Appendix A 32 Point Groups: Notations, Symmetry Elements ...978-1-4419-1417-0/1.pdf · Appendix A...
Transcript of Appendix A 32 Point Groups: Notations, Symmetry Elements ...978-1-4419-1417-0/1.pdf · Appendix A...
Appendix A
32 Point Groups: Notations, Symmetry Elements,
Crystalline Classes, Group Orders, Subgroups,
and Supergroups
Table A.1 Stroganov’s table for 32 point groups
Mono niccli
Triclinic
Orthorhombic
Trigonal
Hexagonal
Tetragonal
Cubic
polar centrosymmetric
2 m 2/m
mm2 mmm222
(C1 ) (Ci )
(C2 ) (Cs ) (C2h )
(C2v ) (D2 ) (D2h )
(C3 ) (S6 ) (C3v ) (D3 ) (D3d )
(S4 ) (D2d )(C4 ) (C4h ) (C4v ) (D4 ) (D4h )
(C3h ) (D3h )(C6 ) (C6h ) (C6v ) (D6 ) (D6h )
(T) (Th ) (Td ) (O) (Oh )432m323 43m m3m
6/mmm6mm6 62m 6226/m6
3 323m 3m3
1 1
4mm 4224 4/m 4/mmm4 42m
A.K. Tagantsev et al., Domains in Ferroic Crystals and Thin Films,DOI 10.1007/978-1-4419-1417-0, � Springer ScienceþBusiness Media, LLC 2010
713
Table A.2 Group orders, subgroups, and supergroups among the 32 point groups
The second column gives the order of the group. The asterisk at the top of each verticalcolumn indicates the supergroup. The �’s vertically below it indicate the subgroups whichbelong to this supergroup. Adapted from Bloss (1971).
714 Appendix A
Appendix B
Ferroic Species
Table B.1 Ferroic species
1 Species no. 2 Species designation 3 nF 4 q 5 qP 6 q" 7 qd 8 qs 9
001 m�3m� d� �43m 1 2 0 1 2 1
002 m�3m� 432 1 2 0 1 0 1 "[V2]
003 m�3m�m3 1 2 0 1 0 1 {V2V2}
004 m�3m� d� 23 1 4 0 1 2 1 RR
005 m�3m� "s� �3xyzm�xy 4 4 0 4 0 4
006 m�3m� P"ds� 3xyzm�xy 4 8 8 4 8 4
007 m�3m� "ds� 3xyz2�xy 4 8 0 4 8 4
008 m�3m� "s� �3xyz 4 8 0 4 0 8
009 m�3m� Pds� 3xyz 4 16 8 4 16 8 RR
010 m�3m� "s� 4z=mxmymz 3 3 0 3 0 3
011 m�3m� "ds� �4z2xymx 3 6 0 3 6 3 1
012 m�3m� "ds� �4z2xmxy 3 6 0 3 6 3 1
013 m�3m� P"ds� 4zmxmxy 3 6 6 3 6 3
014 m�3m� "ds� 4z2x2xy 3 6 0 3 6 3
015 m�3m�"s�4z/mz 3 6 0 3 0 6
016 m�3m� "ds� �4 3 12 0 3 12 6 RR
017 m�3m� P"ds� 4 3 12 6 3 12 6
018 m�3m� "s�mxmymz 1 6 0 6 0 6
019 m�3m� "s�mxym�xymz 3 6 0 6 0 6
020 m�3m� P"ds�mxmy2z 3 12 6 6 12 6
021 m�3m� P"ds�mxym�xy2z 3 12 6 6 12 6 RR
022 m�3m� P"ds�m�xymz2xy 6 12 12 6 12 6 RR
023 m�3m� "ds� 2x2y2z 1 12 0 6 12 6
024 m�3m� "ds� 2xy2�xy2z 3 12 0 6 12 6 RR
025 m�3m� "s� 2xy=mxy 6 12 0 12 0 12 2 IRs
026 m�3m� "s� 2z=mz 3 12 0 12 0 12 RR
027 m�3m� P"ds�mz 3 24 24 12 24 12 2 IRs
028 m�3m� P"ds�mxy 6 24 24 12 24 12
029 m�3m� P"ds� 2xy 6 24 12 12 24 12
030 m�3m� P"ds� 2z 3 24 6 12 24 12 RR
031 m�3m� "s� �1 1 24 0 24 0 24 2 IRs
032 m�3m� P"ds� 1 1 48 48 24 24 24 2 IRs
Table B.1 (continued)
1 Species no. 2 Species designation 3 nF 4 q 5 qP 6 q" 7 qd 8 qs 9
033 �43m� 23 1 2 0 1 1 1 "[V2]
034 �43m� P"ds� 3xyzm�xy 4 4 4 4 4 4
035 �43m� P"ds� 3xyz 4 8 4 4 4 8
036 �43m� "ds� �4z2xmxy 3 3 0 3 3 3
037 �43m� "ds� �4z 3 6 0 3 6 6
038 �43m� P"ds�mxym�xy2z 3 6 6 6 6 6
039 �43m� "ds� 222 1 6 0 6 6 6
040 �43m� P"ds�mxy 6 12 12 12 12 12 2 IRs
041 �43m� P"ds� 2z 3 12 6 12 12 12 RR
042 �43m� P"ds� 1 1 24 24 24 24 24 2 IRs
043 432�d�23 1 2 0 1 2 1
044 432� "ds� 3xyz2�xy 4 4 0 4 4 4
045 432�P"ds�3xyz 4 8 8 4 8 8
046 432�"ds�4z2x2xy 3 3 0 3 3 3
047 432�P"ds�4z 3 6 6 3 6 6
048 432�"ds�2x2y2z 1 6 0 6 6 6
049 432� "ds� 2xy2�xy2z 3 6 0 6 6 6
050 432�P"ds�2xy 6 12 12 12 12 12
051 432�P"ds�2z 3 12 6 12 12 12 RR
052 432�P"ds�1 1 24 24 24 24 24 2 IRs
053 m�3� d� 23 1 2 0 1 2 1
054 m�3� "s� �3xyz 4 4 0 4 0 4
055 m�3� P"ds� 3xyz 4 8 8 4 8 4
056 m�3� "s�mmm 1 3 0 3 0 3
057 m�3� P"ds�mxmy2z 3 6 6 3 6 3
058 m�3� "ds� 222 1 6 0 3 6 3
059 m�3� "s� 2z=mz 3 6 0 6 0 6
060 m�3� P"ds�mz 3 12 12 6 12 6
061 m�3� P"ds� 2z 3 12 6 6 12 6 RR
062 m�3� s� �1 1 12 0 12 0 12
063 m�3� P"ds� 1 1 24 24 12 24 12
064 23�P"ds�3xyz 4 4 4 4 4 4
065 23�"ds�222 1 3 0 3 3 3
066 23�P"ds�2 3 6 6 6 6 6
067 23�P"ds�1 1 12 12 12 12 12
068 6=mmm� d� �6zmy2x 1 2 0 1 2 1
6=mmm� d� �6zmx2y069 6/mmm�Pd�6mm 1 2 2 1 2 1
070 6/mmm�d�622 1 2 0 1 2 1
071 6/mmm�6/m 1 2 0 1 2 1 "V[V2]
072 6=mmm� d� �6 1 4 0 1 4 1 RR
073 6/mmm�Pd�6 1 4 2 1 4 1 RR
074 6=mmm� s� �3zmx 1 2 0 1 2 2
6=mmm� s� �3zmy
075 6=mmm� Pds� 3zmx 1 4 2 1 4 2 RR
6=mmm� Pds� 3zmy
716 Appendix B
Table B.1 (continued)
1 Species no. 2 Species designation 3 nF 4 q 5 qP 6 q" 7 qd 8 qs 9
076 6=mmm� ds� 3z2x 1 4 0 1 4 2 RR
6=mmm� ds� 3z2y077 6=mmm� s� �3 1 4 0 1 0 4 RR
078 6/mmm�Pds�3 1 8 2 1 8 4 RR
079 6=mmm� "s�mxmymz 3 3 0 3 3 3
080 6=mmm� P"ds�mxmy2z 3 6 2 3 6 3
081 6=mmm� P"ds�mxmz2y 3 6 6 3 6 3
6=mmm� P"ds�mymz2x082 6/mmm�"ds�222 3 6 0 3 6 3
083 6=mmm� "s� 2x=mx 3 6 0 6 0 6
6=mmm� "s� 2y=my
084 6/mmm�"s�2z/mz 1 6 0 6 0 6
085 6/mmm�P"ds�mx 3 12 12 6 12 6 RR
6/mmm�P"ds�my
086 6/mmm�P"ds�mz 1 12 12 6 12 6
087 6=mmm� P"ds� 2x 3 12 6 6 12 6 RR
6=mmm� P"ds� 2y088 6/mmm�P"ds�2z 1 12 2 6 12 6
089 6=mmm� s� �1 1 12 0 12 0 12
090 6/mmm�P"ds�1 1 24 24 12 24 12 RR
091 �6m2� d� �6 1 2 0 1 2 1
092 �6m2� Pds� 3m 1 2 2 1 2 2
093 �6m2� ds� 32 1 2 0 1 2 2
094 �6m2� Pds� 3 1 4 2 1 4 4 RR
095 �6m2� P"ds�mxmz2y 3 3 3 3 3 3
096 �6m2� P"ds�mz 1 6 6 6 6 6
097 �6m2� P"ds�mx 3 6 6 6 6 6
098 �6m2� P"ds� 2y 3 6 3 6 6 6
099 �6m2� P"ds� 1 1 12 12 12 12 12
100 6mm�d�6 1 2 1 1 2 1
101 6mm�ds�3zmx 1 2 1 1 2 2
6mm�ds�3zmy
102 6mm�ds�3 1 4 1 1 4 4 RR
103 6mm�"ds�mxmy2z 3 3 1 3 3 3
104 6mm� P"ds�mx 3 6 6 6 6 6
6mm� P"ds�my
105 6mm�"ds�2z 1 6 1 6 6 6
106 6mm�P"ds�1 1 12 12 12 12 12
107 622�Pd�6 1 2 2 1 2 1
108 622�ds�3z2x 1 2 0 1 2 2
622��ds�3z2y109 622�Pds�3 1 4 2 1 4 4 RR
110 622�"ds�2x2y2z 3 3 0 3 3 3
111 622� P"ds� 2x 3 6 6 6 6 6
622� P"ds� 2y112 622�P"ds�2z 1 6 2 6 6 6
Appendix B 717
Table B.1 (continued)
1 Species no. 2 Species designation 3 nF 4 q 5 qP 6 q" 7 qd 8 qs 9
113 622�P"ds�1 1 12 12 12 12 12
114 6=m� d� �6 1 2 0 1 2 1
115 6/m�Pd�6 1 2 2 1 2 1
116 6=m� s� �3 1 2 0 1 0 2
117 6/m�Pds�3 1 4 2 1 4 2 RR
118 6/m�"s�2/m 1 3 0 3 0 3
119 6/m�P"ds�m 1 6 6 3 6 3
120 6/m�P"ds�2 1 6 2 3 6 3
121 6=m� "s� �1 1 6 0 6 0 6
122 6/m�P"ds�1 1 12 12 6 12 12 RR
123 �6� Pds� 3 1 2 2 1 2 2
124 �6� P"ds�m 1 3 3 3 3 3
125 �6� P"ds� 1 1 6 6 6 6 6
126 6�ds�3 1 2 1 1 2 2
127 6�"ds�2 1 3 1 3 3 3
128 6�P"ds�1 1 6 6 6 6 6
129 �3m� Pd� 3m 1 2 2 1 2 1
130 �3m� d� 32 1 2 0 1 2 1
131 �3m� s� �3 1 2 0 1 0 2
132 �3m� Pds� 3 1 4 2 1 4 2 RR
133 �3m� "s� 2=m 3 3 0 3 0 3
134 �3m� P"ds�m 3 6 6 3 6 3
135 �3m� P"ds� 2 3 6 6 3 6 3
136 �3m� "s� �1 1 6 0 6 0 6
137 �3m� P"ds� 1 1 12 12 6 12 6
138 3m�ds�3 1 2 1 1 2 2
139 3m�P"ds�m 3 3 3 3 3 3
140 3m�P"ds�1 1 6 6 6 6 6
141 32�Pds�3 1 2 2 1 2 2
142 32�P"ds�2 3 3 3 3 3 3
143 32�P"ds�1 1 6 6 6 6 6
144 �3� Pd� 3 1 2 2 1 2 1
145 �3� "s� �1 1 3 0 3 0 3
146 �3� P"ds� 1 1 6 6 3 6 3
147 3�P"ds�1 1 3 3 3 3 3
148 4=mmm� d� �4z2xmxy 1 2 0 1 2 1
4=mmm� d� �4z2xymx
149 4/mmm�Pd�4mm 1 2 2 1 2 1
150 4/mmm�d�422 1 2 0 1 2 1
151 4/mmm�s�4/m 1 2 0 1 0 2
152 4=mmm� ds� �4 1 4 0 1 4 2 RR
153 4/mmm�Pds�4 1 4 2 1 4 2 RR
154 4=mmm� "s�mxmymz 1 2 0 2 0 2
4=mmm� "s�mxym�xymz
155 4/mmm�P"ds�mymz2x 2 4 4 2 4 2
4=mmm� P"ds�m�xymz2xy
718 Appendix B
Table B.1 (continued)
1 Species no. 2 Species designation 3 nF 4 q 5 qP 6 q" 7 qd 8 qs 9
156 4/mmm�P"ds�mxmy2z 1 4 2 2 4 2 RR
4=mmm� P"ds�mxym�xy2z157 4=mmm� "ds� 2x2y2z 1 4 0 2 4 2 RR
4=mmm� "ds� 2xy2�xy2z158 4=mmm� "s� 2x=mx 2 4 0 4 0 4
4=mmm� "s� 2xy=mxy
159 4/mmm�"s�2z/mz 1 4 0 4 0 4 RR
160 4=mmm� P"ds�mx 2 8 8 4 8 4 RR
4=mmm� P"ds�mxy
161 4/mmm�P"ds�mz 1 8 8 4 8 4
162 4=mmm� P"ds� 2x 2 8 4 4 8 4 RR
4=mmm� P"ds� 2xy RR
163 4/mmm�P"ds�2z 1 8 2 4 8 4 RR
164 4=mmm� "s� �1 1 8 0 8 0 8
165 4/mmm�P"ds�1 1 16 16 8 16 8 RR
166 �42m� ds� �4 1 2 0 1 2 2
167 �42m� P"ds�mm2 1 2 2 2 2 2
168 �42m� "ds� 222 1 2 0 2 2 2
169 �42m� P"ds�mxy 2 4 4 4 4 4
170 �42m� P"ds� 2x 2 4 4 4 4 4
171 �42m� P"ds� 2z 1 4 2 4 4 4 RR
172 �42m� P"ds� 1 1 8 8 8 8 8
173 4mm�ds�4 1 2 1 1 2 2
174 4mm� "ds�mxmy2z 1 2 1 2 2 2
4mm� "ds�mxym�xy2z175 4mm� P"ds�mx 2 4 4 4 4 4
4mm� P"ds�mxy
176 4mm�"ds�2 1 4 1 4 4 4 RR
177 4mm�P"ds�1 1 8 8 8 8 8
178 422�Pds�4 1 2 2 1 2 2
179 422� "ds� 2x2y2z 1 2 0 2 2 2
422� "ds� 2xy2�xy2z180 422� P"ds� 2x 2 4 4 4 4 4
422� P"ds� 2xy181 422�P"ds�2z 1 4 2 4 4 4 RR
182 422�P"ds�1 1 8 8 8 8 8
183 4=m� d� �4 1 2 0 1 2 1
184 4/m�Pd�4 1 2 2 1 2 1
185 4/m�"s�2/m 1 2 0 2 0 2
186 4/m�P"ds�m 1 4 4 2 4 2
187 4/m�P"ds�2 1 4 2 2 4 2 RR
188 4/m�"s��1 1 4 0 4 0 4
189 4/m�P"ds�1 1 8 8 4 8 4 RR
190 �4� P"ds� 2 1 2 2 2 2 2
191 �4� P"ds� 1 1 4 4 4 4 4
192 4�"ds�2 1 2 1 2 2 2
Appendix B 719
Table B.1 (continued)
1 Species no. 2 Species designation 3 nF 4 q 5 qP 6 q" 7 qd 8 qs 9
193 4�P"ds�1 1 4 4 4 4 4
194 mmm� Pd�mxmy2z 1 2 2 1 2 1
mmm� Pd�mxmz2ymmm� Pd�mymz2x
195 mmm� d� 222 1 2 0 1 2 1
196 mmm� "s� 2x=mz 1 2 0 2 0 2
mmm� "s� 2y=my
mmm� "s� 2z=mx
197 mmm� P"ds�mx 1 4 4 2 4 2 RR
mmm� P"ds�my
mmm� P"ds�mz
198 mmm� P"ds� 2x 1 4 2 2 4 2 RR
mmm� P"ds� 2ymmm� P"ds� 2z
199 mmm� "s� �1 1 4 0 4 0 4 RR
200 mmm� P"ds� 1 1 8 8 4 8 4 RR
201 mm2� P"ds�mx 1 2 2 2 2 2
mm2� P"ds�my
202 mm2� "ds� 2 1 2 1 2 2 2
203 mm2� P"ds� 1 1 4 4 4 4 4 RR
204 222�P"ds�2x 1 2 2 2 2 2
222�P"ds�2y222�P"ds�2z
205 222� P"ds� 1 1 4 4 4 4 4 RR
206 2=m� Pd�m 1 2 2 1 2 1
207 2=m� Pd� 2 1 2 2 1 2 1
208 2=m� "s� �1 1 2 0 2 0 2
209 2=m� P"ds� 1 1 4 4 2 4 2 RR
210 m� P"ds� 1 1 2 2 2 2 2
211 2� P"ds� 1 1 2 2 2 2 2
212 �1� Pd� 1 1 2 2 1 2 1
Column 1: sequential number of the species. Column 2: symbol of the species. Column 3:number of equivalent subgroups F. Column 4: total number of domain states. Column 5:number of ferroelectric domain states. Here 0 means that F is nonpolar while 1 means that F ispyroelectric with only one orientation of PS (or better to say, of the pyroelectric coefficient).Column 6: number of ferroelastic domain states. Number 1 means that F is nonferroelastic.Column 7: number qd of domain states differing in the piezoelectric tensor or, more generally,in the tensor of symmetry V[V2] or [V2]V. Here number 0 means that F is not piezoelectricwhile 1 means that F is piezoelectric but has no new components of the appropriate tensor.Column 8: number of ferrobielastic domain states. Here 1 means that the compliance matrixhas no new components in the distorted phase. Column 9: the lowest order tensor in whichsome domain states are distinguished for higher order ferroics; symbols ‘‘2 IRs’’ and ‘‘RR’’indicate that the transition is induced by two different irreducible representations or by areducible representation, respectively
720 Appendix B
TableC.1
Phase
transitionsinto
ferroelectricphases
1Species
no.
2Speciesdesignation
3q
4Data
on
polarization
5qP
6Data
onstrain
7q"
8Shuvalov’s
symbol
Cubicparentphase
d"xx,d"
xx,d"
xx,0,0,0
006
m� 3m�P"ds�3xyzm
� xy
8Px=
Py=
Pz
8d"
xx,d"
xx,d"
xx," y
z," y
z," y
z4
m3m(4)D
3F3m=
009
m� 3m�P"ds�3xyz
16
[Px=Py=Pz]
8d"
xx,d"
xx,d"
xx," y
z," y
z," y
z4
*m3m(4)D
3F3=
013
m� 3m�P"ds�4zm
xm
xy
6Pz
6d"
xx,d"
xx,d"
zz,0,0,0
3m3m(3)D
4F4mm=
017
m� 3m�P"ds�4z
12
[Pz]
6d"
xx,d"
xx,d"
zz,0,0,0
3*m3m(3)D
4F4=
020
m� 3m�P"ds�m
xm
y2z
12
[Pz]
6d"
xx,d"
yy,d"
zz,0,0,0
6*m3m(3)D
2Fmm26¼
021
m� 3m�P"ds�m
xym
� xy2z
12
[Pz]
6d"
xx,d"
xx,d"
zz,0,0," x
y6
*m3m(3)D
2Fmm26¼
022
m� 3m�P"ds�m
� xym
z2xy
12
Px=
Py(P
x=Py)
12
d"xx,d"
xx,d"
zz,0,0," x
y6
m3m(6)D
2Fmm2=
027
m� 3m�P"ds�m
z24
Px,Py(P
x,Py)
24
d"xx,d"
yy,d"
zz,0,0,"xy
12
m3m(12)A
4Fm=
028
m� 3m�P"ds�m
xy
24
Px=�Py,Pz
24
d"xx,d"
xx,d"
zz," y
z,�" y
z," x
y12
m3m(12)A
*2Fm=
029
m� 3m�P"ds�2xy
24
(Py=
Px)
12
d"xx,d"
xx,d"
zz," y
z,�" y
z," x
y12
*m3m(3)D
2F26¼
030
� 3m�P"ds�2z
24
[Pz]
6d"
xx,d"
yy,d"
zz,0,0," x
y12
*m3m(3)D
4F26¼
032
m� 3m�P"ds�1
48
Px,Py,Pz
(Px,Py,Pz)
48
d"xx,d"
yy,d"
zz," y
z," z
x," x
y24
*m3m(24)A
1F1=
034
3m�P"ds�3xyzm
� xy
4Px=
Py=
Pz
4d"
xx,d"
xx,d"
xx," y
z," y
z," y
z4
� 43m(4/2)D
3F3m
035
� 43m�P"ds�3xyzm
� xy
8(P
x=Py=Pz)
4d"
xx,d"
xx,d"
xx," y
z," y
z," y
z4
*� 43m(4/2)D
3F3
038
� 43m�P"ds�m
xyzm
� xy2z
6Pz
6d"
xx,d"
xx,d"
zz,0,0," x
y6
� 43m(3)D
� 4Fmm26¼
040
� 43m�P"ds�m
xy
12
Px=
Py,Pz
(Px=
Py,Pz)
12
d"xx,d"
xx,d"
zz," y
z,�" y
z," x
y12
� 43m(12/2)A
mFm
041
� 43m�P"ds�2z
12
[Pz]
6d"
xx,d"
yy,d"
zz,0,0," x
y12
*� 43m(6/2)D
� 4F2
Appendix
C
Phase
Transitionsinto
FerroelectricPhases
TableC.1
(continued)
1Species
no.
2Speciesdesignation
3q
4Data
on
polarization
5qP
6Data
onstrain
7q"
8Shuvalov’s
symbol
042
� 43m�P"ds�1
24
Px,Py,Pz
(Px,P
y,P
z)
24
d"xx,d"
yy,d"
zz," y
z," z
x," x
y24
� 43m(24/2)A
1F1
045
432�P"ds�
3xyyz
8Px=
Py=
Pz
8d"
xx,d"
xx,d"
xx,"yz," y
z," y
z4
432(4)D
3F3=
047
432�P"ds�
4z
6Pz
6d"
xx,d"
xx,d"
zz,0,0,0
3432(3)D
4F4=
050
432�P"ds�
2xy
12
Px=
Py
(Px=
Py)
12
d"xx,d"
xx,d"
zz," y
z,�" y
z," x
y12
432(6)D
2F26¼
051
432�P"ds�
2z
12
[Pz]
6d"
xx,d"
yy,d"
zz,0,0," x
y12
*432(3)D
2F26¼
052
432�P"ds�
124
Px,Py,Pz
(Px,Py,Pz)
24
d"xx,d"
yy,d"
zz," y
z," z
x," x
y24
432(24/2)A
1F1
055
m� 3�P"ds�3xyz
8Px=
Py=
Pz
8d"
11,d"
11,d"
11," 2
3," 2
3," 2
34
m3(4)D
3F3=
057
m� 3�P"ds�m
xm
y2z
6Pz
6d"
11,d"
22,d"
33,0,0,0
3m3(3)D
mFmm2=
060
m� 3�P"ds�m
z12
Px,Py
12
d"11,d"
22,d"
33,0,0," 1
26
m3(6)A
mFm=
061
m� 3�P"ds�2z
12
[Pz]
6d"
11,d"
22,d"
33,0,0," 1
26
*m3(3)D
2F26¼
063
m� 3�P"ds�1
24
Px,Py,Pz
24
d"11,d"
22,d"
33," 2
3," 3
1," 1
212
m3(12)A
1F1=
064
23�P"ds�
3xyyz
4Px=
Py=
Pz
4d"
11,d"
11,d"
11," 2
3," 2
3," 2
34
23(4/2)D
3F3
066
23�P"ds�
2z
6Pz
6d"
11,d"
22,d"
33,0,0," 1
26
23(3)D
2F26¼
067
23�P"ds�
112
Px,Py,Pz
12
d"11,d"
22,d"
33," 2
3," 3
1," 1
212
23(12/2)A
1F1
Hexagonalandtrigonalparentphases
d"xx,d"
xx,d"
zz,0,0,0
069
6/m
mm�Pd�6mm
2Pz
2d"
xx,d"
xx,d"
zz,0,0,0
16/m
mm(1)
D6F6mm=
073
6/m
mm�Pd�6
4[P
z]
2d"
xx,d"
xx,d"
zz,0,0,0
1*6/m
mm(1)D
6F6=
075
6/m
mm�Pds�
3zm
x
6/m
mm�Pds�
3zm
y
4[P
z]
2d"
xx,d"
xx,d"
zz,0,0,0
1*6/m
mm(1)
D6F3m=
078
6/m
mm�Pds�
38
[Pz]
2d"
xx,d"
xx,d"
zz,0,0,0
1*6/m
mm(1)D
6F3=
080
6/m
mm�P"ds�
mxm
y2z
6(P
z)
6d"
xx,d"
yy,d"
zz,0,0,0
3*6/m
mm(3)
D6Fmm2=
081
6/m
mm�P"ds�
mxm
z2y
6Py
d"xx,d"
yy,d"
zz,0,0,0
36/m
mm(3)
D2Fmm2=
722 Appendix C
TableC.1
(continued)
1Species
no.
