136mA ࡛࡝ࡽ 136MW ࡞┞ᙔࡌࡾಘྒ″࡚࠵ࡾࠊ

1.
࢕ࣤࢪࢵ࣒ࣜࣤࢹ    ୌ᪁ࠉరࡼ࠾ࡡ᪁Ἢ࡚ࢸ࣭ࣖࢷ࢔ 50%࡚ 1V ࡡࣄѸ 䝗䞀䝤ゝῼ䛮䝗䞀䝤Ꮽᏽ ໩䌝࣭࣑ಘྒࢅ 50ȐㇿⲬ࡚ᚋࡒ࡛ࡌࡾ࡛ࠉࡆࡿࡢ

10mW ࡞┞ᙔࡌࡾࠊࡌ࡝ࢂࡔࡐࡡẒࡢࠤ710−11

1. ࡢࡋࡴ࡞
⢇Ꮔࣄ࣭࣑ຊ㏷ჹࡢ㛏ᖳ࡞ῳࡖ࡙㐅໩ࢅ㐑ࡅࠉ࡚࠵ࡽࠉ࡮࡛ࢆ࡜ࣄ࣭࣑ࢅ஗ࡌࡆ࡛ࡢ࡝࠷ࠊࡌ࡝
ࢂࡔࠉ࡮࡛ࢆ࡜ࡡሔྙࠉࣄ࣭࣑ࢅ⌦᝷Ⓩ࡝㞹Ὦ″ࠉ࠵ࡾ࠷ࡢ㞹ᅸ″࡛⩻࠻ࡾࡆ࡛࠿࡚ࡀࡾࠊ

⛸ࠍࡡ┘Ⓩ࡞㐲ྙࡊࡒຊ㏷ჹ࠿㐘㌹ࠉᘋスࠉ࠵ࡾୌ᪁ࠉࣄ࣭࣑ࡢᵕࠍ࡝ᣪິࢅࡌࡾࠊࢤ࣭ࢪࢷ࢔

࠷ࡢゝ⏤ࡈࡿ࡙࠷ࡾࠊࡐࡿࡼ࡞࠽ࡄࡾࣄ࣭࣑ゝῼࣤࢡ࣬ࣄ࣭࣑࡚࠵ࡖࡒࡽࠉࣁࣤࢲࢺ࣬ࣄ࣭࣑ࡓࡖ

ࡵࠉᙔ↓࡝࠿ࡼࠉࡐࡿࡼ࡞ᚺࡋ࡙᭩㐲໩ࠉ⣵ฦ໩ࡒࡽࡌࡾࠊ࣭࣊ࢰࢹࣞࣤᣲິࢅ㉫ࡆࡊࡒࡽࠉࢨࣤ

ࡈࡿࡒᵕࠍ࡝᪁Ἢ࠽ࡻࡦ⿞⨠࠿࠵ࡾࠊࡆࡆ࡚ࡢࡐ

ࢠࣞࢹࣞࣤᣲິࢅ㉫ࡆࡊࡒࡽࡌࡾࠊࡐࡿࡼࡡᣪິ

ࡿࡼࡡ࠹ࡔࠉࣄ࣭࣑࡛ࡡ㞹☚ẴⓏ࡝┞பష⏕ࢅฺ

࡞ࡻࡖ࡙Ⓠ⏍ࡌࡾ㞹☚ሔࡢን໩ࡌࡾࡡ࡚ࠉཱིฝࡈ

⏕ࡌࡾࣄ࣭࣑ゝῼ࡞⤘ࡖ࡙よㄕࢅモࡲࡾࠊࡐ࠹ࡢ

ࡿࡾಘྒࡵᑊᚺࡊ࡙ን໩ࡌࡾࠊᐁ㝷࡞ࡢࠉಘྒࢅ

ゕࡖ࡙ࡵ࡝࠽ᗀ⠂࡝ᑊ㇗࡚࠵ࡾࡡ࡚ࠉⴥ⩽ࡡ㛏ᖳῼᏽࡊ࡙ࡐࡡ᫤ࠍࡡࣄ࣭࣑ࡡᣪິࢅ᥆ᏽࡌࡾࡡ

ᦘࢂࡖ࡙ࡀࡒ㝟Ꮔຊ㏷ჹ㸝ࠔ୯㧏࢙ࢾ࣭ࣜ࢟ࠕ
㸞࡞࡚࠵ࡾࠊୌྋ࡚ࣄ࣭࣑ࡡධᣪິࢅῼᏽ࡚ࡀࡾࡻ࠹

೩㔔ࡌࡾࡆ࡛ࢅᚒ஡ᢆ࠷ࡒࡓࡀࡒ࠷ࠊ

࡝ࡵࡡࡢᏋᅹࡊ࡝࠷ࠊࡐࡿࡑࡿࡡ┘Ⓩ࡞ྙࢂࡎ࡙

ࣄ࣭࣑࡛⿞⨠࡛ࡡ㞹☚ẴⓏ┞பష⏕ࢅ⩻࠻ࡾ࡛㑂ࡄ࡙㏳ࡿ࡝࠷ࡡ࠿ࠉ࠷ࢂࡹࡾࠔ࣭ࢗ࢘ࢠሔ
㸝wake field㸞ࠕࠔ⤎ྙ࢕ࣤࣅ࣭ࢱࣤࢪ㸝coupling
impedance㸞ࠕ࡚࠵ࡾࠊࣄ࣭࣑࠿ࣄ࣭࣑ࢱࢠࢹࠉ

≁ᛮ࠿ኬࡀࡂ␏࡝ࡾࠔ࢕ࣤࢪࢵ࣒ࣜࣤࢹ=ࣄ࣭࣑࣓ࢼࢰ࣭ࠕࢅ⏕ណࡌࡾࡆ࡛࡞࡝ࡾࠊ
ࣄ࣭࣑࣓ࢼࢰ࣭ࡡスゝࡡኬࡀ࡝Ὦࡿࢅࠉㄏよࢅᜅࡿࡍᩊ࠻࡙ゕ࠹࡝ࡼࡣࠉ

RF ✭ὕࡷ࢞ࢴ࣭࢜㞹☚▴࡝࡜ࡡ⛸ࠍࡡ┷✭⿞⨠(0) ῼᏽࡊࡒ࠷ࣄ࣭࣑ࡡ∸⌦㔖ࢅỬࡴࡾ㸯࡛ 㞹 ☚ Ẵ Ⓩ ࡞ ⤎ ྙ ࡊ ࡙ ࠉࠔ 㞗 ᅆ ୘ Ꮽ ᏽ ᛮ∸⌦㔖ࡡኬࡀࡈࡡ⠂ᅑࠉᡜさ⢥ᗐࠉฦよ⬗.

㸝collective instability㸞ࠕࢅᣚࡗ࡞⮫ࡾࡆ࡛࠿࠵

(1a) ࣄ࣭࣑ࡡ㞹☚ሔࢅゝ⟤ࡌࡾ㸯ࡾࠊ᪡࠷リ࠿ࠉࣄ࣭࣑ࡡషࡖࡒ㞹☚ሔ࡚⮤ฦ⮤㌗

㞹ሔࠉ☚ሔࠉ㞹Ὦࡡᙁᗐࠉ࿔ἴᩐࢪ࣋ࢠࢹࣜ.

࠿㋶ࡼࡿ࡙ࠉᣲິࢅጙࡴࠉ㐑࡞ࡢ࢙࣐ࢴࢰࣤࢪቌ

(1b) ಘྒࡡ᫤㛣࣬࿔ἴᩐᵋ㏸ࡡᢍᥩኬ㸝ࢦ࢕ࢫቌኬ㸞ࡷࣄ࣭࣑᥾኶ࢅ㉫ࡆࡊࡒࡽࡌࡾ

(2) ᳠ฝჹᵋ㏸ࡡ᭩㐲໩ࡡ࡚࠵ࡾࠊࡆࡆ࡚ࡢࠉฝᮮࡾࡓࡄ⠾༟࡝ᘟ࡛├វ(3) ฌ⌦ᅂ㊪ࡡ᭩㐲໩Ⓩㄕ᪺࡞ࡻࡽࠉࡐࡡཋᅄ࡞࡝ࡾᶭჹࡡ⤎ྙ࢕ࣤࣅ(4) ࢨࢪࢷ࣑ࡡ᭩㐲໩࣭ࢱࣤࢪࡡ࣊ࣤࢲῼᏽࠉࣄ࣭࣑୘Ꮽᏽᛮࡡふῼࠉ࡛࡝ࢀ࠹ࠊ୕オࡡࢪࢷࢴࣈࡡኣᑛࡡ࢕ࢷ࣭ࣝࢨࣘ

ࣄ࣭࣑୘ᏭᏽᛮࡡᏭᏽ໩࡞ࡗ࠷࡙ㄕ᪺ࡊࡒ࠷ࠊࡆࣤࡢᚪさ࡞࡝ࢀ࠹ࠊ

ࡡࢷ࣭࣏࡞ࡗ࠷࡙ࡵධ࡙ࢅ⥑⨮ࡌࡾࡆ࡛ࡢᅏ㞬࡚࠵ࡖࡒࡆ࡛ࢅ࠽᩷ࡽࡊ࡙࠽ࡂࠊ

2. ࣄ࣭࣑࡞௛㝮ࡌࡾ㞹☚ሔᐁࡢࢪࢷࢴࣈ(0)࠿⤎ᵋ᭜⩽࡚࠵ࡾࠊ᳠ฝჹࡢࠉ
ࣄ࣭࣑ࡡධᣪິࢅῼᏽ࡚ࡀࡢࡊ࡝࠷࠿ࠉࡒ࠷࡙
࠷ࠉ࠷ࡂࡗ࠾ࡡ∸⌦㔖ࢅῼࡾࡆ࡛ࡢྊ⬗࡚࠵ࡾࠊࡱࡒࠉࡐࡡ⢥ᗐࠉິష⠂ᅑࠉ࿔ἴᩐᖈᇡᖕ࡝࡜ࠉ

ࣄ࣭࣑ゝῼࢅࡌࡾࡒࡴ࡞ࡢࠉࣄ࣭࣑࠾ࡼࡡಘྒ

ࣄ࣭࣑࣓ࢼࢰ࣭ࡡ࣭ࣗࢧ࣭ࡡさ᭻ࡢࠉຊ㏷ჹࢸࢧ

࡞ࡗ࠷࡙⌦よࢅࡊ࡙࠽ࡂᚪさ࠿࠵ࡾࠊࣄ࣭࣑࠿┷࢕ࣤࡡṹ㝭ࠉࡈࡼ࡞ࣄ࣭࣑㐘㌹㛜ጙᚃࡡ౐⏕⤊㥺✭ࣂ࢕ࣈ୯ࢅ㉦ࡾ᫤࡞ࡢࡐࡡ࿔ᅑ࡞㞹☚ሔࢅఔ࡛࡛ࡵ࡞Ὦິࡌࡾࠊࡱࡒฝᮮ࡝࠷ࡆ࡛ࡢ࡚ࡀ࡝࠷

࠹ࠊࡐࡡ㞹☚࢙ࢾ࣭ࣜ࢟ࡡୌ㒂ࢅふῼࡡࡒࡴ࡞ಘ

㸝∸⌦Ἢ์࡞⫴ࡂࡆ࡛ࡢฝᮮ࡝࠷ࠊࠔ⮤↓ࡡዥ♼ࠕྒ࡛ࡊཱི࡙ฝࡌࠊౚ࠻ࡣࠉ㐘ິ࢙ࢾ࣭ࣜ࢟㸦GeVࡈࡱ࡛࠹ࡱࡂጂ༝ࡌࡾࡆ࡛ࡢ࡚ࡀࡾࠊ㸞ࠊࡆࡆࡢࠉ࡚⢇Ꮔᩐ 1012 ಴ࡡ㝟Ꮔࣄ࣭࣑࠿ 850kHz ࡚ࣛࣤࢡ᪝㡥࠾ࡼࣄ࣭࣑࣓ࢼࢰ࣭ࡡ౐⏕ᐁ⦴ࢅ⌦よࡊࠉࡱ

ຊ㏷ჹࢅ࿔ᅂࡊ࡙࠷ࡾ࡛ࠉ࿔ᅂࣄ࣭࣑㞹Ὦࡢࡒ࣓ࢼࢰ࣭࣭࣬ࣗࢧ࣭࡛ࡡࢤ࣐ࣖࢼࢢ࣭ࢨࣘࣤࢅ

F * (r," )(ᅒࡾࡆ࡛࠿㔔さ࡞࡝ࡾࠊࡆࡆ࡚ࡢ᭩ิࡡୌṄ࡛ࡊ࡙ࠉࡱࡍࠉࣄ࣭࣑࡞௛

(2 # $ m 0 ) I m (k r0 ' )I m (k a ' )m=0 2 %& 0=# )㝮ࡌࡾ㞹☚ሔࡡୌ⯙Ⓩ⩻ᐳࢅ⾔࡝࠹ࠊ* [ K m (k a ' )I m (k r ' ) # I m (k a ' )K m (k r ' )]* cos m(" # " 0 ),2. 1. ┷✭ࣂ࢕ ࣈ୯ࡡ ࣄ࣭࣑ ࡡ㞹☚ ሔ[1]⠾༟ࡡࡒࡴ࡞ࠉ┷✭ࣂ࢕ࣈࡢᏰධᑙమ࡚ࠉ༖ᙼ!
a ࡡළ➼ࣂ࢕ࣈ࡚ࠉ↋㝀ࡡ㛏ࡈ࡚࠵ࡾ࡛ࡌࡾࠊz㍂ࢅࣄ࣭࣑ࡡ㐅⾔᪁ྡྷ࡛ࡌࡾළᰍᗑᵾ㸝r,θ,z㸞ࢅ౐࠹ࠊࣄ࣭࣑࠿㏷ᗐ v ࡚㸝r0,θ0㸞ࢅ㐅ࢆ࡚࠷ࡾ࡛
!
ࡌࡾ㸝ᅒ㸞ࠊ✭㛣᫤㛣ᵋ㏸࠿ e" jk(z"vt) ࡡୖオࡡࡻ࠹࡝␧ᐠἴࡓ࡛ࡌࡾ1 ࠊk=Ȱ/v ࡛ࡊࡒࠊ# (r !
$ r0 )" = "0# (% $ % 0 ) e$ jk(z$vt) ,!
r0# (r $ r0 )J = v" = J 0# (% $ % 0 ) e$ jk(z$vt) .
r0

!

=#

ࣄ࣭࣑ࡡ㟴Ḿ⣌࡚ࡢࠉࣄ࣭࣑㞹Ὦ࠿ᾐ࠻ࠉࣄ࣭࣑!

㞹Ⲭ㸯" $ (r % r0 )"* = 0$ (& % & 0 ) e% jk z* # ,
#
r0

!

!

$
" * (r,# , z*) = F * (r,# ) 0 e& jk z* % ,%!

#0 k )I (k r & )* (2 ' ( m 0 ) m 0
2$% 0 & m=0
I m (k a & )࡝࡜࠿ᚋࡼࡿࡾࠊțm0=1 (m=0),țm0=0 (m≠0)࡚࠵
!

࣎࢓ࢮࣤ᪁⛤ᘟ࡞࡝ࡾࠊよࡢࠉ➠㸦⛸ࠉ➠ 2 ⛸ንᙟ࣊ࢴࢬࣜ㛭ᩐ Im(z), Km(z)ࢅ౐࠷ࠉሾ⏲᮪௲㸝r=0!
!
࡚᭯㝀ࠉr=a ࡚ F*=0㸞ࢅ‫࡞࠹ࡻࡌࡒ‮‬Ửᏽࡊࠉ
r>

!

=

+ cos m(" ' " 0 ),࡛࠽ࡂ࡛ࠉ F * (r," ) ࡢ㟸᩟ḗࣉ࣑ࣜ࣌ࣜࢵ᪁⛤ᘟ& k )2, (r $ r0 ), (- $ - 0 )!
,"#2 F * $ ( + F * = $. 0 r0'% *!
ࢅ‫࠿࣑࣭ࣄࠊࡌࡒ‮‬ක㏷࡞㎾ࡂ࡝ࡾ࡛㸝" # $ 㸞ࠉ!

+ cos m(" # " 0 ),Er (r," )+ [ K m (k a & ) I ,m (k r & ) ' I m (k a & ) K ,m (k r & )]ࡡࡲ࡞࡝ࡾࠊ㟴㞹ሔ " * (r,# ,z*) ࡡ᪁⛤ᘟࡢࠉ% ( (r $ r0 )" 2#* = $ 0( () $ ) 0 ) e$ jk z* ' .
& 0'
r0
よࢅ

$0 j k )I (k r ' )* (2 # ( m 0 ) m 0
2%& 0 ' 2 m=0
I m (k a ' )+ [ K m (k a ' )I m (k r ' ) # I m (k a ' )K m (k r ' )]!

!

࡛࡝ࡾࠊ㞹☚ሔࡢࠉ
$% *
$% *
$% *
Er = "#, E& = "#, Ez = * ,
$r
$&
$z
v
v
Br = " 2 E# , B# = 2 Er , Bz = 0 ,cc࠾ࡼịࡴࡼࡿࠉ
Ez (r," )r0 ࡚ࡢࡾࠊࣂ࢕ࣈහ㟻࡞ࠉBθ ࡞Ẓౚࡌࡾ㞹Ὦ࠿ὮࡿࠉBv
JW = " =Erµ 0 µ 0c 2
# v
=" 0
2$ a
(1)
(
I (k r0 ' )% ) (2 " & m 0 ) m
cos m(* " * 0 ),I m (k a ' )m=0࡛࡝ࡾࠊࡆࡿࢅባ㞹Ὦ㸝wall current㸞࡛࿣ࡩࠊ
2.1.1. ┞ᑊㄵⓏࣄ࣭࣑㸝 " # $ , v c " 1㸞ࡡሔྙ㞹Ꮔࠉ㝟㞹Ꮔࡢ㈻㔖࠿ᑚࡈ࠷ࡒࡴࠉຊ㏷ṹ㝭ࡡิ᭿࠾ࡼࡌࡃ࡞┞ᑊㄵⓏ࡞࡝ࡾࠊࣄ࣭࣑࠿z ㍂୕!
!
ࢅ㐅ࢆ࡚࠷ࡾ࡛ࡌࡾ࡛ࠉJW ࡚ m=0 ௧አࡡ㡧ࡢᾐ࠻࡙ࠉባහ㟻࡞ୌᵕ࡝㞹Ὦ࠿ࠉ㞹Ὦᐠᗐ# v
1
J W (k) = " 0,
2 $ a I 0 (k a % )(2)࡚Ὦࡿࡾࠊ୕ᘟࡢ࿔ἴᩐࢪ࣋ࢠࢹࣜ࡝ࡡ࡚ z−vt
1

!
!
!

e j" t ࡢ engineering 㛭౿࡚ࡻࡂ౐ࢂࡿࡾࠊ∸⌦㛭౿࡚
!
"i# t
ࡢe
࠿ᬉ㏳࡚࠵ࡾࠊࡆࡆ࡚ࡢゝῼჹ࡞࡝ࡋࡲࡷࡌ࠷
j"teࢅ౐࠹ࠊౚ࠻ࡣࠉ࢕ࣤࢱࢠࢰࣤࢪࡡ࢕ࣤࣅ࣭ࢱࣤࢪ ࡢ ࠉ engineering convention ࡚ ࡢ jωL, physics
convention ࡚ࡢ−iωLࠊࢾࢴࢹ࣭࣠ࢠ࢓ࢻࣚ࢕ࢧࡡῼᏽ
⤎ᯕࡡ➚ྒࡢ engineering convention ࡝ࡡ࡚さἸណࠊࡡ✭㛣࡚ࡢࡆࡿࡡࣆ࣭࢙ࣛ㏣ንᥦ࡛࡝ࡾࠊ࿔ἴᩐ
ࡡᗀ࠿ࡽࡢ~Ț/a ࡡ⛤ᗐ࡚ࠉz ㍂᪁ྡྷ࡞ࡢ㏣ᩐࡡ
~a/Țࡡᗀ࠿ࡽࢅᣚࡗࡆ࡛࠿ࢂ࠾ࡾࠊ

" # $ ࡡᴗ㝀࡚ࡢࠉz ㍂᪁ྡྷࡡᗀ࠿ࡽ࠿↋ࡂ࡝ࡽࠉ㞹☚ሔࡢࠉ࡮࡛ࢆ࡜ᶋ᪁ྡྷ㸝transverse㸞ᠺ࢕ࣈ⾪㟻ࡡ㞹☚ሔ࠾ࡼࣄ࣭࣑㛏ࢅịࡴࡻ࠹࡛ࡌ
ࡾ࡛ၡ㢗࠿⏍ࡋࡾ1 ࠊฦࡡࡲࢅࡵࡔࠉTEM ἴ࡞㎾ࡂ࡝ࡾࠊࡆࡡᴗ㝀࡚!

