Critical behavior of the antiferromagnetic Heisenberg model on a stacked triangular lattice

Texte intégral

(1)Critical behavior of the antiferromagnetic Heisenberg model on a stacked triangular lattice T. Bhattacharya, A. Billoire, R. Lacaze, T. Jolicoeur. To cite this version: T. Bhattacharya, A. Billoire, R. Lacaze, T. Jolicoeur. Critical behavior of the antiferromagnetic Heisenberg model on a stacked triangular lattice. Journal de Physique I, France, 2007, 4, pp.181-186. �hal-00163978�. HAL Id: hal-00163978 https://hal.archives-ouvertes.fr/hal-00163978 Submitted on 19 Jul 2007. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

(2)

(3)

(4)

(5)

(6) !" # $%

(7) !&' ($ )*

(8) + ,

(9) , -/.10325467684:9;254=<?>@4@ABDCE.F0HGJIJILK=GJ<NM:BPOQ.5RS4=9*4=T*MU/4:VDWX-Y2Z.\[:K=IJGL9^]Q_5<7` a M^<?bcGJ9*M"W5M/d32P>ceNGJfg_DMh-Y2jM*i K=<NGJfg_5M/WDM a 4:9ÎJ4*>lk m4^> nogBZn^p:p=q Qr stsNuwvyx{z}|Nv~s*uwxY| |7y;su~?|N {sH";*v~s3|7y {uwx{zj |7v~x;@vys~xv~x|7 =s |78vyv~xs?¤*sv¥DuSrQ7Fsv~8sv|N|7x{*hv~xv~s?~sty;;z} { |NN;¦§x{s?vyjxtsNuwvys?x{z}xuws?|N*v~sNu¤s?¨l~©@ªNzY«"*¬X ^s?^ g®°;¯*,±³²´|}®µu¡¯;v¶*|;=¢«"sN "¬vy®¸~·ºx¹»|N²´;£®µ¯; |7¶^ ¥ ¼1£sNuws§ysNuw vug|7ysZ;z |7v~x {s5¦§xv~j|vy|7½uwx{vyx{83x{3v~£s§rx{ u¡;^¾¿xuws?5ÀÁÃÂ*Ä¿£x¸ ÅÎ suu~|j|N {x;vwÆh½uwsN Ï|;uyus?l|8¤su ¦1 ~ssN *|Nx~s3vysN|N " {^sÆQv~Y|h8ÈvÇ´|Nx{ÉtÐÊ§|Ys?u~|N;v~yxz}uJL|;|7 x{vyËN;|Nyv~Æ x;ÍxÌZvyy~;st 7vy|7 s3¤ y|Nx{v~vyx{x8?@|N ¥ s?½| Å x{8§7Ds?xu¡s?* s~£s? {xz}|78s?vu¤*uw; Å x3v~*uZ| {8 7uwvÑ|7 ^x £Ë?Ë? {sÒ¥ Ó*Ô ¼D¹;¶Õº¯^±Ö 8 Lz}|7v¤ÕN¹8¶;¯;8¯^±¹ Ó zYxv~v~sº \vy¨×Ø£~ ^Æ£uwx?uÙs?v~vys?u Ú~ýwÛþ=Üÿ Ý¡Ý¡ÞÃÜ Ý¡ß?ÿ àgÝwîNáÜâNâ7þ ÜöÝ¡ÞÃgÞyã:å äÈå1æèçé¤êNë?ìlÜÃí¤îNï§ð=ñéNë?òíÞógôáõí¤Þöcá÷àø°í¤ß;áô8ò^áù;í¤ÜLá÷àíÜú?ëQöläé¤û¤ê÷ü¤ü ò*áù;í¤ÜJáàÃí¤ø°ÜÝ¿âNÝ5ôá ¿ø°ÜÝ ¡àø°í¤ßSâNÝwÞgå ¡ø°Ý¡ß ¡ÝwÞcâ7Ýcôá1ätáàÃø Ý¡ÜÝ5â7î þ ó ± . . . . . .

(10) .

(11) .