2Speciesdesignation
3q
4Data
on
polarization
5qP
6Data
onstrain
7q"
8Shuvalov’s
symbol
6/m
mm�P"ds�
mym
z2x
Px
d"xx,d"
yy,d"
zz,0,0,0
085
6/m
mm�P"ds�
mx
12
[Py,Pz]
12
d"xx,d"
yy,d"
zz," y
z,0,0
66/m
mm(6)A
6Fm=
6/m
mm�P"ds�
my
[Px,Pz]
d"xx,d"
yy,d"
zz,�
ffiffiffi 3
p" z
x,
" zx,(ffiffiffi 3
p=2)(d"
xx�d"
yy)
086
6/m
mm�P"ds�
mz
12
Px,Py
12
d"xx,d"
yy,d"
zz,0,0," x
y6
6/m
mm(6)A
6Fm=
087
6/m
mm�P"ds�
2x
12
[Px]
6d"
xx,d"
yy,d"
zz," y
z,0,0
6*6/m
mm(3)D
2F2
6/m
mm�P"ds�
2y
[Py]
d"xx,d"
yy,d"
zz,�
ffiffiffi 3
p" z
x" z
x," z
x,
(ffiffiffi 3
p=2)(d"
xx�d"
yy)
088
6/m
mm�P"ds�
2z
12
(Pz)
2d"
xx,d"
yy,d"
zz,0,0," x
y6
*6/mmmD2 zFmm26¼
090
6/m
mm�P"ds�
124
[Px,Py,Pz]
24
d"xx,d"
yy,d"
zz," y
z," z
x," x
y12
6/m
mm(12)A
1F1=
092
� 6m2�Pds�3m
2Pz
2d"
xx,d"
xx,d"
zz,0,0,0
1*� 6m
2(1)D
� 6F3m=
094
� 6m2�Pds�3
4[P
z]
2d"
xx,d"
xx,d"
zz,0,0,0
1� 6m2(1)D
� 6F3=
095
� 6m2�P"ds�m
ym
z2x
3Px
3d"
xx,d"
yy,d"
zz,0,0,
(ffiffiffi 3
p=2)(d"
xx�d"
yy)
3*� 6m
2(3/2)D2F
mm2
096
� 6m2�P"ds�m
z6
Px,Py
6d"
xx,d"
yy,d"
zz,0,0," x
y6
� 6m2(6/2)A
� 6Fm
097
� 6m2�P"ds�m
x6
(Py,Pz)
6d"
xx,d"
yy,d"
zz," y
z,0,0
6� 6m2(6/2)A
mxFm
098
� 6m2�P"ds�2x
6(P
y)
3d"
xx,d"
yy,d"
zz,ffiffiffi 3
p" z
x," z
x,(ffiffiffi 3
p=2)
(d" x
x-d" y
y)
6*� 6m
2(12/2)A
1F1
099
� 6m2�P"ds�1
12
(Px,Py,Pz)
12
d"xx,d"
yy,d"
zz," y
z," z
x," x
y12
� 6m2(12/2)A
1F1
104
6mm�P"ds�
my
6Py,dP
z6
d"xx,d"
yy,d"
zz," y
z,0,0
66mm(6/2)A
mFm
6mm�P"ds�
mx
Px,dP
zd"
xx,d"
yy,d"
zz,-�
ffiffiffi 3
p" z
x," z
x,
(ffiffiffi 3
p=22)(d"
xx�d"
yy)
106
6mm�P"ds�
112
Px,Py,dP
z12
d"xx,d"
yy,d"
zz," y
z," z
x," x
y12
6mm(12/2)A
1F1
107
622�Pd�6
2Pz
2d"
xx,d"
xx,d"
zz,0,0,0
1622(1)D
6F6||
109
622�Pds�
34
[Pz]
2d"
xx,d"
xx,d"
zz,0,0,0
1*622(12)D
6F3=
111
622�P"ds�
2x
6Px
6d"
xx,d"
yy,d"
zz,�
ffiffiffi 3
p" z
x," z
x,
(ffiffiffi 3
p=2)(d"
xx�d"
yy)
6622(3)D
2F26¼
622�P"ds�
2y
Py
d"xx,d"
yy,d"
zz," y
z,0,0
Appendix C 723
TableC.1
(continued)
1Species
no.
2Speciesdesignation
3q
4Data
on
polarization
5qP
6Data
onstrain
7q"
8Shuvalov’s
symbol
112
622�P"ds�
2z
2(P
z)
2d"
xx,d"
yy,d"
zz,0,0," x
y6
*622(1)D
6F26¼
113
622�P"ds�
112
Px,Py,(P
z)
12
d"xx,d"
yy,d"
zz," y
z," z
x," x
y12
622(12/2)A
1F1
115
6/m�Pd�6
2Pz
2d"
xx,d"
xx,d"
zz,0,0,0
16/m
(1)D
6F6=
117
6/m�Pds�
34
[Pz]
2d"
xx,d"
xx,d"
zz,0,0,0
1*6/m
D6F3=
119
6/m�P"ds�
m6
Px,Py
6d"
xx,d"
yy,d"
zz,0,0," x
y3
6/m
(3)A
6Fm=
120
6/m�P"ds�
26
(Pz)
2d"
xx,d"
yy,d"
zz,0,0," x
y3
*6/m
(1)D
6F2
122
6/m�P"ds�
112
[Px,Py,Pz]
12
d"xx,d"
yy,d"
zz," y
z," z
x," x
y6
6/m
(6)A
1F1=
123
� 6�Pds�3
2Pz
2d"
xx,d"
xx,d"
zz,0,0,0
1� 6ð1ÞD
� 6F3¼
124
� 6�P"ds�m
3Px,Py
3d"
xx,d"
yy,d"
zz,0,0," x
y3
� 6ð3=2ÞA
� 6Fm
125
� 6�P"ds�1
6(P
x,Py,Pz)
6d"
xx,d"
yy,d"
zz," y
z," z
x," x
y6
� 6ð6=2ÞA
1F1
128
6�P"ds�
16
Px,Py,dP
z6
d"xx,d"
yy,d"
zz," y
z," z
x," x
y6
6(6/2)A
1F1
129
� 3m�Pd�3m
2Pz
2d"
xx,d"
xx,d"
zz,0,0,0
1� 3mð1ÞD
� 3F3m¼
132
� 3m�Pds�3m
4[P
z]
2d"
xx,d"
xx,d"
zz,0,0,0
1�� 3mð1ÞD
� 3F3¼
134
� 3m�P"ds�
mx
6Py,(P
z)
6d"
xx,d"
xx,d"
zz," y
z,0,0
3� 3mð3ÞA
mFm¼
135
� 3m�P"ds�2x
6Px
6d"
xx,d"
xx,d"
zz," y
z,0,0
3� 3mð3ÞD
mF2¼
137
� 3m�P"ds�1
12
Px,Py,(P
z)
12
d"xx,d"
yy,d"
zz," y
z," z
x," x
y6
� 3mð6ÞA
1F1¼
139
3m�P"ds�
mx
3Py,dP
z3
d"xx,d"
xx,d"
zz," y
z,0,0
33m(3/2)A
mFm
140
3m�P"ds�
16
Px,Py,dP
z6
d"xx,d"
yy,d"
zz," y
z," z
x," x
y6
3m(6/2)A
1F1
141
32�Pds�
32
Pz
2d"
xx,d"
xx,d"
zz,0,0,0
132(1)D
3F3=
142
32�P"ds�
2x
3Px
3d"
xx,d"
xx,d"
zz," y
z,0,0
332(3/2)D
2F2
143
32�P"ds�
16
Px,Py,(P
z)
6d"
xx,d"
yy,d"
zz," y
z," z
x," x
y6
32(6/2)A
1F1
144
� 3�Pd�3
2Pz
2d"
xx,d"
xx,d"
zz,0,0,0
1� 3ð1ÞD
3F3¼
146
� 3�P"ds�1
6Px,Py,(P
z)
6d"
xx,d"
yy,d"
zz," y
z," z
x," x
y3
� 3ð3ÞA
1F1¼
147
3�P"ds�
13
Px,Py
3d"
xx,d"
yy,d"
zz," y
z," z
x," x
y3
3(3/2)A
1F1
724 Appendix C
TableC.1
(continued)
1Species
no.
2Speciesdesignation
3q
4Data
on
polarization
5qP
6Data
onstrain
7q"
8Shuvalov’s
symbol
Tetragonalparentphase
d"xx,d"
xx,d"
zz,0,0,0
149
4/m
mm�Pd�4mm
2Pz
2d"
xx,d"
xx,d"
zz,0,0,0
14/m
mm(1)D
4F4mm=
153
4/m
mm�Pds�
44
[Pz]
2d"
xx,d"
xx,d"
zz,0,0,0
1*4/m
mm(1)D
4F4=
155
4/m
mm�P"ds�
mym
z2x
4Px
4d"
xx,d"
yy,d"
zz,0,0,0
24/m
mm(2)D
2Fmm2=
4=mmm�P"ds�m
� xym
z2xy
Px=
Py
d"xx,d"
xx,d"
zz,0,0," x
y
156
4/m
mm�P"ds�
mxm
y2z
4[P
z]
2d"
xx,d"
yy,d"
zz,0,0,0
2*4/m
mm(2)D
4Fmm26¼
4=mmm�P"ds�m
xym
� xy2z
d"xx,d"
xx,d"
zz,0,0," x
y
160
4/m
mm�P"ds�
mx
8[P
y,Pz]
8d"
xx,d"
yy,d"
zz," y
z,0,0
4*4/m
mm(4)A
2Fm=
4/m
mm�P"ds�
mxy
[Px=Py,Pz]
d"xx,d"
xx,d"
zz," y
z,�" y
z," x
y
161
4/m
mm�P"ds�
mz
8Px,Py
8d"
xx,d"
yy,d"
zz,0,0," x
y4
4/m
mm(4)A
4Fm=
162
4/m
mm�P"ds�
2x
8[P
x]
4d"
xx,d"
yy,d"
zz," y
z,0,0
4*4/m
mm(2)D
4F2
4/m
mm�P"ds�
2xy
[Px=
Py]
d"xx,d"
xx,d"
zz," y
z,�" y
z," x
y
163
4/m
mm�P"ds�
2z
8[P
z]
2d"
xx,d"
yy,d"
zz,0,0," x
y4
*4/m
mm(1)D
4F2
165
4/m
mm�P"ds�
116
[Px,Py,Pz]
16
d"xx,d"
yy,d"
zz," y
z," z
x," x
y8
4/m
mm(8)A
1F1=
167
� 42m�P"ds�
mm2
2Pz
2d"
xx,d"
xx,d"
zz,0,0," x
y2
� 42m(1)D
� 4Fmm26¼
169
� 42m�P"ds�
mxy
4Px=
Py,Pz
4d"
xx,d"
xx,d"
zz," y
z,�" y
z," x
y4
� 42m(4/2)A
mFm
170
� 42m�P"ds�
2x
4Px
4d"
xx,d"
yy,d"
zz," y
z,0,0
4� 42m(2)D
2F26¼
171
� 42m�P"ds�
2z
4[P
z]
2d"
xx,d"
yy,d"
zz,0,0," x
y4
*� 42m(1)D
� 4F2
172
� 42m�P"ds�
18
Px,Py,(P
z)
8d"
xx,d"
yy,d"
zz," y
z," z
x," x
y8
� 42m(8/2)A
1F1
175
4mm�P"ds�
mx
4Py,dP
z4
d"xx,d"
yy,d"
zz," y
z,0,0
44mm(4/2)A
mFm
4mm�P"ds�
mxy
Px=
Py,dP
zd"
xx,d"
xx,d"
zz," y
z,�" y
z," x
y
177
4mm�P"ds�
18
Px,Py,dP
z8
d"xx,d"
yy,d"
zz," y
z," z
x," x
y8
4mm(8/2)A
1F1
178
422�Pds�
42
Pz
2d"
xx,d"
xx,d"
zz,0,0,0
1422(1)D
4F4=
180
422�P"ds�
2x
4Px
4d"
xx,d"
yy,d"
zz," y
z,0,0
4422(2)D
2F26¼
422�P"ds�
2xy
Px=
Py
d"xx,d"
xx,d"
zz," y
z,�" y
z," x
y
181
422�P"ds�
2z
4[P
z]
2d"
xx,d"
yy,d"
zz,0,0," x
y4
*422(1)D
4F36¼
182
422�P"ds�
18
Px,Py,(P
z)
8d"
xx,d"
yy,d"
zz," y
z," z
x," x
y8
422(8/2)A
1F1
Appendix C 725
TableC.1
(continued)
1Species
no.
2Speciesdesignation
3q
4Data
on
polarization
5qP
6Data
onstrain
7q"
8Shuvalov’s
symbol
184
4/m�Pd�4
2Pz
2d"
xx,d"
xx,d"
zz,0,0,0
14/m
(1)D
4F4=
186
4/m�P"ds�
m4
Px,Py
4d"
xx,d"
yy,d"
zz,0,0," x
y2
4/m
(2)A
4Fm=
187
4/m�P"ds�
24
[Pz]
d"xx,d"
yy,d"
zz,0,0," x
y2
*4/m
(1)D
4F26¼
189
4/m�P"ds�
18
[Px,Py,Pz]
8d"
xx,d"
yy,d"
zz," y
z," z
x," x
y4
4/m
(4)A
1F1=
190
� 4�P"ds�2
2Pz
2d"
xx,d"
yy,d"
zz,0,0," x
y2
� 4ð1ÞD
� 4F2
191
� 4�P"ds�1
4Px,Py,(P
z)
4d"
xx,d"
yy,d"
zz," y
z," z
x," x
y4
� 4ð4=2ÞA
1F1
193
4�P"ds�
14
Px,Py,dP
z4
d"xx,d"
yy,d"
zz," y
z," z
x," x
y4
4(4/2)A
1F1
Orthorhombicparentphase
d"xx,d"
yy,d"
zz,0,0,0
194
mmm�Pd�m
xm
y2z
2Pz
2d"
xx,d"
yy,d"
zz,0,0,0
1mmm(1)D
2Fmm2=
mmm�Pd�m
xm
z2y
Py
d"xx,d"
yy,d"
zz,0,0,0
mmm�Pd�m
ym
z2x
Px
d"xx,d"
yy,d"
zz,0,0,0
197
mmm�P"ds�
mx
4[P
y,Pz]
4d"
xx,d"
yy,d"
zz," y
z,0,0
2mmm(2)A
mFm=
mmm�P"ds�
my
[Px,Pz]
d"xx,d"
yy,d"
zz,0," z
x,0
mmm�P"ds�
mz
[Px,Py]
d"xx,d"
yy,d"
zz,0,0," x
y
198
mmm�P"ds�
2x
4[P
x]
2d"
xx,d"
yy,d"
zz," y
z,0,0
2*mmm(1)D
2F26¼
mmm�P"ds�
2y
[Py]
d"xx,d"
yy,d"
zz,0," z
x,0
mmm�P"ds�
2z
[Pz]
d"xx,d"
yy,d"
zz,0,0," x
y
200
mmm�P"ds�
18
[Px,Py,Pz]
8d"
xx,d"
yy,d"
zz," y
z," z
x," x
y4
mmm(4)A
1F1=
201
mm2�P"ds�
mx
2Py,dP
z2
d"xx,d"
yy,d"
zz," y
z,0,0
2mm2(2/2)A
mFm
mm2�P"ds�
my
Px,dP
zd"
xx,d"
yy,d"
zz,0," z
x,0
203
mm2�P"ds�
14
[Px,y],dP
z4
d"xx,d"
yy,d"
zz," y
z," z
x," x
y4
mm2(4/2)A
1F1
204
222�P"ds�
2x
2Px
2d"
xx,d"
yy,d"
zz," y
z,0,0
2*222(1)D
2F26¼
222�P"ds�
2y
Py
d"xx,d"
yy,d"
zz,0," z
x,0
222�P"ds�
2z
Pz
d"xx,d"
yy,d"
zz,0,0," x
y
205
222�P"ds�
14
[Px,Py,Pz]
4d"
xx,d"
yy,d"
zz," y
z," z
x," x
y4
222(2/2)A
1F1
726 Appendix C
TableC.1
(continued)
1Species
no.
2Speciesdesignation
3q
4Data
on
polarization
5qP
6Data
onstrain
7q"
8Shuvalov’s
symbol
monoclinicparentphase
d"xx,d"
yy,d"
zz,0,0,d"
xy
206
2/m�Pd�m
2Px,Py
2d"
xx,d"
yy,d"
zz,0,0,d"
xy
1*2/m
(1)A
mFm=
207
2/m�Pd�2
2Pz
2d"
xx,d"
yy,d"
zz,0,0,d"
xy
12/m
(1)D
2F2=
209
2/m�P"ds�
14
[Px,Py,Pz]
4d"
xx,d"
yy,d"
zz," y
z," z
x,d"
xy
22/m
(2)A
1F1=
210
m�P"ds�
12
Pz
2d"
xx,d"
yy,d"
zz," y
z," z
x,d"
xy
2m(2/2)A
1F1
211
2�P"ds�
12
Px,Py
2d"
xx,d"
yy,d"
zz," y
z," z
x,d"
xy
22(2/2)A
1F1
Triclinicparentphase
d"xx,d"
yy,d"
zz,d"
yz,d"
zx,d"
xy
212
� 1�Pd�1
1Px,Py,Pz
2d"
xx,d"
yy,d"
zz,d"
yz,d"
zx,d"
xy
1� 1ð1ÞA
1F1¼
Column1:
sequentialnumber
ofthespecies.Column2:symbolofthespecies.Column3:totalnumber
ofdomainstates.Column4:
nonzero
componentsof
polarizationin
thephaseF,specifiedforthesubgrouporientationshownin
thesymbol.dP
k,changeofpolarizationcomponentwhichwasalreadypermitted
bysymmetry
intheparentphase;P
k,n
ewcomponentofspontaneouspolarization.P
kwithoutbrackets—
‘‘proper’’componentofpolarization(assumingno
unit-cellm
ultiplication),i.e.,P
kserves
astheorder
parameter.P
kin
brackets—
‘‘improper’’polarizationcomponent,arisingasasecondary
effect.T
wolines
are
given
inthetableifboth
proper
andim
proper
scenarioscanberesponsibleforthedevelopmentofpolarization.T
hedistinctionbetween(P
k)an
d[P
k]—
seetextin
Sect.2.1.9.Column5:number
offerroelectricdomain
states.Column6:strain
components(inthecoordinate
system
ofG)compatiblewiththe
symmetry
ofthephase
F.Ifthecomponentis‘‘symmetry
breaking’’(notallowed
inG),itiswritten
as" k
l.Ifitisalreadyallowed
inG,ithastheprefixd.
(Rem
ark:twocomponentsd"
kl,d"
mncompatiblewiththesymmetry
ofG,w
hichare
equalinG,m
aybecomedifferentin
thephase
F.Itisthen
thedifference
d"Skl�d"
Smnbetweenthem
which‘breaks’thesymmetry.)Column7:numberofferroelasticdomain
states.Number1meansthatFisnonferroelastic.Column
8:Shuvalov’ssymbol.Topofeach
section:strain
componentscompatiblewiththesymmetry
oftheparentphase,in
thesequence
d"xx,d"
yy,d"
zz,d"
yz,d"
zx,
d"xy
Appendix C 727
Appendix D
Spontaneous Polarization, Spontaneous Strain,
and Orientation of Domain Walls in Ferroic
Species
In the tables of this appendix, one finds the information on the spontaneouspolarization, spontaneous strain, and orientation of domain walls (DW) in allferroic species. These tables are compiled by Dr. J. Erhart based on his paper(2004) with corrections.
Table D.1 gives the general information on the ferroic domain states, types ofthe domain wall separating them, and the number of the table (from Tables. D.2,D.3,D.4,D.5,D.6,D.7,D.8,D.9,D.10,D.11,D.12,D.13,D.14,D.15,D.16,D.17,D.18, D.19, D.20, D.21, D.22, D.23, D.24, D.25, D.26, D.27, D.28, D.29, andD.30) containing the detailed information for a given species. The table numberapplies to the species of the same line and all following ones until a new tablenumber appears. Tables D.2, D.3, D.4, D.5, D.6, D.7, D.8, D.9, D.10, D.11, D.12,D.13, D.14, D.15, D.16, D.17, D.18, D.19, D.20, D.21, D.22, D.23, D.24, D.25,D.26, D.27, D.28, D.29, andD.30 give the wall orientations in the case where theseare conditioned by the mechanical effects, i.e. in the case ofW1 walls no informa-tion on the wall orientation is given. The wall orientation is specified with thecomponents of its normal vector expressed in orthogonal or hexagonal basis(standard crystallographic coordinate choice). The coordinate system is adoptedthe same as in parent phase. In the titles of Tables D.2, D.3, D.4, D.5, D.6, D.7,D.8,D.9,D.10,D.11,D.12,D.13,D.14,D.15,D.16,D.17,D.18,D.19,D.20,D.21,D.22, D.23, D.24, D.25, D.26, D.27, D.28, D.29, and D.30, the change of thecrystalline class at the phase transitions addressed in the table is indicated.
Table D.1 General information on domain states and types of the domain wall
1 Speciesno.
2 Speciesdesignation
3q"
4qP
5 Domain wall pairsCharge/neutrality of domain walls
6 Tableno.
212 �1! 1 1 2 WN1 �WC
1 D.2
207 2=m! 2 1 2 WN1 �WC
1 D.9
206 2=m! m 1 2 WN1 �WC
1211 2! 1 2 2 WN
f � SC D.3
210 m! 1 2 2 WCf � SN
209 2=m! 1 2 4 WN;Cf � SC;N
208 2=m! �1 2 0 Wf � S
194 mmm! mm2 1 2 WN1 �WC
1 D.21
197 mmm! mx 2 4 WN;Cf �WC;N
f D.10
Table D.1 (continued)
1 Speciesno.
2 Speciesdesignation
3q"
4qP
5 Domain wall pairsCharge/neutrality of domain walls
6 Tableno.
198 mmm! 2x 2 2 WNf �WN
f
196 mmm! 2x=mx 2 0 W f �W f
201 mm2! mx 2 2 WNf �WC
f
204 222! 2x 2 2 WNf �WN
f
202 mm2! 2z 2 1 WNf �WN
f D.11
200 mmm! 1 4 8 WN;Cf � SC;N D.4
199 mmm! �1 4 0 Wf � S
203 mm2! 1 4 4 WNf � SC, WC
f � SN
205 222! 1 4 4 WNf � SC
149 4=mmm! 4mm 1 2 WN1 �WC
1 D.28
153 4=mmm! 4 1 2 WN1 �WC
1184 4=m! 4 1 2 WN
1 �WC1
178 422! 4 1 2 WN1 �WC
1154 4=mmm! mxmymz 2 0 W f �W f D.22
155 4=mmm! 2xmymz 2 4 WN;Cf �WC;N
f
156 4=mmm! mxmy2z 2 2 WNf �WN
f
157 4=mmm! 2x2y2z 2 0 W f �W f
168 �42m! 2x2y2z 2 0 W f �W f
174 4mm! mxmy2z 2 1 WNf �WN
f
179 422! 2x2y2z 2 0 W f �W f
167 �42m! mxym�xy2z 2 2 WNf �WN
f D.23
162 4=mmm! 2x 3 4 WNf �WN
f , WN;Cf � SC;N D.12
160 4=mmm! mx 3 8 WN;Cf �WC;N
f , WN;Cf � SC;N
158 4=mmm! 2x=mx 3 0 Wf �W f, W f � S
170 �42m! 2x 3 4 WNf �WN
f , WCf � SN
175 4mm! mx 3 4 WNf �WC
f , WCf � SN
180 422! 2x 3 4 WNf �WN
f , WNf � SC
163 4=mmm! 2z 4 2 WNf �WN
f , SN � SN D.13
161 4=mmm! mz 4 8 WN;Cf �WC;N
f , S� � S�
159 4=mmm! 2z=mz 4 0 W f �W f, S� S
171 �42m! 2z 4 2 WNf �WN
f , SN � SN
176 4mm! 2z 4 1 WNf �WN
f , SN � SN
181 422! 2z 4 2 WNf �WN
f , SN � SN
187 4=m! 2z 2 2 SN � SN
186 4=m! mz 2 4 S� � S�
185 4=m! 2z=mz 2 0 S� S
190 �4! 2z 2 2 SN � SN
192 4! 2z 2 1 SN � SN
169 �42m! mxy 4 4 WNf �WC
f , WNf � SvC D.14
165 4=mmm! 1 8 16 R, WN;Cf � SC;N D.5
164 4=mmm! �1 8 0 R, W f � S
172 �42m! 1 8 8 R, WNf � SC, WC
f � SN
177 4mm! 1 8 8 R, WNf � SC, WC
f � SN
182 422! 1 8 8 R, WNf � SC
189 4=m! 1 4 8 R, WN;Cf � SC;N
730 Appendix D
Table D.1 (continued)
1 Speciesno.
2 Speciesdesignation
3q"
