Interazione Plasma-Fascio di elettroni

HAL Id: hal-00744942

https://hal.archives-ouvertes.fr/hal-00744942

Submitted on 24 Oct 2012

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Alessio Guarino

To cite this version:

Alessio Guarino. Interazione Plasma-Fascio di elettroni. Polysciences Edition. Polysciences Edition,

pp.100, 2012, 978-1291097122. �hal-00744942�

097122 781291

9

ISBN 978-1-291-09712-2

90000

Alessio Guarino è un fisico italiano nato a firenze nel 1970. Attualmente, lavora all'Università della Polinesia francese a Tahiti. Precedentemente ha lavorato all'Università della California a Santa Santa Barbara (UCSB) e con la Scuola Normale di Lione (ENS-Lyon).|nextline>Le sue ricerche riguardano i sistemi non-lineari fuori equilibrio. Ha condotto studi sui Plasmi, i cristalli liquidi, la fratturazione, i materiali granulari e la fisica medica e sociale.

In questo libro si presentano la teoria e gli esperimenti, condotti su un Tubo ad Onde Progressive, del sistema Plasma-fascio di elettroni. Il sisema Plasma-fascio di elettroni costituisce il paradigma della turbolenza nei plasmi. Il plasma, che nell'interazione con gli elettroni gioca soltanto il ruolo di un dielettrico, può essere vantaggiosamente rimpiazzato da una struttura ad onde lente del tipo di quella che si trova in un tubo ad onde progressive (T.O.P.). Sostituire il plasma con una struttura del genere è vantaggioso poiché si eliminano il rumore di fondo intrinseco ai plasmi e perchè si possono creare delle turbolenze dall'esterno. Non si è quindi costretti a studiare quelle generate spontaneamente dall'agitazione termica.

Alessio GUARINO

Interazione Plasma- Fascio di elettroni

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9 !A/!3#L#(=!#-)![/!R*$4*/!RJn$/!1(>&)$!2h=!02!Y.:5BZH!RJn$/!;*L/!3*++/!0B=!2@O=!.:59/!

0 !^/!S/!g\D*&(=!m/Q/!r&-",*n!#-)!m/!Q/!S#(G*,?/!RJn$/!1(>&)$!.9=!.2O9=!.:@./!

h !R&*,%*=!m/!;/=!#82&93%,.:!;&93!15<3'67![/!c#-!D'$+,#-)!K'4<#-n=!6-%/=!.:0O/!

@ !M/!6/!^$>-')#=!1/!['L*&(!#-)!m/!Q/!S#(G*,?7!#"=)32,(3.1&%!13'1!04!1*3!>5&',%,.3&2!1*302+k/!

RJn$&%$!'"!1(>&)$=!2B=!205=!.::./!

5 ! [/! C/! Q#,+4#--/! ial<*,&4*-+$! '-! o#L*8<#,+&%(*! &-+*,#%+&'-$k/! RJ[! +J*$&$=! _-&L*,$&+n! '"!

K#(&"',-&#=!M#-![&*?'=!.::9/!

: ! A/! [&4'-+*=!#?3'125-1,0.! 04! 12&)),.:! 0'-,%%&1,0.'6@! RJ[! +J*$&$=! _-&L*,$&+n! '"! K#(&"',-&#=!

M#-![&*?'=!.:5:/!

.O !P/!r/!A*-+(*=!#-)!K/!r/!;'G*,$'-=!RJn$/!1(>&)$!.9=!29h2=!.:@.!

.. !Q/!a/!Sn-&%V=!#-)!C/!D/!P#>"4#-=!RJn$/!1(>&)$!2.=!h0B=!.:@5!

./.

,#!$%+&-*!'$+#'$(+./-01!'#(!#+2+%%&'$!#

!"!"#$%&'()*$+%'#,-)./)01).2$+#3$#'-'&&(+%$#

! R*,! $+>)&#,*! &(! $&$+*4#! "#$%&'! )&! *(*++,'-&8<(#$4#! >+&(&7&#4'! (#! +*',&#! %&-*+&%#/! M&!

$><<'-*!%J*!&(!<(#$4#!$&#!%'$+&+>&+'!)#!>-!&-$&*4*!)&!&'-&!<'$&+&L&!%J*!%'-$&)*,*,*4'!"&$$&!*!)#!

