Grain Crystallite Size s

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Crystallite size vs grain size Not equal, a grain can contain several

crystallites

Infinite crystal

Electron density can be described to beproportional to (r-rpqr).

Scattering amplitude is then proportional

to (q-r*hkl).

One dimensional


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Shape of the crystallites The electron density of a crystallite can be

presented as a product of the lattice term(r)of the infinitely large crystal and theshape function (r)of the crystallite.

(r)=(r)(r)

(r)=1 inside the crystallite and 0outside.

Amplitude F of crystallite

F(q) =(r) (r) exp(iqr) d3r Using convolution theorem of Fourier

transform: F(q) = F(q)* (q). The star *

denotes convolution. The function (q) = (r) exp(iqr) d3r Intensity I(q) = F*(q) F(q).


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Convolution theoremConsider functions of one variable fand g.Their convolution is f*g(x) =f(u)g(x-u)du.Fourier transformF(f*g) =f*g(x)exp(-ikx) dx

=dx du f(u)g(x-u) exp(-ikx).Change of variable x-u = y.

F(f*g)(k)=dy du f(u)g(y) exp(-ik(y+u))= du f(u) exp(-iky) du g(y) exp(-iku)=Ff(k)Fg(k).

Intensity of small crystallite

Patterson functionP(r) =(u)(u+r) (u) (u+r) d

3u

= V(u)(u+r)>,

where Vis the volume limited by the shapefunction.


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Intensity I N/V F2Z(q) * |(q)|2

F2 unit cell structure factor Z(q) lattice term (q)|2 broadening due to limited size.

Integrated intensity

Z(q) =sin((N1-1)qa) /sin(qa)sin((N2-1)qb) /sin(qb)sin((N3-1)qc) /sin(qc)

Set N1=N2=N3=N.The maximum intensity at a Bragg peak can

be shown to be equal to f2N2. The width ofthe peak is aproximately proportional to1/N. The integrated intensity (the area ofthe peak) increases proportional to N.


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Crystallite size Form factor of a sphere F(x) = 4/3a3

3 (sin x xcos x)/x3, where x = qa. Reflection spot in reciprocal space is

approximated as a sphere, radius dq. The width of F(x)2gives the angular width

of the diffraction peak 2 dq/(cosq0) q0= 2, 2dq = 3.6/R Angular width d(2) = 0.57/(Rcos )

Crystallite size: simple approach

Let mbe the number of parallelplanes

Let 1 present the highest angle(1) and 2 the lowest angle (2)one can go before gettingdestructive interference.

The path differences are (m+1)and (m-1).

Destructive interference withplanes in the middle of thecrystallite with path differences(m+1)/2 and (m-1)/2.

t=md

B


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Crystallite sizeWe assume triangular shape for the diffraction

peak, FWHM = B = (21-22)/2=12.Path length differences between rays scattered

from the front and back planes of the crystallite2 tsin 1 = (m+1)2 tsin 2= (m-1)t(sin 1 -sin 2) =2tcos((1 +2)/2) sin ((1 2)/2) =

By approximating 1 +2)/2 =Band sin xone obtains 2t (1 2)/2 cosB=

t =/Bcos B

The average size of crystallites

Assume a powder sample of crystallites of thesame size N1=N2=N3=N.

In a powder crystallites take all orientations.Thus one may consider only one rocking

crystallite. Let qhklbe the reciprocal lattice vector for the

reflection hkl. Let the directions of the primary and diffracted

beams vary slightly. Name q = q +qand s-s0= s-s0s.


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Scherrer equationApproximate sin2(Nx)/ sin2xN2exp(-(Nx)2)The intensityI sin2(Nqa)/ sin2qasin2(Nqb)/ sin2qbsin2(Nqc)/

sin2qc

becomesI IeF2N6 exp(-(2) N2((sa)2+(sb)2+(sc)2),sincesin2(q

hkl+2s/)Nq/2 = sin2(hN+sNa/)Nq/2 =

sin2sNa).Setting a=b=c,I IeF2N6 exp(-(2) (Na)2 q)2)

Scherrer equationIt is easier to rock the q-vector than the

crystallite. The difference vector is denotedby s = x+y+, and s2 = (x-sin)2+y2+(cos)2

The intensity at fixed departure isproportional to the sum of all values x andy. I IeF2N6 exp(-(2)(Na)2 cos)2)

exp(-(2

)(Na)2

(x-sin)2

)dx exp(-(2

)(Na)2y2)dy.The intensity becomes proportional to

I K exp(-(2)(Na)2 cos)2) , where Kis a constant. This function is a gaussianand and its FWHM is obtained from 1/2 =exp(-(2)(Na)2 cos)2).

The broadening of the peak B(2) is about 2.

y

x

s

s-s0

s

q


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FWHM of Gaussian f(x) =exp(-x2/b2) FWHM in terms of b: exp(-x2/b2)= => x2=ln 2 b2

FWHM = 2x = 2(ln2)1/2b

Scherrer equation

B(2) = 2(ln2/)1/2 /Na cos. Set L = Na =>

Scherrer equation

B(2) = 0.94 /(L cos),where B is the FWHM of the reflection

More details in Warren, X-ray diffraction,chapter 13


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Stokes and Wilson Let P(2) be the measured intensity of a

reflection. Broadening has been also described as (2) = P(2)d(2)/max(P(2)) B(2) = /(L cos)

Experimental determination

1. Subtraction of background2. Determination of maximum intensity:

statistical accuracy must be quite good

3. Determination of the FWHM: the angular(q-scale) step must be small enoughcompared to the width of the peak

4. Instrumental broadening affects also thewidth


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Instrumental broadening Let g(x) be the measured curve. Instrumental broadening is presented as an

integral equationg(x) =h(x,y) f(y) dy + n(x),

where fis the exact result, h theinstrumentalfunction (kernel), and nstatistical errors.