4qP
5 Domain wall pairsCharge/neutrality of domain walls
6 Tableno.
188 4=m! �1 4 0 R, W f � S
191 �4! 1 4 4 R, WNf � SC
193 4! 1 4 4 R, WNf � SC
129 �3m! 3m 1 2 WN1 �WC
1 D.28
132 �3m! 3 1 2 WN1 �WC
1141 32! 3 1 2 WN
1 �WC1
144 �3! 3 1 2 WN1 �WC
1134 �3m! mx 3 6 WN;C
f � SC;N D.15
135 �3m! 2x 3 6 WN;Cf � SC;N
133 �3m! 2x=mx 3 0 W f � S
139 3m! mx 3 3 WCf � SN
142 32! 2x 3 3 WNf � SC
137 �3m! 1 6 12 R, WN;Cf � SC;N D.6
136 �3m! �1 6 0 R, W f � S
140 3m! 1 6 6 R, WCf � SN
143 32! 1 6 6 R,WNf � SC
145 �3! �1 3 0 R
146 �3! 1 3 6 R
147 3! 1 3 3 R
69 6=mmm! 6mm 1 2 WN1 �WC
1 D.28
73 6=mmm! 6 1 2 WN1 �WC
1107 622! 6 1 2 WN
1 �WC1
115 6=m! 6 1 2 WN1 �WC
175 6=mmm! 3zmx 1 2 WN
1 �WC1 D.28
78 6=mmm! 3 1 2 WN1 �WC
192 �6m2! 3zmx 1 2 WN
1 �WC1
94 �6m2! 3 1 2 WN1 �WC
1109 622! 3 1 2 WN
1 �WC1
123 �6! 3 1 2 WN1 �WC
1117 6=m! 3 1 2 WN
1 �WC1
79 6=mmm! mxmymz 3 0 W f �W f D.24
80 6=mmm! mxmy2z 3 2 WNf �WN
f
81 6=mmm! 2xmymz 3 6 WN;Cf �WC;N
f
82 6=mmm! 2x2y2z 3 0 W f �W f
95 �6m2! mx2ymz 3 3 WNf �WC
f
103 6mm! mxmy2z 3 1 WNf �WN
f
110 622! 2x2y2z 3 0 W f �W f
87 6=mmm! 2x 6 6 WNf �WN
f , WN;Cf � SC;N D.16
85 6=mmm! mx 6 12 WN;Cf �WC;N
f , WN;Cf � SC;N
83 6=mmm! 2x=mx 6 0 W f �W f, W f � S
97 �6m2! mx 6 6 WNf �WC
f , WNf � SC, WC
f � SN
104 6mm! mx 6 6 WNf �WC
f , WCf � SN
111 622! 2x 6 6 WNf �WN
f , WNf � SC
88 6=mmm! 2z 6 2 WNf �WN
f , SN � SN D.17
86 6=mmm! mz 6 12 WN;Cf �WC;N
f , S� � S�
Appendix D 731
Table D.1 (continued)
1 Speciesno.
2 Speciesdesignation
3q"
4qP
5 Domain wall pairsCharge/neutrality of domain walls
6 Tableno.
84 6=mmm! 2z=mz 6 0 W f �W f, S� S
96 �6m2! mz 6 6 WNf �WC
f , S� � S�
105 6mm! 2z 6 1 WNf �WN
f , SN � SN
112 622! 2z 6 2 WNf �WN
f , SN � SN
120 6=m! 2z 3 2 SN � SN
119 6=m! mz 3 6 S� � S�
118 6=m! 2z=mz 3 0 S� S
124 �6! mz 3 3 S� � S�
127 6! 2z 3 1 SN � SN
98 �6m2! 2y 6 3 WNf �WN
f , WNf � SC, WC
f � SN D.18
90 6=mmm! 1 12 24 R, WN;Cf � SC;N D.7
89 6=mmm! �1 12 0 R, W f � S
99 �6m2! 1 12 12 R, WNf � SC, WC
f � SN
106 6mm! 1 12 12 R, WNf � SC, WC
f � SN
113 622! 1 12 12 R, WNf � SC
122 6=m! 1 6 12 R, WN;Cf � SC;N
121 6=m! �1 6 0 R, W f � S
125 �6! 1 6 6 R, WCf � SN
128 6! 1 6 6 R, WNf � SC
5 m�3m! �3xyzm�xy 4 0 W f �W f D.30
6 m�3m! 3xyzm�xy 4 8 WN;Cf �WC;N
f
7 m�3m! 3xyz2�xy 4 0 W f �W f
8 m�3m! �3xyz 4 0 W f �W f
9 m�3m! 3xyz 4 8 WN;Cf �WC;N
f
34 �43m! 3xyzm�xy 4 4 WNf �WC
f
35 �43m! 3xyz 4 4 WNf �WC
f
44 432! 3xyz2�xy 4 0 W f �W f
45 432! 3xyz 4 8 WN;Cf �WC;N
f
54 m�3! �3xyz 4 0 W f �W f
55 m�3! 3xyz 4 8 WN;Cf �WC;N
f
64 23! 3xyz 4 4 WNf �WC
f
10 m�3m! 4z=mzmxmy 3 0 W f �W f D.29
13 m�3m! 4zmxmxy 3 6 WN;Cf �WC;N
f
11 m�3m! �4z2xymx 3 0 W f �W f
12 m�3m! �4z2xmxy 3 0 W f �W f
14 m�3m! 4z2x2xy 3 0 W f �W f
15 m�3m! 4z=mz 3 0 W f �W f
16 m�3m! �4 3 0 W f �W f
17 m�3m! 4 3 6 WN;Cf �WC;N
f
36 �43m! �4z2xmxy 3 0 W f �W f
37 �43m! �4 3 0 W f �W f
46 432! 4z2x2xy 3 0 W f �W f
47 432! 4 3 6 WN;Cf �WC;N
f
18 m�3m! mxmymz 6 0 R, W f �W f D.25
20 m�3m! mxmy2z 6 6 R, WNf �WN
f , WN;Cf �WC;N
f
732 Appendix D
Table D.1 (continued)
1 Speciesno.
2 Speciesdesignation
3q"
4qP
5 Domain wall pairsCharge/neutrality of domain walls
6 Tableno.
23 m�3m! 2x2y2z 6 0 R, W f �W f
39 �43m! 222 6 0 R, W f �W f
48 432! 2x2y2z 6 0 R, W f �W f
56 m�3! mmm 3 0 R
57 m�3! mxmy2z 3 6 R
58 m�3! 222 3 0 R
65 23! 222 3 0 R
21 m�3m! mxym�xy2z 6 6 WNf �WN
f , WN;Cf � SC;N D.26
19 m�3m! mxym�xymz 6 0 W f �W f, W f � S
24 m�3m! 2xy2�xy2z 6 0 W f �W f, W f � S
49 432! 2xy2�xy2z 6 0 W f �W f, W f � S
22 m�3m! 2xym�xymz 6 12 WN;Cf �WC;N
f , WN;Cf � SC;N D.27
38 �43m! mxym�xy2z 6 6 WNf �WN
f , WCf � SN
29 m�3m! 2xy 12 12 R, WNf �WN
f , WN;Cf � SC;N D.19
28 m�3m! mxy 12 24 R, WN;Cf �WC;N
f , WN;Cf � SC;N
25 m�3m! 2xy=mxy 12 0 R, W f �W f, W f � S
40 �43m! mxy 12 12 R, WNf �WC
f , WNf � SC, WC
f � SN
50 432! 2xy 12 12 R, WNf �WN
f , WNf � SC
30 m�3m! 2z 12 6 R, WNf �WN
f , WN;Cf � SC;N,
SN � SND.20
27 m�3m! mz 12 24 R, WN;Cf �WC;N
f , WN;Cf � SC;N,
S� � S�
26 m�3m! 2z=mz 12 0 R, W f �W f, W f � S, S� S
41 �43m! 2z 12 6 R, WNf �WN
f , WCf � SN, SN � SN
51 432! 2z 12 6 R, WNf �WN
f , WNf � SC, SN � SN
61 m�3! 2z 6 6 R, WNf �WN
f
60 m�3! mz 6 12 R, WN;Cf �WC;N
f
59 m�3! 2z=mz 6 0 R, W f �W f
66 23! 2z 6 6 R, WNf �WN
f
32 m�3m! 1 24 48 R, WN;Cf � SC;N D.8
31 m�3m! �1 24 0 R, W f � S
42 �43m! 1 24 24 R, WNf � SC, WC
f � SN
52 432! 1 24 24 R,WNf � SC
63 m�3! 1 12 24 R, WN;Cf � SC;N
62 m�3! �1 12 0 R, W f � S
67 23! 1 12 12 R, WNf � SC
1 m�3m! �43m 1 0
2 m�3m! 432 1 0
3 m�3m! m�3 1 0
4 m�3m! 23 1 0
33 �43m! 23 1 0
43 432! 23 1 0
53 m�3! 23 1 0
68 6=mmm! �6zmy2x 1 0
70 6=mmm! 622 1 0
Appendix D 733
Table D.1 (continued)
1 Speciesno.
2 Speciesdesignation
3q"
4qP
5 Domain wall pairsCharge/neutrality of domain walls
6 Tableno.
71 6=mmm! 6=m 1 0
72 6=mmm! �6 1 0
74 6=mmm! �3zmx 1 0
76 6=mmm! 3z2x 1 0
77 6=mmm! �3 1 0
91 �6m2! �6 1 0
93 �6m2! 32 1 0
108 622! 3z2x 1 0
114 6=m! �6 1 0
116 6=m! �3 1 0
130 �3m! 32 1 0
131 �3m! �3 1 0
148 4=mmm! �4z2xmxy 1 0
150 4=mmm! 422 1 0
151 4=mmm! 4=m 1 0
152 4=mmm! �4 1 0
166 �42m! �4 1 0
183 4=m! �4 1 0
195 mmm! 222 1 0
100 6mm! 6 1 1
101 6mm! 3zmx 1 1
102 6mm! 3 1 1
126 6! 3 1 1
138 3m! 3 1 1
173 4mm! 4 1 1
Column 1: sequential number of the species. Column 2: symbol of the species. Column 3:number of ferroelastic domain states. Column 4: number of ferroelectric domain states.Column 5: type of permissible pairs of domain walls (domain wall orientation symbols:W1-arbitrary,Wf -fixed, S-dependent on spontaneous strains,R-no walls). Charge/neutralityof ferroelectric domain walls is labeled by superscript C, N, respectively For centrosymmetricparent phases both superscripts are used in some cases, i.e. the same domain wall is eithercharged or neutral according to the choice of spontaneous polarization pair (antiparallelspontaneous polarizations are allowed in such species). When the charge/neutrality of adomain wall depends on the specific values of spontaneous strains, it is labeled by super-script*. Column 6: number of the table containing the detailed information on domain statesand orientations of domain walls for given species. The table number applies to the species ofthe same line and all following ones until a new table number appears.The last 34 rows: speciesbelonging to high-order ferroics where the domain states are distinguishable neither byspontaneous polarization nor by spontaneous strain. Here, the domain walls are of W1type and electroneutral for any orientation.
734 Appendix D
Table D.2 Triclinic to triclinic
Species strains �1! 1 polarizations
1 "1 "2 "3"4 "5 "6
� �
P1 P2 P3ð Þ
�P1 �P2 �P3ð Þ
Table D.4 Orthorhombic to triclinic
All speciesmmm! 1mmm! �1ðP ¼ 0Þ mm2! 1 222! 1
strains polarizations polarizations polarizations
1 "1 "2 "3"4 "5 "6
� �
P1 P2 P3ð Þ P1 P2 P3ð Þ P1 P2 P3ð Þ�P1 �P2 �P3ð Þ
2"1 "2 "3"4 �"5 �"6
� �
�P1 P2 P3ð Þ �P1 P2 P3ð Þ P1 �P2 �P3ð ÞP1 �P2 �P3ð Þ
3"1 "2 "3�"4 "5 �"6
� �
P1 �P2 P3ð Þ P1 �P2 P3ð Þ �P1 P2 �P3ð Þ�P1 P2 �P3ð Þ
4"1 "2 "3�"4 �"5 "6
� �
P1 P2 �P3ð Þ �P1 �P2 P3ð Þ �P1 �P2 P3ð Þ�P1 �P2 P3ð Þ
Table D.3 Monoclinic to triclinic
All species m! 1
2=m! 12=m! �1ðP ¼ 0Þ2! 1 (upper polarizations only)
strains polarizations polarizations
1 "1 "2 "3"4 "5 "6
� �
P1 P2 P3ð Þ P1 P2 P3ð Þ
�P1 �P2 �P3ð Þ
2"1 "2 "3�"4 "5 �"6
� �
P1 �P2 P3ð Þ �P1 P2 �P3ð Þ
P1 �P2 P3ð Þ
S 1 2
1 N/A ð010ÞN
ð10aÞC
2 ð010ÞCð10aÞN
N/A
a ¼ "4="6
Appendix D 735
mmm! 1, mmm! �1ðP ¼ 0Þ (above diagonal)mm2! 1 (below diagonal)
S 1 2 3 4
1 N/A ð100ÞCð01aÞN
ð010ÞCð10bÞN
ð001ÞCð1c0ÞN
2ð100ÞCð01aÞN
N/A ð001ÞN
ð1�c0ÞCð010ÞN
ð10�bÞC
3ð010ÞCð10bÞN
ð001ÞN
ð1�c0ÞCN/A
ð100ÞN
ð01�aÞC
4ð001ÞN
ð1c0ÞCð010ÞCð10�bÞN
ð100ÞCð01�aÞN
N/A
222! 1
S 1 2 3 4
1 N/A ð100ÞN
ð01aÞCð010ÞN
ð10bÞCð001ÞN
ð1c0ÞC
2 N/Að001ÞN
ð1�c0ÞCð010ÞN
ð10�bÞC
3 N/Að100ÞN
ð01�aÞC4 N/A
a ¼ "5="6; b ¼ "4="6; c ¼ "4="5
Table D.4 (continued)
736 Appendix D
TableD.5
Tetragonalto
triclinic
Allspecies
422!
14!
1(strains1–4only)
4mm!
1
� 42m!
1� 4!
1(strains1–4only)
4=mmm!
1,4=mmm!
� 1ðP¼
0Þ 4=m!
1,
4=m!
� 1ðP¼
0Þ(strains1–4only)
strains
polarizations
polarizations
polarizations
polarizations
1" 1
" 2" 3
" 4" 5
" 6
��
P1
P2
P3
ðÞ
P1
P2
P3
ðÞ
P1
P2
P3
ðÞ
P1
P2
P3
ðÞ
�P1�P2�P3
ðÞ
2" 2
" 1" 3
" 5�" 4�" 6
��
�P2
P1
P3
ðÞ
�P2
P1
P3
ðÞ
P2�P1�P3
ðÞ
�P2
P1
P3
ðÞ
P2�P1�P3
ðÞ
3" 1
" 2" 3
�" 4�" 5
" 6
��
�P1�P2
P3
ðÞ
�P1�P2
P3
ðÞ
�P1�P2
P3
ðÞ
�P1�P2
P3
ðÞ
P1
P2�P3
ðÞ
4" 2
" 1" 3
�" 5
" 4�" 6
��
P2�P1
P3
ðÞ
P2�P1
P3
ðÞ
�P2
P1�P3
ðÞ
P2�P1
P3
ðÞ
�P2
P1�P3
ðÞ
5" 1
" 2" 3
" 4�" 5�" 6
��
P1�P2�P3
ðÞ
�P1
P2
P3
ðÞ
P1�P2�P3
ðÞ
�P1
P2
P3
ðÞ
P1�P2�P3
ðÞ
6" 1
" 2" 3
�" 4
" 5�" 6
��
�P1
P2�P3
ðÞ
P1�P2
P3
ðÞ
�P1
P2�P3
ðÞ
P1�P2
P3
ðÞ
�P1
P2�P3
ðÞ
7" 2
" 1" 3
�" 5�" 4
" 6
��
P2
P1�P3
ðÞ
�P2�P1
P3
ðÞ
�P2�P1
P3
ðÞ
�P2�P1
P3
ðÞ
P2
P1�P3
ðÞ
8" 2
" 1" 3
" 5" 4
" 6
��
�P2�P1�P3
ðÞ
P2
P1
P3
ðÞ
P2
P1
P3
ðÞ
P2
P1
P3
ðÞ
�P2�P1�P3
ðÞ
Appendix D 737
422!
1(alleightstrains),4!
1(strains1–4only)(abovediagonal)
� 42m!
1(alleightstrains),4!
1(strains1–4only)(belowdiagonal)
S1
23
45
67
8
1N/A
Rð001ÞN
ð1a0ÞC
Rð100ÞN
ð01bÞC
ð010ÞN
ð10cÞ
Cð110ÞN
ð1� 1
� dþÞC
ð1� 10ÞN
ð11d�ÞC
2R
N/A
Rð001ÞN
ð� a10ÞC
ð110ÞN
ð1� 1
� d�ÞC
ð1� 10ÞN
ð11
� dþÞC
ð010ÞN
ð10
� bÞC
ð100ÞN
ð01cÞ
C
3ð001ÞN
ð1a0ÞC
RN/A
Rð010ÞN
ð10� cÞ
Cð100ÞN
ð01
� bÞC
ð1� 10ÞN
ð11
� d�ÞC
ð110ÞN
ð1� 1dþÞC
4R
ð001ÞN
ð� a10ÞC
RN/A
ð1� 10ÞN
ð11dþÞC
ð110ÞN
ð1� 1d�ÞC
ð100ÞN
ð01� cÞ
Cð010ÞN
ð10bÞC
5ð100ÞN
ð01bÞC
ð110ÞC
ð1� 1
� d�ÞN
ð010ÞN
ð10� cÞ
Cð1
� 10ÞC
ð11dþÞN
N/A
ð001ÞN
ð1� a0ÞC
RR
6ð010ÞN
ð10cÞ
Cð1
� 10ÞC
ð11
� dþÞN
ð100ÞN
ð01
� bÞC
ð110ÞC
ð1� 1d�ÞN
ð001ÞN
ð1� a0ÞC
N/A
RR
7ð110ÞC
ð1� 1
� d þÞN
ð010ÞN
ð10
� bÞC
ð1� 10ÞC
ð11
� d�ÞN
ð100ÞN
ð01� cÞ
CR
RN/A
ð001ÞN
ða10ÞC
8ð1
� 10ÞC
ð11d�ÞN
ð100ÞN
ð01cÞ
Cð110ÞC
ð1� 1dþÞN
ð010ÞN
ð10bÞC
RR
ð001ÞN
ða10ÞC
N/A
TableD.5
(continued)
738 Appendix D
4=mmm!
� 1ðP¼
0Þ,4=mmm!
1,4mm!
1(alleightstrains),
4=m!
� 1ðP¼
0Þ,
4=m!
1(strains1–4only)
S1
23
45
67
8
1N/A
Rð001ÞN
ð1a0ÞC
Rð100ÞC
ð01bÞN
ð010ÞC
ð10cÞ
Nð110ÞC
ð1� 1
� d þÞN
ð1� 10ÞC
ð11d�ÞN
2N/A
Rð001ÞN
ð� a10ÞC
ð110ÞC
ð1� 1
� d�ÞN
ð1� 10ÞC
ð11
� d þÞN
ð010ÞC
ð10
� bÞN
ð100ÞC
ð01cÞ
N
3N/A
Rð010ÞC
ð10� cÞ
Nð100ÞC
ð01
� bÞN
ð1� 10ÞC
ð11
� d�ÞN
ð110ÞC
ð1� 1dþÞN
4N/A
ð1� 10ÞC
ð11dþÞN
ð110ÞC
ð1� 1d�ÞN
ð100ÞC
ð01� cÞ
Nð010ÞC
ð10bÞN
5N/A
ð001ÞN
ð1� a0ÞC
RR
6N/A
RR
7N/A
ð001ÞN
ða10ÞC
8N/A
a¼" 4=" 5;b¼" 5=" 6;c¼" 4=" 6,d�¼ð"
5�" 4Þ=ð"
2�" 1Þ
TableD.5
(continued)
Appendix D 739
TableD.6
Trigonalto
triclinic
Allspecies
32!
1� 3m!
� 1ðP¼
0Þ
3!
1,
� 3!
� 1ðP¼
0Þ(strains1–3only)
3m!
1� 3m!
1� 3!
1(strains1–3only)
strains
polarizations
polarizations
polarizations
1" 1
" 2" 3
" 4" 5
" 6
��
P1
P2
P3
ðÞ
P1
P2
P3
ðÞ
P1
P2
P3
ðÞ
�P1�P2�P3
ðÞ
2"þ 1
"� 2" 3
"� 4"� 5
"� 6
��
�Pþ 1
P� 2
P3
��
�Pþ 1
P� 2
P3
��
�Pþ 1
P� 2
P3
��
Pþ 1�P� 2�P3
��
3"� 1
"þ 2" 3
�"þ 4
�"þ 5
"þ 6
��
�P� 1�Pþ 2
P3
��
�P� 1�Pþ 2
P3
��
�P� 1�Pþ 2
P3
��
P� 1
Pþ 2�P3
��
4" 1
" 2" 3
" 4�" 5�" 6
��
P1�P2�P3
ðÞ
�P1
P2
P3
ðÞ
P1�P2�P3
ðÞ
�P1
P2
P3
ðÞ
5"� 1
"þ 2" 3
�"þ 4
"þ 5�"þ 6
��
�P� 1
Pþ 2�P3
��
P� 1�Pþ 2
P3
��
�P� 1
Pþ 2�P3
��
P� 1�Pþ 2
P3
��
6"þ 1
"� 2" 3
"� 4�"� 5
�"� 6
��
�Pþ 1�P� 2�P3
��
Pþ 1
P� 2
P3
��
�Pþ 1�P� 2�P3
��
Pþ 1
P� 2
P3
��
"� 1¼ð1=4Þð" 1�
ffiffiffi 3
p" 6þ3" 2Þ;"� 2¼ð1=4Þð3" 1�
ffiffiffi 3
p" 6þ" 2Þ;"� 6¼ð1=2Þð�
ffiffiffi 3
p" 1�" 6�
ffiffiffi 3
p" 2Þ;
"� 5¼ð1=2Þð�" 5�
ffiffiffi 3
p" 4Þ;"� 4¼ð1=2Þð
ffiffiffi 3
p" 5�" 4Þ;P� 1¼ð1=2ÞðP1�
ffiffiffi 3
pP2Þ;P� 2¼ð1=2Þð
ffiffiffi 3
pP1�P2Þ
740 Appendix D
� 3m!
1,� 3m!
� 1ðP¼
0Þ,32!
1(allsixstrains),
� 3!
� 1ðP¼
0Þ,
� 3!
1,3!
1(strains1–3only)(abovediagonal)
3m!
1(belowdiagonal)
S1
23
45
6
1N/A
RR
ð2� 1� 10ÞN
ð01� 1aÞC
ð11� 20ÞN
ð� 110c þÞC
ð1� 21
0ÞN
ð� 101c �ÞC
2R
N/A
Rð11� 20ÞN
ð1� 10c �ÞC
ð1� 21
0ÞN
ð10� 1
� aÞC
ð2� 1� 10ÞN
ð01� 1bþÞC
3R
RN/A
ð1� 21
0ÞN
ð10� 1c þÞC
ð2� 1� 10ÞN
ð01� 1b�ÞC
ð11� 20ÞN
ð� 110
� aÞC
4ð2
� 1� 10ÞC
ð01� 1aÞN
ð11� 20ÞC
ð1� 10c �ÞN
ð1� 21
0ÞC
ð10� 1c þÞN
N/A
RR
5ð11� 20ÞC
ð� 110c þÞN
ð1� 21
0ÞC
ð10� 1
� aÞN
ð2� 1� 10ÞC
ð01� 1b�ÞN
RN/A
R
6ð1
� 210ÞC
ð� 101c �ÞN
ð2� 1� 10ÞC
ð01� 1bþÞN
ð11� 20ÞC
ð� 110
� aÞN
RR
N/A
a¼
ffiffiffi 3
pð"
5=" 6Þ;b�¼
2ffiffiffi 3
pð"
5�
ffiffiffi 3
p" 4Þ=½�
ffiffiffi 3
pð"
1�" 2Þþ
" 6�;c�¼�ð1=2Þb�
TableD.6
(continued)
Appendix D 741
TableD.7
Hexagonalto
triclinicspecies
Allspecies
6=mmm!
1
622!
1
6=mmm!
� 1ðP¼
0Þ
6=m!
1,6=m!
� 1ðP¼
0Þ(strains1–6only)
6!
1(upper
polarizationsonly,strains1–6only)
6mm!
1
� 6m2!
1� 6!