(12) . . ¼1£s?~sxuZ|7vSvys ysNuws^vZv~xzYs|Yu~|Nv~xuJL|;vy;yÆ&£ ^s?Ñuwv|N ^x}75vysj~xv~x?|7 @s ½| Å x{8}7 *Æ£uw x|7 tu¡Æu¡v~s?z}uj¦§s?~sQv~£sÍy;v|Nv~x;XuwÆ*zYzYs?vy~Æ 8~; ÁÈÄjxu £~;¢s, ^N¦§ v~ Á³±NÄt|Nv {N¦ vys?z s?|Nv~ysNu¥ Ó s Å s?|N 5v~s?8~s?vyx?|N 5v~; u3|7ys | Å |7x |N {sSvy sºuwv~xz}|7vysvys~xv~x?|7 :s? 8s?*vu§|7½ &v~£s?~sSxu1;* /|N;~ss?zYs?*v1s v¡¦s?s? vysNuwssNuwvyx{z}|Nv~sNu¤¥/¼1s}rx{ uw;*¡¾¿xuws*sN ;x^v3¦§£x, ^sNuy~x sºuÈv~£s ¤~x{vyx?|N *Æ£uwxu?|N"s3u¡zY;vy {ÆY8 { {N¦sN Y svw¦s?s?}v¡¦1|7 Y;» ^x{zYsuwx;u¤¨ 8 ¶ v~£sjÈÇÉ |7 /Â Éys?8~z}|N {x{Ëº|7v~x;};y; s? |7u¡x{;/zs?~8sx{Q| ¤;*v~x{*8u1z}|7£s??¥ ¼1sHuwxv~|Nv~x{8/xu§z ½÷Yz;~s8z x?|Nv~sN Y¦§s?hv~s~8v|7vyx{8QuwÆ*zYzYs?vy~Æ3xu "!#!%$ ~8¢s?ÍxÍvysY 7¦1Lv~sz s?|Nv~ys |;u¡s¥ Î ~8zYx{£s?^vZs?£|7z {s3xu8 x"vysuw7?|N { {sº &s? x{z}|N;sv~xZuwÆ£uwvys?z}u1¦§s~ss?xuws?*s?~ju x{uZ|7ysx/|ju x|7 |N~Ñ|78s?zYs?*vs? 7¦v~£s"¤~x{vyx?|N 5v~s?z s?Ñ|7vy~s¥'&Jvxu3|N x^vys?~sNu¡v~x/ÏsNu¡v~x{8@ 8v~´v~s?8~s?vyx?|N { Æ,|7 s s?yx{zYs?*v|N { Ælv~ ¢*N¦v~sQ¤;~ysNu 8 ^x{£ £x Å s?¡ uy|7 x{v¡Æ ¤ |;uyu¥ Î ¦§x ^s? Æ u¡v~ ^xsN ~8v~8vwÆ x?|7 §zY* *s? 5xuv~£s"|N^vyx¸s?y~8z}|78s?v~x Zs?xu¡s?^s?y}zY* ^s? 5; |/uwv|8÷¢sº vy~x|N;£ |73 |7vyv~x¤s@¦§xÍxuu¡x{z sj|7 , ^xu ~|?sNÆ/u¡s?^8v5zYvyx{zYzYs?s½8uwv~Ñs?|7yv~sstxuDs x{ {v~x{zvy {sZ|7;£8s?v~uwxsÈuw;Ñ û5s?x{YSvys? sSN¦ x{v~s|Y|7v~vyÑ|7~s§u¡x{yv~s?x;|N ^ x;x{*j)v y(+x{*~v~¥x?|7Î vSv~£ss £;zYs?½|3|;u~u¡*x|Nv~sN }¦§x{vy&vyxu1z* ^s?- ,/± .L¥ ¼1xuSv~ x|8u»s?s/x{ sNuwvyx{|Nv~sN }^Æ}u¡s NF1| 0)¬)Â ÉZ~s;yz}|7 x{ËN|Nv~x{8 8~8 ?|7 |Nv~x;2 , 3 .L¥"¼1£s}Å ;y~sNu ; ^x ÌZx{Ë~N~Ù:|N |N v~£s?;yÆ 8j4| ³ ¤;z ;£s?^v sºv~8@zY* ^s? x 8 sNuv¡¦15 ³;z 8s?*v@ Òuw8x6 s u¤7¥ &Jvgxu 8 vy|7vY 84Å |7y;sQs?;£;´v~Å s/Å vy|7½uwx{vyx{8 xu&u¡sN8 8 ^s}|7 8v"; Å s?ysN *Æ}v~sjZs?xu¡s?^s?y ÁJ8 tÄ§rx{ uw;*¡¾¿xuws9 *sN 8x{*vSv ^Æh|Y ^;x ::s?~s?*<v ^sº 88x{x{**vvj ^¦§xuy|x÷'sNxuj|N|7uY uw|N 8vy^Ls?v~~~sx Å xxu3|7 §Q;= uwvÑ0 |7 >)As Â*¥sN ¾8Y;x{u¡*z}v3|7¦§ {s?x? ÷,Dxu3vy|Nx u3xs? ^@¦ x|7v~^x;sº 83C| Bvy|7vyx{;*Jx ^sºD u¡vwL;Ñ ^s?vy|Nuwxv~x{8@¥/ ¼1s ^x Å x ^x{}£x Å s?Ñu~|7 5 x{s s?v¡¦s?s?tu¡sN;½ J8 ^sl|N EÑuwv¡J8 ^s?gs?| Å x{;£¿xu ; tv~Zs9 * ÁF0jÄD¬ ^±;®HGI 8¶^® Â^ÉÇ/ÀÁwKÉ JÄ¥5¼1sZ| x Å |7yx|7vyx{8 N- * {sN| Å sNu¿½uØÑ|7vys?5s~v|Nx{|7;£v1v~£s L|Nv~sSNgv~s»?|;u¡s 0#¬ ¶^¥§5 {sº|7~ Æ3zY;ys§x{^8~z}|Nv~x;jxuD£s?sN ^sN |7;v§vys * FÁ 0jÄ x{s x{}v~sjÁ^zjs?57¤;z ;£s?^vÑu ^x{zs?uwx;=Ä |7£s¥ Î 0 ¬ YÇÉ~s?8~z}|N {xËN|7vyx{;8~8 uwvy ^Æ|8u&s?s? s?w8~zYsº ;/| u¡Æu¡v~s?z Ngs?xuws?*s?~ u x{u , ¶L.**Æu¡s§7è|8^J x{sº|75u¡x{;z |zY* ^s? * *s/sN 8Y| £;zY8;s?s;u=8ûwÆ*zYzY s?vy~x@¤uwsvgu |8s§ÀÁL¶NÄ M1ÀÁJÄ~ª;ÀÁLÄO¥ &v:¦§|8uè8 Sv~|Nv £sN|75v¡¦1 ^xzYs?u¡x{;½uPvysSu¡Æu¡v~s?z ^s~;sºu1|tuwsN¤; 3;Ñ ^s?5v~Ñ|7u¡x{v~x;j¦§x3xu 8 Å s?~sº \*Æ vys ÁÃÂ*Älu¡|7 @rx u¡;^¾¿xu¡s?- ^sº ;x^v-¥ &J L|;vv~s»uwÆ*zYzYs?vy~Æ .

