Dislocation Model of
Strain Anisotropy
Tamás Ungár
Department of Materials Physics, Eötvös University Budapest
Budapest, Hungary
anisotropy:
Breadths in a
Williamson-Hall plot
are anisotropic
in terms of hkl indices
Ball-milled WC Gillies, D.C. & Lewis, D.
Powder Metallurgy, 11 (1968) 400.
strain-anisotropy:
global increase
of breadths
Ball-milled WC Gillies, D.C. & Lewis, D.
Powder Metallurgy, 11 (1968) 400.
Evaluation of strain by
ignoring
strain anisotropy
100
300
1
Ball-milled WC Gillies, D.C. & Lewis, D.
Powder Metallurgy, 11 (1968) 400.
4 consecutive papers of A.J.C. Wilson and coworkers:
1) A.R Stokes & A.J.C. Wilson:
The diffraction of X rays by distorted crystal aggregates - I
Proc. Phys. Soc. 56 (1944) 174-181
2) A.J.C. Wilson:
The diffraction of X-rays by distorted-crystal aggregates. II.
Diffraction by bent lamellae
Acta Cryst. 2 (1949) 220-222
3) J.N. Eastabrook & A.J.C. Wilson:
The Diffraction of X-Rays by Distorted-Crystal Aggregates III
Remarks on the Interpretation of the Fourier Coefficients
Proc. Phys. Soc. B 65 (1952) 67-75
4) A.J.C. Wilson:
The diffraction of X-rays by distorted-crystal aggregates. IV
Diffraction by a crystal with an axial screw dislocation
Acta Cryst. 5 (1952) 318-322
A.R Stokes & A.J.C. Wilson:
The diffraction of X rays by distorted crystal aggregates - I
Proc. Phys. Soc. 56 (1944) 174-181
. . . It is found that the "apparent strain" is given by η . . .
where
2 = A + BH
and
2222
222222
)( lkh
lklhkhH
Strain is hkl dependent
Unfortunately:
. . . This equation is verified within the rather large experimental error
for metal filings and wire.
Details of the experimental work will be published elsewhere. . . .
2 = A + BH
“Details of the experimental work”
were never published
The next appearance of strain anisotropy:
structure refinement by the Rietveld method using
neutron diffraction data
Caglioti G, Paoletti A, Ricci FP. Nucl. Instrum. 3 (1958) 223
introduce the term: strain anisotropy
strain anisotropy is disturbing in Rietveld structure refinement
First suggestion to make use of strain anisotropy for dislocation analysis:
P. Klimanek & R. Kuzel:
X-ray Diffraction Line Broadening Due to Dislocations in
Non-Cubic Materials. I. General Considerations and the
Case of Elastic Isotropy Applied to Hexagonal Crystals
J. Appl. Cryst. (1988). 21, 59-66
R. Kuzel & P. Klimanek:
X-ray Diffraction Line Broadening Due to Dislocations in
Non-Cubic Materials. II. The Case of Elastic Anisotropy
Applied to Hexagonal Crystals
J. Appl. Cryst. (1988). 21,363-368
R. Kuzel & P. Klimanek:
X-ray Diffraction Line Broadening Due to Dislocations in
Non-Cubic Crystalline Materials. III. Experimental Results for
Plastically Deformed Zirconium
J. Appl. Cryst. (1989). 22, 299-307
Slip systems from X-ray line broadening in hexagonal crystals:
R. Kužel jr., P. Klimanek, J. Appl. Cryst., 22 (1989) 299-307.
Orientation factor
Pioneering wok,
however,
NOT easy to implement
Copper deformed by Equal Channel Angular Pressing (ECAP)
0 4 8 12
0.00
0.02
0.04
K [1/nm ]
FW
HM
[ 1
/nm
]
111 200 220 311 222 400
Strain anisotropy
Williamson-Hall plot
0 20 40 60 80 100 120 140 160-3
-2
-1
0
10
8
6
4
321
L[nm]{400}{222}{311}{220}{200}{111}ln
A
g2 [1/nm]
Warren-Averbach plot of inert-gas condensed copper
Strain anisotropy
0 20 40 60 80 100 120 140 160-3
-2
-1
0
10
8
6
4
321
L[nm]{400}{222}{311}{220}{200}{111}
ln A
g2 [1/nm]
Warren would have suggested to evaluate this plot as:
Structural investigations of submicrocrystalline metals obtained by
high-pressure torsion deformation R. Kužel, Z. Matej, V. Cherkaska, J. Pešicka, J. Cıžek, I. Procházka, R.K. Islamgaliev, Journal of Alloys and Compounds, 2005
Williamson-Hall plot
Structural investigations of submicrocrystalline metals obtained by
high-pressure torsion deformation R. Kužel, Z. Matej, V. Cherkaska, J. Pešicka, J. Cıžek, I. Procházka, R.K. Islamgaliev, Journal of Alloys and Compounds, 2005
Williamson-Hall plot
SPD:
Severe Plastic Deformation
Soft magnetism in mechanically alloyed nanocrystalline materials T. D. Shen, R. B. Schwarz, and J. D. Thompson, PHYSICAL REVIEW B 72, 14431 (2005)
Fe80Cu20 (at. %)
Strain anisotropy
2 4 6 8 10
-2
-1
0
K [ 1/nm ]
log
A(L
)
L [ nm ]
5
14
23
33
42
51
61
111 200 220 311 222 400 420 331 422
Warren-Averbach plot
Revealing the powdering methods of black makeup in Ancient Egypt
by fitting microstructure based Fourier coefficients to the whole
x-ray diffraction profiles of galena (PbS)
T. Ungár, P. Martinetto, G. Ribárik, E. Dooryhée, Ph. Walter, M. Anne, J.Appl.Phys. 91 (2002) 2455
T
g b
T g
b
gb 0
dislocation is visible dislocation is invisible
strong contrast weak contrast
gb = 0
strong line broadening weak line broadening
Analogous with transmission electron microscopy (TEM)
K.Shiramine, Y.Horisaki, D.Suzuki, S.Itoh, Y.Ebiko, S.Muto, Y. Nakata, N.Yokoyama,
Threading dislocations in InAs quantum dot structure, Journal of Crystal Growth, 205 (1999) 461-466
g = 004 g = 222 g = 222
ln An = ln - S
nA 22g2L2 < > 2
,gL
Fundamental equation of line broadening
Warren & Averbach (1952):
size Fourier coefficients
mean square strain
for dislocations [Krivoglaz, Wilkens]:
ln(Re/L)
C is the contrast factor of dislocations
C = C (g,b,l,cij)
4
2bC
2
,gL
2
,gL
1) The dislocation contrast factors can be evaluated numerically.
2) For polycrystals or when almost all slip systems are populated
averaging over the permutations of hkl can be done
3) Average contrast factors for cubic crystals:
= h00 (1-qH2) ,
where H2=(h2k2+ h2l2+ k2l2) / (h2+k2+l2)2
4) q depends on the a) elastic constants, and on the
b) dislocation character, e.g. edge or screw
CC
2.5 5.0 7.5 10.0 12.5 15.0 17.50.0
0.1
0.2
0.3
0.4 screw; edge
Co
ntr
ast
facto
rs
K [ 1/nm ]
110 200 211 220 222 400 411
Ferritic steel
0 2 4 6
0.00
0.02
0.04
{400}
{311}
{222}
{220}
{200}
{111}
FW
HM
[ 1
/nm
]
K C 1/2
[1/nm ]
modified Williamson-Hall plot
Copper deformed by Equal Channel Angular Pressing (ECAP)
0 10 20 30 40 50-2.0