1(strains1–6only)
strains
polarizations
polarizations
polarizations
1" 1
" 2" 3
" 4" 5
" 6
��
P1
P2
P3
ðÞ
P1
P2
P3
ðÞ
P1
P2
P3
ðÞ
�P1�P2�P3
ðÞ
2"� 1
"þ 2" 3
"þ 4"þ 5
"þ 6
��
P� 1
Pþ 2
P3
��
P� 1
Pþ 2
P3
��
�P� 1�Pþ 2�P3
��
�P� 1�Pþ 2�P3
��
3"þ 1
"� 2" 3
"� 4"� 5
"� 6
��
�Pþ 1
P� 2
P3
��
�Pþ 1
P� 2
P3
��
�Pþ 1
P� 2
P3
��
Pþ 1�P� 2�P3
��
4" 1
" 2" 3
�" 4�" 5
" 6
��
�P1�P2
P3
ðÞ
�P1�P2
P3
ðÞ
P1
P2�P3
ðÞ
P1
P2�P3
ðÞ
5"� 1
"þ 2" 3
�"þ 4
�"� 5
"þ 6
��
�P� 1�Pþ 2
P3
��
�P� 1�Pþ 2
P3
��
�P� 1�Pþ 2
P3
��
P� 1
Pþ 2�P3
��
6"þ 1
"� 2" 3
�"� 4
�"� 5
"� 6
��
Pþ 1�P� 2
P3
��
Pþ 1�P� 2
P3
��
�Pþ 1
P� 2�P3
��
�Pþ 1
P� 2�P3
��
7" 1
" 2" 3
" 4�" 5�" 6
��
P1�P2�P3
ðÞ
�P1
P2
P3
ðÞ
�P1
P2
P3
ðÞ
�P1
P2
P3
ðÞ
8"� 1
"þ 2" 3
"þ 4�"þ 5
�"þ 6
��
P� 1�Pþ 2�P3
��
�P� 1
Pþ 2
P3
��
P� 1�Pþ 2�P3
��
�P� 1
Pþ 2
P3
��
9"þ 1
"� 2" 3
"� 4�"� 5
�"� 6
��
�Pþ 1�P� 2�P3
��
Pþ 1
P� 2
P3
��
Pþ 1
P� 2
P3
��
Pþ 1
P� 2
P3
��
10
" 1" 2
" 3�" 4
" 5�" 6
��
�P1
P2�P3
ðÞ
P1�P2
P3
ðÞ
�P1
P2�P3
ðÞ
P1�P2
P3
ðÞ
11
"� 1"þ 2
" 3�"þ 4
"þ 5�"þ 6
��
�P� 1
Pþ 2�P3
��
P� 1�Pþ 2
P3
��
P� 1�Pþ 2
P3
��
P� 1�Pþ 2
P3
��
12
"þ 1"� 2
" 3�"� 4
"� 5�"� 6
��
Pþ 1
P� 2�P3
��
�Pþ 1�P� 2
P3
��
Pþ 1
P� 2�P3
��
�Pþ 1�P� 2
P3
��
"� 1¼ð1=4Þð" 1�
ffiffiffi 3
p" 6þ3" 2Þ;"� 2¼ð1=4Þð3" 1�
ffiffiffi 3
p" 6þ" 2Þ;"� 6¼ð1=2Þð�
ffiffiffi 3
p" 1�" 6�
ffiffiffi 3
p" 2Þ;
"� 5¼ð1=2Þð�" 5�
ffiffiffi 3
p" 4Þ;"� 4¼ð1=2Þð
ffiffiffi 3
p" 5�" 4Þ;P� 1¼ð1=2ÞðP1�
ffiffiffi 3
pP2Þ;P� 2¼ð1=2Þð
ffiffiffi 3
pP1�P2Þ
742 Appendix D
TableD.7
(continued)
6=mmm!
� 1ðP¼
0Þ,6=mmm!
1,622!
1(all12strains),6=m!
� 1ðP¼
0Þ,
6=m!
1,6!
1(strains1–6only)(abovediagonal)
� 6m2!
1(all12strains),
� 6!
1(strains1–6only)(belowdiagonal)
S1
23
45
67
89
10
11
12
1N/A
RR
ð0001ÞN
ð2aþa�0ÞC
RR
ð2� 1� 10ÞN
ð01� 1cÞ
Cð1
� 100ÞN
ð11� 2d�ÞC
ð1� 210ÞN
ð10� 1e �ÞC
ð01� 10ÞN
ð2� 1� 1bÞC
ð11� 20ÞN
ð1� 10
e þÞC
ð10� 10ÞN
ð1� 21
dþÞC
2R
N/A
RR
ð0001ÞN
ð2gþhþ0ÞC
Rð10� 10ÞN
ð1� 21
� d�ÞC
ð2� 1� 10ÞN
ð01� 1e �ÞC
ð1� 100ÞN
ð11� 2bÞC
ð1� 210ÞN
ð10� 1e þÞC
ð01� 10ÞN
ð2� 1� 1dþÞC
ð11� 20ÞN
ð1� 10
� cÞC
3R
RN/A
RR
ð0001ÞN
ð2h�g�0ÞC
ð11� 20ÞN
ð1� 10
� e �ÞC
ð10� 10ÞN
ð1� 21
� bÞC
ð2� 1� 10ÞN
ð01� 1e þÞC
ð1� 100ÞN
ð11� 2dþÞC
ð1� 210ÞN
ð10� 1� cÞ
Cð01� 10ÞN
ð2� 1� 1
� d�ÞC
4ð0001ÞC
ð2aþa�0ÞN
RR
N/A
RR
ð01� 10ÞN
ð2� 1� 1
� bÞC
ð11� 20ÞN
ð1� 10
� e þÞC
ð10� 10ÞN
ð1� 21
� dþÞC
ð2� 1� 10ÞN
ð01� 1� cÞ
Cð1
� 100ÞN
ð11� 2
� d �ÞC
ð1� 210ÞN
ð10� 1� e �ÞC
5R
ð0001ÞC
ð2gþhþ0ÞN
RR
N/A
Rð1
� 210ÞN
ð10� 1� e þÞC
ð01� 10ÞN
ð2� 1� 1
� dþÞC
ð11� 20ÞN
ð1� 10
cÞC
ð10� 10ÞN
ð1� 21
d�ÞC
ð2� 1� 10ÞN
ð01� 1� e �ÞC
ð1� 100ÞN
ð11� 2
� bÞC
6R
Rð0001ÞC
ð2h�g�0ÞN
RR
N/A
ð1� 100ÞN
ð11� 2
� dþÞC
ð1� 210ÞN
ð10� 1cÞ
Cð01� 10ÞN
ð2� 1� 1d�ÞC
ð11� 20ÞN
ð1� 10
e �ÞC
ð10� 10ÞN
ð1� 21
bÞC
ð2� 1� 10ÞN
ð01� 1� e þÞC
7ð2
� 1� 10ÞC
ð01� 1cÞ
Nð10� 10ÞN
ð1� 21
� d�ÞC
ð11� 20ÞC
ð1� 10
� e �ÞN
ð01� 10ÞN
ð2� 1� 1
� bÞC
ð1� 210ÞC
ð10� 1� e þÞN
ð1� 100ÞN
ð11� 2
� dþÞC
N/A
RR
ð0001ÞN
ð2a�aþ0ÞC
RR
8ð1
� 100ÞN
ð11� 2d�ÞC
ð2� 1� 10ÞC
ð01� 1e �ÞN
ð10� 10ÞN
ð1� 21
� bÞC
ð11� 20ÞC
ð1� 10
� e þÞN
ð01� 10ÞN
ð2� 1� 1
� dþÞC
ð1� 210ÞC
ð10� 1cÞ
NR
N/A
RR
ð0001ÞN
ð2hþgþ0ÞC
R
9ð1
� 210ÞC
ð10� 1e �ÞN
ð1� 100ÞN
ð11� 2bÞC
ð2� 1� 10ÞC
ð01� 1e þÞN
ð10� 10ÞN
ð1� 21
� dþÞC
ð11� 20ÞC
ð1� 10
cÞN
ð01� 10ÞN
ð2� 1� 1d�ÞC
RR
N/A
RR
ð0001ÞN
ð2g�h�0ÞC
10ð01� 10ÞN
ð2� 1� 1bÞC
ð1� 210ÞC
ð10� 1e þÞN
ð1� 100ÞN
ð11� 2dþÞC
ð2� 1� 10ÞC
ð01� 1� cÞ
Nð10� 10ÞN
ð1� 21
d�ÞC
ð11� 20ÞC
ð1� 10
e �ÞN
ð0001ÞC
ð2a�aþ0ÞN
RR
N/A
RR
11ð11� 20ÞC
ð1� 10
e þÞN
ð01� 10ÞN
ð2� 1� 1dþÞC
ð1� 210ÞC
ð10� 1� cÞ
Nð1
� 100ÞN
ð11� 2
� d�ÞC
ð2� 1� 10ÞC
ð01� 1� e �ÞN
ð10� 10ÞN
ð1� 21bÞC
Rð0001ÞC
ð2hþgþ0ÞN
RR
N/A
R
12ð10� 10ÞN
ð1� 21
dþÞC
ð11� 20ÞC
ð1� 10
� cÞN
ð01� 10ÞN
ð2� 1� 1
� d�ÞC
ð1� 210ÞC
ð10� 1� e �ÞN
ð1� 100ÞN
ð11� 2
� bÞC
ð2� 1� 10ÞC
ð01� 1� e þÞN
RR
ð0001ÞC
ð2g�h�0ÞN
RR
N/A
Appendix D 743
6mm!
1
S1
23
45
67
89
10
11
12
1N/A
RR
ð0001ÞN
ð2aþa�0ÞC
RR
ð2� 1� 10ÞC
ð01� 1cÞN
ð1� 10
0ÞC
ð11� 2d�ÞN
ð1� 21
0ÞC
ð10� 1e �ÞN
ð01� 10ÞC
ð2� 1� 1bÞN
ð11� 20ÞC
ð1� 10
e þÞN
ð10� 10ÞC
ð1� 21dþÞN
2N/A
RR
ð0001ÞN
ð2gþhþ0ÞC
Rð10� 10ÞC
ð1� 21
� d�ÞN
ð2� 1� 10ÞC
ð01� 1e �ÞN
ð1� 10
0ÞC
ð11� 2bÞN
ð1� 21
0ÞC
ð10� 1e þÞN
ð01� 10ÞC
ð2� 1� 1dþÞN
ð11� 20ÞC
ð1� 10
� cÞN
3N/A
RR
ð0001ÞN
ð2h�g�0ÞC
ð11� 20ÞC
ð1� 10
� e �ÞN
ð10� 10ÞC
ð1� 21
� bÞN
ð2� 1� 10ÞC
ð01� 1e þÞN
ð1� 10
0ÞC
ð11� 2dþÞN
ð1� 21
0ÞC
ð10� 1� cÞ
Nð01� 10ÞC
ð2� 1� 1
� d�ÞN
4N/A
RR
ð01� 10ÞC
ð2� 1� 1
� bÞN
ð11� 20ÞC
ð1� 10
� e þÞN
ð10� 10ÞC
ð1� 21
� dþÞN
ð2� 1� 10ÞC
ð01� 1� cÞ
Nð1
� 100ÞC
ð11� 2
� d�ÞN
ð1� 21
0ÞC
ð10� 1� e �ÞN
5N/A
Rð1
� 210ÞC
ð10� 1� e þÞN
ð01� 10ÞC
ð2� 1� 1
� dþÞN
ð11� 20ÞC
ð1� 10cÞ
Nð10� 10ÞC
ð1� 21
d�ÞN
ð2� 1� 10ÞC
ð01� 1� e �ÞN
ð1� 10
0ÞC
ð11� 2
� bÞN
6N/A
ð1� 10
0ÞC
ð11� 2
� dþÞN
ð1� 21
0ÞC
ð10� 1cÞ
Nð01� 10ÞC
ð2� 1� 1d�ÞN
ð11� 20ÞC
ð1� 10
e �ÞN
ð10� 10ÞC
ð1� 21
bÞN
ð2� 1� 10ÞC
ð01� 1� e þÞN
7N/A
RR
ð0001ÞN
ð2a�aþ0ÞC
RR
8N/A
RR
ð0001ÞN
ð2hþgþ0ÞC
R
9N/A
RR
ð0001ÞN
ð2g�h�0ÞC
10
N/A
RR
11
N/A
R
12
N/A
a�¼�1�
ffiffiffi 3
pð"
4=" 5Þ;b¼
3ð"
4=" 6Þ;c¼
ffiffiffi 3
pð"
5=" 6Þ,
d�¼
ffiffiffi 3
pðffiffiffi 3
p" 5�" 4Þ=½"
1�" 2�ð1=ffiffiffi 3
pÞ"
6�;e�¼ð"
5�
ffiffiffi 3
p" 4Þ=½"
1�" 2�ð1=ffiffiffi 3
pÞ"
6�,
g�¼
2ð"
5�=ð"
5�
ffiffiffi 3
p" 4Þ;h�¼�4" 5=ð"
5�
ffiffiffi 3
p" 4Þ
TableD.7
(continued)
744 Appendix D
TableD.8
Cubicto
triclinic
Allspecies
432!
123!
1(strains1–12only)
� 43m!
1m
� 3m!
1,m
� 3m!
� 1ðP¼
0Þ
m� 3!
1,m
� 3!
� 1ðP¼
0Þ(strains1–12only)
strains
polarizations
polarizations
polarizations
1" 1
" 2" 3
" 4" 5
" 6
��
P1
P2
P3
ðÞ
P1
P2
P3
ðÞ
P1
P2
P3
ðÞ
�P1�P2�P3
ðÞ
2" 1
" 2" 3
�" 4�" 5
" 6
��
�P1�P2
P3
ðÞ
�P1�P2
P3
ðÞ
�P1�P2
P3
ðÞ
P1
P2�P3
ðÞ
3" 1
" 2" 3
" 4�" 5�" 6
��
P1�P2�P3
ðÞ
P1�P2�P3
ðÞ
P1�P2�P3
ðÞ
�P1
P2
P3
ðÞ
4" 1
" 2" 3
�" 4
" 5�" 6
��
�P1
P2�P3
ðÞ
�P1
P2�P3
ðÞ
�P1
P2�P3
ðÞ
P1�P2
P3
ðÞ
5" 3
" 1" 2
" 6" 4
" 5
��
P3
P1
P2
ðÞ
P3
P1
P2
ðÞ
P3
P1
P2
ðÞ
�P3�P1�P2
ðÞ
6" 2
" 3" 1
" 5" 6
" 4
��
P2
P3
P1
ðÞ
P2
P3
P1
ðÞ
P2
P3
P1
ðÞ
�P2�P3�P1
ðÞ
7" 3
" 1" 2
�" 6�" 4
" 5
��
�P3�P1
P2
ðÞ
�P3�P1
P2
ðÞ
�P3�P1
P2
ðÞ
P3
P1�P2
ðÞ
8" 2
" 3" 1
�" 5�" 6
" 4
��
�P2�P3
P1
ðÞ
�P2�P3
P1
ðÞ
�P2�P3
P1
ðÞ
P2
P3�P1
ðÞ
9" 3
" 1" 2
" 6�" 4�" 5
��
P3�P1�P2
ðÞ
P3�P1�P2
ðÞ
P3�P1�P2
ðÞ
�P3
P1
P2
ðÞ
10
" 2" 3
" 1" 5�" 6�" 4
��
P2�P3�P1
ðÞ
P2�P3�P1
ðÞ
P2�P3�P1
ðÞ
�P2
P3
P1
ðÞ
11
" 3" 1
" 2�" 6
" 4�" 5
��
�P3
P1�P2
ðÞ
�P3
P1�P2
ðÞ
�P3
P1�P2
ðÞ
P3�P1
P2
ðÞ
12
" 2" 3
" 1�" 5
" 6�" 4
��
�P2
P3�P1
ðÞ
�P2
P3�P1
ðÞ
�P2
P3�P1
ðÞ
P2�P3
P1
ðÞ
Appendix D 745
TableD.8
(continued)
Allspecies
432!
123!
1(strains1–12only)
� 43m!
1m
� 3m!
1,m
� 3m!
� 1ðP¼
0Þ
m� 3!
1,m
� 3!
� 1ðP¼
0Þ(strains1–12only)
strains
polarizations
polarizations
polarizations
13
" 2" 1
" 3" 5�" 4�" 6
��
�P2
P1
P3
ðÞ
P2�P1�P3
ðÞ
�P2
P1
P3
ðÞ
P2�P1�P3
ðÞ
14
" 2" 1
" 3�" 5
" 4�" 6
��
P2�P1
P3
ðÞ
�P2
P1�P3
ðÞ
P2�P1
P3
ðÞ
�P2
P1�P3
ðÞ
15
" 1" 3
" 2�" 4
" 6�" 5
��
P1�P3
P2
ðÞ
�P1
P3�P2
ðÞ
P1�P3
P2
ðÞ
�P1
P3�P2
ðÞ
16
" 1" 3
" 2�" 4�" 6
" 5
��
P1
P3�P2
ðÞ
�P1�P3
P2
ðÞ
P1
P3�P2
ðÞ
�P1�P3
P2
ðÞ
17
" 3" 2
" 1" 6�" 5�" 4
��
�P3
P2
P1
ðÞ
P3�P2�P1
ðÞ
�P3
P2
P1
ðÞ
P3�P2�P1
ðÞ
18
" 3" 2
" 1�" 6�" 5
" 4
��
P3
P2�P1
ðÞ
�P3�P2
P1
ðÞ
P3
P2�P1
ðÞ
�P3�P2
P1
ðÞ
19
" 2" 1
" 3�" 5�" 4
" 6
��
P2
P1�P3
ðÞ
�P2�P1
P3
ðÞ
P2
P1�P3
ðÞ
�P2�P1
P3
ðÞ
20
" 2" 1
" 3" 5
" 4" 6
��
�P2�P1�P3
ðÞ
P2
P1
P3
ðÞ
�P2�P1�P3
ðÞ
P2
P1
P3
ðÞ
21
" 1" 3
" 2" 4�" 6�" 5
��
�P1
P3
P2
ðÞ
P1�P3�P2
ðÞ
�P1
P3
P2
ðÞ
P1�P3�P2
ðÞ
22
" 3" 2
" 1�" 6
" 5�" 4
��
P3�P2
P1
ðÞ
�P3
P2�P1
ðÞ
P3�P2
P1
ðÞ
�P3
P2�P1
ðÞ
23
" 3" 2
" 1" 6
" 5" 4
��
�P3�P2�P1
ðÞ
P3
P2
P1
ðÞ
�P3�P2�P1
ðÞ
P3
P2
P1
ðÞ
746 Appendix D
TableD.8
(continued)
432!
1,m
� 3m!
1,m
� 3m!
� 1ðP¼
0Þ(all24strains),23!
1,m
� 3!
1,m
� 3!
� 1ðP¼
0Þ(strains1–12only)(abovediagonal)
� 43m!
1(belowdiagonal)
S1
23
45
67
89
10
11
12
1N/A
ð001ÞN
ð1b0ÞC
ð100ÞN
ð01cÞ
Cð010ÞN
ð10aÞC
RR
RR
RR
RR
2ð001ÞN
ð1b0ÞC
N/A
ð010ÞN
ð10
� aÞC
ð100ÞN
ð01� cÞ
C
RR
RR
RR
RR
3ð100ÞN
ð01cÞ
Cð010ÞN
ð10
� aÞC
N/A
ð001ÞN
ð1� b0ÞC
RR
RR
RR
RR
4ð010ÞN
ð10aÞC
ð100ÞN
ð01� cÞ
Cð001ÞN
ð1� b0ÞC
N/A
RR
RR
RR
RR
5R
RR
RN/A
Rð001ÞN
ða10ÞC
Rð100ÞN
ð01bÞC
Rð010ÞN
ðc01ÞC
R
6R
RR
RR
N/A
Rð001ÞN
ð1c0ÞC
Rð100ÞN
ð0a1ÞC
Rð010ÞN
ðb01ÞC
7R
RR
Rð001ÞN
ða10ÞC
RN/A
Rð010ÞN
ð� c01ÞC
Rð100ÞN
ð01
� bÞC
R
8R
RR
RR
ð001ÞN
ð1c0ÞC
RN/A
Rð010ÞN
ð� b01ÞC
Rð100ÞN
ð0� a1ÞC
9R
RR
Rð100ÞN
ð01bÞC
Rð010ÞN
ð� c01ÞC
RN/A
Rð001ÞN
ð� a10ÞC
R
10
RR
RR
Rð100ÞN
ð0a1ÞC
Rð010ÞN
ð� b01ÞC
RN/A
Rð001ÞN
ð1� c0ÞC
11
RR
RR
ð010ÞN
ðc01ÞC
Rð100ÞN
ð01
� bÞC
Rð001ÞN
ð� a10ÞC
RN/A
R
12
RR
RR
Rð010ÞN
ðb01ÞC
Rð100ÞN
ð0� a1ÞC
Rð001ÞN
ð1� c0ÞC
RN/A
Appendix D 747
TableD.8
(continued)
S13
14
15
16
17
18
19
20
21
22
23
24
1R
RR
RR
Rð110ÞN
ð1� 1dþÞC
ð1� 10ÞN
ð11d�ÞC
ð011ÞN
ð� fþ1� 1Þ
Cð101ÞN
ð1� e þ
� 1ÞC
ð10� 1ÞN
ð1� e �
1ÞC
ð01� 1Þ
N
ð� f �11ÞC
2R
Rð01� 1ÞN
ð� f þ11ÞC
ð011ÞN
ð� f �1� 1ÞC
ð10� 1Þ
N
ð1� e þ
1ÞC
ð101ÞN
ð1� e �
� 1ÞC
ð1� 10ÞN
ð11
� d �ÞC
ð110ÞN
ð1� 1dþÞC
RR
RR
3ð110ÞN
ð1� 1
� d�ÞC
ð1� 10ÞN
ð11dþÞC
RR
ð101ÞN
ð1e �
� 1ÞC
ð10� 1ÞN
ð1e þ
1ÞC
RR
ð01� 1ÞN
ðf�11ÞC
RR
ð011ÞN
ðf þ1� 1ÞC
4ð1
� 10ÞN
ð11
� dþÞC
ð110ÞN
ð1� 1d�ÞC
ð011ÞN
ðf �1� 1ÞC
ð01� 1Þ
N
ðf þ11ÞC
RR
RR
Rð10� 1ÞN
ð1e �
1ÞC
ð101ÞN
ð1e þ
� 1ÞC
R
5R
ð101ÞN
ð1f þ
� 1ÞC
Rð110ÞN
ð1� 1e þÞC
ð011ÞN
ð� dþ1� 1ÞC
RR
ð10� 1Þ
N
ð1� f �1ÞC
RR
ð01� 1ÞN
ð� d�11ÞC
ð1� 10ÞN
ð11� e �ÞC
6ð011ÞN
ðeþ1� 1ÞC
Rð101ÞN
ð1dþ
� 1ÞC
RR
ð110ÞN
ð1� 1
� f þÞC
Rð01� 1ÞN
ð� e�11ÞC
RR
ð1� 10ÞN
ð11
� f �ÞC
ð10� 1ÞN
ð1� d�1ÞC
7ð10� 1Þ
N
ð1f þ1ÞC
RR
ð1� 10ÞN
ð11e �ÞC
Rð011ÞN
ð� d�1� 1Þ
Cð101ÞN
ð1� f �
� 1ÞC
RR
ð01� 1Þ
N
ð� dþ11ÞC
Rð110ÞN
ð1� 1� e þÞC
8R
ð01� 1ÞN
ðeþ11ÞC
Rð101ÞN
ð1� d�
� 1ÞC
Rð1
� 10ÞN
ð11f �ÞC
ð011ÞN
ð� e�1� 1Þ
CR
ð10� 1ÞN
ð1dþ1ÞC
Rð110ÞN
ð1� 1f þÞC
R
9ð101ÞN
ð1f þ
� 1ÞC
Rð1
� 10ÞN
ð11� e þÞC
Rð01� 1ÞN
ðdþ11ÞC
Rð10� 1ÞN
ð1� f þ1ÞC
Rð110ÞN
ð1� 1e �ÞC
Rð011ÞN
ðdþ1� 1ÞC
R
10
ð01� 1Þ
N
ðe�11ÞC
RR
ð10� 1ÞN
ð1� dþ1ÞC
ð110ÞN
ð1� 1f �ÞC
RR
ð011ÞN
ð� eþ1� 1ÞC
ð101ÞN
ð1d�
� 1ÞC
ð1� 10ÞN
ð11f þÞC
RR
11
Rð10� 1ÞN
ð1f �1ÞC
ð110ÞN
ð1� 1� e �ÞC
RR
ð01� 1Þ
N
ðdþ11ÞC
Rð101ÞN
ð1� f þ
� 1ÞC
ð1� 10ÞN
ð11e þÞC
ð011ÞN
ðd�1� 1ÞC
RR
12
Rð011ÞN
ðe�1� 1ÞC
ð10� 1Þ
N
ð1� d�1ÞC
Rð1
� 10ÞN
ð11
� f þÞC
Rð01� 1ÞN
ð� eþ11ÞC
RR
ð110ÞN
ð1� 1
� f �ÞC
Rð101ÞN
ð1� dþ
� 1ÞC
748 Appendix D
TableD.8
(continued)
S1
23
45
67
89
10
11
12
13
RR
ð110ÞC
ð1� 1
� d�ÞN
ð1� 10ÞC
ð11
� d þÞN
Rð011ÞC
ðeþ1� 1ÞN
ð10� 1ÞC
ð1f þ1ÞN
Rð101ÞC
ð1f þ
� 1ÞN
ð01� 1ÞC
ðe�11ÞN
RR
14
RR
ð1� 10ÞC
ð11dþÞN
ð110ÞC
ð1� 1d�ÞN
ð101ÞC
ð1f þ
� 1ÞN
RR
ð01� 1Þ
C
ðeþ11ÞN
RR
ð10� 1ÞC
ð1f �1ÞN
ð011ÞC
ðe�1� 1ÞN
15
Rð01� 1ÞC
ð� fþ11ÞN
Rð011ÞC
ðf �1� 1ÞN
Rð101ÞC
ð1dþ
� 1ÞN
RR
ð1� 10ÞC
ð11� e þÞN
Rð110ÞC
ð1� 1� e �ÞN
ð10� 1ÞC
ð1� d�1ÞN
16
Rð011ÞC
ð� f�1� 1ÞN
Rð01� 1ÞC
ðf þ11ÞN
ð110ÞC
ð1� 1e þÞN
Rð1
� 10ÞC
ð11e �ÞN
ð101ÞC
ð1� d �
� 1ÞN
Rð10� 1Þ
C
ð1� dþ1ÞN
RR
17
Rð10� 1Þ
C
ð1� e þ
1ÞN
ð101ÞC
ð1e �
� 1ÞN
Rð011ÞC
ð� dþ1� 1ÞN
RR
Rð01� 1Þ
C
ðdþ11ÞN
ð110ÞC
ð1� 1f �ÞN
Rð1
� 10ÞC
ð11
� f þÞN
18
Rð101ÞC
ð1� e �
� 1ÞN
ð10� 1ÞC
ð1e þ
1ÞN
RR
ð110ÞC
ð1� 1
� f þÞN
ð011ÞC
ð� d�1� 1Þ
Nð1
� 10ÞC
ð11f �ÞN
RR
ð01� 1ÞC
ðdþ11ÞN
R
19
ð110ÞC
ð1� 1dþÞN
ð1� 10ÞC
ð11
� d�ÞN
RR
RR
ð101ÞC
ð1� f �
� 1ÞN
ð011ÞC
ð� e�1� 1ÞN
ð10� 1ÞC
ð1� f þ1ÞN
RR
ð01� 1ÞC
ð� eþ11ÞN
20
ð1� 10ÞC
ð11d�ÞN
ð110ÞC
ð1� 1dþÞN
RR
ð10� 1Þ
C
ð1� f �1ÞN
ð01� 1ÞC
ð� e�11ÞN
RR
Rð011ÞC
ð� eþ1� 1Þ
Nð101ÞC
ð1� f þ
� 1ÞN
R
21
ð011ÞC
ð� fþ1� 1Þ
NR
ð01� 1ÞC
ðf �11ÞN
RR
RR
ð10� 1ÞC
ð1dþ1ÞN
ð110ÞC
ð1� 1e�ÞN
ð101ÞC
ð1d�
� 1ÞN
ð1� 10ÞC
ð11e þÞN
R
22
ð101ÞC
ð1� e þ
� 1ÞN
RR
ð10� 1ÞC
ð1e �
1ÞN
RR
ð01� 1ÞC
ð� dþ11ÞN
RR
ð1� 10ÞC
ð11f þÞN
ð011ÞC
ðd�1� 1ÞN
ð110ÞC
ð1� 1
� f �ÞN
23
ð10� 1Þ
C
ð1� e �
1ÞN
RR
ð101ÞC
ð1e þ
� 1ÞN
ð01� 1ÞC
ð� d�11ÞN
ð1� 10ÞC
ð11
� f �ÞN
Rð110ÞC
ð1� 1fþÞN
ð011ÞC
ðdþ1� 1ÞN
RR
R
24
ð01� 1ÞC
ð� f�11ÞN
Rð011ÞC
ðf þ1� 1ÞN
Rð1
� 10ÞC
ð11� e �ÞN
ð10� 1ÞC
ð1� d�1ÞN
ð110ÞC
ð1� 1� e þÞN
RR
RR
ð101ÞC
ð1� dþ
� 1ÞN
Appendix D 749
TableD.8
(continued)
S13
14
15
16
17
18
19
20
21
22
23
24
13
N/A
ð001ÞN
ð� b10ÞC
RR
RR
ð010ÞN
ð10� cÞ
Cð100ÞN
ð01aÞC
RR
RR
14
ð001ÞN
ð� b10ÞC
N/A
RR
RR