(13) À ÁL¶ÄM,ÀÁJÄ~ª8ÀÁLÄjxu ^Æ*|7zYx?|7 {Æ/s?£ |7y;sN ,|Nv3v~shyx{v~x?|7 ;x^v3vy ÀÁL¶ÄM ÀÁL¶Ä~ª8ÀÁJ¶ÄF|7½ ÐÀÁL¶Ä-MYÀÁJ¶Ä5xu ÁÂ^Ä¥5¼1£xuDzxuwz |7vs?v¡¦s?s?jv~sZs? |7u¡x{8u £sN|75;Z ^xzYs?u¡x{8ul|7½ £sN|7vw ¦3 ^xzYs?u¡x{8ulxu1Ïxv~sZ^½uw|N @|7½ & *sNuD8v ½| s?'x{ vys}¦1s? ¸uwvy ^x{sº ÀFÁ "Ä ÀFÁ ±7ÄHyx{v~x?|7 s?£;zYs?½|^¥ Î u | ¤;u¡sNÏ£s?s5v~£Ds 0 ¬ ¶Í?|8uwsh~s?z}|Nx{uts? {½uwx Å sh|N |Q ^x{ysNvju¡v~ ^Æ,x v~£~s?s *x{zYs?½uwx{8uDxu§?|7 {sN 3;?¥ Ó 8zYs ~s? x{zYx|7yÆ}";*v~sY|N~ {QÁLÍ§ÄZuwxz |7vyx{;½u³| sHuwN¦§Euw;zs3s x *s?s , NÂ .= ; | ¤;*v~x{*8u1v~Ñ|7u¡x{v~x;hx{}v~sH?|;u¡st75vys3Å sxuws^s?y3|7*v~xs?~yÅ 7 z |7;£s?vl;|Su¡v|;¢sN vy~x|N;£ |75 |7v~vyxsÒ¥¿Z7¦s Å s?@vyss ;£s?^vÑuP;£ |N~s§8v ¤;z |Nv~x {s¦§x{v~Qv~£uws7§vys ÁÂ^Ä sNv~8SzY* ^s lx{Qv~ys?sj ^xzYs?u¡x{;½u¥j¼1xu L|8v§ xuZ ^x }£ {v1vy ysN8x{ sS¦§x{vy hÊÌ Å uwvy ^xsNu³¦§£x ~sº ^xvZ| ÁÃÂ*Ä1vy|7½uwx{vyx{8 Á 0¬ÈÇ´ÉÄ¥ &JvSxuSvys u¡sY7Fv~xut|7~vyx svy/uw£sN x{8^vS8 v~£xuS ^xu~ys |NÆj^ÆjzYsN|NuDN5|3Í' u¡v~ *ÆY7cv~s zY* ^s? L¥¿rQst ^x }|t |7y;su~|7 {su¡x{zj |Nv~x{8 ¦§xv~3zjs?v~vys?5u¡v|7vyxu¡v~xu¿v~|N ~s Å x{8ul|7vyv~s?z vu¤¥ Î uD| ;u¡sNÏs?s§¦sZ|7ys |N {sZv~ x{ ^N¦§Yv~s v~|Nuwxv~x;Yvys?z s?|Nv~ys§x{| s?yÆ ~sºxu¡sz}|Ns?Ø|7 Yv~ 8v|Nx{h~ s? x|7£ {sZsNuwvyx{z}|Nv~sNu57yx{vyx?|N s ;£s?^vÑu¥ Å rQsvy^½uØ*u5;Yvys |8u~uwx?|7 :u x{jzY* ^s? : ^s sN 3^Æ3vys |8u~uwx?|7 :sxuws^ s?yY |7zYx {v~8x|N@¨ ¬ ® Á~±7Ä

(14) ¼1s§s|7£;s1x^vys?|8v~x; xu@8Ë?s?y3Á ¬ ±1x¦§|Nv¿; {N¦Zu÷Äè svw¦s?s? £sN|7ysNuwv¡Jsx{;*;Ñu&7\| u¡v|;¢sN vy~x|N;£ |7} |7 v~vyxsQ|N vys u x{½u&|N~sQv~ys?s ¤;z ;£s?^vZxv sNvy;Ñu¥S¼1£sj |8u~uwx?|7 g;y;½ Íu¡v|7vys3xu³8 /^Æ}zYxx{zx¸Ëx¸£ vysj¾ ;yx{s? v~|NuÅ 8~z Á YÄ§N§v~£sj; {x{£u ¥3¼1sju x{½uÈ|8 ^ v3| |7|N |N~Ñ|78s?zYs?*vl; |Èv~£~s?s uw£ |Nv~v~xsZuwvy~¤v~ys¦§x v~j~s? |7v~x Å s§|N; sNu±;¯3 ^s?8~s?sºu¥ . . ! "$#&%(')&*,+. rQsZz}|; ^s§v¡¦u¡?sºu~uwx Å s§u¡s?vul7c"8^vysS|7y {3u¡x{zj |Nv~x;u@7gvyxulzY* ^s J;u¡x{ u¡ {x;*v~ {Æ'|8uwÆ*zYzYs?vy~x |Nv~vyxsNu}ÁÃ¦§xv~ s~x{* *x";½ |7yÆ´; *x{v~x;uÄ37Hu¡| s - J.-0/N;¦§s~1s -hxu@v~£s1 {xsN|Nlu¡x{Ë?sx{u¡x ^s5v~£s |7sNu¤|7½ ,-2/Z¬´ªº¶3-/vys§u¡v|8÷¢*x{ u¡x{Ë?sÒ¥ ¼1s ½uwvØuws?vlNèuwx{zj |7v~x;u=¦F|;u¿y3;j|t&¡ G4z}|8u~uwx s? {Æ |N|7 {s? £8z v~s?¥lrÍsZz}|8 ^syu18Y |Nv~v~xsNul7cu¡x{Ë?s±?NJ GjÁÃ¦§xv~}|vy;v|N =Å N65;¯; ¯3¯8¯;¯3"8^vys |N~ "u¡¦1s?s uS7§vys3 |Nv~vyxs;ÄÑ@7Â J &± 5 Á¦§x{vy |Yv~8v|N DNt±jÖ;¯8¯"¯8¯;¯"u¡¦1s?s u÷ÄZ|N ¶.

