Short History of Powder Diffractionw3.esfm.ipn.mx/~fcruz/ADR/Cristalografia/Roisnel... ·...

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Mérida November 2003 Taller intensivo: El método de Rietveld, FullProf

Short History of Powder Diffraction

1895 Discovery of X-rays by Roentgen 1912 von Laue demonstrates that X-rays can be diffracted by crystals1935 Le Galley constructs the first X-ray powder diffractometer1947 Phillips introduces the first commercial powder diffractometer1950’s Powder diffraction used primarily to study structural 1960’s imperfections, phase identification, … largely by metallurgists

and mineralogists1969 Hugo Rietveld develops a method for whole pattern analysis of

neutron powder diffraction data1977 Cox, Young, Thomas and others first apply Rietveld method

to synchrotron and conventional X-ray data


Commercial Powder Diffractometer


Uses of Powder DiffractionQualitative Analysis Phase IdentificationQuantitative Analysis Quantitative

Lattice Parameter(indexing&refinement)

Phase Fraction AnalysisStructure Determination Structure

Reciprocal Space Methods Real Space Methods

Structure Refinement StructureRietveld Method

Peak Shape Analysis Peak Shape Crystallite Size DistributionMicrostrain AnalysisAnti-phase domains, stacking faults,…


Limitations of Powder Diffraction for solving crystal structures

• Single crystal diffraction allows solving crystal structures by using appropriate mathematical algorithms and accurate peak intensities.•The 3D set of reflections obtained from a single crystal experiment is condensed into 1D in powder diffraction pattern. This leads to both accidental and exact peak overlap, and complicates the determination of individual peak intensities.• Indexing may be a bottleneck for starting to solve a crystal structure. Multiphase mixtures complicates the task.• Crystal symmetry cannot be obtained directly from powder diffraction patterns.•Preferred orientation leads to biased peak intensities.


Applications of powder diffractionPhysics, Chemistry, Geology, Industry,...

• Phase identification (mineralogy)

• Quantitative analysis (chemical and phase analysis)

• Texture determination of polycrystalline materials

• Residual stress analysis

• Structure determination

• Microstructure and defects

• Structural behaviour under complex environments


Neutron/Synchrotron Powder Diffraction for what?

High precision crystallography with powders in complex environments: high pressure, low and high temperatures. Structural studies of phase transition and transformations

Real crystals: defects, diffuse scattering, total scattering analysis.

Magnetic structures and phase diagrams (NEUTRONS)

Time-dependent phenomena: in situ powder diffraction


Solid State Physics and Chemistry: recent hot topics where powder diffraction has played a

fundamental role

• Transition Metal and Rare Earth Mixed Oxides

• High Tc superconductors and related materials

• Metal-Insulator transitions

• Giant Magneto-Resistance materials

• Low-dimensional systems


Neutron and synchrotron Application of powder diffraction

• Hydrogen in metals• Magnetic intermetallics• Molecular solids and fullerenes• Composite and technological materials:

cements, ceramics, zeolites, catalysts, opto-electronic• Chemical reactions: solid-solid, liquid-solid, gas-solid• Pharmaceuticals compounds and bio-compatible materials• Supramolecular magnetic solids• Ionic conductors, solid electrolytes. Chemical processes in

solid state batteries.• Aperiodic materials: quasicrystals, modulated structures.• Diluted magnetic semiconductors.


Instruments


Scheme of a two axis diffractometer

Collimators α1, α2, α3

Mosaicity of monochromator βM

« take-off » angle : 2θM

Parameters determining resolution and intensity


ID31- cryostat


ID31- cryostream


D1B- overview


D1B- collimators


D2B- overview


A powder diffraction pattern can be recorded in numerical form for a discrete set of scattering angles, times of flight orenergies. We will refer to this scattering variable as : T. The experimental powder diffraction pattern is usually given as three arrays :

The profile can be modelled using the calculated counts: yciat the ith step by summing the contribution from neighbouring Bragg reflections plus the background.

1,2,..., ,i i i i n

T y σ=


yi

Position “i”: Ti

Bragg position Th

yi-ycizero

Powder diffraction profile:scattering variable T: 2θ, TOF, Energy


∆(Q)=2π2 j / (Q2 Vo) ≈ p DQ

Average separation (in Å-1) between adjacent reflections

j General multiplicity of the Laue classVo Volume of the primitive cell in Å3

Resolution Function in reciprocal spaceDQ= FWHM in Å-1

of Bragg peaks

Well separated reflections means their

maxima are distant more than pDQ


Constant wavelength neutron powder diffraction

• Resolution function determined by optical diffractometer arrangements and collimators. Coupled to scattering angle and wavelength. • Limited Q-range: depend on the selected wavelength. In a hot source (low lambda) the gain in Q-range is degraded by lower resolution.• Difficult to obtain high resolution in the whole Q-range in a single instrument.• Simple data treatment. Minimal corrections of the raw data before processing.• Simple model of peak shape and faster calculations in data analysis.