ð100ÞN
ð01
� aÞC
ð010ÞN
ð10cÞ
CR
RR
R
15
RR
N/A
ð100ÞN
ð0� c1ÞC
RR
RR
ð001ÞN
ð1� a0ÞC
RR
ð010ÞN
ð10bÞC
16
RR
ð100ÞN
ð0� c1ÞC
N/A
RR
RR
ð010ÞN
ð10
� bÞC
RR
ð001ÞN
ð1a0ÞC
17
RR
RR
N/A
ð010ÞN
ð� a01ÞC
RR
Rð001ÞN
ð� c10ÞC
ð100ÞN
ð0b1ÞC
R
18
RR
RR
ð010ÞN
ð� a01ÞC
N/A
RR
Rð100ÞN
ð0� b1ÞC
ð001ÞN
ðc10ÞC
R
19
ð010ÞN
ð10� cÞ
Cð100ÞN
ð01
� aÞC
RR
RR
N/A
ð001ÞN
ðb10ÞC
RR
RR
20
ð100ÞN
ð01aÞC
ð010ÞN
ð10cÞ
CR
RR
Rð001ÞN
ðb10ÞC
N/A
RR
RR
21
RR
ð001ÞN
ð1� a0ÞC
ð010ÞN
ð10
� bÞC
RR
RR
N/A
RR
ð100ÞN
ð0c1ÞC
22
RR
RR
ð001ÞN
ð� c10ÞC
ð100ÞN
ð0� b1ÞC
RR
RN/A
ð010ÞN
ða01ÞC
R
23
RR
RR
ð100ÞN
ð0b1ÞC
ð001ÞN
ðc10ÞC
RR
Rð010ÞN
ða01ÞC
N/A
R
24
RR
ð010ÞN
ð10bÞC
ð001ÞN
ð1a0ÞC
RR
RR
ð100ÞN
ð0c1ÞC
RR
N/A
a¼" 4=" 6;b¼" 4=" 5;c¼" 5=" 6,d�¼ð"
4�" 5Þ=½2ð"
2�" 1Þ�;
e �¼ð"
6�" 4Þ=½2ð"
3�" 1Þ�;
f �¼ð"
6�" 5Þ=½2ð"
3�" 2Þ�
750 Appendix D
Table D.9 Monoclinic to monoclinic
All species 2=m! m 2=m! 2
strains polarizations polarizations
1 "1 "2 "30 "5 0
� �
P1 0 P3ð Þ 0 P2 0ð Þ
�P1 0 �P3ð Þ 0 �P2 0ð Þ
Table D.10 Orthorhombic to monoclinic
All speciesmmm! mx mm2! mx(upperpolarizations only)
mmm! 2xmmm! 2x=mxðP ¼ 0Þ222! 2x (upperpolarizations only)
strains polarizations polarizations
1 "1 "2 "3"4 0 0
� �
0 P2 P3ð Þ P1 0 0ð Þ
0 �P2 �P3ð Þ �P1 0 0ð Þ
2"1 "2 "3�"4 0 0
� �
0 �P2 P3ð Þ �P1 0 0ð Þ0 P2 �P3ð Þ P1 0 0ð Þ
S 1 2
1 N/A ð010ÞCð001ÞN
2 ð010ÞNð001ÞN
N/A
Table D.11 Orthorhombic to monoclinic
Species mm2! 2z
strains polarization
1 "1 "2 "30 0 "6
� �
0 0 P3ð Þ
2"1 "2 "30 0 �"6
� �
0 0 P3ð Þ
S 1 2
1 N/A ð100ÞNð010ÞN
2 N/A
Appendix D 751
Table D.12 Tetragonal to monoclinic
All species 4mm! mx
4=mmm! 2x4=mmm! 2x=mxðP ¼ 0Þ422! 2x(upper polarizations only) �42m! 2x 4=mmm! mx
strains polarizations polarizations polarizations polarizations
1 "1 "2 "3"4 0 0
� �
0 P2 P3ð Þ P1 0 0ð Þ P1 0 0ð Þ 0 P2 P3ð Þ�P1 0 0ð Þ 0 �P2 �P3ð Þ
2"2 "1 "30 �"4 0
� �
�P2 0 P3ð Þ 0 P1 0ð Þ 0 �P1 0ð Þ �P2 0 P3ð Þ0 �P1 0ð Þ P2 0 �P3ð Þ
3"1 "2 "3�"4 0 0
� �
0 �P2 P3ð Þ �P1 0 0ð Þ �P1 0 0ð Þ 0 �P2 P3ð ÞP1 0 0ð Þ 0 P2 �P3ð Þ
4"2 "1 "30 "4 0
� �
P2 0 P3ð Þ 0 �P1 0ð Þ 0 P1 0ð Þ P2 0 P3ð Þ0 P1 0ð Þ �P2 0 �P3ð Þ
4mm! mx; 4=mmm! mx (above diagonal)4=mmm! 2x, 4=mmm! 2x=mxðP ¼ 0Þ,422! 2x (below diagonal)
S 1 2 3 4
1 N/A ð110ÞCð1�1�aÞN
ð010ÞCð001ÞN
ð1�10ÞCð11aÞN
2 ð110ÞN
ð1�1�aÞCN/A ð1�10ÞC
ð11�aÞNð100ÞCð001ÞN
3 ð010ÞNð001ÞN
ð1�10ÞN
ð11�aÞCN/A ð110ÞC
ð1�1aÞN
4 ð1�10ÞN
ð11aÞCð100ÞNð001ÞN
ð110ÞN
ð1�1aÞCN/A
�42m! 2x:
S 1 2 3 4
1 N/A ð110ÞCð1�1�aÞN
ð010ÞNð001ÞN
ð1�10ÞCð11aÞN
2 N/A ð1�10ÞCð11�aÞN
ð100ÞNð001ÞN
3 N/A ð110ÞCð1�1aÞN
4 N/A
a ¼ "4=ð"2 � "1Þ
752 Appendix D
TableD.13
Tetragonalto
monoclinic
Allspecies
4=mmm!
2z
4=mmm!
2z=m
zðP¼
0Þ
4mm!
2z(upper
polarizationsonly)
4=m!
2z(strains1–2only)
4!
2z(strains1–2only,upper
polarizationsonly)
4=mmm!
mz
4=m!
mz(strains1-2
only)
4=m!
2z=m
zðP¼
0Þ(strains1–2only)
� 42m!
2z
� 4!
2z(strains
1–2only)
422!
2z
strains
polarizations
polarizations
polarizations
polarizations
1" 1
" 2" 3
00
" 6
��
00
P3
ðÞ
P1
P2
0ð
Þ0
0P3
ðÞ
00
P3
ðÞ
00�P3
ðÞ
�P1�P2
0ð
Þ
2" 2
" 1" 3
00�" 6
��
00
P3
ðÞ
�P2
P1
0ð
Þ0
0�P3
ðÞ
00
P3
ðÞ
00�P3
ðÞ
P2�P1
0ð
Þ
3" 1
" 2" 3
00�" 6
��
00
P3
ðÞ
P1�P2
0ð
Þ0
0�P3
ðÞ
00�P3
ðÞ
00�P3
ðÞ
�P1
P2
0ð
Þ
4" 2
" 1" 3
00
" 6
��
00
P3
ðÞ
P2
P1
0ð
Þ0
0P3
ðÞ
00�P3
ðÞ
00�P3
ðÞ
�P2�P1
0ð
Þ
Appendix D 753
4=mmm!
2z,4=mmm!
2z=m
zðP¼
0Þ,
4mm!
2z,
� 42m!
2z,422!
2z(all
fourstrains),4=m!
2z,4!
2z,
� 4!
2z(strains1and
2only)
(abovediagonal)
4=mmm!
mz(allfourstrains),4=m!
mz,4=m!
2z=m
zðP¼
0Þ(strains1and2only)(belowdiagonal)
S1
23
4
1N/A
ð1� a þ
0ÞN
ð1� a �
0ÞN
ð100ÞN
ð010ÞN
ð110ÞN
ð1� 10ÞN
2ð1
� aþ0Þ�
ð1� a�0Þ�
N/A
ð110ÞN
ð1� 10ÞN
ð100ÞN
ð010ÞN
3ð100ÞN
ð010ÞC
ð110ÞN
ð1� 10ÞC
N/A
ð1aþ0ÞN
ð1a�0ÞN
4ð110ÞN
ð1� 10ÞC
ð100ÞC
ð010ÞN
ð1aþ0Þ�
ð1a�0Þ�
N/A
a�¼ðp�
ffiffiffiffiffiffiffiffiffiffiffiffiffi
p2þ4
p
Þ=2;p¼
2" 6=ð"
2�" 1Þ
*N
eutrality
orchargeofDW
dependsonthevalues
ofstrain
componentsandPS
TableD.13(continued)
754 Appendix D
Table D.14 Tetragonal to monoclinic
Species �42m! mxy
strains polarizations
1 "1 "1 "3�"5 "5 "6
� �
P1 �P1 P3ð Þ
2 "1 "1 "3"5 "5 �"6
� �
�P1 �P1 �P3ð Þ
3 "1 "1 "3"5 �"5 "6
� �
�P1 P1 P3ð Þ
4 "1 "1 "3�"5 �"5 �"6
� �
P1 P1 �P3ð Þ
S 1 2 3 4
1 N/A ð010ÞNð10�aÞC
ð001ÞNð�110ÞC
ð100ÞNð01aÞC
2 N/A ð100ÞNð01�aÞC
ð001ÞNð110ÞC
3 N/A ð010ÞNð10aÞC
4 N/A
a ¼ "5="6
Table D.15 Trigonal to monoclinic
All species
�3m! 2x �3m! 2x=mxðP ¼ 0Þ32! 2x(upper polarizations only)
�3m! mx
3m! mx(upperpolarizations only)
strains polarizations polarizations
1 "1 "2 "3"4 0 0
� �
P1 0 0ð Þ 0 P2 P3ð Þ�P1 0 0ð Þ 0 �P2 �P3ð Þ
2"01 "
02 "3
�"04 �"05 �"06
� �
�P01 P00
1 0� �
�P02 �P002 P3
� �
P0
1 �P001 0� �
P0
2 P00
2 �P3
� �
3"01 "
02 "3
�"04 "0
5 "0
6
� �
�P01 �P001 0� �
P0
2 �P002 P3
� �
P0
1 P00
1 0� �
�P0
2 P00
2 �P3
� �
"0
1 ¼ ð1=4Þð"1 þ 3"2Þ; "0
2 ¼ ð1=4Þð3"1 þ "2Þ; "0
6 ¼ ðffiffiffi
3p
=2Þð"1 � "2Þ; "0
5 ¼ ðffiffiffi
3p
=2Þ"4; "0
4 ¼ ð1=2Þ"4P0
1 ¼ ð1=2ÞP1;P00
1 ¼ ðffiffiffi
3p
=2ÞP1;P0
2 ¼ ðffiffiffi
3p
=2ÞP2;P00
2 ¼ ð1=2ÞP2
Appendix D 755
�3m! 2x, �3m! 2x=mxðP ¼ 0Þ, 32! 2x (above diagonal)�3m! mx, 3m! mx(below diagonal)
S 1 2 3
1 N/A ð11�20ÞNð1�10aÞC
ð1�210ÞNð10�1aÞC
2 ð11�20ÞCð1�10aÞN
N/A ð2�1�10ÞNð01�1aÞC
3 ð1�210ÞCð10�1aÞN
ð2�1�10ÞCð01�1aÞN
N/A
a ¼ �ffiffiffi
3p
"4=ð"2 � "1Þ
Table D.15 (continued)
Table D.16 Hexagonal to monoclinic
All species
6=mmm! 2x622! 2x(upper polarizations only)6=mmm! 2x=mxðP ¼ 0Þ
6=mmm! mx
6mm! mx(upperpolarizations only) �6m2! mx
strains polarizations polarizations polarizations
1 "1 "2 "3"4 0 0
� �
P1 0 0ð Þ 0 P2 P3ð Þ 0 P2 P3ð Þ�P1 0 0ð Þ 0 �P2 �P3ð Þ
2"0
1 "0
2 "3"04 "
05 "
06
� �
P0
1 P00
1 0� �
�P002 P0
2 P3
� �
P00
2 �P02 �P3
� �
�P01 �P001 0� �
P002 �P02 �P3
� �
3"01 "
02 "3
�"04 "05 �"06
� �
�P01 P001 0
� �
�P002 �P02 P3
� �
�P002 �P02 P3
� �
P0
1 �P001 0� �
P00
2 P0
2 �P3
� �
4"1 "2 "3�"4 0 0
� �
�P1 0 0ð Þ 0 �P2 P3ð Þ 0 P2 �P3ð Þ
P1 0 0ð Þ 0 P2 �P3ð Þ
5"01 "
02 "3
�"04 �"05 "06
� �
�P01 �P001 0� �
P002 �P02 P3
� �
P002 �P02 P3
� �
P01 P
001 0
� �
�P002 P02 �P3
� �
6"01 "
02 "3
"04 �"05 �"06
� �
P01 �P001 0
� �
P002 P
02 P3
� �
�P002 �P02 �P3
� �
�P01 P001 0
� �
�P002 �P02 �P3
� �
"01 ¼ ð1=4Þð"1 þ 3"2Þ; "
02 ¼ ð1=4Þð3"1 þ "2Þ; "
06 ¼ ð
ffiffiffi
3p
=2Þð"1 � "2Þ; "05 ¼ �ð
ffiffiffi
3p
=2Þ"4; "04 ¼ ð1=2Þ"4
P01 ¼ ð1=2ÞP1;P
001 ¼ ð
ffiffiffi
3p
=2ÞP1;P02 ¼ ð1=2ÞP2;P
002 ¼ ð
ffiffiffi
3p
=2ÞP2
6=mmm! 2x, 6=mmm! 2x=mxðP ¼ 0Þ, 622! 2x (above diagonal)6=mmm! mx, 6mm! mx (below diagonal)
S 1 2 3 4 5 6
1 N/A ð10�10ÞNð1�21�aÞC
ð11�20ÞNð1�10�aÞC
ð01�10ÞNð0001ÞN
ð1�210ÞNð10�1aÞC
ð1�100ÞNð11�2aÞC
2 ð10�10ÞCð1�21�aÞN
N/A ð01�10ÞNð2�1�1�aÞC
ð1�210ÞNð10�1�aÞC
ð0001ÞNð1�100ÞN
ð2�1�10ÞNð01�1aÞC
3 ð11�20ÞCð1�10�aÞN
ð01�10ÞCð2�1�1�aÞN
N/A ð1�100ÞNð11�2�aÞC
ð2�1�10ÞNð01�1�aÞC
ð0001ÞNð10�10ÞN
4 ð01�10ÞCð0001ÞN
ð1�210ÞCð10�1�aÞN
ð1�100ÞCð11�2�aÞN
N/A ð10�10ÞNð1�21aÞC
ð11�20ÞNð1�10aÞC
5 ð1�210ÞCð10�1aÞN
ð0001ÞNð1�100ÞC
ð2�1�10ÞCð01�1�aÞN
ð10�10ÞCð1�21aÞN
N/A ð01�10ÞNð2�1�1aÞC
6 ð1�100ÞCð11�2aÞN
ð2�1�10ÞCð01�1aÞN
ð0001ÞNð10�10ÞC
ð11�20ÞCð1�10aÞN
ð01�10ÞCð2�1�1aÞN
N/A
756 Appendix D
�6m2! mx
S 1 2 3 4 5 6
1 N/A ð10�10ÞN
ð1�21�aÞCð11�20ÞC
ð1�10�aÞNð01�10ÞN
ð0001ÞCð1�210ÞC
ð10�1aÞNð1�100ÞN
ð11�2aÞC2N/A ð01�10ÞN
ð2�1�1�aÞCð1�210ÞC
ð10�1�aÞNð0001ÞC
ð1�100ÞNð2�1�10ÞC
ð01�1aÞN3N/A ð1�100ÞN
ð11�2�aÞCð2�1�10ÞC
ð01�1�aÞNð0001ÞC
ð10�10ÞN4N/A lð10�10ÞN
ð1�21aÞCð11�20ÞC
ð1�10aÞN5N/A ð01�10ÞN
ð2�1�1aÞC6N/A
a ¼ "4=ð"2 � "1Þ
Table D.16 (continued)
Appendix D 757
TableD.17
Hexagonalto
monoclinic
Allspecies
622!
2z
6=mmm!
2z=m
zðP¼
0Þ
6!
2z(strains1–3only)
6=m!
2z=m
zðP¼
0Þ
(strains1–3only)
6=mmm!
2z
6=m!
2z(strains
1–3only)
6=mmm!
mz6=m!
mz
(strains1–3only)
� 6!
mz
(strains1–3only,upper
polarizationsonly)
6mm!
2z
� 6m2!
mz
strains
polarizations
polarizations
polarizations
polarizations
polarizations
1" 1
" 2" 3
00
" 6
��
00
P3
ðÞ
00
P3
ðÞ
P1
P2
0ð
Þ0
0P3
ðÞ
P1
P2
0ð
Þ0
0�P3
ðÞ
�P1�P2
0ð
Þ
2"� 1
"þ 2" 3
00
"þ 6
��
00
P3
ðÞ
00
P3
ðÞ
P� 1
Pþ 2
0�
�
00
P3
ðÞ
�P� 1�Pþ 2
0�
�
00�P3
ðÞ
�P� 1�Pþ 2
0�
�
3"þ 1
"� 2" 3
00
"� 6
��
00
P3
ðÞ
00
P3
ðÞ
�Pþ 1
P� 2
0�
�
00
P3
ðÞ
�Pþ 1
P� 2
0�
�
00�P3
ðÞ
Pþ 1�P� 2
0�
�
4"� 1
"þ 2" 3
00�"þ 6
��
00�P3
ðÞ
00�P3
ðÞ
P� 1�Pþ 2
0�
�
00
P3
ðÞ
P� 1�Pþ 2
0�
�
00
P3
ðÞ
�P� 1
Pþ 2
0�
�
5"þ 1
"� 2" 3
00�"� 6
��
00�P3
ðÞ
00�P3
ðÞ
�Pþ 1�P� 2
0�
�
00
P3
ðÞ
Pþ 1
P� 2
0�
�
00
P3
ðÞ
Pþ 1
P� 2
0�
�
6" 1
" 2" 3
00�" 6
��
00�P3
ðÞ
00�P3
ðÞ
P1�P2
0ð
Þ0
0P3
ðÞ
�P1
P2
0ð
Þ0
0P3
ðÞ
�P1
P2
0ð
Þ
P� 1¼ð1=2ÞðP1�
ffiffiffi 3
pP2Þ;P� 2¼ð1=2Þð
ffiffiffi 3
pP1�P2Þ;"� 1¼ð1=4Þð" 1�
ffiffiffi 3
p" 6þ3" 2Þ;"� 2¼ð1=4Þð3" 1�
ffiffiffi 3
p" 6þ" 2Þ;"� 6¼ð1=2Þð�
ffiffiffi 3
p" 1�" 6�
ffiffiffi 3
p" 2Þ
758 Appendix D
TableD.17
(continued)
6=mmm!
2z=m
zðP¼
0Þ,622!
2z,6=mmm!
2z,6mm!
2z(allsixstrains),
6=m!
2z=m
zðP¼
0Þ,
6!
2z,6=m!
2z(strains1–3only)(abovediagonal)
6=mmm!
mz(allsixstrains),6=m!
mz,� 6!
mz(strains1–3only)(belowdiagonal)
S1
23
45
6
1N/A
ð2kþl þ0ÞN
ð2k�l �0ÞN
ð2mþnþ0ÞN
ð2m�n�0ÞN
ð1� 10
0ÞN
ð11� 20ÞN
ð10� 10ÞN
ð1� 21
0ÞN
ð2� 1� 10ÞN
ð01� 10ÞN
2ð2kþl þ0Þ�
ð2k�l �0Þ�
N/A
ð2pþqþ0ÞN
ð2p�q�0ÞN
ð2� 1� 10ÞN
ð01� 10ÞN
ð11� 20ÞN
ð1� 10
0ÞN
ð1� 21
0ÞN
ð10� 10ÞN
3ð2mþnþ0Þ�
ð2m�n�0Þ�
ð2pþqþ0Þ�
ð2p�q�0Þ�
N/A
ð10� 10ÞN
ð1� 21
0ÞN
ð2� 1� 10ÞN
ð01� 10ÞN
ð11� 20ÞN
ð1� 10
0ÞN
4ð1
� 100ÞN
ð11� 20ÞC
ð2� 1� 10ÞN
ð01� 10ÞC
ð10� 10ÞN
ð1� 21
0ÞC
N/A
ð2qþpþ0ÞN
ð2q�p�0ÞN
ð2l þkþ0ÞN
ð2l �k�0ÞN
5ð10� 10ÞC
ð1� 210ÞN
ð11� 20ÞC
ð1� 10
0ÞN
ð2� 1� 10ÞN
ð01� 10ÞC
ð2qþpþ0Þ�
ð2q�p�0Þ�
N/A
ð2nþmþ0ÞN
ð2n�m�0ÞN
6ð2
� 1� 10ÞN
ð01� 10ÞC
ð1� 21
0ÞC
ð10� 10ÞN
ð11� 20ÞN
ð1� 10
0ÞC
ð2l þkþ0Þ�
ð2l �k�0Þ�
ð2nþmþ0Þ�
ð2n�m�0Þ�
N/A
Appendix D 759
� 6m2!
mz
S1
23
45
6
1N/A
ð2kþl þ0Þ�
ð2k�l �0Þ�
ð2mþnþ0Þ�
ð2m�n�0Þ�
ð1� 10
0ÞN
ð11� 20ÞC
ð10� 10ÞN
ð1� 21
0ÞC
ð2� 1� 10ÞC
ð01� 10ÞN
2N/A
ð2pþqþ0Þ�
ð2p�q�0Þ�
ð2� 1� 10ÞC
ð01� 10ÞN
ð11� 20ÞC
ð1� 10
0ÞN
ð1� 21
0ÞC
ð10� 10ÞN
3N/A
ð10� 10ÞN
ð1� 21
0ÞC
ð2� 1� 10ÞC
ð01� 10ÞN
ð11� 20ÞC
ð1� 10
0ÞN
4N/A
ð2qþpþ0Þ�
ð2q�p�0Þ�
ð2l þkþ0Þ�
ð2l �k�0Þ�
5N/A
ð2nþmþ0Þ�
ð2n�m�0Þ�
6N/A
*Neutrality/chargeofDW
dependsonthevalues
ofspontaneousstrain/polarization
k�¼
2½"
6�
ffiffiffi 3
pð"
1�" 2Þ�
ffiffiffi 3
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð"1�" 2Þ2þ"2 6
q
�=½"
6þ
ffiffiffi 3
pð"
1�" 2Þ�,
l �¼
2½�
2" 6�
ffiffiffi 3
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð"1�" 2Þ2þ"2 6
q
�=½"
6þ
ffiffiffi 3
pð"
1�" 2Þ�
m�¼
2½2" 6�
ffiffiffi 3
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð"1�" 2Þ2þ"2 6
q
�=½�" 6þ
ffiffiffi 3
pð"
1�" 2Þ�,
n�¼
2½�" 6�
ffiffiffi 3
pð"
1�" 2Þ�
ffiffiffi 3
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð"1�" 2Þ2þ"2 6
q
�=½�" 6þ
ffiffiffi 3
pð"
1�" 2Þ�
p�¼
2½�" 6�
ffiffiffi 3
pð"
1�" 2Þ�
ffiffiffi 3
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð"1�" 2Þ2þ"2 6
q
�=" 6,
q�¼
2½�" 6þ
ffiffiffi 3
pð"
1�" 2Þ�
ffiffiffi 3
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð"1�" 2Þ2þ"2 6
q
�=" 6
TableD.17(continued)
760 Appendix D
Table D.18 Hexagonal to monoclinic
Species �6m2! 2y
strains polarizations
1 "1 "2 "30 "5 0
� �
0 P2 0ð Þ
2"1
0"2
0"3
"04 "
05 "
06
� �
P00
2 �P02 0� �
3"1
0"0
2 "3"0
4 �"05 �"06
� �
�P002 �P02 0� �
4"1 "2 "30 �"5 0
� �
0 P2 0ð Þ
5"0
1 "0
2 "3�"04 �"05 "
06
� �
P00
2 �P02 0� �
6"0
1 "0
2 "3�"04 "
05 �"06
� �
�P002 �P02 0� �
"0
1 ¼ ð1=4Þð"1 þ 3"2Þ; "0
2 ¼ ð1=4Þð3"1 þ "2Þ; "0
6 ¼ ðffiffiffi
3p
=2Þð"1 � "2Þ;"0
5 ¼ ð1=2Þ"5; "0
4 ¼ ðffiffiffi
3p
=2Þ"5P0
2 ¼ ð1=2ÞP2;P00
2 ¼ ðffiffiffi
3p
=2ÞP2
S 1 2 3 4 5 6
1 N/A ð10�1�aÞN
ð1�210ÞCð11�2�bÞCð1�100ÞN
ð2�1�10ÞNð0001ÞN
ð1�21�bÞCð10�10ÞN
ð1�10�aÞN
ð11�20ÞC
2 N/A ð01�1�aÞN
ð2�1�10ÞCð1�21bÞCð10�10ÞN
ð0001ÞNð11�20ÞN
ð2�1�1�bÞCð01�10ÞN
3 N/A ð1�10aÞN
ð11�20ÞCð2�1�1bÞCð01�10ÞN
ð0001ÞNð1�210ÞN
4 N/A ð10�1aÞN
ð1�210ÞCð11�2bÞCð1�100ÞN
5 N/A ð01�1aÞN
ð2�1�10ÞC
6 N/A
a ¼ "5=ð"2 � "1Þ; b ¼ 3a
Appendix D 761
TableD.19
Cubicto
monoclinic
Allspecies
m� 3m!
mxy
� 43m!
mxy(upper
polarizationsonly)
m� 3m!
