Coherent current states in mesoscopic four-terminal Josephson junction

A theory is offered for the ballistic 4-terminal Josephson junction. The studied system consists of a mesoscopic two-dimensional normal rectangular layer which is attached on each side to the bulk superconducting banks (terminals). A relation is obtained between the currents through the different te...

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Datum:1999
Hauptverfasser: Omelyanchouk, A.N., Zareyan, M.
Format: Artikel
Sprache:English
Veröffentlicht: Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 1999
Schriftenreihe:Физика низких температур
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Online Zugang:http://dspace.nbuv.gov.ua/handle/123456789/136224
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Назва журналу:Digital Library of Periodicals of National Academy of Sciences of Ukraine
Zitieren:Coherent current states in mesoscopic four-terminal Josephson junction / M. Zareyan, A.N. Omelyanchouk // Физика низких температур. — 1999. — Т. 25, № 3. — С. 240-248. — Бібліогр.: 14 назв. — англ.

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spelling irk-123456789-1362242018-06-17T03:08:57Z Coherent current states in mesoscopic four-terminal Josephson junction Omelyanchouk, A.N. Zareyan, M. Свеpхпpоводимость, в том числе высокотемпеpатуpная A theory is offered for the ballistic 4-terminal Josephson junction. The studied system consists of a mesoscopic two-dimensional normal rectangular layer which is attached on each side to the bulk superconducting banks (terminals). A relation is obtained between the currents through the different terminals, that is valid for arbitrary temperatures and junction sizes. The nonlocal coupling of the supercurrents leads to a new effect, specific for the mesoscopic weak link between two superconducting rings; an applied magnetic flux through one of the rings produces a magnetic flux in the other ring even in the absence of an external flux through the other one. The phase dependent distributions of the local density of Andreev states, of the supercurrents and of the induced order parameter are obtained. The "interference pattern" for the anomalous average inside the two-dimensional region can be regulated by the applied magnetic fluxes or the transport currents. For some values of the phase differences between the terminals, the current vortex state and the two-dimensional phase slip center appear. 1999 Article Coherent current states in mesoscopic four-terminal Josephson junction / M. Zareyan, A.N. Omelyanchouk // Физика низких температур. — 1999. — Т. 25, № 3. — С. 240-248. — Бібліогр.: 14 назв. — англ. 0132-6414 http://dspace.nbuv.gov.ua/handle/123456789/136224 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
language English
topic Свеpхпpоводимость, в том числе высокотемпеpатуpная
Свеpхпpоводимость, в том числе высокотемпеpатуpная
spellingShingle Свеpхпpоводимость, в том числе высокотемпеpатуpная
Свеpхпpоводимость, в том числе высокотемпеpатуpная
Omelyanchouk, A.N.
Zareyan, M.
Coherent current states in mesoscopic four-terminal Josephson junction
Физика низких температур
description A theory is offered for the ballistic 4-terminal Josephson junction. The studied system consists of a mesoscopic two-dimensional normal rectangular layer which is attached on each side to the bulk superconducting banks (terminals). A relation is obtained between the currents through the different terminals, that is valid for arbitrary temperatures and junction sizes. The nonlocal coupling of the supercurrents leads to a new effect, specific for the mesoscopic weak link between two superconducting rings; an applied magnetic flux through one of the rings produces a magnetic flux in the other ring even in the absence of an external flux through the other one. The phase dependent distributions of the local density of Andreev states, of the supercurrents and of the induced order parameter are obtained. The "interference pattern" for the anomalous average inside the two-dimensional region can be regulated by the applied magnetic fluxes or the transport currents. For some values of the phase differences between the terminals, the current vortex state and the two-dimensional phase slip center appear.
format Article
author Omelyanchouk, A.N.
Zareyan, M.
author_facet Omelyanchouk, A.N.
Zareyan, M.
author_sort Omelyanchouk, A.N.
title Coherent current states in mesoscopic four-terminal Josephson junction
title_short Coherent current states in mesoscopic four-terminal Josephson junction
title_full Coherent current states in mesoscopic four-terminal Josephson junction
title_fullStr Coherent current states in mesoscopic four-terminal Josephson junction
title_full_unstemmed Coherent current states in mesoscopic four-terminal Josephson junction
title_sort coherent current states in mesoscopic four-terminal josephson junction
publisher Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate 1999
topic_facet Свеpхпpоводимость, в том числе высокотемпеpатуpная
url http://dspace.nbuv.gov.ua/handle/123456789/136224
citation_txt Coherent current states in mesoscopic four-terminal Josephson junction / M. Zareyan, A.N. Omelyanchouk // Физика низких температур. — 1999. — Т. 25, № 3. — С. 240-248. — Бібліогр.: 14 назв. — англ.
