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!"#$% &'() *#+,-./!0!1!2%2- 3%"45/!,% 67 .%$% 8 7- -4912 ,83 8+ $-+#1/%28+ 2- :-!+4-$ ;<=> ?H = 10@A-B - 48++8+ $-+#1/%28+ ?H = 7.5@A-B' C308+ $-+#1/%28+ %.84/%3 .%$% D$%,% 2-.-42E4,!% ,83 /-3.-$%/#$%' 8+ /%3%4F8+ 2%+ 1!"%GH-+ I#J3!,%+K .%$L3-/$8+ 2- $-2- - .$8.$!-2%2-+ 2- /$%4+.8$/- ;<M>' N3 "-$%1K % -+/$#/#$% 2- 0%42% .$OP!38 %8 4JQ-1 2- -$3! 2-+/-+ 3%/-$!%!+ /-3 8$!"-3 4% F!0$!2!R%GS8 28+ 8$0!/%!+ Tf 28 7 ,83 8+ 8$0!/%!+ 4d ?4= 3, 4, 5B 28+ 3/%!+ 2 /$%4+!GS8 -8$0!/%!+ 4p ?4= 3, 4, 5B 28+ .4!,/!2-8+' U8 7- -4912K 8 .%$L3-/$8 2- $-2- ?a ≈ 7.79 o AB 5 %483%1%3-4/- .-I#-48 ?Q-V% W"#$% &'TK $-.$82#R!2% 2% $-D-$E4,!% ;<X>BK 8 I#- D%Q8$-,- #3% D8$/- F!0$!2!R%GS8 28+ 8$0!/%!+ Tf ?7-BK (d ? -B - (p ?9B - I#- $-+#1/% 48 -48$3- !" VY %.84/%28 4-+/% /-+-' N+/% $-1%GS8K 48 -4/%4/8K 4S8 5 80+-$Q%2% .%$% /82%+ %+ #$%&&'(%)*&'# 0%+-%2%+ -3 7-' A 7-A+4*012K .8$ -P-3.18K +-$!% #3 3-/%1 4-+/- I#%2$8K 3%+ 3-+38 /-428 #3% ,84+/%4/- 2- $-2- ,83.%$%/!Q%3-4/- "$%42- ?a ≈ 9.25 AoBK 5 /%3053 #3 !+81%4/- 2-Z8428 ;<M>' C 2-+,$!GS8 28 7- -4912 -3 /-$38+ 2- #3 +-3!,842#/8$ Z8428K D8! ,#!2%28+%3-4/- !4[ Q-+/!"%2% .8$ -P.-$!3-4/8+ 2- D8/8-3!++S8K %4%1!+%28+ % 1#R 28 382-18 2- C42-$+84 .%$% #3% !3.#$-R% 3%"45/!,% ;<&K <\>' A+ .%$L3-/$8+ 28 382-18 D8$%3 ,%1,#1%28+ - #3 ,-4Y$!8 ,84+!+/-4/- ,83 -+/% 2-+,$!GS8 D8! 80/!28' N3 .%$/!,#1%$K % Q%$!%GS8 2% 2-4+!2%2- $-1%/!Q% 2- -+/%28+ ?DOS(Ef)B ,83 % /-3.-$%/#$% -+,%1% 2- 3%4-!$% 0-3 +-3-1F%4/- ] .%$/- 2-[ .-42-4/- 2% /-3.-$%/#$% 28 383-4/8 3%"45/!,8 -D-/!Q8 ;<&>K /%1 I#%1 80/!28 .8$ 3-2!2%+ 2- +#+,-./!0!1!2%2- 3%"45/!,% ?W"#$% &'(B' :-+38 4-+/- ,-4Y$!8K 48 -4/%4/8K % +#.$-++S8 28 +#.8+/8 -D-!/8 Z8428 8,8$$- 4#3% -+,%1% 2- /-3.-$%/#$% 3#!/8 +#.-$!8$ ]I#-1% 2- 48++8 -P.-$!3-4/8' ^

(16)

4 12

=

f d p

4 12 0

(17)

!" #$%&' ()+,$-.'!

!"#$%&'() *$ +),-)./ $"#%$ 0&*& $-1#%)" -)0&-!2&*) ") 34#!) *$ ,. *&*) 4)" $ &3 0&%5&3 $6#$%"&3 & $3#$ 4)" 1 *&*& 7)% ,. 7)#$"0!&- $-$#%)3#8#!0) V (r)9 :) 0&3) *$ !"#$%$33$ *$3#& #$3$; $-1#%)"3 $. 4)"3 *$ #$%%&3 %&%&3; )3 $-1#%)"3 4f $3#() /-!"*&*)3 7$-&3 0&.&*&3 5s2_5p6

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,.& 3)-,'() *&*& 7)% ,.& $67&"3() $. #$%.)3 *$ N&%.I"!0)3 $3D1%!0)39 +).$"#&.)3 C,$ ,.& $67&"3() *$3#$ 0$"8%!) 1 *&*) 7$-& #$)%!& *$ 0&.7)3 -!5&"#$3; C,$ -$B& $. 0)"#& & *!3#%!/,!'() $37&0!&- *&3 0&%5&3 $"B)-B!*&3 ") 7%)/-$.& $; 7)%#&"#); -$B& $. 0)"3!*$%&'() & 3)/%$7)3!'() *& *!3#%!/,!'() *&3 0&%5&3 $6#$%"&3 0). ) $3#&*) $-$#%I"!0) -)0&-!2&*)9 M&%& ) .)*$-) *$ 0&%5&3 7)"#,&!3; $30%$B$.)3 7&%& V (r)O

V (r, θ, φ) =X

l

X

m

Am_l rlY_lm(θ, φ) @P9GA

:)33) !"#$%$33$ 1 0&-0,-&% )3 $-$.$"#)3 *$ .&#%!2 *& $C P9G $"#%$ $3#&*)3 4f9 Q$"*) Ψ $ η *)!3 *$3#$3 $3#&*)3; $3#$ 08-0,-) 1 *&*) 7$-& 3$5,!"#$ !"#$5%&-O

