antonella guagliardi

C N R

Powder Diffraction Pattern: Origin of Line Broadening & Peak Shape FunctionsAntonella [email protected]

Istituto I tit t di Cristallografia

B a r i Italy

Summer School on Structure Determination from Powder Diffraction Data Paul Scherrer Institute, June 18th - 22nd, 2008

Experimental diffraction pattern of a polycrystalline material

Sample Profile

h=fgInstrumental profile

Paul Scherrer Institute, June 18th-22nd, 2008

Experimental diffraction pattern of a polycrystalline material

Y (2) = f g = f (2) g(2 2)d 2

+

f

crystal structure and volume of the illuminated sample size and shape of coherent domains lattice distorsions and defects size, divergence, energy dispersion of the beam optics (mirrors, monochromators, analyzers, slits, collimators) axial divergence geometrical aberrations (flat sample, transparency,...)

g

f = f1 f 2 f 3 K f n

g = g1 g 2 g 3 K g nPaul Scherrer Institute, June 18th-22nd, 2008

Experimental diffraction pattern of a polycrystalline materialh profile dominated by The sample profile f LaB6 - SRM 660ESRF - European Synchrotron Radiation Facility - Grenoble - FRANCE

(30.93 KeV, 0.4008 )

h profile Au NPs D~4nmLNLS -Laboratorio Nacional de Luz Sincrotron - Campinas, BRASIL

dominated by The Instrumental profile gPaul Scherrer Institute, June 18th-22nd, 2008

(8.040 KeV, =1.542 )

Powder diffraction pattern modeling:Convolution-based Approach vs Analytical Peak FunctionConvolution-based ApproachPhysically based instrument aberration functions + sample broadenings Traditionally related to microstructural analysis, for extracting the sample contribution from the experimental pattern Empirically based functions, to describe shape and width of diffraction peaks. Traditionally related to structural analysis

Analytical Peak Shape Function

Klug & Alexander, 1954(H.P. Klug, L.E. Alexander, X-ray Diffraction Procedures, 2nd ed., 1974, New York, John Wiley)

describe the diffraction pattern as the convolution of functions able to model various contributions of the instrumental and sample profiles

Multiple convolution integrals are highly time consuming operations!Conventional convolution approach determines the g profile from a standard material (LaB6, Na2Ca3Al2F4)Paul Scherrer Institute, June 18th - 22nd, 2008

90 : Convolution-based Profile FittingAdvantages

In principle can be used for any experimental set-up Allows to control the diffractometer Instrument parameters directly refinableAbility to construct a wide variety of profile shapes

Limits the number of (and the correlations among) parameters

Main limitations

Many different experimental set-ups (geometries & optics) Mirrors and monochromators difficult to be dealt with (depending on thermal stability)

Most methods restricted to functions that can be convoluted analytically (Lorentzians, Gaussians, impulse & exponential functions) R.W. Cheary & A. Coehlo (J. Appl. Cryst.,1987, 20, 173) described a number of tricks" for fast calculation of multiple convolution integrals (numerical integrations, integrations based on semianalitycal procedures) Fundamental Parameters Approach, implemented in TOPASPaul Scherrer Institute, June 18th - 22nd, 2008

Fundamental Parameters Approach

Instrumental profile

Sample profile

Final profile

K1 and K2 (99% of intensity) + group of satellite lines in the high energy tail K1 e K2 made of doublets (1,a 1,b 2,a 2,b) 5 Lorentzian profiles Cu-K Emission profileInfo & Figs. from: A. Kern, A.A. Coelho and R.W. Cheary, in Diffraction Analysis of the Microstructure of Materials, 2004, Springer, ISBN 3-540-40519-4Paul Scherrer Institute, June 18th - 22nd, 2008

Emission ProfileWhat can modify it: Parabolic mirror (different separation of K1 and K2 ) Ni filter for K (tails attenuation) monochromators curve of graphite riduces , can modify Int(K1)/ Int(K2) Ge asymmetric produces a very narrow , eliminates satellites and K2 peaks Different Targets (Cr, Mn, Fe, Co and Ni, up to 7 lorentzians! ) All instrumental contributions vary with 2, except the aberration related to the receiving optic and the finite Lab-source width

Geometric instrumental aberrations tend to determine the peak profile at low to medium angles Emission profile tends to become dominant at higher angle W(2) dominates at 2 > 100 Gi(2) dominate up to 2 40-60Paul Scherrer Institute, June 18th - 22nd, 2008

