• Keine Ergebnisse gefunden

Localization and density of states of disordered low-dimensional systems in a magnetic field

N/A
N/A
Protected

Academic year: 2021

Aktie "Localization and density of states of disordered low-dimensional systems in a magnetic field"

Copied!
134
0
0

Wird geladen.... (Jetzt Volltext ansehen)

Volltext

(1)

                 !  "#$ ! $ % &$% % '()*() +),& % -./ 01* ! $ .$#$ . 2     !  3 4 56 7 01* ! $ 899:

(2)

K L BGM K DB K N ABH@>BC OB>PAB Q K L D DB K R =AM>H OA==AP>HH

;<=>?@=AB CAB DEFS<=>=EGHI J

K L BGM K DB K N ABH@>BC OB>PAB Q K L BGM K DB K T >UG VAF?@WA D>=<P CAB DEFS<=>=EGHI XY K XZ K Q XX[ \GBFE=]AHCAB CAF L B<^M<H_F><FF?@<FFAFI L BGM K DB K DG<_`>F a>b \GBFE=]AHCAB CAF L BGPG=EGHF><FF?@<FFAFI L BGM K DB K c G`>HC dEAFAHC>H_AB

DAW>H CAF a>?@eABAE?@F

L @bFEWI L BGM K DB K ;<^H=AB T <eAB

(3)

>HC =@A CAHFE=bGMF=>=AF GMMABPEGHFEH FGPA `GgkCEPAHFEGH>`CEFGBCABAC FbF=APF

K

l

@A jBF= S>B= EF ?GH?ABHAC gE=@ =@A `G?>`E]>=EGH GM A`A?=BGHF EH =@E?Wm CEFGBk

CABAC gEBAF EH =@A SBAFAH?A GM > ?GHF=>H= P>_HA=E? jA`C

K

l@A FA?GHC S>B= CA>`F

gE=@ FGPA SBGSAB=EAF GM nGPSGFE=A aABPEGHF EH =@A aB>?=EGH>` o<>H=<P

T >`` pMk

MA?= >HC gE=@ =@A BA`>=AC SBGe`AP GMMBAA MABPEGHF PGhEH_ EH > QD F=>=E?mB>HCGP

P>_HA=E? jA`C

K

fH S>B= J

gA EHhAF=E_>=A =@A >hAB>_AC `G?>`E]>=EGH `AH_=@ GM MABPEGHFEH > q<>FE J

D FbF=AP >F> M<H?=EGH GM> P>_HA=E?jA`C >SS`EAC SABSAHCE?<`>B=G =@A gEBA

K dA

F@Gg =@>= =@A `G?>`E]>=EGH `AH_=@ EH?BA>FAF gE=@ =@A P>_HA=E? jA`C eb ?>`?<`>=k

EH_ =@A ><=G?GBBA`>=EGH M<H?=EGH GM FSA?=B>` CA=ABPEH>H=F

K

aGB F=BGH_ jA`CFm =@A

`G?>`E]>=EGH `AH_=@ F>=<B>=AF >= =gE?A E=F h>`<A gE=@G<= P>_HA=E? jA`Cm EH >_BAAk

PAH= gE=@ SBAhEG<F >H>`b=E?>` >HC H<PABE?>` gGBW

K

l@A ?BGFFGhAB eA@>hEG<B GM

=@A `G?>`E]>=EGH `AH_=@ EF F@GgH =G eA _GhABHAC eb =@A P>_HA=E? S@>FA F@EM=EH_

B>=A

K

dA CABEhA =@EF q<>H=E=b >H>`b=E?>``b EH > _AHAB>`m?`GFAC MGBP

K

dA ?GPS>BA

G<B BAF<`=F gE=@ BA?AH= ArSABEPAH=>` gGBW

K

fH S>B= Qm ?@>S=AB smgA EHhAF=E_>=A =@A FSEH SG`>BE]>=EGH GM =@A ao

T

p _BG<HC

F=>=AF>=jrAC j``EH_ M>?=GBFgE=@EH > PGCA`GMFSEHM<`mHGHkEH=AB>?=EH_ nGPSGFE=A

aABPEGHF

K

dA F@Gg =@>= =B>HFE=EGHF eA=gAAH CEtABAH=`b SG`>BE]AC _BG<HC F=>=AF

>F > M<H?=EGH GM =@A P>_HA=E? jA`C G??<Bm EH >_BAAPAH= gE=@ BA?AH= ArSABEPAH=F

K

dA >`FG EHhAF=E_>=A =@A AtA?= GM =APSAB>=<BAm CEFGBCAB >HC FSEHkGBeE= F?>==ABEH_

GH =@A =B>HFE=EGHF

K

fH ?@>S=AB Z

gA ?GHFECAB =@A SBGe`AP GM > MABPEGH F<eUA?= =G > F=>=E? B>HCGP

P>_HA=E?jA`CgE=@ `>B_A PA>H

K

l@EFSBGe`AP EF>`FG EPSGB=>H=MGB=@A nGPSGFE=A

aABPEGH CAF?BES=EGH GM =@A aB>?=EGH>` o<>H=<P

T

>`` ptA?=

K

dA EHhAF=E_>=A =@A

SBGSAB=EAFGM=@A `G?>`E]AC F=>=AFg@E?@ CA=ABPEHA =@A FSA?=B<P EH =@A =>E`FGM=@A

CAHFE=b GM F=>=AF

K

dA ?>`?<`>=A =@A DAHFE=b GM

R

=>=AF EH =@A =>E`F >F > M<H?=EGH GM

=@A AHAB_b >HC =@A PA>H h>`<A GM=@A

cu

a gE=@EH =@A MB>PAgGBWGM=@A vS=EP<P

a`<?=<>=EGH u A=@GC K dA F@Gg =@>=mHA>B =@A ?AH=BAF GM V>HC>< e>HCFm=@A Dv R

EF > ;><FFE>H M<H?=EGH GM =@A AHAB_bm g@ABA>F =@A AHAB_b CASAHCAH?A GM=@A Dv

R

EF HGHk>H>`b=E? HA>B =@A e>HC AC_A

(4)

u

VGW>`EFEAB<H_FhAB@>`=AH <HC CEA y<F=>HCFCE?@=A hGH aABPEGHAH EH HEACBE_CEPAHk

FEGH>`AH

R

bF=APAH

K

DAB ABF=A lAE` eAF?@>M=E_=^ FE?@ PE=CAB VGW>`EFEAB<H_ hGH p`AW=BGHAH EH CE?WAHm

<H_AGBCHA=AH DB>^@=AH EH AEHAP WGHF=>H=AH

u

>_HA=MA`C

K

DAB ]gAE=A lAE` @>HCA`=

hGH AEHE_AH pE_AHF?@>M=AH hGH z{|}{~€ ‚ƒ|{„~ EP aB>W=EGH>`AH o<>H=AHk

@>``AtAW=… ao

T

p† <HC CAP ]<_A@GBE_^ AH

L

BGe`AP hGH FE?@ EH AEHAP F=>=EF?@AHm

]gAECEPAHFEGH>`AH y<M>``FP>_HA=MA`C eAgA_AHCAHm MBAEAH aABPEGHAH

K

fH lAE` J

<H=ABF<?@AH gEB CEA _APE==A`=A VGW>`EFEAB<H_`>^H_A hGH aABPEGHAH

EH AEHAP q<>FEkAEHCEPAHFEGH>`AH

R

bF=AP >`F a<HW=EGH CAF FAHWBA?@= ]<P DB>@=

>H_A`A_=AH

u

>_HA=MA`CAF

K

dEB eABA?@AH CEA w<=GWGBBA`>=EGHFM<HW=EGH CAB

R SAWk

=B>`CA=ABPEH>H=AH <HC ]AE_AHm C>FF CEA VGW>`EFEAB<H_F`>^H_A PE= CAP

u

>_HA=MA`C

>HF=AE_=

K

a<B^ F=>BWA aA`CAB F>==E_^ = CEA VGW>`EFEAB<H_F`>H^ _A eAEP ]gAEM>?@AH E@BAF

dAB=AF EP a>`` G@HA

u

>_HA=MA`Cm EH ^

‡eABAEHF=EPP<H_ ]< eAF=A@AHCAH >H>`b=EFk

?@AH <HC H<PABEF?@AH wBeAE=AH K pF gEBC _A]AE_=m C>FF C>F ^ ‡eAB_>H_FhAB@>`=AH hGH CAB P>_HA=EF?@AH L

@>FAHhABF?@EAe<H_ eAF=EPP= gEBC

K

dEB `AE=AH CEAF >H>k

`b=EF?@ EH >``_APAEHABm_AF?@`GFFAHAB aGBP @ABm<HC hAB_`AE?@AH <HFABA pB_AeHEFFA

PE=U<H^ _F=AH ArSABEPAH=A``AH wBeAE=AH

K

f H lAE`

Q

mO>SE=A` s <H=ABF<?@AH gEB CEA

R SEHSG`>BEF>=EGH CAF ao T p ;B<HC]<k F=>HCFeAE MAF=AH a<^``M>W=GBEH AEHAP u

GCA``hGH HE?@=kgA?@FA`gEBWAHCAHmFSEHeAk

@>M=A=AHz{|}{~€ ‚ƒ|{„~ K dEB ]AE_AH EH ^ ‡eABAEHF=EPP<H_ PE=U<^H_F=AH prk SABEPAH=AHmC>FF ^

‡eAB_>^H_A ]gEF?@AH hABF?@EACAH SG`>BEFEAB=AH ;B<HC]<F=>^HCAH

>`F a<HW=EGH CAF u >_HA=MA`CAF ><M=BA=AH K dEB <H=ABF<?@AH AeAHM>``F CAH pEHi<x CAB lAPSAB>=<Bm ‡HGBCH<H_ <HC R

SEHe>@HgA?@FA`gEBW<H_ ><M CEA ^ ‡eAB_>^H_A K fH O>SE=A` Z eA@>HCA`H gEB C>F L BGe`AP AEHAF aABPEGHF EH AEHAP F=>=EF?@AH y<M>``FP>_HA=MA`C PE= _BGFFAH u E==A`gAB= K DEAFAF L

BGe`AP EF= AeAHM>``F gE?@=E_

M<^B CEA z{|}{~€‚ƒ|{„k N AF?@BAEe<H_ CAF ao T p K dEB<H=ABF<?@AH CEA pE_AHk

F?@>M=AH CAB `GW>`EFEAB=AH y<F=>^HCAm gA`?@A C>F

R SAW=B<P EH CAH R ?@g>H^ ]AH CAB y<F=>HCFCE?@=A eAF=EPPAH K

dEB eABA?@HAH CEA y<F=>HCFCE?@=A EH CAH

R

?@g>^H]AH

>`F a<HW=EGH CAB pHAB_EA <HC CAF PE==`ABAH dAB=AF CAF y<M>``FP>_HA=MA`CF EP

c >@PAH CAB ˆ}€|‰| ‚Š‰‹€‰Œ€{„ u A=@GCA K dEB ]AE_AHm C>FF H>@A CAP yAHk

=B<P CAB V>HC><e>^HCAB CEA y<F=>HCFCE?@=A AEHA ;><xM<HW=EGH CAB pHAB_EA EF=m

gG@EH_A_AH CEA pHAB_EA>e@>^H_E_WAE= CAB y<F=>HCFCE?@=A >H CAH

N

>HCB>H^ CABH

HE?@= PA@B >H>`b=EF?@ EF=

(5)

“ ”•–—˜™š—›™•œ ™œ ™ž•Ÿ ¡Ÿ¡  ¢£ž›¡¤ž ¥ J K J dA>W VG?>`E]>=EGH K K K K K K K K K K K K K K K K K K K K K K K K K K Y J K J K J

DE>_B>PP>=E? >H>`bFEF

K K K K K K K K K K K K K K K K K K K K J Q J K J K Q l@A BG`A GM P>_HA=E? jA`CF K K K K K K K K K K K K K K K K K K J s J K J K¦ DAS@>FEH_ K K K K K K K K K K K K K K K K K K K K K K K K K K K JZ J K J K [ u

>_HA=E? EPS<BE=EAF >HC FSEHkGBeE= F?>==ABEH_

K K K K K K K J§ J K Q R =BGH_ VG?>`E]>=EGH K K K K K K K K K K K K K K K K K K K K K K K K K K J¨ J K Q K J R EH_`A L >B>PA=AB R ?>`EH_ K K K K K K K K K K K K K K K K K K K J¨ J K Q K Q l@A ©GHkVEHA>B ªk u GCA` K K K K K K K K K K K K K K K K K K K Q X « ¬—­œ¡›•˜•–—˜™š—›™•œ ™œ ™ž•Ÿ ¡Ÿ¡  ®¯—œ›¯¤ °™Ÿ¡ž «± Q K J l

@A ><=G?GBBA`>=EGH M<H?=EGH GM FSA?=B>` CA=ABPEH>H=F

K K K K K K K Q s Q K Q l

@A P>_HA=E? S@>FA F@EM=EH_ B>=A

K K K K K K K K K K K K K K K K K K ¦ J Q K Q K J

dEBAF gE=@ FSA?<`>B eG<HC>Bb ?GHCE=EGHF

K K K K K K K K K K ¦ Q Q K Q K Q L >B>eG`E? dEBA K K K K K K K K K K K K K K K K K K K K K K K K ¦ [ Q K¦ u

>_HA=G`G?>`E]>=EGH

K K K K K K K K K K K K K K K K K K K K K K K K K K ¦ § Q K [ c AFEF=>H?A GM q<>FEk J D gEBAF K K K K K K K K K K K K K K K K K K K K K [[ Q K s R

bPPA=Bb CASAHCAH?A GM VG?>`E]>=EGH EH DEFGBCABAC dEBAFI prk

SABEPAH=>` wH>`bFEF K K K K K K K K K K K K K K K K K K K K K K K K K K [ s ² ³´¡ ®¯—œ›¯¤ µ—˜˜ ¶·¡–›ž ¸¥ ¦K J l@A fH=A_AB o<>H=<P T >`` ptA?=I e>FE?F M>?=F K K K K K K K K K K K [Y ¦K Q l@A c