VΨη(~r) =

ˆ

η∗(~r′)V (~r′)Ψ(~r′)dτ′ @P9FA

R"*$ ,3&.)3 ~r = (r, θ, φ)9 S$ .)*) 5$%&-; ) 08-0,-) *)3 $-$.$"#)3 *$ .&#%!2 *&*)3 7$-& $C P9F 1 0).7-!0&*)9 Q!.7-!K0&'T$3 *$3#$ 08-0,-) %$3,-#&.; ") $"#&"#); *) ,3) *$ &-5,"3 %$3,-#&*)3 *& 8-5$/%& *$ #$"3)%$3 @.&#%!2$3A C,$ *!30,#!%$.)3 & 3$5,!%9 :)33& $3#%&#15!& 3$%8 #%&#&% ) 7%)/-$.& *$ ,.& .&"$!%& ,. 7),0) .&!3 5$%&- .&3 7)*$.)3 0!#&% C,$ & C,&"#!*&*$ *$ #$%.)3 & 3$%$. 0&-0,-&*)3 "& $C P9F J8 1 $")%.$.$"#$ *!.!",4*& 7)% 3!.7-$3 )%#)5)"&-!*&*$ *$ $3D1%!0)3 N&%.I"!0)39 M&%& $3#&*)3 4f; 7)% $6$.7-); #)*)3 )3 #$%.)3 7&%& l > 6 3() ",-)39

(18)

!"! #$%&'( )(&$+,-.( -, +.)/, -* )0)('+,12(!

!"#!$#% &' '#'(')*%+ &' (,*-./ 0 ')%-('(')*' +.(1#.2!,&% ,% +'-'( $*.#./,&%+ -'+$#*,&%+ &, "#3'4-, &' *')+%-'+5 6+*, "#3'4-, 0 7%-(,#./,&, )% !%)*'8*% &, *'%-., &' -'1-'+')*,9:'+ ;<=>5

?'@, {ψi

λ} $( !%)@$)*% &' 7$)9:'+ 4,+' A$' 3'-, $(, -'1-'+')*,9B% .--'&$*CD'# EFGH Γi

&' $( 3-$1% &' +.('*-.,+ GI A$' &'+!-'D' , +.('*-., &% 1-%4#'(, 7C+.!% '( A$'+*B%5 ?'@, ˆ

T $( %1'-,&%- ,-4.*-"-.%5 6( 3'-,#I ˆT ψi

λ *'-" *%&%+ %+ '#'(')*%+ &% !%)@$)*% {ψiλ},!.(,

&'2).&%5 J,-, .)D'+*.3,- ,+ 1-%1-.'&,&'+ &' ˆTI *%('(%+ , ,9B% &' $( '#'(')*% R ∈ G +%4 ,+ 7$)9:'+ φ' η &'2).&,+ !%(% φ = ˆT η5 ˆ ΘRφ = φ′ ˆ ΘRη = η′ φ′ _{= ˆ}_T′_η′ ˆ ΘRφ = ˆT′ΘˆRη φ = ˆΘ−1 R Tˆ′ΘˆRη EK5=H J'#, 'A$,9B% K5= *'(%+ A$' ˆT−1 _{= ˆ}_Θ RT ˆˆΘ−1R I A$' 0 , -'3-, &' *-,)+7%-(,9B% 1,-, $(

%1'-,&%- ,-4.*-"-.% ˆT +%4 , ,9B% &' $( '#'(')*% R &' G5 L'+*, (,)'.-,I *'(%+ A$' % !%)@$)*% &' !%(1%)')*'+ {Tα}&' ˆT 3'-,( $(, -'1-'+')*,9B% E'( 3'-,#I -'&$*CD'#H &% 3-$1%

G5 6+*, ,2-(,9B% 1%&' +'- ')*')&.&, !%)+.&'-,)&%M+' , ,9B% &% %1'-,&%- ˆΘRTˆ′Θˆ−1R +%4 ,+

7$)9:'+ &% !%)@$)*% {ψi

λ} ,!.(, &'2).&%5 6+*, -'1-'+')*,9B% 1%&' +'- -'&$/.&, '( *'-(%+

&' !%(1%)')*'+ .--'&$*CD'.+ &% %1'-,&%- ˆTI A$' +.3).2!, A$' 1%&'(%+ '+!-'D'-N

ˆ ΘRTˆ′Θˆ−1R = ni X µ=1 ˆ T_µiΓi_µλ(R) EK5OH

J%- !%)+'A$P)!., &, 'A$,9B% K5O I , ,9B% &' $(, !%(1%)')*' .--'&$*CD'# ˆTi

λ '( ψαj

E ˆTi

λψαj = ˆTλiΘˆR−1ΘˆRψjαH +' *-,)+7%-(, !%(% $(, -'1-'+')*,9B% 1-%&$*% Γi× Γj ' % *'%-'(,

&' %-*%3%),#.&,&' &' -'1-'+')*,9:'+ ;<=> 3,-,)*' , +'3$.)*' -'3-, &' +'#'9B%N

hφm_σ| ˆT_λi|ψ_αji = 0

(19)

∵Γm ∈ Γ/ i× Γj

!"! #$#"%&!' %!(#"!) %#*)+, *+ -*.#/,+& 0121 3+,+ #&4.,!*) 4f' +%#*+) 5!"%!*#*.#) (# #).+(!) l ≤ 3 #$-).#"1 6 %,!(7.! (! %!.#*5-+& V (r)' 5!" 5!"%!*#*.# i' 5!" !7.,! #).+(! 4f' .#" 5!"%!*#*.#) n 8n > 09 +%#*+) #*.,# i − 3 ≤ n ≤ i + 31 :#).+ "+*#-,+' %+,+ ;7# + -*.#/,+& )#<+ (-=#,#*.# (# 0' .#"!) ;7# i ≤ 61

!"!" #$%&'()&%* (% +,%-%.* % ) /%)&%0' (% 123.%&4567'&,

>+"!) -*?#).-/+, 5!" "+-) (#.+&@#) !) ,#)7&.+(!) (+ )#AB! +*.#,-!,1 C -(4-+ 4 #).7(+,"!) +) 5!*)#;7D*5-+) (+) ,#/,+) (# .,+*)=!,"+AB! (# !%#,+(!,#)' ;7# +/!,+ (-)57.-"!) (# "+*#-,+ /#,+&1 :#)5!E,-"!) ;7# 7" !%#,+(!, ˆT .,+*)=!,"+ 5!"! ˆT−1_{= ˆ}_Θ RTˆ′Θˆ−1_R ' -))! )-/*-F5+ ;7# %!(#"!) #)5,#?#, ˆ ΘRTˆ′Θˆ−1R = X β ˆ TβΓTβα(R)