ࡢࠉz ㍂࠾ࡼࡍࡿࡒࣄ࣭࣑࡞ࡻࡾባ㞹Ὦࡢࠉ' r *m$ v JW " # 0. (2 # & m 0 )) 0 , cos m(/ # / 0 )(a+
2% a m=0
="

!

#0 v
a 2 " r02,
2
2
2$ a a + r0 " 2a r0 cos(% " % 0 )ᶋ᪁ྡྷ࡚ࡵၡ㢗࠿㉫ࡀࡾࠊᩝ⊡[4]࡚レࡊࡂㄵࡋࡼࡿ࡙࠷ࡾࠊ

(3)

࡛࡝ࡾࠊ
!

(a)

ᅒ㸯ළ➼ࣂ࢕ࣈ୯ࡡࣄ࣭࣑

(b)
2.1.2. 㟸┞ᑊㄵⓏࣄ࣭࣑ࡡሔྙ

" ࡢࠉྜྷ࢙ࢾ࣭ࣜ࢟ࡡ⢇Ꮔ࡚Ẓ㍉ࡌࡾ࡛㈻㔖࡞㏣Ẓౚࡌࡾࠊ㝟Ꮔࣄ࣭࣑ࠉ࢕࢛ࣤࣄ࣭࣑࡝࡜࡚ࡢ
㈻㔖࠿ኬࡀ࠷ࡡ࡚ࠉྜྷ࢙ࢾ࣭ࣜ࢟ࡡ㞹Ꮔ࡞Ẓ࡬࡙!

࠽ࡻࡐ 1/2000 ௧ୖ࡚࠵ࡾࠊౚ࠻ࡣ J-PARC ࡚ࡢ
Energy"LINAC

ӌ0.181 (0.4) MeV

ӌ 1.2 (1.4)

RCS

ӌ3 GeVӌ 4.2!
࡚࠵ࡾࠊࡐࡡࡒࡴࠉ࣓ࢼࢰ࣭ࢅスゝࡌࡾ࡞࠵ࡒࡖ

(c)

࡙Ἰណࢅさࡌࡾࠊ
ባ㞹Ὦࡡฦᕱࡡᘟ(2)࡞ࡻࡾゝ⟤ೋࢅᅒ࡞♟ࡌࠊJ-PARC RCS ࡡࣄ࣭࣑ࣂ࢕ࣈࡡ඼ᆵೋࢅ࡛ࡖ
࡙༖ᙼ 130 mm ࡛ࡊ࡙࠷ࡾࠊࡱࡍࣄ࣭࣑ࡡࣁࣤࢲ
㛏 5 cm ࡡ᫤࡞ࠉባ㞹Ὦࡡᗀ࠿ࡽࡢࣁࣤࢲ㛏࡞Ẓ࡬࡙ࠉ181 MeV ࡚ࡢ⣑ 5.9 ಶࡓࡖࡒࡵࡡ࠿ࠉຊ㏷

ᅒ㸯༖ᙼ PP ࡡළ➼ࣂ࢕ࣈ /5&7(7(8 ࡡ඼ᆵೋ ࢅὮࡿࡾባ㞹Ὦࠊ㉝⥲㸯㛏ࡈ HP FG ࡱࡒࡢ P H ࡡࣁࣤࢲἴᙟࠉ㟯⥲㸯ባ㞹Ὦ㸝ㇿྒࡢን㌹ࡊ࡙࠵ࡾ㸞
ࠊ F 㝟Ꮔ 2H;ࠉ G 㝟Ꮔ ,H;ࠉ
H 㝟Ꮔ ,H;ࠊV"̲[W ࡛ࡊ࡙ᘟ  ࢅࣆ࣭࢙ࣛ㏣ንᥦࡊࡒ

ࡊ࡙ 3 GeV ࡞࡝ࡾ࡛⣑ 1.9 ಶ࡞΅ᑛࡌࡾ㸝༖ೋᖕ࡚Ẓ㍉㸞ࠊࣁࣤࢲ㛏࠿ᗀ࠿ࡖ࡙ 5m ࡚ࡢ 181 MeV࡚ࡵࣁࣤࢲ㛏࡛࡮࡯ྜྷࡋ࡚࠵ࡾࠊࡆࡡࡒࡴ࡞ࠉ఩
࢙ࢾ࣭ࣜ࢟㝟Ꮔࣄ࣭࣑࡚ࡢࠉ▯ࣁࣤࢲࡡሔྙࠉࣂ

1よỬ Ἢ࡛ࡊ ࡙࠷ ࡂࡗ࠾ ࡡࡵࡡ ࠿࠵ࡾ ࠊBunch shapemonitor [2]ࠉlaser profile monitor [3]࡝࡜ࠊ

2. 2. ࣄ࣭࣑ࡡ ᫤㛣ᵋ ㏸࡛࿔ ἴᩐࢪ ࣋ࢠࢹ ࣜࢅࣆ࣭࢙ࣛᒈ㛜ࡌࡾ࡛ࠉ " n = n " 0 = 2# n T 0 ࡛࠽ࣄ࣭࣑ࡢࠉ㏳ᖏࣁࣤࢲᵋ㏸ࢅᣚࡗ࠾ࠉ࠵ࡾ࠷ࡢ

࠷࡙
Tࣄฺ࣭࣑⏕ࡡ┘Ⓩ࡞ࡻࡖ࡙ࡢࠉDC Ⓩ࡝ࢤ࣭ࢪࢷ

v(" n ) =࢔ࣤࢡ࣬ࣄ࣭࣑ࡡᙟឺࢅ࡛ࡾࠊࡱࡒࠉຊ㏷ჹࡡ㐘

2

1 0
v
$ !v(t) e# jn" t dt = 0 ,T 0 #T 2
T0
0

ࢅ౐ࡖ࡙

㌹࿔᭿࡞㛭㏻ࡌࡾ᫤㛣ᵋ㏸ࢅࡵࡗࠊ

$ౚ࡛ࡊ࡙ J-PARC ࡡຊ㏷ჹࡡࢢ࣭ࢪࢅ⣺௒ࡌ!
ࡾࠊLINAC ࡡሔྙࡢᅒࡡࡻ࠹࡝ࡵࡡ࡚࠵ࡾࠊ

v(t) =% v(" n ) e jn" t =n=#$v 0 $ jn" t,%e
T 0 n=#$

ࡱࡒࠉRCS ࠽ࡻࡦ MR ࡢࠉ࠽࠽ࡻࡐᅒࡡࡻ࠹࡝

࡛࡝ࡾࠊࡆࡿࢅᅒ࡞♟ࡊࡒࠊࡆࡡࡻ࠹࡝࿔ἴᩐ

᫤㛣ᵋ㏸࡛࡝ࡖ࡙࠷ࡾࠊ

ࢅ≺❟ንᩐ࡛ࡌࡾಘྒࢅ࿔ἴᩐ㡷ᇡ㸝frequency!

domain㸞ࡡಘྒ࡛࿣ࡩࠊࡆࡆ࡚≁ᚡⓏ࡝ࡆ࡛ࡢࠉ
᫤㛣㡷ᇡ࡚ࡡࣅ࣭ࢠࡡ㛣㝰 T 㸥࠿ࠉ࿔ἴᩐ㡷ᇡ࡞⛛ࡾ࡛࿔ἴᩐ㛣㝰Ǽf =㸦㸤T0 ࡞࡝ࡾࡆ࡛࡚࠵ࡾࠊ

ᅒ㸯LINAC ࣄ࣭࣑ࡡ᫤㛣ᵋ㏸ᅒ㸯࿔᭿ 9 ࡡț㛭ᩐἴᙟࡡ࿔ἴᩐࢪ࣋ࢠࢹࣜࠊ
I"㸤9࠷ࡂࡗ࠾ࡡ඼ᆵⓏ࡝ࣄ࣭࣑ᙟ≟࡞ࡗ࠷࡙ࡡ᫤
㛣㡷ᇡࠉ࿔ἴᩐ㡷ᇡࡡಘྒࢅ♟ࡌࠊࣄ࣭࣑ἴᙟࢅ I(t) = I0 { 1 − cos(2 π t㸤τb) }㸤2࡛ࡊࡒ᫤ࡡ࿔ἴᩐࢪ࣋ࢠࢹࣜࡢ௧ୖࡡࡻ࠹࡞࡝
ࡾࠊI (" ) = 2I 0sin(n" 0# b )1n" 0# b 1 $ ( n" #0 b!
ࡆࡿࡼࡡಘྒࡡᏽ㔖Ⓩ࡝よᯊࡡࡒࡴ࡞ࠉ⠾༟࡝ሔྙࢅ⩻ᐳࡌࡾࠊࢸࣜࢰ㛭ᩐⓏ࡝ࣂࣜࢪಘྒ࠿ࠉ
T0 ࡡ࿔᭿࡚ࡂࡾ࡛ࡌࡾࠊࡆࡡಘྒࡢ$% " (t # nT 0 ) ,ࣄ࣭࣑ἴᙟ࠿▬ᙟࠉhalf-sinࠉcos㸝୕ᘟ㸞࡛ࡀࡡn=#$ἴᙟ࡛࿔ἴᩐࢪ࣋ࢠࢹࣜࢅᅒ࡞♟ࡌࠊࢸࣜࢰ㛭࡛᭡ࡄࡾࠊࡆࡡࡻ࠹࡝᫤㛣ࢅ≺❟ንᩐ࡛ࡌࡾಘྒ
ࢅ᫤㛣㡷ᇡ㸝time domain㸞ࡡಘྒ࡛࿣ࡩࠊࡆࡿ
!

,I 0 : average (circulating) current,n : integer,n" : number of bunches,h : harmonic number,# 0 : revolution anguler frequency,$ b : bunch half width.

ᅒ㸯7(827 ࡡࣄ࣭࣑

v(t) = v 0

sin(n&% n h)2% ) sin(% n h)!

ᩐࡢౚአⓏ࡞࿔ἴᩐ㡷ᇡ࡚ಘྒᣲᖕ࠿ୌᏽ࡚࠵
ࡾ࠿ࠉ᫤㛣㡷ᇡ࡚ᡀࡾ᫤㛣ᖕǼT ࢅࡵࡗಘྒࡢࠉ
࿔ἴᩐ㡷ᇡ࡚ࡢࠉಘྒᙁᗐࡡໜ⤙⥲ࡡᖕࡢ࠽࠽ࡻ

ࡐǼf = 1㸤ǼT ࡛࡝ࡾࠊ▯࠷᫤㛣ἴᙟࢅῼᏽࡌࡾ
࡞ࡢࠉࡻࡽᗀ࠷࿔ἴᩐᖈᇡᖕ࠿ᚪさ࡞࡝ࡾࠊࣄ࣭࣑࠿ࢨࣤࢠࣞࢹࣞࣤᣲິࡌࡾ࡛ࡀࡢࠉ࿔ἴ
ᩐࢪ࣋ࢠࢹࣜࡢ௧ୖࡡࡻ࠹࡞࡝ࡾࠊࣄ࣭࣑ࡡ࿔ᅂ
᫤㛣࠿ࢨࣤࢠࣞࢹࣞࣤᣲິ
T 0" (t) = T 0 + #T cos($ s t ) ,࡞ࡻࡖ࡙ᣲິࡌࡾࡡ࡚ࠉಘྒࡢ%& " (t # nT 0$ (t)) ,!

v(t) = v 0!

ࡡࡻ࠹࡞᭡ࡀᥦ࠻ࡼࡿࡾࠊࡆࡿࡢ
v &v(t) " 0 ' e jn# (t$%T sin(# st))T 0 n=$&n=#%v &= 0 ' e jn# t e$ jn# %T sin(# st)
T 0 n=$&&v &= 0 ' e jn# t ' J m (n# %T ) e$ jm# st
T 0 n=$&m=$&&v &= 0 ' ' J m (n# %T ) e j(n# $m# s )t ,T 0 n=$& m=$&ᅒ㸯᫤㛣ᵋ㏸࡛࿔ἴᩐࢪ࣋ࢠࢹࣜࠊ㉝⥲㸯୔みἴࠉ⥫⥲㸯ḿᘳἴ㸝 ࿔᭿㸞ࠉ㟯⥲㸯ḿᘳἴ㸝༖࿔!
᭿㸞ࠉ⣰⥲㸯▬ᙟἴࠊ

࡛࡝ࡾࠊࡆࡆ࡚㛭౿ᘟe jz sin" =$% J n (z) e jn" ,n=#$ࢅ౐ࡖࡒ[ref]ࠊJm(z)ࡢ m ḗࡡ➠㸦⛸࣊ࢴࢬࣜ㛭ᩐ࡚࠵ࡾࠊn=0 ࡚ J0 ௧አࡢࢭࣞ࡝ࡡ࡚ࠉm=1,2,!

͐ࡡࢦ࢕ࢺࣁࣤࢺࡢ࡝࠷ࠊࣚࢩ࢛ἴࡡ FM㸝࿔ἴᩐንㄢ㸞࡞㢦జࡊ࡙࠷ࡾࠊᶅᘟⓏ࡝࿔ἴᩐࢪ࣋ࢠ
ࢹࣜࢅᅒ࡞♟ࡌࠊᅒ㸭ࡢ⌦よࡡຐࡄࡡࡒࡴ࡞ーᘿࡊ࡙࠵ࡾࠊౚ࠻ࡣ J-PARC ࣛࣤࢡࡡሔྙࢨࣤࢠ
ࣞࢹࣞࣤᣲິᩐࡢᩐ 10Hz ~ ᩐ kHzࠉຊ㏷᫤㛣ࡢ
10 ms ~ 1 s ࡡ࢛࣭ࢱ࣭࡚࠵ࡾࠊຊ㏷୯࡞࿔ἴᩐࡢ
ን໩ࡌࡾࠊᚉࡖ࡙ୖオࡡࡻ࠹࡞㩥᪺࡝ࢪ࣋ࢠࢹࣜࢅぜࡾࡡࡢࡳࡘ࠾ࡊ࠷ࠊ

ᅒ㸯࿔ἴᩐࢪ࣋ࢠࢹࣜࠊ/5&7(27 ථᑏ࢙ࢾࣜ
࣭࢟ࡡሔྙࠊK"ࠊ୕ᅒࡢ  ࣁࣤࢲࡡࡲࠊୖᅒࡢ ࣁࣤࢲ⵫✒ࡊ࡙࠷ࡾࠊ
ᅒ㸯ࢨࣤࢠࣞࢹࣞࣤࢦ࢕ࢺࣁࣤࢺ

ࣄ࣭࣑ࡢ࠷ࡗࡵ z ㍂୕ࢅ㉦ࡾࢂࡄ࡚ࡢ࡝࠷ࠊᶋ
᪁ྡྷ㸝transverse㸞ࡌ࡝ࢂࡔ z ㍂࡞ᆱ├࡝᪁ྡྷ࡞
ንనࡊ࡙㏳㐛ࡌࡾࠉࡊ࠾ࡵንన࠿࿔᭿Ⓩ࡞ን໩ࡌࡾࡆ࡛࠿࠵ࡾࠊࡆࡿࢅ࣭࣊ࢰࢹࣞࣤᣲິ࡛࿣ࡩࠊ
ࣅࢴࢠ࢓ࢴࣈ࠿ス⨠ࡈࡿ࡙࠷ࡾሔᡜ࡚ࠉࣛࣤࢡࢅ
࿔ᅂࡌࡾࣄ࣭࣑ࢅ n ࿔᭿┘࡚ふῼࡊࡒ᫤ࡡࣄ࣭࣑న⨠ࡢ
x P (nT0 ) = x P 0 + A " P cos # " $ 0 nT0 + % ,()ᅒ㸯࣭࣊ࢰࢹࣞࣤࢦ࢕ࢺࣁࣤࢺ㸝/5&7(27㸞ࠊ࡚࠵ࡾࠊxP0 ࡢ᫤㛣Ⓩ࡞ን໩ࡊ࡝࠷ንన㸝㛚㌮㐠㸞ࠉșP ࡢ࣭࣊ࢰࢹࣞࣤ㛭ᩐ࡚ሔᡜ࡞ࡻࡖ࡙␏࡝ࡾೋ
!

ࢅ࡛ࡾࠉȤșࡢ࣭࣊ࢰࢹࣞࣤࢲ࣭ࣖࣤࠉA,ȭࡢิ᭿ᣲᖕࠉన┞࡚࠵ࡾࠊࡆࡡᣲິ࠾ࡼࡡᐞ୙࠿ຊࢂ
ࡾ࡛ࠉv(t) ==*j (' " t+) )# j (' ( " 0 t+) ) v 0 $ jn" 0 t ,
e ( 0
+e
/,
e
1
+
&v
%
,
/
T 0 n=#$
2+.
v 0 $ jn" 0 t%e
T 0 n=#$$ve+ 0 &v %T 0 n=#$[j (n+' ( ) " 0 t+)] + e[ j(n#' ( ) " 0 t#) ]2

2. 3. ࣄ࣭࣑୘ Ꮽᏽᛮ ࡛ಘྒ [5].

ࣄ࣭࣑࠿୘Ꮽᏽ࡞࡝ࡾ࡛ࠉᵕࠍ࡝ᚡು࠿⌟ࡿ࡙" = n" 0 ࡡ୦ഁ࡞ࠉ࣭࣊ࢰࢹࣞࣤᣲິᣲᖕ࡞Ẓౚࡌ()ࡾኬࡀࡈࡡࢦ࢕ࢺࣁࣤࢺ " = " 0 n ± # $ ࠿࡚ࡀࡾ!
!

ࡆ࡛࠿ࢂ࠾ࡾࠊᅒ࡞ J-PARC MR ࡚ῼᏽࡈࡿࡒ
࣭࣊ࢰࢹࣞࣤᣲິࡡ࿔ἴᩐࢪ࣋ࢠࢹࣜࢅ♟ࡌࠊ!
ῼᏽࢸ࣭ࢰ࠾ࡼࢂ࠾ࡾࡡࡢࠉᅒࡡࡻ࠹࡞" = (n# ± $% & )" 0 ࡡࡲ࡚࠵ࡾࠊ"# $ ࢅ " # ࡡ❻ᩐ㒂࡛ࡌࡾ࡛ࠉ" # = n $ % n ± &" # ( n, n " = #$ ~ #1, 0,1, ~ $ )ࢅ‫‮‬!