series Физика низких температур
work_keys_str_mv AT omelyanchoukan coherentcurrentstatesinmesoscopicfourterminaljosephsonjunction
AT zareyanm coherentcurrentstatesinmesoscopicfourterminaljosephsonjunction
first_indexed 2025-07-10T00:53:39Z
last_indexed 2025-07-10T00:53:39Z
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F <�13@�>�.�2 ��4"C�2�2�>+5�W¢<�-"@ 1Y8 W{-R<�- . 4 1">+5 < ρρ ��CD@I.RF�1 5�}I>+5�.Ab�f�2a.�. ®�>�W�m�rc�� w£;mHd��BhYe�>M5 �K4"4 .�5�b">Q�§lim§,�- .�bG>+CiW"1�5GC :oC�5'bH1"}I}Ijvb">+CiW�1 5�CD:w< .A@ 6P291 } 5 6 � →­ ( U ) -'C ^ .P<�- .�}v13@�6P2 < � → ­ = ω Ω + ∆ ¶ - � →­ = ω FD1�2�- (ω + » À + ³ ϕ­¡� - r ) + Ω 2�>v5"- (ω U ­�� + ³ ϕ­�� - r ) Ω ��� Á . (ω U ­¡� + 0 ϕ­�� - r ) + ω 2�>v5"- (ω + ­�� + ³ ϕ­�� - r ) � d h_�Ge = � → ­ =    ∆ ¶ Ω + (Ω − ω)- � →­   .Y�'4yf − r ω( U − U � ) + ³ ϕ � l = ∆¶ .Y��4 (ω U ­�� + 0 (ϕ À + ϕ­ )- r ) Ω F 1"2�- (ω + » À + 0 ϕ» À ½ � ) + ω 2J>�5'- (ω U ­�� + ³ ϕ­�� - r ) . − · ω( � − � F ) m d h3h�e � 5V<J-G.�:+>+6P>�<2%0�D'���� ξ ¶ <J- .�. ��4"@ .�2�2�>�135'27d hS�'e CD5"bOd hDhYeH}v1�@ �z@a.�.�5��°2|}v8�5"F <J>Q135'2~C�@ .©2 >+6B4':Q>M}I>�.Ab�C�5�b N0.�-"C ^". < � → ­ = ωΩ + ì ⁄ � ³ ∆ ¶· 2�>�5 (ϕ­�� ) ω · + ∆ �· F�1"2 · (ϕ­�� - � ) � d h�r'e = � → ­ = ∆ ¶ Ω F 132 (ϕ » À - r ) + ³ ω 2�>�5 (ϕ » À - � ) . � (ϕ� + ϕM ) �"· m d h�kce�O.\FAC�5 1 ["<JC�>M5 <�-".\@ .�<�C�@Jb�.Ab C�5�b C�bG^�C�5GF .Ab�0@I. .A5��?2�}v8'5"F <J>�1"5"2)� 5 6 E î � (ε) ��["x�C�5'CD:=x <J>QFACD:0F 135G<J>+5�8�j C)<J>�1"5¡1"} ?�C <J2 8"[�CD@�C ��@ . .A53�?2�}v8�5 F <J>�1"5 5 6 (ω) d £"2)m dS 'e�ª;d hJkce�eDm�,�- .±4G1":�.�2K1 }�b�>MCiW�1"5"CD:�F 1�6B4 1 5".�5 <K1"}s<�- . @ . <JC�@�b .�b ��@I. .A5��?2�}v8c5 F <J>�1 5�� < E (ε � ρρ : � + ) � bG.�< .A@ 6P>M5 . <�-".y.�5 .A@aW�>�.�201 }B:�1YF�CD:��K5Gb�@a.�. ^g2 <JC <�.�2B>+5q<�- .H2Ix"2 <�.�6ym ,�-G.�:=1DFACD:PbG.A5G2J>�< x�1"}�2�<�C <�.�2y>+5�<�- .g5 1"@�6BCD:�:+C x3.�@y>+2 W{>�^".�5y["xy<�- .±}v1 @�608c:+C � (ε � ρρ) = � ( � ) 〈 �w. < (ω = − ³ ε � ρρ � 4 + ) 〉 m d h�uce ��2�>+5DW�<�-".H. �c4"@I.�2�2�>�1"5"2¨dS �e)�9d h r'e�CD5"b�<�-".�}IC F <~<�-"C < θ � ­ (ρρ) = θ­�� (ρρ) >+5R<J-G.�F�C�2�.�1�}72�6BCD:v:�E�8�5GF�<�>�135 �PN9. 