6*(# ΓT _{4 7"+ ,#%,#)#*.+AB! ,#(7.G?#& (# G' /#,+(+ %#&! !%#,+(!, 8,#(7.G?#&9 ˆ}_T′_{1 H 7"}

,#)7&.+(! /#,+&' (#(7.G?#& (! .#!,#"+ (# !,.!/!*+&-(+(# (# 5+,+5.#,#) # ,#%,#)#*.+AI#)' ;7# %!(#"!) ,#(7J-, #).+ ,#%,#)#*.+AB! #" .#,"!) (# ,#%,#)#*.+AI#) -,,#(7.G?#-)1 ΓT = X i∈RI aiΓi ai = 1 h X g∈G χT(g)∗χi(g) :#).+ "+*#-,+' .#"!) + )#/7-*.# #$%,#))B! 8;7# 4 #$%,#))B! 01K #)5,-.+ +5-"+9L ˆ ΘRTˆ′Θˆ−1R = X β ˆ T_βiΓi_βα(R) M).+ #$%,#))B! (#.#,"-*+ ;7# %!(#"!) ,#(7J-, ;7+&;7#, !%#,+(!, + )7+) 5!"%!*#*.#) -,,#(7.G?#-) ˆTi β1 C %!))-E-&-(+(# (# ,#(7J-, !%#,+(!,#) + )7+) 5!"%!*#*.#) -,,#(7.G?#-) 4 5@+"+(! "4.!(! (!) !%#,+(!,#) (# N.#?#*) O22P1 M).+) 5!"%!*#*.#) -,,#(7.G?#-) +/#" )!E +) E+)#) 8#).+(!) #&#.,Q*-5!)9 (# 7"+ "+*#-,+ ;7# <R (#.#,"-*+"!) *+ S&.-"+ )#AB!' !7 )#<+' 5!"! 7"+ ,#%,#)#*.+AB! %,!(7.!1 C))-" 5!"! ;7+&;7#, !7.,+ ,#%,#)#*.+AB!' #).+) .+"E4" %!(#" )#, ,#(7J-(+)1 3+,+ ! %,!(7.! (# ,#%,#)#*.+AI#) -,,#(7.G?#-) Γi_{# Γ}j_{' #)5,#?#"!)L}

Γi_{× Γ}j ₌X

l

alijΓl 801T9

(20)

!" # $%&'(%) *+,(+-+, ./+ " 0'1-")" alij # 0%1#2,%(" 3"0 '3$%(+0 i + j4 5 0%2/56!" +0*+(%5)

./+ l ∈ RI $+ G7 5 +./56!" 849 $+:3+ 5 0#,%+ $+ ;)+-0(<=>",$53 *5,5 " *,"$/2" Γi_{× Γ}j_{4 ?0}

("+:(%+32+0 $+ ;)+-0(<=>",$53 *"00/+1 1/%250 *,"*,%+$5$+0 + +02!" 500"(%5$"0 5 %1*",2532+0 5*)%(56@+0 +1 &'0%(5 2+A,%(5 BCDE4 F./%7 G51"0 5*+350 3"0 52+, 5" 2+",+15 $+ H%I3+,=J(K5,27 $+ +0*+(%5) %1*",2L3(%5 *5,5 " (M)(/)" $" (51*" (,%025)%3"4 N51"0 $+:3%, a(lij) = (ijl)7 *5,5 &5(%)%25, 5 3"256!"4 ;"30%$+,+ " (M)(/)" $" 0+I/%32+ +)+1+32" $+ 152,%OP

hψ_µi| ˆT_αm|φj_νi Q84RS

? 2+",+15 $+ H%I3+,=J(K5,2 3"0 I5,532+ ./+ 84R # $5$" *",P

=X

si

(mα, jν|isiµ)hψik ˆTmkφji Q84TS

"3$+ " '3$%(+ si = (mji) +U*,+005 " 3V1+," $+ G+O+0 ./+ 5 ,+*,+0+3256!" Γi 5*5,+(+

35 ,+$/6!" $+ Γm_{× Γ}j_{4 ? +)+1+32" $+ 152,%O hψ}i_{k ˆ}_Tm_kφj_i _{# (<515$" +)+1+32" ,+$/O%$" +} $+*+3$+ 5*+350 $"0 '3$%(+0 i + j4 W02" 0%I3%:(5 ./+ *5,5 (5)(/)5, ./5)./+, +)+1+32" $+ 152,%O $5 &",15 hψi µ| ˆTαm|φjνi7 *,+(%051"0 5*+350 $" (M)(/)" *5,5 /1 +)+1+32" +0*+(':(" + "0 $+15%0 +025,!" $5$"0 *+)5 ("+:(%+32+0 $+ ;)+-0(<=>",$534 ? (50" $+ 15%", %32+,+00+ &'0%(" # " $+ 1"1+32" 53I/)5,7 " ./+ 0%1*)%:(5 1/%2" " (M)(/)" *"%0 *5,5 ./5)./+, +025$" $+ 1"1+32" 53I/)5, si = 14

!"!# $%&'(% &' )+,% )-./0(.1%

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

F0 -50+0 $50 YW0 $" I,/*" G 0!" "-2%$50 *+)5 ,+$/6!" $5 0%1+2,%5 $50 -50+0 $5 ,+*,+0+3= 256!" $+ 1"1+32" 53I/)5, 2"25) J = L+S +1 2+,1"0 $"0 0/-I,/*"0 $+ SO3./+ $+0(,+G+1 5

0%1+2,%5 )"(5) $"0 +025$"0 +)+2,[3%("0 +1 0A)%$"04 ? *,"-)+157 3" +32532"7 ,+./+, 5 /2%)%O56!" $+ 5)I/1 1"$+)" ./+ 0+)+(%"3+ " 1/)2%*)+2" $" +025$" &/3$51+325)4 \", +U+1*)"7 *5,5 " ]^3+_Q4f9_{S /15 I,53$+ G5,%+$5$+ $+ +025$"0 $+ 1"1+32" 53I/)5, 2"25) 0!" *"00'G+%04 Y+I,50}

&+3"1+3")AI%(507 ("3<+(%$50 ("1" ,+I,50 $+ _/3$ QBC`E *5,5 /15 -"5 $%0(/00!"S 0+)+(%"351 3+02+ (50" J = 15

27 +1 3"256!" +0*+(2,"0(A*%(5 6H152 QS =

5

27L = 5S4

(21)

