whose fault is it? · the present line-profile analysis bottom-up approach profile-functions are...

$: Whose Fault is it? · the present line-profile analysis Bottom-up approach profile-functions are created by theoretical methods based on real lattice defects diffraction patterns$
Whose Fault is it?

Tamás Ungár

Department of Materials Physics, Eötvös University Budapest, Hungary

The Power of Powder Diffraction

Erice-2011, 2 - 12 June, Erice, Italy

The Effect of Faulting and Twinning on the

Profiles of Diffraction Peaks

Tamás Ungár

Department of Materials Physics, Eötvös University Budapest, Hungary

The Power of Powder Diffraction

Erice-2011, 2 - 12 June, Erice, Italy

Twinned nanocrystalline grains streaking is typical for faults

T. Waitz, V. Kazykhanov, H.P. Karnthaler, Acta Materialia 52 (2004) 137–147

Twinning in NiTi nanograins

T. Waitz, V. Kazykhanov, H.P. Karnthaler:

Martensitic phase transformations in nanocrystalline NiTi studied by TEM

Acta Materialia 52 (2004) 137–147

Bright field image of a twinned grain. HRTEM image of a grain containing

two twin related variants: RT1 & RT2

Perfect matching along the twin plane

T. Waitz, H.P. Karnthaler, Acta Materialia 52 (2004) 5461–5469

Devitrified naocrystalline particles twin related

R1-R2-R3

phases

Dark-Field

with R2 reflection

Dark-Field

with R1 reflection

Twinning in NiTi nanograins

T. Waitz, T. Antretter, F.D. Fischer, N.K. Simha, H.P. Karnthaler: J. Mech. Phys. Solids, 55 (2007) 419–444

Ni-Ti shape-memory alloys deform by twinning

Planar Defects in Magnesium-FluorogermanateP. Kunzmann in J.M. Cowley, Acta Cryst. (1976). A32, 83

Streaking normal to the fault planes

C A B C

B C A A

A B B B

C C C C

B B B B

A A A A

fcc

se

qu

en

ce

Extr

insic

SF

Intr

insic

SF

Tw

ins

Faulting along the close-packed planes in fcc or hcp crystals

What about hcp metals like:

Mg, Ti, Zr, Be, Zn ?

and their alloys

Ti-Al hcp-DO19

K. Kishida, Y. Takahama, H. Inui,

Acta Mater, 52 (2004) 4941-4952

prismatic

planes

pyramidal

planes

optical micrographs

Ti-Al hcp-DO19


Acta Mater, 52 (2004) 4941-4952

pyramidal planes

Ti-Al hcp-DO19


Acta Mater, 52 (2004) 4941-4952Diffraction spots of

mother and twin crystal

never coincide:

non-merohedral

Ti-6Al-4VG.G.Yapici, I.Karaman,Z.P.Luo, Acta Mater, 54 (2006) 3755

Deformation twins along the 10.1 pyramidal plane-

L. Wu, A. Jain, D.W. Brown, G.M. Stoica,

S.R. Agnew, B. Clausen, D.E. Fielden,

P.K. Liaw Acta Materialia 56 (2008) 688–695

{10.2} {10.1}

I. Kim,W.S. Jeong,J. Kim,K.T. Park,

D.H. Shin Scripta Materialia 45 (2001) 575-581

Twinning on pyramidal planes in hcp crystals

non-merohedral

Philosophy of the present line-profile analysis

Bottom-up approach

profile-functions are

created by theoretical methods

based on real lattice defects

diffraction patterns are fitted by

patterns constructued by using

defect-related profile-functions

Philosophy of line profile analysis

Measured pattern

<=> Isize Istrain + BGImeas

Physically modeled

Iplanar

faults

Experimentally determined

Iinstr

Non-linear, least squares fitting

IStrain : , M, q M = Re1/2

ISize : m,

Iplanar : or

Strain-broadening: strain-profile

is given by:

<g,L2>

mean-square-strain

where the strain Fourier coefficients are:

Dislocation-model for <g,L2> : Krivoglaz-Wilkens [1970]:

b : Burgers vector

: dislocation density

C : Contrast factor of dislocations

f() : Wilkens function [1970]

= L/Re

2

,gL

4

2bC

= f()

0 2 4 6 8-2

0

2

4

6

8

f()

The Wilkens function [1970]

f ()

~ 1/

~ ln()

= L/Re

0 1 20,0

0,5

1,0

M << 1, ~Lorentzian

I/IMAX

d*/d*0.5

M >> 1~Gaussian

Strain-profiles normalized

Strain profile:

inverse Fourier transform of the strain Fourier coefficients

ID(s) = exp{ - 22L2g2 <g,L2> }exp(2iLs) dL

<g,L2>= (b/2)

2 C f()

where:

Size profile: IS

m: median

: variance

assuming log-normal size distribution:

shape-anisotropy can be allowed for

Profile functions for

faulting and twinning

Line broadening from streaking

Fg

Lattice planes

Coordinate systems

a3

a2

a1

bl

bkbh

Hi, Ki

tricline

2 3 L

2% twin density

6% twin density

Inte

nsity [a

.u.]