2xy
m� 3m!
2xy=m
xyðP¼
0Þ
432!
2xy(upper
polarizationsonly)
strains
polarizations
polarizations
1" 1
" 1" 3
�" 5
" 5" 6
��
P1�P1
P3
ðÞ
P1
P1
0ð
Þ�P1
P1�P3
ðÞ
�P1�P1
0ð
Þ
2" 1
" 1" 3
" 5" 5�" 6
��
�P1�P1�P3
ðÞ
�P1
P1
0ð
ÞP1
P1
P3
ðÞ
P1�P1
0ð
Þ
3" 1
" 1" 3
" 5�" 5
" 6
��
�P1
P1
P3
ðÞ
�P1�P1
0ð
ÞP1�P1�P3
ðÞ
P1
P1
0ð
Þ
4" 1
" 1" 3
�" 5�" 5�" 6
��
P1
P1�P3
ðÞ
P1�P1
0ð
Þ�P1�P1
P3
ðÞ
�P1
P1
0ð
Þ
5" 1
" 3" 1
" 5" 6�" 5
��
�P1
P3
P1
ðÞ
P1
0P1
ðÞ
P1�P3�P1
ðÞ
�P1
0�P1
ðÞ
6" 1
" 3" 1
" 5�" 6
" 5
��
�P1�P3�P1
ðÞ
P1
0�P1
ðÞ
P1
P3
P1
ðÞ
�P1
0P1
ðÞ
7" 3
" 1" 1
" 6�" 5
" 5
��
P3
P1�P1
ðÞ
0P1
P1
ðÞ
�P3�P1
P1
ðÞ
0�P1�P1
ðÞ
8" 3
" 1" 1
�" 6�" 5�" 5
��
�P3
P1
P1
ðÞ
0P1�P1
ðÞ
P3�P1�P1
ðÞ
0�P1
P1
ðÞ
9" 1
" 3" 1
�" 5�" 6�" 5
��
P1�P3
P1
ðÞ
�P1
0P1
ðÞ
�P1
P3�P1
ðÞ
P1
0�P1
ðÞ
10
" 1" 3
" 1�" 5
" 6" 5
��
P1
P3�P1
ðÞ
�P1
0�P1
ðÞ
�P1�P3
P1
ðÞ
P1
0P1
ðÞ
11
" 3" 1
" 1�" 6
" 5" 5
��
�P3�P1�P1
ðÞ
0�P1
P1
ðÞ
P3
P1
P1
ðÞ
0P1�P1
ðÞ
12
" 3" 1
" 1" 6
" 5�" 5
��
P3�P1
P1
ðÞ
0�P1�P1
ðÞ
�P3
P1�P1
ðÞ
0P1
P1
ðÞ
762 Appendix D
TableD.19
(continued)
m� 3m!
mxy,� 43m!
mxy(abovediagonal)
m� 3m!
2xy,m
� 3m!
2xy=m
xyðP¼
0Þ,432!
2xy(belowdiagonal)
S1
23
45
67
89
10
11
12
1N/A
ð010ÞN
ð10
� aÞC
ð001ÞN
ð1� 10ÞC
ð100ÞN
ð01aÞC
RR
RR
ð011ÞC
ð� c1� 1ÞN
ð01� 1ÞC
ð� b11ÞN
ð101ÞC
ð1� b� 1ÞN
ð10� 1ÞC
ð1� c1ÞN
2ð010ÞN
ð10
� aÞC
N/A
ð100ÞN
ð01
� aÞC
ð001ÞN
ð110ÞC
ð011ÞC
ðb1� 1ÞN
ð01� 1ÞC
ðc11ÞN
RR
RR
ð10� 1ÞC
ð1c1ÞN
ð101ÞC
ð1b� 1ÞN
3ð001ÞN
ð1� 10ÞN
ð100ÞN
ð01
� aÞC
N/A
ð010ÞN
ð10aÞC
ð01� 1ÞC
ð� c11ÞN
ð011ÞC
ð� b1� 1ÞN
ð10� 1ÞC
ð1� b1ÞN
ð101ÞC
ð1� c� 1ÞN
RR
RR
4ð100ÞN
ð01aÞC
ð001ÞN
ð110ÞN
ð010ÞN
ð10aÞC
N/A
RR
ð101ÞC
ð1c� 1ÞN
ð10� 1ÞC
ð1b1ÞN
ð01� 1ÞC
ðb11ÞN
ð011ÞC
ðc1� 1ÞN
RR
5R
ð011ÞN
ðb1� 1ÞC
ð01� 1ÞN
ð� c11ÞC
RN/A
ð100ÞN
ð0� a1ÞC
Rð110ÞC
ð1� 1� cÞ
Nð001ÞN
ð1a0ÞC
ð010ÞN
ð10� 1ÞC
Rð1
� 10ÞC
ð11
� bÞN
6R
ð01� 1ÞN
ðc11ÞC
ð011ÞN
ð� b1� 1ÞC
Rð100ÞN
ð0� a1ÞC
N/A
ð110ÞC
ð1� 1bÞN
Rð010ÞN
ð101ÞC
ð001ÞN
ð1� a0ÞC
ð1� 10ÞC
ð11cÞ
NR
7R
Rð10� 1ÞN
ð1� b1ÞC
ð101ÞN
ð1c� 1ÞC
Rð110ÞN
ð1� 1bÞC
N/A
ð010ÞN
ða01ÞC
Rð1
� 10ÞC
ð11� cÞ
Nð001ÞN
ð� a10ÞC
ð100ÞN
ð0� 11ÞC
8R
Rð101ÞN
ð1� c� 1ÞC
ð10� 1ÞN
ð1b1ÞC
ð110ÞN
ð1� 1� cÞ
CR
ð010ÞN
ða01ÞC
N/A
ð1� 10ÞC
ð11bÞN
Rð100ÞN
ð011ÞC
ð001ÞN
ða10ÞC
9ð011ÞN
ð� c1� 1ÞC
RR
ð01� 1ÞN
ðb11ÞC
ð001ÞN
ð1a0ÞC
ð010ÞN
ð101ÞN
Rð1
� 10ÞN
ð11bÞC
N/A
ð100ÞN
ð0a1ÞC
Rð110ÞC
ð1� 1cÞ
N
10
ð01� 1ÞN
ð� b11ÞC
RR
ð011ÞN
ðc1� 1ÞC
ð010ÞN
ð10� 1ÞN
ð001ÞN
ð1� a0ÞC
ð1� 10ÞN
ð11� cÞ
CR
ð100ÞN
ð0a1ÞC
N/A
ð110ÞC
ð1� 1
� bÞN
R
11
ð101ÞN
ð1� b� 1ÞC
ð10� 1ÞN
ð1c1ÞC
RR
Rð1
� 10ÞN
ð11cÞ
Cð001ÞN
ð� a10ÞC
ð100ÞN
ð011ÞN
Rð110ÞN
ð1� 1
� bÞC
N/A
ð010ÞN
ð� a01ÞC
12
ð10� 1ÞN
ð1� c1ÞC
ð101ÞN
ð1b� 1ÞC
RR
ð1� 10ÞN
ð11
� bÞC
Rð100ÞN
ð0� 11ÞN
ð001ÞN
ða10ÞC
ð110ÞN
ð1� 1cÞ
CR
ð010ÞN
ð� a01ÞC
N/A
a¼" 5=" 6;b¼ð"
6�" 5Þ=ð"
3�" 1Þ;c¼ð"
6þ" 5Þ=ð"
3�" 1Þ
Appendix D 763
TableD.20
Cubicto
monoclinicspecies
Allspecies
m� 3m!
2zm
� 3m!
2z=m
zðP¼
0Þ
432!
2z(upper
polarizationsonly)
strains1–6only
forfollowingspecies:
m� 3!
2z
m� 3!
2z=m
zðP¼
0Þ
23!
2z(upper
polarizationsonly)
� 43m!
2z
m� 3m!
mzm
� 3!
mz
(strains1–6only)
strains
polarizations
polarizations
polarizations
1" 1
" 2" 3
00
" 6
��
00
P3
ðÞ
00
P3
ðÞ
P1
P2
0ð
Þ0
0�P3
ðÞ
�P1�P2
0ð
Þ
2" 3
" 1" 2
" 60
0
��
P3
00
ðÞ
P3
00
ðÞ
0P1
P2
ðÞ
�P3
00
ðÞ
0�P1�P2
ðÞ
3" 2
" 3" 1
0" 6
0
��
0P3
0ð
Þ0
P3
0ð
ÞP2
0P1
ðÞ
0�P3
0ð
Þ�P2
0�P1
ðÞ
4" 1
" 2" 3
00�" 6
��
00�P3
ðÞ
00�P3
ðÞ
P1�P2
0ð
Þ0
0P3
ðÞ
�P1
P2
0ð
Þ5
" 3" 1
" 2�" 6
00
��
�P3
00
ðÞ
�P3
00
ðÞ
0P1�P2
ðÞ
P3
00
ðÞ
0�P1
P2
ðÞ
6" 2
" 3" 1
0�" 6
0
��
0�P3
0ð
Þ0�P3
0ð
ÞP2
0�P1
ðÞ
0P3
0ð
Þ�P2
0P1
ðÞ
7" 2
" 1" 3
�" 6
00
��
00
P3
ðÞ
00�P3
ðÞ
�P2
P1
0ð
Þ0
0�P3
ðÞ
P2�P1
0ð
Þ8
" 2" 1
" 3" 6
00
��
00�P3
ðÞ
00
P3
ðÞ
P2
P1
0ð
Þ0
0P3
ðÞ
�P2�P1
0ð
Þ9
" 1" 3
" 20
" 60
��
0�P3
0ð
Þ0
P3
0ð
ÞP1
0P2
ðÞ
0P3
0ð
Þ�P1
0�P2
ðÞ
10
" 3" 2
" 1" 6
00
��
�P3
00
ðÞ
P3
00
ðÞ
0P2
P1
ðÞ
P3
00
ðÞ
0�P2�P1
ðÞ
11
" 1" 3
" 20�" 6
0
��
0P3
0ð
Þ0�P3
0ð
ÞP1
0�P2
ðÞ
0�P3
0ð
Þ�P1
0P2
ðÞ
12
" 3" 2
" 1�" 6
00
��
P3
00
ðÞ
�P3
00
ðÞ
0P2�P1
ðÞ
�P3
00
ðÞ
0�P2
P1
ðÞ
764 Appendix D
TableD.20
(continued)
� 43m!
2z(abovediagonal)
m� 3m!
2z,m
� 3m!
2z=m
zðP¼
0Þ,432!
2z(all12strains),m
� 3!
2z,m
� 3!
2z=m
zðP¼
0Þ,
23!
2z(strains1–6only)(belowdiagonal)
S1
23
45
67
89
10
11
12
1N/A
RR
ð100ÞN
ð010ÞN
RR
ð1�z þ
0ÞN
ð1� z �
0ÞN
ð1� 10ÞN
ð110ÞN
ð01� 1ÞC
ð� c11ÞN
ð10� 1ÞC
ð1� d1ÞN
ð011ÞC
ð� c1� 1ÞN
ð101ÞC
ð1� d� 1ÞN
2R
N/A
RR
ð010ÞN
ð001ÞN
Rð101ÞC
ð1c� 1ÞN
ð10� 1ÞC
ð1� c1ÞN
ð1� 10ÞC
ð11
� dÞN
ð011ÞN
ð01� 1ÞN
ð110ÞC
ð1� 1dÞN
ð01� z þÞN
ð01�z �ÞN
3R
RN/A
RR
ð100ÞN
ð001ÞN
ð011ÞC
ðd1� 1ÞN
ð01� 1ÞC
ð� d11ÞN
ð10� 1ÞN
ð101ÞN
ð1� 10ÞC
ð11� cÞ
Nð10z þÞN
ð10z �ÞN
ð110ÞC
ð1� 1� cÞ
N
4ð100ÞN
ð010ÞN
RR
N/A
RR
ð1� 10ÞN
ð110ÞN
ð1z þ
0ÞN
ð1z �
0ÞN
ð011ÞC
ðc1� 1ÞN
ð101ÞC
ð1d� 1ÞN
ð01� 1ÞC
ðc11ÞN
ð10� 1ÞC
ð1d1ÞN
5R
ð010ÞN
ð001ÞN
RR
N/A
Rð10� 1ÞC
ð1c1ÞN
ð101ÞC
ð1� c� 1ÞN
ð110ÞC
ð1� 1
� dÞN
ð01z þÞN
ð01z �ÞN
ð1� 10ÞC
ð11dÞN
ð01� 1ÞN
ð011ÞN
6R
Rð100ÞN
ð001ÞN
RR
N/A
ð01� 1ÞC
ðd11ÞN
ð011ÞC
ð� d1� 1ÞN
ð10� z þÞN
ð10�z �ÞN
ð110ÞC
ð1� 1cÞ
Nð10� 1ÞN
ð101ÞN
ð1� 10ÞC
ð11cÞ
N
7ð1
� z þ0ÞN
ð1�z �
0ÞN
ð101ÞN
ð1c� 1ÞC
ð011ÞN
ðd1� 1ÞC
ð1� 10ÞN
ð110ÞN
ð10� 1ÞN
ð1c1ÞC
ð01� 1ÞN
ðd11ÞC
N/A
ð100ÞN
ð010ÞN
RR
RR
8ð1
� 10ÞN
ð110ÞN
ð10� 1ÞN
ð1� c1ÞC
ð01� 1Þ
N
ð� d11ÞC
ð1z þ
0ÞN
ð1z �
0ÞN
ð101ÞN
ð1� c� 1ÞC
ð011ÞN
ð� d1� 1ÞC
ð100ÞN
ð010ÞN
N/A
RR
RR
9ð01� 1ÞN
ð� c11ÞC
ð1� 10ÞN
ð11
� dÞC
ð10� 1Þ
N
ð101ÞN
ð011ÞN
ðc1� 1ÞC
ð110ÞN
ð1� 1
� dÞC
ð10� z þÞN
ð10� z �ÞN
RR
N/A
Rð100ÞN
ð001ÞN
R
10
ð10� 1ÞN
ð1� d1ÞC
ð011ÞN
ð01� 1ÞN
ð1� 10ÞN
ð11� cÞ
Cð101ÞN
ð1d� 1ÞC
ð01z þÞN
ð01z �ÞN
ð110ÞN
ð1� 1cÞ
CR
RR
N/A
Rð010ÞN
ð001ÞN
11
ð011ÞN
ð� c1� 1ÞC
ð110ÞN
ð1� 1dÞC
ð10z þÞN
ð10z �ÞN
ð01� 1ÞN
ðc11ÞC
ð1� 10ÞN
ð11dÞC
ð10� 1ÞN
ð101ÞN
RR
ð100ÞN
ð001ÞN
RN/A
R
12
ð101ÞN
ð1� d� 1ÞC
ð01� z þÞN
ð01� z �ÞN
ð110ÞN
ð1� 1� cÞ
Cð10� 1ÞN
ð1d1ÞC
ð01� 1ÞN
ð011ÞN
ð1� 10ÞN
ð11cÞ
CR
RR
ð010ÞN
ð001ÞN
RN/A
Appendix D 765
m� 3m!
mz(all12strains),m
� 3!
mz(strains1–6only)
S1
23
45
67
89
10
11
12
1N/A
RR
ð100ÞN
ð010ÞC
RR
ð1�z þ
0Þ�
ð1� z �
0Þ�
ð1� 10ÞC
ð110ÞN
ð01� 1Þ
C
ð� c11ÞN
ð10� 1ÞC
ð1� d1ÞN
ð011ÞC
ð� c1� 1ÞN
ð101ÞC
ð1� d� 1ÞN
2N/A
RR
ð010ÞN
ð001ÞC
Rð101ÞC
ð1c� 1ÞN
ð10� 1ÞC
ð1� c1ÞN
ð1� 10ÞC
ð11
� dÞN
ð011ÞN
ð01� 1ÞC
ð110ÞN
ð1� 1dÞC
ð01� z þÞ�
ð01� z �Þ�
3N/A
RR
ð100ÞN
ð001ÞC
ð011ÞN
ðd1� 1ÞC
ð01� 1ÞC
ð� d11ÞN
ð10� 1Þ
ð101Þ
ð1� 10ÞC
ð11� cÞ
N
ð10z þÞ�
ð10z �Þ�
ð110Þ
ð1� 1� cÞ
4N/A
RR
ð1� 10ÞC
ð110ÞN
ð1z þ
0Þ�
ð1z �
0Þ�
ð011Þ
ðc1� 1Þ
ð101ÞN
ð1d� 1ÞC
ð01� 1ÞC
ðc11ÞN
ð10� 1ÞN
ð1d1ÞC
5N/A
Rð10� 1ÞC
ð1c1ÞN
ð101ÞC
ð1� c� 1ÞN
ð110ÞN
ð1� 1
� dÞC
ð01z þÞ�
ð01z �Þ�
ð1� 10ÞC
ð11dÞN
ð01� 1Þ
ð011Þ
6N/A
ð01� 1ÞN
ðd11ÞC
ð011ÞC
ð� d1� 1Þ
N
ð10� z þÞ�
ð10� z �Þ�
ð110Þ
ð1� 1cÞ
ð10� 1ÞN
ð101ÞC
ð1� 10ÞC
ð11cÞ
N
7N/A
ð100ÞC
ð010ÞN
RR
RR
8N/A
RR
RR
9N/A
Rð100ÞN
ð001ÞC
R
10
N/A
Rð010ÞN
ð001ÞC
11
N/A
R
12
N/A
a¼" 2�" 1;b¼�2" 6;c¼" 6=ð"
3�" 2Þ;d¼" 6=ð"
3�" 1Þ;z �¼ð�
b�
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
b2þ4a2
pÞ=2a
*Neutrality/chargeofDW
dependsonthevalues
ofspontaneousstrain/polarization
TableD.20(continued)
766 Appendix D
Table D.21 Orthorhombic to orthorhombic
Species mmm! mxmy2z
strains polarizations
1 "1 "2 "30 0 0
� �
0 0 P3ð Þ
0 0 �P3ð Þ
Table D.22 Tetragonal to orthorhombic
Species4mm! mxmy2z422! 2x2y2zðP ¼ 0Þ
4=mmm! mxmy2z4=mmm! 2x2y2zðP ¼ 0Þ�42m! 2x2y2zðP ¼ 0Þ
4=mmm! 2xmymz
4=mmm! mxmymz
(P¼ 0)
strains polarizations polarizations polarizations
1 "1 "2 "30 0 0
� �
0 0 P3ð Þ 0 0 P3ð Þ P1 0 0ð Þ0 0 �P3ð Þ �P1 0 0ð Þ
2"2 "1 "30 0 0
� �
0 0 P3ð Þ 0 0 �P3ð Þ 0 �P1 0ð Þ0 0 P3ð Þ 0 P1 0ð Þ
4mm! mxmy2z, 422! 2x2y2zðP ¼ 0Þ; 4=mmm! mxmy2z, 4=mmm! 2x2y2zðP ¼ 0Þ,�42m! 2x2y2zðP ¼ 0Þ(above diagonal)4=mmm! mxmymzðP ¼ 0Þ, 4=mmm! 2xmymz(below diagonal)
S 1 2
1 N/A ð110ÞNð1�10ÞN
2 ð110ÞCð1�10ÞN
N/A
Table D.23 Tetragonal to orthorhombic
Species �42m! mxym�xy2z
strains polarizations
1 "1 "1 "30 0 "6
� �
0 0 P3ð Þ
2"1 "1 "30 0 �"6
� �
0 0 �P3ð Þ
S 1 2
1 N/A ð100ÞNð010ÞN
2 N/A
Appendix D 767
Table D.24 Hexagonal to orthorhombic
All species
�6m2! mx2ymz
622! 2x2y2zðP ¼ 0Þ6=mmm! mxmymz (P¼ 0)6=mmm! 2x2y2zðP ¼ 0Þ 6=mmm! 2xmymz
6=mmm! mxmy2z6mm! mxmy2z(upperpolarizations only)
strains polarizations polarizations polarizations
1 "1 "2 "30 0 0
� �
0 P2 0ð Þ P1 0 0ð Þ 0 0 P3ð Þ�P1 0 0ð Þ 0 0 �P3ð Þ
2"01 "
02 "3
0 0 "06
� �
�P002 P02 0
� �
P01 P
001 0
� �
0 0 P3ð Þ�P01 �P001 0� �
0 0 �P3ð Þ
3"01 "
02 "3
0 0 �"06
� �
�P002 �P02 P3
� �
�P01 P001 0
� �
0 0 P3ð ÞP01 �P001 0
� �
0 0 �P3ð Þ
"01 ¼ ð1=4Þð"1 þ 3"2Þ; "
02 ¼ ð1=4Þð3"1 þ "2Þ; "
06 ¼ ð
ffiffiffi
3p
=2Þð"1 � "2ÞP01 ¼ ð1=2ÞP1;P
001 ¼ ð
ffiffiffi
3p
=2ÞP1;P02 ¼ ð1=2ÞP2;P
002 ¼ ð
ffiffiffi
3p
=2ÞP2
6=mmm! mxmymzðP ¼ 0Þ, 6=mmm! 2x2y2zðP ¼ 0Þ, �6m2! mx2ymz,622! 222ðP ¼ 0Þ(above diagonal); 6=mmm! 2xmymz (below diagonal)
S 1 2 3
1 N/A ð10�10ÞCð1�210ÞN
ð11�20ÞCð1�100ÞN
2 ð10�10ÞN
ð1�210ÞCN/A ð01�10ÞC
ð2�1�10ÞN
3 ð11�20ÞN
ð1�100ÞCð01�10ÞN
ð2�1�10ÞCN/A
6=mmm! mxmy2z, 6mm! mxmy2z
S 1 2 3
1 N/A ð10�10ÞNð1�210ÞN
ð11�20ÞNð1�100ÞN
2 N/A ð01�10ÞNð2�1�10ÞN
3 N/A
768 Appendix D
Table D.25 Cubic to orthorhombic
All species
m�3m! mxmy2z, m�3m! 2x2y2zðP ¼ 0Þ m�3m! mxmymzðP ¼ 0Þ,432! 2x2y2zðP ¼ 0Þ�43m! 222ðP ¼ 0Þ Strains 1–3 only for following species:23! 222ðP ¼ 0Þ , m�3! mmmðP ¼ 0Þm�3! 222ðP ¼ 0Þ , m�3! mxmy2z
strains polarizations
1 "1 "2 "30 0 0
� �
0 0 P3ð Þ0 0 �P3ð Þ
2"3 "1 "20 0 0
� �
P3 0 0ð Þ�P3 0 0ð Þ
3"2 "3 "10 0 0
� �
0 P3 0ð Þ0 �P3 0ð Þ
4"2 "1 "30 0 0
� �
0 0 P3ð Þ0 0 �P3ð Þ
5"3 "2 "10 0 0
� �
�P3 0 0ð ÞP3 0 0ð Þ
6"1 "3 "20 0 0
� �
0 �P3 0ð Þ0 P3 0ð Þ
m�3m! mxmymzðP ¼ 0Þ, m�3m! mxmy2z, m�3m! 2x2y2zðP ¼ 0Þ, 432! 222ðP ¼ 0Þ,�43m! 222ðP ¼ 0Þ (all six strains), m�3! mxmy2z , 23! 222ðP ¼ 0Þ , m�3! mmmðP ¼ 0Þ ,m�3! 222ðP ¼ 0Þ (strains 1–3 only)
S 1 2 3 4 5 6
1 N/A R R ð1�10ÞNð110ÞN
ð10�1ÞN
ð101ÞCð01�1ÞN
ð011ÞC
2 N/A R ð10�1ÞCð101ÞN
ð01�1ÞNð011ÞN
ð1�10ÞN
ð110ÞC
3 N/A ð01�1ÞCð011ÞN
ð1�10ÞN
ð110ÞCð10�1ÞNð101ÞN
4 N/A R R
5 N/A R
6 N/A
Table D.26 Cubic to orthorhombic
All species
m�3m! mxym�xymzðP ¼ 0Þm�3m! mxym�xy2zm�3m! 2xy2�xy2zðP ¼ 0Þ upper symmetry operations only for432! 2xy2�xy2zðP ¼ 0Þ
strains polarizations
1 "1 "1 "30 0 "6
� �
0 0 P3ð Þ0 0 �P3ð Þ
2"1 "1 "30 0 �"6
� �
P3 0 0ð Þ�P3 0 0ð Þ
Appendix D 769
Table D.26 (continued)
All species
m�3m! mxym�xymzðP ¼ 0Þm�3m! mxym�xy2zm�3m! 2xy2�xy2zðP ¼ 0Þ upper symmetry operations only for432! 2xy2�xy2zðP ¼ 0Þ
strains polarizations
3 "1 "3 "10 "6 0
� �
0 P3 0ð Þ0 �P3 0ð Þ
4"3 "1 "1�"6 0 0
� �
0 0 P3ð Þ0 0 �P3ð Þ
5"3 "1 "1"6 0 0
� �
�P3 0 0ð ÞP3 0 0ð Þ
6"1 "3 "10 �"6 0
� �
0 �P3 0ð Þ0 P3 0ð Þ
S 1 2 3 4 5 6
1 N/A ð100ÞNð010ÞN
ð01�1ÞN
ð�a11ÞCð101ÞN
ð1�a�1ÞCð10�1ÞCð1�a1ÞN
ð011ÞN
ð�a1�1ÞC
2 N/A ð011ÞCða1�1ÞN
ð10�1ÞCð1a1ÞN
ð101ÞN
ð1a�1ÞCð01�1ÞCða11ÞN
3 N/A ð110ÞCð1�1�aÞN
ð1�10ÞN
ð11�aÞCð100ÞNð001ÞN
4 N/A ð010ÞNð001ÞN
ð1�10ÞCð11aÞN
5 N/A ð110ÞN
ð�11�aÞC
6 N/A
a ¼ "6"3 � "1
Table D.27 Cubic to orthorhombic
All species �43m! mxym�xy2z m�3m! 2xym�xymz
strains polarizations polarizations
1 "1 "1 "30 0 "6
� �
0 0 P3ð Þ P1 P1 0ð Þ�P1 �P1 0ð Þ
2"3 "1 "1"6 0 0
� �
P3 0 0ð Þ 0 P1 P1ð Þ0 �P1 �P1ð Þ
3"1 "3 "10 "6 0
� �
0 P3 0ð Þ P1 0 P1ð Þ�P1 0 �P1ð Þ
4"1 "1 "30 0 �"6
� �
0 0 �P3ð Þ �P1 P1 0ð ÞP1 �P1 0ð Þ
5"3 "1 "1�"6 0 0
� �
�P3 0 0ð Þ 0 P1 �P1ð Þ0 �P1 P1ð Þ
6"1 "3 "10 �"6 0
� �
0 �P3 0ð Þ P1 0 �P1ð Þ�P1 0 P1ð Þ
770 Appendix D
S 1 2 3 4 5 6
1 N/A ð10�1ÞCð1�a1ÞN
ð01�1ÞCð�a11ÞN
ð100ÞCð010ÞN
ð101ÞCð1�a�1ÞN
ð011ÞCð�a1�1ÞN
2 ð10�1ÞCð1�a1ÞN
N/A ð1�10ÞCð11�aÞN
ð101ÞCð1a�1ÞN
ð010ÞN
ð001ÞCð110ÞN
ð1�1aÞC
3 ð01�1ÞCð�a11ÞN
ð1�10ÞCð11�aÞN
N/A ð011ÞN
ða1�1ÞCð110ÞN
ð1�1�aÞCð100ÞN
ð001ÞC
4 ð100ÞNð010ÞN
ð101ÞCð1a�1ÞN
ð011ÞCða1�1ÞN
N/A ð10�1ÞCð1a1ÞN
ð01�1ÞN
ða11ÞC
5 ð101ÞCð1�a�1ÞN
ð010ÞNð001ÞN
ð110ÞCð1�1�aÞN
ð10�1ÞCð1a1ÞN
N/A ð1�10ÞCð11aÞN
6 ð011ÞCð�a1�1ÞN
ð110ÞCð1�1aÞN
ð100ÞNð001ÞN
ð01�1ÞCða11ÞN
ð1�10ÞCð11aÞN
N/A
a ¼ "6=ð"3 � "1Þ
Table D.27 (continued)
Table D.28 Tetragonal to tetragonal, trigonal to trigonal, hexagonal to trigonal, and hex-agonal to hexagonal. Structure of tensor components is the same in all species listed, but thecoordinate system is chosen for parent phase of each species
All species
4=mmm! 44=mmm! 4mm4=m! 4422! 4
�3m! 3�3m! 3m�3! 332! 3
6=mmm! 3zmx
6=mmm! 36=m! 3�6m2! 3zmx�6m2! 3 �6! 3622! 3
6=mmm! 6mm6=mmm! 66=m! 6622! 6
strains polarizations polarizations polarizations polarizations
1 "1 "1 "30 0 0
� �
0 0 P3ð Þ0 0 �P3ð Þ
Table D.29 Cubic to tetragonal
All species
m�3m! 4zmxmxy
m�3m! �4z2xmxyðP ¼ 0Þm�3m! �4z2xymxðP ¼ 0Þm�3m! 4z2x2xyðP ¼ 0Þm�3m! �4ðP ¼ 0Þm�3m! 4 m�3m! 4z=mzmxmyðP ¼ 0Þm�3m! 4z=mzðP ¼ 0Þ
432! 4�43m! �4z2xmxyðP ¼ 0Þ�43m! �4ðP ¼ 0Þ432! 4z2x2xyðP ¼ 0Þ
strains polarizations polarizations
1 "1 "1 "30 0 0
� �
0 0 P3ð Þ 0 0 P3ð Þ0 0 �P3ð Þ 0 0 �P3ð Þ
2"3 "1 "10 0 0
� �
P3 0 0ð Þ P3 0 0ð Þ�P3 0 0ð Þ �P3 0 0ð Þ
3"1 "3 "10 0 0
� �
0 P3 0ð Þ 0 P3 0ð Þ0 �P3 0ð Þ 0 �P3 0ð Þ
Appendix D 771
S 1 2 3
1 N/A ð10�1ÞCð101ÞN
ð011ÞN
ð01�1ÞC
2 N/A ð110ÞN
ð1�10ÞC
3 N/A
Table D.29 (continued)
772 Appendix D
TableD.30
Cubicto
trigonalspecies
Allspecies
m� 3m!