!"#$%&'( ()#"( * ( "&+,-.# +( /0'&1"0( SO3 &' /&,/ /,$)",!#/ +& #"+&' 2301(4 5,&

&6!"&//(' ( /0'&1"0( %#7(% +# &/1(+# &%&1"8307#9 :/ "&%(-;&/ +& 7#'!(10$0%0+(+& <=>? ),0(' &/1& !"#7&+0'&31#9 @"(1(A/&4 $(/07('&31&4 +( 7#'!("(-.# &31"& (/ 1($&%(/ +& 7("(71&"&/ +# )",!# 2301# +& 031&"&//& & +# )",!# SO39 "&/,%1(+# 23(% * 5,& # ',%10!%&1# 7#' J 7#3/1(31&

&/1("B 7%(//027(+# &31"& (/ "&!"&/&31(-;&/ Γi _{+# )",!# !#31,(%9}

:!"&/&31('#/ &' +&1(%C& # !"#$%&'( !("( # D&3+_{0%,/1"(3+# ( 7#3/1",-.# +& ,' !#1&370(%}

+& 7('!# 7"0/1(%03# & /,( ,10%0E(-.# 3# 7B%7,%# +& )"(3+&E(/ FG/07(/9 H('#/ F(E&" # 7B%7,%# &' /0'&1"0( 7I$07( Oh9 D&3+1&' 7#32),"(-.# &%&1"8307( <J&?4f19 : "&)"( +& K,3+ 3#/

+0E 5,&4 +& 1#+#/ &/1(+#/ (310//0'*1"07#/ 5,& !#+&' /&" F#"'(+#/4 # ',%10!%&1# F,3+('&31(% * +(+# !&%# &/1(+# 2_F₅

29 L' 1&"'#/ +& C("'8307#/ &/F*"07#/4 # !#1&370(% +& 7('!# 7"0/1(%03#

* +(+# !#"M V (r) = D4[Y40+ r 5 14(Y 4 4 + Y4−4)] + D6[Y60− r 7 2(Y 4 6 + Y6−4)] NO9PQ

L' 1&"'#/ +& #!&"(+#"&/ &5,0R(%&31&/4 &/7"&R&'#/M

V (r) = B₄c[O₄0+ 5O₄4] + B₆c[O0₆− 21(O₆4+ O−4₆ )] NO9OSQ / &/1(+#/ "&/,%1(31&/ /.# +(+#/ !&%( +0()#3(%0E(-.# +( '(1"0E O9OS4 5,& !#+& /&" 7(%A 7,%(+( F(70%'&31& 3( $(/& |JMi ,/(3+# (/ 1($&%(/ TB &60/1&31&/ <==?9 L/1&/ &/1(+#/ /.# 7%(//027(+#/ &' 1&"'#/ +(/ "&!"&/&31(-;&/ +# )",!# +& !#31# Oh 9 D#'# # 3I'&"# +&

&%*1"#3/ * G'!("4 (/ "&!"&/&31(-;&/ /&".# "&!"&/&31(-;&/ +& /!034 #, "&!"&/&31(-;&/ +#$"(+(/9 ',%10!%&1# J = 5

2 /& +0R0+& &' ,' +,$%&1# Γ7 & ,' 5,("1&1# Γ89

Γ8 = 0.912871|5₂ ±5₂i + 0.408248|5₂ ∓3₂i |5 2 ±12i Γ1₇ = −0.40828|5 2± 5 2i + 0.912871| 5 2∓ 3 2i NO9OOQ

U&/1( '&/'( '(3&0"(4 1('$*' !#+&'#/ &37#31"(" #/ (,1#R(%#"&/ N&3&")0(/Q +# /0/1&'(M

Γ8 = 2B4c

Γ7= −4Bc4 NO9OVQ

&/1(+# F,3+('&31(% +&!&3+&"B4 !#"1(31#4 +# /03(% +# !("W'&1"# Bc

49 F(1# +&/1&

7B%7,%# +&!&3+&" (!&3(/ +# !("W'&1"# +& 5,("1( #"+&' * ,' 7(/# !("107,%(" +&R0+# (# F(1# 5,& #/ &%&'&31#/ +& '(1"0E +# !("W'&1"# +& /&61( #"+&' /.# 3,%#/ !("( # ',%10!%&1# J = 5

29

(22)

!"! # $%&%4 12' B4c = −720(144)() *+, % # %-.!/# 012/!(%2.!3 4 /!/# 5%3# 61!".%.# Γ8' 1( %-.!/# 7#( /%8%2%"%-7927:! 4; <-.! /%8%2%"%-7927:! 4 1( !-5%7.# 012/!(%2.!3 /# -:-.%(!; <( 61!361%" -:.1!=># "%!3' 2# %2.!2.#' # 7!(5# 7":-.!3:2# 2># -%"? # @2:7# .%"(# /# A!(:3.#2:!2# /# -:-.%(!; <( 5!".:713!"' 2# 7!-# /% 1( %B5%":(%2.# /% <CD' .%(#- ! :2.%"!=># E%%(!2 61%' (%-(# 61% 0"!7!(%2.%' 61%F"! %-.! /%8%2%"%-7927:! /# 61!".%.# Γ8 %( 61!."# %-.!/#-; <0%:.#- /% .%(5%"!.1"!' 2# %2.!2.#' 0!7:3(%2.% 3%G!( H (:-.1"! /%-.%-%-.!/#-;

!"!# $%&'()+%,-. ( /%0.1 (23(4564. (7 /%73. /12)%0*'.!