*(*++,'-&! L&-%'(#+&! #! 4>'L*,$&! $'(+#-+'! (>-?'! (T#$$*! 7/! K'-$&)*,&#4'=! &-'(+,*=! )&! <'+*,!

+,#$%>,#,*! ?(&! >,+&! +,#! &! %'$+&+>*-+&! )*(! <(#$4#/! 3*! *E>#7&'-&! >$#+*! -*((#! +*',&#! %&-*+&%#! $'-'!

(T*E>#7&'-*!)&!c(#$'L!*!E>*((#!)&!R'&$$'-/!D*(!-'$+,'!%#$'!(T*E>#7&'-*!)&!c(#$'L!d]!

∂1

∂+ + L ∂1

∂7 − *a 4

∂1

∂L = O

! ././.!

)'L*! *=! 4=! L! *)! 1 = 1YL= 7=+Z @! $'-'! ,&$<*++&L#4*-+*]! (#! %#,&%#= (#! 4#$$#! (#! L*('%&+N! *! (#!

)&$+,&G>7&'-*!)*((*!L*('%&+N!)*?(&!*(*++,'-&/!"!d!&(!%#4<'!*(*++,&%'/!

3T*E>#7&'-*!)&!R'&$$'-!&-L*%*!d]!

ε O ⋅ ∂"

∂ A = ρYA=1Z

././2!

%'-! ε

!%'$+#-+*!)&*(*++,&%#!)*(!L>'+'!*! ρY7=+Z !)*-$&+N!)&!%#,&%#!)*"&-&+#!%'4*]!

ρY7=+Z = −-* 1YL=7=+Z)L ( ! −. )

! ././B!

&-! %>&!.! d! (#! )*-$&+N! )*?(&! &'-&=! %J*! $><<'-&#4'! *$$*,*! %'$+#-+*! *)! &$'+,'<#/! 64<'-&#4'! #(!

$&$+*4#! >-#! <*,+>,G#7&'-*! )&! <>($#7&'-*! ω! *! $L&(><<&#4'! 1YL=7=+Z *)! &(! %#4<'! *(*++,&%'! &-!

$*,&*!)&!1'>,&*,]!

./2 1YL=7=+Z = 1 _' YL=7Z + 1 _ω

YL=7Z!

ω

≠ '

" * ^&ω

⁺

! ././9#!

aY7=+Z = a _ω

Y7Z!

ω

≠'

" * ^&ω

⁺

! ././9G!

%'-]!

ω - = 2π-

^ ././0!

&-!%>&!^!d!&(!<*,&')'!)*((#!<*,+>,G#7&'-*!*! 1 _' YL=7Z!d!(#!%'4<'-*-+*!)*((#!$*,&*!)&!1'>,&*,!<*,!

ω-sO=! %&'I! d! (#! )&$+,&G>7&'-*! $+#7&'-#,&#! %J*! $&! J#! E>#-)'! &(! $&$+*4#! -'-! d! <*,+>,G#+'/!

K'-$&)*,#-)'!%J*]!

1 _ω

YL=7Z = 1 _ω

YLZ!* ^−&V

⁷

! ././h!

*)!&-$*,*-)'!(*!Y././9/Z!-*((T*E>#7&'-*!)&!c(#$'L!Y./././Z!'++*-&#4']!

&Yω _- − V _- LZ1 _ω

+ *

4 Y ∂1 _ω

-T

∂L Za _Yω

-

− ω

Z

#

$ %

&

' (

ω

≠'

" ⁺ ₄ ^{* a} ω

∂ 1 _'

∂L = O

!././@!

['L*!V-!d!&(!->4*,'!)\'-)#/!

6(! +*,4&-*! )*((#! $'44#+',&#! d! )'L>+'! #((T#%%'<<&#4*-+'! +,#! &! L#,&! 4')&/! M&! $><<'-*! %J*! $&#!

#GG#$+#-7#! <&%%'('! )#! <'+*,('! +,#$%>,#,*/! a(&4&-#-)'! E>*$+'! +*,4&-*! )#((T*E>#7&'-*! *)!

*$<(&%&+#-)'!&-!">-7&'-*!)&!1 _ω

!$&!+,'L#]!

1 _ω

= * 4

a _ω

-

&Yω - − V _- LZ

∂1 _'

∂L ! ././5!

./B

3T*E>#7&'-*!)&!R'&$$'-=!>+&(&77#-)'!(#!Y././9Z!<>`!*$$*,*!$%,&++#]!