ᅒ㸯/5&7(27 ථᑏ᫤ࡡ࣭࣊ࢰࢹࣞࣤᣲິ㸝ᆱ├᪁ྡྷ㸞ࠊ

ࡒࡌ n, n'ࡢ↋ᩐ࡞࠵ࡾࡡ࡚ࠉ " # ࡡ❻ᩐ㒂ࡊ࠾ࢂ!
!
࠾ࡼ࡝࠷ࠊ" # ࡡᩒᩐ㒂ࢅ▩ࡾ࡞ࡢࠉౚ࠻ࡣ├᥃Ⓩ!
!
࡞ࡢࠉࣛࣤࢡධ࿔ࡡ࣭࣊ࢰࢹࣞࣤᣲິࢅ㏛ࡖ࡙ᣲ!
ິᩐࢅᩐ࠻ࡿࡣⰃ࠷ࠊ
!
ᅒ
㸯/5&7(27 ථᑏ᫤ࡡ࣭࣊ࢰࢹࣞࣤᣲິ㸝ᆱ├᪁ྡྷ㸞ࠊ࡞ J-PARC MR ࡡථᑏ࢙࣭࡚ࣚⓆ⏍ࡊࡂࡾࠊࡆࡆ࡚ࡢࠉࣄ࣭࣑ࡡన┞✭㛣࡚ࡡ㞗ᅆ㐘ິࢅ࣓࣭ࢺ࡞ฦよࡊ࡙⩻࠻ࡾ᪁Ἢࢅ⠾༟࡞⣺௒ࡌࡊࠉ๑⟿ࡡ᫤㛣࣬࿔ἴᩐ㡷ᇡࡡἴᙟ࡛㛭㏻ࡘࡄࡒ࠷ࠊࣄ࣭࣑ࡡ㐘ິࡢ㸨ḗඔ࡚࠵ࡾ࠿ࠉ㐅⾔᪁ྡྷ㸝⦢᪁ྡྷ㸞ࡡᣲິᩐ࡛ࠉᶋ᪁ྡྷࡡᣲິᩐ࠿ኬࡀࡂ␏࡝
ࡾࡡ࡚ࠉ୦᪁ྡྷࡡ㐘ິ࠿⤎ྙࢅ㉫ࡆࡊ࡙࠷ࡾࡻ࠹
࡝≁ื࡝ሔྙࢅ㝎࠷࡙ࠉࡐࡿࡑࡿࢅฦࡄ࡙⩻࠻ࡾࡆ࡛࠿࡚ࡀࡾࠊౚ࠻ࡣ J-PARC ࣛࣤࢡࡡሔྙࢨࣤ
ࢠࣞࢹࣞࣤᣲິᩐࡢᩐ 10Hz ~ ᩐ kHz㸝๑ฝ㸞ࠉ࣭࣊ࢰࢹࣞࣤᣲິᩐࡢᩐ MHz ࡚ࠉ࠽ࡻࡐ㸨᰾ࡵ㞫ࡿ࡙࠷ࡾࠊ
2.3.1. ⦢᪁ྡྷࡡ࣓࣭ࢺ

ࡒᆱ├᪁ྡྷࡡ࣭࣊ࢰࢹࣞࣤᣲິࢅࣛࣤࢡୌ࿔࡞

ࡱࡍ㐅⾔᪁ྡྷ㸝⦢᪁ྡྷ㸞ࡡ࣓࣭ࢺࢅ⩻࠻ࡾࠊᶋࢂࡒࡖ࡙ῼᏽࡊࡒ⤎ᯕࢅ♟ࡌࠊᣲິ࠿ 21 ࡞ࡷࡷ᪁ྡྷࡢᣲິࡊ࡝࠷࡛ࡌࡾࠊࡆࡡሔྙࡡ⦢᪁ྡྷࡡన

㊂ࡽ࡝࠷⛤ᗐ࡚࠵ࡾ࡛ㄖࡲཱིࡿࡾࠊᐁ㝷ࠉスゝೋ┞✭㛣࡚ࡡࣄ࣭࣑ฦᕱࢅ⩻࠻ࡾࠊᏭᏽࣄ࣭࣑ฦᕱ
࠿ " 0 (r) ࠉࡐࡆ࠾ࡼࡡ᚜ᑚ࡝ࡍࡿ࠿ "1 (r,# ) ࡓ࡛ࡌࡢ~20.7 ௛㎾࡚࠵ࡾࠊ࣭࣊ࢰࢹࣞࣤ ࢲ࣭ࣖࣤ࠿ࠉౚ࠻ࡣ 20.7 ࡝ࡡ࠾
21.3 ࡝ࡡ࠾ࡢࠉ㸩ᴗ㞹☚▴ࡡ཭ᮨງࢅ࠾࠻࡙ࡲ
!
࡙ࠉࢲ࣭ࣖࣤࡡቌ΅ࡡྡྷࡀࢅㄢ࡬ࡿࡣึื࡚ࡀࡾࠊ!

ࡾ㸯" (r,# ) = " 0 (r) + "1 (r,# ) e j$s / c .
!

charge density

z

z

z

Phase spacequadrupole

dipolel=1

sextupole

l=2

…..

l=3

ᅒ㸯1RQJLWXILQFOPRIHV

dipole mode density l(z)xzzz

superimposedmonopole
l=0

dipolel=1quadrupole …..
l=2ᅒ㸯9UFQV[HUVHPRIHV̲ ㍂㸝W ㍂㸞࡞Ἒࡖ࡙న⨠ಘྒࢅぜ࡙࠷ࡾ

᚜ᑚ࡝ንᙟࢅ⾪ࡌฦᕱࡢࠉ
"1 (r,# ) = & $ l Rl (r) e% jl# ,l࡛ࠉన┞✭㛣හࡡິᙼ᪁ྡྷ࡛みᗐ᪁ྡྷࡡ࣓࣭ࢺࡡ
㔔ࡠྙࢂࡎ࡚᭡ࡂࡆ࡛࠿࡚ࡀࡾࠊ࣓࣭ࢺε =1~3!

ࢅᶅᘟⓏ࡞ᅒ࡞♟ࡊࡒࠊୖṹ࠿న┞✭㛣හࡡฦᕱࡡንᙟ࡚ࠉ୕ṹ࠿ࡐࡡ z ㍂㸝t ㍂㸞࡫ࡡᑏᙫ
࡚࠵ࡾࠊ㏳ᖏࣄ࣭࣑࣓ࢼࢰ࣭࡚ࡢࡆࡡᑏᙫ࠿ふῼࡈࡿࡾࠊ

2.3.2. ᶋ᪁ྡྷࡡ࣓࣭ࢺ
ᶋ᪁ྡྷ࣓࣭ࢺࡢࠉỀᖲࡱࡒࡢᆱ├᪁ྡྷࡡన┞✭㛣࡚ࡡࠉ⦢᪁ྡྷ࡛ྜྷᵕࡡᣲິ࣓࣭ࢺ࡚࠵ࡾࠊࡊ࠾
ࡊࠉᶋ᪁ྡྷ࡚ࡢࠉ㐅⾔᪁ྡྷࡡ㐘ິࡵ⩻៎ࡊ࡝ࡄࡿ
ࡣ࡝ࡼ࡝࠷ࠊ୦᪁ྡྷࡡ㐘ິ࠿⤎ྙࡌࡾౚ࡚࠵ࡾࠊ
ᶋ᪁ྡྷࡡฦᕱࢅ f0(q,ȟ)ࠉf1(q,ȟ)ࠉ⦢᪁ྡྷࡡฦᕱ
ࢅ g0(r,ȭ)ࠉg1(r,ȭ)࡛ࡊ࡙ࠉ㎾జⓏ࡞ࡢ
" (r,#; q,$ ) % f 0 (q)g 0 (r) + f1 (q,$ )g1 (r,# ) e j&s / c࡛࡝ࡾࠊȐ=Ȱβ+ℓȰs−ȥȰβ㸤Ȟ࡚࠵ࡾࠊᅒ࡚ࡢࠉᶋ᪁ྡྷ࡞ dipoleࠉ⦢᪁ྡྷ࡞ε =0,1,2!

ᅒ㸯ᐁ㝷ࡡ <FOO(XUUHQW2RQLWRU

ࡡ࣓࣭ࢺࡡሔྙࡡᣲິࣂࢰ࣭ࣤࢅ♟ࡊࡒࠊࡐࡡ⤎ᯕࠉ఩ᇡ㐵᩷࿔ἴᩐࡢࠉࢤ࢓ࡡ࢕ࣤࢱࢠ

3. ࣄ࣭࣑ゝῼࣄ࣭࣑㞹Ὦࢅῼᏽࡌࡾ⿞⨠࡞ࡢࠉ㧏ᇡ㐵᩷࿔ἴ!
ᩐࡡኬࡀ࠷㡨࡞ࠉWCMࠉCTࠉࣆ࢔࣭ࢺࣁࢴࢠᆵ

ࢰࣤࢪࢅ LࠉㇿⲬᢤᢘࢅ R ࡛ࡊ࡙R"L = ,
L఩ᇡࡡ࿔ἴᩐ≁ᛮࡢj" / " L
V (" ) =RI B ,
1+ j" / " LCTࠉDCCT ࠿࠵ࡾࠊ࡛࡝ࡾࠊ

3. 1. 㞹Ὦೋࠉ ࣁࣤࢲ 㛏ࡡゝ ῼࢤ࢓ࡷᢤᢘࡢࠉΰὯ࡝┷✭ࢅಕࡗࡒࡴ࡞ኬẴഁ
3.1.1. Wall Current Monitor (WCM)!

࡞⨠࠾࡝ࡄࡿࡣ࡝ࡼ࡝࠷ࠊࡐࡡࡒࡴ㏳ᖏࠉࣂ࢕ࣈ

2.1⟿࡚㏑࡬ࡒࡻ࠹࡞ࠉࣄ࣭࣑ࣂ࢕ࣈࡡහഁ࡞

ࡡวࡿ┘࡞ࡢࣛࣤࢡ≟ࡡࢬ࣐ࣚࢠࢪࢅ᥃ྙࡊ࡙

ࡢࠉࣄ࣭࣑㞹Ὦ IB ࡛ྜྷࡋ᪁ྡྷ࡚㏣➚ྒࡡባ㞹ὮẴᐠࢅಕࡗᵋ㏸࡞ࡌࡾࠊහ㟻ࡡࢲ࣭ࣔࢩ࢓ࢴࣈࢅ

㸝wall current㸞I W = −IB ࠿Ὦࡿࡾࠊࣄ࣭࣑ࣂ࢕ࣈ

㑂ࡄࡾࡒࡴ࡞ࠉTiN ⷟⭯࡝࡜ࡡࢤ࣭ࢷ࢔ࣤࢡࢅ᪃

ࢅୌ࢜ᡜ࡚วࡖ࡙ࠉᢤᢘࢅཱི௛ࡄࡾ࡛㞹ᅸ RIW ࠿ࡌࡆ࡛ࡵ࠵ࡾࠊࡐࡡሔྙࡢㇿⲬᢤᢘࡻࡽ༎ฦ㧏࠷

Ⓠ⏍ࡌࡾࡆ࡛࠿஢᝷࡚ࡀࡾ㸝ᅒ㸞ࠊᐁ㝷࡞ῼᏽ

ᢤᢘೋ࡞ࡌࡾᚪさ࠿࠵ࡾࠊࡌࡾ࡞ࡢ㞹☚ࢿ࢕ࢫࡡࢨ࣭ࣜࢺࡡࡒࡴ࡞ࠉᑙమ࡚

J-PARC MR ࡚ࡡ᳠ฝἴᙟࡡౚࢅᅒ࡞♟ࡌࠊᅑ࠹ࡆ࡛࠿ᚪさ࡞࡝ࡾࠊࡌࡾ࡛ࠉባ㞹Ὦࡢࡻࡽ఩
࠷࢕ࣤࣅ࣭ࢱࣤࢪࡡࢨ࣭ࣜࢺࢅὮࡿ࡙ࡊࡱ࠹ࡡ
࡚ࠉࡐࡿࢅ㜴Ḿࡌࡾࡒࡴ࡞☚ᛮమࢤ࢓ࢅ᣼ථࡌ
ࡾࠊ᭩⤂Ⓩ࡞ᅒࡡࡻ࠹࡝ࢬࢴࢹ࢓ࢴࣈ࡞࡝ࡾࠊ

ᅒ㸯ባ㞹Ὦࡡῼᏽ

(a)ᅒࠉᅒࠉᅒ࡞ KEK-PS MR ࡡ CT ࡡౚࢅ♟ࡌࠊ఩ᇡ㐵᩷࿔ἴᩐ࠿ 800 Hzࠉ᫤ᏽᩐ~0.2
ms ࡚ࠉ100 ns ࡡࣁࣤࢲ㛏࡞ᑊࡊ࡙ࡢ༎ฦ࡚࠵ࡾࠊ

ᅒ㸯(9 3WXUQHRLO (b)ᅒ㸯/5&7(27 ࡡ <(2 ࡞ࡻࡾࣁࣤࢲἴᙟࡡふῼࠊF ථᑏ᫤ G ࣆࣚࢴࢹࢹࢴࣈ㸝,H;㸞
ࠊ

3.1.2. Current Transformer (CT)
Current Transformer (CT)ࡢࠉࣄ࣭࣑㞹Ὦࢅῼᏽࡌࡾᵾ‵Ⓩ࡝チ᩷⿞⨠࡚࠵ࡾ㸝ᅒ㸞ࠊࣄ࣭࣑㞹Ὦ I B ࠽ࡻࡦ㸧ḗᕬ⥲࡞Ὦࡿࡾ㞹Ὦ I!

࠿☚ᛮమࢤ࢓හ࡞ࡗࡂࡾ☚ሔࡢµIµNI,
B= B +!
2"r!2"r࡚࠵ࡾࠊ㸧ḗᕬ⥲࡞Ⓠ⏍ࡌࡾㄇᑙ㞹ᅸࡢd
V = (NBS) .
dt㸧ḗᕬ⥲࡞ࡢㇿⲬᢤᢘ࠿᥃⤾ࡈࡿ࡙࠷ࡾ㸝ࡒ࠷࡙ᅒ㸯(9 ࡡᵋ㏸

࠷ 50Ȑ㸞ࡡ࡚
!

!

!

!
!

!

!
!

V = "RI ,࡛࡝ࡾࠊ㞹ὮࡡὮࡿ㎲ࡳྡྷࡀࢅḿ࡞࡛ࡖ࡙࠷ࡾࠊ

ࡱ࡛ࡴࡾ࡛ࠉdµ (I B + NI)(NS) = #R I ,dt
2"r
L = µN 2 S 2"r ࢅࡗ࠾࠹࡛ࠉ
dI R
d I+ I = " ( B ),
dt L
dt N
࡝ࡾ᚜ฦ᪁⛤ᘟ࠿ᚋࡼࡿࠉよࡢ$& t dI (t") e R t" L (&V (t) = RI (t) = R % # Bdt" ) e+R t L .
&' 0 dt&*N࿔ἴᩐ㡷ᇡ࡚ࡢࠉ
j" / " L RI B
V=,
1+ j" / " L N
R"L = ,
L࡛࡝ࡾࠊᅒ㸯(9 ࡡ࿔ἴᩐ≁ᛮᅒ㸯(9࡞ࡻࡾࣄ࣭࣑ಘྒἴᙟ

㧏࿔ἴഁࡢࠉㇿⲬᢤᢘ࡞୩าࡡᾃ㐗ᐖ㔖 Cs ࠿Ꮛᅹࡌࡾ࡛ࡊ࡙࣓ࢸࣜ໩࡚ࡀࠉ௧ୖࡡ࿔ἴᩐ≁ᛮࢅᚋ
ࡾ㸯
RI B
.
#" + j" /C s R + 1/ LC s N
J-PARC ࡚ࡢࠉኬᙁᗐࣄ࣭࣑ࢅ᝷ᏽࡊ࡙ࢤ࢕ࣜ
V=j" /C s R

2$ᕬᩐࢅኬࡀࡂࡊࡒࠊスゝࣄ࣭࣑㞹Ὦࡢࣁࣤࢲࡡࣅ
!

࣭ࢠೋ࡚ 200A ࡞ࡵ㐡ࡌࡾࠊ㸧ḗഁ㸝ㇿⲬᢤᢘ㸞࡞Ὦࡿࡾ㞹Ὦࡢ I B N ࡝ࡡ࡚ࠉㇿⲬᢤᢘࢅ 50Ȑࠉ(a)࠵ࡾ࠷ࡢཬᑏྺ཭⏕ࡡᢤᢘ 50Ȑࢅ୩า࡞ථࡿ࡙
25Ȑ࡛ࡊ࡙ࡵࠉฝງ㞹ᅸࡢ 5 N ~ 10 N kVࠉᾐ㈕ࣂ
!
࣭࣠ࡡ█᫤ೋࡢ
1 N 2 ~ 2 N 2 MW ࡞ࡵ㐡ࡌࡾࠊୌ᪁ࠉスゝᙁᗐࡡ 1%ࡡࣄ࣭࣑ࡵふῼࡌࡾᚪさ࠿࠵!
ࡾࡡ࡚ㇿⲬᢤᢘࡢ΅ࡼࡈ࡝࠷ࡆ࡛࡞ࡊࡒࠊ఩࿔ἴ
!
㐵᩷࿔ἴᩐࡢࠉ
" L = R L , L = µN 2 S 2"r ࡝ࡡ࡚ࠉ఩
࿔ἴ㐵᩷࿔ἴᩐࡵᾐ㈕ࣂ࣭࣠ࡵᕬᩐࡡ㸧஋࡚ᨭၻࡌࡾࡆ࡛࡞࡝ࡾࠊMR ࡚ࡢ N=50 turn, R=25Ȑ!
㸝50Ȑ,2!para.㸞࡛ࡊࡒࠊᕬᩐࢅቌࡷࡌ࡛ᾃ㐗ᐖ㔖࡞ࡻࡾ㧏࿔ἴ≁ᛮࡡᝇ໩࠿ၡ㢗࡛࡝ࡾࠊࡈࡼ࡞⌟ᐁⓏ࡝ఎ㏞⥲㊪࣓ࢸ࡚ࣜよᯊࡊࡒ⤎ᯕࠉv(l) =Z c ( jZ 0 (1" cos kl) + Z c sin kl)" jZ c (Z 0 + Z l ) cos kl + (Z c2 + Z 0 Z l ) sin kl࡚Ⰳࡂ⾪ࡈࡿࡾࡆ࡛࠿ࢂ࠾ࡖࡒࠊ

Z I
# l b
N

(b)
ᅒ㸯(9 ࡡ࿔ἴᩐ≁ᛮࡡᐁῼ࡛ゝ⟤ࠊ F ᣲᖕ
G న┞ࠊ
ᅒ࡞౐ࡖࡒ࣓ࢸࣜࢅ♟ࡌࠊᅒ࡞ࠉf~220
MHz ࡡ✭ὕභᣲࡡຝᯕࡵྱࡴ࡙ࣈࣞࢴࢹࡊ࡙࠵
ࡾࠊ├ᙼ 300mm ⛤ᗐࠉࢤ࢓࡞Ἒࡖࡒ㛏ࡈ࡚~1m⛤ᗐ࡚࠵ࡾࠊ㞹☚ἴࡢࠉ┷✭୯࡚ࡢ 10 MHz ࡚ἴ㛏 30m ࡚࠵ࡾࡡ࠿ࠉCT ࡡᵋ㏸୯࡚ࡢཬභᣲ࠿㉫ࡆࡾ⛤ᗐࡱ࡚ἴ㛏▯⦨࠿㉫ࡀ࡙࠷ࡾ࡛⩻࠻ࡼࡿ!