G`A GM DEFGBCABI VG?>`E]>=EGHkDA`G?>`E]>=EGH

lB>HFE=EGHF K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K s Q ¦K¦ l@A DAHFE=b GM R

=>=AFI SAB=<Be>=EhA >HC HGHSAB=<Be>=EhA ?>`?<`>k

=EGHF K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K s ¦ ¦K [

VG?>`E]>=EGH EH > F=BGH_ P>_HA=E? jA`C

K K K K K K K K K K K K K K K s § ¦K [ K J

aEA`C =@AGBA=E? >SSBG>?@

K K K K K K K K K K K K K K K K K K K s § ¦K [ K Q v=@AB >SSBG>?@AF K K K K K K K K K K K K K K K K K K K K K K K s ¨ ¦K s l @A aB>?=EGH>` o<>H=<P T >`` ptA?= K K K K K K K K K K K K K K K K s Y ¦K s K J L

BGSAB=EAF GM g>hAM<H?=EGHF EH =@A VV `AhA`

K K K K K K K K K s Y ¦K s K Q l@A V><_@`EH F=>=A K K K K K K K K K K K K K K K K K K K K K K ZX

(6)

¦K s

n@>B_AC pr?E=>=EGHFI aB>?=EGH>` n@>B_AF >HC aB>?=EGH>`

R =>=EF=E?F K K K K K K K K K K K K K K K K K K K K K K K K K K K K Z Q ¦K s K [ d@>= >eG<= DEFGBCABÀ K K K K K K K K K K K K K K K K K K K K Z ¦ ¸ Á•¤Â•ž™›¡ ៤™•œž ı [ K J l@A n@ABHk R

EPGHF=B>HFMGBP>=EGHI > FEH_<`>B;><_A =B>HFMGBP>=EGH Z s [ K Q l@A PA>H jA`C >SSBGrEP>=EGH K K K K K K K K K K K K K K K K K K K K Z§ [ K¦ l@A c >HCGP L

@>FA wSSBGrEP>=EGHI ?GHC<?=EhE=b >HC A`A?=BGP>_k

HA=E? BAFSGHFA M<H?=EGH K K K K K K K K K K K K K K K K K K K K K K K K Z¨ [ K [

l@A BG`A GM CEFGBCABI BA>` B>HCGP A`A?=BE? jA`CF >HC j?=E=EG<F B>Hk

CGP P>_HA=E? jA`CF K K K K K K K K K K K K K K K K K K K K K K K K K K § Q ± Á•¤Â•ž™›¡ ៤™•œž ř›´ žÂ™œ ÆÆ s K J

prSABEPAH=F GH FSEH SG`>BE]>=EGH EH =@A ao

T p K K K K K K K K K K K §§ s K Q L

G`>BE]>=EGH =B>HFE=EGHF >HC ?BGFFEH_F GM naF V>HC>< `AhA`F

K K K K ¨X

s

K¦ R

PGG=@AHEH_ GM =@A =B>HFE=EGHFI CEFGBCAB >HC FSEHkGBeE= AtA?=F

K K ¨ ¦ s K¦K J l@A BG`A GMj?=E=EG<FmEH@GPG_AHAG<FP>_HA=E?jA`CFI FAPEk ?`>FFE?>` =@AGBb K K K K K K K K K K K K K K K K K K K K K K K K ¨ ¦ s K¦K Q R SEHkGBeE= AtA?=F K K K K K K K K K K K K K K K K K K K K K K K ¨[ s K [

lAPSAB>=<BA F?>`EH_ GM =@A SG`>BE]>=EGH >HC =@A

FSEHkiES _>S K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K ¨ s Ä ¡œž™›£ •Ç ¢›—›¡ž •Ç ៤™•œž ™œ — ȗœ •¤ ¬—­œ¡›™– Ù¡˜  É¥ Z K J

a>E`<BA GM H>EhA SAB=<Be>=EhA =A?@HEq<AF

K K K K K K K K K K K K K K K ¨Y Z K Q R APE?`>FFE?>` BAF<`=F K K K K K K K K K K K K K K K K K K K K K K K K K K Y Q Z K¦ V>HC>< VAhA` =>E`F K K K K K K K K K K K K K K K K K K K K K K K K K K Y[ Z K¦K J

o<>`E=>=EhA SE?=<BAI VEMF@E=] >B_<PAH=

K K K K K K K K K K K Y¨

Z

K¦K Q

DABEh>=EGH GM =@A F>CC`A SGEH= Aq<>=EGH

K K K K K K K K K K K JX[

Z

K¦K¦

DEF?<FFEGH GM =@A BAF<`=F

K K K K K K K K K K K K K K K K K K K JX§ Æ Á•œ–˜¯ž™•œž ““² Ê ¶Ë—˜¯—›™•œ •Ç ›´¡ ¡™­¡œË—˜¯¡ž •Ç ›´¡ ”—˜—–¡ÌÍ¡˜›Ÿ—¤™ џ—›•Ÿ““± Í Ï•œÌ™œ›¡Ÿ—–›™œ­ Ÿ¡žÂ•œž¡ ǯœ–›™•œ •Ç ÁÞ ž¯ÐÑ¡–› ›• — –•ž›—œ› Í ““¥ Á ʖҜ•Å˜¡ ­¤¡œ›ž “«²

(7)

l@A S@bFE?FGMCEFGBCABAC FbF=APF gE=@ BAC<?AC CEPAHFEGH>`E=b@>FeAAH =@A F<ek

UA?= GM EH=AHFA =@AGBA=E?>` >HC ArSABEPAH=>` BAFA>B?@ EH =@A S>F=

¦ X

bA>BF

K

‡H=E` BA?AH=`bm CEFGBCAB g>F >`g>bF >FF<PAC =G eA GM A`A?=BGF=>=E? H>=<BA >HC

?GHHA?=AC gE=@ =@A SBAFAH?A GM EPS<BE=EAF >HC EPSABMA?=EGHF EH =@A ?BbF=>`

K

o<>H=<P EH=ABMABAH?A AtA?=F EH > CEFGBCABAC A`A?=BGF=>=E? SG=AH=E>`CB>F=E?>``b>Mk

MA?= =@A SBGSAB=EAF GM =@A A`A?=BGHE? g>hAM<H?=EGHF

K

w??GBCEH_ =G =@A ?A`AeB>=AC

F?>`EH_ =@AGBb GM `G?>`E]>=EGHm CEFGBCABACm HGHkEH=AB>?=EH_ GHAkCEPAHFEGH>` … J

>HC =gGkCEPAHFEGH>` …QD† FbF=APF >= =@A >eFG`<=A ]ABG GM =APSAB>=<BA >BA EHk

F<`>=GBF FEH?A >`` =@A GHAkS>B=E?`A g>hA M<H?=EGHF >BA `G?>`E]ACm HG P>==AB @Gg

gA>W =@A CEFGBCAB EF

K

l@A ArSGHAH=E>` `G?>`E]>=EGH GM>`` =@A AE_AHF=>=AF @>F eAAH

SBGhAH BE_GBG<F`b MGB J D >HC q<>FE J D FbF=APF K l@A Q D ?>FA EF PGBA CEÙ?<`= =G

=>?W`A FEH?A E= `EAF >= =@A AC_A eA=gAAH PA=>``E? >HC EHF<`>=EH_

K

vH =@A ?GH=B>Bbm

=@BAAkCEPAHFEGH>` FbF=APF Ar@EeE= > CEFGBCAB EHC<?AC PA=>`kEHF<`>=GB =B>HFE=EGH

… u fl† K fH =@A Q D >HC ¦

D ?>FAm =@A GH`b >h>E`>e`A BE_GBG<F BAF<`=F ?GH?ABH =@A

ArSGHAH=E>` `G?>`E]>=EGHGMA`A?=BGHFEH =@A =>E`FGMq<>H=<PkPA?@>HE?>``b >``GgAC

AHAB_b e>HCF

K

l@A >SS`E?>=EGH GM > P>_HA=E? jA`C EHC<?AF ?@>H_AF EH =@A S@>FA GM =@A g>hA

M<H?=EGHF >HC =@<F >tA?=F EH=ABMABAH?A AtA?=F

K c

G<_@`b FSA>WEH_m F<?@ AtA?=F

>BA BAC<?AC C<A =G =@A eBA>WEH_ GM =EPA BAhABF>` FbPPA=Bb

K

wF > ?GHFAq<AH?Am

EH =@A SBAFAH?A GM > F=BGH_ P>_HA=E? jA`Cm =@A `G?>`E]>=EGH `AH_=@ GM A`A?=BGHF EH

q<>FE J

D FbF=APF EF CG<e`AC

K

wF M>B >F QD FbF=APF >BA ?GH?ABHACm =@A EH=ABS`>b

GM CEFGBCAB >HC @E_@ P>_HA=E? jA`CF EF BAFSGHFEe`A MGB =@A fH=A_AB o<>H=<P

T >``

ptA?=

K

fH =@EF BA_EPAm V>HC>< `AhA`F >BA eBG>CAHAC EH=G e>HCF >HC F=>=AF >BA

`G?>`E]AC EH =@A =>E`F GM =@A e>HCF

K

©A>B =@A ?AH=BAF GM V>HC>< e>HCFm =@ABA

>BA FEH_<`>BE=EAF EH =@A `G?>`E]>=EGH `AH_=@ >F > M<H?=EGH GM =@A AHAB_b >HC =@<F

`G?>`E]>=EGHkCA`G?>`E]>=EGH =B>HFE=EGHF EHC<?AC eb =@A P>_HA=E? jA`C G??<B

K

fH =@A >eGhA F?AH>BEGm A`A?=BGHkA`A?=BGH EH=AB>?=EGHF AH=AB GH`b >= HGH ]ABG

=APSAB>=<BA >F > CAS@>FEH_ FG<B?A g@E?@ <SFA=F q<>H=<P EH=ABMABAH?A >HC `Gk

?>`E]>=EGH

K T

GgAhABm =@AEB AtA?=F ?>H eA PGBA CB>P>=E?

K

aGB EHF=>H?Am E= EF gA``

WHGgH =@>= nG<`GPe EH=AB>?=EGHF >`GHA ?>H EHC<?A >

u flm =@A FG ?>``AC u G==k T <ee>BC =B>HFE=EGH K

wH <HCABF=>HCEH_ GM P>Hb ArSABEPAH=>``b GeFABhAC

u flFm

(8)

CEFGBCAB >HC EH=AB>?=EGHF EH=G >??G<H=

K

wF BA_>BCF QD FbF=APF EH > F=BGH_ P>_HA=E? jA`Cm =@A EH=ABS`>b GM nG<`GPe

EH=AB>?=EGHF >HC CEFGBCAB `A>CF =G =@A aB>?=EGH>` o<>H=<P

T

>`` ptA?=

K

fH =@EF

BA_EPAm C<A =G EH=AB>?=EGHF >HC @E_@ P>_HA=E? jA`CFm A`A?=BGHF MGBP HAgm ?G`k

`A?=EhA _BG<HC F=>=AF

K ‡

H=E` BA?AH=`bm P<?@ GM G<B <HCABF=>HCEH_ GM =@EF AtA?=

g>F e>FAC GH`b GH =@A F=<Cb GM =BE>` _BG<HC F=>=A g>hAM<H?=EGHF >HC HG= GH >Hb

FbF=AP>=E? SAB=<Be>=EhA >SSBG>?@

K

c

A?AH=`bm > HAg =@AGBb GM =@A ao

T

p @>F eAAH CAhA`GSACm=@A FG ?>``AC nGPk

SGFE=A aABPEGH =@AGBb

K

dE=@EH =@A MB>PAgGBW GM =@EF =@AGBbm > HAg GeUA?=m =@A

nGPSGFE=A aABPEGH …na†m P>CA G<= GM GHA A`A?=BGH >HC >H AhAH H<PeAB GM i<r

q<>H=> >==>?@AC =G E=m EF EH=BGC<?AC

K

f= =<BHF G<= =@>= naF eA@>hA EH P>Hb BAk

FSA?=F `EWA gA>W`b ?GBBA`>=AC MABPEGHF gE=@ F=BGH_`b BAHGBP>`E]AC q<>FES>B=E?`A

AtA?=EhA S>B>PA=ABFC<A =G nakna EH=AB>?=EGHF

K

w= =@A HGHEH=AB>?=EH_ na q<>FEk

S>B=E?`A `AhA`m =@A ao T p GM A`A?=BGHF EF P>SSAC GH=G =@A fo T p GM naF K u >Hb EPSGB=>H= MA>=<BAF GM =@A ao T p ?>H eA >CCBAFFAC >= =@EF `AhA` K

D<A =G =@AEB ?GPSGFE=A H>=<BAm naF ArSABEAH?A > j?=E=EG<F B>HCGP P>_HA=E?

jA`C …

cu

a† EH >CCE=EGH =G =@A Ar=ABH>` GHA >HC =G =@A F?>`>B CEFGBCAB SG=AHk

=E>` C<A =G EPS<BE=EAF

K

vgEH_ =G E=F BA`Ah>H?A EH =@EF AtA?=EhA CAF?BES=EGH GM =@A

ao T pm=@A SBGe`AP GM> MABPEGH PGhEH_ EH > Q D cu

a @>F >==B>?=AC ?GHFECAB>e`A

EH=ABAF= BA?AH=`b

K

a<B=@ABPGBAmF=G?@>F=E?>``b EH@GPG_AHAG<F P>_HA=E? jA`CF ?>H

HGg eA ArSABEPAH=>``b BA>`E]AC EH h>BEG<F g>bF

K

l@EF SBGe`AP EF CEBA?=`b >PAH>e`A =G =@A =@AGBb GM HGHEH=AB>?=EH_ S>B=E?`AF >HC

E= @>F eAAH F@GgH =@>= >

Q

D A`A?=BGH _>F EH > F=>=E?

cu

a eA`GH_F =G =@A ?GHk

hAH=EGH>` <HE=>Bb <HEhABF>`E=b ?`>FF

K

l@ABAMGBAm >`` =@A F=>=AF >BA ArSA?=AC =G eA

`G?>`E]AC EH F<?@ FbF=APF g@AH =@A

cu a @>F ]ABG PA>H h>`<A >HC =@A o<>H=<P T >`` ptA?= G<_@= =G G??<B g@AH =@A >hAB>_A h>`<A GM =@A cu a EF `>B_A K

fH =@EF =@AFEF gA EHhAF=E_>=A =@A `G?>`E]>=EGH GM A`A?=BGHF EH q<>FE J

D gEBAF EH

=@A SBAFAH?A GM > P>_HA=E? jA`C >HC =@A `G?>`E]>=EGH GM MABPEGHF EH QD F<eUA?=AC

=G > B>HCGP P>_HA=E? jA`C gE=@ `>B_A PA>H h>`<A EH =@A =>E`F GM V>HC>< e>HCF

K

dA ?>`?<`>=A =@A `A>CEH_ ArSGHAH=E>` M>?=GB GM =@A CAHFE=b GM F=>=AF EH =@AFA =>E`F

K

dA >`FG F=<Cb =@A FSEH SG`>BE]>=EGH GM =@A ao

T

p _BG<HC F=>=AF EH =ABPF GM HGH

EH=AB>?=EH_ FSEHM<` naF

K

f

H?@>S=AB J

mgA _EhA>H EH=BGC<?=EGH=G FGPAGM=@A PGF=F<??AFFM<` >SSBG>?@AF

=G `G?>`E]>=EGH EH CEFGBCABAC FbF=APF

K

wM=AB CEF?<FFEH_ =@A gA>W `G?>`E]>=EGH

?GBBA?=EGHF =G =@A =B>HFSGB= SBGSAB=EAF GM F<?@ FbF=APFm gA =<BH =G =@A F?>`EH_

=@AGBb GM`G?>`E]>=EGH >HC =@AH =G > jA`Ck=@AGBA=E? >SSBG>?@m=@A F<SABFbPPA=BE?