ε ' V _- a _ω

= −-* ( ! 1 _ω

Y,=LZ ⋅ )L )

! ././:!

*!)#((#!Y././5Z!$&!,&%#L#!(#!,*(#7&'-*!)&!)&$<*,$&'-*!)*(!$&$+*4#]!

. − ω <

2

V _-

∂1

∂L

V _- L − ω _- ⋅

! ^{)L = O}

! ././.O!

)'L*]!

ω < = -* ² ε ' 4

././..

d!(#!<>($#7&'-*!)&!<(#$4#/!

M><<'-&#4'! %J*! (#! )&$+,&G>7&'-*! )*((*! L*('%&+N! )*(!$&$+*4#! $&#! %'4*! E>*((#!,#<<,*$*-+#+#!&-!

"&?>,#! ././! M&! &<'+&77#! %&'I! %J*! &-! >-! <(#$4#! &-! *E>&(&G,&'=! ,#<<,*$*-+#+'! )#((#! ?,#-)*!

?#>$$&#-#!%*-+,#+#!$>(('!7*,'=!$&#!$+#+'!&-&*++#+'!>-!"#$%&'!)&!*(*++,'-&!)&!L*('%&+#T!4*)&#!>=!(#!

<&%%'(#!?#>$$&#-#=!%J*!=!%'4*!L*),*4'=!&-)>,,N!)*((*!&-$+#G&(&+N!-*(!$&$+*4#/!M&!"#!(T&<'+*$&!%J*!

&(!"#$%&'!$&#!)*G'(*=!'LL*,'!%J*!(#!$>#!)*-$&+#T!$&#!4'(+'!<&%%'(#!,&$<*++'!#!E>*((#!)*(!<(#$4#/!

= ! ?;@

; 6 ⁵ I,' ^!

6 ! I,' 9

ω J K 7

∆;

;

./9 1&?>,#!./.!

R'$$&#4'!)>-E>*!$%,&L*,*]!

1 _' YLZ = " _< YLZ + - _G - _' " _' YLZ

! ././.2!

K'-! "<YLZ! *! "OYLZ! ,&$<*++&L#4*-+*! ">-7&'-*! )&! )&$+,&G>7&'-*! )*(! <(#$4#! *! )*(! "#$%&'! )&!

*(*++,'-&=! -'! )*-$&+N! )*(! <(#$4#! *! -G! )*-$&+N! )*(! "#$%&'! )&! *(*++,'-&/! R'$$&#4'! $%,&L*,*! (#!

,*(#7&'-*!)&!)&$<*,$&'-*!%'4*]!

. − ω <

2

V _-

∂"

∂L

V _- L − ω -

! ⋅ ^{)L = ω} _V ^{2 <}

-

- _G - _'

∂"

∂L

V _- L − ω -

! ⋅ ^)L

! ././.B!

[*"&-&#4'!ε Yω _. = B _. Z!">-7&'-*!)&*(*++,&%#!)*(!<(#$4#!%'4*]!

εYω - =V _- Z = .− ω <

2

V _-

∂"

∂L

V _- L − ω -

! ⋅ ^)L

! ././.9!

aT! (#! ">-7&'-*! )&*(*++,&%#! %J*! #L,*GG*!&(!$&$+*4#!$*!-'-!%&!"'$$*!&(!"#$%&'!)&!*(*++,'-&!Y-GsOZ/!

D*(!-'$+,'!%#$']!

εYω - =V _- Z = ω <

2

V _- - _G - _'

∂"

∂L

V _- L − ω -

! ⋅ ^)L

! ././.0!

[&$+&-?>&#4'!#)*$$'!)>*!%#$&!)&""*,*-+&=!"#$%&'!%#()'!'!"#$%&'!",*))'!)T*(*++,'-&/!3#!$'(>7&'-*!

)*((T&-+*?,#(*! *! E>&-)&! &(! %'4<',+#4*-+'! "&$&%'! )*(! $&$+*4#! $#,N! )&L*,$'! <*,! &! )>*! %#$&/! 3#!

)&""*,*-7#! +,#! &! )>*! "#$%&! $+#! -*((#! (#,?J*77#! )*((#! )&$+,&G>7&'-*! )*((*! L*('%&+N=! E>&-)&! )#((#!

)&""*,*-+*!+*4<*,#+>,#=!)#!E>&!(#!)*-'4&-#7&'-*!"#$%&'!%#()'!'!",*))'/!M&!)&%*!"#$%&'!",*))'!

