Diffraction Lineshapes(From “Transmission Electron Microscopy and Diffractometry of

Materials”, B. Fultz and J. Howe, Springer-Verlag Berlin 2002. Chapter 8)

Peak form for X-ray peaks:Gaussian

LorentizianVoigt,

Psudo-Voigt:

2

20 )(

exp)0(),(

xxIxI

Gaussian function

x0

)0(I

eI /)0(GB

2/)0(I

2

20)(

exp)0(2

)0(

xx

II

2

20)(

2ln

xx

2ln2GBFWHM

2ln0 xx

Lorentzian function or Cauchy form

20 )(1

)0(),(

xx

IxI

2CBFWHM

x0

)0(I

GB2/)0(I

20 )(1

)0(

2

)0(

xx

II

1)( 20

xx 0xx

Voigt: convolution of a Lorentzian and a Gaussian

)(erfiRe)0(),,( zIxI 2ix

z

Complex error function

)(erfc)(erfi2

izez z

)5145.41186.21245.21( 2 GV BBFWHM

most universal; more complex to fit.

Lorentzian function or Cauchy form20 )(41

)0()(

C

C

Bxx

IxI

CBFWHM

2

20)(4

2lnexp)0()(G

GB

xxIxI

Gaussian functionFWHM

pseudo-Voigt:

GB

)()1()()0(),( xIxIIxI GCp

: Cauchy content, fraction of Cauchy form.

2ln22ln2 GG BB

22 CC BB

2 = FWHM

FWHM2ln2

Lineshapes: disturbed by the presence of K1 and K2.

Decouple them if necessary:

Rachinger Correction for K1 and K2 separation:

Assume: (1) K1 and K2 identical lines profiles (notnecessarily symmetrical); (2) Ip of K2 = ½ Ip of K1.

tan22

tan)/(22

)/sin2)(cos/(2

cos/2

cos2sin2

d

dd

)()( 12

313

212

111

10

)(

)(

)(

0)(

II

II

II

I

2/)()( 11414 III

2/)()( 12515 III

2/)()( 131 iii III…

…

2/)()( 11 miii III General form

Example: Separated by 3 unitIi: experimental intensity at point iIi(1): part of Ii due to due to K1

Diffraction Line Broadening and Convolution

Sources of Broadening: (1) small sizes of crystalline (2) distributions of strains within individual crystallites, or difference in strains between crystallites (3) The diffractometer (instrumental broadening)

Size Broadening:

Interference function

3

12

222

sin

sin

i i

iitotal

NFAI

a

a

Define deviation vector

332211 bbb

3

12

222

sin

sin

i ii

iiitotal a

aNFAI

)()()()( 332211 IIII

11

2111

2

11 sin

sin)(

a

aNI

22

2222

2

22 sin

sin)(

a

aNI

…

11

2111

2

11 sin

sin)(

a

aNI

I

21N2

11 )0(0 NI

Half width half maximum (HWHM): particular

k

'1

2)( 21

'1 NI

'1

'1 usually small

2)(

sin

sin

sin)(

21

21

'1

11'1

2

1'1

211

'1

2'11

N

a

aN

a

aNI

11'111

'1 sin2)( aNaN

Solve graphically

Define 11'1 aNx

~ 1.392

Solution: x = 1.392

392.111'1 aN

LaNaN

443.0443.0392.1

1111

'1

Define1

k0k k

0kksink

sin2

k

kddd

kd

cos2

cos2'1

L

443.0

cos2

FWHM '12 kd

cos2

89.0

L

cos2

89.0

L

In X-ray, 2 is usually used, define 2B

cos

89.0

BL B in radians

If the is used instead of 2, K should be divided by 2.

cosB

KL Scherrer equation, K is Scherrer constant

Strain broadening:Uniform strain lattice constant change Bragg peaksshift.Assume strain = d0 change to d0(1+ ).