HGHk`EHA>B ªkPGCA`

K

fH ?@>S=AB

Q

gA EHhAF=E_>=A =@A EHi<AH?A GM > P>_HA=E? jA`C GH =@A `G?>`E]>=EGH

SBGSAB=EAF GM A`A?=BGHF EH q<>FE J

D FbF=APF eb PA>HF GM > FSA?=B>` ?GBBA`>=EGH

M<H?=EGHm=@A ><=G?GBBA`>=EGH GM FSA?=B>` CA=ABPEH>H=

K

l@A >hAB>_AC `G?>`E]>=EGH

`AH_=@ EF CABEhAC >F =@A ?BGFFGhAB `AH_=@ F?>`A MBGP ?GBBA`>=AC =G <H?GBBA`>=AC

AHAB_b `AhA` F=>=EF=E?F

K l

(9)

`AH_=@ EH =@A ?BGFFGhAB BA_EGH eA=gAAH =@A GB=@G_GH>` AHFAPe`A …HG P>_HA=E?

jA`C† >HC =@A <HE=>Bb AHFAPe`A …PGCAB>=A`b F=BGH_ P>_HA=E? jA`C†

K

dA CABEhA

=@A P>_HA=E? S@>FA F@EM=EH_ B>=A >H>`b=E?>``b EH > hABb _AHAB>`m?`GFAC MGBP

K l@EF

q<>H=E=b EF F@GgH =G _GhABH =@A ?BGFFGhAB eA@>hEG<B GM =@A `G?>`E]>=EGH `AH_=@

K

aGB F=BGH_ jA`CFm gA F@Gg =@>= =@A `G?>`E]>=EGH `AH_=@ F>=<B>=AF >= =gE?A E=F

h>`<A gE=@G<= P>_HA=E? jA`CmEH >_BAAPAH=gE=@ SBAhEG<F >H>`b=E?>`gGBW

K

DEBA?=

q<>H=E=>=EhA ?GPS>BEFGH gE=@ BA?AH= ArSABEPAH=F EF _EhAH

K fH ?@>S=AB ¦ gA CEF?<FF =@A _AHAB>` =@AGBb GM =@A fo T p >HC =@A ao T p g@E`A EH ?@>S=AB [

gA EH=BGC<?A =@A =@AGBb GM FSEH`AFF nGPSGFE=A aABPEGHF

K

fH =@A

`>==AB ?@>S=AB gA >`FG ?>`?<`>=A ArS`E?E=`b =@A ?GBBA`>=GB GM =@A B>HCGP P>_HA=E?

jA`C ArSABEAH?AC eb naF

K

fH ?@>S=AB s gA EHhAF=E_>=A =@A FSEH SG`>BE]>=EGH GM =@A ao

T

p _BG<HC F=>=AF

>= jrAC j``EH_ M>?=GBF gE=@EH > PGCA` GM FSEHM<`m HGHkEH=AB>?=EH_ nGPSGFE=A

aABPEGHF

K

dA F@Gg =@>= =B>HFE=EGHF eA=gAAH CEtABAH=`b SG`>BE]AC _BG<HC F=>=AF

>F > M<H?=EGH GM =@A P>_HA=E? jA`C G??<Bm EH >_BAAPAH= gE=@ BA?AH= ArSABEPAH=F

K

l@A AtA?= GM =@A j?=E=EG<Fm F=>=E?

cu

a GH =@A FSEH SG`>BE]>=EGH =B>HFE=EGHF EF

>`FG EHhAF=E_>=ACm >F gA`` >F =@A AtA?= GM =APSAB>=<BA >HC FSEHkGBeE= F?>==ABEH_

K

f

H ?@>S=AB Z

gA ?GHFECAB > MABPEGH …A`A?=BGH GB nGPSGFE=A aABPEGH† EH

Q D

F<eUA?= =G > F=>=E?

cu

a gE=@ `>B_A PA>H h>`<A >HC gA EHhAF=E_>=A =@A SBGSAB=EAF

GM =@A `G?>`E]AC F=>=AF g@E?@ CA=ABPEHA =@A FSA?=B<P EH =@A =>E`F GM =@A CAHFE=b

GM F=>=AF

K

l@EF >H>`bFEF AH>e`AF <F =G ?>`?<`>=A =@A CASAHCAH?A GM =@A DAHFE=b

GM

R

=>=AF GH =@A AHAB_b >HC =@A PA>H h>`<A GM =@A

cu

a EH =@A =>E`F gE=@EH

=@A MB>PAgGBW GM =@A vS=EP<P a`<?=<>=EGH

u A=@GC K dA F@Gg =@>=m HA>B =@A ?AH=BAF GM V>HC>< e>HCFm=@A Dv R EF > ;><FFE>H M<H?=EGH GM=@A AHAB_bmg@ABA>F =@A AHAB_b CASAHCAH?A GM =@A Dv R EF HGHk>H>`b=E? HA>B =@A e>HC AC_A K

(10)
(11)

ߎ×àáØâàØŽ ؏ ãؒŽÔՑԑÕ

ä咐‘æ ’

fH =@EF ?@>S=AB gA >CCBAFF =@A q<>H=<PkPA?@>HE?>` PG=EGH GM > S>B=E?`A EH >

B>HCGPm F=>=E? SG=AH=E>`

K

D<A =G q<>H=<P EH=ABMABAH?A SBG?AFFAFm CEFGBCAB `Gk

?>`E]AF >`` =@A A`A?=BGHE? g>hAM<H?=EGHF EH GHAkCEPAHFEGH>` >HC =gGkCEPAHFEGH>`

FbF=APF

K

fH =@BAA CEPAHFEGHFm > =B>HFE=EGH MBGP PA=>``E? =G EHF<`>=EH_ eA@>hEG<B

G??<BF eb ?@>H_EH_ =@A F=BAH_=@ GM CEFGBCAB

K

dA F=>B= =@EF ?@>S=AB eb CEF?<FFEH_ =@A jBF= q<>H=<P ?GBBA?=EGHF =G =@A ?`>Fk

FE?>` eA@>hEG<B GM CEFGBCABAC FbF=APFm =@A gA>W `G?>`E]>=EGH ?GBBA?=EGHF

K l

@AH

=@A F<SSBAFFEGH GM =@EF AtA?= C<A =G > P>_HA=E? jA`C EF CEF?<FFACm >F gA`` >F =@A

AtA?= GM FSEHkGBeE= F?>==ABEH_ >HC FSEHkiES F?>==ABEH_

K

aEH>``bmgA MG?<FGH PGBA FGS@EF=E?>=ACmHGHkSAB=<Be>=EhA =@AGBEAFGMq<>H=<P

`G?>`E]>=EGH

K

wM=AB SBAFAH=EH_ =@A F?>`EH_ =@AGBb GM`G?>`E]>=EGHmgA eBEAib CEF?<FF

=@A F<SABFbPPA=BE? HGHk`EHA>B ªkPGCA`mg@E?@ @>F `AC =G FE_HEj?>H= >Ch>H?AF EH

=@A F=<Cb GM =@EF SBGe`AP

K

çèç éêëì íîïëðñòëóñîô

fH =@EF FA?=EGH gA gE``?GHFECAB > _>FGMHGHEH=AB>?=EH_ A`A?=BGHF g@E?@ ArSABEAH?A

> F?>`>BSG=AH=E>`õ …Ÿ†?BA>=AC eb > B>HCGP CEF=BEe<=EGH GMF=>=E? EPS<BE=EAF

K l@A

T

>PE`=GHE>Hö÷øQùúûõ …Ÿ†EF=EPAkBAhABF>` EHh>BE>H= …l

c f†

K

dA gE`` >FF<PA =@>=

=@A EPS<BE=EAF >BA CE`<=Am E

K A

K

eG=@ =@A aABPE g>hA`AH_=@ üý >HC =@A B>H_A þ GM

=@A EPS<BE=b SG=AH=E>` >BA P<?@ FP>``AB=@>H =@A PA>H MBAA S>=@ ÿ ýmg@ABA

ý EF=@A aABPEhA`G?E=b >HC  EF=@A =B>HFSGB==EPAmCAjHAC >F=@A ?@>B>?=ABEF=E?

=EPA >H A`A?=BGH ?>H =B>hA` eAMGBA =@A CEBA?=EGH GME=FhA`G?E=b EFB>HCGPE]AC …MGB>

k?GBBA`>=AC B>HCGP SG=AH=E>`  ?GEH?ECAF gE=@ =@A PGPAH=<P BA`>r>=EGH =EPA

m FAA F<eFA?=EGH J K J K J † K R

APE?`>FFE?>``bm=@A F=G?@>F=E? PG=EGH GMA`A?=BGHFEH > CEFGBCABAC PA=>`>= X EF CAF?BEeAC eb =@A N G`=]P>HH Aq<>=EGH  J  K fH =B>HFSGB= =@AGBb GM PAFGF?GSE?

(12)

FbF=APFm =@A `>B_AkCEF=>H?A >HC `GH_k=EPA eA@>hEG<B GM =@A S>B=E?`A …?GPS>BAC

=G ÿ >HC  BAFSA?=EhA`b† EF GM EH=ABAF=I EH =@EF `EPE= =@A

N

G`=]P>HH Aq<>=EGH

FEPS`EjAF CB>F=E?>``bm bEA`CEH_ =@A CEt<FEGH Aq<>=EGH

   ÷ …Ÿ  † X! … J K J †

l@A ?`>FFE?>`=B>HFSGB=GMCEFGBCABAC PA=>`FEF=@ABAeb ?@>B>?=ABE]AC eb =@A CEt<k

FEGH ?GHF=>H=



m g@E?@ EF BA`>=AC =G =@A ?GHC<?=EhE=b ª eb =@A pEHF=AEH BA`>=EGH

ª Q "÷#  m g@ABA  " EF =@A A`A?=BGH ?@>B_A >HC # =@A CAHFE=b GM F=>=AF K fH

=@EF SE?=<BAmq<>H=<P PA?@>HE?F GH`b CA=ABPEHAF =@A F?>==ABEH_ ?BGFF FA?=EGH MGB

A`A?=BGHkEPS<BE=b F?>==ABEH_ EH =@A

N

G`=]P>HH Aq<>=EGH

K

f= g>F BA>`E]AC EH Qm

¦

 =@>= M<HC>PAH=>` q<>H=<P EH=ABMABAH?A AtA?=F bEA`C

?GBBA?=EGHF =G =@A ?`>FFE?>` ?GHC<?=EhE=b

K l

@AFA AtA?=F ?>H eA q<>`E=>=EhA`b <Hk

CABF=GGC eb ?GHFECABEH_ =@A BA=<BH SBGe>eE`E=b GM > S>B=E?`Am E

K A

K

=@A SBGe>eE`E=b

ö$ MGB >H A`A?=BGH g@E?@ F=>B=AC >= > SGEH=

Ÿ >= =EPA X =G >BBEhA >= =@A F>PA SGEH= >= =EPA  K w??GBCEH_ =G aAbHP>Hm =@EF q<>H=E=b EF =@A PGC<`<F Fq<>BAC GM =@A

EH=A_B>` GhAB >`` S>=@F g@E?@ F=>B= >=

Ÿ >= X >HC AHC >= Ÿ >=  I ö$ %& ' ( ' %÷ & ' % ( ' %÷ û & ')*'+ ( ' (, '+ … J K Q † g@ABA ( ' m =@A >PS`E=<CA MGB =@A S>=@ -m EFI ( ' ArS./ 0 1 $ 2 3 … J ø QùúŸ4 ÷ ' …† û õ …Ÿ ' …†††5 … J K¦ † d@AH üý 6 ÿm GH`b ?`>FFE?>` S>=@F @>hA =G eA EH?`<CAC K T AH?Am =@A jBF= =ABP EH Aq<>=EGH J K Q