Streak -21

Twinning on 10-12

Intensity distribution along the (Hi,Ki)=(-2 1) streaksDIFFaX, Treacy MMJ, Newsam JM, Deem MW, Proc. Roy. Soc. London A, (1991) 433, 499-520

NOT line-profiles yet

twinning on

pyramidal planes

Overlapping peaks along the same streak: asymmetry

L

L

almost symmetric peaks

strongly asymmetric peaks

interference

-1.1 -1.0 -0.9 -0.8

0.0

0.5

1.0

Inte

nsity

A.U

.

-12-14 -3030

L

Ti: pyramidal twin plane:

{11.4} reflection: sub-reflection (parent crystal)

{31.0} reflection: sub-reflection (twin crystal)

11 2

2 4

030

= 6 %

-1,2 -1,0 -0,8

0

1

L

Inte

nsity [

a.u

.]

(11-22)

twin plane -12-14

sub-reflection

Ti


only the sub-reflection in the {11.4} reflection

11 2

2 4

= 6 %

Intensity distribution in the streaks

twinning probability or twin boundary frequency:

fraction of twin lamellae of thickness of n crystal-layers: Wn

1)1( n

nW

0 15 30 45 60

0.00

0.05

0.10 = 10%

Wn

n

Intensity distribution in the streaks: I(L)

where:

I(L) : Lorentzian type function

0

2

2

0

0 14

1

LLA

FWHM

LL

ILI asym

L

1

2

tri

LD

FWHM

L. Balogh, G. Tichy and T. Ungár, J. Appl. Cryst. 42, (2009) 580-591.

It can been shown that:

L. Balogh, G. Tichy and T. Ungár, J. Appl. Cryst. 42, (2009) 580-591.

is a universal relation for twinning,

where Dtri = 5.775

1

2

tri

LD

FWHM

-1.1 -1.0 -0.9 -0.8

0.0

0.5

1.0

Inte

nsity

A.U

.

-12-14 -3030

L

0

2

2

0

0 14

1

LLA

FWHM

LL

ILI asym

L

I(L) : the sum of

a symmetrical + an antisymmetrical

Lorentzian type function

Intensity distribution in the streaks: sub-profiles

g

2ko

k

sz= sg

s

s

Intensity distribution in reciprocal-space is 3 dimensional

I=I(s,s,sg)

Line profile:

intensity distribution along the diffraction vector: g

scattered radiation is integrated normal to the g vector

I(sg) = I(s,s,sg) ds ds

Polycrystal sphere

Polycrystal

scattering cone base

k0k

g

F

Crystal

Streaking

bh

bl

Hi, Ki

bk

integration

Correlation between streaking and line broadening

Polycrystal sphere

Polycrystal

scattering cone base

k0k

g

F

Crystal

Streaking

bh

bl

Hi, Ki

bk

integration normal to the g vector

It can been shown that:

transformation between

L and g is linear

to a very good approximation

2

1( ) ( )g L

h k lFWHM FWHM

g a

1

2

tri

LD

FWHM

L. Velterop, et. al., J. Appl. Cryst. (2000). 33, 296-306

E. Estevez-Rams, et. al., J. Appl. Cryst. 34, (2001) 730

L. Balogh, G. Ribárik, T Ungár, J.Appl.Phys. 100 (2006) 023512

L into g transformation for twinning on

close-packed planes

L. Balogh, G. Tichy and T. Ungár J. Appl. Cryst. 42, (2009) 580-591.

L into g transformation for twinning on

pyramidal planes in hcp crystals


{11.4} reflection: sub-reflection (parent crystal)

{31.0} reflection: sub-reflection (twin crystal)

11 2

2 4

030

= 6 % 10 11 12

0.0

0.5

1.0In

ten

sity

A.U

.

g [ 1/nm ]

-12-14 -3030

along the streaks there is NO hkl dependence,

up to the asymmetry

Profile functions for twinning

L into g transformation

produces the hkl dependence

Resultant or measured profiles are

the sum of sub-profiles

Summation of sub-profiles

311

111

311

111

111

111

311 311

either or

the 311 type sub-reflections in fcc crystals

31-1

h+k+l=3

function

3-1-1

h+k+l=1

311

h+k+l=5

g Sg

2

kk0

intrinsic stacking faults on the (111) plane

8,8 9,0 9,2 9,4

0,0

0,5

1,0

1,5

Complete

h+k+l=+5

component: h+k+l=+3

h+k+l=+1

Arb

itra

ry U

nits

g [ 1/nm ]

Intrinsic SF

311

Subreflections according to different hkl permutations

= 4 %

10,85 10,90 10,95

0

100

200

300

g [1/nm]

Rel. Inte

nsity

3 % twins on the

{10.1} planes {11.4}

1-214

11-24

-1-124

10,85 10,90 10,95

0

10

20

30

Re

l. I

nte

nsity

g [1/nm]