Cu-target

Synthesis of (100)-LaB6 (SRM 660) line profile Lab-DiffractometerSample Size & microstrain

fgWCuK W GEqEquatorial plane Target width Divergence slit angle Receiving slit angle Sample thickness

W Geq GAxAxial plane Target length Soller slits Receiving slit length Sample length

g

Paul Scherrer Institute, June 18th - 22nd, 2008

Rietveld MethodRietveld, 1969(H.Rietveld, J. Appl. Cryst, 1969, 2,65)

introduces a phenomenologic approach, aiming at refining a structural model from neutron powder diffraction data. The method describes the powder diffraction peaks through analytical functions

Y c (i ) = S IhLp hP (i , h) A(i, h) + Y b (i )h

Lecture focused on

WPPDS = scale factor Ih = reflection intensity (peaks area) Lph = Lorenz and polarization factors P(i,h) = analytical peak shape function (symmetric & normalized) A(i,h) = peak asymmetry function Yb(i) = backgroundPaul Scherrer Institute, June 18th - 22nd, 2008

Analytical Peak Shape Functions

Y c (i ) = S IhLp hP (i , h ) A (i , h ) + Y b (i )h

Rietveld Neutrons Gaussian Peaks

Gaussiana Lorentziana

G (i,h) =C = 4 ln 2

( )C0

1 2

2 C0 2 i 2 h exp 2 Hh Hh 1

2 = scan angle 2 = Braggs angle of reflection h Hh = Full Width at Half Maximum (FWHM)hi

0

X-Ray Voigtian Peaks

L(i, h ) =

V=GL

2 1+ 4 ( 2 i 2 h )2 2 H h Hh

1


Analytical Peak Shape FunctionsPseudo-Voigth =01 h =1 h =0

PV (i , h ) = h L (i , h ) + (1 h )G (i , h )mixing parameter (fraction of L)

completely Lorentzian peak shape completely Gaussian peak shape

Pearson VIImh = 1 mh =1 mh = mixing parameter

PVII(i,h) =

C0 1 + C1 (2 i 2 h) Hh Hh

mh

completely Lorentzian peak shape completely Gaussian peak shape

2(m) 21 m 1 2 C0 = 1 (m 0.5)( ) 2 1

(

)

C1 = 4 21 m 1

(

)

Hh, h, mh variables dependent on hPaul Scherrer Institute, June 18th-22nd, 2008

Peak shape/width angular dependencePearson VII2 2 H h = U tan h+V tan h+W h

Cagliotis equation

mh = m0 + m1 /2h + m2 /2h2U, V, W, m0 , m1, m2 Parameters to be refined Voigt Pseudo-Voigt (TCH) Pseudo-VoigtAllow to combine the G and L components so that they can be related to instrumental and sample broadenings

(P. Thompson, D.E. Cox, J.B. Hastings, J. Appl. Cryst., 1987, 20, 79)

2 2 2 H G = U tan h+V tan h+W + P cos h

Hh = F(HG, HL)

Cagliotis equation

(R.A. Young & P. Desai, Arch. Nauk Mater., 1989, 20, 79)

h = F(Hh, HL)

H L = X cos h + Y tan hU, V, W, P, X, Y Parameters to be refinedPaul Scherrer Institute, June 18th-22nd, 2008

Cagliotis equationHG dependence on through W,V,U

Caglioti et al., 1958, proposed to model the instrumental function of neutron diffraction patterns with gaussian peaks. The Cagliotis model relies on the hypothesis that, for each optical elements of the diffractometer (crystal and/or collimator) the probability of a neutron to reach the detector is described by a gaussian distribution. (G. Caglioti, A. Paoletti, F.P. Ricci, Nuclear Instrum., 1958, 3, 23).

Sabine, 1987, generalised the Cagliotis approach to any system of alternating crystals and collimators, to deal with the more complex intrumental configuration of synchrotrons and high resolution lab diffractometers, still within a gaussian approximation.(T.M. Sabine, J. Appl. Cryst.,1987, 20, 173)

Gozzo et al., 2006, extended the Caglioti/Sabines approach by including the effect of collimating and refocusing mirrors, still within a gaussian approximation.(F. Gozzo, L. De Caro, C. Giannini, A. Guagliardi, B. Schimtt, A. Prodi, J. Appl. Cryst.,2006, 39, 347)Paul Scherrer Institute, June 18th-22nd, 2008