3xyzm
� xy
m� 3m!
� 3xyzm
� xyðP¼
0Þ
m� 3m!
3xyz2
� xyðP¼
0Þ
m� 3m!
� 3xyzðP¼
0Þ
m� 3m!
3xyz
432!
3xyz2
� xyðP¼
0Þ
432!
3xyz
� 43m!
3xyzm
� xy
� 43m!
3xyz
m� 3!
3xyzm
� 3!
� 3xyzðP¼
0Þ23!
3xyz
(upper
polarizationsonly)
strains
polarizations
polarizations
polarizations
polarizations
1" 1
" 1" 1
" 6" 6
" 6
��
P1
P1
P1
ðÞ
P1
P1
P1
ðÞ
P1
P1
P1
ðÞ
P1
P1
P1
ðÞ
�P1�P1�P1
ðÞ
�P1�P1�P1
ðÞ
�P1�P1�P1
ðÞ
2" 1
" 1" 1
" 6�" 6�" 6
��
�P1
P1
P1
ðÞ
�P1
P1
P1
ðÞ
P1�P1�P1
ðÞ
P1�P1�P1
ðÞ
P1�P1�P1
ðÞ
P1�P1�P1
ðÞ
�P1
P1
P1
ðÞ
3" 1
" 1" 1
�" 6�" 6
" 6
��
�P1�P1
P1
ðÞ
�P1�P1
P1
ðÞ
�P1�P1
P1
ðÞ
�P1�P1
P1
ðÞ
P1
P1�P1
ðÞ
P1
P1�P1
ðÞ
P1
P1�P1
ðÞ
4" 1
" 1" 1
�" 6
" 6�" 6
��
P1�P1
P1
ðÞ
P1�P1
P1
ðÞ
�P1
P1�P1
ðÞ
�P1
P1�P1
ðÞ
�P1
P1�P1
ðÞ
�P1
P1�P1
ðÞ
P1�P1
P1
ðÞ
Appendix D 773
TableD.30(continued)
m� 3m!
� 3xyzm
� xyðP¼
0Þ,m
� 3m!
3xyzm
� xy,m
� 3m!
3xyz2
� xyðP¼
0Þ,m
� 3m!
� 3xyzðP¼
0Þ,m
� 3m!
3xyz;432!
3xyz2
� xyðP¼
0Þ,
432!
3xyz(abovediagonal);� 43
m!
3xyzm
� xy,� 43m!
3xyz;m
� 3!
� 3xyzðP¼
0Þ,m
� 3!
3xyz,23!
3xyz(belowdiagonal)
S1
23
4
1N/A
ð100ÞC
ð011ÞN
ð001ÞN
ð110ÞC
ð010ÞC
ð101ÞN
2ð100ÞN
ð011ÞC
N/A
ð010ÞC
ð10� 1ÞN
ð001ÞN
ð1� 10ÞC
3ð001ÞN
ð110ÞC
ð010ÞN
ð10� 1ÞC
N/A
ð100ÞC
ð01� 1ÞN
4ð010ÞN
ð101ÞC
ð001ÞN
ð1� 10ÞC
ð100ÞN
ð01� 1ÞC
N/A
774 Appendix D
Appendix E
Piezoelectric Coefficients in Ferroelectric Phases
of BATiO3-Type Perovskites
Tables of this appendix give the d-piezoelectric coefficients in ferroelectric
phases of BaTiO3-type perovskites calculated as a result of linearization of
the electrostrictive equation. Rows stand for domain states (DS) represented
in Fig. 2.3.5. Piezoelectric coefficients are related to the cubic coordinate
system. The meaning of symbols a, b, c, d, and e in each phase is shown
below each table.
TableE.1
Tetragonalphase
(speciesm
� 3m–4mm)
DS
PS
d11
d12
d13
d15
d16
d21
d22
d23
d24
d26
d31
d32
d33
d34
d35
1100
cb
b0
00
00
0a
00
00
a
2� 100
–b
–c
–b
00
00
00
–a
00
00
–a
3010
00
00
ab
cb
00
00
0a
0
40
� 10
00
00
–a
–b
–c
–b
00
00
0�a
0
5001
00
0a
00
00
a0
bb
c0
0
600
� 10
00
–a
00
00
–a
0–b
–b
–c
00
a¼
2Q
44PSw 3
3
b¼
2Q
12PSw 1
1
c¼
2Q
11PSw 1
1
Componentsofthepermittivitycorrespondto
thematrix
ofpermittivityforDS-1
given
below
w 11
w 33
w 33
0 @
1 A
776 Appendix E
TableE.2
Orthorhombicphase
(speciesm
� 3m–mm2)
DS
PS
d11
d12
d13
d15
d16
d21
d22
d23
d24
d26
d31
d32
d33
d34
d35
1110
dc
b0
ec
db
0e
00
0a
a
2� 1
� 10
–d
–c
–b
0–e
–c
–d
–b
0–e
00
0–a
–a
31
� 10
dc
b0
–e
–c
–d
–b
0e
00
0–a
a
4� 110
–d
–c
–b
0e
cD
b0
–e
00
0a
–a
5011
00
0a
ab
Dc
e0
bc
de
0
60
� 1� 1
00
0–a
–a
–b
–d
–c
–e
0–b
–c
–d
–e
0
701
� 10
00
–a
ab
Dc
–e
0–b
–c
–d
e0
80
� 11
00
0a
–a
–b
–d
–c
e0
bc
d–e
0
9101
db
ce
00
00
aa
cb
d0
e
10
� 10
� 1–d
–b
–c
–e
00
00
–a
–a
–c
–b
–d
0–e
11
� 101
–d
–b
–c
e0
00
0a
–a
cb
d0
–e
12
10
� 1d
bc
–e
00
00
–a
a–c
–b
–d
0e
a=
ffiffiffi 2
pQ
44PSw 1
1
b=
ffiffiffi 2
pQ
12PS(w
33+
w 23)
c=
ffiffiffi 2
pPS(Q
12w 3
3+
Q11w 2
3)
d=
ffiffiffi 2
pPS(Q
11w 3
3+
Q12w 2
3)
e=
ffiffiffi 2
pQ
44PS(w
33+w 2
3)
Componentsofthepermittivitycorrespondto
thematrix
ofpermittivityforDS-5
given
below
w 11
w 33
w 33
w 33
w 33
0 @
1 A
Appendix E 777
TableE.3
Rhombohedralphase
(speciesm
� 3m–3m)
DS
PS
d11
d12
d13
d14
d15
d16
d21
d22
d23
d24
d25
d26
d31
d32
d33
d34
d35
d36
1111
ab
bd
cc
Ba
bc
dc
bb
ac
cd
2� 1
� 1� 1
–a
b–b
–d
–c
–c
–b
–a
–b
–c
–d
–c
–b
–b
–a
–c
–c
–d
311� 1
ab
b–d
–c
cB
ab
–c
–d
c–b
–b
–a
cc
–d
4� 1
� 11
–a
–b
–b
dc
–c
–b
–a
–b
cd
–c
bb
a–c
–c
d
51� 11
ab
b–d
c–c
–b
–a
–b
c–d
cb
ba
–c
c–d
6� 11
� 1–a
–b
–b
d–c
cB
ab
–c
d–c
–b
–b
–a
c–c
d
7� 111
–a
–b
–b
–d
cc
Ba
bc
–d
–c
bb
ac
–c
–d
81� 1
� 1a
bb
d–c
–c
–b
–a
–b
–c
dc
–b
–b
–a
–c
cd
a¼ð2=ffiffiffi 3
pÞP
SðQ
11w 1
1þ2Q
12w 1
2Þ
b¼ð2=ffiffiffi 3
pÞP
S½ðQ
12w 1
1þðQ
12þQ
11Þw
12�
c¼ð2=ffiffiffi 3
pÞP
SQ
44ðw
11þw 1
2Þ
d¼ð4=ffiffiffi 3
pÞP
SQ
44w 1
2
Componentsofthepermittivitycorrespondto
thematrix
ofpermittivityforDS-1
given
below
w 11
w 12
w 12
w 12
w 11
w 12
w 12
w 12
w 11
0 @
1 A
778 Appendix E
Appendix F
Tensors: Properties and Notations
Transformation Laws for Tensors
Transformation law for a second-rank tensor:
T 0ip ¼ aijapqTjq:
Transformation law for an nth-rank tensor:
M 0ij...ptn
¼ aii 0ajj 0 . . . app 0att 0
nMij...pt
n:
Here n under the lines indicate the number of a’s in the product or totalnumber of suffices. Summation over repeated (dummy) suffices from 1 to 3 isimplied (Einstein convention). The tensor in the original orthogonal referenceframe X1 X2 X3ð Þ is written without the prime. The tensor in the trans-formed orthogonal reference frame X01 X02 X03ð Þ is written with the prime.The table of the direction cosines for transformation of the reference frameX1 X2 X3ð Þ ) X1
0 X20 X3
0ð Þ is denoted as
aij ¼ cosðX 0i ^ XjÞ:
Transformation law for an nth-rank pseudotensor:
M0ij...ptn
¼ aii0ajj0 . . . app0att0
nMij...pt
ndetðaÞ
Voight Notations for Tensors
Stress vector: sn ¼ s11 s22 s33 s23 s13 s12ð ÞStrain vector: "n ¼ "11 "22 "33 2"23 2"13 2"12ð ÞHook’s law: sn ¼ snm"m and "n ¼ snmsm with the summation over dummy
suffices: n;m ¼ 1� 6.
Stiffness and compliance symmetric matrices:
cmn ¼
c1111 c1122 c1131 c1123 c1113 c1112
c2222 c2233 c2223 c2213 c2212
c3333 c3323 c3313 c3312
c2323 c2313 c2312
c1313 c1312
c1212
0
B
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
C
A
;
smn ¼
s1111 s1122 s1131 2s1123 2s1113 2s1112
s2222 s2233 2s2223 2s2213 2s2212
s3333 2s3323 2s3313 2s3312
4s2323 4s2313 4s2312
4s1313 4s1312
4s1212
0
B
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
C
A
:
Direct and converse piezoelectric effects: Pi ¼ dinsn and "n ¼ dinEi , with thesummation over dummy suffices: n ¼ 1� 6; i ¼ 1� 3:
Matrix of piezoelectric d-coefficients:
din ¼d111 d122 d133 2d123 2d113 2d112
d211 d222 d233 2d223 2d213 2d212
d311 d322 d333 2d323 2d313 2d312
0
B
@
1
C
A
:
Notation for Symmetry of Tensors
½V2� – symmetric second-rank tensor: qij ¼ qji.fV2g – antisymmetric second-rank tensor: qij ¼ �qji.V2 – second-rank tensor."½V2� – symmetric second-rank pseudotensor: qij ¼ qji."fV2g – antisymmetric second-rank pseudotensor: qij ¼ �qji."V2 – second-rank pseudotensor.½V3� – third-rank tensor symmetric with respect to all indices.V½V2� – third-rank tensor symmetric with respect to two indices: qijk ¼ qikj.fV2gV – third-rank tensor antisymmetric with respect to two indices:
qijk ¼ �qjik.V3 – third-rank tensor."V½V2� – third-rank pseudotensor symmetric with respect to two indices:
qijk ¼ qikj."fV2gV – third-rank pseudotensor antisymmetric with respect to two
indices: qijk ¼ �qjik.
780 Appendix F
½V4� – fourth-rank tensor symmetric with respect to all indices.V½V3� – fourth-rank tensor symmetric with respect to the last three indices.½½V2�2� – fourth-rank tensor symmetric with respect to two pairs of indices
under permutations: qijkl ¼ qjikl ¼ qijlk ¼ qklij.½ðV2Þ�2 – fourth-rank tensor symmetric with respect to the permutations of
the pairs of indices: qijkl ¼ qklij.fV2 V2g – fourth-rank tensor antisymmetric with respect to the permutations
of the pairs of indices: qijkl ¼ �qklij.½V2�V2 – fourth-rank tensor symmetric with respect to two indices:
qijkl ¼ qjikl.V4 – fourth-rank tensor.