I %2.%2/:(%2.# /#- %0%:.#- /% 7!(5# 7":-.!3:2# %( (%/:/!- (!7"#-7J5:7!- 5#/% -%" #F.:/# 5#" 7?3713# /:"%.#- F!-%!/#- %( (#/%3#- .%J":7#- -:(53%-; <-.%- (#/%3#- -># :(5#".!2.%- 5#:-5%"(:.%( 7#(5!"!=># /:"%.! 7#( %B5%":(%2.#-; K!(#- 7#2-:/%"!" ! -%81:2.% A!(:3.#2:!2! %( 2#--# 7?3713#' %-7":.! %( #"/%( /%7"%-7%2.% /% :(5#".L27:! /#- .%"(#-M

H = Hintra+ Hsp−orb+ Hcryst+ HZ NO;OPQ

<( O;OP Hintra /%-7"%G% ! :2.%"!=># /% $#13#1(F; <-.% 4 # .%"(# (!:- :(5#".!2.% /# A!(:3.#2:!2# 61% /%-7"%G% ! :2.%"!=># $#31(F:!2! /#- %34."#2- /! 7!(!/! 4f; I -%812/# .%"(# 61%F"! ! /%8%2%"%-7927:! /# 5":(%:"# % -%3%7:#2! # (13.:53%.# 2_F₅ 2 7#(# # %-.!/# 012/!(%2.!3 N"%8"! /% A12/Q; I .%"7%:"# .%"(#' /:G:/% # (13.:53%.# 2#- 2 %-.!/#- 61% 7!3713!(#- 2! -%=># !2.%":#" % # @3.:(# .%"(#' ! :2.%"!=># E%%(!2' 5"#(#G% ! -%5!"!=># 7#(53%.! /% .#/#- #- 2RG%:- /% %2%"8:!; S61: G!(#- 2#- !.%" !5%2!- !# (13.:53%.# /# %-.!/# 012/!(%2.!3 2_F₅ 2' /% (!2%:"! 61% O;OP -% "%/1T !M H ≈ Hcryst+ HZ = Bc 4[O40+ 5O44] + gJµBJ.H NO;OUQ I 7?3713# /!- %2%"8:!- /% O;OU 4 0%:.# %B!.!(%2.% 5!"! Hcryst % 1-!V-% ! .%#":! /% 5%"V .1F!=># %( 5":(%:"! #"/%( 5!"! 7!3713!" #- 2RG%:- W2!:- /% %2%"8:! /!/#- 5#" HZ; I-"%-13.!/#- -># !5"%-%2.!/#- /% (!2%:"! %-61%(?.:7! 2!- W81"!- O;X! % O;XF; I G!3#" Bc 4 = −720(144)() /% 7!(5# 7":-.!3:2# *+, 0#: 1-!/# 2%-.% 7?3713# $%3+_; I 7#2Y%7:(%2.# /#- 2RG%:- /% %2%"8:! 2#- 5%"(:.% 5"#--%81:" 7#( # 7?3713# /!- 8"!2/%T!-.%"(#/:2L(:7!-; S "%-5#-.! /# -:-.%(! H !=># /% 1( 7!(5# (!824.:7# %-.?.:7# H 4 /!/!' 2! /:-.":F1:=># 7!2Z2:7!' 5#" *P[,M OU

(23)

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(24)

M (T, H) = P2J+1=6 n=1 mn(H) exp(−βEn) P2J+1=6 n=1 exp(−βEn) !"!#$ %&'( mn = − ∂En ∂Hn ( β = 1 kBT

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≈ 0.1% '%6 F( 6( :%87%?-+&'% :%8% % F(3+

%3 +.&'+A 83.-% 8+.6 7?%5E5(<A 23( + 636:(7-.=.<.'+'( '% :%87%6-% 4 -%-+<8(&-( .&'(7(&'(&-( '+ -(87?+-3?+ 8+6 23( +<,38+ 7(23(&+ .873?(/+ '( -(??+ ?+?+A &%-+'+8(&-( K'3+_{A 7%'(}

6(? ?(67%&6E5(< 7(<% :%87%?-+8(&-( '( =+.B+ T "

L3-?% 7%&-% .&-(?(66+&-(A 4 23( + 8+,&(-./+01% +J(-+'+ 7%? (J(.-%6 '( :+87% :?.6-+<.&%A 6+-3?+ (8 :+87%6 ?(<+-.5+8(&-( 8+.6 =+.B%6 23( + 8+,&(-./+01% &+ +36M&:.+ '(6-(6 (J(.-%6" L=6(?5( 23( &+ D,3?+ !"H+ 23( (8 =+.B+ -(87(?+-3?+ (J(.-%6 '( :+87% :?.6-+<.&% ?(<(5+&-(6$ + 8+,&(-./+01% NE :%8(0+ + 6+-3?+? 7%? 5%<-+ '( H = 5O"

L3-?+ ,?+&'(/+ 23( 7%'(8%6 :+<:3<+? '.?(-+8(&-( + 7+?-.? '%6 &;5(.6 '( (&(?,.+ '+ D,3?+ !"P 4 % :+<%? (67(:;D:%" ) :%87+?+01% :%8 % (B7(?.8(&-%A &% (&-+&-%A &1% 4 '.?(-+A 7%.6 &38 6G<.'% -(?(8%6 5E?.+6 :%&-?.=3.0Q(6 7+?+ % :+<%? (67(:;D:% -%-+< 23( 4 ,(?+<8(&-( %=-.'% (8 (B7(?.8(&-%6 RIPS" C% (&-+&-%A 7?%:('(? :%8 % :E<:3<% 4 =+6-+&-( .&6-?3-.5% 7+?+ -(?8%6 38+ ?(J(?M&:.+ 7+?+ %6 (J(.-%6 '( :+87% :?.6-+<.&% &(6-+ ,?+&'(/+ ( +.&'+ '.6:3-.? %6 ?(63<-+'%6 '+ D,3?+ !"#" L :+<%? (67(:;D:% 4 %=-.'% '.?(-+8(&-( '+ 6(,3.&-( (B7?(661%T CV(T ) = ∂ ∂T( P2J+1=6 n=1 Enexp(−βEn) P2J+1=6 n=1 exp(−βEn) ) !"!P$

L :E<:3<% 7%'( 6(? J(.-% &+ 7?(6(&0+ %3 +36M&:.+ '( (J(.-% U((8+&A % 23( .&-?%'3/ 38+ '(7(&'M&:.+ (8 J3&01% '( H 7+?+ CV" V(+<./+8%6 % :E<:3<% ?(&%?8+<./+&'% %6 &;5(.6 '(

(25)

3+

h M (2K, H) M (12K, H)

(26)

! "#$% &% '#("% )*+ & ,%! $"% -( . /0%&. 1(!&%, !0%2 / ,3" 0 "4 ! "#$% E0 = 0* 5

'#("% )*6 ,./0"% ./ " /(20%&./ & /0 7427(2.8 3%"% 7%,3. ,%#!90$7. H = 0 :'# )*6%; 3%"% %2#(!/ <%2." / & 7%,3. %32$7%&. :'# )*6=;*