(15) Â G8J§¶8YÁÃ¦§x{vyY|vy;vÑ|7 =N±SÖ·N¯3¯;¯8¯3uw¦s?s uÄ¥&JYvys<u¡v§?|8uwsS¦sSuwsº v~s sN|Nvw §|Nv~ |7 {8;yx{v~£z/lx{,v~£s" |Nv~s?3v¡¦ ?|8uwsNu¦s"u¡sN ,|7´Æ*yx ZÅ s?y~s? |7£|7v~x; |N {8;~xv~Dz , Ö .¿¦§£s?~ssN|;/sº|7vw §|7vy/u¡¦1s?s xu1; {N¦1sN }^Æh|N/s?s?y;Æh;½uws? Å xuw¦1ss ¥¼1ss^vZu¡s?vNuwxz £ |7vyx{8uD¦F|;u§~h;ÍÊ Î Å sNv~8Z;z v~su 83 |Nv~v~xsNu=N@u¡x{Ë?sN"Â JN± 5 5¶88¯ J8¯ ¶ 5NJNÂt|7½ "Â G8J¶8¦§x{vyÂSvy G±¯ 5Æ*yx ZÅ s?¡ ys? |N|Nv~x;Euw¦1ss u¥ 1;vy,uwxz £ |7vyx{8uS8s?*v~Ñ|7v~sjx{Qv~£s xzYzYsN ^x|7vys Å x¤x{xvwÆ N5v~£sÈvy|Nuwxv~x{8@¥ Ùs?vuD£;v~s ^Æ vysSvy;v|N @u x{ s?Fuwxv~s§;§u¡ |7v~vyxs YÁ ,¸± Ñ ¶ .ÃÄ¦s zsN|;u¡~sN jv~stz}|N;s?vyx{ËN|Nv~x;@¨ @Á DÄl¬ ¶± > ÁJÄ vysuwuys vyx{x {x{v¡Æ:¨ Á 5Äl¬ - J.¶ -0/ > J > J ÁJ¶Ä |N &vysS;yv~^L;Ñ ^s? zj |N^v¨ Á 5Äl¬± ¶± > > J J ® ÁÃÂ^Ä u ½uwvv~hsNuwvyx{z}|Nv~s3v~£s Huwv~Ñ|7vys?;ÆQv~hs?^vy|8vtv~s~xv~x?|7 ls ;£s?^vÑuÈ¦F|;E Å *|Ns? {^vst|N7c Yv~£v~ss?¤j~x{v~vyx?sN|Nuw vyv~x{z}s?z |Nv~ s§s?Ñvy|7vysSs?~s ;*~£s?|;*u§vv~uls ~8 z ;xv~^v§s¦§£8 {s? ~7s ¦§xÁ 5uws?Älvx71u g- sNÏx{½ ^|Nv~s x{ 8s?û ¨ Á * Ä ÁJÖÄ Á *wÄ Á 5Ä Á DÄ - ® Áw·;Ä ¼1sZs?*v~Ñ| ; |7v~x;@;y;z v~£s Å |7 s½uwsN ;5v~£sZuwx{zj |7v~x;@Nv~ * xu5 ^;£s ½uwx{£Èvys ¾½s~~s^s?y7 Ó ¦1s?½ uws?jv~sº÷£xÏs , 5L.J¥c¼1xulvysN÷£xÏs§xugx{ Å |N {|N {sZv~ s^vy| 8 |7vys3x{Ð|\sx{;*;y* Í7Duwx{Ës ±7ª - ÁÃrQs *s?u¡v|7½ "v~|Nv?=¦§s ½uwsN , {x ^ {ÆE;vÑuwx *s/uw½÷ | vyx{;*vj|7£;s1xv z |ºÆQ;x Å s}¦§~;£ ysNuw£ {vuÄ¥¼1£s u¡v|Nv~xu¡v~x?|7 |7|N {Æ£uwxu xuj ^;£s/¦§x{v~ysNu sN¤v vy,;¯Í£x{u¤³½uwx{£ u¡vwL;Ñ ^s?Yx|;uJ ¤;~ysNvysN |8÷¢*¢^£x¸sYÁLuws?ss¥ £¥Ê§sy¥ , · .l|N /~ss?ys?sºu³vys?~sx{=ÄÑ¥t¼1su¡vt;¯ N5sN|8÷Yy"xu§ ^xuy?|7Ñ ^sN ;§vys?yz}|7 x{ËN|Nv~x{8@¥ Â . . . . .

(16). . . . . . 2. &%. ')(+*,. . &%. .-0/ (1* . 3. 2 . 3. %. 3 3. . $. . .-98 (+*. 3 /54/76. :. . ;%. <. . . . . . #. . . . . "!. .

(17) . 8 (1*. >=.

(18) ¾ ~8zu¡x{zj |Nv~x;u=N |Nv~v~xsNu@7xysN|;u¡x{ uwxË?sNu- > - J >®?®?®8 Å s~;x sºuwv~xz}|7vy;ÑuD7 * |N~st;vÑ|7xsN &*Æ}u¡; Å x{£3v~stsNÏ|7vyx{8u¨ Á l Ä5¬ JÁ 5 Ä® ÁGÄ 8. . :. * %(*)""),+-". %(*)""7,+4..