./0 E>#-)'! d! L*,&"&%#+#! (#! %'-)&7&'-*! V _&-

V _- >> V _- ∆c <

ω -

! %'-! ∆c < ! (#,?J*77#! #! 4*+#T! #(+*77#! )*((#!

)&$+,&G>7&'-*! )*((*! L*('%&+N! )*(! "#$%&'! *! B _,. ! <#,+*! &44#?&-#,&#! )*(! L*++',*! )T'-)#/! W>*$+#!

%'-)&7&'-*!&4<(&%#!%J*!(#!)&$+,&G>7&'-*!$&#!4'(+'!$+,*++#=!#(!(&4&+*!>-#!)*(+#!)&![&,#%/!D*(!%#$'!

'<<'$+'=!%&'I!$*! V _&-

V _- << V _- ∆c <

ω -

!$&!J#!E>*(('!%J*!$&!%J&#4#!>-!"#$%&'!%#()'!)&!*(*++,'-&/!

1ABCDE#CAFGE#GD#HFHIIJEKD#

M><<'-&#4'! %J*! &(! "#$%&'! )&! *(*++,'-&! $&#! %#()'=! %&'I! %J*! $'))&$"&! (#! %'-)&7&'-*!

V _&-

V _- << V _- ∆c <

ω -

/!

R'-&#4'! B _. = B _2. + ,B _,. %'-! B _2. !<#,+*!,*#(*!*! B _,. <#,+*!&44#?&-#,&#!)*(!L*++',*!)T'-)#/![#((#!

Y././.0/Z!'++*-&#4']!

B _. ² εYω _. =B _. Z = ω ₎ ² . _<

. ₀

∂ 4 ₀

∂9 9 − ω _. B _2. + , B _,. ω _.

B _2. ² )

* + ,

- . 9 − ω _.

B _2.

)

*

+ ,

- .

2

+ B _,. ω _. B _2. ² )

*

+ ,

- .

! 2 ^/9 _{! ././.h!}

6$'(#-)'!(#!<#,+*!&44#?&-#,&#]!

64 V [ _- ² εYω _- = V _- Z ] ^{= ω} <

2 - _G - _'

∂" _'

∂L

V

ω

V

)

* + +

, - . .

L− ω

V

)

* + +

, - . .

+ V

ω

V

)

* + +

, - . .

#

$

%

&

'

! ( ^)L _{! ././.@!}

3T&<'+*$&! )&! "#$%&'! %#()'! %&! <*,4*++*! )&! <'+*,! %'-$&)*,#,*! %'$+#-+*! ∂ " _'

∂L ! &-! E>#-+'! (#! $>#!

L#,&#7&'-*!I!+,#$%>,#G&(*!&-!%'-",'-+'!#!E>*((#!)*(!+*,4&-*!)*((T*E>#7&'-*!Y././.@Z!,#%%J&>$'!&-!

<#,*-+*$&!E>#),#/!;&$'(L*-)'!(T&-+*?,#(*!'++*-&#4']

64 V [ _- ² ε Y ω - =V _- Z ] ⁼ πω <

2 - _G - _'

∂" '

∂L ^{L =}

! ././.5!

./h

6-!?*-*,#(*=!(#!">-7&'-*!)&*(*++,&%#!<'$$&*)*!>-#!<#,+*!,*#(*!*)!>-#!<#,+*!&44#?&-#,&#]!

ε Yω _. = B _. Z = ε ₂ Yω _. = B _. Z + ,ε _, Yω _. =B _. Z

! ././.:!

)'L*! ε _, Yω _. =B _. Z=! (#! <#,+*! &44#?&-#,&#=! )*$%,&L*! (T*""*++'! 3#-)#>! )*(! <(#$4#/! M*=! %'4*! -*(!

-'$+,'!%#$'=!&(!"#$%&'!)T*(*++,'-&!$&!+,'L#!-*((#!%')#!)*((#!)&$+,&G>7&'-*!)*((*!L*('%&+N=!%&'I!&-!

>-#!7'-#!&-!%>&!%&!$'-'!<'%J&!*(*++,'-&!)*(!<(#$4#=!#((',#!(T*""*++'!3#-)#>!d!<&%%'('!*!<'$$&#4'!

+,#$%>,#,('/!M><<'-&#4'!&-'(+,*!%J*!&(!<(#$4#!-'-!$&#!$',?*-+*!)&!&-$+#G&(&+N=!'LL*,'!%J*!$*,L#!