5 '#("% )*6% ,./0"% 7., 72%" >% % %!.,%2$% [email protected] & <$&. % 3" / !C% &. /0%&. DE 7$0%&. & 7%,3. 7"$/0%2$!.* 5 %!.,%2$% 9 =%/0%!0 2%"#%8 <$& !7$%!&. % #"%!& / 3%"%CF. & /0 /0%&. D7$0%&. &. /0%&. 1(!&%, !0%2* 5 '#("% )*6= %3" / !0% . 7%2." /3 7G'7. & =%$D% 0 ,3 "%0("% 7., 7%,3. %32$7%&. :H = 18 38 5 7 H;* 5 %32$7%CF. & /0 / 7%,3./ 3".,.< (,% / 3%"%CF. &. -(%"0 0. (,% !.<% %!.,%2$% [email protected] / & / !<.2< * I.0 -( . 3$7. &% %!.,%2$% / & /2.7% 3%"% ,%$/ %20% 0 ,3 "%0("% 7., . %(, !0. &. 7%,3.* J!0 " //%!0 !.0%" -( 3%"% (, 7%,3. & 5 H8 0%2 7.,. %3" / !0%&. !% '#("% )*K8 ./ 1 $0./ L4 /F. =%/0%!0 <$/G< $/* M/0% 9 ,%$/ (,% <$&N!7$% -( 8 1 0$<%, !0 8 % <%2N!7$% &. O , O P 4Q12 9 0 0"%<%2 !0 * 5 3%"0 & %20% 0 ,3 "%0("% &. 7427(2. 9 =%/0%!0 /$,$2%" R '#)*6%8

,./0"% %3 !%/ % 2%"#% %!%.,%2$% & [email protected] 3." 1 $0. &. 7%,3. 7"$/0%2$!.*

!"!# $%&'() *+,)-%.,/( 0 123!

I% S20$,% / CF.8 <$,./ % 3".1(!&% $!T(N!7$% &./ 1 $0./ & 7%,3. 7"$/0%2$!. 3%"% &(%/ $,3."0%!0 / #"%!& >%/ 1G/$7%/* I.//% %!42$/ 1.$ 3.//G< 2 3.$/ 1.,./ 7%3%> / & & 0 ",$!%" . /0%&. 1(!&%, !0%2 &. /$/0 ,%* I /0% / CF. <%,./ &$/7(0$" 7.,. -( D3 "$, !0./ & M?U 7.!0"$=( , 3%"% % & 0 ",$!%CF. &. /0%&. 1(!&%, !0%2*

Q%"% !0 !& ",./ , 2@." /0 3.!0.8 !.0 ,./ -( (, D3 "$, !0. & M?U 7.!/$/0 =%/$7%, !0 !% %32$7%CF. & (, 7%,3. H 7.!/0%!0 :H = H0~r;8 3 "3 !&$7(2%" % (, 7%,3.

& ,$7"..!&% -( .3 "% !(,% 7 "0% 1" -(N!7$% 'D% ν8 R (, /$/0 ,% & /3$! 2 0"V!$7. , (, /0%&. |ψi* ? . D3 "$, !0. 9 1 $0. % 0 ,3 "%0("%/ /('7$ !0 , !0 =%$D%/8 |ψi / "4 . /0%&. 1(!&%, !0%2 & /0 /$/0 ,% .( 0 "4 %3 !%/ (,% 3 -( !% 7.!0"$=($CF. & /0%&./ D7$0%&./*

5 %32$7%CF. &. 7%,3. H 3.& -( ="%" % & # ! " /7N!7$% & /0 /0%&. 3".,.< " 0"%!E /$CW / !0" ./ !G< $/ & ! "#$% " /(20%!0 / & /0% -( ="%* M/0% -( ="% / ,3" .7."" -(%!&. /0%&. |ψi /04 %//.7$%&. % (, G.! X"%, "* Q%"% =%$D. <%2." & H08 % / 3%"%CF. !0"

/0 / !G< $/ 9 !7.!0"%&% 3." 3"$, $"% ."& , , 0 ."$% & 3 "0(=%CF. 3%"% . /0%&. |ψi & 3 !& "4 2$! %", !0 & H0* U 0.,%!&. !.//. D ,32. 3%"% . O 3+ , /$, 0"$% 7S=$7%8

3 !/ ,./8 3." /$,32$7$&%& 8 -( . /0%&. 1(!&%, !0%2 9 . &(=2 0. Γ7 -)*))* O.,. /%= ,./8

/0 /0%&. /0%"4 =%/0%!0 %1%/0%&. &. 3"$, $". /0%&. D7$0%&.8 & ,%! $"% -( 3.& ,./ /0(&%" % $!0 "%CF. Y ,%! 7.!/$& "%!&. %3 !%/ . /0%&. 1(!&%, !0%2* Q%"% . &(=2 0. Γ7

0 ,./ % / #($!0 ,%0"$>Z

(27)

h 3+ H = 0

(28)

−gJµBH(5₂(−0.40828)2− 3₂(0.912871)2) 0

0 gJµBH(5₂(−0.40828)2−3₂(0.912871)2)

! !"!#$ %&'() *)+&,-). ) /,0&-&+1) /& &+&-2,) &+(-& 3' /3,' &'()/3' 45& 03-*)* 3 /567&(3 /3 &'()/3 05+/)*&+()7 8 /)/) 93-: ∆E = 2gJµBH(− 5 2(−0.40828) 2₊ 3 2(0.912871) 2₎ . = gef fµBH !"!;$

<&-=&6) 45& gef f /&>+,/3 )=,*) gef f = 2gJ(−5₂(−0.40828)2 + 3₂(0.912871)2) ≈ 1.428$

/&9&+/& 05+/)*&+()7*&+(& /3 &'()/3 Γ7 )(-)?8' /& '&5' =3&>=,&+(&'" @* 5* &A9&-,*&+(3

/& @BC 93/&*3' 0)=,7*&+(& 36(&- 3 ?)73- /& g 9&73 =)*93 /& -&''3+D+=,). 93,':

g = hf

µBHRes !"!E$

F+/& HRes8 3 ?)73- /3 =)*93 /& -&''3+D+=,)" G*) =3*9)-)1H3 /3' ?)73-&' /)' &45)1I&'

!"!; & !"!E 93/& '&7&=,3+)- 3 Γ7 =3*3 3 &'()/3 05+/)*&+()7 /3 J&3+ &* ',*&(-,) =K6,=)" %&

5*) *)+&,-) 2&-)7.