(19) "$#&%' "89#:$". . . (*)!)/0+1%2".3+4)$". (*)!550+1%98.3+1%".. . ! (*)!5),+4)6.3+4)6!. %(*/!/8,+4)/.3+4;!.. (*78,+4)!.3+4)). (*66,+4)!5.3+4) %'.. g¼ |7£ {s}±;¨& yu¨t¤~x{vyx?|N ¿vys?z s?Ñ|7v~£~s|7 Qyx{v~x?|7 gs? ;s?*vuZsNuwvyx{z}|Nv~sNu¤¥ ¼1£sAÑuwv}8 {z 8x Å sNuvys" x{sº|7uwx{ËsNu7v~£s"v¡¦1Q |Nv~v~xsNuu¡sN =¥X¼1s"v¡¦1 *zjs?u1xuwx ^s |N~s?*v~£sNuwxu|N~s3v~£ssNu¡v~x{z |7v~sºu17v~sHuwvÑ|7v~xuwvyx?|N ¿s?y~8uè ^x{ysNv |N &x ^¤sN =^8"vyst |;u¡vZ ^x{8x{vÑu175vyst^zjs?5;hv~s?x1 sv?¥ * %(*)"5/,+-". %(*)""%=+-". %(*)""8,+45. %(*)""5,+45. %(*)""5,+1%'..

(20) 5!)9#:$" 5!79#:5) "9#:57 "89#?" "89#:$". %(*/!/$"<+45!6.3+1%'. %(*/!880+4;!;.3+1%'7. (*)!850+47!8.3+1%". %(*/!870+47@%'.3+1%'8. (*)@%%A+1%!%'.3+1%';.. ! (*)!,+1%'5.3+1%'7!. (*)!),+1%'8.3+4 %9. %(*/!8$">+4).3+45 %9. (*)!)$">+4.3+45/!. (*)@%'),+4)$".3+4;!.. (*6/,+4)!6.3+4) %'. (*6$">+4)!/.3+4). (*/),+1%2".3+4). (*6),+4)!8.3+4) %'. (*68,+4)!.3+4) %'.. g¼ |7£ {s^¨§Ê Î ~£u¨~xv~x?|7 :v~s?z s?Ñ|7vy~st|N "¤~x{vyx?|N =s? ;s^vÑu1sNuwvyx{z}|Nv~sNu¤¥ ¼1£sAÑuwv}8 {z 8x Å sNuvys" x{sº|7uwx{ËsNu7v~£s"v¡¦1Q |Nv~v~xsNuu¡sN =¥X¼1s"v¡¦1 *zjs?utx{u¡x ^s |N~s?*v~sºuwxu3|N~shv~shsNuwvyx{z}|Nv~sN u¡v|Nv~xu¡v~x?|7 §s?y~8uF ^x{ysNv3|N x ^¤sN =^8"vyst |;u¡vZ ^x{8x{vÑu175vyst^zjs?5;hv~s?x§ {sv¥ ¼1s ~sNu¡ {vÑuD|7Ys8 xYv~s uwsº; ; zY37c¼g|7è¥{±3ÁÃysNu ¥l¼g|7@¥µÄl; vys3&¡YÁ~sNu ¥1Ê Î Ä |7vÑ|^¥5rx{v~£x{Y8ZuwvÑ|7v~xuwvyx?|N g|;?£|;¤Æ|N { @sNu¡v~x{z |7v~sºu |N~s}¤;z |Nv~x {sgv~s~sjxus?xv~s|N^ÆE sN|7 |7vyv~xsYu¡x{Ë?s ^s s? ^ssg;j|N^Æ *xu~¤~s |NÆjs?v¡¦1s?s?}vystvw¦juws?vÑu175yu¤¥ £9 ½|7 @sNu¡v~x{z |7v~sZxu§v~*u¤¨ ÁJ¹Ä * ¬ ±;®µ¯7Â;ÂÒ¶È²´®°¯8¯;¯8^® . . . Ö.

(21) Qr s½uwsZv~xu Å |N {sZv~8z vys Á *wÄÑ Á +*¡Äl|N ^x{38 ^s1v~ sºuwv~xz}|7vys5©Pªº«= 5ªN«=Ò|7 3«xvuws ¸y¥@Ê§sNuw£ {vuc?|7 s18 x3¼g|7@¥±SÁJ¼g|7@¥µÄ=8lv~£s &"Á¡Ê Î Ä5yu¥§rQs3Ï;vys3u¡s |7Ñ|7vys? {Æ}v~£ssNu¡v~x{z |7v~sº Ñ ^x{ysNv

