PSFC/JA-01-10

Equilibrium and Stability Properties of Self-Organized Electron Spiral Toroids

C. Chen, R. Pakter and D. C. Seward*

April 2001

Plasma Science and Fusion Center Massachusetts Institute of Technology

Cambridge, MA 02139, USA

*Electron Power Systems, Inc. 42 Washington Drive

Acton, MA 01720, USA This work was supported in part by Ballistic Missile Defense Organization, Contract No. DSWA01-97-M-0537. Reproduction, translation, publication, use and disposal, in whole or part, by or for the United States government is permitted. Submitted for publication in Physics of Plasmas.

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CD02UK/T-:<GHFu.-@H3 )PA:*_0B*B<,TL?_*/:<, 0:*GM/:<,.-02W.B2-/.F"-LKR602/NVr02m.m.@HGH-3:<LK=<LKGH3 02@+CD02U;/.-:<GHFu.-"@M3 XN:<,.-

` E?S5-"yNJ.GM@HGMW.=<GQ002=<-*B<:?02W.@H-0: ,.GHUK,-"@M-"F":Q=<L;/3.-/.B2GM:<GH-B6nkGQ) -N)HXN0: ,TGMU;,:<L;=<LKGH3 02@NB<-"@HRkgiC0UK/.-":<GHFu.-"@M3.B?p<X

02@H:<,.LKJ.U;,:<,.-V02=<-*J./.B<:?02W.@H-0: @HL?_5-"@M-"F:<=<L;/3.-/.B2GM:<GH-B6nkGQ) -N)HX.02:@ML?_5:<LK=<L;GM3 0@NB<-"@HRHgAC0UK/.-":<GMF*u.-"@M3.B?p<)

[K

6*$%a2*%

o sx/.3.-"=:<,T-6-"vN:Q=?0LK=<3.GH/ 02=<VNgiCaLN3.-6m8-"=Q:<JT=<W 02:<GHLK/.BU;GMdK-"/GH/ ` yNB?)Tn<[;fKpkg2n<Z;YKp<X;:<,.-6F"LK/TB<-=<d.0:<GML;/LKR

-"@M-"F":<=<LK/F, 02=<UK-GHB*CD02GH/N:?02GH/.-3X;_6,.GMF,GMB63.-"B<F"=<GMW8-3W;V:<,.-F"LK/N:<GH/;JTGM:QV-yNJ 0:<GMLK/

99:;

n# ; p .=<?> n# ; p)* n# ; p sY n [Kp

NGH/.-N02=<GHlGH/.U ` y+)8n e [Kp<XN_-LKW.:?0GM/

8 . [ n * p . * + . * + sY n ZKp

IK=<LKC ` y+)+n e Z;p<X;:<,T-3T-/.B2GM:QVmT-"=<:<J.=<W 02:<GHLK/ GMB6UKGHdK-"/WNV

[ 8 8

n * p .

* + n \Kp

_*,.-"=<-:<,.--"y;J.GH@HGMW.=<GHJ.Ct-@H-"F:<=<L;/102/.U;J.@A0=*h.L?_sdK-"@MLNF"GM:QV GMB63.-"u./.-3W;V

2n p *,+ n p n b p

_*,.GHF2,GMB06RkJ./.F":<GHLK/L;R )S*,.-@HGH/.-;0=<GMl"-"3RkLK=<F"--yNJ 0:<GML;/GMB6UKGHdK-"/WNV

99#; )*0. n * + )- + >< p )* . n )* >< p * + )- + 8 -

n ) .0* + )- + ) . )* 1 )- 1"p n Kp

` yNJ 02:<GHLK/5n e ;p F;0/W8--vNm.=<-"B<B<-"3GH/:<,.-F"LKCamTL;/.-"/;:*RkL;=<C

8 n 8 2p * 8 Z * + . * + 8 -n .0* + 1"p6 n Kp

8 n 8 2p * + . [ n

2p 8 * 8 -

+ n fKp

_*,.-"=<- *n p -

1 n p . n Kp

GHB:<,.--@H-"F:<=<L;/F"V;F"@HLK:<=<LK/Rk=<-"yNJ.-"/.FVaGH/:<,.--"y;J.GH@MGHW.=<GHJ.C :<L;=<LKGH3 02@8CD02UK/.-":<GHF6uT-@H3 1 n p . )- 1 )[