1 ["<�C�>+5 � (ε � ρρ � * ϕ � 1 ) = � ( � ) ∑� ≠ ­ 〈 �w. < (ω = − ³ ε � ρρ : 4 + ) 〉θ� = = � ( � )∑� ≠ ­ θ � ­ (ρρ) �w. < � → ­ (ω = − ³ ε) = = � ( � )∑� < ­ θ À » (ρρ) �K.Bf < � → ­ (ω = − ³ ε) + < ­ → � (ω = − ³ ε) l = = π∆ ¶ � ( ; )∑� < » θ � ­ (ρρ)  2�>+5 ϕ » À r  δ ( ε � − ∆ ¶ ��� Á ϕ­��� ) m d h��'e �R.¢F�CY5�CD:+2I1¥8c2I.1 �£"2)m©dS 'eXC�5'b dYh�r'eO< 1¥13[G<JC�>+5 〈 � + < 〉 C <gCO4 13>+5G<q13}�<J-�.�³v<J-¡2�>Mb". ρρ � mP,�-".�5;�P<�- . @�.�2I8Z:=<�>+5 Wg. �c4"@ .A2 2J>�1"5�FAC�5�[".7@I.�4c:+C F�.�b�>+5 w£;m�d�uce�<�1 }I>+5"b|<J- .9F 8{@ @ .A5 <�bG.A5"2�>�<�x C (ρρ � ) m�,�-".9F�CD:=F 8c:+C <J>�1"5�1�} <J- . F�8�@�@a.A5G<Kb".�5"2�>�< x�C�<�<�-".BC�@�['>�<J@ CY@IxH4 1">�5G<�1�}¤<J-�.±5 1"@ 6PCD: @�. F <JC�5DW�8Z:QC�@�NP>v:v:KF�1"69.q>+5�<�- .�5G.��G<�2I. F <J>�1 5�C�5�bL<J- . �K4"4 .�5�b">Q�wm2 B.A@a.gN0.�FACD:�F�8Z:QC)<�.g<�- .g< 1 <JCD:PF 8�@�@I.�5G<yµ �} :=1 N0>+5�W¨>+5G<�1|<J- . ³ <�-y[�CD5"U;m�{. <98�292 <JC�@I<9NP>�<�-H<�-".BFACD2I.�1 }KC�2�6PCD:v:;E�8{5 F <J>Q135�da%0� '� � ξ ¶ eDmc��5|13@�b .�@�< 1~}I>+5"b�µ � ��NT.P-"C ^".0< 1�FAC�:=F 8c:+C <�. <J-G.©>+5G<�.iW�@�C�:Kµ � = ∫ ( � F ) C (ρρ � ) ⋅ � � � �KN0-G.A@a.q<J-G.©>+5�<�.iW{@�CD:P>+2 <JC�UG.A5y1D^".�@�<J-�.±³M<J-H2�>�bG.01"}o@ . F�<�C�5 W"8c:+C�@)m �K}v<�.�@BFACD:�F�8Z:QC)<J>�1"5g1 }eC (ρρ � ) }I@a1"6­d��skce~C�5'b�d��K�'e~C�5'b <JC�U'>=5�WH<J- .�>�5"< .�W�@�CD:�1 ^ .A@ � � � ��N0.T13[G<JC�>+5 µ � = ¹ /#+ ∆ ¶ �r π ∑­ = � ô γ � ­ 2�>M5    ϕ � − ϕ » r    × × <JCY5G-    ∆����� Á�� (ϕ � − ϕ­ ) - r'l r ¸    � d h �Ge NP- .�@�. � = √� � + ' � C�5'b γ � ­ = γ­�� A γ ìQí = h − 4 √h + 4 � A γ · ô = h − h √h + 4 � A γ ìI· = γ ìUô = γ ·�í = γ í�ô = hr    ¼ + 4 √ h + 4 � − h    � d hi�Ge C�@�.LW�.�1 60. <J@ >�F�CD:¨}v1 @J6 }IC F�< 1�@ 2q<�-�C <gbG.A4�.A5'bR1�5¦<J- . NP>+b <�-�jM<�1 jI:�.A5�W"<J-O@ C <J>�1 4 = ' ½ � m ,�-G.g4 1"2�>�<�>�^ .g2�> Wc5 1�}�µ � F�1 @J@a.A2�4 1 5�bG2q<�1O<�- .�b'>�@�. F <J>�1 5X1"}y<�-".�F 8{@ @ .A5 < }I@�1 6­<�-".q5 1"@�6BCD:�:+C x3.�@|<�1�<J-G.©³M<J-�[�C�5'U;m���1Y< .�<�-�C < ∑� = ì ô µ � = �Zm ,�-".¥}v13@�6T8c:QC�d h��Ge }v1�@VF�8'@J@a.A5G<Jjv4"-'C�2I.¥@ .Y:+C <J>�1 5�2 W�.�5G.A@ C :+>��i.A2 <J- .0. �c4"@I.