Characterization of Elasticity-Tensor Symmetries Using SU(2)

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Transcript of Characterization of Elasticity-Tensor Symmetries Using SU(2)

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0 e∓ιπ/3

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Z6V9�Ó5$ZWl&V1V�#IZ\f35mp658��5mY%Z6V�46psh � f35%0uα = sin(θα/2)e1 − cos(θα/2)e2

S�lnf35%465θα ∈ {0,±π/2, π}

h � f35 YmV�4\465T5p�(¤V�032co0�� p6X�����46V�X�( V�#�46V�Z[;�Z\ceV�03pÔcep

G̃c = {I3, R±π/2,e3 , Rπ,e3 ,−Ruα , α ∈{1, 2, 3, 4}} SÓlnf,coge5�Z\f35ºYmV�4\465mp-(ÓV�032co0��æp\X�����46V�X�(dV�#

SU(2)cep

Hc =

{±I2,±

(e±ιπ/4 0

0 e∓ιπ/4

),±(ι 00 −ι

),±(

0 ιι 0

),

±(

0 1−1 0

),±(

0 ιe±ιπ/4

−e∓ιπ/4 0

)} h<v�>��R

ª coZ\f�465mp�(¤5mY%ZÈZ6V¦Z\f35ºV�4\Z\f3V�03V�4� ;�g¹� ;�p\cep {e1, e2, e3}V�#R3 SZ\f,cep15%gk;�p\Z\ceY%coZ$�

Z65%03p6V�4nf ;�p(p\c��æco032,5�(¤5%032,5%0CZÔYmV) "(ÓV�035%0AZ6p"h � f35mp[5ºYmV) "(ÓV�035%0AZ6p1;�465c1111 = c2222, c3333,c1122, c1133 = c2233,c1212, c1313 = c2323.

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Gg =

±

cos θ − sin θ 0sin θ cos θ 00 0 1

cos θ sin θ 0sin θ − cos θ 00 0 1

, θ ∈ (−π, π]

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� f35B�346p\Zn5%ge5� �5%0CZ6pÈV�#�Z\f351p��+ " �5%Z\4�����46V�X�(=;�465 ±Rθ,e3 SI0 ;� �5%g���Z\f35Ô46V�Z[;�5Z\ceV�03pA�!� ;�0���ge5

θ;��ÓV�X,ZC��5mY%Z6V�4

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h� f35ÔYmV�4\465mp�(¤V�032co0��¦p\X�����46V�X�(�V�#�46V�Z[;�Z\ceV�03p9cep

G̃c = {Rθ,e3 ,−Ru(θ), θ ∈(−π, π]} S,lnf,coge5�Z\f35ºYmV�4\465mp�(¤V�032co0�� p\X�����46V�X�( V�#

SU(2)cep

Hc =

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0 e−ιθ/2

),±(

0 eιθ/2

−e−ιθ/2 0

), θ ∈ (−π, π]

}.

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c1111 = c2222, c3333,c1122, c1133 = c2233,c1212 = 1

2(c1111 − c1122), c1313 = c2323.

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cep

Hc =

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),±(

0 1−1 0

),±(ι 00 −ι

),

±(e±ιπ/4 0

0 e∓ιπ/4

),±(

0 e±ιπ/4

−e∓ιπ/4 0

),

±(

cos π/4 ±ι sinπ/4±ι sinπ/4 cos π/4

),±(

cos π/4 ∓ sinπ/4± sinπ/4 cos π/4

),

±( ∓ sinπ/4 ι cos π/4ι cos π/4 ∓ sinπ/4

),±( ±ι sinπ/4 cosπ/4− cos π/4 ∓ι sinπ/4

),

±(ι cos π/4 ∓ sinπ/4± sinπ/4 −ι cos π/4

),±( ±ι sinπ/4 ι cos π/4ι cos π/4 ∓ι sinπ/4

),

±( ± sinπ/4 cos π/4− cos π/4 ± sinπ/4

),±(ι cos π/4 ∓ι sinπ/4∓ι sinπ/4 −ι cos π/4

)}.

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e1, e2, e3;�465È2,5%Z65%4� ¦co035m2��'�ÕZ\f35nX,0,coZ[;�4��"(,gk;�035mp&V�#¤465TÌ 5mY%Z\ceV�03psS�Z\f35È5%gk;�p\Z\ceY%coZ,�¦Z65%03p6V�4f ;�pnZ\f,465m5ºco032,5�(Ó5%032,5%0AZ1YmV) "(¤V�035%0CZ6p"h � f35mp65�YmV) "(ÓV�035%0AZ6pÔ;�465

c1111 = c2222 = c3333,c1122 = c1133 = c2233,c1212 = c1313 = c2323.