EF =@A ?`>FFE?>` BA=<BH SBGe>eE`E=b >HC =@A FA?GHC BASBAFAH=F =@A

EH=ABMABAH?A GMh>BEG<F>PS`E=<CAF

K

aGBPGF=S>=@F=@A EH=ABMABAH?A EFHG=AFFAH=E>`

FEH?A =@AEB S@>FAF CEtAB F=BGH_`b >HCm=@ABAMGBAm=@A EH=ABMABAH?A =ABPF >`FG CEtAB

F=BGH_`b

K T

AH?Am =@A >hAB>_A h>`<A GM =@A EH=ABMABAH?A =ABP gE`` h>HEF@ g@AH

F<PPEH_ GhAB >`` S>=@F Qm [  K T GgAhABm GgEH_ =G =@A l c f GM G<B PGCA`m =@A

>PS`E=<CAFGM=EPAkBAhABFAC =B>UA?=GBEAF >BA Aq<>` >HCm=@ABAMGBAm=@A EH=ABMABAH?A

eA=gAAH =@AP ?>HHG= eA HA_`A?=AC

K

wF > BAF<`=m =@A ?GH=BEe<=EGH =G ö$ GM =gG

=EPAkBAhABFAC S>=@F EF

% ( ' %÷ û % (7 ' %÷ û Q8" ( ' (, 7 ' [ % ( ' %÷ … J K [ † E K A K =gE?A E=F ?`>FFE?>` h>`<A K fH ?GBBAFSGHCAH?A =G =@EF AH@>H?APAH= GM `G?>`E]>k

=EGH GM A`A?=BGHFm =@ABA EF > CA?BA>FA EH ?GHC<?=EhE=b

K

VA=9F HGg AF=EP>=A =@A P>_HE=<CA GM =@EF ?GBBA?=EGH

K R

EH?A =@A q<>H=<P PAk

?@>HE?>` gEC=@ GM >H A`A?=BGH S>=@ EF : üým =@A hG`<PA ?GhABAC eb > S>=@ EH >H

EH=ABh>` 3  EF _EhAH eb ü; 7< ý ý 3  K vH =@A G=@AB @>HCm >H A`A?=BGH FSBA>CF GhAB > hG`<PA …  

†;=÷ >M=AB > =EPA EH=ABh>` 

K

(13)

dt

λ

F

F

v

?@ABCD EFEG HDIJDKCL IM N ODPMQ@RKDCODSK@RA TNKUF

=<eA GM gEC=@ : üý =G EH=ABFA?= E=FA`M EF =@A B>=EG GM =@AFA =gG hG`<PAFEH=A_B>=AC

GhAB >`` =EPAF …FAA j_

K J K J †I > : 1 VW =X Y 3  ü; 7< ý ý …   † ;=÷ ! … J K s†

l@A `GgAB `EPE= GM EH=A_B>=EGH EF  FEH?A =@A CEt<FEGH Aq<>=EGH P>WAF FAHFA GH`b

MGB =EPAF  Z



K

wF BA_>BCF =@A <SSAB `EPE=mE=9F =@A =EPA E= =>WAF MGB >H A`A?=BGH

=G FSBA>C GhAB =@A g@G`A F>PS`A … >FF<PEH_ =@>= [ EF =@A `EHA>B CEPAHFEGH GM =@A

F>PS`A†

K T

AH?Am =@A gA>W `G?>`E]>=EGH ?GBBA?=EGHF =G ª >BA  [  ª :  ª>  Q"÷ 0 \ ] ^ … [  ÿ†ø_ 3 J `H…[øÿ†ø_÷ 3 Q … ÿ 7<  [ 7< †ø_` 3 ¦ … J K Z † w=  Z X

m EHA`>F=E? F?>==ABEH_ SBG?AFFAF …"

7  " 7 >HC " 7  S@GHGH EH=AB>?k

=EGHF† =AHC =G CAF=BGb =@A ?G@ABAH?A GM =EPAkBAhABFAC S>=@F

K

fM gA EH=BGC<?A

> S@AHGPAHG`G_E?>`m kCASAHCAH= S@>FA ?G@ABAH?A =EPA a…† g@E?@ >??G<H=F

MGB >`` =@A S@>FA eBA>WEH_ SBG?AFFAFm > S@>FA ?G@ABAH?A `AH_=@ ?>H eA CAjHAC >F

[a…† b

 a

K

d@AH [a…†EFFP>``AB=@>H =@A >?=<>`CEPAHFEGH GM=@A F>PS`A

[ m [ P<F= eA BAS`>?AC eb [a…† EH MGBP<`>F J K s >HC J K Z …FAA FA?=EGH J K J K¦ † K dA g>H= =G F=BAFF @ABA >_>EH =@>= MGBP<`> J K Z EF h>`EC SBGhECAC =@>= üý 6 ÿ GBm Aq<Eh>`AH=`bm cýÿ d J … cý aABPE g>hA hA?=GB† K u GBAGhABm EH GHA >HC =gG

CEPAHFEGHFm E= M>E`F >= F<Ù?EAH=`b `>B_A `AH_=@ F?>`AF eA?><FA =@A BAC<?=EGH EH

?GHC<?=EhE=b _BGgF >F [ EH?BA>FAF

K

l@EF eA@>hEG<B F=BGH_`b F<SSGB=F =@A F?>`EH_

=@AGBb GM `G?>`E]>=EGHmg@E?@ SBACE?=F =@>= >`` =@A g>hAM<H?=EGHF >BA `G?>`E]AC EH J

C >HC QC FbF=APF C<A =G CEFGBCAB Qm s …FAA FA?=EGH J K Q K J † K

(14)

“f“f“ ™—­Ÿ—¤¤—›™– —œ—˜£ž™ž

l@A q<>H=<P ?GBBA?=EGHF =G =@A ?GHC<?=EhE=b >HC =@A BA=<BH SBGe>eE`E=b C<A =G

EH=ABMABAH?A ?>H eA ?>`?<`>=AC EH > PGBA BE_GBG<Fg>b eb =@A <FA GMCE>_B>PP>=E?

=A?@HEq<AF

K

dA gE`` ?GHFECAB =@A HGHEH=AB>?=EH_

T >PE`=GHE>H g  ÷ Qùú û õ …Ÿ† … J K § † g@ABA ùú EF =@A A`A?=BGH AtA?=EhA P>FF >HC õ …Ÿ† EF =@A B>HCGP A`A?=BGF=>=E? SG=AH=E>` K

dA gE`` >FF<PA =@>= õ …Ÿ† EF ;><FFE>H >HC k?GBBA`>=AC >HC =@>= E=F

PA>H EF ]ABGI hõ …Ÿ†i X hõ …Ÿ†õ …Ÿj†i k l …Ÿ  Ÿj † m %õ %÷ …Ÿ  Ÿj † … J K ¨ † g@ABA m EF =@A EPS<BE=b ?GH?AH=B>=EGH K l

@A CEF=BEe<=EGH M<H?=EGH GM =@A CEFGBk

CAB SG=AH=E>` EF > …õ† n ArS  …m %õ %÷† 7<o 3 Ÿ õ …Ÿ†÷ 

>HC =@A CEFGBCAB >hAB>_A EF

CAjHAC >F h !!! ip o 3 õ …Ÿ†> õ  !!! K fH =@A `EPE= GM gA>W CEFGBCABm üý 6 ÿm=@A A`A?=BGHE? FA`MkAHAB_b q ?>H eA >CAk

q<>=A`b AF=EP>=AC EH =@A FA`Mk?GHFEF=AH=

N GBH >SSBGrEP>=EGH R n N w …FAA j_ K J K Q † K

fH =@A BA_EGH ?`GFA =G =@A aABPE AHAB_bm %r

 rý%6 rým =@A EP>_EH>Bb S>B= GM q EF _EhAH eb  Z  I fPq…r  † k 0  Q_ m %õ %÷ #ý … J K Y † g@ABA #ý EF =@A DAHFE=b GM R =>=AF >= =@A aABPE AHAB_b >HC  EF =@A PGPAH=<P

BA`>r>=EGH =EPA >= =@A aABPE AHAB_b

K

aGB k?GBBA`>=AC SG=AH=E>`Fm EF ECAH=E?>`=G

=@A =B>HFSGB= =EPA 

K

l@A BA>` S>B= GM qm =@G<_@ CEhAB_AH=m ?>H eA >eFGBeAC

EH=G >H EBBA`Ah>H= BAHGBP>`E]>=EGH GM =@A AHAB_b

K

f= ?>H eA F@GgH =@>=m EM üý 6 ÿm CE>_B>PF g@ABA EPS<BE=b `EHAF ?BGFF ?>H eA

HA_`A?=AC …MGB =@A FA`M AHAB_b†  Z  K l @ABAMGBAm MGB `>B_A AHAB_EAF r : rým =@A >hAB>_AC BA=>BCAC >HC >Ch>H?AC

;BAAH9F M<H?=EGHF stuvw >BA _EhAH eb

stuvw …Â r† J r  rx y / 0 øQ … J K JX † g@ABA rx ö÷ø Qùú K aG<BEAB =>BHFMGBPEH_ pq K … J K JX †m gA _A= s tu vw …Ÿ  Ÿjz r† s tu vw 2 …Ÿ  Ÿjz r†" 7{ | }|+ { W~ … J K JJ † g@ABA s tuvw 2

EF =@A MBAA A`A?=BGH BA=>BCAC …BAFS

K

>Ch>H?AC† ;BAAH9F M<H?=EGH

K

s CA?>bF GH `AH_=@ F?>`AF GM =@A GBCAB GM ÿ C<A =G A`>F=E? EPS<BE=b F?>==ABEH_

B>HCGPE]EH_ =@A S@>FA GM =@A A`A?=BGHE? g>hAM<H?=EGH >PS`E=<CA …Ÿ 

†

K l@A

;BAAH9F M<H?=EGH s ?>HHG= =@<F _B>FS =@A q<>H=<P AtA?=F CAF?BEeAC EH FA?=EGH J

K J

mg@E?@ G??<B >= `>B_AB `AH_=@ F?>`AF

K

(15)

E

p,

p,

E

p’,

E

?@ABCD EFG ‚UDDPDSKCIR@SODPMQDRDCAL@RKUDƒ„…†F‚UDKU@S‡@RKDCRNPP@RDCDTCDODRKOKUDDˆNSK

MDCJ@IR@SHCDDR‰O MBRSK@IR NRŠ KUDŠNOUDŠ P@RD CDTCDODRKOKUD @JTBC@KL SICCDPNKIC ‹ŒŽF

[ Z

ÿ …GBm Aq<Eh>`AH=`bm GH =EPA F?>`AF  Z

 ÿøý†m GHA HAACF =G ?GHFECAB

q<>H=E=EAF ?GH=>EHEH_ @E_@AB PGPAH=F GM sm F<?@ >F =@A CEFGBCAB >hAB>_A GM =@A

A`A?=BGH CAHFE=b  …Ÿ † k h% …Ÿ  †%÷i GB =@A >hAB>_AC ?GHC<?=EhE=b ª K fH =@A

MG``GgEH_m gA gE`` ?GHFECAB =@A >hAB>_AC A`A?=BGH CAHFE=b

K R EH?A …Ÿ  † 1 3 Ÿjs t …Ÿ  zŸj  j† , …Ÿj  j† … J K J Q† …g@ABA  Z  j

†m =@A >hAB>_AC CAHFE=b ?>H eA gBE==AH >F

 …Ÿ  † 1 3 Ÿj 3 Ÿjj‘ …Ÿ  zŸj Ÿjj  j †…Ÿj  j †…Ÿjj  j† … J K J ¦ † g@ABA ‘ …Ÿ  zŸj Ÿjj  j† hst …Ÿ  zŸj  †s v …Ÿjj  jzŸ  †i ! … J K J[ †

DE>_B>PP>=E?>``bm =@EF >hAB>_A EF _EhAH eb =@A F<P GM =gG ?`>FFAF GM CE>_B>PFI

> ?`>FF g@ABA =@A =gG S>B=E?`A `EHAF >BA HG= ?GHHA?=AC eb EPS<BE=b `EHAFmbEA`CEH_

=@A SBGC<?= GM =@A EPS<BE=bk>hAB>_AC SBGS>_>=GBF >HC > ?`>FF g@ABA =@Ab >BA

?GHHA?=AC

K

l@A FA?GHC ?`>FF ?GH=>EHF =@A EH=ABAF=EH_ S@bFE?F

K

n`>FFE?>` CEt<FEGH … J K J[ † EF ’?GH=>EHAC’ EH =@A q<>H=<P ?GBBA`>=GB … J K J[ † K R

EH?A ?`>FFE?>``b =@A AhG`<=EGH GM =@A >PS`E=<CA …Ÿ 

† >HC E=F ?GHU<_>=A >PS`Ek

=<CA 

, …Ÿ

>`GH_ =@A CEt<FEGH S>=@ >BA HG= EHCASAHCAH=m=@A ?`>FFE?>` CEt<FEGH

EF CAF?BEeAC eb =@A `>CCAB CE>_B>PF F@GgH EH j_ J

K¦K

fH =@AFA CE>_B>PF =@A

EPS<BE=b `EHAF ?GBBA`>=A =@A AhG`<=EGH GM  >HC 

,

>= Aq<>` SGFE=EGHF >HC Aq<>`

=EPAF

K

l@A ?GH=BEe<=EGH GM =@A _AHABE? `>CCAB CE>_B>P gE=@  EPS<BE=b `EHAF ?>H eA gBE==AH >F “” …• –† … J K J s† g@ABA “ … • – † m %õ %÷ & • s t …Â j rj †s v …Â r †

(16)

   

,E

p

,E

p’

p

,E’

p’,E’

p’

,E’

p ,E

+

+

...

?@ABCD EF—G ‚UD ˜@™BOIR PNŠŠDC Š@NACNJOF

,E

p

,E

p’

p

,E’

p’,E’

p’

,E’

p ,E

+

+

...

