The Effect of Faulting and Twinning on the Profiles of ... Ungar/III-Planar-defects... · on the...
74
The Effect of Faulting and Twinning on the Profiles of Diffraction Peaks Tamás Ungár Department of Materials Physics, Eötvös University Budapest, Hungary
Transcript of The Effect of Faulting and Twinning on the Profiles of ... Ungar/III-Planar-defects... · on the...
The Effect of Faulting and Twinning
on the
Profiles of Diffraction Peaks
Tamás Ungár Department of Materials Physics, Eötvös University Budapest, Hungary
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-Fluorogermanate P. 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
sequence
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
K. Kishida, Y. Takahama, H. Inui,
Acta Mater, 52 (2004) 4941-4952
pyramidal planes
Ti-Al hcp-DO19
K. Kishida, Y. Takahama, H. Inui,
Acta Mater, 52 (2004) 4941-4952 Diffraction spots of
mother and twin crystal
never coincide:
non-merohedral
Ti-6Al-4V G.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 + BG Imeas
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
F g
Lattice planes
Coordinate systems
a3
a2
a1
bl
bk bh
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) streaks DIFFaX, 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
Ti: pyramidal twin plane:
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
2 ko 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
L
integration normal to the g vector
Polycrystal sphere
Polycrystal
scattering cone base
k0k
g
F
Crystal
Streaking
bh
bl
Hi, Ki
bk
L
It can been shown that:
transformation between
L and g is linear
to a very good approximation
streaking is along the L direction
line profile is along the g direction
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
Ti: pyramidal twin plane:
{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
tensity
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
k k0
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]
Re
l. I
nte
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
Two different twin planes: {10.1} and {10.2}
{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
Two different twin planes: {10.1} and {10.2}
{11.4}, {20.1}, {21.0} reflection
Philosophy of line profile analysis
Measured pattern
<=> + BG Imeas
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 copper G. Sanders, G. E. Fougere, L. J. Thompson, J. A. Eastman, J. R. Weertman, Nanostruct. Mater. 8, (1997) 243.
Inert-Gas condensed nanocrystalline copper T.Ungár, S.Ott, P.G.Sanders, A.Borbély, J.R.Weertman, Acta Materialia, 10, 3693-3699 (1998)
L. Balogh, G. Ribárik, T Ungár, J.Appl.Phys. 100 (2006) 023512
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
L. Balogh, G. Ribárik, T Ungár, J.Appl.Phys. 100 (2006) 023512
Tsinter 1800 oC
Sintering nanocrystalline SiC J. 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 GPa No strain
Sintering nanocrystalline SiC J. Gubicza, S. Nauyoks, L. Balogh, J. Lábár, T. W. Zerda, T. Ungár, J. Mater. Res. 22 (2007) 1314
Equiaxed grains with
twin boundaries
Sintering nanocrystalline SiC J. Gubicza, S. Nauyoks, L. Balogh, J. Lábár, T. W. Zerda, T. Ungár, J. Mater. Res. 22 (2007) 1314
1800 oC at
2 GPa
40 60 80 100 120
0
4000
8000
422
420331
400222
311
220
200
2 [degree]
co
un
ts1111800 oC at
2 GPa
Sintering nanocrystalline SiC J. Gubicza, S. Nauyoks, L. Balogh, J. Lábár, T. W. Zerda, T. Ungár, J. Mater. Res. 22 (2007) 1314
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]
Sintering nanocrystalline SiC J. Gubicza, S. Nauyoks, L. Balogh, J. Lábár, T. W. Zerda, T. Ungár, J. Mater. Res. 22 (2007) 1314
eng = 33% eng = 42%
Twinning in tensile deformed TWIP steel L. Balogh, D.W. Brown, T. Holden, in preparation
Twinning in tensile deformed TWIP steel L. 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
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 min G. 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