gef f = |2gJhΓi|Sz|Γii| !"LM$

8 5*) &A9-&''H3 ?N7,/) 9)-) 3 =N7=573 /3 ?)73- /& gef f 45)+/3 3 &'()/3 05+/)*&+()7

/3 ','(&*) 8 5* /567&(3. ',(5)1H3 /& *5,(3' O3+' /& (&--) -)-) &* ',*&(-,) =K6,=)" @* &'9&=,)7. 8 ) ',(5)1H3 /) *),3- 9)-(& /3' O3+' P-)*&-'" Q3 =)9O(573 2. )9-&'&+()*3' ) +3'') =3+(-,65,1H3 9)-) 3 &'(5/3 /3 &'()/3 05+/)*&+()7 /& ,*95-&R)' /& (&--)' -)-)' &* !"##$%"&'#$ 6&* =3*3 +) /&(&-*,+)1H3 /3' 9)-D*&-3' /& =)*93 =-,'()7,+3. 45& /&9&+/& /3 =3+S&=,*&+(3 /3' *&=)+,'*3' /& -&7)A)1H3 /3 ','(&*) & /3 &'(5/3 /) /&9&+/T+=,) /3 &'9&=(-3 =3* ) (&*9&-)(5-)"

!" !""#$%& '()*+,

U,*3' 45&. &* !"##$%"&'#$ 9-&&+=S,/)'. 3' O3+' +3 ,+(&-,3- /) &'(-5(5-) /) 2),37) &'(H3 '5V&,(3' ) 5* 93(&+=,)7 7,2)+(& 0-)=3 45&. '593'()*&+(&. (&* =)-)=(&-O'(,=)' /& 5* 93(&+=,)7 S)-*W+,=3" @'(& 93(&+=,)7 /N 3-,2&* ) 5*) /,+D*,=) /& ?,6-)1H3 /3' O3+' 45& 8 =3+S&=,/) =3*3 %(##)'*+ ,-&$ . Q&'() '&1H3 ?)*3' )9-305+/)- ) /,'=5''H3 /&'(& =3+=&,(3"

(29)

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

I (0!#*,-% &# >"%$%! #$ A+(%1+! &# !J/""#*/&("#! $%!"*%/3!# $/("% )#$ !/.#&(&+ #$ 2*%$%'#* "+1 .#0>*(%: I '()*+,-% &#!"#! >"%$%! 5= !/2%!"+$#0"#= 2%/.% .%**#1+.(%0+&+ .%$ + 2+*"# #1#"*K0(.+ # 2*%$%'# + 4%*$+,-% &# $%&%! 0-% &(!2#*!('%! # 1%.+1(E+&%!= $/("% #?.(#0"#! +% #!2+16+* 4H0%0! &+ *#&#: L $%&#1% $+(! !($21#! 2+*+ "+(! 4H0%0! 5 /$ $%&#1% &# (0!"#(0= /$ $%&#1% %0&# %! 4H0%0! !-% 0-% &(!2#*!('%! # 0-% (0"#*+A#0"#!: ;#!"# $%&#1%= + &#0!(&+&# M0%*$+1(E+&+N &# 4H0%0! 5 !($21#!$#0"#O

g(ω) = 3N δ(ω − ωE) MP:8PN

%0&# N 5 0Q$#*% "%"+1 &# >"%$%! '()*+0&% #$ /$ Q0(.+ 4*#D/@0.(+ ω: L #4#("% &# "+(! $%&%! 0+ .%0&/"('(&+&# "5*$(.+ 5 +"5 6%G# #0"#0&(&% +2#0+! !%) /$ 2%0"% &# '(!"+ D/+1("+"('%: I 2*#!#0,+ &#!"#! $%&%! 1%.+1(E+&%! &($(0/( % 1('*# .+$(06% $5&(% &%! 4H0%0! &# F#)R# #= 2%* .%0!#D/@0.(+= &($(0/( % C/<% &# .+1%* &% !H1(&%: S+)#3!# #<2#*($#0"+1$#0"# D/#= #$ +1A/0! !(!"#$+!= % #4#("% 5 2*%2%*.(%0+1 + α−1

3= %0&# α 5 + .%0.#0"*+,-% &%! >"%$%!

D/# 2*##0.6#$ +! A+(%1+! 789 MD/# '+*(+ #0"*# 0 # 1N: L #4#("% &# "+(! $%&%! 0% .+1%* #!2#.T?.% &%! !(!"#$+! 2*##0.6(&%!= 0% #0"+0"%= 5 4+.(1$#0"# &#"#*$(0+&% /$+ '#E D/# + #D P:8P 4+.(1("+ #0%*$#$#0"# % .>1./1% &+ (0"#A*+1 2+*+ % .+1%* #!2#.T?.% &# /$ !H1(&%O

C = ∂ ∂T ˆ ∞ 0 g(ω) ~ω exp(β~ω) − 1dω MP:88N

L *#!/1"+&% &# P:88 5 % .+1%* #!2#.T?.% CE &# /$ !H1(&% &# (0!"#(0O

CE = 3N kB exp(θE T )( θE T )2 (exp(θE T ) − 1) MP:8UN L 1($("# &# )+(<+! "#$2#*+"/*+! &+ #<2*#!!-% P:8U 5 )+!"+0"# &(4#*#0"# &+ /!/+1 &# F#)R# MC ∝ βT3_{N 7PV9: F#!"+ $+0#(*+= + .%$2+*+,-% &% .+1%* #!2#.T?.% &# !(!"#$+! 2*##0.6(3}

&%! # 0-% 2*##0.6(&%! 2%&# !#* /!+&+ +"5 2+*+ &#"#*$(0+* % 2+*W$#"*% θE D/# .+*+."#*(E+

(30)

! "#$%& % !' $!(!' (% )*&' %*&+

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5!. %A%$5#!6 3 (%./*%"* % '% (D &7$" H.%<7O&-*" -"."- %.=' *-" ωEinstein ≈ 3.12 × 1012 s−1

FθEinstein = 150 WG+ )' " H.%<7O&-*" 3 -%.-" (% 1000 2%/%' $"*!. <7% " H.%<7O&-*" (% )LM %

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(31)