$'(+#-+'!)#!$><<',+'!)&!<,'<#?#7&'-*!<*,!(*!'-)*=!E>&-)&]!

εYω - =V _'- Z = O

././2O!

['L*! V _'- !I!&(!->4*,'!)T'-)#!%J*!#L,*GG*!&(!4')'! ω - !$*!-*(!$&$+*4#!-'-!%&!"'$$*!&(!"#$%&'!)&!

*(*++,'-&/!

ML&(><<&#4'!#(!<,&4'!',)&-*!(#!">-7&'-*!)&*(*++,&%#!)*(!<(#$4#]!

ε Yω _. = B _. Z ≈ ( δB _2. + ,B _,. ) ^{∂ε 2}

∂ B ^B

! ././2.!

%'-]!

!! B _. = B _0. + δB _{2 .} + ,B _,.

! ././22!

^,#$%>,#-)'!?(&!',)&-&!$><*,&',&!#(!<,&4'!(#!Y././.5Z!)&L*-+#]!

64 V [ _- ² εYω - = V _- Z ] = V _&- V _'- ² ∂ε _,

∂V ^V

! ././2B!

*!E>&-)&]!

./@

V &- = πω <

2 - _G - _'

. V _'- ² ∂ε ,

∂V ^V

∂" _'

∂L ^{L =}

! ././29!

M&! L*)*! %J*! +>++&! &! 4')&! %J*! J#--'! (#! L*('%&+N! )&! "#$*! c _φ = ω -

V _'- ! $>((#! <#,+*! #! <*-)*-7#!

<'$&+&L#= ∂" _'

∂L c

> O =)*((#!)&$+,&G>7&'-*!)*((*!L*('%&+N!)*(!"#$%&'!)&!*(*++,'-&!$'-'!&-$+#G&(&/!A(&!

*(*++,'-&! %J*! J#--'! >-#! L*('%&+N! 9! <,'$$&4#! #! E>*((#! )&! "#$*! )*((T'-)#! C _ϕ ! ,&$>'-#-'! %'-!

E>*$+T>(+&4#/!C((#!,&$'-#-7#=!?(&!*(*++,'-&!,*$+#-'!&-+,#<<'(#+&!-*((T'-)#!*!$&!J#!>-'!$%#4G&'!)&!

*-*,?&#/! W>#-)'! (#! L*('%&+N! &-&7&#(*! )*((#! <#,+&%*((#! I! )&! <'%'! $><*,&',*! #! C _ϕ ! I! ,#((*-+#+#!

)#((T'-)#! *! E>&-)&! ?(&! %*)*! *-*,?&#=! 4*-+,*! $*! (T*(*++,'-*! I! &-&7&#(4*-+*! >-! <'T! <&F! (*-+'!

)*((T'-)#!I!#%%*(*,#+'=!%&'I!#%E>&$+#!*-*,?&#/!j!%J&#,'!%J*=!$*!-*((T&-+',-'!)*((#!L*('%&+#!)&!"#$*!

)*((T'-)#=! $'-'! <&F! ->4*,'$&! ?(&! *(*++,'-&! %'-! 9 = C _ϕ + δ9 ,&$<*++'! #! E>*((&! %'-! L*('%&+N!

9 = C _ϕ − δ9=%&'I! $*! ∂" '

∂L c

> O =! (T'-)#! $#,N! #4<(&"&%#+#! #! )&$%#<&+'! )*((T*-*,?&#! %&-*+&%#! )*?(&!

*(*++,'-&! )*(! <(#$4#/! CGG&#4'! E>&-)&! (#! %,*$%&+#! )*&! 4')&! %J*! J#--'! L*('%&+N! )&! "#$*! $>((#!

<#,+*!#!<*-)*-7#!<'$&+&L#!)*((#!)&$+,&G>7&'-*!)*((*!L*('%&+N!)*?(&!*(*++,'-&=!%&'I!)&!>-'!$<*++,'!

)&! (#,?J*77#! <,'<',7&'-#(*! #((#! )&$<*,$&'-*! )*(! "#$%&'=! "&?>,#! ./2#/! R*,! (#! %'-$*,L#7&'-*!

)*((T*-*,?&#=! E>*$+#! %,*$%&+#! -'-! <>`! *$$*,*! &-"&-&+#/! K&! $#,N! )>-E>*! >-#! $#+>,#7&'-*=! %J*!