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Gc = O(3)h � f35 YmV�4\465mp�(¤V�032co0�����46V�X�( V�#�46V�Z[;�Z\ceV�03p cep

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Hc = SU(2)h

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c1111 = c2222 = c3333 = 2c1212 + c1122,c1122 = c1133 = c2233,c1212 = c1313 = c2323.

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r2

r3

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G̃c = {A ∈ SO(3), P (AX) = P (X), PV (AX) = PV(X), PD(AX) = P (X),∀X ∈ R3}.ª 5 l�;�0AZiZ6V�(,X,gog.� ;�Y6q Z\f35mp65 Z\f,465m57(ÓV�g��I03V) ¦ck;�gepiZ6V

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§vòCîJ�Pñi�hïΠ[ö�¥�¦�ÿ�¤

φ :

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Re((z2)2 − (z1)2

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r2

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ϕ : (z1, z2) 7→ (ξ1(z1, z2) = (z1)2−(z2)2, ξ2(z1, z2) = −ι((z1)2+(z2)2), ξ3(z1, z2) = 2z1z2)<£3Î��R� f357(,X,gog� ;�Y6qp¦V�#1Z\f35æZ\f,465m5æf3V) �V)��5%035mV�X3pJ(¤V�g��A03V) ¦ck;�gepsS

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P (z1, z2) = P ((z1)2 − (z2)2,−ι((z1)2 + (z2)2), 2z1z2),

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z1 ;�032z2 S

PV(z1, z2) = PV((z1)2 − (z2)2,−ι((z1)2 + (z2)2), 2z1z2)

;�032

PD(z1, z2) = PD((z1)2 − (z2)2,−ι((z1)2 + (z2)2), 2z1z2),

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Hc = {U ∈ SU(2), P (UZ) = P (Z), PV(UZ) = PV(Z), PD(UZ) = P (Z),∀Z ∈ C2}.

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P (8)(t) = P (z1/z2, 1),

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t = z1/z2 ©

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lnf35%465q

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P (8)(t) = a8t8 +a7t

7 +a6t6 +a5t

5 +a4t4−a5t

3 +a6t2−a7t+a8,

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a6 = 4[−c1111 + c2222 − 4c2323 + 4c1313 − 2c2233 + 2c1133 + 2ι(c1112 + c1222 − 2c1233 − 4c1323)],

a5 = 8[−3c1113 − c1322 + 4c1333 − 2c1223 + ι(c1123 + 3c2223 − 4c2333 + 2c1213)],

a4 = 2(3c1111 + 3c2222 + 8c3333 − 16c2323 − 16c1313 + 4c1212 + 2c1122 − 8c1133 − 8c2233).<£3�;�RÅ'V�g��A03V) ¦ck;�g

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2 − aV3 t+ aV4 .<£3κ�R

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aV3 = 4[c1113 + c1223 + c1333 − ι(c1213 + c2223 + c2333)],

aV2 = 2(−c1111 − c2222 + 2c3333 − 2c1212 + c1313 + c2323).

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aD3 = 4[c1113 + c1322 + c1333 − ι(c1123 + c2223 + c2333)],

aD2 = 2(−c1111 − c2222 + 2c3333 − 2c1122 + c1133 + c2233).

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a7 = a5 = aV3 = aD3 = 0.� f35���;�0,cep\f,co0���V�#nZ\f35mp65�#OV�X,4iYmV) "(,ge5T��YmVA5T� Y%ce5%0AZ6pÕcepÆ5RÍAX,c���;�ge5%0AZ � X3p\co0��YmV�032coZ\ceV�03p <£3�;�R�SÈ<£3Î��R¦;�032+<£�2�sR � Z6V ;�p��p\Z65� V�# 5%c���fAZ 5RÍAX ;�Z\ceV�03p¦lncoZ\f5%c���fAZÈX,0,qA03Vsln03p"h � f35�p[V�goX,Z\ceV�0dV�#'Z\f,cep1p��p\Z65� cepc1113 = c1223 = c1333 = c1213 = c2223 = c2333 = c1322 = c1123 = 0.