�2�2�>�1"5�d hYe�<�1|<J- .0F�C�2 .91�}sCB2�6BCD:�: OQP , ¹SRi¹ST#UWVSXYRZRi¹ST#U\[SU ¯ U ¹�[ ³ T^]9¹_[ P [�V P / ³ V = P X�Ra`�U ¹"R8] ³ T ¯Yb�c P [�¹ /�, [ P Ted�X�TfV.U ³ P T gah ijh É ÆwÑ h i É h É�Õ3k�È�Óml ÈAË�Æ8n+×�ËvØ ÙIÚ�Ú�Ú Øpo�ÐjqSrDØ�Ñ"Ö�s qat"s 6T.A2 1 2�F 134G>�F u"j+< .A@�60>M5�CD:qE�8'5"F <J>�1 5�m���<X}v1':�:�1YNB2X}I@I1"6 d h��'e�<J-'C <�<J-�.�}v13@�6�}IC F�< 1"@ γ � ­ F�C�5�5�13<B['.¨}IC F <�1"@ >���.�b;� > m .Gm��§4"@ .�2I.�5"< .Abq>M57<J- .y}v13@�6 γ � ­ = γ � γ­ �§>+57F 1"5 <J@�C�2I<T<�1 <�-".|FAC�2 .~1"}9@I.D:+C�<J>�1"5�dDhYeD�§NP-�.A@ . γ � ­ = ( h - ¾ � ) ( h - ¾ ­ ) m,�-'>+2�.�2�2I.�5"<�>+CD: }v.�C <�8c@I.B1"}s<J-G.PF 8�@�@I.�5G<Jjv4"-"CY2a.±@ .D:QC <�>�135'2 @ .�} :=. F�<�29<J-�.~5"1 5c:�1YF�C :o5"C)<�8�@a.~1 }�<J-G.~2 8�4 .�@ F�8�@ @ .A5 <�20>+5 <�-".P6T.�2�1 2 F�1 4">�F�698�:�<J>�jM<�.�@�6P>�5'CD:�/Y1"2I.�4"-'2I1"5|E�8�5GF <J>�1"5;m ,�- .�F 8{@ @�.�5 <�j�4"-'C�2 .�@I.D:+C�<J>�1"5"2Ld h �GeSCD@I.�^'CD:Q>+b¡}v1"@ CD@I[�>�<�@�C�@IxX< .A6B4 .�@�C)<�8�@a. ¸ mP��5O<�-".p:+>+60>�<�>+5DW¢F�C�2I.�2713} ¸ = �pC�5"bV<�.�6P4 .�@�C)<�8�@a.gFD:=1"2a.g<�1 ¸ � <J-G.q.��c4"@ .A2 2�>?1"5 d h��'eK<�C�U .�2�<J-G.P}v1"@�6B2 µ � = ��� + ∆¶ �� ∑­ = � ô γ � ­ 2 >+5    ϕ À − ϕ­�    }v1"@ ¸ = �Z� d h��'e µ � = ��� + ∆¶· �u π ¸ � ∑» = ì ô γ À » 2J>M5 (ϕ � − ϕ­ ) }v1�@ ¸ −� º��Tm d h_ Ge �a5�<�- .HF�C�2I.H1 }PC�@ [�>�<�@�C�@ x�:�.�5DW"<J-'2&%0�e'¡�oN9.�@I.�2 <J@ >�F�< <J-G.�F 135'2�>Mb".�@�C <�>�1"5g}v1"@B<J- .y< .A6B4 .�@JC < 8�@ .yFY:�132 .y<�1 ¸ � m �a5�<J-'>�2yF�C�2�.qF 8�@ @�.�5 <�j�4'-"C�2 .g@I.D:QC)<J>�1"5"2H2 >+6P> :QC�@|<�1�<J- . .��c4"@I.�2�2�>�1"5|d ha �e¤FACD5B[".�1 ["<JC�>M5".�b;m3,�-".Kb'>+}I}v.�@I.�5"F .K>+2w>M5 W�. 1"60. <J@ >�F�CD:Z}v1 @�6X}IC F <�1"@�2)m��I5~}IC F <��"N9.0-"C ^".K<�-".9@I.�2 8c:=< µ � = ��� + ∆¶· �u π ¸ � ∑» = ì ô γ� � ­ ( 4 : % � ' ) 2 >�5 (ϕ � − ϕ­ ) � d�r �Ge NP- .�@�.T<J-G.9W�.�5 .A@�CD:+>���.Ab�}v1�@ 6_}IC F <�1 @�2KC�@ .9W�>�^".�5y["x γ � ô-ì = γ � ·-ì = γ � · í = γ � ô í = u π · √h + 4 · ∫ − ¾ �"· ¾ � · � 8 ∫ − ×�æ<ßUä<×zÞ    ¾· +    π � · � θ F 132 θ ∑ = ¶ ∞ .D��4Hf − % ( 4 - r + 8 )( r T + h )- (ξÉ F�1"2 θ) � ( r � + h ) · � γ � ô�· = γ � ·�ô ≡ u π · √h + 4 · ∫ − ¾ ��· ¾ � · ��� ∫ − ×zæQßUä�×�Þ    ¾· + Ç    ×�æ�ßUä<×�Þ    ¾ � −    � θ F 1�2 θ ∑� = ¶ ∞ . ��4 � −% ( rfT + h )- (ξ É F�1 2 θ) l ( � � + h ) · � d�rch�e γ � ìI· = γ � ìUô = γ � í�· = γ � í�ô = γ � ô�ì ( % → '¡��' → %T� 4 → h - 4 ) � γ � ìUí = γ � í-ì = γ � ô�· ( % → � : ' → % � 4 → h - 4 ) B.A@a. ξÉ = � � ½ � π ¸ m0�I5�<�-".