(22) 3uwv|Nv~xuwv~x?|7 s~~8u§8z vysN ~;zv~£st ^xu s?u¡x{; }Ngv~sSysNuw£ {vuDy;zv~s»;¯xu£|7 Yv~£s s~~8uSx{½ ^sN }^Æhv~s3£s?yv|7x^v¡Æ};Qv~£sj ^s?v~s~zYx|7vyx{;}N * @8z vysN "|8u vysjÁJ|Nuw8 {vys Å |7 sN5v~s8Ä ^x ::s?ys?sts?v¡¦1s?sYv~s Å |7 {£sNu1;vÑ|7xsN &½uwx{£ 8 sNu¡v5sNuwvyx{z}|Nv~s1N * |7½ \vys Å |N {sNuguwxtv~sZ;sZu¡v|N |7Ñ Y ^s Å x|Nv~x;3sNu¡v~x{z |7v~sÒ¥ ½|7 *zj sug|7~sZ©@ªN«Y¬´^®µ¯^±=²Q®µ¯;¶Î 0 } Dªº« ¬ ^®µ¯^±=²Q®µ¯;¶^*«Y¬ ®¸·º¹¿²Q®°¯8¶^ 8 v|Nx{sº Ðy;z x{sº|7 vÑuS7§vys}Ê Î |Nv| ;t {èÁ Á * Ä~ÄÑ@ {@Á =Á * Ä~ÄZ|N @Á Ä@~sºu sNvyx Å s? Æ¥=¼1£s§~sNu¡ {vÑuPN=vy6s ½vul|N~s§u¡v|N {sZ|N|7xuwvc;zYxv~v~x vysuwz}|N { sNuwv5 |Nv~vyxs |Nv|^¥ rQs};u¡s? s}v~½|7v3*Æ s?Ñu~?|N {x{£/xu s?yx sN /¦§x{vyx{,s?y~8u|7½ |;u3u¡|7 §x vy~ssY ^xzYs?u¡Å x{8u"xu Å s?yÆ/uwz |7 { FÁ Å )®µ¯^±7ÄÑ¥Y¼1s Å |7 s\83«/xut8z |7vyx{ s ¦§xv~hv~s Å |7 sS;Zv~sjÀÁÂ^Ä ½^sN ;x^3v , ¹ .5ÁJ« ®¸·NÖÄ¥5rÍs| Å stv~*u³8v|Nx{sº ss { {s^vs Å x ^s?¤s 8h| ;*v~x^£;u vy|Nuwxv~x{8x{vys 0 ¬¶,?|;u¡sQ|7 ¦s ½| Å sZu¡N¦§Yv~½|7v1vys ~xv~x|7 *s? 8s?*vu5|7ys s?¡sNv~ Æ38z |Nv~x{£ {s1¦§xv~jv~s ÊÌ *~sNst ^v~xjv~xv~;hst7 |N§~vy8s sjS|7 ÇstÉZ7s?Ø «= |7 ½x{¢uwx{sZ8@x{}¥v~¼1£sÈsj½uwu |NsN l;x :sxuwssN^|NvSs?s?y zY;*s? ^*sv 5,ÁÃ8xu ^¡|7*v~sN Ä x" !È¬ ¶^¥ ysNuwÅ vu3|N~sYx z}|7y¢sN Q ^xuy|7;ys?s?zYs^vt¦§x{vy v~£sY {N¦1s3uwvÑ|7vyxuwvyx?u "8^vys |N~ {~sNu¡ {vÑu³NÊ³s¤y¥ , Â .ÁJ«"¬®°Ö8¹È²´®°¯8Ä¥ ¼1szs?v~* £u³N|7|N {Æ£uwxuS|7ys *x ::s?~s^v¥¿Ê§sy¥ , Â .u¡sNu5 |7vÑ|»vÑ|7¢s?x{j|S¦§x *s§~s?8x{;j|N~8 3v~s§vy|7½uwx{vyx{8 ;x^v |N }|; ~½uwv§v~£s Å |7 {£sNuDNgv~st¤~x{vyx?|N s? 8s?*vuD½uwx{£3v~sS8 \L|8uwx;sº Ñ |7vÑ| ¤; { | u¡x{

(23) zYs?vy* =¥l¼1£xu1zYs?vy* |8u1v~s ^xu~|8 |7*v|78stv~j;x sZzj÷j¦1sx{;*v vy ; x^vÑuD¦§xv~j |7y;s Å |N {sZ7Á +*¡ Ä - ¥¿rQsSs?*Å v~Ñ|;v5vyss Å ;£s?^vÑu1 ^x~sN¤v~ {Æ y;z |7vÑ| ys?¦1sx{;*v~sº &v~ *~¥trÍs3½| Å sH|7 uwh|Yz }x{8s?tu¡v|Nv~xu¡v~x?u¨Z¦1s3u¡s G4M'±?¯ £sN|7v¡J|Nv~Çs?s?y;Æ¡¤;u¡s? Å x{Ðuw¦s?s ut¦§s?~sº|;= u , NÂ .1u¡sNu 5?¡8¯ vyx{zYsºu 8¯;¯8¯;¯Yu¡¦1s?s u17Ø|\ sNu~u5.s }¤x{s?*v|7 {8;yx{v~£z/¥ §sNuwvyx{z}|Nv~sN juwvÑ|7vyxuwvyx?|N =s?y~8§; vys ^s?v~s~zYx|7vyx{;/N * xuS;£sY;Ñ ^s?NZz}|78x{vy ^suwz}|N { s?Sv~½|7 vys 8sYx ysy¥ , Â .J¥ . . . /. 3 3. /. 3. 3 /,4/ 6. . . :. . ,. . /. /. . 3. 1 . 3 3. 3 /,4/76. %. %. <. . . 8 (+*. . . . #. *,+%$1#!"$&*,+. J Å x{s?¦ N§v~£s ~sNu¡s?*vSs Å x ^s?½sÈ8 ÁÂ^Ä§s?| Å x{8|7v0¬ ¶Y¦1sj|7ys sN "v~ u¡;8sNuwvZv~£s |;u¡s3 ^x|7;Ñ|7zu¡N¦§"x"¾gx{¥±Ss?v¡¦1s?s?vw¦j|7 Y;Z *x{zYs?½uwx{8u |N &; zY* ^s? u1¦§x{v~/vy~ss; zY;ys¤;z ;£s?^vÑu¥SÙs?vS½u9u¡vtuw£zYz}|7yx{Ë?sZv~£s &. 5.