?@ABCD EFšG ‚UD „IITDCIR SCIOODŠ Š@NACNJOF

J û /  –   › ÷ … J K JZ † >HC  j Âû •mr j r û– K R <PPEH_ <S =@A _AGPA=BE? FABEAFGM`>CCABCE>_B>PF

…=@A FG ?>``AC DEt<FGH†mgA _A=

 …• – † J ø / – û › ÷ … J K J§ †

g@E?@ EF EHCAAC =@A aG<BEAB=B>HFMGBP GM=@A SBGS>_>=GB GM =@A CEt<FEGH Aq<>=EGH

… J K J †I  …Ÿ  zŸj  j† œ …    j† … [ _  …   j †† ;=÷ " 7  |}| +ž W Ÿ   ¡ } ¡ + ž … J K J¨ †

d@>= >eG<= EH=ABMABAH?A AtA?=FÀ fM gA =gEF= =@A `>CCAB CE>_B>PF GM j_

K J

>HC =@AH ArS`GE= =@A =EPAkBAhABF>` EHh>BE>H?A GM G<B FbF=APm gA _A= =@A FG ?>``AC

P>rEP>``b ?BGFFAC CE>_B>PF CASE?=AC EH j_

K J

K [

K

l@AFA CE>_B>PF ?`A>B`b CAF?BEeA

=@A ?GBBA`>=EGH GM =EPA BAhABFAC S>=@F >HC >BA =@<F BAFSGHFEe`A MGB =@A gA>W

`G?>`E]>=EGH ?GBBA?=EGHF =G =@A BA=<BH SBGe>eE`E=b

K

l@A F<P GM P>rEP>``b ?BGFFAC

CE>_B>PF EF ?>``AC > nGGSABGH >HC EF _EhAH eb ¢…Â

Âj – † k  …Â û Âj – †  [ m Z  K

dA ?>H HGg =>WA > ;><FFE>H g>hA S>?WA= GMgEC=@ þ SA>WAC >= =@A GBE_EH >= =EPA

 X

>HC ?>`?<`>=A =@A =APSGB>` AhG`<=EGH GM =@A >hAB>_AC CAHFE=b  …Ÿ  † K fM gA

EH?`<CA eG=@ `>CCAB CE>_B>PF >HC ?BGFFAC CE>_B>PF EH pq

K … J K J ¦ †mgA _A=  …Ÿ  † J … [ _   † ; =÷ …" 7 £ W Ÿ  ¡ û " 7 £ W W † œ …  † … J K JY †

(17)

g@ABA œ …  † EF =@A F=AS M<H?=EGH K f= MG``GgF MBGP … J K JY † =@>= =@A >hAB>_AC CAHFE=b >= Ÿ X

EF =gE?A >F `>B_A >F ArSA?=AC MBGP =@A CEt<FEGH Aq<>=EGH C<A =G =@A

nGGSABGHm EH >_BAAPAH= gE=@ =@A q<>`E=>=EhA ?GHFECAB>=EGHF GM FA?=EGH J

K J

K

“f“f« ³´¡ Ÿ•˜¡ •Ç ¤—­œ¡›™– ¦¡˜ ž

n`>FFE?>``bm =@A =B>HFSGB= SBGSAB=EAF GM A`A?=BGHF EH > PA=>` >BA GH`b gA>W`b >Mk

MA?=AC eb > ?GHF=>H= P>_HA=E? jA`C §m >F `GH_ >F =@A EHAq<>`E=b

–¨  © J …g@ABA –¨ "§ø ùª EF =@A ?b?`G=BGH MBAq<AH?b† @G`CF  J  K aGB `>B_AB § …F<?@ =@>= – ¨ « J

†m =@A eAHCEH_ GM ?`>FFE?>` =B>UA?=GBEAF GH =@A F?>`A GM =@A PA>H MBAA S>=@

ÿ EF q<E=A CB>P>=E? >HC EH=ABAF=EH_ S@AHGPAH> G??<B

K

o<>H=<P PA?@>HE?>``bm > gA>W P>_HA=E? jA`C ?>H F=BGH_`b >tA?= =@A dVk

?GBBA?=EGHF=G ª FEH?A E=eBA>WF =@A =EPA BAhABF>` FbPPA=Bb  § m ¨  K l@A >PS`E=<CAF

GM =gG =EPAkBAhABFACm FA`MkEH=ABFA?=EH_ S>=@F - >HC



- >?q<EBA >CCE=EGH>` S@>FA

M>?=GBFI ( ' ¬ ( ' ArS  / " 0ª ­ ' Ê ® 3 Ÿ  ( ' ArS  / _§¯ ° 2  (7 ' ¬ ( ' ArS  / _§¯ ° 2  … J K Q X † g@ABA ° 2 EF =@A q<>H=<P GM P>_HA=E? i<r >HC ° EF =@A P>_HA=E? i<r =@BG<_@ =@A `GGS -K l

@A S@>FA CEtABAH?A eA=gAAH =gG =EPAkBAhABFAC S>=@F EF ±²„€m

³´ Q_

° ø

°

2

m e<= h>BEAF ?GHFECAB>e`b MGB CEtABAH= `GGSF

K l

@ABAMGBAm =@A gA>W

`G?>`E]>=EGH =ABP >?q<EBAF > B>HCGP S@>FA CASAHCEH_ GH =@A `GGS FE]A

K l@A

B>HCGPHAFF GM =@A S@>FA <`=EP>=A`b `A>CF =G =@A F<SSBAFFEGH GM nGGSABGHF >HC

=@<F =G =@A CAF=B<?=EGH GM =@A q<>H=<P EH=ABMABAH?A AtA?=F BAFSGHFEe`A MGB dVI

=@A P>_HA=E? jA`C >?=F >F >H µ‹€¶ CAS@>FEH_ FG<B?A

K

l@A gA>W `G?>`E]>=EGH

?GBBA?=EGHF =G ª >BA HGgI

ªøª : 1 VW =X Y 3  ü; 7< ý ý …  † ; =÷ h "·¸ a u $ w i … J K Q J † g@ABA h !!!

i EHCE?>=AF >H >hAB>_A GhAB >`` ?`GFACm CEt<FEhA S>=@F

K ‡SGH >hAB>_EH_m gA _A= hA rS… / ³´ …††i k ArS…  

ø¹†m g@ABA ¹ EF >H AtA?=EhA CAS@>FEH_ =EPA

C<A =G §

K

l@A `>==AB q<>H=E=b ?>H eA AF=EP>=AC eb >B_<EH_ =@>= EH=ABMABAH?A EF

CAF=BGbAC g@AH =@A P>_HA=E? i<r =@BG<_@ > CEt<FEhA `GGS GM FE]A …

 ¹† < =÷ EF GM =@A GBCAB GM ° 2 I §  ¹ : ° 2 º ¹ : ÿ÷ ¹ ø  ! … J K QQ†

f= EF HGg F=B>E_@=MGBg>BC =G AF=EP>=A =@A P>_HA=G?GHC<?=EhE=bI

ª…§†  ª¹ * 2 : "÷ 0 » `H ¼ ¹ VW ½¾ 3 Q ¿ ¼ ¹ ½¾ 3 ¦ … J K Q ¦ †

(18)

   

l@A EH?BA>FA GM ª >F > M<H?=EGH GM § …HA_>=EhA P>_HA=GBAFEF=>H?A† @>F eAAH Gek

FABhAC MGB > `GH_ =EPA K w= Z X m[ P<F= eA BAS`>?AC eb [a >HC =@A ?@>B>?=ABEF=E?

P>_HA=E? jA`CF >BA CA=ABPEHAC eb =@A ?GHCE=EGH ÿ¹ : [am E

K A K m § : 0ªø"[÷ a K l@EF ?GHCE=EGH EF Aq<Eh>`AH= =G –¨  : 0øaÀý © 0øÀý K fH G<B PGCA` 0øÀý 6 J >HC

=@ABAeb =@A ?@>B>?=ABEF=E? jA`CF >BA FG gA>W =@>= =@A ?GHhAH=EGH>` P>_HA=GBAFEFk

=>H?A EF SB>?=E?>``b ]ABG

K

aGB gEBAF gE=@ CEt<FEhA gEC=@ l d ÿm=@A P>_HA=E? B>=A h>BEAF q<>CB>=E?>``b

gE=@ =@A P>_HA=E? jA`Cm J øa… §†  ¼ W ½ W ¯§÷øÁÂmg@ABA ¯ EF =@A ?BGFF FA?=EGH GM

=@A gEBAm>HC =@A ?GHF=>H= ÁÂ CASAHCF GH =@A _AGPA=Bb GM=@A gEBAm=@A CEBA?=EGH

GM =@A P>_HA=E? jA`C >HC =@A F?>==ABEH_ PA?@>HEFPF  Y



K

aGB Ar>PS`Am MGB > Qk

CEPAHFEGH>` gEBA GM CEt<FEhA ?BGFF FA?=EGH EH > SABSAHCE?<`>B P>_HA=E? jA`Cm E=

bEA`CF ÁÂ

¦K

aGB > gEBA gE=@ e>``EF=E? ?BGFF FA?=EGHm l à ÿm>HC > P>_HA=E? jA`C SABSAHCE?k

<`>B =G E=F ?BGFF FA?=EGHm =@A P>_HA=E? jA`C CASAHCAH?A GM =@A gA>W `G?>`E]>=EGH

?GBBA?=EGH =G =@A ?GHC<?=EhE=b EF gA>WAHAC eb i<r ?>H?A``>=EGH AtA?=F C<A =G

eG<HC>Bb F?>==ABEH_  JX



K f

M =@A P>_HA=E? jA`C EF FG FP>`` =@>= `AFF =@>H GHA

i<r q<>H=<P ´

2

Īø" EF SAHA=B>=EH_ >H >BA> lÿm =@A AtA?=EhA CAS@>FEH_ B>=A J

øa… §† EF q<>CB>=E?>``b EH?BA>FEH_ >F MGB CEt<FEhA ?BGFF FA?=EGHF

K T

GgAhABm E=F

F`GSA EF FP>``ABeb >=`A>F= > M>?=GB løÿm>F > ?GHFAq<AH?A GM=@A i<r ?>H?A``>=EGH

AtA?= GM AC_A =G AC_A FWESSEH_ GBeE=F  JX m JJ  K d@AH §lÿ d ´ 2

m =@A AtA?=EhA CAS@>FEH_ B>=A J

øa…§† g>F MG<HC eb > FAPEk

?`>FFE?>` PA=@GC =G EH?BA>FA GH`b `EHA>B`b gE=@ =@A P>_HA=E? jA`C §  JX

m JJ



K

lG ?GH?`<CAm > gA>W P>_HA=E? jA`C ?>H?A`F =@A `A>CEH_ GBCAB dV ?GBBA?=EGHF

=G =@A CEt<FEGH ?GHF=>H= >HC =@A ?GHC<?=EhE=b

K T

GgAhABm @E_@AB GBCAB ?GH=BEe<k

=EGHF MBGP =@A ±µ‰~{„ |{±~ >`GHA `A>C =G > BAHGBP>`E]>=EGH GM =@A CEt<FEGH

?GHF=>H= AhAH EH =@A <HE=>Bb AHFAPe`AF

K

f= EF CEÙ?<`= =G Ah>`<>=A @E_@AB GBCAB

?GH=BEe<=EGHF CE>_B>PP>=E?>``bm FEH?A =@A U<H?=EGHF eA=gAAH CEt<FGH `>CCAB CEk

>_B>PF …FG ?>``AC ’

T

EW>PE eGrAF’† HAAC =G eA CBAFFAC eb > H<PeAB GM EPS<BE=b

`EHAF

K

l@A SBGe`AP EF AhAH @>BCAB =G =>?W`A gE=@EH FAPE?`>FFE?>` F?@APAF

K fH FA?=EGH J K Q K

Q gA gE`` EH=BGC<?A > jA`C =@AGBA=E? >SSBG>?@ g@E?@ =<BHF G<= =G eA

ECA>` MGB FG`hEH_ SBGe`APF g@E?@ _G eAbGHC `Gg GBCAB SAB=<Be>=EhA ?GBBA?=EGHF

K

u

GBAGhABm gA gE`` FAA =@>= @E_@AB GBCABm F<e=`A q<>H=<P EH=ABMABAH?A AtA?=F CG

`A>C =G `G?>`E]>=EGH GM >`` AE_AHF=>=AF EH `GgkCEPAHFEGH>` FbF=APFm AhAH EH =@A

SBAFAH?A GM > P>_HA=E? jA`C

K

“f“f² ¡Â´—ž™œ­

w= HGH]ABG `Gg =APSAB>=<BAFm =@A EH=AB>?=EGH GM =@A A`A?=BGH gE=@ =@A AHhEBGHk

PAH= S>B=E>``b CAF=BGbF =@A gA>W `G?>`E]>=EGH AtA?=F

K

pHAB_b Ar?@>H_A eA=gAAH

=@A A`A?=BGH >HC =@A AHhEBGHPAH= <SFA=F =@A ?GHF=B<?=EhA EH=ABMABAH?A eA=gAAH

=EPAkBAhABFAC S>=@Fm =@<F F<SSBAFFEH_ =@A nGGSABGH PGCAF

K

(19)

    EH>H= CAS@>FEH_ PA?@>HEFP >HC a EF _EhAH eb  7Å mg@ABA Æ Qø ¦ MGB > J kC gEBA >HC Æ J MGB > QkC j`P  J Qm J ¦  K

D<A =G A`A?=BGHkS@GHGH F?>==ABEH_m =@A ArSGHAH= ?BGFFAF GhAB =G Æ [

>=

@E_@ABme<=HG==GG @E_@m=APSAB>=<BAFmM<`j``EH_=@A EHAq<>`E=b cÇ 6 …0÷

øÀý†ÈÂm

g@ABA ÈÂ EF=@A GS=E?>`DAebA S@GHGH MBAq<AH?b  J[



K l

@EF SGgAB ?>H eA FP>``ABm

C<A =G =@A ?GHjHAPAH=m EH q<>H=<P gEBAF

K w= AhAH @E_@AB =APSAB>=<BAFm =@ABA EF > ?BGFFGhAB =G Æ ¦ …EH FGPA >PGBS@G<F PA=>`Fm =G Æ Q † g@E?@ EF >`FG C<A =G A`A?=BGHkS@GHGH EH=AB>?=EGHF  J[  K L

@>FA ?G@ABAH?A EF >`FG gA>WAHAC eb EPS<BE=EAF gE=@ EH=ABH>` CA_BAAF GM MBAAk

CGPmF<?@ >F P>_HA=E? EPS<BE=EAF

K

dA gE`` eBEAib CEF?<FF P>_HA=E? EPS<BE=EAFEH

=@A HAr= FA?=EGH

K

fH J

D >HC QD FbF=APFm =@A S@AHGPAHG`G_E?>` CAjHE=EGH GM a _EhAH EH FA?=EGH J

K J

EF HG= >SS`E?>e`A >= Ar=BAPA`b `Gg =APSAB>=<BAFm g@AH =@A S@>FA ?G@ABAH?A

`AH_=@ [a eA?GPAF ?GPS>B>e`A =G =@A `G?>`E]>=EGH `AH_=@ [

¨

GM A`A?=BGHE? g>hAk

M<H?=EGHF

K

fM =@A FbF=AP FE]A [ EF >`FG `>B_AB=@>H [¨m=@A SAB=<Be>=EhA =BA>=PAH=

GMCEFGBCAB G<=`EHAC EH =@A SBAhEG<F FA?=EGHF eBA>WF CGgH >HC > HGH SAB=<Be>=EhA

=@AGBb EF ?>``AC MGB

K

“f“f¸ ¬—­œ¡›™– ™¤Â¯Ÿ™›™¡ž —œ  žÂ™œÌ•ŸÐ™› ž–—››¡Ÿ™œ­

fH =@A SBAFAH?A GM P>_HA=E? EPS<BE=EAFm q<>H=<P EH=ABMABAH?A EF S>B=E>``b CAk

F=BGbAC FEH?A =@A A`A?=BGHE? S>=@ EF S>B=E>``b WAS= =B>?W GMm C<A =G =@A iESSAC

EPS<BE=b FSEHF BAhA>`EH_ =@A hEFE= GM >H A`A?=BGH

K

w S@AHGPAHG`G_E?>` P>_HA=E?