S*,.-m8-=<:<J.=<W8-35-@H-F":<=<GHFD0/.31CD02UK/.-":<GHFau.-"@H3.B ) n ; p02/.3 ) n ; p*B?02:<GHB<RkVr@MGH/.-N02=<GHl-"3O 02vN_-@H@ B6-yNJ 02:<GHLK/.B

< > ) n# ; p 8 - 2n# ; p6 n ]Kp

< ) n# ; p . 99#; ) n# ; p sY( n [;YKp

< > ) n# ; psY, n [;[Kp

< ) n# ; p 8 [ 99:; ) n ; p 8 # - n p )* n ; p .0* +Nn p )- + 2n# ; p n [;ZKp

LK:<-:<, 02:6RHL;= ) n# ; p 1 n p - 2 + )- 1 X ` y8)+n e [K[KpGMB02J.:<L;C0:<GMFN02@H@MVB?0:<GMB2u.-3X;GQ) -N)HX

<?> ) n# ; p [ 9

9 n p 1?n p - 2 + sY, n [;\Kp

_*,.-"=<- GMB:<,.-CD0?wkL;= ` E?S'=?03.GMJ.BKX02/.3 GHB:<,.-3.GHB<m.@Q02F"-"C-"/N:a02@HLK/.U:<,.-:<LK=<LKGH3 )PA:B<,.LKJ.@H3WT-a=<-"CGH/.3.-"3r:<, 02::<,.-aJ./.GH:dK-"F:<L;= )- 1 GMBFL;/.B<GH3.-=<-"302B0F"LK/.B<:?0/N:J./.GM:d;-F":<LK=GH/r:<,.-am.=<-B2-/N:

02/ 0@MVNB<GHB?) ` y;J 0:<GML;/.Bn e ]Kp<X+n e [KYKp602/.31n e [KZ;p FN02/W8--vNm.=<-"B<B<-"3GH/:<,.-F"LKCamTL;/.-"/;:6RkLK=<C

[ n p . + 8 -

n [ b p

[ n +Np 8 1 n [;Kp

1 . 8 # - * n [ Kp

8 1 .

+ 8 - n *(+/.0*,+ p n [;fKp

c,.GH@M-:<,.-"=<-02=<-RkLKJ.=-"yNJ 02:<GHLK/.Bn e [ b pkg2n e [KfKpRkL;=:<,.=<-"-u.-"@H3Dd.02=<GQ02W.@H-"B X + 02/.3 1 XT_-_LK=<9_*GH:<, ` y;B?)+n e [K;p*02/.35n e [ Kp :<LaLKW.:?0GM/

8 [[ 8

.

-

* n [;Kp

02/.3

1 [[ 8

.

-

* n [;]Kp

[Kf

_*,.-"=<- 6n p GHB*3.-"u./.-"3GH/ ` y+)+n<Z b p<) L;:Q-:<, 0:*_*,.-"/ * * [KXN_-, 0<dK-0m.m.=<L?vNGMCD02:<-"@HV

8 # + 8 1 Y, n Z;YKp

F"LK=<=<-"B<mTL;/.3.GH/.U:<L *m8-=<:<J.=<W 0:<GML;/.B?)S LaB<J.CaCD02=<GHl-W.=<GH-"h.V;XN-"yNJ 02:<GHLK/.Bn<Z b pQX+n e \Kp<X+n e Kp<X+n e fKp<X+n e [;Kp*0/.31n e [K];p =<-"@A0:<-:<,.-mT-"=<g

:<J.=<WT-"3ayNJ 0/;:<GH:<GM-"B X * X *,1 X X + X802/.3 1 ) S*,.-"B<-6-"y;J 0:<GML;/.B:<L;UK-":<,.-=_*GM:<,D7TL;GMB<B2LK/ B

-"y;J02:<GML;/5n e [ b p_*GH@M@+W8-J.B2-3GH/ e m.m8-"/.3.GMv11:<La3.-"=<GHdK-0/-GHUK-"/Nd.02@HJ.--"yNJ 02:<GHLK/GH/;d;LK@Hd;GH/.UaL;/.@MV

6n pjn # p +Nn p<)