(24) ¢*N¦§ {sº ^;st;£v|7xsN ~8z s?~vy~½|7v~x Å sÊÌ uwvy ^xsNu¨ xÄ¼1s?~sQxu}|,x Å su~|N t {x{£s * Á 0 Ähu¡s |7Ñ|7vyx{ | ÑuwvwL;Ñ ^s?h~s?8x{;~8z | u¡sN8 L; *s?t~s?8x{;/£sN|70)¬ÂÉ¤¥ &JvuSu¡ { sjxu§¢*N¦§ y;zÊÌ uwvy ^xsNu , 3. |8u³¦s? { è|;u§v~syx{vyx?|N |N {s * ; 0¬Â¥ xxÃÄ3Ù:|N~;s¤ )uwv~½ ^x{sNu , G .§Å | Å s}uwN¦§vy|7v3vys/u¡v|7£ {sC*sN 8x{*v\8 '|7 Å s * ÁF0jÄ s?Ñuwxu¡vu3u¡zY;vy {Æ x,v~s}~s;x{8 ¬ |7 ' >0 >Â¨}v~s}vy 8 |N~Æ} {xst7Ø¾¿x{£¥{±;¥ xx{xÄ Ó zY;vysNuyu |N {8 v~£s 0 ¬ Å s?yv~x|7 §|N^xu |7s/u¡N¦§^Æ'uwvy ^Æ*x{ vysj; x{sº|7tu¡x{;z | zY* ^s? Øuwxv~sN /v~}v~£s ³ Å sºv~8SzY* ^s? L¨xvÈxuSx{ vS;Qv~£s £;zY8;s?s;uSu |;¤s&ÀFÁ hÄ MQÀÁLÄyª;ÀFÁ Ä MÍÀÁLÄ ,±?8¯ .L¥ s ½ ut|Yuwx; s u¡v|N {s ½^sN 8x{*vD8 !#! Å |N {sNu57- ´¶^-¥ &JYvysS |N~8s ´ {x{zx{vlvysSs? ;s?*vu y;z v~£xuguwx{8z}|1zY* ^s |7ys1vys§uy|7zYs|;u@v~£uws7:v~£sD x{sº|7@zY* ^s J¨=vyxu@| £ su |Nvvyst s?§ sv§8~sN5¾¿x¥±;¥ &Ã¦s1s? {xs Å slxv~£s s?~vy~½|7v~x Å sÊÌ,~sNu¡ {vÑu;vys?£sNsNuyu~|7yx{ ÆSv~sx Å s?uy|7 x{Cs E*~FÁ 0jÄt|7/;£ {Æ/x{*v~suwsºvvysj;yx{Ë?8^vÑ|7 5|N^x?u ¬ ¶Ys?v¡¦1s?s !È¬ &|N !t¬Â£¥5¼1sZuwxz sNuwvg^Æ ;vysNuwxu¿xugv~s?3vy|7v5vys |NsÁ !ÈÄgxul *x Å x ^sN tx{3v¡¦1 ys?;x;uZ^Æhv~s {x{£s * FÁ 0jÄS|8utuwN¦§/x{4 ½¥{±8¥t¼1xuZ {x{£sx{*v~suwsºvu1vyCs %¬¶ |N^xu1|Nv§|3yx{v~x?|7 ½ ^x{zYsuwx;C 0 * ¥ §"8^vys~|N~ ~sºuw vuDvy^½uDx{z {Æv~|Nv 0 * xuPs?v¡¦s?s?tv~ys?s§|N »85 ^xzYs?u¡x{8u@uwxs5¦1s;u¡s? Å s|S8^vyx{*;½uPvy|7½uwx{vyx{8 |Nv!È¬ ¶^¥ Ó xstx{Q|8 ^x{vyx{8/¦1s½| ss x ^s?½sS; ÁÃÂ*Ä§s?| x{;£Sv~xuzYsN|Nu vy|7v;vyx{£ *|7z |7v~x1| £ suDs?Å v¡¦1sÅ s? vysZs?x{8^ 8~* Y7Å !È¬ |N !È¬ ¶^¥ rQs/8v~sQv~|Nv&vysQuwvÑ|7 4s *sN ;x{*v}v~|Nv}; Å s?~½u v~s/¦§£; {sQu¡sN8 L; *s? *;z}|Nx{}~sN *sNu§v~}|j¢^N¦§= ^sº ;x^vÁÃN1vysrx u¡;^¾¿xu¡s?tÀFÁ "ÄL|7zYx {ÆÄ ¼1 £!;x$ uxxÍuvy8=s uwxÈuwvy¬ s?^¶"v ¦§|;uwx{sjv~h¦§v~s?sysYL|8v~£vZs?v~~sj|NxvÈutvy| s3 sNu¡v¤|N {x {|7suw*Æ*sNzY zYs?8vyx{*~Æ}vs?;£ |7y ";s?zYsN|Ns^3v !È,¸±?¬Z¯ .JÂ ¥ |N Å s/vysQ {x{£s * Á 0 Ä ^sNuY8v} s {;£ v~,vys/rx{ uw; *¡¾¿xuws L|7zYx {Æ¥ ysNuw£ {vuZuwss?zv~js?£ {½ ^sSv~£sx{*v~yx{8x{ ~ Òu~|7 , 8¶ .-0 * ¬X¶ ; Å s?yÆY u¡sv~¶^ |S ;;x?|7 uyuwxx{ x{v¡ÆÈ8D¦§£x3v~s?ysxul;z s? {xS|N~8zYs?*v?¥grÍs§8v~sZv~|Nv vys |8uws ^x|N;Ñ|7z¦s ~ uws xu¿xj|7;ys?s?zYs^v@ ¦§xv~j|7 ¢*N¦§jÊÌ ysNuw vul|N *sNu1;v§ysNÏx{ysvyst|; £ ^x{vyx{;7Ø|7*Æ s¦ ^sº ;x{*v?*u¡s?s?Yx s?~vy~|Nv~x; vys?8~Æ¥ Z~sNu¡ {vÑu|N uw;x Å su¡ ;yvÈvy vys Å |N {x *x{v¡ÆY7Øv~sH»Ç É§s |Nuwx;@ |SL|;v§vy|7vxulL|7Dy;zs?x{38 Å x{8u1u¡x{¤s s?yv~y|7vyx Å sSvy~sN|Nv~zYs?*vlNc|3uwx{8z}| z* ^s? ls; {sºvuS; ;|N §|8u sNvÑu¥ Ô ~s Å x{8uu¡v~ *x{sNuSx{½ ^s?sN /| Å s s?¡;yzYsN =@|8u ½uw|N J;£ {Æ s?yv~y|7vyx{;hv~s;~Æj; u x^J¦F| Å sSs?£x{vÑ|7v~x;u , ¶ .J¥ rQs"| shv~*u ;£v|7xsN '| uwxz {s xv~£~s}7v~£s/yx{v~x?|7 §s?½| x{8YN3| z |7;£s?v¦§Åxv~,|/?|N^vysN ;y; ¡u¡v|7vys¥ &J v~£s ^Æ£uwx?|7 ?|8uws5¦1s ~Å sN ^xvtv~|Nv Zs?xu¡s?^s?y"s {x{z |7;£s?vut¦§x{ { s? {8Ívy/v~s ÁÃÂ*Ä£x Å s?Ñu~|N {x{v¡ÆQ |;uyuYÁÃv~£s&s ·.