ࡾࠊୌᏽ࿔ἴᩐ㛣㝰࡚ཬභᣲࢅ⧖ࡽ㏁ࡌࠊ⌟ᅹࡡCT ࡚ࡢᢤᢘ࡞ࡻࡾࢱࣤࣅࣤࢡ࡚භᣲࢅ⥾࿰ࡊ࡙࠷ࡾ㸝3.3.1⟿㸞ࠊභᣲ࿔ἴᩐࢅ୕ࡅࡾ࡞ࡢࠉౚ࠻ࡣࠉ㸧ḗࢤ࢕ࣜࢅ୕ୖᕞྎࡡ㸩࢜ᡜ࡞ᑻᡜⓏ࡞ᕬࡄࡣࡻ࠷ࠊ㸩࢜ᡜ࠾ࡼࡡಘྒࡡ࿰ಘྒࢅ࡛ࡿࡣࠉࣄ࣭࣑ᙁᗐ࡞Ẓౚࡌࡾಘྒࢅᚋࡾࡆ࡛࠿࡚ࡀࡾࠊ୕ୖᕞྎࡡ࿰ࢅ
࡛ࡾࡆ࡛࡚ᶋ᪁ྡྷࡡ㸩ᴗ࣓࣭ࢺ௧ୖࡡኣ㔔ᴗᠺฦࡡᙫ㡢ࢅᢒโ࡚ࡀࡾࠊୌ᪁ࠉᡜ᭻ࡡ࿔ἴᩐᖈᇡ
࡚ಘྒྙᠺࢅ⾔࡝࠹ᚪさ࠿࠵ࡾࠊ
3.1.3. ࣆ࢔࣭ࢺࣁࢴࢠᆵ CT๑⟿࡚㏑࡬ࡒ CT ࡚ࡢࠉ఩ᇡ㐵᩷࿔ἴᩐ࡞㝀ᗐ࠿࠵ࡾࠊ⵫✒᫤㛣 ᩐ 10msࠤᩐ s ࢅࡆ࠻ࡾࣄ࣭࣑
ࡡᙁᗐࢅ┞ᑊ⢥ᗐ 1%௧ୖ࡚࣓ࢼࢰ࣭ࡌࡾࡆ࡛ࡢ
ᅒ㸯(9 ࡡఎ㏞⥲㊪࣓ࢸࣜᅏ㞬࡚࠵ࡾࠊࡐࡆ࡚ࠉࣆ࢔࣭ࢺࣁࢴࢠ࣬ࢤ࢕ࣜࢅ
㏛ຊࡊ࡙఩ᇡ㐵᩷࿔ἴᩐࢅኬᖕ࡞ᨭၻ㸝ᑚࡈࡂ㸞ࡌࡾࡆ࡛ࢅ⩻࠻ࡾ[6]ࠊ

ᅒ㸯ࣆ࢔࣭ࢺࣁࢴࢠᆵ (9ࠊ

!

!
!
!

!

!

ࢤ࢓ࢅ㈇ࡂ☚ሔࡢµB=(I B + N 1I 1 + N 2 I 2 ) ,2"r྘㒂ࡡ㞹ᅸࡢ
d
V1 = (N 1BS) = "R1I 1 ,dtd("A)V1 " V2 = (N 2 BS) ,dt
V2 = R2 I 2 .
࢕ࣤࢱࢠࢰࣤࢪࡢ
µSN 12µSN 2 2
µSN 1N 2
L1 =, L2 =
,M =
.
2"r
2"r
2"r
ࡆࡿࡼࢅ౐ࡖ࡙ฝງ㞹ᅸࡢࠉ
j# (AM / R2 + L2 / R2 )R2 I B
Vout = R2 I 2 = "1+ j# (AM / R2 + L1 / R1 + L2 / R2 ) N 2
࡛࡝ࡾࠊ఩ᇡ㐵᩷࿔ἴᩐࡢࠉࢣ࢕ࣤฦ఩ࡂ࡚ࡀࡾ㸯1"L =AM / R2 + L1 / R1 + L2 / R2R
# 2 , A >> 1.
AM
KEK-PS MR ࡚ࡢࠉࣆ࢔࣭ࢺࣁࢴࢠᆵ CT ࡞᫤ᏽᩐ⿭ർᅂ㊪ࢅ㏛ຊࡊ࡙ࠉ2 s ࡚ࡡࢺ࢖࣭ࣜࣈࢅ

!

1%௧ୖ࡞ᢒ࠻ࡒ[7]ࠊᅒࠉᅒ࡞࿔ἴᩐ≁ᛮࡡゝ⟤ೋࠉῼᏽἴᙟࢅ♟ࡊࡒࠊᅒ㸯0*058,H;2FLQ7LQJ ࡡࣄ࣭࣑ᙁᗐ㸝୯ṹࡡࢹ࣭ࣝࢪ㸞;]SIL[-PVIL[ࠊ3.1.4. DCCT (Direct-current CT)ࣄ࣭࣑ࢅ㛏᫤㛣⵫࠻ࡾ⵫✒ࣛࣤࢡࠉ⾢✲ࣛࣤࢡ࡝࡜࡚ࡢࠉ๑⟿ࡡ᪁Ἢ࡚ࡵࢦࢡ࠿ၡ㢗࡛࡝ࡖ࡙ࡊ
ࡱ࠹ࠊࡐࡆ࡚ DC ࡱ࡚ῼᏽ࡚ࡀࡾ CTࠉࡌ࡝ࢂࡔ
DCCT ࠿㛜Ⓠࡈࡿ࡙࠷ࡾࠊ⌟ᅹࠉ௧ୖࡡࡻ࠹࡝ࡵ
ࡡ࠿ຊ㏷ჹࡡ DCCT ࡛ࡊ࡙ᐁ⏕໩ࡈࡿ࡙࠷ࡾ㸯ձ ࣂ࣒ࣚࢹࣛࢴࢠ DCCT,ղㇿᢤᢘࢅ౐ࡖࡒ DCCT[8],
ճ

SQUID

(Superconducting

QuantumInterference Device)ࢅ౐ࡖࡒ DCCT[9].ࡆࡆ࡚ࡢձࢅ⠾༟࡞⣺௒ࡌࡾ[10]ࠊᅒࡡࡻ࠹
࡞☚ᛮమࢤ࢓ࡡ 1 ḗഁ࡞ທᣲࢤ࢕ࣜࢅᕬࡀࠉ㸧ḗഁ࡞᳠ฝࢤ࢕ࣜࢅᕬ࠷ࡒᵋ㏸࡛࡝ࡖ࡙࠷ࡾࠊ1 ḗഁࡡ㸧ࡗࡡࢤ࢓࠿༎ฦ㣤࿰ࡌࡾࡻ࠹࡝ᣲᖕ࡚ࠉ㏣᪁ྡྷ࡞ທᣲࡌࡾࠊ࿔ἴᩐࡢ 1kHz ⛤ᗐ࡚࠵ࡾࠊࡐ
ࡆ࡞㸝ࣄ࣭࣑࡝࡜ࡡ㸞㞹Ὦ࠿Ὦࡿࡾ࡛ࠉࡐࡿࡑࡿ
ࡡࢤ࢓࡞ࡢࠉඔࡡທᣲ☚ሔ࡛㏣᪁ྡྷ࡞☚ሔ࠿ຊࢂࡾࡡ࡚ࠉ㣤࿰ࡌࡾࢰ࢕࣐ࣤࢡ࡞᫤㛣ᕣ࠿⏍ࡋࡾࠊ᳠ฝࢤ࢕࡚ࣜࡆࡡ☚ᮨን໩ࢅ᳠ฝࡌࡾ࡛ᅒࡡv Ѹ Ѹ v+ ࡚♟ࡈࡿࡾࡻ࠹࡞ທᣲ࿔ἴᩐࡡഃᩐḗ㧏ㄢἴࢅኣࡂྱࡳಘྒ࠿ᚋࡼࡿࡾࠊࡆࡡಘྒ࠾ࡼທ
ᣲ࿔ἴᩐࡡ 2 ಶἴࢅ᳠ฝࡌࡾࠊࣛࢴࣈࣜ㝎ཡࡡࡒࡴࡡ࣭ࣞࣂࢪࣆ࢔ࣜࢰ㸝ທᣲ࿔ἴᩐࡡ࠽ࡻࡐ

ᅒ㸯ࣆ࢔࣭ࢺࣁࢴࢠᆵ (9 ࿔ἴᩐ≁ᛮ㸝ゝ⟤ೋ㸞ࠊ

1/100 ௧ୖ: 1~10 Hz㸞ࢅ㏳ࡊ࡙ฝງࡈࡿࡾࠊᚺ➽ࡢ DC~10 Hz ⛤ᗐ࡛㟸ᖏ࡞㐔࠷ࡡ࡚ࠉ๑⟿ࡡࣆ࢔࣭ࢺࣁࢴࢠ௛ࡀ CT ࡛⤄ࡲྙࢂࡎ࡙ࠉᗀᖈᇡ໩࠿
ᅒࡼࡿࡾࠊᅒ㸯ࣂ࣒ࣚࢹࣛࢴࢠ )((9@Bࠊᅒ㸯୩าࣆ࢔࣭ࢺࣁࢴࢠᆵ )((9@Bࠊࣂ࣒ࣚࢹࣛࢴࢠ CT ࡛ࣆ࢔࣭ࢺࣁࢴࢠᆵ CT ࢅ⤄ࡲྙࢂࡎࡒᅂ㊪ࢅᅒࠉᅒ࡞♟ࡊࡒࠊࣂࣚ
࣒ࢹࣛࢴࢠ CT 㒂ࡡࣆ࢔࣭ࢺࣁࢴࢠ࣬ࢤ࢕࡛ࣜࣆ
࢔࣭ࢺࣁࢴࢠᆵ CT ࡡࣆ࢔࣭ࢺࣁࢴࢠ࣬ࢤ࢕࡛ࣜ
࠿├า࠾୩า࠾ࡡ㐢࠷࡚࠵ࡾࠊ├า᥃⤾࡚ࡢࠉࣛࢴࣈࣜ఩΅ࡡࡒࡴ࡞ࢤ࢓ࡡ☚Ẵ≁ᛮࢅ㧏࠷⢥ᗐ࡚ࣁࣚࣤࢪࡈࡎࡾᚪさ࠿࠵ࡖࡒࠊࡆࡿࢅ୩า᥃⤾࡞ࡌࡾࡆ࡛࡚ࠉࣛࢴࣈࣜ㞹Ὦ࠿ࣆ࢔࣭ࢺࣁࢴࢠᆵ
ᅒ㸯᳠ฝࢤ࢕ࣜ࡞ㄇᑙࡈࡿࡾಘྒ@Bࠊ

CT ࡡࣆ࢔࣭ࢺࣁࢴࢠ࣬ࢤ࢕ࣜࢅὮࡿ࡝࠷ࡻ࠹࡞
࡚ࡀࠉ↋㐽ืࢤ࢓ࡡ౐⏕࡚ࡵ༎ฦ࡝ᛮ⬗࠿ᐁ⌟ࡊ࡙࠷ࡾ[10,11]ࠊࡆࡡ୩าࣆ࢔࣭ࢺࣁࢴࢠᆵ࡚⿿ష
ࡊࡒ J-PARC MR ࡡ DCCT ࡡῼᏽౚࢅᅒ࡞♟ࡌࠊᅒ㸯├าࣆ࢔࣭ࢺࣁࢴࢠ᪁ᘟ )((9@Bࠊᅒ㸯/5&7(27 ࡡ )((9 ฝງἴᙟࠊ

3. 2. న⨠ࡡゝ ῼ࡛࡝ࡾࠊࡆࡿࢅみᗐȟ࡚วࡽཱིࡖ࡙ゝῼࡌࡾࠊ⠾ࣄ࣭࣑ࡢ┷✭ࢱࢠࢹࡡ୯ࢅ㉦ࡖ࡙࠷ࡂࠊࣄ࣭࣑
ࡢ᭯㝀࡝ฦᕱࢅᣚࡖ࡙࠷ࡾ࠿ࠉࡐࡡࣄ࣭࣑న⨠࡛
ࡢࠉ㏳ᖏࠉࣄ࣭࣑ࡡ㞹Ⲭฦᕱࡡ㔔ᚨࡡన⨠ࢅᣞࡌࠊᐁ㝷࡞ࡢⰅࠍ࡝ࠔన⨠ࠕ࠿▩ࡽࡒࡂ࡝ࡾ㸯࣭ ࣁࣤࢲࡡన⨠ࣁࣤࢲẎࡡన⨠ᖲᆍన⨠!

= ࣛࣤࢡ࡚ࡢ COD࣭ ࣁࣤࢲහࡡ྘᫤็ࡡన⨠!

࣭ ࢤ࣭ࢪࢷ࢔ࣤࢡ࣬ࣄ࣭࣑ࡡన⨠ࢤ࣭ࢪࢷ࢔ࣤࢡ࣬ࣄ࣭࣑ࡡన⨠ῼᏽ࡞ࡢࠉ࿔ᅂ
ࣄ࣭࣑࡞ࡗ࠷࡙ࡢࣆࣚ࢕࢕ࣤࢡ࣬࣠࢕࣭࣓ࣕࢼࢰ
!
࣭ࠉṟ⏻࢝ࢪࣈࣞࣆ࢒࢕࣓ࣜࢼࢰ࣭࡝࡜ࡡ‵㟸◒༟ࡡࡒࡴ࡞ࠉᕞྎȟ"ȧ࡚ᖕࢭࣞࡡ㞹ᴗ࡚ࡡ㞹Ⲭ㔖ࢅ⩻࠻ࡾࠊ
x
x2 $ y2" L = # (1+ 2 + 2+ L) 㻏㻃
aa2
xx2 $ y2" R = # (1$ 2 + 2+ L) 㻏
aa2
࡝ࡡ࡚ࠉ
" L #" R2x ax%x2 # y2 (x''1# 2** $ 2 㻏
$
$
2
2
2
2
" L + " R 1+ 2x a
a&
a
a
)

ࢅゝ⟤ࡌࡾࡆ࡛࡞ࡻࡽన⨠ࢅᚋࡾࡆ࡛࠿࡚ࡀࡾࠊ
ᐁ㝷࡞ࡢࠉᅒࡡࡻ࠹࡝ᙟ≟ࡡ㞹ᴗࢅ㒼⨠ࡌࡾࠊቪᆵࣈࣞࣆ࢒࢕࣓ࣜࢼࢰ࣭ࠉཱིฝࡊࣄ࣭࣑࡞ࡗ࠷
࡙ࡢ◒ቪᆵࡡ྘⛸ࣈࣞࣆ࢒࢕࣓ࣜࢼࢰ࣭࡚⾔࡝࠹ࡆ࡛࠿ኣ࠷ࠊࡒࡓࡊࠉࡆࡿࡼࡡ࣓ࢼࢰ࣭࡞ࡗ࠷
࡙ࡢ๪យࡊࠉࡆࡆ࡚ࡢ㞹☚ሔࢅ᳠ฝࡌࡾ࣓ࢼࢰ
࣭㸯㟴㞹ᆵࠉࢪࢹࣛࢴࣈࣚ࢕ࣤᆵ᳠ฝჹ࡞ࡗ࠷࡙
ㄕ᪺ࡌࡾ㸝ࢤ࣭ࢪࢷ࢔ࣤࢡ࣬ࣄ࣭࣑ࡡన⨠ῼᏽ࡞ࡢ౐࠻࡝࠷㸞
ࠊ
3.2.1. 㟴㞹ᆵ᳠ฝჹࣄ࣭࣑ࣂ࢕ࣈ⾪㟻ࡡ㞹ሔ E ࡱࡒࡢྜྷࡋࡆ࡛࡚
࠵ࡾ࠿㞹ⲬȪࢅ᳠ฝࡌࡾ᪁Ἢ࡚ࠉኣ᪁㟻࡚౐ࢂࡿᅒ㸯ළ➼ࣂ࢕ࣈ୯ࡡࣄ࣭࣑࡞ࡻࡾㄇᑙ㞹Ⲭࠊ࡙࠷ࡾ᪁Ἢ࡚࠵ࡾࠊ
2.1⟿࡚ぜࡒࡻ࠹࡞ࠉࣄ࣭࣑࠿┞ᑊㄵⓏࡡሔྙࡢ㸧ḗඔࡡ㟴㞹ሔၡ㢗࡞ᖉ╌࡚ࡀࡾࠊ㟴㞹ᆵ᳠ฝჹ㸝electrostatic monitor㸞࡛࿣ࡣࡿࡾᡜ௧࡚࠵ࡾࠊ
ࡐࡆ࡚ࠉ⥲ᐠᗐȢ[C/m]ࡡ⥲㞹Ⲭ࠿న⨠ (x, y) [m]࡞࠵ࡾ࡛ࡌࡾࠊࡆࡿ࠿ࢱࢠࢹ⾪㟻ࡡȟࡡన⨠࡞ㄇ!

ᑙࡌࡾ㞹ⲬฦᕱȪ[C/m2]ࡢࠉᘟ  ࠾ࡼ%
a2 ' r2" (a,# , r,$ ) =2&a a 2 + r 2 ' 2a r cos(# ' $ )2- $ '2" /r
11+ . 2& ) cos n(* + , )4=
2#a 10 n=1 % a (
43
$x
'
" /
y
=
11+ 2& cos * + sin * )
%a
(
2#a 0
a
2
$ x2 + y2'
2xy)) + L4,
+ 2&&
cos
2
*
+
sin
2
*
2
43
a2
% a
(

ᅒ㸯ᖲ⾔ฦ๪㞹ᴗࠊ⢥ᗐⰃࡂన⨠ࢅῼᏽࡌࡾࡒࡴ࡞ࡢࠉ㧏ḗࡡ㟸⥲(4)ᙟຝᯕ࠿↋っ࡚ࡀ࡝࠷ࠊࣄ࣭࣑ࢦ࢕ࢫࡡᑚࡈ࠷㞹Ꮔࠉ㝟㞹Ꮔࣛࣤࢡ࡝࡜࡚ࡢࠉ๑ࡵࡖ࡙࣠࢕࣭ࣕᰧ
ḿ࡞ࡻࡖ࡙ࠉ࠵ࡾ࠷ࡢゝ⟤࡞ࡻࡖ࡙ᑊᚺ㛭౿ࢅịࡴ࡙࠽࠷࡙ࠉῼᏽ᫤࡞ࡐࡡ࣏ࢴࣅࣤࢡࢸ࣭ࢰࢅ⏕࠷ࡾࡆ࡛࠿⾔࡝ࢂࡿࡾࠊ

!