{11.4}

1-214

11-24

-1-124

3 % twins on the

{10.2} planes

subreflections and {11.4} powder diffraction profiles

Twin planes: {10.2} Twin planes: {10.1}

hkl dependence in the I(g) profiles: Ti

10,88 10,92 10,96

0,0

0,5

1,0

Re

l. I

nte

nsity

g [1/nm]

{11.4}

10.1 Twin

10.2 Twin

Two different twin planes: {10.1} and {10.2}

{11.4}, {20.1}, {21.0} reflection

8,08 8,12 8,16

0,0

0,5

1,0

g [1/nm]

Re

l. I

nte

nsity

{20.1}

10.1 Twin

10.2 Twin


{11.4}, {20.1}, {21.0} reflection

10,32 10,36 10,40

0

300

600

g [1/nm]

Re

l. I

nte

nsity

{21.0}

10.1 Twin

10.2 Twin


{11.4}, {20.1}, {21.0} reflection

Philosophy of line profile analysis

Measured pattern

<=> + BGImeas

Physically modeled

theoretical pattern: Ith(g)

Isize Istrain IPF

Experimentally determined

Iinstr

Profile-functions for planar faults:

n

j

j

hklL

j

Lhkl

PF

hkl gIwgIwgI1

, )()()(

functions sub-profiles

w: fractions, ( permutations of hkl)

Sub-profiles:

symmetrical + antisymmetrical

Lorentzian functions

easy to parameterize

0 10 20

-0.2

-0.1

0.0

0.1

shift

s [1

/nm

]

5-3-3

5-33

533

0 10 20

0.0

0.2

0.4

533

5-33

5-3-3

fwhm

[1/

nm]

Shifts

Breadths

Sub-reflections of the {533} reflection for intrinsic SF

5th order polinomials

Inert-Gas condensed nanocrystalline copperG. Sanders, G. E. Fougere, L. J. Thompson, J. A. Eastman, J. R. Weertman, Nanostruct. Mater. 8, (1997) 243.

Inert-Gas condensed nanocrystalline copperT.Ungár, S.Ott, P.G.Sanders, A.Borbély, J.R.Weertman, Acta Materialia, 10, 3693-3699 (1998)


40 80 120

0

25000

50000

2o

70 80 90

0

6000

12000

220 311 222

111

200

220 311 222 400

Cou

nts

eCMWP

TEM and X-ray grain size

Twin density vs. crystallite size: in Cu

0 25 50 75 100

0

4

8

O2 - IS

P2 - IS

P2 - T

N2 - IS

N2 - C

ECAP(a)

ECAP(b)

Tw

in D

ensi

ty

[%

]

<x>area

[ nm ]

Twinning in Cu:

below ~ 40 nm


Tsinter 1800 oC

Sintering nanocrystalline SiCJ. Gubicza, S. Nauyoks, L. Balogh, J. Lábár, T. W. Zerda, T. Ungár, J. Mater. Res. 22 (2007) 1314

2 GPa 8 GPa

4 6 8 10 120,04

0,08

0,12

K [1/nm]

FW

HM

[1

/nm

]

111

200

220 311

222 400331

420

422

1800 oC at

2 GPaNo strain


Equiaxed grainswith

twin boundaries


1800 oC at

2 GPa

40 60 80 100 120

0

4000

8000

422

420331

400222

311

220

200

2 [degree]

counts

1111800 oC at

2 GPa


eCMWP

procedure

23

45

67

8

1400

1500

1600

1700

1800

0

2

4

6

8

10

12

Tw

in d

ensi

ty

[%]

Temperature [K]

Pressure [GPa]


eng = 33% eng = 42%

Twinning in tensile deformed TWIP steelL. Balogh, D.W. Brown, T. Holden, in preparation

Twinning in tensile deformed TWIP steelL. Balogh, D.W. Brown, T. Holden, in preparation

TOF neutron diffraction

at SMARTS

Lujan Neutron Scatt. Cent.

Los Alamos Nat. Lab.

eCMWP

procedure

Access to the software package

- CMWP

- eCMWP

- ANIZC

- planar-defect parameter-files

http://www.renyi.hu/cmwp

http://metal.elte.hu/anizc/

http://metal.elte.hu/levente/stacking

http://www.renyi.hu/cmwp






Thanks to my coworkers:

Levente Balogh

Gábor Ribárik

Géza Tichy

Thank you

for your

attention

Dislocations in the pyramidal twin boundaries in Mg

B. Li, E. Ma, AM 2004

CdF2 ball milled for 12 minG. Ribárik, N. Audebrand, H. Palancher, T. Ungár and D. Louër, J. Appl. Cryst. (2005). 38, 912–926

T. Waitz, V. Kazykhanov, H.P. Karnthaler:

Martensitic phase transformations in nanocrystalline NiTi studied by TEM

Acta Materialia 52 (2004) 137–147

Bright field image of a twinned grain. HRTEM image of a grain containing

two twin related variants: RT1 & RT2

Perfect matching along the twin plane

whose fault is it? · the present line-profile analysis bottom-up approach profile-functions are...

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