�:Q>M6P>�<^%0�&'� �� ξ ¶ � γ � � ­@ .�b 8�F .0< 1 γ � ­ m �c��� '#�Y!i* �f��%�*���! #�*��"&"!i*Q$���$�� �i&/'��)#�(�&"# #��)�"!Z� �Y�G%*��"%"&�(f��%7$ #�% ��# '�� #j��� ��!8��# � 5�<J-">M2~2 .�F <J>Q135�N�.7NP>v:v:91Y['<JC�>+5�<�- .q2�8'4 .A@ F 8�@�@ .�5 <bG.A5G2J>Q<�x²C�5'b²<�- . >+5'b 8"F .Ab�1"@�bG.A@p4'C�@�CY6z.�< .A@OC <XC�5CD@I[�>�<�@�C�@Ixq4 13>+5G<913} <J-G.~5 1"@�6BCD:�:QC x".�@9>+5�<J-G.PF�CY2a.�13}KC2�6BCD:�:9E�8c5 F <J>�1 5�mK��<y<J-G.7W�>�^ .A5O4�1�>M5 <y1�}�<J-G.g5 1"@�6BCD::+C�x .�@ ρρ = � * + 8 C 〈 4 + < 〉 = ∑� > ­ ( 〈 4 + < 〉θ � + 〈 4 + < 〉θM�F ) = = ³ ∑� > ­ 〈 4 + 〉θ F M ∆ �� 2�>M5 ϕ­�� ω · + ∆ �· F�1 2 · (ϕ­�� - r ) � d�r r'e NP-G.�@I.\N0.`-"C)^ .\8�2 .Ab 〈 4 � < 〉θ F M = 〈 4 + 〉θ F M < � → » � 〈 4 � 〉θM�F = − 〈 4 + 〉θ F M C�5"b < ­ → � = < � → ­∗ m ,�-".tF�8�@�@a.A5G<obG.A5"2�>�<�x~>M2�1 ["<JC�>+5G.�bB["x~@I.�4':+C�F�>=5�Wyd�r3rGe >+5y<J-�.� §£ m�d�u'eD´ C (ρρ) = r π ¹&� ( � )∆ ¶ × × ∑� > ­ 〈 � + 〉θ F M 2 >+5 ϕ­��r ��� à .    ∆ � F�1 2 (ϕ­�� - r )� ¸    m d�rYk'e ,§-�.0.��c4"@a.A2�2�>Q1�5�d�rDk�e9b .�2 FA@ >�[".�2�<J-�.P2�4'C <J>+CD:�b">+2I<�@�>�["8�j <J>�1 5713}�<�- .�F 8�@�@I.�5"<0b .�5�2 >�<�x7>+5"2�>�b".B<J-G.±5G13@�6BCD:o:+C�x .�@ m �a5 1 @�b .A@L<�1_}I>+5Gb <J-G.O.��c4':+>�FA>Q<S. �c4"@I.�2�2�>�1"5 }v1�@L<J- . F�1 .�}I}I>?F�>�.�5"<�2 〈 4 + 〉θ F M >M5D £;m'd rDkceD�YN0.9-"C�^ .K< 10F 135'2�>+bG.A@ }v138'@�b">+}I}v.�@I.�5"<�@ .�W�>�1 5�2s>+5|<J- .B5 1"@�6BCD:Z@ . F�<�C�5 W"8c:+C�@�C�5'b 13[G<JC�>+5 C (ρρ) >�5V.AC F�-O@ .�W�>�1 5X2I.�4"CD@�C <�.D:�xXd�2 . .q�K4"4".�5"j b�>���eDmZ,�-'>+2tFACD:�F�8;:+C <J>�1 5�-'C�2t[�. .A57b 1"5 .~>+5�<J-G.B�K4"4".�5"j b�>��¢C�5'bO<�- .S@ .�2I8;:=<7}v13@ C (ρρ) >M2|W{>�^".�5p[�xRd��okceqC�5'b d��K�"eDm 0.A@ .9N0.9NP@ >�<�.& w£;mcd�rYk'eK>+5|<J-�.P691"@I.9<J@�C�5'2�4"CD@�j .A5G<K}v1"@�6¨m0�c. <K8�2t>+5 <J@ 1 b"8"F . θ 6 � ­ (ρρ) C�2t<J-�.P8�5G>�<K^ . F�< 1�@ >+5¥<�-".¡b�>+@a.