ࡊ࠾ࡊࠉࣄ࣭࣑ࢦ࢕ࢫࡡኬࡀ࠷ຊ㏷ჹ㸝ౚ࠻ࡣJ-PARC RCS, MR,͏͏㸞࡚ࡢࠉࡆࡡ㟸⥲ᙟຝᯕ࠿↋っ࡚ࡀ࡝ࡂ࡝ࡾࠊࣄ࣭࣑᩷㟻ᙟ≟ࡢࠉ㐘㌹ࣂ࣒࣭ࣚࢰࠉࣛࣤࢡࡡሔᡜ㸝ș㛭ᩐ࠿ንࢂࡾ㸞ࠉ࢙ࢾ࣭ࣜ࢟ࠉỀᖲ㟻࠾ᆱ├㟻࠾࡚ንࢂࡾࠊࡐࡿࡼ࡞
ࡻࡾన⨠ῼᏽೋࡡࢫࣝࡡኬࡀࡈࡢࠉJ-PARC ࡡࣛࣤࢡࡡሔྙࠉᩐ 100ȣm ࡞࡝ࡾࠊన⨠⢥ᗐ 100ȣm ௧ୖࢅ┘ᣞࡌ J-PARC ࡚ࡢన⨠ᚺ➽࡞㟸⥲ᙟ≁ᛮ࠿ฝ࡝࠷᪁Ἢࢅ᤿⏕ࡊࡒࠊ┷✭ࣂ࢕ࣈ࡞ㄇᑙࡈࡿࡾ㞹Ⲭࡡฦᕱࡢࠉᘟ㸝4㸞ᅒ㸯┷✭ࣂ࢕ࣈ⾪㟻ࡡ㞹Ⲭࡡวࡽཱིࡽ᪁ࠊ

ࡡࡻ࠹࡞ࣆ࣭࢙ࣛᒈ㛜ࡡᙟ࡞࡝ࡖ࡙࠷ࡾࠊࡆࡡ࠹
ࡔࡡ cosȟ, sinȟࡡ㡧ࡡ౿ᩐࡡࡲࢅཱིฝࡎࡿࡣࡻ
࠷ࠊ௑ z ㍂᪁ྡྷࡡ㞹Ⲭฦᕱ࡞ን໩ࡢ↋࠷࡛ࡊ࡙࠷

ࡆࡡ㞹ᴗࡡ㸨ḗඔ✭㛣࡚ࡡᙟ≟ࡢࠉᅒࡡࡻ࠹࡞ᑊみ⥲࢜ࢴࢹ࡚࠵ࡾࠊ

ࡾࡡ࡚㸝㸧ḗඔ࣓ࢸࣜ㸞ࠉ㞹Ⲭࢅวࡽཱིࡾ᪁Ἢ࡛
ࡊ࡙ z ㍂᪁ྡྷ࡞ cosȟࡱࡒࡢ sinȟࡡ౪Ꮛᛮࢅࡵ
ࡖ࡙ࡌࡿࡣࡻ࠷ࠊᅒ࡞♟ࡊࡒࡻ࠹࡞ࠉᶋ㍂ࢅ aȟࠉ⦢㍂ࢅ z ࡞࡛ࡾ࡛ࠉࡆࡿࡢ┷✭ࣂ࢕ࣈࡡᒈ㛜ᅒ࡚࠵ࡾࠊᩫ⥲㒂ฦࡡㄇᑙ㞹Ⲭࢅวࡽཱིࡾࡆ࡛ࢅ
⩻࠻ࡾࠊ
2#

QL = $ a d"0

l 2(cos"$ % (a," ,r,& ) dz

'l 2

2#

!
!

l
= $ a d" ( cos " + 1) % (a," , r,& )20"l # x &=%1+ ( ,
2 $ a'QR = $ a d"
0

!

l2

2#

$ % (a," ,r,& ) dz

ᅒ㸯ᑊみ⥲࢜ࢴࢹ㞹ᴗࠊJ-PARC RCS ࠽ࡻࡦ MR ࡡ BPM ࡡスゝࢅᅒ࡞♟ࡌࠊኬࡀࡈࡢ࡮࡯ࢪࢢ࣭ࣜࡌࡾࠊRCS BPM
ࡡහᙼ࠿࠽ࡻࡐȭ260 mmࠉMR BPM ࡚ࡢ࠽ࡻࡐ
ȭ130 mm ࡚࠵ࡾࠊl 2'cos"2#l(1 % cos " ) & (a," , r,' )20"l $ x '=&1# ) .
2 % a(= $ a d"!
!

!

ࡐࡿࡑࡿࡡᕣ࡛࿰ࡡẒࢅ࡛ࡾ࡛ࠉన⨠ࢅᚋࡾ㸯
QL " QR x# .
QL + QR aᆱ├᪁ྡྷࡵࠉ㞹ᴗࢅ z ㍂ࡱࢂࡽ࡞ 90 ᗐᅂ㌹ࡊ࡙!

ࡷࡿࡣ
QU " QD y# ,QU + QD aࢅᚋࡾࠊ!

(a) RCS BPM

" I1 %
" V1 %$ '$ '$ I 2 ' = 1 $V2 ' ,$I3 'R $ V3 '$ '$ '# I 4 &R
#V4 &" I1 %" I1 %" )1 %$ '$ '$ '$ I 2 ' + $ I 2 ' = j( $ ) 2 ' .
$I3 '
$I3 '
$)3 '
$ '
$ '
$ '
# I 4 &C # I 4 &R#) 4 &

ࡆࡆ࡚Ȱࡢみ࿔ἴᩐࠉI ࡢㄇᑙ㞹Ⲭ࡚࠵ࡾࠊ㞹Ὦ!

(b) MR BPM
ᅒ㸯/5&7(7LQJ ࡡ '52ࠊన⨠ࡡ⥲ᙟᛮ࠿ᐁ⌟࡚ࡀࡾ㞹ᴗ࡛ࡊ࡙ࡢ࠷ࡂ

) " q11 q12" V1 %+ $$ '$V2 ' = j( +* j( $ q21 q22$ V3 '+ $ q31 q32$ '+, $# q41 q42#V4 &-01" 11 %q13 q14 %+ $ ''q23 q24 ' I + $ 1 2 ',+ .
q33 q34 ' R + $ 1 3 ''+/ $# 1 4 '&q43 q44 &ࡗ࠾ࡡࡵࡡ࠿࠵ࡾࠊᅒ࡞㸦ౚࢅᣪࡅࡾ[13]ࠊୌ

࠿ᚋࡼࡿࡾࠊ I ࡢ༟న⾔า࡚࠵ࡾࠊྎ㎰ࡡ(͐)−1⯙Ⓩ࡞ࠉ௴ណ᩷㟻ᙟ≟࡚ࡵࠉᑊみ⥲࢜ࢴࢹ࡞ࡻࡽ
!
ฦ๪ࡈࡿࡒ㸧㞹ᴗࡢࣄ࣭࣑న⨠࡞ᑊࡊ࡙⥲ᙟᚺ

ࡢ㏣⾔าࡡណࠊ྘㞹ᴗ࡞ㄇᑙࡈࡿࡾ㞹Ⲭࡢࠉළ➼➽ࢅࡵࡗࡆ࡛࠿♟ࡈࡿࡾࠊ

ࡼࡿࡾ㸯
# x&
%1 + (
a : inner radius of the electrode# "1 & % a (x
I B : beam current% ( %1 ) (% " 2 ( = % a (I gl ,g " 0.5 : geometrical factor%"3 ( % y ( B v
1+l : electrode length% ( % a($" 4 '% y(v : speed of the beam
%1 ) ($ a'ᚉࡖ࡙ࠉ྘㞹ᴗࡡฝງ㞹ᅸ࠿ࠉࣄ࣭࣑న⨠㸝x, y㸞࠽ࡻࡦ࿔ἴᩐࡡ㛭ᩐ࡛ࡊ࡙ᚋࡼࡿࡾ㸯
!
" x%
1+ '
01$
) " q11 q12 q13 q14 %- $ a'" V1 %+ $+ $10 x '$ ''$V2 ' = j( +* j( $ q21 q22 q23 q24 ' + I +. $ a 'I gl .
B
$ V3 'v+ $ q31 q32 q33 q34 ' R + $1+ y '
$ '
$
'
$
'
a
+
+
#V4 &
, # q41 q42 q43 q44 &/ $ y'$10 '# a&ᅒ࡞ࠉ㸧㞹ᴗ࣋࢓ࡡࡲᢜࡀฝࡊࡒ࡛ࡀࡡ➴౮ᅂ㊪ࢅᥝ࠷ࡒࠊ

!

ᅒ㸯న⨠ࡡ⥲ᙟᚺ➽ࡡ࠵ࡾ㞹ᴗࠊ୕༖ฦࡢȟ"cȧࢅし࠹࡬ࡒ㞹ᴗࠉన⨠ゝ⟤ࡡฦẍ࡛ࡊ࡙౐࠹ࠊ᫤㛣㡷ᇡ࡚ࡡᚺ➽ࡢࠉ௧ୖࡡᵕ࡞࡝ࡾ[14]ࠊ྘
㞹ᴗࡡ㞹న Viࠉ㞹Ὦ Ii ࢅࠉ྘㞹ᴗࡡ㟴㞹ᐖ㔖 qii ࣬!
ㄇᑙ౿ᩐ qij㸝iӆj㸞ࡡᏽ⩇ᘟࢅ᫤㛣᚜ฦࡌࡾࡆ࡛
࡞ࡻࡽࠉ" I1 %" q11$ '$$ I 2 ' = j( $ q21
$I3 '
$ q31
$ '
$
# I 4 &C
# q41

q12
q22
q32
q42

q13
q23
q33
q43

ᑊみ⥲ฦ๪ࡡሔྙ࡞ࡢ௧ୖࡡࡻ࠹࡞࡝ࡾ࡛⩻࠻

q14 %" V1 %'$ 'q24 '$V2 ',q34 '$ V3 ''$ 'q44 &#V4 &ࢅᚋࡾࠊୖ௛ࡀ῟Ꮚ C ࡢ㟴㞹ᐖ㔖ࢅὮࡿࡾ㞹ὮࡡណࠊㇿⲬᢤᢘ R ࡞ࡗ࠷࡙ࡡ࢛࣭࣑ࡡἪ์ࠉ࠽ࡻࡦ
!

ࢅᾐཡࡊ࡙྘㞹ᴗࡡࣄ࣭࣑࡞ࡻࡾㄇᑙ㞹Ⲭ࡛ฝ!
ງ㞹ᅸࡡࡲࡡᘟ

ࣄ࣭࣑࡞ࡻࡖ࡙㞹ᴗ࡞ㄇᑙࡈࡿࡾ㞹Ⲭࡡ᫤㛣᚜ฦ࠿㞹Ὦ࡚࠵ࡾࡆ࡛࠾ࡼࠉ௧ୖࡡᘟࢅᚋࡾࠊ

ᅒ㸯㸧㞹ᴗ࣋࢓ࡡ࡛ࡀࡡ➴౮ᅂ㊪ࠊ୕オࡡ࣓ࢸࣜࢅ J-PARC MR ࡡ BPM ࡞㐲⏕ࡊࡒౚࢅࡊࡴࡌࠊහ༖ᙼ a=130/2 mm ࢅ౐࠷ࠉᑊみ⥲ฦ๪㒂ฦ࡛༟⣟ළ➼㒂ࡡẒࢅ 0.171 ࡛ࡊ࡙⿭ḿࢅຊ࠻ࡒࠊ qij 㸝ᐖ㔖౿ᩐࠉㄇ㞹౿ᩐ㸞࡛ࡊ࡙ࡢࠉ

Network analyzer ࡞ࡻࡾῼᏽೋࢅ౐ࡖࡒࠊPort1,
Port2㸝50Ω⣌㸞ࢅ㸩㞹ᴗࡡ࠹ࡔ㸧ࡗ࡞⦽ࡁࠉ௙ࡡ
࣭࣎ࢹࢅ 50Ω࡚⤂❻ࡊࡒࠊࡐࡡ㒼⨠࡚ᩋ஗⾔าࢅῼᏽࡊࠉࡈࡼ࡞ [qij] ࡞ንᥦࡊࡒ⤎ᯕࠉ௧ୖࡡೋ
ࢅᚋࡒࠊ༟నࡢ pF ࡚࠵ࡾࠊ!
"197.535 -7.457-0.004-0.018 %$'-7.457 205.356 -0.020-0.066 '.
qij = $
$ -0.004-0.020 207.689 -7.301 ' !
$'-0.066-7.301 210.604 &# -0.018!
࣠࢕࣭ࣕἪࢅ౐ࡖࡒన⨠វᗐࡡῼᏽೋ࡛୕オࡡ2#

QL = $ a d"2#

= $ a d"0ᘟ࡞ࡻࡾゝ⟤ೋࢅᅒ࡞♟ࡌࠊⰃ࠷ୌ⮬࠿ᚋࡼ!Ѹࡿ࡙࠷ࡾࠊฝງ㞹ᅸࡡ఩ᇡ㐵᩷࿔ἴᩐ㸝@
3dB㸞㸯⣑ 16.5 MHz ࡡୌ⮬ࡵⰃ࠾ࡖࡒࠊ
!
㟴㞹ᆵ BPM ࡡᚺ➽≁ᛮࡢࠉᚃ㏑ࡡࢪࢹࣛࢴࣈ

$ % (a," ,r,& ) dz

0'l 2
l 2&cos"' ,-) x + x(z + y(z$ dz 2#a .1+ 2+*

%l 2

+

( )

!

l 2(cos"

a

cos "

$
y + x"z + y"z
sin # &%a%( x + x"z + y"z ) 2 # ( y + x"z + y"z) 2 cos 2$
+ 2'
'
a2
&
'
$
)
sin
2
#
+
L
&
)(
a2
%
#l $ x ' #l l( x+ + y+) x #l l( x+ + y+) y
"
*
* ,
*
* .
&1+ ) +
2 % a( 2
4a
a 2
4a
a
+

2( x + x"z + y"z )( y + x"z + y"z )QR ࡞ࡗ࠷࡙ࡵྜྷᵕ࡞ࡊ࡙ࠉx ᪁ྡྷ㸝Ềᖲ᪁ྡྷ㸞࡞ࡗ࠷࡙ࠉ
" x l( x% + y%) x l( x% + y%) y
$ +
& '
& ,
# a
4a
a
4a
a
ࢅᚋࡾࠊy ᪁ྡྷࡵྜྷᵕ࡚࠵ࡾࠊౚ࠻ࡣࠉa = 65 mm,

ࣚ࢕ࣤ BPM ࡡ఩࿔ἴᴗ㝀࡛ࡊ࡙ࡵᚋࡼࡿࡾࠊ
!

ε = 100 mm, x' = 2 mrad ࡚ࠉ┞ᑊㄏᕣ 0.08 %࡚
!

࠵ࡾࠊࡱࡒ y ࡡన⨠࠿ x ࡡῼᏽ࡞ᙫ㡢ࡌࡾࠊḗ࡞ࠉ┷✭ࣂ࢕ࣈ࡞ṹᕣ࠿࠵ࡾሔྙ࡜࠹࡝ࡾ
࠾ࠊ⠾༟࡝ౚ࡛ࡊ࡙ළ➼ࣂ࢕ࣈ࡞ྜྷ㍂ࡡṹᕣ࠿࠵ࡾሔྙࢅ⩻ᐳࡌࡾ[15]ࠊᅒࡡࡻ࠹࡝ᙟ≟ࢅ⩻࠻ࡾࠊṹᕣ㎾ࡂ࡚ z ౪Ꮛᛮࡡ㞹Ⲭฦᕱ࠿⏍ࡋࡾ㸝ᅒ(a) ࣠࢕࣭ࣕἪ࡞ࡻࡾῼᏽೋ.

㸞ࠊᅒ㸯┷✭ࣂ࢕ࣈ࡞ṹᕣ࠿࠵ࡾሔྙࠊ(b) ୕オࡡ࣓ࢸࣜ࡞ࡻࡾゝ⟤ೋ.
ᅒ㸯'52 ᳠ฝჹࡡన⨠វᗐࡡ࿔ἴᩐ≁ᛮࠊ௧ୖࠉ࠷ࡂࡗ࠾ࡡၡ㢗Ⅴ࡞ゕཀྵࡊ࡙࠽ࡂࠊ୕オࡡᑙฝ࡚ࡢ⣌ࢅ㸧ḗඔ࡚⩻࠻࡙࠷ࡾࠊ
ࡱࡍࠉࣄ࣭࣑࠿ z ㍂࡞ᖲ⾔࡚࡝࠷ሔྙ࡜࠹࡝ࡾ
࠾ࠊ஦ḗඔ࣓ࢸࣜࡡᯗහ࡚ぜ✒ࡵࡽࢅ⾔࡝࠹ࠊࣄ࣭࣑న⨠㸝x, y㸞࠿ z ᪁ྡྷࡡ౪Ꮛᛮ
x = x 0 + x" z + y" z,
y = y 0 + x" z + y" z,ࢅᣚࡗ࡛ࡌࡾࠊᕞ㞹ᴗ࡞ㄇᑙࡈࡿࡾ㞹Ⲭ㔖ࡢ!

ᅒ㸯ࣂ࢕ࣈ⾪㟻㞹Ⲭ㸝҃*U㸞ࠊࣄ࣭࣑࠿ z ㍂୕ࢅ㉦ࡾሔྙࠉࡌ࡝ࢂࡔ୯ᚨ࡞࠵ࡾ

々ᩐࡡ⥲㊪ࡡ⤄ࡲྙࡎ࡞ࡻࡖ࡙⤎ྙ࡞᪁ྡྷᛮࢅ

ሔྙࠉวࡽཱིࡾ㞹Ⲭࡢᣚࡗ㸝Ѝ᪁ྡྷᛮ⤎ྙჹ㸞ࠊࡆࡿࢅࣄ࣭࣑࣓ࢼࢰ࣭

z1 (" )

2&

Q+ = # a d"0z 2 (" )2&(l/2)cos" +z 0

= # a d"
0
2'

# dz

0

ࡡࡻ࠹࡝ࡵࡡ࡚࠵ࡾࠊ
$ (a,0,0,0,z),$ dz % (a,# ,r,& ,z)z 2 (# )l/2+z 02'

= $ a d#0!

ࡽࠉ୕オࡡᛮ㈻ࢅങ࠻࡙࠷ࡾࠊୌ⯙Ⓩ࡝ᙟ≟ࡢᅒ-l/2+z 0
z1 (# )

Q" = $ a d#!

࡞ᚺ⏕ࡊࡒࡵࡡ࠿ࢪࢹࣛࢴࣈࣚ࢕ࣤࠕ᳠ฝჹ࡚࠵

# dz $ (a," ,r,% ,z)$ dz% (a,0,0,0,z),(l/2)cos# +z 0ᅒ㸯ୌ⯙Ⓩ࡝ࢪࢹࣛࢴࣈࣚ࢕ࣤࡡ᩷㟻ᙟ≟ࠊ࡛࠾ࡄࡾࠊㄏᕣฦࢅྱࡴ࡙
Q+ = Q0 + "Q+ ,Q# = Q0 + "Q# ,l
Q0 = $ ,
2
࡛࠾ࡂ࡛ࠉ
"Q+

!

&

= 2#b$ 0 ' Fq

% qb

q=0

zq

&

% qb

J 0 (z q )I 0 (% q l/2) e(% q z 0(e(% q (z 0 (l/2),%qᅒ㸯ࢪࢹࣛࢴࣈࣚ࢕ࣤ '52"Q(
= 2#b$ 0 ' Fq
q=0zq

J 0 (z q )e(% q (l/2+ z 0 )( I 0 (% q l/2) eᅒ࡞♟ࡌࡻ࠹࡝ PU㸦ᮇࡡሔྙࢅ࣓ࢸࣜ໩

(% q z 0,%qࡌࡾࠊࣄ࣭࣑ࡢක㏷࡚㉦ࡖ࡙࠷ࡾ࡛௫ᏽࡌࡾࠊ
2.1.1⟿࡚ぜࡒࡻ࠹࡞ 2D 㟴㞹ၡ㢗࡛⩻࠻ࡼࡿࡾࠊ

࡞࡝ࡾࠊࡆࡆ࡚ࠉF q ࡢ Field matching ࡡሾ⏲᮪ࣄ࣭࣑ࡡన⨠ࢅ(x0,y0)࡛ࡊ࡙ࠉࣄ࣭࣑㟴Ḿ⣌࡚ࠉ௲࡞ࡻࡖ࡙ᏽࡱࡾᒈ㛜౿ᩐ࡚࠵ࡾࠊ
!