(25) vy8|7v5s?*xvlv ¦1"8 s? ;sZ8x{x{*}v~s?yvysN/uwvy|Yx{v~Zs?½v~uwt;;t£yvs |7 x~sºvyuws?*sZv|Nyv~x{x{v~8x2?|7,¸± *¯L.s?Ä¥3| rÍx{s3;££l;7v~ès3v~;sv~s?|7£*v~v~sºy s u¡Æu¡v~s?z u^zj z;~s ~s? s Å |7*v1v~3vysts? s~x{zs?^vÑ|7 :uwxv~Å |Nv~x;@¥ NUMBER OF COMPONENTS. . 21.8 2ND ORDER. 1ST ORD. 3. 2. 3. 4. DIMENSION. ¾gx{;£~sh±;¨ Ô ½|;uws ^x|N;|Nz%N§vys !§¡ ^xzYs?u¡x{8|7 1ÀÁ "Ät|7*v~sN /zY* ^s? L=x{Qv~£s Á !tÄÃ |Ns¥ $ . +* #& ( (+ '). rQs3¦18 / {x¢svyYv~|N¢ ÌZÆ !sN|7 |Nx{@ !S| Å x ÍÙ 7Æ£ ¦1s?èØÌZÆ*|N 1|78v |N 3Ê§*÷ 18~;£|7xu¿7è¼§Í/;lvys?xguw 8~v?¥5rQs§|7 uwSvy|7£¢ Î ¥ Ô ¥ ;£ |N !¥ Ô ¥ 1s? |78s?D8 Å |7yx{8u³x^;yz}|7vyx{8u1|N 8vSvyxu§u¡ ~sNv¥ . . <.

(26) )(+%$ % ( ( !#"$ (#% & (' )* ( $,+ + ( .-0/ (#132 ) ;/7@% +1%'// %9.( *43 5 (76 !#" ( 8 '#9 / )#* (.-0/ ( %: ) $";<+ %'/67.<;==( > & ?@!A)7$ ( 0!#"B ( $ ( # )#* (C-/ (%,D32 ) %%';/,+1%'/!66.( G.

(27) *54 ( $ ' ) > ( = 9 9 !#" 5 - ( * ? ) .-0/ ( ( 9 9 ( ) 5 %'6; +1%'//)!.( "&

(28) ( ) * ( ,$ + + ( .0- / ( ) 5)87 +1%'/88!.; * (.-0/ (# ( *#+ ! D ) %'// +1%'//!.(. *;4 $ ( = = ( ! !" / ) . " @! &8 9 - !9 ! &9 ! $ #/ + !, 9 9 " 5- / A$ 55 % / ) +#9 %';9#&%9/0%9// $, 9 " ( *74 $ ( (# !@ , !#" A( ( !#" !A) .-0/ ( ( 9 9 ( AD>7!5;,+1%'/88!.; $ $ ( ( !@ !#" ( ( !#" !A) .-0/ ( ( 9 9 ( %%'/; +1%'/8/!.(. *64 $ (> ) A( ' ) 0!#"B$ ( )! (A.-0/ ( >" 132 665,+1%'//!.( *84

(29) ( 4 ) .-0/ ( ( >#$ ) "/ %97<+ %'/88.2( */4

(30) (# (# 9& ! / @!&% C- $ 9 ! 9 !, !#"' ?9 (' .- ! ! ))#* ( +, ) %3( = 0 !#"B ( (76, ! " ( ) $' &" C ) ! "- /. ) %'/6$"; * ( %3( 6 # # > !#" * (0 @! ! # * 9 ! - ' ! $ #9& &@!" ! + - " 9 &9 &8 9 - 1 +-". 23+ ! !9 3 /54 " # 9 - # 9 9 -# " 9 - &9= , ! / 9 9 - > 9 # &9 ( = %')4 ( $' ) > (= 9 9 ) (= " !#" 5- ( * ( ) / + + @!09 .- 56 /!59#:)$"" ) & ?9 9 " 9 ! ' C-/ > (. ¹.

(31)