Group-theoretical Applications in the Study of...
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C. Capillas, M. I. Aroyo, J.M. Perez-Mato
Group-theoretical Applications in the Study of Reconstructive Phase
Transitions
Satellite Conference of AsCA’06/CrSJTheoretical Crystallography and Materials Science
Tsukuba, Japan, November 2006
Depto. de Física de la Materia CondensadaUniversidad País Vasco, Bilbao, Spain
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G2
G1
G2
G1
Symmetry and Phase Transitions
Can symmetry relations be useful in the description of phase transitions with no group-subgroup relations
between the phases ?Satellite Meeting AsCA’06/CrSJNovember 18-19,Tsukuba, Japan
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Examples of TP
! !"#$%&'()*+"$,-%,(%#+#$%,./0%12%"$*"$1%3"$140+
! 5))%,(%6))%12%"$*"$1%70,"$+%"+%#+
! 812)6$0240%,(%'()*+"$,%12%91#
! :;',<1,0%,(%!"#$%&'()*+"$,-%12%612"'.%+071)(24;),('+%%
===>?%"24%==>?=%"+%@"!A%=2!A%B$!A%82CA%DECA%#49A%#490FFF
! :;',<1,0%,(%812)6$0240
! 3)/%,(%6))%/3"+0%12%,'"2+1,1(2%70,"$+%"+%G1A%8'F
! 3)/%,(%6))%12%#(A%H0A%G$A%B7FFF
RECONSTRUCTIVE phase transitions
Satellite Meeting AsCA’06/CrSJNovember 18-19,Tsukuba, Japan
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How to describe this transformation?!! Wurtzite (Z=2, Zp=2)
G1 = P63mc!! Rocksalt (Z=4, Zp=1)
G2= Fm-3m
No group-subgroup relation between the phasesSatellite Meeting AsCA’06/CrSJNovember 18-19,Tsukuba, Japan
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FERROELECTRIC phases in the system PbZr1-xTixO3
Example:
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Transition Paths (TP)
! transient state
space group H
displacive distortion
!! Structure 1space group G1
!! Structure 2space group G2
Only a single symmetry break and symmetry gain for the whole transition path
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group-subgrouptransformation
group-subgrouptransformation
affine transformation
Transformation Mechanism: Group-theoretical Model
symmetry break symmetry gain
G1
G2
H1 H2
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R3mR3mZ = 1
Na 1a 0 0 0
Cl 1b 1/2 1/2 1/2
! = 90 !" 60
Pm3m
Fm3m
Cesium Chloride
Z = 1
Rocksalt
Z = 4
Na 4a 0 0 0
Cl 4b 1/2 1/2 1/2
Na 1a 0 0 0
Cl 1b 1/2 1/2 1/2
[ 4 ] [ 4 ] 1/2(b+c),1/2(a+c),1/2(a+b)
Buerger Mechanism: Group-theoretical Scheme
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PmmnPmmn
Pm3m
Fm3m
Cesium Chloride
Z = 1
Rocksalt
Z = 4
Z = 2
-a+c,a+c,b+1/2 1/2(a+b)+1/4,c,1/2(a-b)+1/4
[ 12 ] [ 12 ]
Na 4a 0 0 0
Cl 4b 1/2 1/2 1/2
Na 1a 0 0 0
Cl 1b 1/2 1/2 1/2
Cl 2b 1/4 3/4 z2
Na 2a 1/4 1/4 z1
z2 : 1 ! 3/4
z1 : 1/2 ! 1/4
Watanabe Mechanism: Group-theoretical Scheme
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The general problem: to find and analyze the most (energetically) favourable transition paths
!!Sowa Hatch & Stokes
What can be derived only from symmetry arguments?
Our problem:
"#$%&'$%&(&)*+
!!P. Tolédano, V. Dimtriev&%!,*''$-.*!/,"$0-,+
Previous work:
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AIMs Description and classification of possible transition paths based on:
-SYMMETRY conditions
-STRUCTURAL conditions
Bilbao Crystallographic Serverwww.cryst.ehu.es
Development of the corresponding computer tools
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group-subgrouptransformation
group-subgrouptransformation
affine transformation
Transformation Mechanism: Group-theoretical Model
symmetry break symmetry gain
G1
G2
H1 H2
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Symmetry Conditions
• The description of the intermediate state involves a common subgroup pair (H1, H2) of the symmetry groups of the two stable phases such G1 > H1 and G2 > H2.