!"! !""#$%& '()*+ # $%&' '("$(%!# %!,,(-$%&

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ω) = T r ˆ +∞ −∞ µ(t) exp−iωtdt J/90LK I' ,% 5%+'($ "') %F2+%$$>( 2)+) ( ,#2(3( ,% +),#)G>( %' ;"-G>( ,( 5%'2(@ 2+%4#$)'($ %$5",)+ ) M)'#35(-#)-) )2+(2+#),) 2)+) ( $#$5%')9 A) +%2+%$%-5)G>( ,% M%#$%-*%+&@ %$5) M)'#35(-#)-) ,D ) %<(3"G>( 5%'2(+)3 ,( ,#2(3(9 6' &%+)3@ %$5) M)'#35(-#)-) B ,#<#,) %' N 2)+5%$ !"% ->( -%4%$$)+#)'%-5% 4('"5)' %-5+% $#9 O('%'($ H = H0+ Hp+ Hm J/90PK 6' /90P H0 B ) M)'#35(-#)-) ->( 2%+5"+*),)9 6$5) M)'#35(-#)-) ,D (+#&%' Q$ 3#-E)$ (*$%+<),)$@ 2(#$ B %3) !"% ,%$4+%<% ($ ,#;%+%-5%$ -=<%#$ ,% %-%+&#) ,( $#$5%')9 $%&"-,) M)'#5(-#)-) B "') 2%+5"*)G>(@ "$")3'%-5% ) #-5%+)G>( ,#2(3)+@ !"% 2+(<(4) "') )3)+&)R '%-5( ,)$ 3#-E)$ ,% +%$$(-S-4#)9 T35#') M)'#35(-#)-) B )!"%3) !"% 4)++%&) ($ %;%#5($ !"% %$5",)'($ -%$5) $%G>(9 6$5) M)'#35(-#)-) 4('"5) 4(' H0 % 4(' µ %@ 2(+5)-5(@ ) 2+#(+#@ ->( 4)"$) %;%#5( )3&"' $(*+% ( %$2%45+(9 7%" %;%#5(@ -( %-5)-5(@ B #-,#+%5(@ 2(#$ %3) ->( 4('"5) 4(' Hp9 ( '(,#I4)+ Hp J#-5%+)G>( ,#2(3)+K@ %$5) M)'#35(-#)-) 5%' "' 2+(;"-,( %;%#5( $(*+% )$ 3#-E)$ ,( %$2%45+(9 6$5) M)'#35(-#)-) ,%$4+%<% ($ '(<#'%-5($ ,($ $2#-$ 3(4)3#U),($ %' +%3)GH%$ )$ 4((+,%-),)$ ,( %$2)G( +%)39 V') ("5+) E#2C5%$% 4(' ) !")3 5+)*)3E)'($ B !"% ($ %3%'%-5($ ,% Hp $>( 2%!"%-($ ( $"I4#%-5% ,% '(,( ) ->( 4)"$)+%' 4(-%F>( %-5+% ($ ,#;%+%-5%$ -=<%#$ ,% %-%+&#) ,% H09 63#'#-)-,( ($ %3%'%-5($ ->( ,#)&(-)#$ ,% Hp I4)'($ 4(' "') %F2+%$$>( *)$5)-5% $#'23%$ 2)+) ( ,#2(3( +),#)-5% %' ;"-G>( ,( 5%'2(W uij′(t) = u0_ijexp i ˆ t 0 ∆ωij(t′)dt′ J/90XK (-,% ,%I-#'($W 0N

(32)

∆ωij(t) =

Hp(t)ii− Hp(t)jj

~

! "#$%&'! ()*+ ,-! -'!$. /&0 1 !203!4 !5$!614 70 40& 0"085. 3! 9!%8'5.38!3! Hp /&0 .

%.68%035. 7.4 :5.%.4 -!&4! !';&% 0"085. 4.<$0 . %.%035. 8$$!78!7.) = 06.'&>?. 50%2.$!' 70 Hp 1 7!7! 2.$

i~ ˙Hp = [H0, Hp] + [Hm, Hp] @()*AB

C!%.4 !/&8 4&2.$ /&0 . 0"085. 7. %.68%035. 7.4 :5.%.4 0% Hp 1 !'0!5#$8.) .44. 2.35.

4&45035!D40 !. 60$8,-!$%.4 /&0 3! %!8.$8! 7.4 -!4.4E .4 0"085.4 %!;3158-.4 3. %.68%035. 7.4 :5.%.4 1 20/&03.E 70 %!308$! /&0E 7. 2.35. 70 6845! 7!4 /&!3587!70 %!;3158-!4E . %.68%035. 7.4 :5.%.4 1 704-.$$0'!-8.3!7.) = /&!3587!70 ∆ωij 3?. 1 7!7! !3!'858-!%0350)

F$!5!D40 70 &%! "&3>?. /&0 40 -.345$#8 -.% <!40 0% !';&34 !$;&%035.4 70 2'!&4!<8'87!70) G!4 20$-0<! /&0 1 045! "&3>?. /&0 -!$$0;! .4 0"085.4 70 Hp 0E -.340/&0350%0350E .4 0"085.

70 Hm 4.<$0 Hp)

= 4&<4585&8>?. 7! "#$%&'! ()*+ 0% ()*H $04&'5! 3! 0I2$044?. /&0 /&0$0%.4 2!$! ! ".$%! 70 '83J!) Iij(ω) ∼ ˆ +∞ −∞ exp(−i(ω − ω(0)_ij )t + i ˆ t 0 ∆ωij(t ′ )dt′) 2

K45! 0I2$044?. 1 48%2'8,-!7! -.% ! 70,38>?. 70 7&!4 8%2.$5!3504 /&!3587!704)

ω = ω − ω_ij(0) η(t) = ˆ t 0 ∆ωij(t ′ )dt′

L&0 4?.E $0420-586!%0350E . 70468. 70 "$0/&M3-8! 7. -035$. 7! $044.3N3-8! 0 . 70468. 70 "!40 7. -035$. 7! $044.3N3-8!) .44! 0/&!>?. 5.%! ! ".$%!O I(ω) = ˆ +∞ −∞ exp(−iωt + iη(t)) 2 @()*PB = 0/&!>?. !-8%! 1 2$.35!%0350 5$!34".$%!7!E 3&%! ".$%! /&0 ! 708I! 0% 50$%.4 70 &%! Q"&3>?. 70 -.$$0'!>?.R

ϕ(τ ) =ei(η(t+τ )−η(t)) =448% ,-!%.4 -.% ! 0/&!>?.O