<£�Î��R� fAX3p"S�;�0 5%gk;�p\Z\ceY%coZ$� Z65%03p6V�4 f ;�p�;� �V�03VAY%goco0,ceY�p��+ " �5%Z\4��±c�#¡;�032 V�0,g��±c�#coZ¡f ;�p � lncoZ\f 465mp�(¤5mY%ZÆZ6V ;�0ÒV�4\Z\f3V�03V�4� ;�g� ;�p\cep � Z\f,co4\Z65m5%0 co032,5�(Ó5%032,5%0AZYmV) "(¤V�035%0CZ6pB��c���5%0��'��5T�2(,465mp6p\ceV�0�<£º�R�h

K�L-M�NTO�QTSTU>NyV>QTWYX[ZRQPS�\@]P_R`Ta�bR`T]Ta�aTc�\"d-e+f,d�g2\BV�X,dvc

¬ x nv}[r[{ q { r�­®rt¯°m�}[l�u�nvm�± SU(2)�>�

� /�����ûB�*bü�h 3h�ýE¦&¥��������¡ ¡¢R t�¡¤7�$¥T¦���§Î¨Cõ+���^§)¨��tõ+§Î�¡¨�§Pñi £¢^�P¤)î?î�¥��¡¨T¤J  ò=�Φió§)¦��÷¤�  òJ�tõ2¥7�tõ!¨�¥-¥@ñ2§)�³¤)¦i§)î? ¡�Î�<�

P (8) ú P (4)V

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P (2q)(t) = εqP (2q)(t/ε), q ∈ {2, 4},

P (2q)(t) = εqt2qP (2q)(−ε/t), q ∈ {2, 4},<£�Î��R

ô8õ2¥�¨�¥ε2 = 1

ú�)¦ióP (4) ¨-¥tò>¥T¨y�@�°§7ÿ�§��tõ^ñ�§)�÷¤)¦i§Îî@ £�)�÷� P (4)

V�)¦�ó

P(4)Dø

¨v§>§�òRø�/(0å5%gk;�p\Z\ceY%coZ,� Z65%03p6V�4�f ;�p�V�4\Z\f3V�Z\46V)(,ceY p��! " �5%Z\4�� c�#9;�032�V�0,g��dc�#P (z1, z2)

SPV(z1, z2)

;�032PD(z1, z2)

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(z1, z2) 7→

±(z1, z2)±(ιz2, ιz1)±(z2,−z1)±(ιz1,−ιz2).

<£��;�R

� f351co0'��;�4\ck;�03Ym5�V�#P (z1, z2)

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P (8)(t)SP

(4)V (t)

;�032P

(4)D (t)

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t = z1/z2 7→{ ±t = εt±1/t = −ε/t.

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Im(a8) = Im(a6) = Im(aV4 ) = Im(aD4 ) = 0.

� f35mp65�# V�X,4(5RÍAX ;�Z\ceV�03pÈYmV) =�,co035m2 lncoZ\f=YmV�032coZ\ceV�03pÕ<£�Î��R�c� "(,g�� Z\f ;�Z

c1112 = c1222 = c1233 = c1323 = 0.<£�Î��R

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P (8) ú P (4)V

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P (2q)(t) = ωqP (2q)(t/ω), q ∈ {2, 4},

P (2q)(t) = ωqt2qP (2q)(−ω/t), q ∈ {2, 4},<t�'È�R

ô8õ2¥�¨�¥ω3 = 1

ú��)¦ióP (4) ¨�¥¡òR¥T¨P�=�°§:ÿ�§Î�tõ@ñ2§)�³¤)¦i§)î? £�)�<� P (4)

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(z1, z2) 7→

±(z1, z2)±(z2,−z1)

±(e±ιπ/3z2,−e∓ιπ/3z1)

±(e±ιπ/3z1, e∓ιπ/3z2).

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P (8)(t)SP

(4)V (t)

;�032P

(4)D (t)

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t = z1/z2 7→{

ωt−ω/t .

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� f35C�346p\ZÈYmV�032coZ\ceV�0dco0=5T�2(,465mp6p\ceV�0Ò<t�'È�R&cep(5RÍCX,c���;�ge5%0CZ1Z6V

P (8)(ωt) = ωP (8)(t),

P(4)V (ωt) = ω2P

(4)V (t),

P(4)D (ωt) = ω2P

(4)D (t).