#,,&L*,N!E>#-)'!?(&!*""*++&!-'-!(&-*#,&!-'-!$'-'!<&F!+,#$%>,#G&(&/!1&?>,#!./2G/!3#!$#+>,#7&'-*!d!

)'L>+#!#((#!)&"">$&'-*!)*?(&!*(*++,'-&!)*(!"#$%&'=!(#!%>&!)&$+,&G>7&'-*!<*,)*!E>&-)&!(#!$>#!"',4#!

&-&7&#(*/!

/KIω

:I%

=!? ; @

; 7

,45&%%0$,-),8$-),)'1%*9)+) /;$+7()$'&,-),7',8$-$,)'1%*9)+&

=)>70*,* =)>70*,9

1&?>,#!./2!

./5

K'-$&)*,&#4'! >-! *(*++,'-*! )*(! "#$%&'! )&! L*('%&+N! > _. =! "&?>,#! ./BC/! 6-&7&#(4*-+*="&?>,#! ./BX=!

*-+,#! &-! ,&$'-#-7#! %'-! &(! 4')'! %J*! J#! (#! L*('%&+N! )&! "#$*! <&F! L&%&-#! #((#! $>#! L ₂ /! 6-! $*?>&+'!

<*,'T=! "&?>,#! ./BK=! E>#(%J*! #(+,'! 4')'! &-$+#G&(*=! L _B =! #%E>&$+*,N! #GG#$+#-7#! *-*,?&#! )#! <'+*,!

,&$>'-#,*! %'-! (T*(*++,'-*! %'-$&)*,#+'/! C! E>*$+'! <>-+'=! (#! <#,+&%*((#! d! $'??*++#! #)! >-! %#4<'!

*(*++,&%'!%#'+&%'=!)#!E>&!(#!)&"">$&'-*!)*(!"#$%&'!)&!*(*++,'-&/!

:

;

:

:

;

;

;

;

;

=?;@

=?;@

=?;@

;

; 7 _A

2

B

; B

; ₂ 7 _A

3

C

D

1&?>,#!./B!

./:

S#+*4#+&%#4*-+*=! %&`! $&! +,#)>%*! -*((#! )*"&-&7&'-*! )&! >-! %'*""&%&*-+*! )&! )&"">$&'-*=! [YL=7Z =!

%J*!$'))&$"#!(#!$*?>*-+*!*E>#7&'-*]!

∂" _'

∂7 = ∂[YL=7Z

∂L

∂" _'

∂L ! ././20!

C!<#,+&,*!)#((*!*E>#7&'-&!Y././@Z!*!Y././5Z=!$L'(?*-)'!E>#(%J*!%#(%'('=!$&!+,'L#]!

[YL=7Z = * ² 4 ²

a _ω

-

2

& V ( _- L − ω - )

-

"

! ././2h!

∂[YL=7Z

∂L = 2V _&- [YL=7Z

! ././2@!

6-+*?,#-)'! ->4*,&%#4*-+*! E>*$+*!)>*!>(+&4*!*E>#7&'-&!*!(#!Y././29Z=!,&%#L&#4'!(T#-)#4*-+'!

&-!"&?>,#!./9/!

./.O

< A <

2

< 2

< A L?;I(@

;

;

=?;@

L?;I$@

L?;I @

=?;I @

=?;I$@

1&?>,#!./9!

3#! )&"">$&'-*! *! (#! $#+>,#7&'-*! (#! $&! J#! #-%J*! E>#-)'! &(! "#$%&'! &-+*,#?&$%*! %'-! >-#! $'(#! '-)#!

&-$+#G&(*=! &-! E>*$+'! %#$'=! <*,'T=! &(! 4*%%#-&$4'! "&$&%'! %J*! (*! ?*-*,#! -'-! I! #-%',#! %J&#,'/! 3*!

$<&*?#7&'-&! )#+*! &-! <,*%*)*-7#! L#(?'-'! &-"#++&! $'(+#-+'! -*(! %#$'! %&! $&#-'! #(4*-'! )>*! '-)*!

&-$+#G&(&!%J*!&-+*,#?&$%'-'!%'(!"#$%&'/!

[*"&-&#4'! ',#! >-#! ?,#-)*77#! %J*! %&! $#,N! >+&(*! <*,! )&4'$+,#,*! (T*E>&L#(*-7#! "',4#(*! +,#! &!