Sabines generalization of Cagliotis equationThe probability of a foton to reach the detector Is the product of the gaussian probability of each optical element Product of esponential terms2 exp ' m

Double crystal monochromator, sample, crystal analyzer - Setting (+,-,+,-) Total reflection probability 2 2 b + 2 I ( ) = dd exp ' + 2 ' + ' a m m

=

finite detector acceptance

=

divergence of the initial beam

= - m = difference between the monochromator Braggangles of an individual () and the central (m) ray

b=

tan a tan 2 tan m tan m

Monochromator crystal

m = Bragg angle of

a = Bragg angle ofAnalyzer crystal

m = width of the beamdivergence probability distribution function

m = width of the monochromatorcrystal mosaic block probability distribution function (Darwin width )

a =

width of the analyzer crystal

mosaic block probability distribution function (Darwin width )Paul Scherrer Institute, June 18th-22nd, 2008

Instrumental Resolution Functiontan tan a tan 1 2 tan a 2 2 1 + + m + 2a H G (2 ) = m 2 2 tan m 2 tan m tan m tan m m = FWHM of the divergence initial beam distribution function i = FWHM of the mosaic-block distribution function ( Darwin width) of the monochromator / analyzer crystal2 2

IRF

2 2 4 m 8 m 22m 22m 2 1 2 2 tan tan + 4 m + 2m + 2a HG = + + tan 2 tan tan m 2 tan m m m

U

V

W


IRF at SLS-MS

2 m 2 tan a tan 2 + 2a + 2f ,eff 2 H G (2 ) = p + tan 2 tan m m 2

2 2 4 p p 22m 2 22m 2 tan tan + p + 1 2m + 2a + 2 ,eff + + H = f tan 2 tan tan 2 m tan m 2 m m 2 G

U

V

WPaul Scherrer Institute, June 18th-22nd, 2008

Sample Profile: Scattering from a crystal sin Ma r * sin Nb r * sin Pc r * G (r *) = f i ( S ) exp 2i (hxi + kyi + lzi ) sin a r * sin b r * sin c r * i =1q

Structure Factor Reciprocal vector

Interference Function

r * = ha * + kb * +lc* =

s s0

Laue equations

a

s s0

=h

b

s s0

sin 2 [N a a (s - s0 )] sin 2 [N b b (s - s0 )] sin 2 [N c c (s - s0 )] I = GG* = F sin 2 [a (s - s0 )] sin 2 [b (s - s0 )] sin 2 [c (s - s0 )]2

a

b c Ncc

sin 2 (Nh ) I ( h) F sin 2 (h )2

Column of unit cells


Naa

=k

c

s s0

=l

Sample Profile: Scattering from a powder sampleFor the periodicity of the Interference Function sin 2 (Nh ) sin 2 (Nh ) = 2 2 sin (h ) h = h h '

(

)

sin 2 (Nh ) (h )2Translated in h=h h = -1 h = +1

Reciprocal Lattice plane of a single crystal cubic in shapeFig Diffraction along r* = d* = (100)

-3

-2

-1

0

+1

+2

+3

Powder diffraction sphere

Powder sampleDiffraction over a sphere of radius |r*| = |d*| Interference Function should be integrated over the sphere to obtain the powder diffraction profileTangent plane

Integration depends on shape of crystallites and on point crossing direction (Miller indices of the reflection) Good approximation simplifying calculations: integration over the tangent plane(domains not too small / shape not strongly anisotropic!).

(Fig. 2a from: P. Scardi, in Analisi di Materiali Policristallini, 2006, Insubria University Press, ISBN 978-88-95362-04-5)Paul Scherrer Institute, June 18th-22nd, 2008

Sample Profile: size broadeningCubic unit cell (lattice parameter = a) Crystallites cubic in shape, all with same size D=Na Diffraction profile along h00 constant section of tangent plane & diffraction sphere intersection

sin 2 (Nas ) I h 00 ( s) (as )2Integral Breadth

s = r * rh*00

sin 2 (Nas ) ds Peak Area Na 1 I (s)ds = (as )2 h 00 ( s ) = = = = sin 2 (Ns ) ( Na ) 2 D Peak Maximum I (0) Lim s 0 (s )2

Integral Breadth inversely proportional to the domain size General relationship, independent of domain shape, lattice symmetry and hkl

( r*) =

K D

(2 ) =

K D cos

Scherrer formula(P. Scherrer, Gott. Nachr., 1918, 2, 98)