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ª 5ÔY";�0æY[f35mY6qæ2co465mY%Z\g���Z\f ;�Z9Z\f35h�346p\Z9YmV�032coZ\ceV�0Öco0æ5T�2(,465mp6p\ceV�0å<t�'��R�465mp\X,goZ6pco0

P (8)(t) = a7t7 + a4t

4 − a7t = t(a7t6 + a4t

3 − a7).<t�Î�AR

jAc� ¦cogk;�4\g���S*l&5=Y";�0 p65m5ÖZ\f ;�Z Z\f35�gk;�p6Z¦Z�l&V YmV�032coZ\ceV�03p co0 5T�+(,465mp6p6ceV�0 <t�'��R465mp\X,goZnco0

P (4)(t) = a2t2.

<t�'��R�,46V) 5T�2(,465mp6p\ceV�0+<t��AR=# V�4

P (8) ;�032±5T�2(,465mp6p\ceV�0+<t�'��R=# V�4?�ÓV�Z\fP

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a8 = a6 = a5 = aV4 = aV3 = aD4 = aD3 = 0.<t�'��R

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¬ x nv}[r[{ q { r�­®rt¯°m�}[l�u�nvm�± SU(2)��;

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q = 4S,Z\f35ºV�0,g��æ035%lø5RÍAX ;�Z\ceV�0=l&5C��5%Z(cep

Re(a7) = 0h

jAco03Ym5i;�gog�Z\f35Æp[5���5%0dYmVA5T� Y%ce5%0AZ6pÔco0d5T�+(,465mp[p\ceV�0 <t�'��R(;�465ºYmV) "(,ge5T�¤S Z\f35%co4��;�0,cep\f,co0��?��c���5mpnX3p.# V�X,4\Z65m5%0�co032,5�(Ó5%032,5%0AZ(5RÍCX ;�Z\ceV�03psh,U°#�l�5º;�2,2

Re(a7) =0SIl&5�f ;>��5���#^Z65m5%0=5RÍAX ;�Z\ceV�03p"h38.�æp6V�g��Aco0��ÕZ\f35mp65�5RÍAX ;�Z\ceV�03pnl�5�p65m5 Z\f ;�Z(;�0

5%gk;�p\Z\ceY%coZ,�³Z65%03p6V�4�f ;�p�Z\4\c���V�0 ;�gÔp��+ " �5%Z\4�� c�#º;�032 V�0,g��³c�# � lncoZ\f 465mp�(¤5mY%ZZ6V ;�0 V�4\Z\f3V�03V�4� ;�g¹� ;�p\cep � coZ(f ;�pÈp6c��Öco032,5�(¤5%032,5%0CZ YmV) "(¤V�035%0CZ6pE��c���5%04�'�5T�2(,465mp6p\ceV�0�<v�>3�R�h� /�����ûB�*bü�h �3hÚýE¦ß¥T���)�T�¡ £¢� t�¡¤Ç�$¥T¦��R§)¨Jõ+�)���$¥��¡¨v��ï�§)¦i�)���P¤)î?î�¥��¡¨P¤�  ò��Φió§)¦��÷¤�  òJ�tõ2¥7�tõ!¨�¥-¥@ñ2§)�³¤)¦i§)î? ¡�Î�<�

P (8) ú P (4)V

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(4)D

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P (2q)(t) = αqP (2q)(t/α), q ∈ {2, 4},P (2q)(t) = αqt2qP (2q)(−α/t), q ∈ {2, 4},

<t�!;�R

ô8õ2¥�¨�¥α4 = 1

ú.�)¦ióP (4) ¨�¥tò>¥T¨P�=�°§7ÿ-§Î�tõ?ñ2§)�³¤)¦i§)î? ¡�Î�<� P (4)

V�Φió

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(z1, z2) 7→

±(z1, z2)

±(e±ιπ/4z1, e∓ιπ/4z2)±(ιz1,−ιz2)±(ιz2, ιz1)±(z2,−z1)

±(ιe±ιπ/4z2,−ιe∓ιπ/4z1).

<t�'º�R

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P (4)(t)SP

(4)V (t)

;�032P

(4)D (t)

p[;�Z\cep�#t�Ico0��iYmV�0252coZ\ceV�03pÆ<t�!;�R9X,032,5%4ÈZ\f35A# V�gogeV"lnco0�� Z\4[;�03p�#OV�4� ;�Z\ceV�0Ý©

t = z1/z2 7→{

αt−α/t .

<t�'��R� f35��346p\Z1YmV�032coZ\ceV�0=co0d5T�2(,465mp6p\ceV�0�<t�!;�R�cepÈ5RÍAX,c���;�ge5%0CZ(Z6V

P (8)(αt) = P (8)(t),

P(4)V (αt) = α2P

(4)V (t),

P(4)D (αt) = α2P

(4)D (t).

