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)


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



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



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



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



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.


3q"

4qP


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


1 Speciesno.


3q"

4qP


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


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


1 Speciesno.


3q"

4qP


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


1 Speciesno.


3q"

4qP


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


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


38 �43m! mxym�xy2z 6 6 WNf �WN

f , WCf � SN

29 m�3m! 2xy 12 12 R, WNf �WN


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


1 Speciesno.


3q"

4qP


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


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


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!


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


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


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!


� 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


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


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


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)


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


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


� 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)


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.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


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



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


4 ð01�10ÞCð0001ÞN

ð1�210ÞCð10�1�aÞN

ð1�100ÞCð11�2�aÞN

N/A ð10�10ÞNð1�21aÞC


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


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


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


ð"1�" 2Þ2þ"2 6

q

�=½"

6þ

ffiffiffi 3

pð"

1�" 2Þ�

m�¼

2½2" 6�

ffiffiffi 3


ð"1�" 2Þ2þ"2 6

q

�=½�" 6þ

ffiffiffi 3

pð"

1�" 2Þ�,

n�¼

2½�" 6�

ffiffiffi 3

pð"

1�" 2Þ�

ffiffiffi 3


ð"1�" 2Þ2þ"2 6

q

�=½�" 6þ

ffiffiffi 3

pð"

1�" 2Þ�

p�¼

2½�" 6�

ffiffiffi 3

pð"

1�" 2Þ�

ffiffiffi 3


ð"1�" 2Þ2þ"2 6

q

�=" 6,

q�¼

2½�" 6þ

ffiffiffi 3

pð"

1�" 2Þ�

ffiffiffi 3


ð"1�" 2Þ2þ"2 6

q

�=" 6


760 Appendix D

Table D.18 Hexagonal to monoclinic

Species �6m2! 2y


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


ð2�1�1�bÞCð01�10ÞN

3 N/A ð1�10aÞN

ð11�20ÞCð2�1�1bÞCð01�10ÞN


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


766 Appendix D

Table D.21 Orthorhombic to orthorhombic

Species mmm! mxmy2z


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)


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


1 "1 "1 "30 0 "6

� �

0 0 P3ð Þ

2"1 "1 "30 0 �"6

� �

0 0 �P3ð Þ

S 1 2


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)


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


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


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


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


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


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


ð01�1ÞN

ð�a11ÞCð101ÞN

ð1�a�1ÞCð10�1ÞCð1�a1ÞN

ð011ÞN

ð�a1�1ÞC

2 N/A ð011ÞCða1�1ÞN

ð10�1ÞCð1a1ÞN

ð101ÞN

ð1a�1ÞCð01�1ÞCða11ÞN


ð1�10ÞN

ð11�aÞCð100ÞNð001ÞN


ð1�10ÞCð11aÞN

5 N/A ð110ÞN

ð�11�aÞC

6 N/A

a ¼ "6"3 � "1


All species �43m! mxym�xy2z m�3m! 2xym�xymz


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.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


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


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


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)


thematrix


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


thematrix


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

appendix a 32 point groups: notations, symmetry elements ...978-1-4419-1417-0/1.pdf · appendix a...

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