F?>==ABEH_ B>=A É ?>H eA EH=BGC<?AC =G >??G<H= MGB =@EF HAg F?>==ABEH_ PA?@>k

HEFP

K

l@A EH=AB>?=EGH AHAB_b GM >H A`A?=BGH gE=@ > F@GB=kB>H_A P>_HA=E? EPS<BE=b

`G?>=AC >= È · ?>H eA gBE==AH >F g É õÉÊ Ë ®¢ · …Ÿ  È · †mg@ABA Ê Ë >HC ¢ · CAHG=A =@A

A`A?=BGH >HC EPS<BE=b FSEHm BAFSA?=EhA`bm >HC õÉ EF =@A F=BAH_=@ GM =@A EPS<BE=bk

A`A?=BGH P>_HA=E? EH=AB>?=EGH

K

wFF<PEH_ =@>= =@A P>_HA=E? EPS<BE=EAF >BA CE`<=A

…E

K A

K

 6 Ɇm=@A F?>==ABEH_ B>=A 0øÉ =>WAF =@A MGBP  J s I 0 É Q_#ý …Ìm %õÉ %÷h ¢Í ÷ i† … J K Q [ †

g@ABA Ìm EF =@A P>_HA=E? EPS<BE=b ?GH?AH=B>=EGH >HC h !!!

i CAHG=AF >H >hAB>_A

GhAB =@A … >FF<PAC† B>HCGP CEBA?=EGHF GM=@A EPS<BE=b FSEHF

K

fMÉ d a…†m=@ABA

EF HG =APSAB>=<BA CASAHCAH?A GM =@A ?GHC<?=EhE=b

K

R

SEHkGBeE=F?>==ABEH_mEH ?GH=B>F= =G P>_HA=E?kEPS<BE=bF?>==ABEH_mCGAFHG=hEk

G`>=A =EPAkBAhABF>`FbPPA=Bb

K T

GgAhABmE=FAtA?= GH =@A ?GHC<?=EhE=b EFHGH =BEhk

E>` FEH?A E= BAhABFAF =@A FE_H GM =@A q<>H=<P ?GBBA?=EGH >HC BAC<?AF =@A F=BAH_=@

eb > M>?=GB GM =gG  J s K l@A ?GHC<?=EhE=b EF =@AH ŠŒƒ Î ƒ =@>H ?`>FFE?>``b ArSA?=AC K

dA gE`` HGg SBGhECA > S@bFE?>` SE?=<BA MGB =@A FSEHkGBeE= AtA?=F MG``GgEH_

BAMABAH?A  JZ



K

(20)

e Ï    Ð

>FF<PEH_ > B>H_A GM =@A EPS<BE=b SG=AH=E>` 6 ÿ†I

õ û / õÉÑ…Ò Ò Òj† ® Ê Ë k õ … J û /Ó ® Ê Ë † … J K Qs† g@ABA Ó õÉÑøõ …Ò ÒÒ j †m Ò >HC Òj

>BA =@A A`A?=BGH PGPAH=> eAMGBA >HC >M=AB

F?>==ABEH_

K l

@EFGSAB>=EGHbEA`CF> BG=>=EGHGM Ê

Ë eb >H_`AF Ó · >BG<HC =@A FSEH >rEF K

d@AH > S>B=E?`A EFF?>==ABAC MBGP F=>=A %Òi%Ê

Ë i=G F=>=A %  Ò i%Ê Ë j im=@A >??<P<`>=EhA

AtA?=FGH =@A FSEH >CC <S =G > jHE=A BG=>=EGH

8 F<?@ =@>= %Ê Ë j i 8 %Ê Ë i K l@A P>=BEr GM =@A FSEH BG=>=EGH EFI 8  ?GF… œ øQ†" · uÔÕ a w=÷ / FEH… œ øQ†" 7 · uÔ 7 a w=÷ / FEH… œ øQ†"·uÔ 7 a w=÷ ?GF… œ ø Q †" 7 · uÔÕ a w=÷  … J K Q Z † g@ABA ° m œ >HC ´ >BA p<`AB >H_`AF …FAA  J§ † K

aGB =@A =EPAkBAhABFAC F?>==ABEH_

SBG?AFFm =@A jH>` F=>=A EF %  Ò i%Ê Ë jj im g@ABA %Ê Ë jj i 8 7< %Ê Ë i K l @ABAMGBAm =@A

EH=ABMABAH?A =ABP gE`` ?GH=>EH >H >CCE=EGH>`M>?=GB h Ê Ë jj %Ê Ë j i h Ê Ë %8÷%Ê Ë i K wFF<PEH_ =@>= %Ê Ë i % û J ø Q

im=@A `>==AB P>=BEr A`APAH= eA?GPAF

h Ê Ë jj %Ê Ë j i  FEH ÷ … œ ø Q † û ?G F÷ … œ øQ†ArS… / ° û / ´† ! … J K Q § †

fM =@ABA EF HG FSEH iESm =@AH >`` >H_`AF >BA ]ABG >HC =@A P>=BEr A`APAH= EF J

I gA

@>hA =@A BA_<`>BgA>W `G?>`E]>=EGH ?GBBA?=EGH

K

fM=@A FSEHkGBeE= ?G<S`EH_ EFF=BGH_m

=@A FSEH F=>=A CEt<FAF GH =@A <HE= FS@ABA >HC =@A GBEAH=>=EGHF GM =@A jH>` FSEH

F=>=AF >BA F=>=EF=E?>`

K

l@ABAMGBA =@A >hAB>_A GM … J K Q § † bEA`CF =@A M>?=GB  J øQ …GM

?G<BFAm =@EF BAF<`= CGAF HG= CASAHC GH =@A EHE=E>` F=>=A %Ê

Ë i†

K

wF CEF?<FFAC >eGhAm >H Ar=ABH>` P>_HA=E? jA`C CAF=BGbF =EPA BAhABF>` EHh>BEk

>H?A >HC >?=F >F >H AtA?=EhA CAS@>FEH_ MGB?A

K T

AH?Am EH =@A ?>FA GMFbF=APF gE=@

F=BGH_ FSEHkGBeE= F?>==ABEH_m=@A ?GHC<?=EhE=b F@G<`C CA?BA>FA g@AH =@A P>_HA=E?

jA`C EF =<BHAC GH  J ¦  K l

@EF SBACE?=EGH @>F eAAH ?GHjBPAC ArSABEPAH=>``b  JZ



K

çèÖ ×óØîôÙ íîïëðñòëóñîô

“f«f“ ¢™œ­˜¡ ڗŸ—¤¡›¡Ÿ ¢–—˜™œ­

l@A CEPAHFEGH`AFF?GHC<?=>H?A GM> CkCEPAHFEGH>`P>?BGF?GSE?>``b@GPG_AHAG<F

@bSAB?<eA GM `EHA>B CEPAHFEGH V EF CAjHAC >F

Û … [† k s… [† "÷ ø0 ! … J K Q ¨ † g@ABA s… [† ª[ ; 7 ÷ K

w??GBCEH_ =G =@A GHAkS>B>PA=AB F?>`EH_ =@AGBb GM `Gk

?>`E]>=EGH Qm s

m >=  X

=@A =B>HFSGB= SBGSAB=EAF GM > CEFGBCABAC FbF=AP >BA

CA=ABPEHAC eb

Û

…[† >`GHA

K u

GBA FSA?Ej?>``bmEM gA ?GHFECAB > @bSAB?<eA GM `EHA>B

(21)

e Ï    Ý ?GHC<?=>H?A GM =@A `>B_A ?<eA CASAHCF GH`b GH Ü >HC =@A ?GHC<?=>H?A GM A>?@ e`G?WI Û …Ü[† Þ…Ü Û …[†† … J K Q Y †

aGB > ?GH=EH<G<F h>BE>=EGH GM =@A `EHA>B CEPAHFEGH GM =@A FbF=APm gA _A= eb

CEtABAH=E>=EH_ Aq<>=EGH … J K Q Y † Qm sm Z I 3 `H Û 3 `H[ [ Û 3 Û … Ü[† 3 [ J Û 3 Û … Ü[† 3 Ü J Û 3 ޅ Ü Û † 3 Ü k ߅ Û † ! … J K¦ X †

l@ABAMGBAm =@A `G_>BE=@PE? CABEh>=EhA GM

Û EF FG`A`b > M<H?=EGH GM Û E=FA`M K l@EF >B_<PAH= EF GM ?G<BFA h>`EC GH`b EM [ EF `>B_AB =@>H >Hb G=@AB PE?BGF?GSE? `AH_=@

F?>`A GM =@A FbF=AP …üým ÿ†

K fM ߅ Û † Z X

m =@A ?GHC<?=>H?A EH?BA>FAF gE=@ =@A FE]A GM=@A F>PS`Am >HC hE?Ahk

ABF> MGB HA_>=EhA ß

K l

@ABAMGBAm =@A F?>`EH_ M<H?=EGH ß FSA?EjAF =@A =B>HFSGB=

SBGSAB=EAF MGB > FbF=AP EH =@A =@ABPGCbH>PE? `EPE=

K

l@A `EPE=EH_ eA@>hEG<B MGB

=@A ßkM<H?=EGH ?>H eAA>FE`bGe=>EHACI EH=@A `EPE=GMgA>WCEFGBCABmE

K A K MGB Û d J m

=@A FbF=AP F@G<`C F@Gg PA=>``E? eA@>hEG<B >HC =@A ?GHC<?=>H?A EF CAF?BEeAC eb

?`>FFE?>` =B>HFSGB= =@AGBbI s… [† ª[ ; 7 ÷ K T AH?A ߅ Û † 3  Q CASAHCF GH`b GH

=@A CEPAHFEGH>`E=b GM =@A FbF=AP

K fH =@A `EPE= GM F=BGH_ CEFGBCAB … Û 6 J †m =@A ?GHC<?=>H?A @>F =@A MGBP Û …[† n ArS…  [øà†m g@ABA à EF =@A `G?>`E]>=EGH `AH_=@m

>HC =@A ßkM<H?=EGH CASAHCF `G_>BE=@PE?>``b GH

Û m ߅ Û † `H Û û ª ! … J K¦ J †

f= EF =@ABAeb HA_>=EhA EH >Hb CEPAHFEGH MGB FP>``

Û

K

f= EF HGg BA>FGH>e`A =G >FF<PA =@>= =@A F?>`EH_ M<H?=EGH EF > ?GH=EH<G<Fm

PGHG=GHE?>``b EH?BA>FEH_ M<H?=EGH GM

Û

>HC =G CB>g E= eb EH=ABSG`>=EH_ MBGP =@A

>FbPS=G=E? eA@>hEG<B >= `>B_A >HC FP>`` Û …FAA j_ K J K s† K l@EF >FF<PS=EGH EF ?GBk

BGeGB>=AC eb =@A =@AGBb GMgA>W `G?>`E]>=EGHmg@E?@ SBACE?=F HA_>=EhA ?GBBA?=EGHF

=G =@A ?`>FFE?>` ?GHC<?=>H?A EH >Hb CEPAHFEGH K aGB EHF=>H?Am MBGP MGBP<`> … J K Z †

gA Ge=>EH MGB =@A >FbPS=G=E? F?>`EH_ M<H?=EGH EH =@A =gGkCEPAHFEGH>` ?>FAI

߅ Û †  J _ ÷ Û ! … J K¦ Q† l@A jBF= BAF<`= GM =@A F?>`EH_ =@AGBb EF =@>= EH ¦ D =@ABA EF > jrAC SGEH= m CAjHAC eb ߅ Û ¨ † X

m g@E?@ ?GBBAFSGHCF =G =@A PA=>`kEHF<`>=GB =B>HFE=EGH CAF?BEeAC eb

wHCABFGH

K

fM gA F=>B= gE=@ > FbF=AP gE=@ ?GHC<?=>H?A `>B_AB =@>H

Û ¨

m <SGH

EH?BA>FEH_ =@A FE]A GM =@A F>PS`A =@A ?GHC<?=>H?A EH?BA>FAF >HC =@A FbF=AP iGgF

=G =@A PA=>``E? BA_EPA K vH =@A ?GH=B>Bbm EM =@A ?GHC<?=>H?A GM G<B F>PS`A EF FP>``AB =@>H Û ¨ m =@A FbF=AP eA?GPAF >H EHF<`>=GB EH =@A `EPE= [ ¬ á K fH =@A J D >HC Q