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P (8)(t) = a8t8 + a4t

4 + a8.<£�2�sR

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P (4)(t) = a2t2.

<£�Î3�R� f35&gk;�p\Z$YmV�032coZ\ceV�0¦co0i5T�2(,465mp6p\ceV�0�<t�!;�R�S�lnf35%465

α = 1;�032

q = 4S�c� "(,goce5mp

Z\f ;�ZIm(a8) = 0

h��,46V) YmV�032coZ\ceV�03pi<£�2�sR�# V�4P (8) ;�032�<£�Î3�R.# V�4h�¤V�Z\f

P(4)V;�032

P(4)D

Ssl�59YmV�03Y%goX32,59Z\f ;�Z$;�0¦5%gk;�p\Z\ceY%coZ,�ÆZ65%03p6V�4'f ;�p'Z65%Z\4[;���V�0 ;�g3p��! " �5%Z\4��c�#1;�032³V�0,g�� c�# � lncoZ\f 465mp�(Ó5mY%ZÕZ6V�;�0 V�4\Z\f3V�03V�4� ;�g9� ;�p6cep � Z\f35Æ#OV�gogeVslnco0��YmVI5T� Y%ce5%0CZ6pB��;�0,cep\fÝ©

Im(a8) = a7 = a6 = a5 = aV4 = aV3 = aD4 = aD3 = 0.<£�Î��R

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Ð � � �Ñ�h���h�ݧΨ��tõ2¥��¡¨T �ï'§)¦i�)�^�ΦióÖ�$¥��¡¨��>ï�§)¦i�Î�=�T¤�î?î�¥��¡¨T £¥T�yúJô�¥Ãõ+�)ö)¥��¢-§Îî@îJ§)¦��R¥R�.§vò*�tõ�¥®òR§)�t��§ÎôH [¦!ïhò�§Îð!¨��$¥-¥T¦�¥ �Rð2�Î�¡ £§)¦����

Im(a8) = Re(a7) = a6 = a5 = aV4 = aV3 = aD4 = aD3 = 0.

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Im(a7) = 0òR§)¨*�$¥��¡¨��>ï�§)¦��)��ø

� /�����ûB�*bü�h �hàýE¦�¥T�����T�¡ £¢� ¡�¡¤��$¥T¦���§Î¨1õ+���:�¡¨v�)¦Y�Pö�¥T¨P�R¥��÷¤� [��§��¡¨v§Pñ� ¡¢:�P¤)î��î�¥��¡¨T¤4  ò7�)¦�óǧΦY�÷¤Ç  ò��tõ2¥7�tõ'¨-¥-¥^ñ�§)�÷¤)¦i§Îî@ £�)�÷�

P (8) ú P (4)V

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P (2q)(t) = zqP (2q)(t/z), q ∈ {2, 4},P (2q)(t) = zqt2qP (2q)(−z/t), q ∈ {2, 4},

<£���AR

ô8õ2¥�¨�¥z = eιθ

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P (z1, z2)SPV(z1, z2)

;�032PD(z1, z2)

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(z1, z2) 7→{±(eιθ/2z1, e−ιθ/2z2)

±(eιθ/2z2,−e−ιθ/2z1).

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� f351co0'��;�4\ck;�03Ym5�V�#P (z1, z2)

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t = z1/z2 7→{eιθt−eιθ/t .

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P(4)V (t) = a

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;�032P

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h � f,cepÈc� "(,goce5mpÈZ\f35A# V�gogeV"lnco0�� 5%c���fAZ15RÍAX ;�Z\ceV�03p>©

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a8 = a7 = a6 = a5 = 0.<£�κ�R

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P (2q)(t) = αqP (2q)(t/α),

P (2q)(t) = αqt2qP (2q)(−α/t),2qP (2q)(t) = (t− α)2qP (2q)(βa

t+αt−α ),

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(z1, z2) 7→

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±(e±ιπ/4z1, e∓ιπ/4z2)

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Im(a8) = a7 = a6 = a5 = 0.<£�Î3�R