• The compatibility between the occupied Wyckoff orbits in the intermediate state.
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group-subgrouptransformation
symmetry break
Phase 1
Phase 2
G
H
H
affine transformation
Group-subgroup related transformations
transformation matrix (P,p)
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index [i]index [i]
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Hermann, 1929:
For each pair G>H, there exists a uniquely defined intermediate subgroup M, G ≧M ≧ H, such that:
M is a t-subgroup of G
H is a k-subgroup of M
G
M
H
it
ik
PROBLEMS:GROUP-SUBGROUP RELATIONS
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Symmetry conditions for TP
! !"#$"%"&'((')"$*+,-'*.
G2G1
!/ !0
!1!2!3
!4 !5 !6
!7
Symmetry conditions for TP
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COMMON SUBGROUPSCOMMON SUBGROUPS
i2 = i1 ·Z1
Z2
·|PG2
|
|PG1|·fG2
fG1
the set of common subgroup types is finite if a maximum k-index is defined
index condition
ZH1= ZH2
i2 =
|PG2|
|PH2|·Z
p
H2
Zp
G2
i1 =
|PG1|
|PH1|·Z
p
H1
Zp
G1
G1
H
G2
[i1]
i2 = i1 ·Z1
Z2
·|PG2
|
|PG1|·fG2
fG1
the set of common subgroup types is finite if a maximum k-index is defined
index condition
ZH1= ZH2
i2 =
|PG2|
|PH2|·Z
p
H2
Zp
G2
i1 =
|PG1|
|PH1|·Z
p
H1
Zp
G1
[i2][i1]
H1 H2!
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The method: Common Subgroups
!" #$%%!&'()*(+$,!+$-(./012)/3.()*(%4'(%5)(-$%%!&'.
!!" 6-$..!*!&$%!)7(!7(&)78/1$%'(./012)/3.
!!!"97-:($(2'32'.'7%$%!;'(*)2('$&4(&-$..
<)2('$&4(&)++)7(./012)/3(=5!%4(>'%'2+!7'>(!7>!&'.?(@
SUBGROUPGRAPH program at www.cryst.ehu.es
Branch 1 Branch 2
Imm2
F-43m Fm-3m
Imm2 Imm2
Class 13 subgroups
Class 13 subgroups
Class 29 subgroups
COMMON SUBGROUPSCOMMON SUBGROUPS
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Symmetry conditions for TP
! !"#$"%"&'((')"$*+,-'*.
! /'(.%0#+#1#02"'3"045"$.1#00#),$"'3"045"'&&*.#56"78"
'3"045"09'"$0-*&0*-5$"#)"045"&'((')"$*+,-'*."!:
H
G1 G2
Order – disorder mechanisms not included!
A B4a 4c
B A 4b 4a
2b2a
H1 H2
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(P1,p1) (P2,p2)
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Symmetry conditions for TP
Maximal-symmetry transition paths are candidates for energetically favourable transient states!
G2G1
!"
!#!$
%&'()**+,-)'./012,2/0('3-+'&455&255+1'6)'!7'
&455&255+1'6)'355'2,('(468-/49(
Y1
H1 H2
Z2Z1
Symmetry conditions for TP
Y2
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Maximal symmetry transition paths
!"#"$%&"'&$(%&)%**&+,*$"-*").$"/#&"#&0&"1&*"+"$%2
3%$&/'&-.$(1&',*'"**"#4&$(%&$5/&16++%$76&)/#2"$"/#1&.#2&
(.8"#4&+.9"+.*&16++%$76
!!!!!
G1 G2
0: 0; 0<{ }0=
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Minimum deformation strain in the transformation
Minimum distance between the corresponding atoms in the initial structures described in the subgroup reference frame
Structural Conditions
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Strain Calculation
X1 = R1 · x1 X2 = R2 · x2 Ri =
!