�F <J>Q135�13}�<�- . ³ → � <�@�CiE�. F�< 1�@ax²4"CY2 2�>=5�W 3�¯Yb ¹5476 ¯ R�¹&8 ¯ T ¯ T �9-2: � :<; ]9¹ b 8 ¯ TfV , P X=4 q t t gah ijh É Æ¤Ñ h i É h É�Õ3k ÈAÓml ÈAË�Æ8n+×�ËvØ ÙIÚ�Ú Ú3Øpo�Ðjq rYØ�Ñ"Ö�s <�-�@a138�WG-7<�-".�[c>�2 .�F <�1 @91"} θ � ­ (ρρ) A <J-G.�5 〈 4 + 〉θ F M F�C�57[". NP@�>�@�@I.�5HCY2 〈 4 + 〉 θ À » = ∫¶ θ F M � θr π 4 + = � + π 2�>M5    θ � ­r    θ 6 � ­ m d�rDuce nw1"60[�>M5">+5DW w£"2 m�d rDkce9C�5�b7d�r)uceD��N0.B1Y["<�CY>�5 C (ρρ) = ¹ / � ∆ ¶r π ∑� < ­ 2�>�5    θ � ­r    θ 6 À » (ρρ) × × 2�>M5    ϕ » − ϕ À r    <�C�5�-    ∆ ¶ 2�>M5�f (ϕ­ − ϕ � ) - rGl r ¸    m d�r �'e ��5HCTNPC xH2�>+6B>M:QCD@s<�1|NP-"C�<�NT.P-"C ^".Pb�1�5�.P}v1"@\C (ρρ) ��<�- . b'>+2I<�@�>?['8"<�>�135�13}t<J-G.�>�5'b 8�F .�b71�@ b".�@04"CD@�C�6T.�< .A@TF�C�57[". 1 ["<�C�>+5".�b;mH��5 <J-'>+2�FAC�2 .ON0.X5 . .�b <�1¡FACD:�F 8�:+C <�.O<�- . C)^3.�@�CiW".­13} 13}I}Ijvb�>MCiW�1"5"CD:R.Y:�.A6T.�5 <¥13} <J- .�60C�<�@�>���0@I. .A5��?2K}v8�5 F <J>Q1�5§� = ω( 4 + � ρρ) �{>+5¨<J- .�b">+@ . F�<�>?1"5y1 } 4 + ´ 〈 � 〉 = ∑À ≠ ­ 〈 = 〉θ� = ∑� ≠ ­ θ � ­ (ρρ) = � → » m d�r �'e �s.A4':+C�F�>�5 Wqd�r ��e9>+57d�k'e0C�5"bHCD}v<�.�@�<J-�.PF�CD:=F 8c:+C <J>�1"5;�cN9. 1 ["<�C�>+5H}v1"@ ψ( � � 8 ) = ∆( � : 8 ) - λ ´ ψ(� � 8 ) = ∆ ¶ λ ∑� = ì � θ � ( � : � ) . À ϕ� m d�r �'e B.A@a. θ � (� � 8 ) >+29<�- .�C�53W�:=.�[Gx�N0-'>�FA-7³v<J-72�>+b .~>+292 . .A5 }I@ 136�<�-".H4 1">�5"< ρρ ≡ ( � � 8 ) m ,�-G.HC�5 Wc:�.A2 θ � ( � � 8 ) �°2�C�@a. 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]9¹ b 8 ¯ TfV , P X=4 q t�� gah ijh É Æ¤Ñ h i É h É�Õ3k ÈAÓml ÈAË�Æ8n+×�ËvØ ÙIÚ�Ú Ú3Øpo�Ðjq rYØ�Ñ"Ö�s θ ìa· = hr (θ ì + θ � − θ í − θô ) � θ ì¡í = h r (θ ì − θ � + θ í + θô ) � θ ·�í = hr (−θ ì + θ · + θí − θ ô ) � θ· ô = θ ô m d��Bh�e �9:+2 1~N9.PF�C�5y8�2 .0<�-".B@�.D:+C <J>�1 5 〈 4 + 〉θ F M = ∫ (θ F M ) � θr π � + ( � F31'2 θ + C 2 >+5 θ) < 1~1 ["<�C�>+5 〈 4 + 〉 θ ��� ( � �/8 ) = � � � π * Á�ÂIà f α(−� � − 8 ) l − 2 >�5|f α(� : 8 ) l�l � + f¤F�1"29f α(� �/8 ) l − F 1329f α(−� � − 8 ) lJl�C 1 � 〈4 + 〉θ ��� ( � �/8 ) = � +r π * 2 >+57f α(−� � 8 ) l − 2 >+57f α(−� � − 8 ) l�l * + f¤F�1"29f α(−� � 8 ) l + F�1 29f α(−� � − 8 ) l�l�C 1 � d��KrGe 〈 4 + 〉θ ��� ( � � 8 ) = � +r π * 2�>+57f α(� � − 8 ) � − 2 >+5 � α(−� � 8 ) l�l * + f¤F�1"29f α(� � − 8 ) � − FY1�2¤f α(−� � � ) l�l3C , � 〈 4 + 〉θ ��� (� � 8 ) = � +r π * − Á ÂIà f α(� � 8 ) � + 2�>+5 � α(� � − 8 ) � � * + f¤F�1 29f α(� � � ) l − F�1"29f α( � � − 8 ) l�l�C , � NP-G.