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t = 1;�032

α = −1 � Z\f ;�Z

a4 = 14a8.<£�Î��R

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P (8) ú P (4)V

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P (2q)(t) = εqP (2q)(t/ε) ωqt2qP (2q)(−ω/t) 2−q(t− α)2qP (2q)(βat+αt−α )

b V�03VAY%goco0,ceYε2 = 1�14\Z\f3V�Z\46V)(,ceYε2 = 1 ω2 = 1� 4\c���V�0 ;�gε3 = 1 ω3 = 1� 5%Z\4[;���V�0 ;�gε4 = 1 ω4 = 1� 4[;�03psh,cep6V�Z\46V)(,ceY |ε| = 1 |ω| = 1�&X��,ceYε4 = 1 ω4 = 1 α4 = 1, β2 = 1U�p6V�Z\46V)(,ceY |ε| = 1 |ω| = 1 |α| = 1 |β| = 1

� f35Æ5RÍCX ;�Z\ceV�03pnco0dZ\f35A�346p\ZÈYmV�goX� ¦0 YmV�4\465mp-(ÓV�032�Z6VÕ46V�Z[;�Z\ceV�03p ;��¤V�X,Ze3

h� f35n5RÍAX ;�Z\ceV�03p�co0¦Z\f35np65mYmV�032 YmV�goX� ¦0 YmV�4\465mp�(¤V�032¦Z6V 465TÌ 5mY%Z\ceV�03p&;��¤V�X,Z�Z\f35(,gk;�035mp9Z\f ;�ZnYmV�0CZ[;�co0

e3h � f35 5RÍAX ;�Z\ceV�03p9co0ÖZ\f351gk;�p6Z9YmV�goX� ¦0�YmV�4\465mp�(¤V�032ÖZ6V

46V�Z[;�Z\ceV�03pÔ;��ÓV�X,Ze1

;�032e2�!�

π/2h

U$#¹(¤V�g��I03V) ¦ck;�gP (2q) (t)

cepÈV�#$Z\f35A# V�4�

P (2q) (t) = a0 + a1t+ · · ·+ a2qt2q,

Z\f35%0�Z\f35A�346p6Z(5RÍCX ;�Z\ceV�0dZ\4[;�03p\gk;�Z65mpÈZ6V

a0 = εqa0

a1 = εq−1a1hhhan = εq−nan;�032ÖZ\f35ºp[5mYmV�032=5RÍCX ;�Z\ceV�0dZ\4[;�03p\gk;�Z65mpÈZ6V

a0 = ω−qa2q

K�L-M�NTO�QTSTU>NyV>QTWYX[ZRQPS�\@]P_R`Ta�bR`T]Ta�aTc�\"d-e+f,d�g2\BV�X¡]�d

3Î3 j�kl�mRnTo j2p�q nsrtnvu p o)wyx n$z�{|m�}t~y{

a1 = −ω1−qa2q−1hhhan = (−1)nωn−qa2q−n.

�,46V) Z\f35mp65 5T�+(,465mp[p\ceV�03pJ# V�4 Z\f35 YmVI5T� Y%ce5%0CZ6p"S&l&5�Y";�0 p[5m5dZ\f354#OV�gogeVslnco0��p\Z[;�Z65� �5%0AZ6p"h��� V�4(;= �V�03VIY%goco0,ceYA �5m2coX� =S3V�0,g�� Z\f35��346p\Zn5RÍCX ;�Z\ceV�0�f ;�p9Z6V?�¤5 p[;�Z\cepv� 5m2

lncoZ\fε2 = 1

h jAco03Ym5q

cepÈ;�0=5���5%0�0CX� ^�Ó5%4"S,l�5�p65m5ÔZ\f ;�Z(;�gog�VA2,2�YmVI5T� Y%ce5%0CZ6pf ;>��5�Z6V?�Ó5*%"5%46V,h� V�4$;�0iV�4\Z\f3V�Z\46V)(,ceY �5m2coX� =SsZ\f35��346p\Z�5RÍAX ;�Z\ceV�0Õf ;�p�Z6VE�¤5*p[;�Z\cepv� 5m2ÆlncoZ\f

ε2 = 1;�032 Z\f35ip65mYmV�032 5RÍCX ;�Z\ceV�0 f ;�pÔZ6V��¤5ÆpT;�Z\cepv� 5m2dlncoZ\f

ω2 = 1hi�346V)

Z\f357�346p\Z¦5RÍAX ;�Z\ceV�0 l&5�p[5m5ÖZ\f ;�Z�V�0,g��ÒZ\f35�5���5%0 YmVI5T� Y%ce5%0CZ6pæ;�465Ö03V�0�%"5%46V,h�,46V) Z\f35ºp65mYmV�032�5RÍAX ;�Z\ceV�0=l&5¡p65m5 Z\f ;�Z

a2k = a2q−2kh

� V�4 ;dZ\4\c���V�0 ;�g9 �5m2coX� =S�Z\f357�346p\Z¦5RÍCX ;�Z\ceV�0±f ;�pÕZ6V��¤5Öp[;�Z\cepv� 5m2 lncoZ\fε3 = 1