D ?>FAm ß EF >`g>bF HA_>=EhA >HC =@ABAMGBA >H EHF<`>=EH_ BA_EPA F@G<`C

>`g>bF eA BA>?@AC EH =@A =@ABPGCbH>PE? `EPE=mHG P>==AB @Gg gA>W =@A CEFGBCAB

EF

(22)

e Ï    e

d=3

d=2

d=1

β( g)

ln g

g

c

?@ABCD EFâG ‚UD OSNP@RA MBRSK@IR ãŒ

ä

 NO N MBRSK@IR IM PR

ä F

R

@GB=`b >M=AB=@A gGBWFGH =@A F?>`EH_ =@AGBb GM`G?>`E]>=EGHm> jA`C =@AGBA=E?>`

>SSBG>?@ g>F CAhA`GSAC EH GBCAB =G _EhA > MGBP>`U<F=Ej?>=EGH =G =@A >FF<PS=EGHF

P>CA EH Qm s

K

l@A HAr= FA?=EGH gE`` CA CAhG=AC =G =@EF >SSBG>?@

K “f«f« ³´¡ ϕœÌ”™œ¡—Ÿ ª ̬• ¡˜ wF gA CEF?<FFAC EH FA?=EGH J K J K J m =@A >hAB>_A GM =@A SBGC<?= st…r û – †sv…r†

?GH=>EHFEHMGBP>=EGHGH EH=ABMABAH?A AtA?=F>HC =@<Fq<>H=<P `G?>`E]>=EGH

K fH =@A

CE>_B>PP>=E? GB FAPE?`>FFE?>` =@AGBb GM gA>W `G?>`E]>=EGHm =@>= SBGC<?= ?>H eA

?>`?<`>=AC GH`b SAB=<Be>=EhA`b …EH =@A S>B>PA=AB J øcýÿ† K fH =@A R <SABFbPPA=BE? HGH`EHA>B ªkPGCA` >SSBG>?@m =@A >hAB>_A hs t å sv å 7æ i EF ?>BBEAC G<= Ar>?=`b eb

ArSBAFFEH_ E= >F > M<H?=EGH>` EH=A_B>` GhAB ?`>FFE?>` >HC ;B>FFP>HHE>H jA`CF  J¨  K VA=9F BA?>`` =@A F=>HC>BC EH=A_B>`F 1 ç è é * < 3 ÷ê é ArS…  & · ë ê , · ( ·ë ê ë † _ ç CA= ( … J K¦¦ † 1 ç è é * < 3 ÷ê é êìê , í ArS…  & · ë ê , · ( · ë ê ë † _ ç CA= ( … ( 7< †ì í … J K¦ [ † g@ABA 3 ÷ê é 3 c A ê 3 fP ê 3 ê , 3 ê øQ / >HC î ( P<F= @>hA > SGFE=EhAm BA>` S>B= MGB

(23)

e Ï    e ?GHhAB_AH?A K fM gA =>WA î ( / …r û / - î g†m =@A FA?GHC EH=A_B>` bEA`CF 1 ç è é * < 3 ê , é 3 ê é ê í ê , ” ArS… / & · ë ê , · …r û / - g · ë †ê ë † / … / _† ç s í ” CA=…r û / - g ·ë † … J K¦ s† g@ABA -Z X K R

EH?A =@A EPS<BE=b SG=AH=E>` AH=ABF =@A `AM=k@>HC FECA GM … J

s† GH`b hE> =@A

ArSGHAH=E>` M<H?=EGHm E= EF A>Fb =G =>WA =@A >hAB>_A GM =@A EH=A_B>` MGB ;><FFE>H

CEFGBCAB

K T

GgAhABm SABMGBPEH_ =@EF >hAB>_A CGAF HG= bEA`C =@A >hAB>_A GM =@A

A`A?=BGHE? ;BAAH9F M<H?=EGHm FEH?A =@A BE_@=k@>HC FECA GM … J K¦ s† >`FG ?GH=>EHF =@A CA=ABPEH>H= GM î ( K

lG _A= BEC GM =@EF >gWg>BC CA=ABPEH>H=m pMA=Gh A= >`

K  JY m Q X 

EH=BGC<?AC >H=E?GPP<=EH_ ;B>FFP>HH h>BE>e`AF

K

f= ?>H eA F@GgH =@>= =@A

;B>FFP>HH >H>`G_<AF GM =@A ;><FFE>H ’eGFGHE?’ EH=A_B>` … J K¦¦ m J K¦ [ † _EhA 1 <ïïï ç è é 3ð , é 3ð é ArS…  & · ë ðñ · ( · ë ð ë † CA= ( ! … J K¦ Z † 1 <ïïï ç è é 3ð , é 3ð é ð ì ð , í ArS…  & · ë ðñ · ( · ë ð ë † … ( 7< †ì í CA= ( ! … J K¦ § †

l@ABAMGBAm eb EH=BGC<?EH_ > F<SABFbPPA=BE? ;><FFE>H EH=A_B>` GhAB > FA= GMFbPk

PA=BE? >HC >H=EFbPPA=BE? jA`CFm GHA ?>H A`EPEH>=A =@A <Hg>H=AC CA=ABPEH>H=

K

wH>`G_G<F`bm GHA ?>H CAjHA =@A >hAB>_A GM =@A SBGC<?= GM =gG ;BAAH9F M<H?k

=EGH EH =ABPF GM > PGBA ?GPS`E?>=AC M<H?=EGH>` EH=A_B>` GM =gG FA=F …BA`>=AC =G

=@A st >HC sv BAFSA?=EhA`b† GM ;B>FFP>HH >HC eGFGHE? jA`CFm ?GPSGFEH_ >H

AE_@=k?GPSGHAH= F<SABFbPPA=BE? jA`C hA?=GB …FG ?>``AC F<SABjA`C† ò

K

wM=AB >hk

AB>_EH_m GHA Ge=>EHF >H EH=AB>?=EH_ òók=@AGBbm g@ABA =@A EH=AB>?=EGH F=BAH_=@ EF

SBGSGB=EGH>` =G =@A h>BE>H?A GM =@A B>HCGP SG=AH=E>` …m %õ %÷ MGB =@A SG=AH=E>` … J K ¨ †† K l@A M<H?=EGH>` EH=A_B>` CAjHEH_ hs tsvi EF EHh>BE>H= <HCAB > _BG<S GM =B>HFMGBP>=EGHF ò ¬ ò z

=@A P>=BE?AF  CAjHA > FbPPA=BE? FS>?A >HC ?GHFEF=F

GM=gG e`G?WF GM [

Ò [

P>=BE?AF BA`>=AC =G =@A ?GPS>?= >HC HGH ?GPS>?= FA?=GB GM

=@A S>B>PA=AB FS>?A BAFSA?=EhA`b

K l@A GtkCE>_GH>` e`G?WF >BA S>B>PA=BE]AC eb ;B>FFP>HH h>BE>e`AF K l@A EH=AB>?=EGH =ABP ?>H eA CA?G<S`AC hE> > T <ee>BCk R =B>=GHGhE?@ =B>HFMGBk

P>=EGH eb EH=BGC<?EH_ > HAg ;><FFE>H EH=A_B>` GhAB ¨ Ò ¨ ô P>=BE?AF K l@A HAg

jA`C P<F= ?>S=<BA =@A M<`` FbPPA=Bb GM =@A M<H?=EGH>` EH=A_B>` EH GBCAB =G F<?k

?AAC EH CAF?BEeEH_ q<>H=<P `G?>`E]>=EGH AtA?=F

K

l@ABAMGBAm =@A jA`C ô

P<F= `EhA

EH =@A FbPPA=BE? FS>?A CAjHAC eb =@A P>=BE?AF  >HC EF =@<F ?GPSGFAC GM eG=@

?GPP<=EH_ >HC >H=E?GPP<=EH_ jA`CF

K

wM=AB EH=A_B>=EH_ G<= =@A jA`C òm=@A >?=EGH GM =@A F<SABP>=BE?AF ô eA?GPAF ¯ ô  _0 [ ³ 1 3 Ÿ R =B ô ÷ …Ÿ† û J Q 1 3 Ÿ h Ÿ % R =B`H õ 7< … î Ÿ î  †%Ÿi … J K¦ ¨ †

(24)

e Ï    ee g@ABA =@A SBGS>_>=GB õ EF _EhAH eb õ… î Ÿ î  †   î ö÷ Qùú û …– û / †ö ` Q û / 0 Q ô … î Ÿ † 7< … J K¦ Y † >HC ³ Q_ m %õ %÷ø0 EF =@A PA>H `AhA` FS>?EH_ K ö ` EF > ¨ Ò ¨ CE>_GH>` F<SABP>k

=BErmg@GFA <SSAB >HC `GgAB [ Ò [ CE>_GH>` e`G?WF >BA ÷ >HC  ÷ BAFSA?=EhA`b K w FP>`` EP>_EH>Bb S>B= /  … Z X † EF >CCAC =G –

=G >FF<BA =@A ?GHhAB_AH?A GM =@A

EH=A_B>`F

K

R

EH?A EH =@A FAPE?`>FFE?>` `EPE= üý 6 ÿ EH=ABMABAH?A AtA?=F G??<B >= `AH_=@

F?>`AF P<?@ `>B_AB =@>H ÿm =@A S@bFE?F GM `G?>`E]>=EGH EF _GhABHAC eb `GH_ g>hAk

`AH_=@ PGCAF GM ô K l@A AtA?=EhA >?=EGH GM =@AFA PGCAF … J K¦ ¨ † ?>H =@AH eA

Ge=>EHAC eb ArS>HCEH_ >BG<HC =@A @GPG_AHAG<F F>CC`A SGEH= FG`<=EGH GM … J K ¦ ¨ †m g@E?@ F>=EFjAF MGB – X ô / _#ý h Ÿ %…r  î ö÷ Qùú û / 0 Q ô † 7< %Ÿi ! … J K [X † w FG`<=EGH GM =@EF Aq<>=EGH EF ô 2 ö ` m g@E?@ ?GBBAFSGHCF =G =@A R n N w … J K Y † MGB =@A A`A?=BGHE? FA`MkAHAB_b K aGB – X

m =@A BG=>=EGHF ø g@E?@ `A>hA ô

EH

=@A FbPPA=BE? FS>?A bEA`C =@A ?GPS`A=A P>HEMG`C GM F>CC`A SGEH= FG`<=EGHF >F ô

øö

` ø

K

l@A BG=>=EGHFg@E?@ `A>hA ö

`

EHh>BE>H=?>H eA M>?=GBE]AC G<=m`A>hEH_

=@A F>CC`A SGEH= FG`<=EGHF ô

=G eA A`APAH=F GM > FAPEFEPS`A F<SABFbPPA=BE?

FS>?A

K

u

GBAGhABmMGB üý 6 ÿ =@ABA >BA P>FFEhA `GH_E=<CEH>` PGCAF g@E?@ GH`b ?@>H_A

=@A F@GB= CEF=>H?A S@bFE?F …>= `AH_=@ F?>`AF GM =@A GBCAB GM ÿ† >HC ?>H =@<F eA

EH=A_B>=AC G<=

K

l@ABAMGBAm GH`b =B>HFhABFA PGCAF S`>b >H EPSGB=>H= BG`A EH =@A

S@bFE?F GM `G?>`E]>=EGH

K

l@AFA PGCAF SBAFABhA =@A HGHk`EHA>B ?GHF=B>EH= ô ÷…Ÿ† ù K fM GHA ArS>HCF =@A >?=EGH ¯ ô 

>BG<HC =@A F>CC`A SGEH= FG`<=EGHm GHA _A=F

¯ ô  _#ý ¨ 1 3 Ÿ R =B  …  ô †÷ û Q / …– û / †ö ` ô  ! … J K [J †

l@EF >?=EGH CAF?BEeAF =@A F`Gg FS>=E>` i<?=<>=EGHF GM ô

>= jHE=A MBAq<AH?b

–

K

l@AFA PGCAF >BA U<F= =@A CEt<FEGH PGCAFm =@A DEt<FGHF >HC =@A nGGSABGHFm

g@E?@ gA CEF?<FFAC EH FA?=EGH J K J K J K

l@AEB ArEF=AH?A EF =@A ?GHFAq<AH?A GM =@A

FSGH=>HAG<F eBA>WEH_ GM =@A F<SABFbPPA=Bb

K l@A MBAq<AH?b – S`>bF =@A BG`A GM >H Ar=ABH>` jA`C K

l@A jA`C =@AGBb CAF?BEeAC eb =@A >?=EGH … J K [J † >HC =@A ?GHF=B>EH= ô øö ` ú øm ô ÷…Ÿ† ù

…g@ABA ø EF >H >BeE=B>Bb <HE=>Bb F<SABP>=BEr†

eA`GH_F =G =@A ?`>FF GM HGH`EHA>B ªkPGCA`F K w= – X m ¯ ô 

EF EHh>BE>H= gE=@

BAFSA?= =G =@A =B>HFMGBP>=EGH ô ¬ ø ô ú ø ! … J K [ Q†

w ?GHF=>H= P>_HA=E? jA`C ?>H eA =>WAH EH=G >??G<H= eb P>WEH_ =@A BAS`>?Ak

PAH= î  ¬ î  û …"øª† î  ` ʅ î Ÿ † EH MGBP<`> … J K ¦ Y † K l@A CE>_GH>` L ><`E P>=BEr  `

(25)

e Ï    e€

@>F =G eA EH=BGC<?AC FEH?A A>?@ F<SABhA?=GB ?GH=>EHF eG=@ HGBP>` >HC ?GPS`Ar

?GHU<_>=A h>BE>e`AF …FAA  J¨



†

K

fM =@A jA`C EF hABb gA>Wm ÿ 6 ÿ¹m E=F AtA?= ?>H eA

?GHFECABAC >F > FP>``SAB=<Be>=EGH >HC > HAg AtA?=EhA >?=EGH MGB =@A F`Gg PGCAF

GM ô

?>H eA Ge=>EHAC eb P>WEH_ >H ArS>HFEGH EH eG=@

 ô

>HC Ê >BG<HC =@A

F>CC`A SGEH= FG`<=EGH GM =@A >?=EGH … J K ¦ ¨ † K l @EF ArS>HFEGH bEA`CFI ¯ ô  _#ý ¨ 1 3 Ÿ R =B   …  ô û / " ª Ê ô  ` †÷ û Q / …– û / †ö ` ô  ! … J K [ ¦ † w??GBCEH_ =G =@EF BAF<`=m =@A CA_BAAF GM MBAACGP GM ô =@>= ?GPP<=A gE=@  ` BAk