Ⅴ (x,y) ࡞࠽ࡄࡾ࣎ࢷࣤࢨࣔࣜࡢࠉࣄ࣭࣑ࠉPU୕ࡡ⥲㞹Ⲭᐠᗐࢅࡐࡿࡑࡿ " 0 , "1 ࠉ㞹న౿ᩐࢅJ-PARC MR ࡚ࡢࠉථᑏཱི࣬ฝࡊ㒂ࡡࠉኬࡀ࡝
࢓ࣂ࣭ࢲࣔ࠿ᚪさ࡝㒂ฦ࡚ࡆࡡၡ㢗࠿⏍ࡋ࡙࠷

p10 , p11 ࡛ࡌࡾ࡛ࡾࠊ10 mm ๑ᚃࡡ⣌⤣Ⓩ࢛ࣆࢬࢴࢹㄏᕣ࠿⏍ࡋ࡙

V1 (x, y, x 0 , y 0 )= p10 (x, y, x 0 , y 0 ) " 0!(x 0 !
, y 0 ) + p11 (x, y) "1,࠷ࡾࠊṹᕣࡡよᾐ࠿᰷ᮇⓏᑊ➿࡞࡝ࡾ࠿ࠉ㝼᥃ࡌࡾ BPM ࡚┞பᰧḿࡌࡾࡆ࡛ࡵ࡚ࡀࡾࠊ

!

!

࡛᭡ࡄࡾࠊᐁ㥺ᐄ⣌࡞⛛ࡾ࡛ࠉTEM ἴࡡᛮ㈻࠾
ࡼ㞹ሔࠉ☚ሔࡢ[16]E" = #$" V1 e m jk0 z&(#$ (p % + p % ) e# jk0 z
11 1+= ' " 10 0+()#$" (p11%1# ) e+ jk0 z

3.2.2. ࢪࢹࣛࢴࣈࣚ࢕ࣤ᳠ฝჹ[16]!
ࢪࢹࣛࢴࣈࣚ࢕ࣤ㸝stripline㸞࡛ࡢࠉୌ⯙Ⓩ࡞

ࡢࠉ⾪⿤㟻࡞ᑙమ⟡ࢅᙟᠺࡊࡒᯀ≟ㄇ㞹మᇱᯀࡡහ㒂࡞⥲≟ࡡᑙమ⟡ࢅᙟᠺࡊࡒᵋ㏸ࢅᣚࡔ㞹☚H " = ±Y0 zˆ # $" V1 e m jk 0 z')+Y zˆ # $ ( p % + p % ) e& jk 0 z" 10 0+11 1+=( 0&Y0 zˆ # $" ( p11%1& ) e + jk 0 z*)ἴࢅఎ㐡ࡌࡾఎ㏞⥲㊪࡚࠵ࡾ[17]ࠊንᙟ∟࡛ࡊ࡙ࠉ⿤㟻ࡡࡲ࡞ᑙమ⟡ࢅᙟᠺࡊࠉ⾪㟻࡞⥲≟ࡡᑙమ⟡!
ࢅᙟᠺࡊࡒᵋ㏸ࡡఎ㏞⥲㊪㸝࣏࢕ࢠࣞ࣬ࢪࢹࣛࢴ
ࣈࣚ࢕ࣤ㸞ࡵ࠵ࡾ㸝ᅒ㸞ࠊ≁ᚡ࡛ࡊ࡙ࡢࠉฦᕱ

ࡆࡆ࡚ Y0 = " 0 µ 0 , "# ࡢᶋ᪁ྡྷࡡ gradient ࡚࠵ᏽᩐᅂ㊪࡚࠵ࡽᇱᮇ࣓࣭ࢺࡢ TEM ἴ࡚࠵ࡾ㸝࢜!
ࢴࢹ࢛ࣆ࿔ἴᩐ௧୕࡚ࡢ㧏ḗ࣓࣭ࢺ࠿Ⓠ⏍㸞ࠉ㧏ࡾࠊࣄ࣭࣑ࢱࢠࢹ⾪㟻࠾ࡼ PU ⾪㟻ࡱ࡚ࡡ⥲✒ฦ࿔ἴ≁ᛮ࠿Ⰳ࠷ࡡ࡚࣏࢕ࢠࣞἴᅂ㊪࡞౐ࢂࡿࡾࠉ

!

࡞ࡻࡖ࡙ PU ࡡ㞹న࠿ịࡴࡼࡿࡾࠊ
!

$& (p " + p " ) e# jk0 z
11 1+
V1± e m jk0 z = % 10 0+&'
p11"1# e+ jk0 z㞹ᅸ″ࢅ➴౮㞹Ὦ″࡞⨠ࡀᥦ࠻ࡿࡣࠉⰃࡂ౐ࢂࡿ

(1)ࡾ➴౮ᅂ㊪࠿ᚋࡼࡿࡾࠊ

ࡱࡒࠉPU ࢅὮࡿࡾ㞹Ὦࡢࠉ☚ሔࡡ᥃⥲ᠺฦࢅ!

PU ᩷㟻ࢅᅑࡳ⥲࡞Ἒࡖ࡙✒ฦࡊ࡙ᚋࡼࡿࡾࠊࣄ
࣭࣑࠾ࡼࡡᐞ୙㸝 p10 㸞ࡢࢭ࡚ࣞࠉPU ⮤㌗ࡡᐞ୙
㸝 p11 㸞ࡡࡲ࠿ṟࡾࡆ࡛࠿ࢂ࠾ࡾࠊ
I 1± e m jk 0 z = $ H " # ds = ±c% 1± e m jk 0 z .
!
C

!
!
!

(2)
V1+ ,V1" , I 1+ , I 1" ࢅୌណⓏ࡞ᏽࡴࡾࡒࡴ࡞ࡢሾ⏲ᅒ㸯➴౮ᅂ㊪ࠊ᮪௲࠿ᚪさ࡚࠵ࡾࠊ୕Ὦ❻ࠉୖὮ❻࡞࡙ࡐࡿࡑࡿ
R0 , Rl ࡚⤂❻ࡈࡿࡒሔྙࢅ⩻࠻ࡾ㸯
V1+ + V1" = "R0 (I 1+ + I 1" ) ,V1+ e" jk 0 l + V1" e jk 0 l = Rl (I 1+ e" jk 0 l + I 1" e jk 0 l ).

!
!

!

!

!

!

!

!
!

୕Ὦ❻ࠉୖὮ❻࡚ࡡ㞹ᅸࢅịࡴࡾ࡛ࠉࡐࡿࡑࡿ
V1 (0) = V1+ + V1"
cp10 R0 (p11 + cRl ) sin(k 0 l) # 0=
2 + c 2 R R ) sin(k l)" jcp11 (R0 + Rl ) cos(k0 l) + ( p110 l0ᅒ㸯ࢪࢹࣛࢴࣈࣚ࢕ࣤ࣬ࣅࢴࢠ࢓ࢴࣈࡡ㸦ౚࠊ㞹Ⲭࠉ㞹Ὦ࡞ࡻࡾ㒂ฦ㸝 Vs 㸞࠾ࡼ࡝ࡾࠊࡆࡿࢅ౐

=180mm࡛ࡌࡾࠊࣄ࣭࣑࠿ z ㍂୕࡞࠵ࡾሔྙࢅ⩻࠻ࡾࠊ㞹

රమⓏ࡞ᅒ࡞ࡗ࠷࡙ゝ⟤ࡊ࡙ࡲࡾࠊp10, p 11

ࡢ Green ࡡ┞ཬᏽ⌦ࠉ㙶ാἪࠉ➴み෕ാ࠵ࡾ࠷ࡢ
V1 (l) = V1+ e" jk 0 l + V1" e + jk 0 lሾ⏲さ⣪Ἢ࡝࡜ࡡᩐೋゝ⟤࡞ࡻࡖ࡙ịࡴࡾࡆ࡛"2 jcp10 (p11 " cR0 )R2 sin(k 0 l) # 0="(p11 " cR0 )(p11 " cRl ) + e 2 jk 0 l ( p11 + cR0 )(p11 + cRl ) ࠿࡚ࡀࡾࠊ
p10 (x, y, d, 0)࡛࡝ࡾࠊ Z c = p11 c = 1 cC1 ( C1 ࡢ㟴㞹ᐖ㔖)ࢅ⩻៎
1 %'(x $ d 2 ) 2 + y 2
d 2 $ b )'ࡌࡾ࡛ࠉ p11 " cR0 = c(Z c " R0 ) ࡛࡝ࡽࠉ୕Ὦ❻࡚࢕=$ 2 Log& Log*
4 "# 0 '(b $ d $ d1 '+(x $ d $ d1) 2 + y 2
ࣤࣅ࣭ࢱࣤࢪᩒྙࡊࡒሔྙࡢୖὮ❻࡚ฝງ࠿↋
,
!
!
!
2
2
2
࠷ࡆ࡛࠿ࢂ࠾ࡾࠊ
1
a +b $ dp11 (d, 0) =cosh$1!
2"# 02ab㞹ᅸࠉ㞹Ὦ࠿‫ࡌࡒ‮‬᪁⛤ᘟࢅịࡴࡾ࡛ࠉ.
!
dV1
k 0 p10#jkz
0+ j"L1I 1 = # j
I 0e
ࡒࡓࡊࠉ
dz
c2
d1 (d 2 " d) = a 2 , (d + d1)d 2 = b 2 ,
dI 1
k p
!
+ j"C1V1 = j 0 10 I 0 e# jk 0 z
dz
cZ c
࡚࠵ࡾࠊ
I
=
c
"
p
=
1
C
Z
=
cL
=
1
cC
0 , 111,c1
1 ࢅ౐ࢅᚋࡾࠊ 0!
(ౚ㸦) PU!ࡡ≁ᛮ࢕ࣤࣅ࣭ࢱࣤࢪ 50Ȑࠉ୦❻ᮆ࡚ࡖࡒࠊᩒྙࡊ࡙࠷ࡾሔྙ
PU ࡞Ⓠ⏍ࡌࡾ㞹ᅸࡢࠉ୕オࡡ㏳ࡽࠉ㞹ᅸࡢࠉ!
!
!
V
a=1.5mm, b=67mm, d=65mm, Zc=50 Ȑ , εࣄ࣭࣑㞹Ⲭ࡞ㄇᑙࡈࡿࡾ㒂ฦ㸝 i 㸞࡛ࠉPU ୕ࡡࡖ࡙᪁⛤ᘟࢅ᭡ࡀ├ࡌ࡛!
dV1s+ j"L1I 1 = 0
!
dz
dI 1
+ j"C1V1s = 0dz࡛࡝ࡾࠊࣄ࣭࣑ࡡᙫ㡢ࡢሾ⏲᮪௲ࡡࡲ࡞⌟ࡿࡾࠊ!
V1s (0) = "V1i (0) " R0 I 1 (0) ,!
V (l) = "V (l) + R I (l) .
1s

1i

l 1

న౿ᩐࡢࡐࡿࡑࡿ
p10 (0,0,d,0)= 3.88 " 10 8
p11 (d,0) = 1.50 " 1010

࡛࡝ࡾࠊTransfer impedance ( Z // = V1 (0) I 0 )ࢅịࡴࡒ⤎ᯕࢅᅒ㸧࡞♟ࡌࠊࡐࡿࢅ[Shafer],[Barry]ࡡゝ⟤ᘟ
Z up (" ) = jZ c!
!

!

# % j&!
esin&
2$Z Barry (" ) =µ 0cbLog( )R0 sin$
2#Z cd

" = #l c࡞௥ථࡊࡒ⤎ᯕࡵ♟ࡊࡒࠊ[Shafer]ࡢළ➼㟻ࡡୌ!

㒂ࢅวࡽཱིࡖࡒᙟ≟ࡡ㞹ᴗࢅ௫ᏽࡊ࡙࠷ࡾࠊ

!

[Barry]ࡢኯࡈࢭࣞࡡ࣠࢕࣭ࣕࢅ௫ᏽࡊ࡙࠷ࡾࠊౚ
㸦ࡡᙟ≟ࡢࠉ࡜ࡔࡼ࡞ࡻࡖ࡙ࡵ㎾జ࠿ᝇ࠷ࡒࡴࠉ
ᅒࡡ⤎ᯕ࡞࡝ࡖࡒ࡛⩻࠻ࡼࡿࡾࠊ(ౚ㸧) PU ࡡ≁ᛮ࢕ࣤࣅ࣭ࢱࣤࢪ 202ȐࠉୖὮ❻ᮆ࡚ࢨ࣭ࣘࢹࡊ࡙࠷ࡾሔྙ(J-PARC MR ࡞ス⨠
ࡊࡒ"Single Pass Monitor (SPM)"ࡡࣂ࣒࣭ࣚࢰ)a=1.5mm, b=92mm, d=67mm, Zc=202 Ȑ , ε=180mm ࡛ࡌࡾࠊ㞹న౿ᩐࡢࡐࡿࡑࡿ

!

!

ࣄ࣭࣑ゝῼ࡞඙❟ࡖ࡙ࠉࣄ࣭࣑࣓ࢼࢰ࣭ࡡ≁ᛮࢅ(d,0) = 6.05" 1010

ᢍᥩࡌࡾࡆ࡛ࡢ㔔さ࡚࠵ࡾࠊ௧ୖ࡞඼ᆵⓏ࡝᪁Ἢ

࡛࡝ࡾࠊTransfer impedance ࢅịࡴࡾ࡛ᅒ

ࢅาᣪࡌࡾࠊ

ࡡࡻ࠹࡞࡝ࡾࠊࡐࡿࢅ[Barry]ࡡゝ⟤ᘟZ Barry (" )3.3.1. CT ࡡᰧḿ#b&µ c= 0 Log% (R0 sin)
$d '
2"Z c

# R &22
2% 0 ( cos ) + sin )$ Zc '

࡞௥ථࡊࡒ⤎ᯕࡵ♟ࡊࡒࠊࡆࡡሔྙࠉ[Barry]ࡡኯ
ࡈࢭࣞࡡ࣠࢕࣭ࣕࡡ㎾జ࠿Ⰳ࠷࡛⩻࠻ࡼࡿࡾࠊࡱ

!

3. 3. ࣊ࣤࢲᰧ ḿ

p10 (0,0,d,0)= 5.70 " 10 9
p11!

ᅒ㸯ౚ㸧ࡡ᩷㟻ࡡሔྙࡡ WUFQVIHULPSHIFQHH
ࡡ࿔ἴᩐ≁ᛮࠊ

ࡒࠉ఩࿔ἴ≁ᛮ࡚ࡢ J-PARC MR ࡡ SPM ࡡ࣠࢕࣭ࣕᰧḿࢸ࣭ࢰ࡛Ⰳࡂྙࡖ࡙࠷ࡾࠊ

ྜྷ㍂ᵋ㏸ࡡ୯ᚨᑙమࡡⰃᑙమ࣠࢕࣭ࣕ࠵ࡾ࠷
ࡢࣂ࢕ࣈ࡞㞹ὮࢅὮࡊࠉࣛࢰ࣭ࣤ㞹Ὦࡢአ㒂ᑙమࢅὮࡌࡆ࡛࡚ࠉࣄ࣭࣑㞹Ὦࢅᶅ᧻ࡌࡾࡆ࡛࠿ୌ⯙Ⓩ࡞⾔࡝ࢂࡿ࡙࠷ࡾࠊ⣌࠿㞗୯ᏽᩐ࡛ぜ࡝ࡎࡾ⛤ᗐࡡ఩࠷࿔ἴᩐࡡῼᏽ࡚ࡢࠉCT ࢤ࢓හ࡞࣠࢕ࣕ
࣭㸝ᑚࡈ࠷㞹″࡚㞹Ὦೋࢅኬࡀࡂࡌࡾᚪさ࠿࠵ࡾሔྙ࡞ࡢ々ᩐࢰ࣭ࣤ㸞ࢅ㏳ࡌࡡࡲ࡚Ⰳ࠷ࠊࡱࡒࠉ
㏭୯ࡡࢢ࣭ࣇࣜࠉฌ⌦ᅂ㊪ࡡᙫ㡢ࢅ㎲ࡲ࡚ᰧḿࡌ

Barry [2]ࡾࡆ࡛࠿ᚪさ࡞࡝ࡾࠊ࢛ࣤࣚ࢕࡚ࣤᰧḿࡌࡾࡻ࠹࡞ࡊࡒ Feedback ௛ࡀ CT ࡡᵋᠺࢅᅒ࡞♟ࡌࠊThis paper

Shafer [1]ᅒ㸯ౚ㸦ࡡ᩷㟻ࡡሔྙࡡ WUFQVIHULPSHIFQHH
ࡡ࿔ἴᩐ≁ᛮࠊ

ᅒ㸯8(9 ࡡᰧḿ⏕ᕬ⥲ࠊ㧏࿔ἴࡱ࡚ᰧḿࡌࡾሔྙ࡞ࡢࠉῼᏽ⿞⨠࠾ࡼࡡಘࢷ࣭ࣂ࣭⟮ࡡ㒂ฦࡢᐁ㝷ࡢࠉ❻ᯀࡡࡲࠉኣ㔔Ȣ/4ྒ㸝ኬᢤࠉ≁ᛮ࢕ࣤࣅ࣭ࢱࣤࢪ 50Ȑ࡛࡝ࡖ࡙࠷ࡾ㸞⟮ࠉࡱࡒࡢࢷ࣭ࣂ࣭⟮࡞࡙⾔࡝ࡖࡒࠊ࠿ཬᑏࡊ࡝࠷ࡻ࠹࡞ࡌࡾࡒࡴ࡞ࠉࠔࢷ࣭ࣂ࣭⟮ࠕ࡝࡜ࢅ౐⏕ࡌࡾ㸝࢕ࣤࣅ࣭ࢱࣤࢪᩒྙࠉimpedance
matching㸞ࠊᅒࡡࡻ࠹࡝⛸㢦࠿࠵ࡾࠊ㞹ẴⓏ≁
ᛮ࡛౐⏕᪁Ἢࢅ⩻៎ࡊ࡙Ửࡴࡾᚪさ࠿࠵ࡾࠊᅒ㸯<(2 ࡡ࿔ἴᩐ≁ᛮࠊ(a) හᑙమ࡛አᑙమࡡᙼࡡẒࢅୌᏽ࡞ಕࡗሔྙ.

(b) හᑙమᙼୌᏽࠉአᑙమᙼࡢࢷ࣭ࣂ࣭.