�@I. α( � �/8 ) >+2zWG>Q^3.�5q['x £;msd�r� 'eDm ,�- .yF 13@�@I.�2�4 135'b">+5�Wg@�.D:+C <J>�1 5�2A�§^�CD:+>+bg}v1 @01D<J-G.A@B@I.iW�>�1"5"2)�§F�C�5q[".|1 ["<JC�>+5G.�b }I@ 136«d��Bh)e9C�5'b7d���r'eD��8c2�>M5DWH<J-G.PC�4�4G@I1"4"@�>�C�<�.B@�8Z:=.�2�1 }�>+5"bG.���C�5GbyF�1 1 @JbG>+5�C < .0. ��F�-"CY53W�.|d�2I. .B[".Y:�1YN|eDm �s.A4':+C�F�>�5 W §£G2�m;d��BhAe0C�5"b7d��Kr'e9>+57d�rDk{eD�'N0.01 ["<JC�>M5H}v1"@�F 8�@�@ .�5 <�b .�5'2�>�< x¨>M5HCP4G13>M5 <K1 }�@I.iW{>�1�5©��� C �U� ( � �/8 ) = f − � (� � 8 ) + � (� � 8 ) l�� ì�í + f − � (−� � − 8 ) − � (� �/8 ) l�� ìI· − − f � (� � 8 ) + � (−� � − 8 ) ��� ·Ìô + f � (−� : − 8 ) − � (� � 8 ) l � · í � d���k'e NP-G.�@I. � ( � � 8 ) = 2�>+5 α( � � 8 ) * − F 132 α( � � 8 ) CP� � ( � � 8 ) = 2�>�5 α(−� � 8 ) * + F 1�2 α(− � � � ) CB� d���u'e CD5"b � � ­ = ( ¹ � + ∆ ¶ - r π) 2�>+5 (ϕ­�� ½ r ) <JCY5G-�f ∆ ¶ FY1c2 (ϕ­�� - r ) - r ¸ lim�,�- .0F 8�@�@ .�5 <Kb .�5�2 >�<�x~>M5~1D<J-G.A@s@ .�W�>�1 5�2s>+2¤1 ["<JC�>+5G.�b }I@ 136 C �U� ["xHC�4"4':�x�>�5�W¨<�- .�}v1":v:=1 N0>+5�W¨@ 8c:�.A2 1�}s4"-'C�2 .PC�5�b¨F�1D1�@ b�>M5"C <�.B.��GFA-'C�5 W".G´ C � = C �z� (� → − 8 - 4 � 8 → � - 4 � 4 → h - 4 A * → − C � C → − * A ϕ ì → ϕ ô � ϕ · → ϕ ì � ϕ í → ϕ·�î ϕ � → ϕ í ) � C �U�U� = � �<� (� → 8 - 4c�/8 → −� - 4 : 4 → h - 4 A * → � �fC → − *pA ϕ � → ϕ · � ϕ · → ϕ í � ϕ í → ϕ ô�î ϕ ô → ϕ ì ) � C �/½ = C � � (� → −� � 8 → −� ��4 → 4 A ( * → − � � � → − C ) A ϕ ì → ϕ í � ϕ · → ϕ ô � ϕ í → ϕ ì�î ϕ ô → ϕ · ) m d��K�Ge ,�-G.�2�C�60.y@ .Y:+C <J>�1 5�2TCY2Bd��K�'eBFAC�5�[".~8'2�.�bg}v1 @ θ � ­ C�5"b 〈 4 � 〉 θ F M d�<J- .H4"-"CD2I.|.��'FA-GCY5DW".�2P-'C ^ .�< 1H['.�@I.�4Z:QC F .Ab�["x F 1"@�@ .A2 4"1 5�bG>+5DW�>+5"bG.���.��"F�-'C�5 W".�2�eDm OQP , ¹SRi¹ST#UWVSXYRZRi¹ST#U\[SU ¯ U ¹�[ ³ T^]9¹_[ P [�V P / ³ V = P X�Ra`�U ¹"R8] ³ T ¯Yb�c P [�¹ /�, [ P Ted�X�TfV.U ³ P T gah ijh É ÆwÑ h i É h É�Õ3k�È�Óml ÈAË�Æ8n+×�ËvØ ÙIÚ�Ú�Ú Øpo�ÐjqSrDØ�Ñ"Ö�s qat�� ì�å��"å��±å���Ø â�à ×�æ<ÜUç�î E���� ¼��,����¼ Å4ÆzÇ��¿¼���� î�ì�ô ��þ�ì�ï zï-ÿ�å·-å Uå�ê¶å��!