Time of Flight neutron powder diffraction

• Resolution function determined by the pulse width and the flight path.• Large Q-range with excellent resolution for long flight path instruments.• Simple to obtain high resolution in the whole Q-range in a single instrument, using several banks. • Spectro-diffractometers allows to study the dynamics together with the structural aspects.• Complex data treatment. Important corrections of the raw data before processing.• Complex models of peak shape and lengthy calculations in data analysis.


Neutrons versus Synchrotron: advantages and drawbacks for Powder Diffraction

Constant scattering length. Contrast.Low absorption: easy sample environmentMagnetic structuresHigh precision in structure refinementModerate resolution

Extremely high resolutionSubtle distortionsIndexing and Structure determinationAnomalous scatteringTexture effects

Neutrons Synchrotron X-rays


How and where finding information about powder

diffraction?


Downloading of SoftwareThe above software and resources can be downloaded via the CCP14

site:http://www.ccp14.ac.uk

Graphical tutorial run-through of most of this software is located via (“look before you try”):

http://www.ccp14.ac.uk/tutorial/

Freely available softwaretools for crystallography

Lachlan M.D. Cranswick ([email protected])CCP14 Project for Single Crystal and Powder Diffraction

http://www.ccp14.ac.uk [email protected]


Armel Le Bail page: http://sdpd.univ-lemans.fr


Some equations and formalism


In scattering experiments, the incident particle (neutron, electron, photon...) experiences a change in its momentum and energy.

(h/2π)Q =(h/2π)(kF-kI)= h s hν=EF-EI

In the following we shall be concerned with elastic scattering (hν=0) for which kF=kI= 2π/λ and Q= Q = (4π/λ) sinθ , θ being half the scattering angle.

“crystallographic scattering vector”: s = Q/2π.

Some equations and formalism (1)

kI=2π/λ uI

kF=2π/λ uFQ= kF - kI

2θ


In the kinematics theory (first Born approximation), the amplitude of the wave scattered by an object is the Fourier transform (FT) of its scattering density (SD) ρ(r) measured in cm-2. Any object can be considered as constituted by atoms of SD ρaj(r) centred at positions Rj; the SD and the corresponding scattered amplitude and intensitycan be written as:Scattering density: ρ(r) =∑j ρaj(r-Rj)Scattered amplitude: A(s)=FT[ρ(r) ]

A(s)= ∑jexp2πisRj∫ρaj(u) exp2πisud3u

A(s) = ∑ j fj(s) exp2πi s Rj

I(s) = A(s)A(s)* = ∑ i ∑ j fi(s) fj(s)* exp2πi s (Ri-Rj)



Magnetic scattering of neutrons

Dipolar interaction (µn , m): vector scattering amplitude

( ) ( ) ( )2

12

Q m Qa Q Q mM er f

Qγ

= −

kI=2π/λ uI

kF=2π/λ uF

Q= kF - kI


Magnetic scattering of neutrons

( ) ( ) ( ) ( )2

12

Q m Qa Q Q m Q mM er f p f

Qγ ⊥

= − =

( ) ( ) 3exp( )Q r Q r rmf i dρ= ∫m

m⊥

Q=Q eOnly the perpendicularcomponent of m to Q=2πh

contributes to scattering

p=0.2696 10-12 cm


For powders, we have to average the intensity for all possible orientations of an object with respect to the incident beam. Theintensity depends on the length, s, of s and the whole set of inter-atomic distances Rij=Ri-Rj; and is given by the Debye formula in terms of Q=2πs:

I(Q) = ∑i ∑ j fi fj sinQRij /(QRij)


Debye Formula


Total scattering density of the infinite object can be decomposed in the following way:

ρ∞(r) =∑n ρc(r-Rn) = ρc(r) * ∑n δ(r-Rn) = ρc(r) * z(r)

For a finite crystal z(r) must be replaced by z(r)g(r) , where g(r) is the shape factor of the crystal defined as g(r)=1 for r inside the crystal, and g(r)=0 for r outside. We define G(s)=FTg(r). The scattered amplitude for a finite crystal is:

A(s)=FTρf(r) Z(s) = FTz(r) = 1/Vc ∑H δ(s-H)

A(s)= FTρc(r) * z(r)g(r) = F(s)Z(s) * G(s) = F(s)/Vc∑HG(s-H)

Laue conditions:s=H ⇒Bragg Law (s=s=2sinθ/λ, H=1/dhkl)

Some equations and formalism (4)Scattering Amplitude of a Finite Crystal


I(s)=F2(s)/V2c ∑ HG(s-H) ∑ H'G*(s-H') ≈ NF2(s)/(VVc)∑HG2(s-H)

G2(s) is Fourier transform of the auto-correlation function:

V.η(r)=∫g(u)g(r+u)d3u

rThe interpretation of η(r) is straightforward: it represents the fraction of the total volume shared in common between the object and its "ghost" displaced by the vector r. Obviously, η(0)=1 and decreases as r increases.