Appendix F 781
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References 811
Index
A
A1/a2-pattern, 525f, 533, 536, 575, 579, 585,587–588, 592, 668, 670
A1/a2-variant, 524–525, 565–566, 570–572,575, 579, 584
Ab initio calculations, 317, 319, 604A/c- pattern, 532–533, 569, 572, 589,
591–592, 594–598, 668–669Activated regime, 392–403, 418, 425, 427,
455, 457Activation
energy, 372, 394, 453, 454f–455f, 464, 611,613, 636, 652, 679
field, 372, 397–403, 417, 430, 433, 439f,442, 611, 693
A-domains, 123, 130, 152, 154–155, 158–160,254, 259, 263–264, 389, 524,527–529, 532, 563–565, 571, 575,668, 694–697
AFM, 143, 203AgCl, 135–136, 186, 274, 288AgNbO3, 286Aizu strain, 25–28, 65, 68, 78a-alanine, 127, 442, 513AlPO4, 114t, 267Ambivalent pair, 53, 57Ammonium chloride, 112t, 350Anisotropy of the wall velocity, 369, 417Antiferrodistortive, 15Antiphase boundaries, 126t–127t, 129,
240–241, 261, 311, 313–314, 322A-SNOM, 201, 290Aurivillius phase, 544Average spontaneous strain, 223, 262
B
Ba2NaNb3O15, 285Backscattered electrons, 183–184
Backswitching, 333, 340, 345, 375–376, 383,414, 435, 476–477, 480, 614, 695,698, 703
(Ba,Pb)TiO3, 285Barium titanate, see BaTiO3
Barkhausen pulses, 169, 389, 434–435(Ba,Sr)TiO3, see BSTBaTiO3, 6, 13–14, 24, 39, 47, 59–60, 62, 66, 68,
70–71, 75, 93–97, 112t, 114t, 123–125,128–132, 136, 152, 155–156, 159–160,165, 167, 174, 178, 181–183, 193,195–196, 199, 201, 203–205, 211, 214,223–224, 227, 253, 255–260, 262, 275t,281, 283, 285, 300, 307, 309–310, 313,315–317, 319, 322, 336, 338, 341, 343,345, 349, 360–371, 378–380, 386–394,397–402, 410, 412, 417, 430–438,441–442, 456, 467, 484, 486, 488–492,497–498, 512–514, 518, 521–524,532–534, 545, 554, 556, 558–560, 575,580–582, 585, 614–615, 646, 649, 658,661, 665, 667, 693, 774–780
Beam coupling topography, 180–181Bending contribution, 498–502Bi4Ti3O12, 58, 116t, 126t, 169, 204, 522–523,
542–543BiVO4, 116t, 237, 313Bloch line, 312Bloch wall, 272, 300Branching, 218, 219f, 231Brazilian twins, 266Bright field image, 188–190BST, 524Burgers vector, 240, 250, 550–551, 638
C
C9H18NO, 116t, 385Ca2Sr(C2H5COO)6, 132
813
CCD camera, 163, 180, 187CCD microscope, 163, 169Cd2(NH4)2(SO4)3, 113t, 263Cd5(PO4)3Cl, 113t, 356C-domains, 130, 149, 152–154, 157, 159–160,
254, 258, 263–264, 488–489, 525,527–528, 530–535, 546, 563, 565,569–571, 575–577, 580–582, 589,591–597, 599, 602, 668, 670, 694–697
(CH3NH3)5Bi2Br11, 117t, 142, 376Charge injection, 624, 626f, 688Checkerboard pattern, 124, 260Circular domains, 233, 416–417Circulation line, 312Clamping approximation, 68, 164, 167, 169,
237, 243, 261Clapping angle, 66–68, 164, 167–169, 173,
221t, 249, 528Coercive field, 85, 89, 92, 157, 295, 333, 337,
339, 348, 356, 358, 406, 408–410,432–434, 438–440, 443, 447–449,451–452, 454–456, 466, 468, 476–477,480, 484, 488, 498, 504, 512, 522, 608,611, 614–620, 622–627, 630–631,643–644, 679, 683, 700
Cole-Cole plot, 503–507, 510Compatibility condition, 62–63, 68–69, 189,
208, 221–223, 238, 241, 243, 262, 264,267, 353, 382
Compensation by free charges, 255Composition, 2, 5, 11, 66, 72, 111, 236, 268,
391, 443–445, 530, 589, 591, 597, 617,620, 648, 651
Conditions of mechanical compatibility, 301,305, 311, 382, 524
Confocal scanning optical microscopy, 201Contact mode, 149, 151fCorrelation
coefficient, 220, 287f, 292, 308, 605energy, 292, 294, 303, 305, 307, 310–311length, 293–294, 296, 306, 658radius, 293, 656, 658, 662
Creep, 321, 354, 418–429, 507–508,510–511, 693
exponent, 425–426, 428–429Critical nucleus, 395–396, 398, 400–401, 451,
453, 456–457, 617Critical thickness, 214, 548, 550–551, 559,
573, 575, 585, 587–588, 590, 594,607, 669
Crystal family, 20, 50Crystallization temperature, 522, 546, 582,
590, 592–594, 651
CsCuCl3, 58, 113t, 350CsH2PO4, 118t, 196–197CsHSeO4, 76, 116tCSOM, see Confocal scanning optical
microscopyCsPbCl3, 112t, 205Curie-Weiss constants, 84, 102–105, 217,
604, 665–667, 706Curie-Weiss temperatures, 104, 296,
556–557, 604, 664–667, 705–707A/c- variant, 524–528, 564, 566, 569,
575–583, 669Czochralski-grown crystals, 260
D
Dark field imaging, 187–190, 285–286Dauphine twinning, 265, 350Dauphine twins, 46, 73, 123, 174, 177, 196,
266–267Dead layer, 605, 607, 660Debye relaxation, 487Debye screening length, 220, 309, 414Decoration technique, 8, 131–142, 228, 230,
234, 361, 370, 372Defect-stimulated nucleation, 452Degeneracy factor, 39, 46Degenerate domain states, 38Delamination, 678, 690Dense domain pattern, 210, 213, 217–218,
240, 263, 267, 489, 516, 560, 573,579–580, 584–585, 594, 601, 603, 671,674–677, 709f, 710
Depinning, 418–419, 422–424, 428, 500,504, 513
Depletion layer, 619–620, 643–644,662–664
Depolarization energy, 208, 211–212, 215,226, 228, 256, 326–327, 395–396, 422,425–426, 457, 482–483
Depolarizing field, 104, 199, 209, 211, 217,219–220, 226, 231, 257, 262, 284, 326,359, 367, 376, 412–414, 451, 456, 482,516, 559, 561, 599, 604, 629, 632,634–635, 655–656, 671
Dielectric permittivitybackground, 103, 104–105, 217
Diffusion of free carriers, 374Dinamical contrast, 281Dip technique, 123, 125, 128Dislocation, 25, 39
density, 547–550, 552DKDP, 246, 251, 252f, 280, 291, 446
814 Index
Domaincoalescence, 353, 363, 457–459, 464, 610contribution, 447, 464, 471, 479, 484–485,
491, 630, 654, 668–671, 710–711freezing, 447–448, 505, 511–514nucleation, 122, 183, 430, 441f, 449,
451–457, 611–613, 679, 689, 695, 702pair, 52–60, 62–64, 66–69, 74, 77t, 95, 99,
123, 125t, 129, 162, 164, 167, 171, 175,183, 197, 236, 254, 256, 260, 262–263,266, 278, 291, 311, 315, 370, 388–389,542–543
period, 210, 217, 219–220, 560–562,567–568, 572, 577, 579, 584, 586–587,596–604, 675, 707
wall thickness, 190, 196, 205, 218,272–273, 280f, 281, 284–287, 291, 293,314–315, 535, 573, 595–596
900 domain walls, 152, 191, 192f, 199,259–260, 273, 283f, 285f, 285–286,287f, 307, 309f, 386, 435, 515, 541,543f, 544, 693–697
Double hysteresis loops, 87, 334, 336,436–437
Dynamic transition, 470DyScO3, 667f
E
EBBA, 138, 140Effective dimension of the wall, 324Effective Hamiltonian, 317, 319Effective interface dimension, 505Effective mass, 496–498, 515–516, 518EFM, 143–147, 149, 622Elastic anisotropy, 303, 562, 574Elastic Green functions, 560, 575Elasto-optics, 173–174, 266Electrical conductivity, 128, 185, 256–257,
336, 361, 374Electrocaloric effect, 485–488Electrolyte electrodes, 362Electron
emission effect, 204holography, 275t, 283–285, 291mirror microscopy, 182, 193–194
Electro-optics, 173–175Emission
cold-field, 635Emission of the acoustic waves, 516EPR, 204–205Equilibrium residual misfit strain, 548Equitranslational transitions, 43
Etchant, 58, 123, 125, 127t, 128–129, 241fEtch hillocks, 122Etching rates, 122–123, 125, 127t, 129Etch pits, 122–123, 126t–127tEuler equation, 293, 297, 301, 314, 404, 605Ewald sphere, 187–188Extrapolation length, 658, 662Extrinsic contribution, 481, 483–490, 502,
504–506, 508, 513, 667–677
F
Ferrobielastic switching, 266, 350Ferrobielectric, 356Ferroelastic
domain pairs, 53–58, 62–63, 66, 69, 74,129, 162, 164, 171, 183, 197
domain states, 4f, 45, 47, 50–51, 53, 68,76, 94–95, 125, 162, 164, 205, 222,235–243, 262, 443, 527, 552–553
loops, 345, 348Ferroelastoelectrics, 29, 31, 35–36, 44, 47, 58,
90, 92, 99, 123, 195, 208, 265–267,349–350, 356
Ferroelectric breakdown, 481–483Ferroelectric/electrode interface, 104,
410–414, 452–453, 456, 458, 599, 628,639, 641, 643, 660, 680, 682, 686–687
Ferroic, 1, 3, 5–107, 109–121, 123, 129, 143,161, 164–166, 170–172, 193–195, 197,203–208, 220–225, 247, 253, 255, 262,266, 268, 271–274, 279, 285, 288,291–293, 296, 305, 310, 313, 315,319–322, 328, 349, 351, 353–358, 361,382, 391–392, 418–419, 430, 521, 548,560, 574, 704, 713, 728–770
Fictitious dislocations, 560, 573, 575Flexible defects, 419, 427–428Flexoelectric coupling, 304, 639Flexoelectric effect, 304, 318, 559,
637–638Forsbergh pattern, 260fForward growth, 353, 378, 430, 441,
445–446, 483, 677, 702Free-charge density, 211, 359Frequency dependence of the polarization
loop, 468Frequency modulation, 144Friction force, 143, 150–151, 159, 409, 515Frozen disorder, 419, 421Frozen polarization, 480–481Full ferroelectric, 39, 51Fused quartz, 532
Index 815
G
GaAs, 532Gadolinium molybdate, see GMOGASH, 19, 114t, 132–133, 135–136, 145,
150–153, 185–186, 234–235, 289,361, 372
GdDy(MoO4)3, 195, 282GdF3, 136, 288Gibbs energy, 82–84, 86, 89, 91, 93–94, 96,
307, 355, 359, 705GMO, 6–8, 30–31, 34–35, 74–75, 97–99,
116t, 126t, 136, 169, 193, 203, 214,240–241, 247, 253, 261, 273, 276t,278, 282–283, 288, 311, 314, 383–385,410, 414
Gold electrodes, 365Growth dimensions, 442, 447Growth pyramids, 225–226, 228,
234, 255Guanidinium uranyl sulphate trihydrate,
see GASHGyration tensor, 170–172
H
Hard-ferroelectric approximation, 217–219,359, 482, 600, 603–608, 629, 632
Head-to-head, 60, 71–72, 233, 255,258–259, 358–359, 413–414, 446, 639,686, 694
HF, 123–128, 136Hg2Br2, 115t, 313–314High resolution transmission electron
microscopy, 187, 191–193, 285HNO3, 118, 126–128H3PO4, 118t, 124, 125t–127tHRTEM, see High resolution transmission
electron microscopyHysteresis loop, 6–9, 19, 85, 87, 253,
331–340, 344–351, 368, 375, 387,431–439, 442–444, 447, 450, 455,466–471, 475–480, 613–653
I
Illumination, 131, 163, 169, 180, 187, 189,199, 360, 375, 435, 627, 652, 685
Imprint, 427, 627–628, 631, 634–636, 645,647–649, 652–653, 680, 685–686, 688,690, 700, 704
Improper ferroelectrics, 34, 38–39, 46–48, 60,81, 97–99, 210, 214, 217–218, 240, 263,296, 310, 414, 604
Injection, 624–627, 631, 635, 685–686,688–689
Injection model for imprint, 631–637Internal bias, 333, 336, 442, 627–629, 631,
637, 641–642, 645–649, 695, 698Intrinsic contribution, 483, 501, 654,
662–664, 672, 675Inverse exponential law, 368–369, 371–372,
376, 378–379IrO2, 541, 687, 698Irreducible representation, 44, 46, 48, 720
J
Jonscher universal relaxation law, 505
K
K2Cd2(SO4)3, 113t, 264K2NaAlF6, 356K2ZnCl4, 142KAlSi3O8, 268KCl, 123KD2PO4, 116t, 276t, 279, 391KDP, 30–31, 35, 68, 74, 91–92, 97, 99–100,
105, 116t, 137, 169, 203–204, 214, 224,235, 239, 245–253, 276t, 279–280, 315,384–385, 388–391, 403, 446–448, 462,488, 511–513, 705–707, 709–711
KFe(MoO4)2, 115t, 242, 244, 346, 348KH3(SeO3)2, 7, 117t, 348Kinematical contrast, 281KIn(WO4)2, 115t, 244Kittel-Mitsui-Furuichi, 210KNbO3, 75–76, 112t, 126t, 179, 193, 197,
224, 253, 261, 276, 285, 523, 542–543Kolmogorov-Avrami, 442, 447, 459,
462, 464, 466, 468, 492, 610, 613, 700KPFM, 143KSc(MoO4)2, 115t, 205KSCN, 115t, 237–240, 313–314KTaO3, 253, 524–525, 530–532, 589–592, 596K(Ta,Nb)O3, 257KTiOPO4, 117t, 127t, 134, 141, 174, 186,
197, 199, 505–506
L
(La AlO3)0.29 (SrAl0.5Ta0.5O3)0.71, 667LaAlO3, 152–153, 280, 291, 524, 530–531,
591, 593, 595Lagbeinites, 168Lamella domain patents, 249
816 Index
Lamellar twins, 1LaNbO4, 116t, 345Landauer theory, 451–452, 613, 617Landau theory, 11, 34, 44, 61, 81, 100–101,
105, 107, 310, 404, 440, 456, 476–477,496, 498, 514f, 544, 554, 560, 580, 605,655, 669, 705
LaP5O14, 117t, 235–236LAS, 113t, 135–136Laser tomography, 203Lateral piezoresponse, 159–160Lead germanate, 58, 72–73, 114t, 126, 131,
172, 214, 233, 275, 291, 328, 361, 365,374–375, 382–384
Left coset, 41–43, 45Lens-shaped domains, 218, 238–240, 373Lenticular domains, 226–227, 381, 417LFM, 143, 150–151Li2Ge7O15, 105, 107, 113t, 139LiCl, 362, 380, 436LiCsO4, 345–346LiH3(SeO3)2, 118t, 141, 492LiNbO3, 8–9, 72, 114t, 126t, 128, 131, 152,
155, 174, 179, 185–186, 201, 203–204,443–446, 483
Linear dichroism, 172Linear wall, 298, 311, 503LiNH4SO4, 113t, 282Liquid crystal, 131–132, 137–142, 225, 228,
361, 372–373Liquid electrode, 177, 338, 362, 365–369,
379, 381, 436LiTaO3, 70, 72, 114t, 126, 186, 201–202, 204,
275t, 285, 290f, 443–446Lithium niobate, see LiNbO3
Logarithmic charge relaxation, 634, 636
M
MASD, 113t, 262Matthews-Blakeslee criterion, 548Maximum polar subgroups, 33, 48MBBA, 138, 140–141Mean strain approach, 560, 562, 573–578,
581–582, 586–588, 598Mechanical twinning, 2Mercury, 115t, 128, 226, 371Mg3B7O13Cl, 75, 113tMgO, 125t, 445f, 524, 527–534, 542, 582,
590–593, 595Micro-Raman spectroscopy, 205Microstresses, 561, 566–570, 575–576, 595Miller-Weinreich theory, 394–397, 417, 693
Minor species, 45–46Misfit dislocations, 522, 544–545, 547–548,
552, 559, 582, 590–591, 595–596,637–638, 641
Misfit strain, 544–548, 550–554, 556–559,563–567, 574, 580–582, 585, 587–596,638–639, 665–667, 670, 696, 705–707
Mixed boundary conditions, 657Mobility law, 366, 383–385, 390, 405–406,
408–410, 412, 415, 427–428,502–503, 507
Monte-Carlo simulations, 319Morphic birefringence, 28, 166Morphic components, 165, 356Morphotropic phase boundaries, 65,
451, 455Multiaxial ferroelectrics, 51, 59, 208, 253,
260, 307
N
NaNbO3, 75, 78–80, 253, 261NaNO2, 14–15, 117t, 127t, 129, 134, 138,
141–142, 178, 185, 196, 199, 274t,281–282, 288, 306
Natural spontaneous strains, 20–28, 50, 65,90, 165, 222, 237, 545, 585
NdP5O14, 74, 75, 117t, 131, 197, 203–204,235–236
Near-field scanning optical microscopy, 201Needle-like domains, 389–391, 445Neel type structure, 313Nematic liquid crystal, see NLC(NH2CH2COOH)3�H2SO4, see TGS(NH4)2BeF4, 117t, 137(NH4)2SO4, 113t, 127t(NH4)3H(SO4)2, 205NH4Cl, 74, 112t, 173NH4LiSO4, 135NLC, 137–142, 373–374NMR, 205, 237Non-activated regime, 392, 394,
402–410, 425Nondestructive visualization, 157Nonferroics, 16Nonlinear dielectric constant, 154Non-linear dielectric response, 153–155,
359, 683Nucleation
barrier, 454, 455f, 457, 613problem, 396, 451, 456rate, 395, 397–401, 404, 442, 451–452,
455, 461, 464, 466, 468–469, 494time, 461
Index 817
Nucleation-limited-switching model,610–613
Nuclei of reverse domains, 353, 449
O
Oernstein-Zernicke function, 294Optical activity, 57–58, 119t, 161, 170–172,
233, 361, 375Optical impermittivity, 163, 166Optical indicatrix, 162–163, 174, 235Order-disorder, 13, 14, 105, 237–238, 268Order-parameter vortex, 312Orientational domain states, 39–40, 42, 46,
48, 237, 249Oxide electrode, 615–616, 627, 651–652,
661–662, 687Oxygen vacancies, 437, 618, 652,
686–687Oxygen vacancy, 322, 437, 652, 686–687
P
Parent misfit strain, 545, 547, 554, 556–559,580–582, 585, 588–591, 594–595, 665,670, 705–707
Partial ferroelastic, 39, 51, 57–58, 95, 253Partial ferroelectric, 39, 47Partial switching, 121, 344, 369, 371, 441,
470, 473, 477–481Passive layer, 247, 484, 599–602, 604, 617,
620–626, 628–636, 639, 664,671–677, 682
Pb3(PO4)2, 6, 114t, 222, 242–244, 277,313, 348
Pb3(PxV1-xO4)2, 348Pb3(VO4)2, 114t, 244–245Pb5Ge3O11, see Lead germanatePbCl2, 136Pb(Fe1/2Nb1/2)O3, 505–506(Pb,La)TiO3, see PLTPbTiO3, 66, 94, 96, 112t, 125t, 152–153,
159–160, 189–190, 223, 257–259,285–287, 523–525, 527–532, 551,558–559, 572–573, 590–597, 602,669, 703
PbZrO3, 15, 46, 78–80, 112t, 125t,253, 261
Pb(Zr,Ti)O3, see PZTPeierls barrier, 317–318, 394–395, 398,
402–403, 516, 518Peierls potential, 321, 323, 392, 425, 490–493,
496–498, 502Penetration twins, 1–2
Permissible wall, 62–64, 66–68, 74, 80, 243,250, 254, 262–264, 315
Permittivity, 7, 9, 28, 47, 61, 103–105, 134,165–166, 170, 210, 212, 217, 263,295–296, 326, 334, 387, 406, 417, 455,482, 484–497, 501–514, 653–659,662–675, 705, 709–771
PFM, 143, 150, 158–161, 328, 691–698,700–704
Phasecoexistence, 87–88distorted, 485field calculations, 560, 582, 584–585high symmetry, 13, 17, 268, 529high temperature, 13, 110, 139, 266, 546,
552, 644incommensurate, 10, 109–110, 120,
265, 267low symmetry, 13, 18, 20, 22, 25, 38, 40,
92, 240, 269, 544, 584low temperature, 13, 58, 110, 135, 163,
205, 247, 253, 260, 270, 296, 304, 345,513, 522, 546
matching, 177–178transformation, 3–4, 6, 15, 20, 52, 88, 110,
165, 268, 442, 457transition in the wall structure, 312
Piezoelectric clamping, 485, 488–490Piezoelectric force microscopy, see PFMPiezoelectric hysteresis loops, 475–481, 698Pinning, 157, 231, 309, 321, 354, 371,
418–429, 500–502, 507–509, 597,617–618, 669, 680–689, 707
Pinning pressure, 418–428Pinning of wall by charged defects, 309PLT, 524, 531–532PLZT, 336–337, 647Polarization fatigue, 677–690, 704Polarization reversal, 6–9, 58, 71, 141–142,
153, 183, 262, 333, 340, 349, 363–364,382, 404–406, 443, 449–450, 468,481–483, 610, 626, 677, 680–682,690–704
Polarizing microscope, 75, 137, 170, 172,233, 243–244, 255–256, 360–361, 369,375, 389
Poling back technique, 341–342, 608–609Prefractals, 215, 248Preisach model, 472, 474–475Pressure acting on a wall, 357–358Primary ferroics, 28–29Proper ferroelastics, 34, 235, 364, 704–706,
709–710
818 Index
Proper ferroelectrics, 34, 39, 42, 47–48,50, 60, 98–100, 105, 107, 170,210–211, 217–218, 414, 604,704–706, 710
Protonic conductivity, 338Prototypic, 13, 557Pseudo-proper ferroelectrics, 217Pt, 112t, 126t–127t, 171, 204, 308–309, 374,
541, 615–616, 626–627, 648, 650,687–688, 698–699, 701, 704, 778
PUND test, 645Pyroelectric coefficient, 17–21, 30, 45, 48, 84,
175, 197, 199, 331, 486, 720Pyroelectric probe technique, 199, 361, 371PZT, 146, 152–153, 156–158, 201, 342,
358, 451, 453, 455, 465, 529–531,539–542, 593–594, 596–598, 611–612,614–617, 620, 627, 648–652, 678–679,684–685, 687–689, 697–698, 700–701,703–704
Q
Quartz, 2, 46, 58, 73, 114t, 123, 172, 174, 177,195–196, 265–267, 349–350
R
Random bond, 321–322, 325, 419, 499, 502Random field, 321–322, 499, 508, 642–643Rayleigh constant, 685Rayleigh loops, 470–475RbCaF3, 205RB defect, 321–322, 325, 329, 427, 429RbH2PO4, 116t, 391R-cases, 64Real and imaginary parts of the
susceptibility, 485Relative coherency strain, 546, 563, 565, 567,
569, 571, 576–577, 581–582Remeika method, 193, 255–256, 365Reorientable defects, 642, 648Residual misfit strain, 548, 590, 592–593,
595–596Retardation effects, 376–377RF defect, 321–322, 324–326, 419–420,
427–428, 437, 443Rochelle Salt, 31, 118t–119t, 133, 161,
168, 213–214, 224, 241–242,332, 335–336, 348, 360, 363,382–384, 511
Rotational wall, 300, 311Roughening, 319–329, 498
Roughness exponent, 321, 323–325,327–328, 425, 509–510
Rub technique, 125RuO2, 660, 687
S
SAED, see Selected area electron diffractionSample surface, 5, 121, 124, 137, 141–143,
147, 149, 154, 156, 158, 162,182–183, 185, 199–201, 203, 205, 210,218, 254, 258, 284, 288–289, 315, 344,360, 481
Sawyer-Tower circuit, 332–333, 336, 432SBN, 115t, 157SbSI, 71, 126, 134Scanning acoustic microscopy, 203Scanning electron acoustic microscopy, 203Scanning force microscopy, see SFMScanning nonlinear dielectric microscopy,
143, 153, 524, 533Screening, 102, 140, 143, 147, 219–220, 231,
309, 359, 412–414, 554, 599, 603, 624,626, 633, 636, 640–641, 655, 659–662,685–686, 690
Secondary electrons, 183–186Secondary ferroics, 29, 31Secondary order parameter, 38Second optical harmonics, see SHSeed inhibition, 680–683, 685–687Selected area electron diffraction,
187–191, 539Selective crystallization, 134SFM, 142–144, 146, 150–153, 160–161, 231,
236, 288–289, 481–483, 524, 526–527,530, 533, 541, 544, 596, 690–692, 694,696–698
SH, 176–179Short-circuited capacitor, 209, 554, 599,
602–605Size effects, 422, 569, 571, 613–627Slater model, 315–317, 319Sliding regime, 508, 510Sn2P2S6, 118, 142, 182SNDM, 143, 150, 153–155, 524SNOM, see Near-field scanning optical
microscopySoft domains, 606Spectrum of waiting times, 455,
611–613Spinodal decomposition, 247, 268Spinodals, 449Spontaneous birefringence, 28, 165–166,
168, 170, 174
Index 819
Spontaneous polarization, 7–8, 16–18, 21,33, 35, 38, 47–48, 50–51, 59, 83–84,89–90, 94, 99–100, 102, 104–107, 119t,143, 146–147, 154–157, 184, 189–190,210–211, 284, 295, 311, 318, 322,326–327, 340–341, 351–352, 386, 388,402, 436–437, 449, 481, 515–517, 536,538, 559, 584–585, 590–593, 600, 641,668, 670, 673, 707, 727
Spontaneous strain, 8, 20–28, 47, 50–51,62–63, 68, 76, 78, 90–91, 130, 165–166,168, 221–223, 236–237, 240, 243, 257,262, 270, 280, 311, 322, 343, 364, 536,545, 585–586
Square-root thickness dependence, 562Square-shaped domain, 364–365, 380Sr0, 61Ba0, 39Nb2O6, see SBNSrBi2Ta2O9, 51, 116t, 522–523, 544, 608–609,
614–616, 698–699SrRuO3, 539, 616, 661SrTiO3, 112t, 253, 261, 303, 313, 505, 522,
524, 526, 529–531, 533–535, 542–543,557, 591, 593, 595–597, 603, 658,666–667, 694
S-SNOM, 201–202, 290SSPM, 143, 147–149Stochastic regime, 508Stoichiometric, 443–445Strain gradient, 268, 304, 559, 637–639, 648Stress-free walls, 62–63Stress release, 522, 538, 544–553, 564–565,
575, 590–591, 594–595, 637–638Stroboscopic illumination, 163, 169,
360, 375Strong pinning, 419–423, 426–428,
500–501, 509Suborientational domains, 68–69Substrate, 129, 131–132, 135–137,
152–153, 315, 484, 521–522, 524–535,538, 542, 544–576, 579–593, 603,637–638, 649, 653, 664–669, 692–696,704–707, 710
Super-ionic conduction, 505Surface boundary conditions, 373Surface layer, 1, 154, 184, 226, 228, 232, 256,
260, 275, 338, 365, 367, 370, 373, 376,411–413, 434, 625, 640–641, 662,688–689
Surface-stimulated domain nucleation,613
S walls, 63, 65, 68, 75–80, 237, 243–245, 260,311, 353
Swiss cheese technique, 181–182
Switching current, 340–341, 343, 349,361–363, 365, 372–373, 378–379, 389,434, 443–447, 465–466, 492–496,646–647
Switching regime, 429, 441, 504,510–511, 646
Switching time, 8, 341–343, 348, 373–374,379, 430–431, 441, 447, 608
Symmetry breaking strain, 23, 90
T
T*-effect, 343–344Tail-to-tail, 72, 255, 686, 694Tanane, 30, 116t, 385–386TANDEL effect, 339–340Tanh-type solution, 293, 298–299, 302Temperature
autostabilization, 339gradient, 72, 199, 223, 233, 247, 251–252,
255, 257–258hysteresis, 88
Tertiary ferroics, 29, 356Tetragonality
ratio, 546, 563strain, 546, 586, 596, 668
TGFB, 118t–119t, 137, 233, 438–442, 462–463TGS, 6, 70–73, 88–90, 118t, 125–128,
131–142, 145, 150–153, 177–178,183–186, 197, 199–200, 202–204,224–233, 253, 256, 274, 281–282,288–290, 295, 305–306, 335, 339–340,343, 349, 355, 358, 361, 364, 370–374,378, 381, 410, 415, 417, 438–443,446–448, 467, 470, 488, 492, 495,505–506, 512–513
TGSe, 118t–119t, 227, 233, 306, 381, 415, 417Thermal expansion, 11, 27, 90, 532, 552, 554,
559, 586, 590, 666Thermal fluctuations, 236, 272, 292, 294,
322, 325, 392, 394, 404, 418–419, 424,427, 457, 490, 497
Thermally activated regime, 13, 391, 394,418, 422–423, 425, 427–428, 455–457,461, 490, 636, 693
Thermodynamic coercive field, 85, 89,92, 295, 356, 406, 408–410, 434,440, 448–449, 451–452, 466,476–477, 611
Thiourea, 117t–118t, 133, 137, 282Thomas-Fermi screening length, 655,
659–660, 662Threshold conduction, 621, 624–627, 679
820 Index
Through domains, 353, 441, 449, 460, 462,467, 483, 671, 702
Time of thermal relaxation, 486Topological defects, 312, 320, 418, 421Top-seeded solution growth, 255Transformation twins, 3, 75Transient current method, 341–342, 608Transition
displacive, 13distortive, 11–13, 43ferrodistortive, 14–15reconstructive, 11–12, 41
Translational domain states, 39–41,44, 69
Transmission electron microscopy,121, 182, 186–193, 268, 283,285–287, 524
Tricritical point, 82, 294, 296Triferroelastic, 119t, 356Triglycine sulphate, see TGSTriple hysteresis loops, 437Turmaline, 172Tweed patterns, 267–270Twin boundary, 2, 5Twin components, 1–3Twinning complex, 52–53, 57, 64Twinning dislocations, 250, 391Twinning operation, 1, 52–53, 57–58, 64, 66,
164, 166, 168, 170, 173, 255Two – beam diffraction conditions,
189–190, 285Two-dimensional nucleation,
398–400, 442
U
Uniaxial ferroelectrics, 51, 225–235, 374,443, 558
UV light illumination, 627, 652
V
Velocity anisotropy, 373, 379Vertical piezoresponse, 158–160
Vicinal substrate, 529Virtual ground, 333Voltage offset, 442–444, 627–632, 634–635,
637, 639–653, 686
W
Waiting time for nucleation, 464Wall mobility, 85, 228, 366, 383,
393–394, 397–408, 411, 415, 502–503,675, 692
900 wall motion, 360Wandering exponent, see Roughness
exponentWeak beam dark field image, 187–190,
285–286Weak ferroelectricity, 100–107Weak pinning, 421–426, 428–429, 507–509Wedge-shaped domains, 74–75, 131, 236,
254, 382, 386, 388–391Wedge-type domains, 430Wf walls, 68, 74–78, 80, 237, 240, 243–244,
260–262WO3, 115t, 183
X
X-raydiffraction, 23–24, 88, 175, 194, 205, 237,
247, 249, 266, 268, 277, 280diffractometry, 280scattering, 276, 278, 280, 535topography, 70, 121, 194–196, 236,
255, 266, 274–275, 277, 281–282, 291
Y
YBa2Cu3O7–x, 172, 270, 280, 286
Z
Zig-zag domain wall pattern, 256Zig-zag shape, 374
Index 821