$&$+*4&! R(#$4#81#$%&'! )T*(*++,'-&! *! M+,>++>,#! #)! '-)*! (*-+*8"#$%&'! )T*(*++,'-&/! M&! )*"&-&$%*!

&4<*)*-7#!)T&-+*,#7&'-*!<(#$4#8"#$%&']!

./..

; _< = < a _ω

-

2 >

2V _'- ² R _ω

././25!

)'L*!< a _ω ²

> !d!(#!4*)&#!)*(!E>#),#+'!)*(!%#4<'!*(*++,&%'!%#(%'(#+'!$>((#!$*7&'-*!)*(!"#$%&'!)&!

*(*++,'-&=!R _ω

!d!(#!<'+*-7#!)*((T'-)#=!%J*!-*(!%#$'!>-&)&4*-$&'-#(*!d!)*"&-&+#!%'4*]!

R _ω

= c _?

ω-

ζ _ω

! ^)σ

././2:!

)'L*! c _?

ω-

! d! (#! L*('%&+N! )&! ?,><<'=! ζ _ω

! (#! )*-$&+N! )T*-*,?&#! )*((T'-)#/! 3T&-+*?,#(*! d! "#++'! $>(!

<&#-'!-',4#(*!#((#!)&,*7&'-*!)&!<,'<#?#7&'-*!Y#$$*!7Z/!

3#!)*"&-&7&'-*!)*((#!)*-$&+N!)T*-*,?&#!)*((T'-)#]!

ζ ω

= .

.hπ ω - a _ω ²

∂ε

∂ω )

* ,

- ω

=V

! ././BO!

%&!<*,4*++*!)&!%#(%'(#,-*!(#!<'+*-7#]!

R _ω

= .

.hπ ω -

)ω )V )

* ,

-

ω- = V '-

∂ε

∂ω )

* , - _ω

-

=V

a _ω ²

! ^)σ

! ././B.!

*!$><<'-*-)'!%J*!a _ω

!$&#!%'$+#-+*!$>(!<&#-'!)T&-+*?,#7&'-*]!

R _ω

= .

.hπ ω -

∂ε

∂V )

* , - _ω

-

=V

C _G < a _ω ²

>

! ././B2!

%'-! C _G ! $*7&'-*! )*(! "#$%&'! )T*(*++,'-&/! 6-$*,*-)'! E>*$+T>(+&4#! *E>#7&'-*! -*((#! )*"&-&7&'-*! )&!

&4<*)*-7#!)T&-+*,#7&'-*!<(#$4#8"#$%&'!'++*-&#4']!

; _< = 5π

V _'- ² ω -

∂ε

∂ V )

* , - ω

= V

C _G ! ././BB!

./.2 [*"&-&#4'!&-'(+,*!(T&4<*)*-7#!)*(!"#$%&'!%'4*]!

D ₄ = C ₀ E ₀ =

− (. ₀ 5 ₀ ² 23

−. < 35 ₀ F _< = 2π5 0 . ₀

ω ₎ ² . _< F _< ! ././B9!

*!&(!<#,#4*+,'!)&!?>#)#?-'!)*(!<(#$4#!K _< ]!

K ^B _< = 6 ' ; <

9 c _' = ; <

9; _" = ω <

2

ω _- ² - _G - _'

.

> _' ω -

V '- 2 ∂ε

∂V )

* + , - .

ω

=V

! ././B0!

['L*! > ' ! d! >-#! L*('%&+N! %J*! $'))&$"#! 6 ' = - G *> ' ! *! C ₀ = . 2

(5 ₀ ²

3 =! %'-! 6 ' ! *! c ' ! ,&$<*++&L#4*-+*!

%',,*-+*!*!<'+*-7&#(*!)*(!"#$%&'!)&!*(*++,'-&/! 4 !d!(#!4#$$#!)*((T*(*++,'-*/!

R'$$&#4'!E>&-)&!&-+,')>,,*!E>*$+#!->'L#!?,#-)*77#!-*((#!Y././29Z!%J*!<*,+#-+'!)&L&*-*]!

V &- = K <

B π> ' ω -

∂" _'

∂L )

* + , - .

L= ω

V

! ././Bh!

W>*$+#! d! )>-E>*! (\*$<,*$$&'-*! )*(! +#$$'! )&! %,*$%&+#! %J*! '++*-&#4'! <*,! (T&-+*,#7&'-*! <(#$4#8

"#$%&'!)&!*(*++,'-&!-*((T&<'+*$&!)&!"#$%&'!%#()'!*!)*G'(*/!K'4*!?&#T!#%%*--#+'!<&F!L'(+*!L*),*4'!