Some equations and formalism (5)Scattered Intensity of a Finite Crystal


Defects: average lattice + structure factor depending on unit cell

I(s) =∑n(∑ m FmF*m+n) exp2πi s Rn

Fm is the structure factor of the cell m. Taking into account the long range homogeneity of the object, the average value pn=⟨FmF*

m+n⟩ is independent of m. The number of terms in the inner sum is given by Vη(Rn)/Vc and the equation can be transformed to:

I(s) = V/Vc ∑ nη(Rn) ⟨FmF*m+n⟩ exp2πi s Rn=

= N ∑nη(Rn) pn exp2πi s Rn

Some equations and formalism (6)Scattered Intensity of a Real Crystal


Average structure factor: F= ⟨F⟩ =1/N ∑mFm

local fluctuation of the structure factor: φn=F-Fn

pn=F2+⟨φmφ*m+n⟩ = F2+Φn,

The intensity formula can be transformed to:

I(s) = IBragg + IDiffuse =

N F2 ∑ nη(Rn) exp 2πi s Rn + N ∑ nη(Rn) Φn exp2πi s Rn =

N F2/(VVc) ∑ HG2(s-H) + N ∑ n Φn exp2πi s Rn

In the last expression, we have made the approximation η(Rn)=1 because

Φn decreases with n faster than η(Rn).

Some equations and formalism (7)Scattered Intensity of a Real Crystal


Fm=F exp2πi s um, with the constraint: ∑ mexp2πi s un=0

Defining: ζ(Rn, s)= ⟨exp2πis(um-um+n)⟩.

The intensity formula can be transformed to:

I(s) = I(s) = N F2 ∑ nη(Rn) ζ(Rn, s) exp2πi s Rn

If s=H+∆s the scattered intensity around a Bragg peak is given by:

IH(∆s) =N F2H ∑nη(Rn) ζH(Rn) exp2πi ∆s Rn ≈ F2

H Ωx(∆s)

size strain

Some equations and formalism (8)Strained Crystals


Reciprocal space associated to an Ideal Powder: set of spherical

shells whose intersections with the Ewald sphere produces scattering

distributed in cones having as a common axis the incident beam.

Single crystal intensity around a single Bragg peak:

IH(∆s) ≈ F2H Ωx(∆s) 3D

For a powder (after spherical averaging), set of equivalent nodes :

IH(∆s) ≈ ∑H F2H ΩH (∆s) 1D

Some equations and formalism (9)Powder Average


The Rietveld Method for refinement of crystal and

magnetic structures


profile

The model to calculate a powder diffraction pattern is:

( )h hh

ci i iy I T T b= Ω − +∑( ) 1x dx

+∞

−∞Ω =∫

Profile function characterized by its full width at half maximum (FWHM=H)and shape parameters (η, m, ...)

( ) ( ) ( )x g x f x instrumental intrinsic profileΩ = ⊗ = ⊗

The profile of powder diffraction patterns


The profile of powder diffraction patterns

( )h hh

c i iiy I T T b= Ω − +∑Contains structural information: atom positions, magnetic moments, etc( )h h II I= β

( , )h PixΩ = Ω β Contains micro-structural information: instr. resolution, defects, crystallite size, ..

( )Bi ib b= β Background: noise, diffuse scattering, ...


, ,h

( )h hci i iy s I T T bφ φ φφ

= Ω − +∑ ∑

Several phases (φ = 1,nφ) contributing to the diffraction pattern

, ,h

( )h h

p p p p pci i iy s I T T b

φφ φφ

= Ω − +∑ ∑

Several phases (φ = 1,nφ) contributing to several (p=1,np) diffraction patterns


The Rietveld Method consist of refining a crystal (and/or magnetic) structure by minimising the weighted squared difference between the observed and the calculated pattern against the parameter vector: β

22

1( )

n

i i cii

w y yχ β=

= −∑

21i

iwσ

=2iσ : is the variance of the "observation" yi


Least squares: Gauss-Newton (1)Minimum necessary condition:

A Taylor expansion of around allows the application of an iterative process. The shifts to be applied to the parameters at each cycle for improving χ2 are obtained by solving a linear system of equations (normal equations)

2

0∂=

∂χβ

( )icy β 0β

0

0 0

0

( ) ( )

( )( )

A b

ic ickl i

i k l

ick i i ic

i k

y yA w

yb w y y

β β

β

=

∂ ∂=

∂ ∂∂

= −∂

∑

∑

βδ

β β

β


Least squares: Gauss-Newton (2)

The new parameters are considered as the starting ones in the next cycle and the process is repeated until a convergence criterion is satisfied. The variance of the adjusted parameters are calculated by the expression:

The shifts of the parameters obtained by solving the normal equations are added to the starting parameters giving rise to a new set

01 0= + ββ β δ

1( ) ( )Ak kk

N - P+C

2 2ν

22

ν

σ β χ

χχ

−=

=


Least squares: a local optimisation method

• The least squares procedure provides (when it converges) the value of the parameters constituting the local minimum closest to the starting point

• A set of good starting values for all parameters is needed

• If the initial model is bad for some reasons the LSQ procedure will not converge, it may diverge.