-1.5
-1.0
-0.5
0.0 {400}{311}{200}
{220}{222}
{111}
10
8
6
4
3
L [nm]
ln A
+
L
' W(g
)
g2 C [ nm-2 ]
modified Warren-Averbach method
Inert-gas condensed copper
Average crystallite size in the
inert-gas condensed copper specimen
classical Warren-Averbach analysis
200 – 400 reflections
<Lo> 7 nm
modified Warren-Averbach analysis
all reflections
<Lo> 18 nm
TEM size: 18 nm
T.Ungár, S.Ott, P.G.Sanders, A.Borbély, J.R.Weertman, Acta Materialia, 10, 3693-3699 (1998)
Soft magnetism in mechanically alloyed nanocrystalline materials T. D. Shen, R. B. Schwarz, and J. D. Thompson, PHYSICAL REVIEW B 72, 14431 (2005)
Fe80Cu20 (at. %)
0 5 10 15 20 25
-2
-1
0
422331
420
222311
220
400
111
200
L [nm]
5
9
14
19
23
28
33
37
42
47
51
56
61
log
A(L
)
K2C [ nm
-2 ]
modified Warren-Averbach method
Ball-milled galena (PbS) T. Ungár, P. Martinetto, G. Ribárik, E. Dooryhée, Ph. Walter, M. Anne, J.Appl.Phys. 91 (2002) 2455
TEM Dislocations structure in ball-milled PbS (galena) P. Martinetto, J. Castaing, and P. Walter, P. Penhoud and P. Veyssiere, J. Mater. Res., Vol. 17, No. 7, Jul 2002
1014 m-2
nanocrystalline SiC
sintered at 1800 oC by 5.5 GPa J. Gubicza, S. Nauyoks, L. Balogh, J. Labar, T.W. Zerda, T. Ungár,
J. Mater. Res. 22, 1314-1321 (2007)
0 4 8 120.00
0.01
0.02
422
331
420
400222
311
220
200
111
FW
HM
[1/n
m]
K [1/nm]
0 5 10 15 20 250.00
0.01
0.02
K2C [1/nm]
FW
HM
[1/n
m]
422400
420331
311
220 222
200111
Twinning Twins + dislocations
Williamson-Hall plot
modified Williamson-Hall plot
100 nm
in single crystals the concept of
average contrast factors
does NOT work
each reflection needs an individual contrast factor
Diffraction on a single grain in an
MgSiO3 perovskite
P. Cordier, T. Ungár, L. Zsoldos, G. Tichy,
Dislocations creep in MgSiO3 perovskite
at conditions of the Earth's uppermost lower mantle,
Nature, 428 (2004) 837-840.
Lower
mantle
Transition
zone
Upper
mantle
3 GPa
1100°C
13 GPa
1400°C
23 GPa
1600°C
135 GPa
3500°C
Depth
100 km
410 km
520 km
670 km
2900 km
P, T
Olivine (Mg, Fe)2SiO4
Wadsleyite (Mg, Fe)2SiO4
Ringwoodite (Mg, Fe)2SiO4
Pyroxenes (Mg, Fe)SiO3
(Ca, Mg, Fe)2Si2O6
Perovskite (Mg, Fe, Al)(Si, Al)SiO3-x
CaSiO3
Magnesiowustite (Mg, Fe)O
Garnets (Mg, Fe, Ca)3
Al2Si3O12
Garnets (Mg, Fe, Ca)3(Al, Si)2Si3O12
Schematic composition of Earth's mantle
-0,05 0,00 0,05
1E-3
0,01
0,1
1
I/IMax
K [1/nm]
hkl K [1/nm]
110 2.92
120 4.57
121 4.79
022 4.99
Strain anisotropy
2 4 6 8 10
0.004
0.006
0.008
0.010B
read
ths
[1/
nm]
K [ 1/nm ]
FWHM
Integral Breadth
120
K
[ 1
/nm
]
K [ 1/nm ]
006
123
023
211
121
022112
021
111
110
Strain anisotropy
Williamson-Hall plot
Measured dislocation contrast factors: Cm
2 4 6 8 100,00
0,08
0,16
121
211
006
123
023
022021
120
112
111
110
Cm
K [ 1/nm ]
Measured and calculated dislocation contrast factors: Cm , Ccalc
2 4 6 8 10
0.00
0.08
0.16
006
121
123
023
022
021
211
120
112
111
110
Dis
loca
tion
Con
tras
t Fac
tors
K [ 1/nm ]
Ccalc
Cm
The only Burgers vectors
that survived the search-match
by comparing the measured and calculated
dislocation contrast factors
[100]
[010]
0,0 0,5 1,0 1,5 2,0 2,5
0,0
0,2
0,4
0,6 BE
PrE
PYE
PR2E
PR3E
PY2E
PY3E
PY4E
S1
S2
S3
Cav
2l2/3(ag)