(c) "multiple quarter wavelength section
transformer"[16]zᅒ㸯࣊ࣤࢲᰧḿ᫤㛣㡷ᇡࡡಘྒἴᙟࠊ㉝⥲㸯ථງಘྒࠉ㟯⥲㸯<(2 ฝງಘྒࠊ

ᅒ㸯ࢷ࣭ࣂ࣭ྜྷ㍂ఎ㏞⥲㊪ࠊ
J-PARC MR ࡡ CT, WCM ࡡ࣊ࣤࢲᰧḿࡢᅒ
ࡡࡻ࠹࡝ᵋᠺ࡚⾔࡝ࡖࡒࠊ⤎ᯕࢅᅒ 49~54 ࡞ࡊࡴࡌࠊᅒ㸯(9 ࡡ࣊ࣤࢲῼᏽ⣌ࠊᅒ㸯ྜྷ㍂ንᥦჹ࡞ࡻࡾ <(2 ࣊ࣤࢲᰧḿࠊࡼịࡴࡒࠊ࣊ࣤࢲᰧḿ࡚ࡢࠉ㧏࿔ἴ㐵᩷࿔ἴᩐ㸯⣑ 150 MHz @ −3 dBࠉࢦࢡ㸯ȫ= 9.81 msࠉ఩ᇡ㐵᩷࿔ἴᩐ㸯fL = 1㸤2 πȫ = 16 Hz ࡚࠵ࡖࡒࠊ
ᅒ㸯(9 ࡡ࿔ἴᩐ≁ᛮࠊ

3.3.2. BPM ࡡᰧḿ
BPM ࡡ࣊ࣤࢲᰧḿ࡚ࡢࠉన⨠វᗐ౿ᩐࠉ࢛ࣆ
ࢬࢴࢹೋࢅịࡴࡾࠊࣄ࣭࣑ࡡ௥ࢂࡽ࡞ z ㍂࡞ᖲ⾔࡞ᘿࡼࡿࡒ㸦ᮇࡡ࣠࢕࣭ࣕ࡞㞹Ẵಘྒࢅථງࡊࠉ
BPM ࠾ࡼࡡฝງࢅῼᏽࡌࡾࠊ࣠࢕࣭ࣕࢅ୕ୖᕞ
ྎ㸝x, y ㍂᪁ྡྷ㸞࡞⛛ິࡌࡾ࠾ࠉ࣠࢕࣭ࣕࢅᅖᏽ
ࡊ ࡙ BPM ࢅ ⛛ ິ ࡊ ࡙ ⾔ ࡝ ࢂ ࡿ ࡾ ࠊ ᅒ   ࡞
J-PARC MR ࡡౚࢅࡊࡴࡌࠊ

ᅒ㸯࣊ࣤࢲᰧḿ᫤㛣㡷ᇡࡡಘྒἴᙟࠊ㉝⥲㸯ථງಘྒࠉ(9 ࡡฝງ㸯(9 ฝງἴᙟࠊ

ᅒ㸯'52 ࡡ࣠࢕࣭ࣕᰧḿࡡࢬࢴࢹ࢓ࢴࣈࠊ
ୌ᪁ࠉࣄ࣭࣑࡛࣠࢕࣭ࣕࡢࠉࡐࡿࡑࡿࠉ㞹Ⲭ࣬㞹Ὦࢮ࣭ࢪࠉ㸝࡮࡯Ᏸධ㸞ᑙమሾ⏲࡚࠵ࡽࠉజ࡙
㟸࡝ࡾࡵࡡ࡚࠵ࡾࠊ⌦ㄵⓏ࡞⢥ᐠ࡞スゝࡊࡒ BPM
ࢅᇱ‵ჹ࡛ࡊ࡙ࡆࡿ࡛ࡡẒ㍉ᰧḿࢅ⾔࡝࠹᪁Ἢࠉ
࣠࢕࣭ࣕ࡞ࡻࡾ࣊ࣤࢲᰧḿࡢ᳠ฝჹࡡฝᮮරྙ
ࡡ☔ヾ࡞࡛࡜ࡴ࡙ BPM ࡡᰧḿࡢᚃ㏑ࡡࣄ࣭࣑࣊
࣭ࢪᰧḿ࡞ࡻࡾ࡛࠷࠹᪁ྡྷࡵ⩻࠻ࡼࡿ࡙࠷ࡾࠊ
ᚃ㏑ࡡ⤎ྙ࢕ࣤࣅ࣭ࢱࣤࢪࡡ࣠࢕࣭ࣕ࡞ࡻࡾ
ᅒ㸯ࢪࢷࢴࣈᚺ➽ࠊ࣊ࣤࢲῼᏽ࡛ࡵ୩ࢆ࡚ࠉᚉᮮ࠾ࡼ౼ฺ࡚࠵ࡾ࡛࠷
࠹ࡆ࡛࡚⾔࡝ࢂࡿ࡙࠷ࡾ࠿ࠉཋ⌦Ⓩ࡞ḿᙔᛮࢅッ᪺ࡌࡾࠔᵾ‵⌦ㄵࠕࡢ↋࠷ࡡ࠿ᐁ᝗࡚࠵ࡾ࡛ᛦࢂ

఩ᇡ㐵᩷࿔ἴᩐ࠿ࢾࢴࢹ࣭࣠ࢠ࢓ࢻࣚ࢕ࢧࡡ࿔

ࡿࡾࠊࡆࡆ࡚ࡢࠉ⠾༟࡞㟴㞹࣓ࢸࣜࢅ౐ࡖ࡙ࠉࣄ

ἴᩐ⠂ᅑ࡞ථࡼ࡝࠷ࡆ࡛ࡵ࠵ࡾࠊJ-PARC ࡚ࡢࠉ࣭࣑࡛࣠࢕࣭ࣕἪ࡞ࡻࡾẒ㍉᳠ゞࢅモࡲࡾࠊࢪࢷࢴࣈᚺ➽㸝༎ฦ㛏࠷▬ᙟἴࢅථງ㸞ࡡࢦࢡ࠾఩࿔ἴᴗ㝀(DC)ࡡሔྙ࣠࢕࣭ࣕ࡞ࡒ࠷ࡌࡾᚺ
➽ࡢ௧ୖࡡᵕ࡞⩻࠻ࡼࡿࡾࠊ

࣠࢕࣭ࣕࡡ㞹ẴⓏ୯ᚨ࡛ᶭ᲌Ⓩ୯ᚨࡢࠉࡍࡿࡾ
ࡡ࡚ࠉࡍࡿ࠿ၡ㢗↋࠷⛤ᗐࡡኯࡈࡡ࣠࢕࣭ࣕࢅ⏕࠷ࡾ[18]ࠊ
ղᏽᅹἴ࠿❟ࡒ࡝࠷ࡻ࠹࡞ࡌࡾ≁࡞ᑊみ⥲ฦ๪ࡡ BPM ࡢࠉz ㍂᪁ྡྷฦᕱ࡞୘
ᆍୌ࠿࠵ࡾ࡛ㄏࡖࡒฝງࢅࡌࡾࠊ3.2.1⟿ࡡṹᕣࡡຝᯕ࡛ྜྷᵕࡡཋ⌦࡞ࡻࡾࠊᡜ᭻ࡡ⢥ᗐࡡ⠂ᅑ࡞࡝ࡾࡻ࠹࡞࢕ࣤࣅ࣭ࢱࣤࢪᩒྙࢅࡌࡾࠊ

ᅒ㸯࣠࢕࣭ࣕᰧḿࠊ
ᅒ 56 ࡡࡻ࠹࡞ࠉ㞹ᅸ″࠿᥃⤾ࡈࡿ࡙࠷ࡾ࡛ࡌ
ࡾࠊࡆࡡሔྙࠉ࣠࢕࣭ࣕ⾪㟻࡞ㄇᑙࡈࡿࡾ㞹ⲬࡢȢ[C/m]࡛࡝ࡾࠊࡆࡡ㞹Ⲭ㔖Ȣࡢࠉ࣠࢕࣭ࣕࠤࣄ

!

!

࣭࣑ࢱࢠࢹ㛣ࡡ㟴㞹ᐖ㔖 Cw(x,y)ࢅ౐ࡖ࡙ࠉ
" = C w (x, y)V ,
2"# 0
CW =
㻏
% 2
2
2
2 (
r
+
a
$
x
+
y
w
'
*
cosh$1'
*
2rw a
'
*
&
)

(

)

ᅒ㸯㞹ᴗẎࡡฝງಘྒ[V࣠࢕࣭ࣕన⨠ࠊῼᏽࡢࠉᅒ࡞♟ࡊࡒࡻ࠹࡝ࢬࢴࢹ࢓ࢴࣈ࡚
࠽ࡆ࡝ࡖࡒࠊBPM ᮇమࢅࠉx, y న⨠⢥ᗐs50ȣm

࡛᭡ࡂࡆ࡛࠿ฝᮮࡾࠊᚉࡖ࡙ࠉ྘㞹ᴗࡡฝງࡢࠉ

௧හࡡ⛛ິᯣྋ࡞㍍ࡎ࡙⛛ິࡊࡒ[19]ࠊῼᏽ⢥ᗐ࣠࢕࣭ࣕἪ࡞ࡻࡾሔྙࡢࠉౚ࠻ࡣ# x&
L(x, y) = g L " %1 + (
$ a'
x
1+
a
= gLV
,
# 2
2
2
2 &
r
+
a
)
x
+
y
w
%
(
cosh)1 %
(
2r
a
%
(
w
$
'

ࢅ☔ಕࡌࡾࡒࡴ࡞ࢾࢴࢹ࣭࣠ࢠ࣬࢓ࢻࣚ࢕ࢧࢅ౐(⏕ࡊࡒࠊJ-PARC MR ࡡ BPM ࡡᰧḿ⤎ᯕࡡౚࢅᅒࠉᅒ࡞♟ࡌࠊ)ࡡࡻ࠹࡞᭡ࡂࡆ࡛࠿࡚ࡀࡾ㸝gL ࡢ left 㞹ᴗࡡ gain
factor㸞ࠊᅒ 57 ࡞࣠࢕࣭ࣕᰧḿࡡ⤎ᯕ࡛୕オࡡᘟ!

࡞ࡻࡾࣆ࢔ࢴࢷ࢔ࣤࢡ⤎ᯕࢅ♟ࡌࠊ㟸ᖏ࡞Ⰳࡂྙ
ࡖ࡙࠷ࡾࠊᐁ㝷ࡡࣄ࣭࣑࡚ࡢ౿ᩐ CW ࠿࡝࠷࡛஢
᝷ࡈࡿࡾࠊࡱࡒࠉǼ㸤Ȋࡡゝ⟤ࢅࡌࡾ࡛ࠉCW ࡡ౿ᩐࡢ࡮࡯┞ṽࡊ⤎ᯕ࡞ࡢ࡮࡛ࢆ࡜ᙫ㡢ࡊ࡝࠷࡛⩻࠻ࡼࡿࡾࠊ
࣠࢕࣭ࣕᰧḿࢅ⾔࡝࠹୕࡚ࡡἸណⅤࢅาᣪࡌࡾ㸝J-PARC ࡡ⤊㥺ࡻࡽ㸞㸯ձ࣠࢕࣭ࣕ㸝࢓ࣤࢷࢻ㸞ࡡኯࡈᅒ㸯/5&7(27'52 ࡡᰧḿ⤎ᯕࠊ㟯Ⅴ㸯ῼᏽ⤎
ᯕࠉ㉝Ⅴ㸯࢛ࣆࢬࢴࢹࠉᅂ㌹⿭ḿᚃࡡࢸ࣭ࢰⅤࠊࣤࢡ㞹☚▴ࡡທ☚ࠉQ 㞹☚▴ࡡ࢛ࣆࢬࢴࢹ࠾ࡼࡂ
ࡾ࢞ࢴࢠ㸞࠿࠵ࡾ࡛
nm n + m22 n12x 3 = " 12 x1 + 12 11
x2 + C ,m12
m12ࡡࡻ࠹࡞㏛ຊࡡ㡧 C ࠿ຊࢂࡾ[18]ࠊ㌮㐠ࡡ᫤㛣Ⓩ
ንິさᅄ࠿୕オࡡ BPM㸨ྋࡡአ㒂࡞࠵ࡾ࡛ࡌࡿ
!

ࡣࠉふῼ⤎ᯕࡡ᫤㛣Ⓩንິࡢ BPM ࢨࢪࢷ࣑ࡡಘ
ྒࡡࣁࣚࢵ࢞࠾ࡼࡂࡾࡵࡡࠉࡌ࡝ࢂࡔ BPM ฦよ⬗ࡡᐞ୙ࡡࡲ࡚࠵ࡾࠊᚉࡖ࡙
N{($ x 3n # Ax1n + Bx 2n + C

"=

1 n=1
N #1

1+ A 2 + B 2

)}

2

,

ࡻࡽ BPM ࡡฦよ⬗࠿ᚋࡼࡿࡾࠊ
ᅒ㸯/57('9'52 ࡡᰧḿ⤎ᯕࠊ㉝Ⅴ㸯ῼ!
ᏽ⤎ᯕࠉ㟯⥲㸯ࣆ࢔ࢴࢷ࢔ࣤࢡࠊ

3.4.2. ࢛ࣆࢬࢴࢹ
Beam based alignment ࡛࿣ࡣࡿ࡙࠷ࡾ᪁Ἢ

3. 4. ࣄ࣭࣑࣊ ࣭ࢪᰧ ḿ୕オࡡࡻ࠹࡞࣠࢕࣭ࣕ࡞ࡻࡾ࣊ࣤࢲᰧḿࢅ࠽ࡆ࡝ࡖ࡙ࡵࠉຊ㏷ჹࡡ⌟ሔ࡚ࡡス⨠ㄏᕣࠉࢢ࣭ࣇ
࡚ࣜࡡ΅⾮ࠉฌ⌦ᅂ㊪࡚ࡡࢣ࢕࢛ࣤ࣬ࣆࢬࢴࢹㄏ
ᕣࠉ᳠ฝჹ࣬ࢢ࣭ࣇࣜ࣬ฌ⌦ᅂ㊪㛣ࡡᏽᅹἴ࠿ࠉ
BPM ࡡῼᏽ⤎ᯕ࡞ㄏᕣࢅᣚࡔ㎲ࡳࠊࡆࡆ࡚ࡢࠉBPM ࡡᛮ⬗ࢅࣄ࣭࣑⮤㌗ࡡಘྒ࡞ᇱࡘ࠷࡙Ửᏽࡌࡾ᪁Ἢࢅ㏑࡬ࡾࠊ
3.4.1. న⨠ฦよ⬗㸝3BPM ┞㛭Ἢ㸞ྜྷୌࡡࣄ࣭࣑࠿ࠉ㎾എ࡞ス⨠ࡈࡿࡒ㸨ࡗࡡ BPM
ࢅ㏳㐛ࡌࡾࡆ࡛࡞╌┘ࡊ࡙ࠉBPM ࡡฦよ⬗ࢅフ౮࡚ࡀࡾࠊ㸨࢜ᡜ࡞ 1,2,3 ࡛␊ྒࢅ௛ࡄࡾ࡛ࠉࡐ
ࡿࡑࡿࡡሔᡜ࡚ࡡన⨠ xࠉലࡀ x'ࡢࠉ㏭୯࡞㟸⥲ᙟ☚ሔ࠿↋࠷ሔྙࡢ
# x 2 & # m11 m12 &# x1 &% (=%(% ( ,$ x"2 ' $ m21 m22 '$ x"1 '# x 3 & # n11 n12 &# x 2 &% (=%(% ( ,$ x"3 ' $ n 21 n 22 '$ x"2 '࡚ࠉ௧ୖࡡࡻ࠹࡞㸩ᴗ㞹☚▴ࡡ☚ሔ୯ᚨ࡛ BPM
ࡡ㞹ẴⓏ୯ᚨࡡ࢛ࣆࢬࢴࢹࢅࠉᇱᮇⓏ࡞ࡢຊ㏷ჹ
ࣚࢷ࢔ࢪࡡ࣓ࢸࣜ࡞౪Ꮛࡊ࡝࠷࡚ịࡴࡾ᪁Ἢ࡚
࠵ࡾࠊຊ㏷ჹධమ࠾ࡼぜࡾ࡛ࠉ"alignment"ࡢ㞹☚▴ࡡス⨠࢓ࣚ࢕࣒ࣤࣤࢹࢅ㏻᝷ࡊ࡙ࡊࡱ࠹ࠊ
ࠔ࡜࠹ࡷࡖ࡙㞹☚▴ࡡ࢓ࣚ࢕࣒ࣤࣤࢹࢅ࡚ࡀࡾࢆ
ࡓࠕ࡛㈻ၡࢅུࡄࡒ⤊㥺ࡢࠉⴥ⩽ࡡࡲ࡚ࡢ࡝ࡈࡐ
࠹࡚࠵ࡾࠊBPM ࡡ alignment ࡛࠷࠹୹᪠࡚࠵ࡾࠊ
ཋ⌦ࡢ௧ୖࡡ㏳ࡽ࡚࠵ࡾࠊࡱࡍࠉBPM ࠿㸩ᴗ㞹☚▴හ࠵ࡾ࠷ࡢ㎾എ࡞ス⨠ࡈࡿ࡙࠷ࡾ࡛ࡌࡾࠊ
ࣄ࣭࣑࠿㸩ᴗ㞹☚▴ࡡ୯ᚨࢅ㏳ࡾ࡛ࡀࠉࣄ࣭࣑㔔ᚨࡢ࢞ࢴࢠࢅུࡄ࡝࠷ࠊᚉࡖ࡙㸩ᴗ㞹☚▴ࡡᙁࡈࢅࣄ࣭࣑࠿ࣞࢪࡊ࡝࠷⠂ᅑ࡚ን໩ࡈࡎ࡙ࡵࠉ㌮㐠
࡞ን໩ࡢぜࡼࡿ࡝࠷ࠊୌ᪁ࠉࣄ࣭࣑࠿㸩ᴗ㞹☚▴ࡡ୯ᚨ࠾ࡼࢫ࡙ࣝ࠷ࡾ࡛㸩ᴗ㞹☚▴ࡡᙁࡈ࡞ࡻ
ࡖ࡙㌮㐠࠿ንࢂࡾࠊࡐࡆ࡚ࠉ㌮㐠࡞ን໩ࡡぜࡼࡿ࡝࠷᫤ࡡ BPM ࡡన⨠ῼᏽೋ࠿ BPM ࡡ࢛ࣆࢬࢴ
ࢹ࡛࡝ࡾࠊLINAC ࡡሔྙࡢ 1 ᅂ㏳㐛ࡡࣄ࣭࣑ࠉ
ࣛࣤࢡ࡚ࡢ㛚㌮㐠㸝COD㸞ࡓ࠿ࠉᇱᮇⓏ࡞ࡢྜྷᵕ࡚࠵ࡾࠊ!

࡚⤎ࡦࡗࡄࡼࡿࡾࠊῼᏽ࡚ࡀࡾ㔖 x1, x2, x3 ࡡࡲࡡ!

㛭౿ᘟࢅࡗࡂࡾ࡛
nm n + m22 n12x 3 = " 12 x1 + 12 11
x2 ,
m12
m12

྘ࣁࣤࣈ㌮㐠࡚ࠉ㸩ᴗ㞹☚▴ࡡ㞹Ὦೋࢅን࠻࡙

ࢅᚋࡾࠊࡈࡼ࡞ࠉBPM ࡡ࢛ࣆࢬࢴࢹㄏᕣࠉࢱ࢕

࡙࠷ࡾࠊ྘ࣁࣤࣈ㌮㐠Ẏ࡞∂x㸤∂IQ ࢅ⟤ฝࡊࠉ࣭࣎ࣜ࢞ࢴࢠ㸝೩ྡྷ㞹☚▴ࡡㄏᕣ☚ሔࠉࢪࢷ࢓ࣛ

BPM#89 న⨠࡛∂x㸤∂IQ ࢅࣈࣞࢴࢹࡊࡒࡡ࠿(d)࡚

!

J-PARC MR ࡡౚࢅᅒ࡞♟ࡌࠊ(a)~(c)࡚ࡢࠉ
BPM#89 ࡚ࡡ COD ῼᏽࢅ࠽ࡆ࡝ࡖࡒ⤎ᯕࢅ♟ࡊ࠵ࡾࠊ∂x㸤∂IQ=0 ࡡ x ࠿࢛ࣆࢬࢴࢹ࡚࠵ࡾࠊ

(a)⦢᪁ྡྷࠉᶋ᪁ྡྷࡡ࣭ࢗ࢘ࢠ࣎ࢷࣤࢨࣔࣜ W(s)㸯

(b)

L/2

F// =

# F// ds = "eqW // (s),"L / 2
L/2

F$ =# F$ ds = "eqyW$ (s),"L / 2

(c)

(d)

!

!