Profile R-factors used in Rietveld Refinements


R-factors and Rietveld Refinements (1)

, ,

,

100obs i calc i

ip

obs ii

y yR

y

−=

∑∑

1/ 22, ,

2,

100i obs i calc i

iwp

i obs ii

w y yR

w y

−

=

∑

∑

R-pattern

R-weighted pattern

1/ 2

2,

( )100expi obs i

i

N P CRw y

− + =

∑

Expected R-weighted pattern



Reduced Chi-square

Goodness of Fit indicator

2

2 wp

exp

RRνχ

=

wp

exp

RS

R=



Two important things:

• The sums over “i” may be extended only to the regions where Bragg reflections contribute

• The denominators in RP and RWP may or not contain the background contribution


Crystallographic R-factors used in Rietveld Refinements

, ,

,

' '100

' '

obs k calc kk

Bobs k

k

I IR

I

−=

∑∑

Bragg R-factor

, ,

,

' '100

' '

obs k calc kk

Fobs k

k

F FR

F

−=

∑∑

Crystallographic RF-factor.


Crystallographic R-factors used in Rietveld Refinements

Provides ‘observed’integrates intensities for calculating Bragg R-factor

In some programs the crystallographic RF-factor is calculated using just the square root of ‘Iobs,k’

,, ,

,

( )( )' '

( )i k obs i i

obs k calc ki calc i i

T T y BI I

y B Ω − − = −

∑

,,

' '' ' obs k

obs k

IF

jLp=


The structural information contained in the integrated

intensities


2h h

I L pO ACF=

Integrated intensities are proportional to the square of the structure factor F. The factors are: Lorentz-polarization (Lp), preferred orientation (O), absorption (A), other “corrections” (C) ...


The Structure Factor contains the structural parameters

( ) ( ) 1

2h h h t rn

j j j jsj s

F O f T exp i Sπ=

= ∑ ∑

( , , ) ( 1, 2, ... )r j j j jx y z j n= =

( ) ( ) ( ), , , , 1, 2, ...h Ts s Gs

h k l h k l S s N= = =

sinexp( )j jT B2

2

θλ

= −


Structural Parameters(simplest case)

( , , )r j j j jx y z= Atom positions (up to 3nparameters)

jO Occupation factors (up to n-1 parameters)

jB Isotropic displacement (temperature) factors (up to n parameters)


Structural Parameters(complex cases)

As in the simplest case plus additional (or alternative) parameters:

• Anisotropic temperature (displacement) factors

• Anharmonic temperature factors

• Special form-factors (Symmetry adapted spherical harmonics ), TLS for rigid molecules, etc.

• Magnetic moments, coefficients of Fourier components of magnetic moments , basis functions, etc.


Magnetic scattering:

Intensity (non-polarised neutrons)

Magnetic interaction vector:

*hhhhh MM ⊥⊥ ⋅+= *NNI

( )( ) ( ) ( )( )hMeehMehMeM h ⋅−=××=⊥

kHh += ⇐ Scattering vector


( ) ( ) ( ) 1

2k kM h h S H k t rn

j j j js j j jssj s

p O f T M exp i Sπ ψ=

= + − ∑ ∑

The magnetic structure factor:

Magnetic structures

2kk

m S kRlj j lexp iπ= −∑


Electronic crystallization in a Li battery material: columnar ordering of electron and

holes in the spinel LiMn2O4

J. Rodríguez-Carvajal, G. Rousse, Ch. Masquelier and M. HervieuPhysical Review Letters, 81, 4660 (1998)

LiTd [Mn2]Oct.O4 : Mn3.5+

High temperature: mixed valence state


LiMn2O4 : Electron Diffraction

320 K 230 K

J. Rodríguez-Carvajal et al, PRL, 81, 4660 (1998)

Cubic Fd3ma = 8,248 (1) Å

Orthorhombic Fddda = 24.7435(5) Åb = 24.8402(5) Åc = 8.1989(1) Å

x 9

0 12 0

12 0 0

040

400


30 40 50 60 70 80

230K

350K

2 theta (°)

LiMn2O4 : Neutron Diffraction, 3T2 (LLB)

Charge ordered state

Orthorhombic Distortion Superstructure reflections


<Mn-O> = 1,915(4) ÅMn(5) = 3.90+

∆ = 6.1

a

bc

<Mn-O> = 1,903(4) ÅMn(4) = 4.02+

∆ = 4.6

64 « Mn4+ »80 « Mn3+ -like»