á-Ù�Ø�â�î�ñ.å��¸å�ê�Ö¸Ü�٣ݠ×�Þ ß�àÌÛ á â�î�×zÞ/é�Ô�å�ñ�å �!Ü�Ù�� Ö.×�Þ�î + � ��¼É � ��¾ ¼�Â���Ã�Ä�¼���î�ì�ìUð þ�ì�ï zï-ÿ��M������¼���¼�À���ÁeÂ���Ã4Äg¼�Å4ÆÊÇ �¿¼ �-î�� �þ�ìUï� �ï�ÿ��¿åí-å�Ô�å L å � Û Ù�×�Þ�é ñ�å��¸å�ê�Ö¸Ü�Ù Ý/×�Þ�ßUà�Û�á�â�î�À���Á_Â���Ã�Ä-¼�Å4ÆÊÇ �¿¼G���-îð þ�ì�ï ï�ô-ÿ�åô-å�ñ¸å �f×�æ<Û ÞÌܶ×zÞ/é H(å ��×�ä<ÜUæ�Þ/Û-î&Å�ÆzÇÌ�¿�G���¸� � � ��Ä�ÄgËG���U�-�I�£� �z�¸�¡�¶�IÆ�� �-�z�Q��Ä�Æz�Q���"È��M�Q�U���Qî� Ø�Ù�Ü¡Ý&î���ÜUû��!Û�æ<â�þ�ì¡ï�ði·/ÿ�å �-å�è�å"é Ü���æ�á�Ý/Þ\ê�á�ë Û äMÜzæ�î.ñ¸å��.å�ê�Ö.Ü�Ù Ý/×zÞ/ßUà�Û�á/â�î.×�Þ�éaÔ!å L å � Û Ù<î Å4ÆzÇ��¿���U��� � ���-î�ì ��í�þ�ì�ï ï �-ÿJ�-Ø�ë�Ø�é�����!-î�· ¶ í þ�ì�ï ï� �ÿ�å ��å"�!å�õ-å � Ù�ÜUÜUÖ±Ø�ÞÌü�î�ñ�å � å #�×zâ/×zæQØ ×�Þ�î�ñ.å��¸åfê�Ö.Ü�٣ݠ×�ÞÌß�à Û áÌâ î�×zÞ/éè.å-é�Ü$�fæ<áÌÝ/Þ�ê�á�ë/Û�ä<ÜUæzî-Å4Æ�Ç/�¿�£�U� � �i�&%�î�· � í�þ�ìUï�ï �ÌÿJ� ' ���U��Æ�¼���¼Å�Æ�Ç � å�( %�îY� �s· ð�· í�þ�ì¡ï�ï�� ÿ�å �å���åÌõ�å � ÙGÜUÜUÖ±Ø Þ�ü�î�Å4Æ G �MÆ��U�¿� ��î���Ü�Ø é�ÜUÞ*).Þ-Ø ç ÜUæ<òQØ£äGÝùþ�ìUï�ï ð�ÿ�å ð-å±è.å.é/Ü+��æ�á Ý/Þ´ê�á ë�Û�ä�ÜUæ�î.ñ.å��¸å�ê�Ö.Ü�Ù Ý�×�Þ�ßUà/Û�á�â î.×zÞ/éaÔ�å L å � Û/Ù�î Àf�ÊÁ_Â���Ã�Äg¼)Å�ÆzÇ �¿¼ ��(-î, -�& þ�ìUï�ï ð�ÿJ� À��z���¸Ë�� Ã����.� �<��Æ����Ì��U� ���-©MÃ!�-�/.�ï ð ¶ �-ì0 zô�åï-å�õgåB��å21.Ü�Ø é�×/î3��å õ�å�ç ×�Þ �ÜUÜUò�îùø!å54,å5��Ù�×zö/û.Ø ÷<â î�×�Þ éH±å���Û�æ<üÌà/ò�îgÅ4ÆzÇ��¿¼.E¸����¼ � ����î�è ���-ì�ð þ�ìUï�ï ð�ÿ�åì ¶ å§��å���å���×zü û�Ü�ÙMÙ<î.è.å�è!Ø Üzé Ü�Ù�î�×zÞ/é6��å �à/×zÞ/ü�î Å�ÆzÇÌ�¿�G�U�+� ������îô � þ�ì�ï ï�ô-ÿ�åì ì�åfH¸å�Ô�ØGÙ�ÜUÞ�ë/ÜUæ<üÌÜUæ�î�«�¼�Å4ÆzÇ��¿¼D���7(�î�ì�ï � þ�ìUï���ð�ÿ�åìI·-å Uå�ê�å"�!á-Ù�Ø â�×�Þ�é�ñ¸å8�¸åfê�Ö.Ü�Ù Ý�×�Þ�ßUà/Û�á�â�î + � ��¼fÉ�� ��¾ ¼!Â&��Ã�Ä-¼ (�î· ï�� þ�ì�ï zð-ÿ�� ������¼���¼�À���ÁeÂ���Ã�Äg¼ Å4ÆzÇ��¿¼ (�î�ì�ô�· þ�ìUï� �ð�ÿ0�UåìUí-å�ñ.å3��å�ñ�Þ�é�æ<ÜUÜUç�î «�Æ ¼)È�¾z��Äg¼�Â��U� ��¼ + � ��¼ (-%gî�ìUð�· í�þ�ìUï���ô�ÿ9� ��� � ¼Å4ÆÊÇ �¿¼8��È4Â&Å��7!�î�ìI·�· ð�þ�ì�ï �zô-ÿ7�¿åìUô-å±ñ.å��¸å ê�Ö.Ü�Ù Ý/×zÞ/ßUà�Û�á/â�î¶è.å"é/Ü��fæ<áÌÝÊÞúê�á�ë/Û�ä<ÜUæzî±×�Þ é� �å.õgå 4�á-Ù�ÜUæzî + � ��¼�É�� ��¾ ¼�Â���Ã�Äg¼ � ��î � ¶ ì�þ�ìUï�ï ô�ÿ:�IÀf�ÊÁýÂ-��Ã4Äg¼�Å4ÆÊÇ �¿¼���-î-í�ï ð þ�ì�ï ï�ô-ÿ7�Uå 3�¯Yb ¹5476 ¯ R�¹&8 ¯ T ¯ T �9-2: � :<; 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