;�032 Z\f35ip65mYmV�032 5RÍCX ;�Z\ceV�0 f ;�pÔZ6V��¤5ÆpT;�Z\cepv� 5m2dlncoZ\fω3 = 1

hi�346V) Z\f35B�346p\Zn5RÍCX ;�Z\ceV�0Öl�5Ôp[5m5(Z\f ;�Z9Z\f35 V�0,g�� 03V�0�%"5%46VÕYmVI5T� Y%ce5%0CZ6p

ak;�465ÔZ\f3V�p65

# V�4(lnf,ceY[fq − k cep12c��Icep\c��,ge5^�'�=Z\f,465m5�h��,46V) Z\f35¡p[5mYmV�032 5RÍAX ;�Z\ceV�0 l�5¡p[5m5

Z\f ;�Z# V�4ÈZ\f35mp65 03V�0�%"5%46V YmVA5T� Y%ce5%0AZ6pak = (−1)kωk−qa2q−n

h� V�41;ÕZ65%Z\4[;���V�0 ;�g� �5m2coX� =S Z\f35A�346p\ZÈ5RÍAX ;�Z\ceV�0df ;�pnZ6VJ�Ó5�p[;�Z\cep�� 5m2�lncoZ\f

ε4 = 1;�032 Z\f35æp65mYmV�032 5RÍCX ;�Z\ceV�0 f ;�pÆZ6V��¤5 p[;�Z\cepv� 5m2�lncoZ\f

ω4 = 1h¹ª 5

p65m5 Z\f ;�Z¡Z\f35æV�0,g�� 03V�0�%"5%46V�YmVI5T� Y%ce5%0CZ6pak

;�465 Z\f3V�p65�# V�4¡lnf,ceY[fq − k cep

2c��Icep\c��,ge5H�!�A#OV�X,4"h>�346V) øZ\f35*p[5mYmV�032º5RÍCX ;�Z\ceV�0¡l&5&p65m5KZ\f ;�ZÝ# V�4�Z\f35mp65*03V�0�%"5%46VYmVI5T� Y%ce5%0CZ6p

ak = (−1)kωk−qa2q−nh

� V�4Ö; Z\4[;�03p���5%46p65%g�� cep6V�Z\46V)(,ceY� �5m2coX� =S�Z\f35��346p6Z 5RÍAX ;�Z\ceV�0 f ;�pæZ6V��¤5p[;�Z\cep�� 5m2 lncoZ\f |ε| = 1

h2ª 5�p65m5ÔZ\f ;�Z(;i03V�0�%"5%46V�YmVA5T� Y%ce5%0AZÈcep(¤V�p6p\c��,ge5 V�0,g��c�#n = q

Slnf35%0εn−q = 1

h� V�4K;ÔY%X��,ceY. �5m2coX� =S�Z\f35��346p\ZK5RÍCX ;�Z\ceV�0¦f ;�p'Z6V��¤5&p[;�Z\cep�� 5m2¡lncoZ\f

ε4 = 1;�032 Z\f351p[5mYmV�032æ5RÍAX ;�Z\ceV�0Öf ;�p�Z6V=�¤5(p[;�Z\cep�� 5m2 lncoZ\fω4 = 1

hI/(gep6V¡Z\f351Z\f,co4625RÍAX ;�Z\ceV�0 f ;�p Z6V��Ó5 p[;�Z\cepv� 5m2 lncoZ\f

α4 = 1;�032

β2 = 1h�ª 5 p65m5dZ\f ;�Z

Z\f35ÔV�0,g���03V�0�%"5%46VÕYmVA5T� Y%ce5%0AZ6pak

;�4651Z\f3V�p65B#OV�4�lnf,ceY6fq− k cep92c��Icep\c��,ge5E�'�

# V�X,4"h��,46V) Z\f35&p65mYmV�032¡5RÍAX ;�Z\ceV�0Õl&5�p65m5�Z\f ;�Z¹# V�4�Z\f35mp65&03V�0�%"5%46V YmVA5T� Y%ce5%0AZ6pak = (−1)kωk−qa2q−n