"
aisin(!i)sin("!
i) 0 0
!aisin(!i)cos("!
i) bisin(#i) 0
aicos(!i) bicos(#i) ci
#
$
! =1
2(e + eT + eT e) =
1
2R!T
1(M1 ! M2)R
!1
1
Mi =
!
"
#
!ai · !ai !ai ·!bi !ai · !ci
!bi · !ai!bi ·
!bi!bi · !ci
!ci · !ai !ci ·!bi !ci · !ci
$
%
&
X2 ! X1 = e · X1 with e = R!1
2R1 ! I
standard root tensor
metric tensor
Valid for linear and non-linear strains!
Strain definition:
Degree of lattice distortion: S =1
3
!
"
"
#
3$
i=1
!2i
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!max < !tol
Establishes the connection between the two end phases in the subgroup reference frame
(defines the transition path)
Structural condition (maximum atomic distance in the transformation):
Mappings of the Atoms
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!! Wurtzite (Z=2, Zp=2)G1 = P63mc
!! Rocksalt (Z=4, Zp=1)G2= Fm-3m
Example: Wurtzite to rocksalt in GaN
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!! Rocksalt (Z=4, Zp=1)G2= Fm-3m
Example: Wurtzite to rocksalt in GaN
!! Wurtzite (Z=2, Zp=2)G1 = P63mc
What is the full set of symmetries, candidates for energetically favorable intermediate transient states?
!
"###!"##$%&##'%&###(
)###!"##$%&##'%&####*#+&%,
"####$##(#####(######(
)####%##$%'##$%'##$%'
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!"""!""""#"""$%""&%'#
("""!""""#"""$)""""&)
!! Wurtzite (Z=2, Zp=2)G1 = P63mc
!! Rocksalt (Z=4, Zp=1)G2= Fm-3m
Zp=2H = Cmc21
$%'"%*+
$)'"%*+
&),"+*-
%*.
%*.
%*)
-a-b, a+b, c a, b-1/4, c
Cmc21 TP: Displacements
!"""!#""%*+"")*+"""#
("""!#""%*+"")*+""""&"/+*-
!"""$"""#"""""#""""""#
("""$%""%*)""%*)""%*)
y1(A), y2(B) and z2(B)coordinates are
the internal degreesof freedom
A 2a 0 y1 z1=0
B 2a 0 y2 z2
A 2a 0 y1 z1=0
B 2a 0 y2 z2
-a-b,a+b,c a, b-1/4,c
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Cmc21 TP: Cell parameters
!"#$%&!"#$$'"$ !(#$!(#$!(
!)$$$$$$!"$$$$$$$$$!($
*)$$%&!"$$$$$$$$$$$$$$$!(')$$$$$$$'"$$$$$$$$$!(
!#*#'
!! Wurtzite (Z=2, Zp=2)G1 = P63mc
!! Rocksalt (Z=4, Zp=1)G2= Fm-3m
Zp=2H = Cmc21
-a-b, a+b, c a, b-1/4, c
+$$$!"$$,-&$$.-&$$$/0$$$!"$$,-&$$.-&$$$$1$2&-3
+$$$#$$$/$$$$$/$$$$$$/0$$$#%$$,-.$$,-.$$,-.
Cell parametersa,b,c are the
external degreesof freedom
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Maximal Transition Paths!! Rocksalt (Z=4, Zp=1)
G2= Fm-3m!! Wurtzite (Z=2, Zp=2)
G1 = P63mc
!"#$#%
"&'()#%**+
",)-./0123#03#)45#%**+
1044#-643#$#+ 1044#-643#$#7
G1 G2
!"#$% &'
% &'$
!"#$#8
1044#-643#$#7
!"#$#8
{ }
Pna21
?
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solution
P63mc Fm3m
Cmc21 Cmc21
Pna21 Pna21 Pna21 Pna21 Pna21
Maximal Transition Paths
i=48i=6
Class 1 Class 2 Class 3 Class 4
3 new Pna21 transition paths
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Maximal symmetry transition paths: S < 0.25 !tol < 2A
Example: Wurtzite to Rocksalt
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