%J*! >-T&)*-+&%#! *$<,*$$&'-*! $#,N! +,'L#+#! <*,! &(! $&$+*4#! $+,>++>,#! #)! '-)*! (*-+*8"#$%&'! %#()'! *!

)*G'(*!)&!*(*++,'-&/!

1ABCDE#LJHGGE#GD#HFHIIJEKDM#&HEJDA#FDKHAJH#

! M><<'-&#4'! #)*$$'! %J*! &(! "#$%&'! )&! *(*++,'-&! &-+,')'++'! -*(! <(#$4#! $&#! ",*))'/! K&'I!

V _&-

V _- >> V _- ∆c <

ω -

/!1&?>,#!./0/!

./.B

7

=?;@ M

;

1&?>,#!./0!

C<<,'$$&4&#4'!(#!)&$+,&G>7&'-*!)*((*!L*('%&+N!)*(!"#$%&'!#)!>-#!)*(+#!)&![&,#%]!

" ' = δYL − >Z

././B@!

6-!E>*$+'!%#$'=!)#((#!Y!././.0Z!'++*-&#4']!

εYω - =V _- Z = ω <

2

V _- - _G - _'

. V _- L − ω -

∂

∂L δYL − >Z!

! ^)L

! ././B5!

;&$'(L*-)'!(T&-+*?,#(*!<*,!<#,+&!$&!+,'L#]!

εYω - =V _- Z = ω <

2 - _G - _'

. YV _- > − ω - Z ²

! ././B:!

M&#!ω ' !(#!<>($#7&'-*!)*(!4')'!%J*!&-!#$$*-7#!)*(!"#$%&'!)&!*(*++,'-&!$&!<,'<#?#!%'-!L*('%&+N!)&!

"#$*!>/!R*,!+#(*!4')'!L#(?'-'!(*!,*(#7&'-&]!

./.9 V _' = ω '

>

εYω ' = V _' Z = O

!! ././9O!

R'-&#4'!#)*$$'!ω - = ω ' + δω - !*!$L&(><<&#4'!(#!Y././B:Z!#(!<,&4'!',)&-*]!

∂ε

∂ω _. δω _. + ∂ε

∂B _. δ B _. = ω ₎ ² . _<

. ₀ . YB _. 9 − ω _. Z ²

! ././9.!

64<'-*-)']!

64 )Yδω - Z

)YδV - Z

#

$ %

&

' ( = O!!!!!! / !!δω - = O

! ././92!

L*)&#4'!%J*!ω = ω ₀ !I!&(!4')'!<&F!&-$+#G&(*/!

M&!<>`!)&4'$+,#,*!%J*!<*,!>-!"#$%&'!$>""&%&*-+*4*-+*!L*('%*!(#!(#,?J*77#!#!4*+N!#(+*77#=! δω

=!)*(('!$<*++,'!)*&!4')&!&-$+#G&(&!)&4&->&$%*!4'(+'!,#<&)#4*-+*/!M&!+,'L#!&-"#++&]!

δω

= 2

B

ω <

2 - _G - _'

.

> ² ∂ε

∂V )

* , - ω

= V

)

* + + +

,

- . . .

. h

ω '

V7

! ././9B!

M&!<>`!E>&-)&!%'-$&)*,#,*!('!$<*++,'!%'4*!"',4#+'!)#(!$'('!4')'! ω ' =!E>*(('!<&F!&-$+#G&(*/!6-!

<,#+&%#=! I! &(! $'('! %J*! &-+*,#?&$%*! %'(! "#$%&'/! A(&!#(+,&!$&!#-->((#-'!<,&4#!%J*!('!<'$$#-'!"#,*/!

R'$$&#4'!E>&-)&!"#,*!>-!4')*(('!#)!'-)#!$&-?'(#/!

K'-$&)*,&#4'!E>&-)&!%J*!-*(!$&$+*4#!$&!<,'<#?J&!$'('!&(!4')'!ω = ω ₀ /!

R'-&#4'! ^{V = V} ' + δV =! )'L*! δV ! ,#<<,*$*-+#! (#! <*,+>,G#7&'-*=! $><<'$+#! <&%%'(#=! )'L>+#! #((#!

<,*$*-7#!)*(!"#$%&'/!