8 delocalised holes

<Mn-O> = 1,996(4) ÅMn(2) = 3.27+

∆ = 19.4

<Mn-O> = 2,020(5) ÅMn(3) = 3.12+

∆ = 36.6

<Mn-O> = 2,003(2) ÅMn(1) = 3.20+

∆ = 20.6

LiMn2O4 : Partial Charge Ordering


b

222

33

33

33

33

22

22

22

22

1 133

33

33

33

22

22

1

22

22

33

33

33

33

22

22

122

22

33

33

33

33

1

22

22

1

a

2

1 1 1 1 1

1 1

1 1

1 1

1 1 1 1 1

111

x

x x

x xx

x

x

x xx x

xxx x x x x

x x

x

x

xxxx

x x x

xx

x

xx x

x x

x

x xx

xxxxxx

x x x x x xx x x x x x

x

x x

xx

xx

x

x

x

x

x

Mn3+

Mn4+

X/ Li+

c

Mn(3) Mn(2)

LiMn2O4 : Partial Charge Ordering


Nature of the phase transition in LiMn2O4

Nature of the phase transition in Nature of the phase transition in LiMnLiMn22OO44

The structural transition is driven by a charge ordering process. The Coulomb repulsion is however of secondary importance compared to electron-lattice coupling via theJahn-Teller effect

The structural distortions and average distances around the Mn ions support the atomic scale charge ordered state below the transition temperature.

The distorted pyrochlore lattice of Mn-ions is then half doped with a partial charge ordering at low temperature. The magnetic ground state is incommensurate and very complex.


Behaviour of Rietveld R-factors, and other indicators, versus counting statistics for

perfect and biased models


0

5

10

15

20

25

30

35

40

2,0 2,5 3,0 3,5 4,0 4,5 5,0 5,5

R-factors (RP, R

WP, R

Bragg and R

F)

cRp(B)cRwp(B)RbRF

R(%

)

Log(Counting Time)

NIntdp=16, Nprof=8, Nref=393Neff=(173, 244, 301)

Correct model

RM (systematic errors): Behaviour of R-factors versus counting time (correct model)


RM (systematic errors): Behaviour of RWP factors versus counting time (biased peak shape)

0

10

20

30

40

50

2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5

R-factors (RWP

) RwpcRwpRwp(B)cRwp(B)R

WP(%

)

Log(Counting Time)

NIntdp=16, Nprof=8, Nref=393Neff=(173, 244, 301)Biased peak shape


0

2

4

6

8

10

12

14

2,0 2,5 3,0 3,5 4,0 4,5 5,0 5,5

R-factors (RBragg

and RF)

RbRF

R(%

)

Log(Counting Time)


RM (systematic errors): Behaviour of RBragg and RFfactors versus counting time (biased peak shape)


RM (systematic errors): Behaviour of reduced Chi-square versus counting time (correct model)

0,600

0,800

1,000

1,200

1,400

2,0 2,5 3,0 3,5 4,0 4,5 5,0 5,5

R-factors(χ 2)

Chi2Chi2(B)

Red

uced

χ2

Log(Counting Time)

NIntdp=16, Nprof=8, Nref=393Neff=(173, 244, 301)

Correct model


0

10

20

30

40

50

60

2,0 2,5 3,0 3,5 4,0 4,5 5,0 5,5

R-factors(χ2)

Chi2Chi2(B)

Red

uced

χ2

Log(Counting Time)


RM (systematic errors): Behaviour of reduced Chi-square versus counting time (biased peak shape)


The peak shape function of powder diffraction patterns


The peak shape function of powder diffraction patterns contains the Profile Parameters

(constant wavelength case)

( , ) ( , )h P h Pi ix T Tβ βΩ = Ω −

( ) 1x dx+∞

−∞Ω =∫

The cell parameters are included, through Th, within the profile function. They determine the peak positions in the whole diffraction pattern.


2( ) exp( )G GG x a b x= −

2

2 ln2 4ln2G Ga b

H Hπ= =

Gaussian function

Integral breadth:1

2 ln 2G

Ha

πβ = =


2( )1

L

L

aL xb x

=+

2

2 4L La b

H Hπ= =

Lorentzian function

Integral breadth:1

2L

Ha

πβ = =


Fwhm

BG

x0

x

I

Comparison of Gaussian and Lorentzian peak shapes of the same peak height “I” and same width “Fwhm”


Convolution properties of Gaussianand Lorentzian functions

1 2 1 2

2 21 2 1 2

( , ) ( , ) ( , )

( , ) ( , ) ( , )