[K

*66 o # 6 S*,.--"vN:<=?02LK=<3.GH/ 02=<VNgiCaLN3.--GHUK-"/Nd.02@HJ.--"yNJ 02:<GHLK/DGHB3.-"=<GMd;-3rGM/D:<,.=<-"-C0?wkL;=B<:<-"m.B?) e Ba0u.=<B<:

B<:<-"m XN_--@HGMCaGH/ 02:<-:<,.-3.-"/.B<GH:AVm8-=<:<J.=<W 0:<GML;/ GH/ ` y+)+n e [ b p WNVB<J.W.B2:QGH:<J.:<GH/.U ` y;BK)8n e \Kp*0/.3

n e [KKpGM/N:<L7TL;GMB<B2LK/ B*-"yNJ 02:<GHLK/1n e [ b p<) S*,.GHB*=<-N023.GH@MVU;GMd;-B

[

[ 8

8

- 8

8 [ n * p 8 [

[ 8 n * p . 8 * + n [Kp

PQ:GHB-"dNGM3.-"/N:GH/ ` y+)*n<[;p6:<, 0::<,.-u.-"@H35mT-"=<:<J.=<W 0:<GML;/ #sGHB3.=<GHdK-"/qB<-@HRkgiF"LK/.B2GMB<:<-"/N:<@MV1WNV1:<,.--"@M-"F":<=<LK/3.-"/.B<GH:QVmT-"=<:<J.=<W 02:<GHLK/_*,.GHF2,GHB6/.L?_j-"vNm.=<-"B<B<-"3GH/:<-"=<CB6LKR:<,.-dK-"@MLNF"GM:QVm8-=<:<J.=<W 0:<GML;/.B

* 0/.3 *,+ )e B06B<-"FL;/.3B<:<-"m X;_--vNm.=<-"B<B*:Q,T-6d;-"@MLNFGH:QVamT-"=<:<J.=<W 02:<GHLK/.B * 02/.3 * +GM/:Q-"=<CaBL;R+:<,.-u.-@H3

mT-"=<:<J.=<W 0:<GML;/ DJ.B2GM/.U:<,T-*@HGM/.-N02=<GHl"-3aRkL;=<F-6-"y;J 0:<GML;/.Bn e ;p02/.3Dn e fKp02/T3a@MGH/.-;0=<GMl"-"3 O 0vN_-"@H@ B-"y;J02:<GML;/.Bn<Z b p<X8n e [K;p*02/.31n e [K];p<) PA/.3.-"-"3 X;B2J.W.B<:<GH:<J.:<GM/.U ` yNB?)8n e [K;p*02/.31n e [K];p+GH/N:<L ` y+)+n e Kp<X

B<L;C-B<:<=?0GMU;,;:<RkL;=<_02=<310@MU;-W.=?0VNGM-"@M3TB

n 8 2p [ .

[ 8 * . n 8 Z p * + -

[ 8 *(+ # [ 8

# n ZKp

02/.3B<J.W.B2:QGH:<J.:<GH/.U ` y8)+n<Z b p GH/;:<L ` y+)+n e f;p VNGM-"@M3.BKX

[ n

2p 8 * . n 8 2p * + 8 -

n \Kp

E?LK@HdNGM/.U ` yNB?)+n<*ZKp*0/.31n<*\Kp RkLK= * 02/.3 * +NXN_-u./T3:<, 02:

* -

n 8 Z p . n 8 p [ 8 * +# [ 8

n b p

*,+ -

[ 8 * +# [ 8

[ n

p 8

8 n 8 p [ .

[ 8

n Kp

_*,.-"=<-:<,.-yNJ 02/N:<GM:QV n p GHB*3T-u./.-"3GH/ ` y8)+n<ZKf;p<)

[K]

e B06:<,.GM=<3B<:<-"m X;_-B2J.W.B<:<GH:<J.:<- ` y;BK)Tn< b p*02/.3rn<*Kp+GH/N:<L ` y+)8n<[;p+:<LLKW.:?02GH/:<,.--"GMU;-"/;d.0@MJ.-

-"y;J02:<GML;/

[

[ 8

# 8

[ 8 [ .

[ 8 #

[ 8

[

n 8 Z p # . n 8 p [ 8 *(+ #

[ 8 # n Kp

. [

[ 8 n 8 Z p . n 8 p [ 8 * +#

[ 8

.