Diffraction condition:

)1(1

)1(

1

00

dd

Gk

Gdd

kd

0

1

Gdkd

sin2

kIn terms of d

kdcos2

Gdd

cos2

Peak shift

Gd

d

cos2 k

tansin2

cos2

d

d Larger shift for the diffractionpeaks of higher order

Distribution of strains diffraction peaks broadeningStrain distribution relate to

k

2

||

'12

G

'1 is the HWHM of the diffraction G along x̂

Instrument broadening:

Main Sources:

Combining all these broadening by the convolution procedure asymmetric instrument function

convolution

The Convolution Procedure:instrument function f(x) and the specimen function g(x)the observed diffraction profile, h(). The convolution steps are * Flip f(x) f(-x) * Shift f(-x) with respect to g(x) by f(-x) f(-x) * Multiply f and g f(-x)g(x) * Integrate over x

0 1 2-1-201234

0 1 2-1-201234

f(x)

g(x)

)()()( hdxxgxf

Assume f and g are the functions on the right, the h() that we will get is

1 2-1-201234

f(-x)

0

01234

0 2-2

= -1

01234

0 2-2

= 1

01234

0 2-2

= -2

01234

0 2-2

= 0

01234

0 2-2

= 2

01234

2-2 0

31/616/3

07/6

0h()

)()()()()( xgxfdxxgxfh

56

Convolution of Gaussians:

2

20 )(

exp)0(),(

xxIxI

2ln2B

Two functionsf(): breadth Bf

g(): breadth Bg h() = f()*g(); breadth Bh

222gfh BBB

http://www.tina-vision.net/docs/memos/2003-003.pdf

B

Convolution of Lorentzians:

Two Lorentzian functions: f(): breadth Bf g(): breadth Bg

h() = f()*g(); breadth Bh

gfh BBB

20 )(1

)0(),(

xx

IxI

2B

Fourier Transform and Deconvolutions:

Remove the blurring, caused by the instrument function:deconvolution (Stokes correction).

Instrument broadening function: f(k) (*k is function of )True specimen diffraction profile: g(k)Measured by the diffractometer: h(K)

n

linkenFkf /2)()(

'

' /2' )()(n

lkinenGkg

''

'' /2'' )()(n

lKinenHKh

l: [1/length], the range in k ofthe Fourier series is the interval–l/2 to l/2.

Fourier transform the above three functions (DFT)

dkkgkKfKh )()()(

dkenGenFKhl

ln

lkin

n

lkKin

2/

2/

/2'/)(2

'

'

)()()(

The function f and g vanished outside of the k range Integration from - to is replaced by –l/2 to l/2

dkeenFnGKhl

l

lknni

n n

linK

2/

2/

/)(2/2' '

'

)()()(

Orthogonality condition

nn

nnldke

l

l

lknni

'

'2/

2/

/)(2

if 0

if '

dklknnilknndkel

l

l

l

lknni

2/

2/

''2/

2/

/)(2 )/)(2sin()/)(2cos('

vanishes by symmetrydklknn

l

l

2/

2/

' )/)(2cos(

))](sin())([sin()(2

'''

nnnnnn

l

nn ' if 0

ldkl

l

2/

2/

n

linKenFnGlKh /2)()()( ''

'' /2'' )()(n

lKinenHKh

)()()( nHnFnlG Convolution in k-space is equivalent to a multiplicationin real space (with variable n/l). The converse is alsotrue. Important result of the convolution theorem!

Deconvolution:)(

)()(

nlF

nHnG

{G(n)} is obtained from n

linkenGkg /2)()(

Data froma perfect specimen

Data fromthe actualspecimen

RachingerCorrection(optional)

RachingerCorrection(optional)

f(k)Stokes

CorrectionG(n)=

H(n)/F(n)h(k)

F.T.

Correcteddata free

of instrumentbroadening

F.T.-1

g(k)

“Perfect” specimen: chemical composition, shape,density similar to the actual specimen ( specimenroughness and transparency broadening are similar)* E.g.: For polycrystalline alloy, the specimen is usually obtained by annealing

f(k), g(k), and h(k): asymmetric F.T. complex coeff.