;�032�Z\f35mp65ºYmVI5T� Y%ce5%0CZ6pE ÆX3p6Z1;�gep6V�p[;�Z\cep�#t� Z\f35�Z\f,co4625RÍAX ;�Z\ceV�0�S lnf,ceY[f Y";�0��¤5i5";�p\cog��=Y6f35mY[q�5m2��'� Y[f3VAV�p6co0��3SY# V�41co03p6Z[;�03Ym5�S

t = 1;�032α = −1

h

è�ö����t�"ÿCþ¹�>���^ü�ù¹�´Ôp\co0��¦Z\f35ºp\X� " ;�4���V�#$465mp\X,goZ6pnco0 jI5mY%Z\ceV�0�;¦;�032�Z\f35C(,46VIV�#OpÈco0 jI5mY%Z\ceV�0Ç�Sl�5&Y";�0Õp\Z[;�Z659;�gog!(¤V�p6p\c��,ge5*46V�X,Z65mp'V�#3co03Y%465";�p6co0�� p-�! " �5%Z\4\ce5mp¹# V�4';�0Õ5%gk;�p\Z\ceY%coZ$�

��� ½T·�¶'·�Ç�¼[·�·�Ç�¶���¸�²O·�¶ �mÍ�¼[·�ÀÁ½�¿ÔÄ�½m¶L²�¿�½[·¤´sÀÁ¶L»¢ÄÔ¼T¿m´È¸�¶L²O· ¸�ÀÁ¹�·�ÀÁ½�¿Ô½�¿1·�Ç�¶'¹L½m¶ �n¹LÀÁ¶L¿m·an¾�½[¸�Ë�Ç�ÀÁ¹WÇ

n = qÃ�² ÀÁ¿�¹�¶KÀ¢¿1·�Ç�¼[·�¹L¼T² ¶

εn−q = 1 Ð

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¬ x nv}[r[{ q { r�­®rt¯°m�}[l�u�nvm�± SU(2)3Î�

Z65%03p6V�4"S�;�p�p\f3Vsln0åco0åZ\f35^#OV�gogeVslnco0��d2ck;���4[;� =h � f35ÕZ�l�V�0CX� ^�Ó5%46p�Z\f ;�ZA# V�g�5geVsl+Z\f35�0 ;� �5mp(V�#8��c���5%0 p-�! " �5%Z\4\ce5mpÔ;�465�Z\f35�0CX� ^�Ó5%4ÔV�#'Z\f35ºco032,5�(Ó5%032,5%0AZYmV) "(¤V�035%0CZ6pÕV�#(Z\f35æYmV�4\465mp�(¤V�032co0���5%gk;�p6Z\ceY%coZ$� Z65%03p6V�4¦;�032ÒZ\f35 0AX� =�¤5%4ÕV�#Z\f35�5%ge5� �5%0AZ6p(co0�Z\f35ºYmV�4\465mp-(ÓV�032co0��æp��! " �5%Z\4��7��46V�X�(�h

Tr. Isotropic 5, 8

Isotropic 2, 8

Orthotropic 9,8

Cubic 3,48

Monoclinic 13,4

Trigonal 6,12

Anisotropic 21,2

Tetragonal 6,16

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V�462co0 ;�Z65ip��p\Z65� 5T�,5�(,Z�jI5mY%Z\ceV�0&�h � f35¡Z\4[;�03pv# V�4� ;�Z\ceV�03p1co0åjI5mY%Z\ceV�0��æ;�4655T�2(,465mp6p65m2 co0ÖYmVIV�462co0 ;�Z65mp�Z\f ;�ZÈ;�465(0 ;�Z\X,4[;�g�# V�4&Z\f35E��c���5%0�p��! " �5%Z\4�� Y%gk;�p6p"h� f,cep�2,VA5mp&03V�Z. �5";�0�SCf3Vsl&5���5%4"SZ\f ;�Z�l&51l&V�X,ge2æ03V�Z�V)�,Z[;�co0æZ\f35Ôp[;� �51Y[f ;�4v5;�Y%Z65%4\c�%s;�Z\ceV�0±X3p\co0��å2c�Ä�5%465%0CZ�YmVIV�462co0 ;�Z65mp"h0ª 5æl&V�X,ge2±V)�,Z[;�co0±5T�+(,465mp[p\ceV�03pZ\f ;�Z�Y6f ;�4[;�Y%Z65%4\c�%"5 5";�Y6f Y%gk;�p6p"S��,X,Z Z\f35��dl�V�X,ge2��Ó5@ �V�465Õ2c�� Y%X,goZ�Z6Væl&V�4\qlncoZ\f�h

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