[ 8

[ 8 *(+ # [ 8

[ n

2p 8

_*,.-"=<- [, Z, >>> X802/.3 n p - n p6 ! GMB*:<,.-ank@MLNF;0@Ap+-"@H-F":<=<LK/m.@A0B<C06Rk=<-yNJ.-"/.F"V;)ITGH/ 02@H@MVNX0Rk:<-=0@M-"/.UK:<,NV50@MU;-W.=?0GMFa-vN-"=<FGHB<-NXT_-FN02/r-"v;mT=<-B<B:<,.--"GHUK-"/;dT02@MJT--yNJ 0:<GML;/qn<Kp

GH/:<,.-RkLK@H@ML?_6GM/.UaRkLK=<C

[

[ 8 M[ . n p 8

H[ . +Nn p sY( n fKp

_*,.-"=<-*:<,.-6B<J.B<F"-"m.:<GMW.GH@MGH:QVRkJ./.F":<GML;/.B n p0/.3 + n p02=<-*3.-"u./.-"3aGM/ ` yNB?).n<Z;Kp02/.3Dn<Z Kp<X=<-"B<mT-"F":<GMd;-@HVN)

` yNJ 0:<GMLK/5n<*fKpCJ.B<:6WT-B<LK@HdK-"3B2J.W+wk-F":*:<LD0m.m.=<LKm.=<GQ02:<-WTLKJT/.3 02=<VF"LK/T3.GM:<GHLK/.B?)I?L;=0/ ` E?S

GH/LKm8-/B<m 0F-nk-N) U8)MXNGH/102GH=?p<XN:<,.-WTLKJT/.3 02=<VF"LK/.3.GH:<GML;/.B0=<-

#*n7sYKpsY 02/.3 #*n7 p ! n Kp

( L;/.B<GMB2:<-/N:_*GH:<,r:<,.-aW8LKJ./.3 0=<VDFL;/.3.GM:<GHLK/.BGM/ ` y+)n<*Kp<X+:Q,T-W8LKJ./.302=<VDFL;/.3.GH:<GML;/.B02: "!02/.3 _6GM@H@TW8-63.-"=<GHdK-"3GM/ e m.mT-"/.3.GHv ^ 02Rk:<-"=0*W.=<GH-R+3.GHB<F"J.B<B<GHLK/L;R+:<,.-6B2LK@MJT:<GMLK/TB:<L ` y+)8nQ*fKp+GH/

:<,.-d.02F"J.J.Ct=<-"UKGHLK/.B6GM/ e m.m8-/T3.GMvr(*)

ZKY

% 2& 6a% #cr-B<LK@HdK-:<,.--GHUK-"/Nd.02@HJ.--"yNJ 02:<GHLK/'n<*fKpGM/q:<,T-d.0FJ.J.C&=<-"UKGHLK/.Ba_*GH:<,0 % "! 02/.3q_*GH:<,

XN_6,.-=<- n p 2n p * +Nn p * n p 'YK)IKLK= % "! L;= X ` y+) n<f;pFN02/-"v;mT=Q-"B<B2-3502B

[

[ 8

8

sY n [Kp

LK= # . [

[ . [ 8

# .

8

# sY n ZKp

c,.GH@M-/.-GH:<,.-"= ` y+)8n<(6[Kp+/TLK= ` y8)+n<(*ZKp B<-"-CaB*:<LWT-02/ 02@HVN:<GMFN02@H@MV:<=?02F":?02W.@H-;XN_-FN02/B<LK@HdK-RkL;= # 6WNV=<-:<J.=</.GH/.Ua:<L ` yNB?)+n e [ b pkg2n e [;fKp<)6E?-":<:<GH/.U 2n p n p * +Nn p * n p sYKX

_--vNm.=<-"B<B ` yNB?)+n e [ b pkg2n e [Kf;p*02B

[ n p . + jY( n \Kp

[ n + p 8 1 n b p

1 . sY( n Kp

8 1.