2
I. C. Dragomir and T. Ungár J. Appl. Cryst., 2002, 35, 556-564.
The parabolas for the eleven sub-slip-systems in Ti
vs. x=(2/3)(l/ga)2
Mg deformed at different temperatures between RT and 300oC
Williamson-Hall plot for deformation at 200oC:
2 4 6 8 100,002
0,004
0,006
0,008
0,010
20.3
11.2
20.110.310.210.100.210.0
Bre
adth
s [1
/nm
]
K [1/nm]
Integral breadths
FWHM
The q parameters for Ti
Sub-Slip
System
hk.0 q1 q2
BE 0.20227 -0.101142 -0,102625
PrE 0.35387 -1.19272 0.355623
Pr2E 0.04853 3.6161928 1.2264112
Pr3E 0.10247 2.017177 -0.616631
PYE 0.3118 -0.894009 0.1833109
PY2E 0.09227 1.299046 0.3972469
PY3E 0.09813 1.894120 -0.365739
PY4E 0.09323 1.5270212 0.146150
S1 0.1444 0.59492 -0.710368
S2 0.41873 1.25714 -0.94015
S3 3.61x10-6 165366 -98611
C
- Two experimental parameters: q1(m) and q2
(m)
- Volume fractions of active slip systems: hi
= 1 ih
a) Edge dislocations:
Major slip
systems
sub-slip-systems Burgers vector Slip plane Burgers vector
types
Basal BE <2-1-10> {0001} a
Prismatic PrE <-2110> {01-10} a
PrE2 <0001> {01-10} c
PrE3 <-2113> {01-10} c + a
Pyramidal PyE <-12-10> {10-11} a
Py2E <-2113> {2-1-12} c + a
PyE3 <-2113> {11-21} c + a
PyE4 <-2113> {10-11} c + a
b) Screw dislocations:
sub-slip-systems Burgers vector Burgers vector
types
S1 <2-1-10> a
S2 <-2113> c + a
S3 <0001> c
Three Burgers vector types
The measuremets provide 3 equations:
(1) =
(2) =
(3) = 1
i = 1, 2 or 3 for <a>, <c> or <c+a> dislocations,
respectively
)(1mq
3
1
)(1
2)(0.
1
i
ii
ihki qbCh
P
)(2mq
3
1
)(2
2)(0.
1
i
ii
ihki qbCh
P
3
1iih
Fitting of measured and theoretical q parameters
For a particular Burgers vector type or i value:
the weights of the sub-slip-systems are:
1 or 0 in all possible combinations
accepted solution: if all hi > 0
not accepted solution: if any of hi < 0
0 50 1000
100
200
[%]
<a><c>
<c+a>
Num
ber
of solu
tio
ns
hi
Bar diagram of the solution matrix of the hi fractions:
in the as cast Mg specimen
The volume fractions of the Burgers vector types,
<a>, <c> and <c+a>,
vs. the temperature of deformation in Mg:
0 100 200 300
0
50
100
as cast
[10
-13 m
-2 ]
[%]
<a>
<c>
<c+a>hi
T [°C]
0
20
40
60
Mathis, K., et al. Acta mater. 52, 2004, 2889–2894.
Below about 100 oC:
<a> type dislocations in basal slip
+ twinning
Above about 100 oC:
twinning is replaced by <c+a> dislocations
(energetically more favorable than twinning)