If the effect of the instrument is the same in allpoints, the integral equation is of convolutiontype: g(x) =h(x-y) f(y) dy.

Special solution for this case

Sometimes it can be assumed that both thediffraction peak and the instrumental functionare Gaussians. For the FWHM of a convolutionof two Gaussians fand gholds:

f*g

2=f

2+g

2

For some other setups the peaks might beapproximated as Lorenz (Cauchy) functions1/(1+a2x2).

Then f*g=f+g.


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Convolution of Gaussians f(x) =exp(-x2/b2) FWHM in terms of b:exp(-x2/b2)= => x2=ln 2 b2

FWHM = 2x = 2(ln2)1/2b g= exp(-x2/d2)

f*g= constant exp(-y2/(b2+d2))

Effect of strain on diffraction peak


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Broadening of reflection: strain No strain, the position of the peak q =

/d, where dis the distance of the latticeplanes

Uniform strain: dincreases => qdecreases.

Non-unform strain: dvaries: If =d, theposition may not change, but the FWHM ofthe peak increases.

Broadening

Bragg law 2dsin = Differentiate 2dsin + 2dcos = 0 d/dtan = 2


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Size distribution for crystallites By fitting a simulated powder pattern,

derived from an appropriate physicalmodel, to experimental data.

dislocations and other lattice distortions Model shapes for size distributions

Gaussian G(x) = (2b2)-1/2exp(-(x-a)2/2b2)

Lognormalf = 1/(r(2ln(1+c))1/2exp(ln(r/(1+c)1/2)2/(2ln(1+c)));

Size-strain line broadening

For analysing line broadening, correction forinstrumental broadening must be done first. Wavelength dispersion Slits

Deconvolution, ill-posed problem: goodmeasurements are essential!

Voight function is usually assumed for the lineshape in the powder diffraction refinementprograms.


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Broadening: size, strain Size (Scherrer): FWHM proportional to

1/cos Strain: FWHM proportional to tan Measure peaks, e.g. 00l. Determine the widths

Plot the width as a function of l(or angle) Study the angular dependence.

Profile fitting

Lorenzian (Cauchy function)FWHM = A/cos + B tan + C Voight functionFWHM = U tan2 + V tan + W + P/cos2


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Good references J. I. Langford, D. Lour and P. Scardi. Effect of a

crystallite size distribution on X-ray diffractionline profiles and whole-powder-pattern fitting. J.Appl. Cryst. (2000). 33, 964-974

D. Balzar et al. Size-strain line-broadeninganalysis of the caria round-robin sample. J. Appl.Cryst. 2004, 37, 911-924.

Ungar et al. Crystallite size distribution and

dislocation structure determined by diffractionprofile analysis: principles and practicalapplication to cubic and hexagonal crystals. J.Appl. Cryst. 2001, 34, 298-310.

Diffraction pattern of wood

Reflection 200

Reflection 004

003

002

102

-110

110

q=4 sin

q


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Monoclinic Cellulose I004

200

C-axis

In-situ X-ray diffraction and tensiletesting

Stress-strain curves from tensile testing giveinformation on the deformation of themacroscopical sample

X-ray diffraction (XRD) gives the the behaviour of

crystalline cellulose under tension.


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Comparison of ESRF and HASYLAB

stretching rate 0.2 m/s,measuring time


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Warren-Averbach Profile of Bragg reflection P(q) is assumed as a

convolution of profiles presenting strain and sizedistortion.

P(dqi) = const A(l,dqi) cos(qL) + B(L,dqi) sin(Lq),where L gives the distance perpendicular to thediffracting planes and dq is the deviation from thereflection position.

Fourier transform of the profile gives for the Fouriercoefficients A(L) and B(L). Usually only A(L) areconsidered.

ln A(L) = ln ALs - 22 L2 g2 , where ALs are sizecoefficients, g the absolute value of the scattering vector,and is the mean square strain. The parameter L isdefined as L=na3, where a3 = (sin2-sin1), n integer.

Warren-Averbach

To obtain information on size and distortioncoefficients, at least two orders of reflection haveto be measured.

See e.g. Warren: X-ray diffraction, Berkum et al.Applicabilities of the Warren-Averbach analysisand an alternative analysis for separation of sizeand strain broadening. J. Appl. Cryst. 1994, 27,345-357.


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XRD and elastic properties 1-d case, stretched cylinder Force Fin y-direction Tension F/A Increase of length y=L/L, where L

is the length of the cylinder. Hookes law F/A = Ey, where Eis

Youngs modulus and A the area ofthe cross section of the cylinder.

For a cylinder x=y=D/D = - y,where Dis the diameter and is thePoisson ratio.

F

L

y

Elastic properties

Determining relative increase of lengthusing x-rays. Determine change of latticespacing d/dof lattice planes

perpendicular to the force F. F/A = E/ d/d.


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Residual stress When metal bar is deformed (applied

stress) it causes inner stress. Microstrain: Lattice planes are deformed, d

varies. Macrostrain: Distance of lattice planes

increases uniformly.

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Grain Crystallite Size s