+ jY n Kp

cr-, 0<d;-Rk=<L;C ` y+)8n<(6\Kp<X

+ 8 [ n p6 n fKp

02/.3Rk=<LKC ` y+)+n<(*Kp<X

1 8

n Kp

E?J.W.B2:QGH:<J.:<GH/.U ` y;BK)8n<(6fKp*0/.31n<(*;p GM/N:<L ` y+)8n<( b p VNGH-@H3.B -"B<B<-"@ B*-"yNJ 02:<GHLK/

[

n p . 8

n p sY n ]Kp

RkLK=6:Q,T-d.02=<GA0W.@M- )-"/.-"=?02@+B<L;@MJ.:<GHLK/.B6:<L ` y8)+n<(*];p*02=<-

n p @ n # p . n # p" n [;YKp

ZK[

_*,.-"=<- @ n p60/.3 n p602=<-:<,.-u.=<B<:<g0/.3B<-"FL;/.3.gi9NGM/T35+-"B<B<-"@ RHJT/.F:<GHLK/.BLKRLK=<3.-"= X+0/.3 0/.3

0=<-aF"LK/.B2:?02/N:<B:<LW8-a3.-":<-"=<CGH/.-"3rWNVD:<,.-WTL;J./.3 02=<VrFL;/.3.GM:<GHLK/.BU;GMd;-/1GH/ ` y+)n<*Kp*:<LKU;-:<,.-"=

_*GH:<,qB<L;@MJ.:<GHLK/.B:<L ` y8)6n<*fKp_*GH:<,.GM/:<,.- ` E?St-"@H-F":<=<LK/@A0<V;-=Dn"! % % p<)'E?J.W.B<:<GH:<J.:<GH/.U

` yNB?)8n<(*fKp02/.3rn<(6[KYKp+GH/;:<L ` y+)Tn<Z b p<XK_-LKW.:?0GM/U;-/.-"=?02@8B<LK@HJ.:<GHLK/.B:<L:<,.--"GMU;-/Nd.02@HJ.--yNJ 02:<GHLK/GH/

:<,.-d.02F"J.J.Ct=<-"UKGHLK/.B6LKJ.:<B<GH3.-:<,.- ` E?Ss-@H-"F:<=<L;/@Q0<VK-"=*_*GH:<, '%&"! 02/.3_*GM:<, X

6n p 8 @ " n # p . " n p n [;[Kp

_*,.-"=<- @#" n p @ n p6 02/T3 " n p n p6 )cr-D/.L?_ u./.3jB<mT-"F"GA0@B<LK@HJ.:<GML;/.BRHL;=r0/ ` E?S GH/LKm8-"/B<m 0F-DWNVq:?09;GH/.U5GM/N:<Ls02F"F"LKJ./N::<,.-

WTL;J./.3 0=QVF"LK/.3.GH:<GML;/.BUKGHdK-"/aGH/ ` y+).n<;p<)+PQ/a:<,.-6GM/./.-"=d.02F"J.J.C =<-U;GML;/a_*GM:<,DY '%&"! X sYKX

WT-"FN02J.B2-0@M@8LKR+:<,.-u.-"@M3mT-"=<:<J.=<W 02:<GHLK/.BCJ.B<:W8-6u./TGM:<-a02:7sYK) S*,T-=<-"RkLK=<-NX;RkLK=Y '%&"! X;:<,.--"GMU;-/.RkJ./.F":<GHLK/ #*n p FN0/WT--"vNm.=<-B2B<-"3102B

#*n p 8 @ " n # p6 n [;ZKp

02/.3:<,.-F"LK=<=<-"B<mTL;/.3.GM/.Uam8-=<:<J.=<W8-3u.-@H3.B02=<-UKGHdK-"/WNV

1 n p 8

@ n # p6

n p @ n p6 n [;\Kp

+ n p

@ " n # p~-=<-NX

GHB0F"LK/.B<:?0/N:*:<LW8-3.-":<-=<CaGM/.-"3W;V:<,.-W8LKJ./.3 0=<VFL;/.3.GH:<GMLK/TB02: 7 "! )

PQ/:<,.-L;J.:<-"=*dT02F"J.J.Ct=<-"UKGHLK/_6GM:<, X X.WT-"FN02J.B2-:<,.-mT-"=<:<J.=<WT-"3u.-"@M3TB6RkLK=<C 02/

LKJ.:<gAUKLKGH/.Ua_0<dK-N) S*,.-"=<-"RkLK=<-NX;RkL;= XN:<,.--"GMU;-"/.RkJ./.F:<GHLK/ *n pF;0/W8--vNm.=<-"B<B<-"310B

6n p 8 $ " n # p6 n [ b p

_*,.-"=<- $ n p GHB*:<,.-u.=<B2:<gi9NGM/.35~60/.9K-"@8RkJ./TF:<GHLK/LKR LK=<3T-= 3.-"u./.-"3WNV