L x H L x H L x H H

G x H G x H G x H H

⊗ = +

⊗ = +

( , ) ( , ) ( , , )L G L GL x H G x H V x H H⊗ =


( ) ( ) (1 ) ( )pV x L x G xη η′ ′= + −The pseudo-Voigt function

The Voigt function

( ) ( ) ( ) ( ) ( )V x L x G x L x u G u du+∞

−∞= ⊗ = −∫( ) ( , , ) ( , , )L G L GV x V x H H V x β β= =

( ) ( , , )pV x pV x Hη=


Mapping: Pseudo-Voigt ⇔ VoigtThomson-Cox-Hasting formulation

( , ) ( , )G LH F H Hη =

1( , ) ( , )G LH H F H η−=

5 4 3 2 2 3 4 5( 2.69269 2.42843 4.47163 0.07842 )G G L G L G L G L LH H H H H H H H H H H= + + + + +2 3

1.36603 0.47719 0.11116L L LH H HH H H

η = − +

2 30.72928 0.19289 0.07783LHH

η η η= + +

2 3 1/ 2(1 0.74417 0.24781 0.00810 )GHH

η η η= − − −


2 22tan tan

cosG

GIH U V Wθ θ

θ= + + +

tancosL

YH X θθ

= +

Profile Parameters(simple cases)

Parameters controlling the Full-Width at half maximumU, V, W, IG, X, Y


Microstructural effectsAnisotropic peak

broadening


2 2 2 22( (1 ) ) tan tan

cosG

G STIH U D V Wξ θ θ

θ= + − + + +

[ ( )]( ) tancos

ZL ST

Y F SH X Dξ θθ

+= + +

Modeling the Gaussian and Lorentziancomponents of the profile function in terms of microstructural parameters


xy

LhΩ

h

The intrinsic profile of a particular reflection due to size effect has an integral breadth , the Scherrer formula:

gives the volume-averaged apparent size of the crystallites in the direction normal to the scattering planes. This apparent size has a perfectly defined physical interpretation:

in terms of the normalized column-length distributionpV(L):

*

1cosV

S S

D λβ θ β

= =

( ) 3

1,...

1 1 ,i

Vi N i C

D L x y dN V=

= ∑ ∫∫∫ h r

0

( )VVD L p L dL

∞

= ∫

Sβ

Anisotropic broadening due to size effects


Special expressions for anisotropic broadening due to size effects:

Microstructural effects are simulated in the Rietveld Method using the Voigt approximation(Langford & Louër).

• Infinite needles • Infinite platelets• Finite cylinders


Spherical harmonics to simulate the average form of crystallites

( )cos1 cos ; /sinlmp lm

lmp

ma P p

mDΦ

= Θ = + − Φ ∑ h

hhh

( ),Θ Φh h : Polar angles of reciprocal vector h w.r.t. crystal frame

( )cos

cossincos lmp lm

lmp

mkFWHM a Pm

λθ

Φ = Θ Φ

∑ hh

h


The intrinsic profile of a particular reflection due to a straineffect has an integral breadth , the apparent strain is defined as

We shall use the so called maximum strain, that is derived from the apparent strain as:

cotDη β θ=Dβ

14 2 D

de dd

η β∆= = =


Anisotropic strain broadening

Phenomenological model: strains considered as fluctuations and correlation between metric parameters J. Rodríguez-Carvajal et al (J. Phys. Cond. Matt. 3, 3215 (1991)

( )2

1 ;hkl ihkl

M M hkld

α= =

The metric parameters αi (direct, reciprocal or any combination) are considered as stochastic variables with a Gaussian distributioncharacterized by :• the mean ⟨ αi⟩ and • the variance-covariance matrix Cij


The mean and the variance of the function Mhklare given by (JRC et al , J. Phys. Cond. Matt. 3, 3215 (1991) ):

( );hkl iM M hklα=

( )2

,hkl ij

i j i j

M MM Cσα α

∂ ∂=

∂ ∂∑If the metric parameters are taken as the coefficients of the quadratic form: 2 2 2

2

1

hkl

Ah Bk Cl Dkl Ehl Fhkd

= + + + + +

( )

2

4

H K Lhkl HKL

HKLH K L

M S h k lσ+ + =

= ∑P. W. Stephens,J. Appl. Cryst. 32, 281 (1999)

Cij contains 21 parameters, 15 independent

αi=A,B,C,D,E,F


S_400 S_040 S_004 S_22022.04(78) 17.74(57) 0.016(2) -38.8(1.2)Lorentzian Parameter: 0.093(2)

Nd2NiO4, LT

A-strain h k l43.4585 0 1 248.1172 1 0 2 7.1018 1 1 0 5.9724 1 1 1 4.1383 1 1 2 9.7952 0 0 4 4.0162 1 1 379.5271 0 2 087.5578 2 0 0


Portion of the neutron diffraction pattern of Pd3MnD0.8 at room temperature obtained on 3T2 (LLB, λ = 1.22 Å). On top, the comparison with the calculated profile using the resolution function of the instrument. Below the fit using IsizeModel = -14. Notice that only the reflections with indices of different parity are strongly broadened. An isotropic strain, due to the disorder of deuterium atoms, is also included for all kind of reflections.


