J.C. Taylor, Powder Diffraction 6, 2-9 (1991)Computer programs for standardless quantitative analysis of minerals
using the full powder diffraction profile
R.J. Hill, Powder Diffraction 6, 74-77 (1991)
Expanded use of Rietveld method in studies of phase abundance inmultiphase mixtures