P>EH <HSAB=<BeAC eb > P>_HA=E? jA`Cmg@E`A =@GFA CA_BAAF GM MBAACGP g@E?@ CG

HG= ?GPP<=A >?q<EBA > P>FF

K

l@A MGBPABCAHG=A =@A CEt<FEGH PGCAFg@ABA>F =@A

`>==AB BASBAFAH= =@A jA`C FAHFE=EhA nGGSABGH PGCAF

K

l@ABAMGBAm EM =@A jA`C § EF

F=BGH_ AHG<_@ …e<= F=E``F>=EFjAF ÿ 6 ÿ¹†m =@A EHi<AH?A GM =@A nGGSABGHF eA?GPA

HA_`E_Ee`A >HC ?>H eA HA_`A?=ACm EH >_BAAPAH= gE=@ =@A CEF?<FFEGH SBAFAH=AC EH

FA?=EGH J K J K Q K

fH =@EF`EPE==@A FbF=AP ?>H eA AtA?=EhA`b CAF?BEeAC eb [ Ò [ P>=BEk ?AF K

l@A F>CC`A SGEH= FG`<=EGHF >BA =@AH A`APAH=F GM > CEtABAH= FbPPA=BE? FS>?Am

FEH?A =@A =ABP ?GH=>EHEH_ =@A hA?=GB SG=AH=E>` EH … J K [ ¦ † eBA>WF =@A FbPPA=Bb … J K [ Q† K l

@EF AtA?= EF <`=EP>=A`b C<A =G =@A jA`C § eBA>WEH_ =@A =EPAkBAhABF>`

FbPPA=Bb GM =@A

T

>PE`=GHE>H …<HE=>Bb AHFAPe`A†

K

d@AH ÿ¹ eA?GPAF ?GPS>B>e`A =G ÿm =@A EHi<AH?A GM GBeE=>` AtA?=F eA?GPAF

FE_HEj?>H=

K

fH =@EF ?>FA gA AH=AB =@A q<>H=<P

T

>`` BA_EPAmg@E?@ EF HG= ?GBBA?=`b

CAF?BEeAC eb =@A >?=EGH … J K [ ¦ † K

dA gE`` CEF?<FF =@EF BA_EPA EH ?@>S=AB

¦K

fM GHA g>H=F =G F=<Cb FSEH EH=AB>?=EGHF …P>_HA=E? EPS<BE=EAF >HC FSEHkGBeE=

EH=AB>?=EGH†mGHA @>F =G EH=BGC<?A =gG AE_@=k?GPSGHAH=F F<SABjA`CFm ?GBBAFSGHCk

EH_ =G FSEHk<S >HC FSEHkCGgH S>B=E?`AF K l@ABAMGBAm =@A P>=BE?AF ô P<F= @>hA JZ Ò JZ ?GPSGHAH=F K

fH =@A SBAFAH?A GM P>_HA=E? EPS<BE=EAFm =@A FbPPA=Bb GM ô

EF =@A F>PA >F EH =@A SBAFAH?A GM > P>_HA=E? jA`C …=@G<_@ =@A CEPAHFEGH GM

=@A P>=BE?AF EF @E_@AB†m g@ABA>F > FbF=AP gE=@ FSEHkGBeE= EH=AB>?=EGHF …>HC HG

P>_HA=E? EPS<BE=EAF† eA`GH_F =G > HAg FbPPA=Bb ?`>FFm =@A FbPS`A?=E? ?`>FFm EH

g@E?@ =@A =EPAkBAhABF>` FbPPA=Bb GM î

g

EF SBAFABhAC e<= =@A ?AH=B>` FbPPA=Bb EF

eBGWAH

K

f= EF SGFFEe`A =G F@Gg =@>=m EH =@A `EPE= g@ABA =@A CAhE>=EGHF GM =@A P>=BEr ô

MBGP =@A F>CC`A SGEH= FG`<=EGH ö

` >BA FP>``m > SAB=<Be>=EhA ArS>HFEGH GM … J K [J †

bEA`CF=@A q<>H=<P ?GBBA?=EGHF =G =@A ?GHC<?=EhE=b CEF?<FFAC EH SBAhEG<FFA?=EGHF

K

f= @>F eAAH SBGhAH =@>= =@A F<SABFbPPA=BE? HGH`EHA>B ªkPGCA` …©Vªk PGCA`†

EF BAHGBP>`E]>e`A K u GBAGhABm E= @>F eAAH F@GgH =@>= > BAHGBP>`E]>=EGH _BG<S =BA>=PAH= GM … J K [J † EH …Q û

ˠ CEPAHFEGHF bEA`CF =@A F?>`EH_ M<H?=EGH ?GH UA?=<BAC

eb weB>@>PF A= >`

K EH Q



K T

GgAhABm E= EF HG= ?`A>B bA= g@A=@AB =@A ©Vª PGCA`

>HC =@A GHA S>B>PA=AB F?>`EH_ =@AGBb CG ?GBBA?=`b CAF?BEeA =@A PA=>`kEHF<`>=GB

=B>HFE=EGH EH CEFGBCABAC PA=>`F

K R

EH?A =@A F=>=EF=E?>` CEF=BEe<=EGH GM =@A ?GHC<?k

=>H?A

Û

CGAF HG= M<`j`=@A ?AH=B>` `EPE==@AGBAPmGHA ?G<`C _<AFF =@>= =@A >hAB>_A

(26)

e Ï    e PGPAH=F GM =@A CEF=BEe<=EGH GM Û F@G<`C eA =>WAH EH=G >??G<H= K l@EF EPSGB=>H=

EFF<A EF ?<BBAH=`b =@A F<eUA?= GM EH=AHFA BAFA>B?@

K

f

H =@A ?>FA GM q<>FEk J

D FbF=APF gE=@ CEt<FEhA gEC=@ l d ÿm =@A >?=EGH … J

K [J

†

?>H eA FEPS`EjACeb =>WEH_EH=G >??G<H=GH`b=@A ]ABG=B>HFhABF>`FS>?A @>BPGHE?F

GM =@A F<SABP>=BEr ô I ¯ ô  _#ýl ¨ 1 3û R =B   … 3 ô 3û †÷ û Q / …– û / †ö ` ô  ! … J K [[ †

l@A >eGhA >SSBGrEP>=EGH EF h>`EC SBGhECAC =@>= =@A H<PeAB Ü

¨ GM =B>HFhABF>` ?@>HHA`F GM =@A gEBA EF `>B_A K f= =<BHF G<= =@>= =@A q<>FEk J D ©VªkPGCA` ?>H

eA FG`hAC Ar>?=`b eb <FEH_ =@A =B>HFMABkP>=BEr PA=@GC

K

l@A e>FE? ECA> eA@EHC

=@EF PA=@GC EF =G BAC<?A =@A M<H?=EGH>` EH=A_B>` GhAB =@A F<SABP>=BE?AF ô =G > CEtABAH=E>` Aq<>=EGH K l@EF EF ?GPS`A=A`b >H>`G_G<F =G CABEhEH_ > R ?@BG^CEH_AB

Aq<>=EGH MBGP =@A ?GBBAFSGHCEH_ q<>H=<PkPA?@>HE?>` aAbHP>H S>=@ EH=A_B>`

K

fH =@A SBAFAH= ?>FAm =@A BG`A GM =@A S>B=E?`A ?GGBCEH>=A EF S`>bAC eb =@A P>=BEr ô

m g@E`A =@A BG`A GM =@A =EPA EF S`>bAC eb =@A ?GGBCEH>=A

û

>`GH_ =@A gEBA

K dA

gE`` CEF?<FF =@EF PA=@GC PGBA Ar=AHFEhA`b EH =@A MG``GgEH_ ?@>S=AB

K fH Q J  m =@A `G?>`E]>=EGH `AH_=@ [¨ GM q<>FEk J

D CEt<FEhA gEBAF g>F Ge=>EHAC

MBGP =@A FS>=E>` CA?>b GM =@A CAHFE=b ?GBBA`>=EGH M<H?=EGHm g@E?@ g>F ?>`?<`>=AC

eb =@A >eGhA PA=@GC

K l@A BAF<`= g>FI [ ¨ ß_0#¯  … J K [ s† g@ABA ¯ EF=@A gEBA ?BGFFFA?=EGH K fH … J K [ s†mß J Q [ ?GBBAFSGHCF=G HG P>_HA=E?

jA`C …GB=@G_GH>` AHFAPe`A†m jHE=A P>_HA=E? jA`C …<HE=>Bb AHFAPe`A† >HC F=BGH_

FSEHkGBeE= F?>==ABEH_ …FbPS`A?=E? AHFAPe`A† GB P>_HA=E? EPS<BE=EAFmBAFSA?=EhA`b

K

fH =@A HAr= ?@>S=ABm gA gE`` CABEhA MGBP<`> … J

K [

s† …<S =G > H<PABE?>` ?GAÙ?EAH=†

eb ?>`?<`>=EH_ > FSA?=B>` ?GBBA`>=EGH M<H?=EGH

(27)

8

àü‘ŽáŽ ×àáØâàØŽ ؏

ãؒŽÔÕ‘Ô‘Õ ÖàÖæ Øԑ’

fH =@EF ?@>S=AB gA ?GHFECAB =@A P>_HA=E? jA`C CASAHCAH= `G?>`E]>=EGH EH > CEFGBk

CABAC q<>H=<P gEBA EH > HGHSAB=<Be>=EhA M>F@EGH

K

dA EH=BGC<?A > ?GBBA`>=EGH M<H?=EGHm =@A ><=G?GBBA`>=EGH M<H?=EGH GM FSA?=>`

CA=ABPEH>H=F …w

R

D†mg@E?@ CEtABF MBGP =@GFA EH=BGC<?AC EH =@A SBAhEG<F ?@>S=AB

EH =@>= E= CGAF HG= HA?AFFE=>=A =@A <FA GM =@A M<`` F<SABFbPPA=BE? PA=@GCm e<=

F=E`` SBGhECAF EHMGBP>=EGH GH q<>H=<P `G?>`E]>=EGH

K

dA jHC > ?GH=EH<G<F EH?BA>FA GM =@A `G?>`E]>=EGH `AH_=@ gE=@ =@A P>_HA=E?

jA`CmF>=<B>=EH_ >==gE?A E=F h>`<AgE=@G<= P>_HA=E? jA`C

K

l@A ?BGFFGhAB eA@>hEGB

EF F@GgH =G eA _GhABHAC eG=@ EH =@A gA>W >HC F=BGH_ `G?>`E]>=EGH BA_EPA eb =@A

P>_HA=E? S@>FA F@EM=EH_ B>=A 

7< ¹

K l

@EF q<>H=E=b EF CABEhAC >H>`b=E?>``b EH ?`GFAC

MGBP >F > M<H?=EGH GM =@A B>=EG GM =@A PA>H MBAA S>=@ ÿm=@A gEBA =@E?WHAFF l >HC

=@A P>_HA=E? `AH_=@ ÿ¹

K N

G=@ S>B>eG`E? gEBAF >HC gEBAF gE=@ FSA?<`>B eG<HC>Bb

?GHCE=EGHF >BA ?GHFECABAC

K

v<B BAF<`=F _AHAB>`E]A SBAhEG<F CABEh>=EGHF GM ¹ MGB

CEFGBCABAC gEBAF

K

aEH>``bm gA CEF?<FF =@A >SS`E?>eE`E=b GM =@A >H>`b=E?>` MGBP<`>F =G BAFEF=>H?A

PA>F<BAPAH=F EH =@A F=BGH_ `G?>`E]>=EGH BA_EPA >HC gA ?GPS>BA G<B BAF<`=F gE=@

BA?AH= ArSABEPAH=>` gGBW eb ;ABF@AHFGH A= >`

K Q

Q 

K

w ?GPS>BEFGH gE=@ BA?AH=

H<PABE?>` gGBW EF >`FG P>CA

K

l@A BAF<`=F SBAFAH=AC EH =@EF ?@>S=AB @>hA eAAH S<e`EF@AC EH Q

¦ 

K

Öèç ýþê ëÿóîïîØØêðëóñîô ÿôïóñîô î êïóØëð êóêØñôëôó

fH =@A SBAhEG<F ?@>S=ABm gA @>hA FAAH =@>= =@A F=<Cb GM F=BGH_ A`A?=BGH `G?>`E]>k

=EGH EH > CEFGBCABAC SG=AH=E>` HA?AFFE=>=AF > HGHSAB=<Be>=EhA >hAB>_EH_ GM SBGCk

<?=F GM;BAAH9F M<H?=EGHF

K

l@EF?>H eA >?@EAhAC eb PA>HF GM=@A F<SABkFbPPA=Bb

Referenzen

ÄHNLICHE DOKUMENTE

[Rit08] Ritz W., Über eine neue Methode zur Lösung gewisser Variationsprobleme der mathema- tischen Physik. and Kittel C., Indirect Exchange Coupling of Nuclear Magnetic Moments

in a sufBciently strong magnetic field — where the spin- density waves have a gap — or at half-filling — where the Hubbard model is insulating and the charge excitations are massive

In summary, the strength of this thesis lies in the provision of systematic and compre- hensive investigation of the interaction and surface-driven modifications of

GGA+U density functional calculations of the spin-exchange parameters, as well as magnetic susceptibility and isothermal magnetization measurements, reveal that

Ground state properties of materials are strongly influenced by the dimensionality and geometry of the network, as well as the environment in which its components, atoms or

Usually it is more convenient to use the action than the Hamiltonian of a quantum system in quantum field theory. Instead of explicitly constructing the Green’s function structure

This serves to validate the analytical asymptotic formulas which turn out to be accurate in the high and low reflectivity regions with dimensionless wave number K &lt; 2and K &gt;

The main goal of the conference is a better understanding of the interplay between low-dimensional topology and. (higher) representation theory and of their applications to