Simulated Annealing ….


Classically, crystal structure determination is considered as a process to determine the “phases” of the structure factors

( )

( ) ( ) ∑

∑Φ+−=

−=

hhh

hh

hrr

hrr

iexpF

iexpF

πρ

πρ

2

2

For a centrosymmetric structure is 0 or 1/2

The knowledge of all phases for the measured structure factors provides a density map from which the structure is derived (chemically recognised).

hΦ


The resolution of the “phase problem” is the goal of the crystal structure determination methods

Direct Methods tackle the problem looking for phase relations (tangent formula) between

structure factors of different reflections

Direct methods need a high number of reflections and good resolution (powders)

Direct methods are generally very efficient


But …sometimes direct methods fail in

solving particular structuresor

cannot be applied because poor data quality (low resolution)


Structure factor calculation

If the chemical composition and the space group are known we have to determine just the 3n variables

( ) ( ) [ ] ∑∑=

=s

js

n

jjjj SiexpTfOF rthhh π2

1

( ) ( )n,...,jz,y,x jjjj 21==r

( ) [ ] rs

js

n

jjjjrobs SiexpTfOF ∑∑

=

≈ rthh π21

( ) ( )N,...,rl,k,h rr 21==h


Direct space methods:

•Look directly for atom positions explaining the experimental data

•Minimize a reliability factor with respect to the

“configuration vector” or “chromosome”

nnn z,y,x,...z,y,x,z,y,x 222111=ϖ

( ) ( ) ( )∑=

−=N

rrcalcrobs,FFcR

1

22 ϖϖ hh


Solving Crystal structures by Powder diffraction and direct space methods(1) Indexing the

powder diffraction pattern: DICVOL, TREOR, ITO,…(WinPLOTR)

(2) Extracting integrated intensities (FullProf)

Space Group determination

(4) If ExPo fails thenUse Simulated

Annealing (FullProf) or Genetic Algorithms

(3) Use ExPo to solve the structure


Simulated Annealing (SA):

The SA method is a general purpose optimization techniquefor large combinatorial problems introduced by:

Kirpatrick, Gelatt and Vecchi, Science 220, 671-680 (1983).

The function, E(ω) to be optimized with respect to the configuration described by the vector state ω is called the “cost” function.


Simulated Annealing (SA):

The SA method applied to structural problems:

•J. Pannetier, J. Bassas-Alsina, J. Rodríguez-Carvajaland V. Caignaert, Nature 346, 343-345 (1990)

•J.M. Newsam, M.W. Deem and C.M. Freeman, Accuracy in Powder Diffraction II. NIST Special Publ. No. 846, 80-91 (1992)

•J. Rodríguez-Carvajal, Physica B 192, 55-69 (1993)


beginInitialise (set to zero useful quantities, do preliminary calculations )τ = 1do

doPerturb the system:ωold → ωnew, ∆=E(ωnew)-E(ωold)

if ∆ ≤ 0 then accept, elseif exp(-∆/Ττ) > random[0,1] then acceptif accept then Update (replace ωold by ωnew)

until equilibrium is approached closely enough (Ncyc)Tτ+1 = f(Tτ) (decrease temperature, usually Tτ+1 = q Tτ, q≈0.9)τ = τ + 1

until stop criterion is true (maximum τ, convergence, low % accepted...)end

The Simulated Annealing Algorithm


COMM Ab initio structure solution of PbSO4 (Simulated Annealing, data D1A-ILL)! Files => DAT-file: pb_san, PCR-file: pb_san!Job Npr Nph Nba Nex Nsc Nor Dum Iwg Ilo Ias Res Ste Nre Cry Uni Cor Opt Aut

1 0 1 0 0 0 0 0 0 0 0 0 0 12 3 0 0 0 0!Ipr Ppl Ioc Mat Pcr Ls1 Ls2 Ls3 Syo Prf Ins Rpa Sym Hkl Fou Sho Ana

0 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 0!!NCY Eps R_at R_an R_pr R_gl Thmin Step Thmax PSD Sent0

1 0.10 1.00 1.00 1.00 1.00 15.0000 0.0200 120.0400 0.000 0.000!

12 !Number of refined parameters!-------------------------------------------------------------------------------! Data for PHASE number: 1 ==> Current R_Bragg for Pattern# 1: 7.86!-------------------------------------------------------------------------------PbSO4

!!Nat Dis Mom Pr1 Pr2 Pr3 Jbt Irf Isy Str Furth ATZ Nvk Npr More

5 0 0 0.0 0.0 1.0 0 4 0 0 0 0.00 0 7 0!P n m a <--Space group symbol!Atom Typ X Y Z Biso Occ In Fin N_t Poi /CodesPb PB 0.81174 0.23348 0.83479 1.42124 0.50000 0 0 0 0

11.00 21.00 31.00 0.00 0.00S S 0.93358 0.23348 0.32454 0.41603 0.50000 0 0 0 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