$ n p @ n p . n p n [;Kp

ZKZ

S*,.-F"LK=<=<-"B<m8LK/.3.GH/.UamT-"=<:<J.=<WT-"3u.-"@M3TB02=<-U;GMd;-/W;V

1?n p 8

$ n # p"

n p

$ n # p6 n [ Kp

+Nn p $ " n # p

~-=<-NX GHB0F"LK/.B<:?0/N:*:<LW8-3.-":<-=<CaGM/.-"3W;V:<,.-W8LKJ./.3 0=<VFL;/.3.GH:<GMLK/TB02: 7 )S*,.-=<CaB6=?023.GQ02@7TL?V;/N:<GH/.Uh.JTvGMB

n p [Z + n p 1 n p?-"v;m n<ZPAC ; p 8 Z

* * * *

$ " n p $ n # p?-vNm n<ZPAC ; p6n [;fKp

RkLK= )PAR:<,.- ` E?SsGMB6B<:?0W.@M-_*GH:<,PQC YKX.:<,.-/:<,.-7.L?VN/N:<GM/TUah.J.vGHB6B<CD02@H@+J./.3.-"=6B<CD02@M@m.J.=<:<J.=<W 0:<GML;/.B?)/:<,.-LK:<,.-"=*,02/.3 XNGHR :<,.- ` E?SsGMB6J./.B<:?0W.@M-_6GM:<,PAC YKXN:<,.-"/:<,.-7.LKV;/N:<GH/.U

h.J.vGH/.F"=<-;0B<-"B -"vNmTLK/T-/N:<GA0@M@HV02B60RkJ./.F":<GML;/LKRT:<GMCa-02Bm.J.=<:<J.=<W 0:QGHLK/TB U;=QLK_X"=<-"3.J.F"GH/.U6:<,.-@HGHRH-":<GHC-

LKR:<,.- ` E?S)

ZK\

*66 oa%S LB<LK@HdK-:<,.--GHUK-"/Nd.02@HJ.--"yNJ 02:<GHLK/qn<*fKp/NJ.Ca-=<GHF;0@M@HVGM/D:<,.--"@M-"F:<=<L;/D@A0<V;-"=6Rk=<L;C = "! :<L

X _-CJ.B<:B<-"::<,.-W8LKJ./.3 0=<VrFL;/.3.GM:<GHLK/.B02::<,.-GH/./.-"=02/.35LKJ.:<-"=B<J.=<R<0F-"BL;R02/ ` E?S)IKLK=

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29

Fig. 1

( )φρ,ρer

φer

2br

1br0

φφφeV r

( )φρ,ρer

φer

2br

1br0

φφφeV r

30

Fig. 2

1.000 1.002 1.004 1.006 1.008 1.010ρ / rb1

-1.6

-1.2

-0.8

-0.4

0.010

4 Ω s /ω

p(r b1

)

(a)

1.000 1.002 1.004 1.006 1.008 1.010ρ / rb1

0.0

0.4

0.8

1.2

1.6

102 ω

b /ω

p(r b1

)

(b)

31

Fig. 3

1.000 1.002 1.004 1.006 1.008 1.010ρ / rb1

-3.0

-2.0

-1.0

0.010

4 Ω s /ω

p(r b1

)

(a)

1.000 1.002 1.004 1.006 1.008 1.010ρ / rb1

0.0

0.4

0.8

1.2

103 ω

b /ω

p(r b1

)

(b)

32

Fig. 4

0 2 4 6 8 10l

0.07

0.08

0.09

0.10

0.11

0.12R

e ω

/ωp(

r b1)

(a)

0 2 4 6 8 10l

-0.006

-0.003

0.000

0.003

0.006

Im ω

/ωp(

r b1)

(b)

33

Fig. 5

0.0 0.5 1.0 1.5 2.0 2.5α

0.000

0.002

0.004

0.006

0.008

Im ω

/ωp(

r b1)

l=5l=6

l=7l=8

34

Fig. 6

0 2 4 6 8 10l

0.0

0.2

0.4

0.6R

e ω

/ωp(

r b1)

(a)

0 2 4 6 8 10l

-0.03

-0.02

-0.01

0.00

0.01

Im ω

/ωp(

r b1)

(b)