It holds also for crystallites polydisperse in size!Paul Scherrer Institute, June 18th-22nd, 2008

Sample Profile: lattice distorsion effectsReal materials often include defects limiting the coherence of their crystal lattice non-homeneous strain field effetcs on position, size and shape of reciprocal space points/ powder peak profile complex problem

Heuristic approach to obtain a useful relationship between peak broadening & lattice distorsions

d d =strain

macrostrain (macroscopic homogeneous strain) microstrain (non-homeneous strain field - on the length scale of crystallite size or smaller)microstructure

zero strain (no macro/microstrain)

macrostrain

Peak shift

microstrain

Peak broadening

macrostrain & microstrain(Fig. 13.4 from: P. Scardi, in Powder Diffraction Theory and Practice, 2008, Royal Society of Chemistry, ISBN 9780-85404-231-9)

Peak shift & broadening


Sample Profile: strain broadeningDifferentiating Braggs law at constant wavelength

0 = 2d sin +2d cos microstrain

2 = -2 tan d d = -2 tan (2) -2 < 2 >1/2 tan

macrostrain

A.R. Stokes, A.J.C. Wilson, Proc. Phys. Soc., London, 1944, 56, 174

peak profile (width & shape) depending on the strain field distribution within the sample

FWHM

1 2 = H G 2 ln 2

G

1

LHL

=

2

(2 ) r.m.s. strain Root mean square deformation

K D cos

+2 2

12

tan

Combined size-strain

< 2 >1/2

P/cos X/cos

(HG) (HL)

U tan Y tan

(HG) (HL)

(r*) =cos (2) (r*) < 2 >1/2 r*

strain broadening affects the peak in a r*-dependent wayPaul Scherrer Institute, June 18th-22nd, 2008

Sample Profile: size broadening vs

Size broadening (D = 2 m)

IRF


Sample Profile: size & strain broadening

IRF Size broadening (D = 2 m)

Strain broadening ( = 510-5)


2 2 2 H G = U tan h+V tan h+W + P cos h

H L = X cos h + Y tan hInstrumental contributions Sample contributions

GOpticsCaglioti, Sabine (1958, 1987)

L

G, Lsize

G, Lstrain

tan(Klug & Alexander ,1954)

1/cos(Scherrer, 1918)

tan(Stokes & Wilson, 1944)

W,V,U

Y

P, X

U, Y

Same parameters include effects from different sources Difficult separation of different contributions Correlation problems & instability during refinementPaul Scherrer Institute, June 18th-22nd, 2008

Asymmetry Correction

Y c (i ) = S IhLp hP (i, h) A(i, h) + Y b (i )h

Intensity (a.u.)

FWHM=0.8, m=2

FWHM=0.4, m=2

13

14

15

16

17

18

19

20

21

22

Axial Divergence (up to ~ 50 in 2 ) Flat Sample Transparency Beam divergence in the axial plane (soller slits)

2 (deg)

Semi-empirical analytical functions (1-2 parameters to be refined) Split-Functions (PV, PVII)Paul Scherrer Institute, June 18th-22nd, 2008

Axial Divergence

Detector aperture

Detector 2 scan across successive D-S cones

Incoming beam Perpendicular to the slide plane

Perfectly monochromatic and horizontally & vertically collimated incident beam Diffracted beam exhibits an axial divergence simply because it describes a Debye-Scherrer cone

0 < 2 < 90

At low angle the detector intercepts first the corners A and B, then goes across the arc AB At high angles, AB tends to a straight line and the detector intercepts the arc AB all at oncePaul Scherrer Institute, June 18th-22nd, 2008

Correction of axial divergence asymmetryConvolution-based model(FPA R.W. Cheary, A. Coelho, J. Appl. Cryst., 1998, 31, 862)

(L.W. Finger, D.E. Cox and A.P. Jephcoat, J. Appl. Cryst., 1994, 27, 892)

Rietveld analytical PSF

Inte ntensity (a.u.)

H = 10 mm H = 5 mm H = 3 mm

L = sample-to-RS distance S = illuminated sample length H = RS aperture S/L & H/L parameters to be refined

Si (111)

FWHM=0.04 =0.2 =0.40 L=800 mmS =2.75 mm

7.44

7.52

7.60

2 (deg)

AD effects reduced by controlling the aperture of the receiving slitPaul Scherrer Institute, June 18th-22nd, 2008

antonella guagliardi

Documents