! Sc1 Sc2 Sc3 Sc4 Sc5 Sc61.531 0.000 0.000 0.000 0.000 0.000

0.00 0.00 0.00 0.00 0.00 0.00! a b c alpha beta gamma

8.485130 5.402066 6.964059 90.000000 90.000000 90.0000000.00000 0.00000 0.00000 0.00000 0.00000 0.00000

! Limits for selected parameters (+ steps & BoundCond for SA):1 0.0000 1.0000 0.0500 1 x_Pb2 0.0000 1.0000 0.0500 1 y_Pb_SO13 0.0000 1.0000 0.0500 1 z_Pb4 0.0000 1.0000 0.0500 1 x_S5 0.0000 1.0000 0.0500 1 z_S6 0.0000 1.0000 0.0500 1 x_O17 0.0000 1.0000 0.0500 1 z_O18 0.0000 1.0000 0.0500 1 x_O29 0.0000 1.0000 0.0500 1 z_O210 0.0000 1.0000 0.0500 1 x_O311 0.0000 1.0000 0.0500 1 y_O312 0.0000 1.0000 0.0500 1 z_O3

! T_ini Anneal Accept NumTemps NumThCyc InitConf8.000 0.900 0.008 60 0 0

! NCyclM Nsolu Num_Ref Nscalef Algor150 1 71 1 2


Extracting Magnetic Intensities


0.00

0.20

0.40

0.60

0.80

1.00

0 5 10 15 20 25

LiMn2O

4

Ph_Mn2a1Ph_Mn2a2Ph_Mn2a3Ph_Mn2a4Ph

ases

(mod

2π)

1/T

Behavior of parameters in Simulated Annealing runs


Average step ...

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 10 20 30 40 50 60

LiMn2O

4

<Step>

<Ste

p>

t


Examples


Crystal structure of DFe2(P2O7)2 (D:Sr,Pb)

Pb

O22

O12

O22

O14

O23

Pb

O13

O14

O22

O12

O21O23

O13

O13

O23

O14

O12

O22

O12

O21O23

O13

O14

O22

O12

O13

a

b

c

Pb +2

Fe+3

P+5

O-2

_Triclinic P 1

a=4.7982(2) Å

b=7.1125(2) Å

c=7.8360(3) Å

α=89.816(2) Å

β=87.590(2) Å

γ=73.132(2) Å


Magnetic susceptibility of PbFe2(P2O7)2

0

10

20

30

40

50

60

-100 -50 0 50 100 150 200 250 300

1/χ

(g/e

mu)

Temperature

PbFe2(P

2O

7)2

0

0.01

0.02

0.03

0 10 20 30 40 50

χ

Temperature


Observed versus calculated neutron powder diffraction patterns of SrFe2(P2O7)2 on 3T2

Room temperature, 3T2λ= 1.225 Å

1228 reflections, 55 free parameters


Observed versus calculated neutron powder diffraction patterns of PbFe2(P2O7)2 on G42

Room temperature, G4.2λ= 2.343 Å

285 reflections, 46 free parameters


Observed neutron powder diffraction patterns of SrFe2(P2O7)2 on G41 as a function of temperature

(001) (010) (011) & (0 1 -1)

(100)T= 1.4 KT= 6.0 KT=11.0 KT=16.0 KT=19.0 KT=22.0 KT=25.0 KT=28.0 KT=34.0 KT=36.0 KT=45.0 K

Strongest magnetic reflections

FePO4impurity


Observed neutron powder diffraction patterns of PbFe2(P2O7)2 on G41 as a function of temperature

(001)(010) (011) & (0 1 -1)

(100)T= 1.4 KT= 5.0 KT=10.0 KT=14.5 KT=17.0 KT=20.0 KT=22.0 KT=24.5 KT=25.5 KT=26.5 KT=27.5 KT=39.0 K

Strongest magnetic reflections

FePO4impurity


Observed versus calculated neutron powder diffraction patterns of SrFe2(P2O7)2 on G4.1

T=1.4 K, G4.1λ= 2.43 Å

Nuclear reflections of FePO4Magnetic reflections of FePO4


Observed versus calculated neutron powder diffraction patterns of PbFe2(P2O7)2 on G4.1

T=1.5 K, G4.1λ= 2.43 Å

Nuclear reflections of FePO4Magnetic reflections of FePO4


Magnetic structure of SrFe2(P2O7)2 and PbFe2(P2O7)2

SrFe2(P2O7)2 at 1.4K PbFe2(P2O7)2 at 1.5K

Atom Atomic position Magnetic moment components in triclinic cell (µB)

x y z Mx My Mz Mx My Mz

Fe(1) 1 e ½ ½ 0 –4.75(4) 1.5(1) 0 –4.49(5) 1.5(1) 0

Fe(2) 1 b 0 0 ½ 4.75(4) –1.5(1) 0 4.49(5) –1.5(1) 0


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