)()(

)()(1)()(

niFnF

niHnH

lniGnG

ir

irir

)()(

)()(

)()(

)()(1)()(

niFnF

niFnF

niFnF

niHnH

lniGnG

ir

ir

ir

irir

)()(

)()()()(1)(

22 nFnF

nFnHnFnH

lnG

ir

iirrr

)()(

)()()()(1)(

22 nFnF

nFnHnFnH

lnG

ir

irrii

nir l

nknG

l

nknGkg

2sin)(

2cos)()( real part

g(k) is real and can be reconstructed as

n

linkir eniGnGkg /2)]()([)(

nir l

nki

l

nkniGnG

2sin

2cos)]()([

Simultaneous Strain and Size Broadening:

True sample diffraction profile: strain broadening and size broadening effect

Take advantage of the following facts:Crystalline size broadening is independent of GStrain broadening depends linearly on G

Usually, know one to get the other

Both unknown

Williamson-Hall MethodEasiest way!Requires an assumption of the shape of the peaks:

)exp(1

)(sin

)(sin)(

2

2

2

2

GGa

NaI

Kinematical crystal shape factor intensity

Gaussian functioncharacteristic of thestrain broadening

convolution

dd

2

2

exp)( 22 to relate G

)1(0 Gk 00 GkG GG

0

Assume a Gaussian strain distribution (quick falloff forstrain larger than the yield strain) ()

dGG

dd

22

2

exp1

)()(

2GG

Approximate the size broadening part with a Gaussian function

Good only when strain broadening >> size broadening

)exp(1

))(

exp()(2

2222

GG

NaNI

NaB

392.1

22ln2 (see page 9)

NaNa 1

2ln

392.1 characteristic width

22 GG

The convolution of two Gaussians

2

22

)(exp)(

kG

NI

222

2222

2 11)(

G

LG

aNk

hkldGk

1sin2

Plot k2 vs G2

(k)2

G2

2

1

L

2Slope =(HWHM)

Approximate the size broadening and strain broadening: Lorentzian functions

LNaB

443.0392.122

Size:

2)443.0

(1

)0()(

LI

I

Strain:

22

2

)(1

11

)443.0

(1)(

G

GLN

I

2

22

2

)(1

11

)(1

11

G

GG

2GG

The convolution of two Lorentzian

22

2

)(1

11

)443.0

(1)(

G

GLN

I

2

2

)(1

1)(

kG

NI

2443.0 GL

k

Plot k vs G

k

GL

443.02Slope =

(HWHM)

The following pages are from:

http://www.imprs-am.mpg.de/nanoschool2004/lectures-I/Lamparter.pdf

Ball-milled Mo from P. Lamparter

(FWHM)

GL

2

2

Nanocrystalline CeO2 Powder from P. Lamparter

Nb film, WH plot from P. Lamparter

from P. Lamparter

anisotropy of shape or elastic constants, strains. and sizes k2 vs G2 or k vs G not linear

Using a series of diffraction e.g. (200), (400){(600) overlap with (442), can not be used} provide a characteristic size and characteristic mean-square strain for each crystallographic direction!

Ek fit better than k in this case elastic anisotropic is the main reason for the deviation of k to G.

Ball-milled bcc Fe-20%Cu

Warren and Averbach MethodFourier Methods with Multiple Orders

Q)2exp()()Q( diQLLAI

)()()( LALALA D

size strain

How to interpret A(L)?

GQ

from P. Lamparter

from P. Lamparter

from P. Lamparter

from P. Lamparter

from P. Lamparter

from P. Lamparter

Williamson-Hall MethodEasy to be doneOnly width of peaks needed

Warren-Averbach MethodMore mathematicsPrecise peak shapes neededDistributions of size and microstrainRelation to other properties(dislocations)