ᅒ㸯'52 ࡡ 'HFPGFVHIFOLJQPHQWࠊ

4. ⤎ྙ࢕ࣤࣅ࣭ࢱࣤࢪゝῼ[5]!

ࣄ࣭࣑࠿┷✭ࣂ࢕ࣈ୯ࢅ㉦ࡾ࡛ࡀࠉ2.1⟿࡚㏑࡬ࡒࡻ࠹࡞࿔ࡽ࡞㞹☚ሔࢅఔࡖ࡙࠷ࡾࠊᏰධᑙమࡡ
!
ୌᵕ࡝ࣂ࢕ࣈ࡚࠵ࡿࡣၡ㢗↋࠷࠿ࠉᐁ㝷ࡡຊ㏷ჹ
࡚ࡢࠉ┷✭ࣂ࢕ࣈࡡහഁ࡞ࠉṹᕣ࣬ࢷ࣭ࣂ࣭ࠉࣄ

࡛ࡐࡡࢸ࢔࣒ࣤࢨࣘࣤ㸯[W // ] = [V /C ],[W" ] = [V /Cm].
⦢᪁ྡྷࠉᶋ᪁ྡྷࡡ⤎ྙ࢕ࣤࣅ࣭ࢱࣤࢪ Z(s)㸯
$ dz "i#z / c
Z // = %"$e
W // (s),ci $ dz "i#z / c
Z& = %"$eW& (s),'c࡛ࡐࡡࢸ࢔࣒ࣤࢨࣘࣤ㸯[Z // ] = ["],
[Z# ] = [" / m].
࣭ࢗ࢘ࢠ࣎ࢷࣤࢨࣔࣜࠉ⤎ྙ࢕ࣤࣅ࣭ࢱࣤࢪࡡౚ
ࢅ௧ୖ࡞♟ࡌࠊ

Longitudinal wake potential࣭࣑ゝῼᶭჹࠉRF ຊ㏷✭ὕ㸤✭⬏ࠉ࣏ࢡࢾࢴࢹࠉ
ࢬࣈࢰ࣑㞹ᴗ࡝࡜࠿Ꮛᅹࡌࡾࠊࡱࡒࠉ┷✭ࣂ࢕ࣈ⮤మ࠿᭯㝀ࡡᢤᢘࢅࡵࡖࡒᑙమ࡚࠵ࡾࠊරమౚ࡛
ࡊ࡙ࠉ┷✭ࣂ࢕ࣈ࡞ṹᕣ࠿࠵ࡾሔྙࠉࣄ࣭࣑࠿㉦ࡾ㍂୕࡞㐅⾔㸝z ㍂㸞᪁ྡྷ࡞㞹ሔ࠿ฝᮮࡾ㸝ᅒ㸞ࠊ

Transverse wake potentialᅒ㸯┷✭ࣂ࢕ࣈṹᕣࡡ࣭ࢗ࢘ࢠሔࠊ
඙⾔ࡌࡾⲬ㞹⢇Ꮔࡢࠉ㊝㞫 s ᚃࢀࢅ㉦ࡾࢷࢪࢹ⢇Ꮔ࡞ງࢅ୙࠻ࡾࠊࡆࡿࢅࣄ࣭࣑㍂࡞Ἒࡖ࡙✒ฦࡊࡒࡵࡡ࠿ࠉࣄ࣭࣑ࡡུࡄཱིࡾ㐘ິ㔖࡚ࠉ࣭ࢗ࢘ࢠ
࣎ࢷࣤࢨ࡛ࣔࣜ࿣ࡣࡿࡾࠊࡆࡿࡡࣆ࣭࢙ࣛንᥦ࠿
⤎ྙ࢕ࣤࣅ࣭ࢱࣤࢪ࡚࠵ࡾࠊ௧ୖ࡞㛭౿ᘟࢅࡱ࡛
ࡴࡾࠊᅒ㸯ᢤᢘᛮባ㸝UHVLVWL[H\FOO㸞ࡡ࣭ࢗ࢘ࢠ࣬
࣎ࢷࣤࢨࣔࣜࠊ㉝⥲㸯ࣄ࣭࣑࣬ࣁࣤࢲࠉ㯦⥲㸯࣭ࢗ࢘ࢠ࣎ࢷࣤࢨࣔࣜࠊ

ࡆࡿࡼࡡ࣭ࢗ࢘ࢠሔࡢࣄ࣭࣑ࢅ୘Ꮽᏽ࡞ࡊ࡙ࠉ
ࣄ࣭࣑ࢦ࢕ࢫࢅᣉࡅࡒࡽࠉࣄ࣭࣑ࣞࢪࢅᘤࡀ㉫ࡆࡊࡒࡽࠉ࣐ࣜࢿࢨࢷ࢔ࢅᝇ໩ࡈࡎࡒࡽࡌࡾࠊ୘Ꮽ
ᏽⓆ⏍ࡡὮࡿࡢᴣ␆௧ୖࡡࡻ࠹࡝ࡵࡡ࡚࠵ࡾࠊ

ᅒ㸯ᢤᢘᛮባࡡ⤎ྙ࢕ࣤࣅ࣭ࢱࣤࢪࠊ

ᅒ㸯ࣄ࣭࣑୘ᏭᏽࡡⓆ⏍ࠊࡆࡡࡻ࠹࡝୘Ꮽᏽᛮ࠿㉫ࡆࡼ࡝࠷ࡻ࠹࡞ୖオࡡࡻ࠹࡝ᑊ➿ࢅࡌࡾࡡ࠿ࠉ᭩㎾㸝⣑ 1980 ᖳ௥௧ᅒ㸯✭ὕࡡ࣭ࢗ࢘ࢠ࣬࣎ࢷࣤࢨࣔࣜࠊ୕ᅒ㸯6"ࠉୖᅒ㸯6"ࠊ㜾ᘋス㸞ࡡຊ㏷ჹࡡᏽ▴࡞࡝ࡖ࡙ࡀ࡙࠷ࡾࠊ
ձ஥๑࡞ຊ㏷ჹࡡ⤎ྙ࢕ࣤࣅ࣭ࢱࣤࢪࢅフ౮ࡊ࡙࠽ࡀࠉᚪさ࡝ࡵࡡ࡞ࡗ࠷࡙ࡢ⤎ྙ࢕ࣤࣅ࣭ࢱࣤ
ࢪࡡᨭၻࢅ࠽ࡆ࡝࠹ࠊ㸲࢕ࣤࣅ࣭ࢱࣤࢪ࣬ࣁࢩ࢘
ࢴࢹ㸝impedance badget㸞ࡡషᠺղ୘Ꮽᏽᛮフ౮ࡡ⤎ᯕࠉᏭᏽᛮࡡ☔ಕ࡚ࡀ࡝࠷
instability ࡞ ࡗ ࠷ ࡙ ࡢ ࠉ ื ㏭ ᑊ ➿ ࢅ ㅦ ࡋ ࡾ 㸯
feedback

dampers,

Landaudamping⏕

nonlinear magnets etc.
୕オࡡࠉ஥๑ࡡ⤎ྙ࢕ࣤࣅ࣭ࢱࣤࢪフ౮࠿ᮇ⟿ࡡ୹㢗࡚࠵ࡾࠊࢤࣤࣅ࣭ࣖࢰ࡞ࡻࡾ㞹☚⏲ࢨ࣐࣭ࣖࣝࢨࣘࣤ࠿Ⓠ㐡ࡊࡒ௑᪝࡚ࡢࠉ㸨ḗඔࢨ࣐ࣖࣝ
࣭ࢨࣘࣤ࡞ࡻࡽ⤎ྙ࢕ࣤࣅ࣭ࢱࣤࢪࢅịࡴࡾࡆ㸣࡛࠿ฝᮮࡾ࠿ࠉ࢞ࢴ࣭࢜ࡷຊ㏷ RF ✭⬏ࡡࡻ࠹࡝々㞟࡝ᙟ≟࣬ᛮ㈻ࡡ☚ᛮమࠉ❟మⓏ࡞㒼⥲ࡈࡿࡒᑙమ࡝࡜࠾ࡼᵋᠺࡈࡿࡾᶭჹࡡ⤎ྙ࢕ࣤࣅ࣭ࢱ
ᅒ㸯✭ὕࡡ⤎ྙ࢕ࣤࣅ࣭ࢱࣤࢪࠊ୕ᅒ㸯6"ࠉ
ୖᅒ㸯6"ࠊ

ࣤࢪࡡࢨ࣐࣭ࣖࣝࢨࣘࣤ࡞ࡢᅏ㞬ࢅఔ࠹ࠊᐁᶭࡡ
ᨭⰃࡡࣈࣞࢬࢪ࡚ࠉࢬࢴࢹ࢓ࢴࣈࡡን᭞ࠉ࠵ࡾࢤ
࣭ࣤ࣎ࢾࣤࢹࡡ⬲╌࡝࡜࠿ࠉ⤎ྙ࢕ࣤࣅ࣭ࢱࣤࢪ
࡞࡜࠹ᙫ㡢ࡌࡾࡡ࠾ࢅ༳ᗑ࡞᳠ッࡌࡾ᪁Ἢࡵ᭻

ࡱࡿࡾࠊࡐࡡࡻ࠹࡝ࢂࡄ࡚ࠉ྘⛸ࡡ⤎ྙ࢕ࣤࣅ࣭
ࢱࣤࢪࡡ࣊ࣤࢲῼᏽ࠿㛏ᖳ࠽ࡆ࡝ࢂࡿ࡙ࡀ࡙࠷
ࡾࠊࡆࡆ࡚ࡢẒ㍉Ⓩ Q ೋࡡ఩࠷ᶭჹ࡞㐲ࡌࡾ࣠࢕࣭ࣕࢅ౐࠹᪁Ἢࢅ⣺௒ࡌࡾࠊ
4. 1. ⦢᪁ྡྷ࢕ ࣤࣅ࣭ ࢱࣤࢪ ࡡ࣊ࣤ ࢲῼᏽ [20]࠾ࡼ Z ࢅịࡴࡾࠊ
DUT ࠿㞗୯ᏽᩐ࡛ぜ࡝ࡎ࡝࠷ሔྙࡢࠉZ ࠿ᑚࡈ࠷࡛ࡊ࡙ịࡴࡼࡿࡒࠉ
#S&
Z = "2ZC ln%% 21,DUT (( ,$ S21,REF 'ࢅ⏕࠷ࡾࠊࣄ࣭࣑ࣂ࢕ࣈࡡ୯ᚨ࡞⣵࠷㔘ᒌ࣠࢕࣭ࣕࢅᘿ
!
ࡖ࡙ࠉྜྷ㍂ᵋ㏸ࢅᙟᠺࡌࡾࠊࡆࡆࢅఎᦑࡌࡾ㞹☚

4. 2. ᶋ᪁ྡྷ࢕ ࣤࣅ࣭ ࢱࣤࢪ ࡡ࣊ࣤ ࢲῼᏽ [20]ἴࡢ࢜ࢴࢹ࢛ࣆ࿔ἴᩐ௧ୖ࡚ࡢ TEM ἴࡡࡲ࡚࠵

㸧ᮇࡡ㔘ᒌ࣠࢕࣭ࣕࢅᖲ⾔࡞ᘿࡖ࡙ࠉࣄ࣭࣑ࡡ

ࡾࠊࡆࡿࡢක㏷࡚㉦ࡾࣄ࣭࣑࡞ఔ࠹㞹☚ἴ࡞㎾ᶋ᪁ྡྷࢱ࢕࣭࣓࣭࣎ࣜ࣬ࢺࢅᶅ᧻ࡌࡾࡵࡡ࡚࠵

࠷ࠊ࣠࢕࣭ࣕࡡᙼ࠿↋㝀ᑚ࡞࡝ࡾᴗ㝀㸝ྜྷ㍂ᵋ㏸ࡾࠊ㸧ᮇࡡ㔘ᒌ࣠࢕࣭࡛ࣕࣄ࣭࣑ࣂ࢕ࣈࡢ

ࡡ≁ᛮ࢕ࣤࣅ࣭ࢱࣤࢪ ZcЍ҄㸞࡚ࡢḿࡊ࠷ೋࢅ୙

"shielded pair" ఎ㏞⥲㊪ࢅᙟᠺࡌࡾࠊࡆࡡ⣌࡞ࡢ࠻ࡾ࡛⩻࠻ࡼࡿࡾ࠿ࠉᐁ㝷୕ࡢྊ⬗࡝⠂ᅑ࡚⣵࠷

even(㸠㸠)࣓࣭ࢺ࡛ odd(㸠̿)࣓࣭ࢺ࠿ఎᦑ࡚ࡀ

࣠࢕࣭ࣕࢅ㐽ᢝࡌࡾࠊࡆࡡ᪁Ἢ࡚ࡢࠉ⦢᪁ྡྷࡡ⤎ࡾ࠿ࠉࣄ࣭࣑ࡡᶋ᪁ྡྷࢱ࢕࣭࣓࣭࣎ࣜ࣬ࢺࢅᶅ᧻

ྙ࢕ࣤࣅ࣭ࢱࣤࢪࢅῼᏽࡌࡾࠊ

ࡌࡾࡡࡢ odd ࣓࣭ࢺࡡࡲ࡚࠵ࡾࠊࢾࢴࢹ࣭࣠ࢠ࢓
ࢻࣚ࢕ࢧࡡ࣭࣎ࢹ 1 ࠾ࡼࡡಘྒࢅ odd ࣓࣭ࢺ࡞ᩒ
ྙࡈࡎࡾࡒࡴࠉ㧏࿔ἴᩐᖈᇡࡡࣀ࢕ࣇࣛࢴࢺ࠿౐
⏕ࡈࡿࡾࠊࡆࡿࡢ఩࿔ἴ࡚ࡢᢇ⾙Ⓩ࡞ᅏ㞬࡝ࡡ࡚ࠉJ-PARC ࡚ࡢ࣏ࣜࢲ࣭࣎ࢹ࣬ࢾࢴࢹ࣭࣠ࢠ࢓
ࢻࣚ࢕ࢧࢅ౐⏕ࡊ࡙࠷ࡾࠊ

ᅒ㸯⦢᪁ྡྷ࢕ࣤࣅ࣭ࢱࣤࢪῼᏽࠊྜྷ㍂࣠࢕࣭ࣕἪࠊῼᏽࡢࠉ⢥ᗐࡡふⅤ࠾ࡼ࣊ࢠࢹࣜ࣬ࢾࢴࢹ࣭࣠ࢠ

ᅒ㸯ᶋ᪁ྡྷ࢕ࣤࣅ࣭ࢱࣤࢪῼᏽࠊ ᮇࡡ࣠࢕ࣕ
࣭ࢅᘿࡖ࡙ RIIPRIH ࡡ㏩㐛౿ᩐࢅịࡴࡾࠊ࢓ࢻࣚ࢕ࢧࢅ⏕࠷࡙࠽ࡆ࡝ࢂࡿࡾࠊῼᏽᑊ㇗
㸝DUT, Device under test㸞࠿ᑻᅹࡌࡾ࢕ࣤࣅ࣭ࢱ

DUT ࡡ࢕ࣤࣅ࣭ࢱࣤࢪ Z ࡢࠉS21 ࢅ odd ࣓࣭ࢺࡡࣤࢪ Z ࡡሔྙࠉ㏩㐛౿ᩐ㸝S21㸞ࢅῼᏽࡊࠉ
1" S21
Z = 2ZC,
S21㏩㐛౿ᩐ࡛ࡊ࡙ࠉ๑⟿ࡡᘟ࠾ࡼịࡴࡼࡿࡾࠊࡐࡡ

࡞ࡻࡽ Z ࢅᚋࡾࠊ㏳ᖏࡢࠉࣝࣆ࢒ࣝࣤࢪ࣬ࣂ࢕ࣈ
ࡡ㏩㐛౿ᩐ㸝S21,REF㸞࡛ࠉDUT ࡡ㏩㐛౿ᩐ㸝S21,DUT㸞!

!
!

ࡡῼᏽࢅ࠽ࡆ࡝࠷ࠉ
1" S21,NZ = 2ZC,
S21,N
S
S21,N = 21,DUT ,S21,REFZ ࢅࡵࡔ࠷࡙ࠉǼࢅ࣠࢕࣭ࣕ㛣㊝㞫࡛ࡊ࡙ࠉ
cZ
,
(5)
Z" =
#$2
࡞ࡻࡽᶋ᪁ྡྷ࢕ࣤࣅ࣭ࢱࣤࢪ࠿ᚋࡼࡿࡾࠊ
࢞ࢴ࣭࢜ࡡᶋ᪁ྡྷ࢕ࣤࣅ࣭ࢱࣤࢪࡢࠉኬࡀ࡝࢕ࣤ

!

ࣅ࣭ࢱࣤࢪ″ࡡು⿭࡚࠵ࡾ࠿ࠉ఩࿔ἴ≁ᛮࡡῼᏽ
ࡢࠉ㏩㐛ಘྒࡡ SN ࠿ᝇ࠷ࡒࡴ୕オࡡ᪁Ἢ࡚ࡢᅏ
㞬࡚࠵ࡾࠊJ-PARC ࡚ࡢࠉ఩࿔ἴ࡞㝀ࡾ࡛࠷࠹๑
ᥞ࡚ࠉ∞❻ࢅࢨ࣭ࣘࢹࡊཬᑏ౿ᩐ㸝S 11㸞࡞ࡻࡽῼᏽࢅ࠽ࡆ࡝ࡖ࡙࠷ࡾ㸝ᅒ㸞[20]ࠊࢨ࣭ࣘࢹࡈࡿ
ࡒ࣋࢓⥲ࡢࢤ࢕࡛ࣜぜ࡝ࡎࡾࠊS11 ῼᏽࡢࢤ࢕ࣜ
ࡡ࢕ࣤࢱࢠࢰࣤࢪࡡῼᏽ࡞࡝ࡖ࡙࠷ࡾࠊDUT ࠿࠵ࡾ࡛ྺ཭ࠉన┞ንన࠿㉫ࡀࡾࡒࡴࠉDUT ࡡᐞ୙ࡢ REF ῼᏽ࡚ࡡ࢕ࣤࢱࢠࢰࣤࢪ࠾ࡼࡡࢫ࡛ࣝࡊ࡙ふῼࡈࡿࡾࠊ

ᅒ㸯8 ࡞ࡻࡾῼᏽ㸝∞❻ࢨ࣭ࣘࢹ㸞ࠊ
Z in,REF = ZC!
!

ᅒ㸯7(8ཱིฝࡊ࢞ࢴ࣭࢜㸝ࣈࣞࢹࢰ࢕ࣈ㸞ࠊ1+ S11,REF
1+ S11,DUT, Z in,DUT = ZC,
1" S11,REF
1" S11,DUT"Z in = Z in,DUT # Z in,REF ,
"Z in
,
Z=
l
!
࡛ࡊ࡙ࠉᘟ㸝5㸞࡞௥ථࡊ࡙ᶋ᪁ྡྷ࢕ࣤࣅ࣭ࢱࣤ

ࢪࢅịࡴࡾࠊ!

㛏ࡈε , ≁ᛮ࢕ࣤࣅ࣭ࢱࣤࢪ ZC ࡡ࣋࢓ఎ㏞⥲㊪ࢅ z=ε ࡚ࢨ࣭ࣘࢹࡊ࡙ࠉz=0 ࠾ࡼぜࡒ࢕ࣤࣅ࣭

ᅒ㸯7(8ཱིฝࡊ࢞ࢴ࣭࢜ࡡᵋᠺࠊ

ࢱࣤࢪࡢ
Z in = ZC

1" e"2 j#l / c

1+ e"2 j#l / c$'$ #l ' 3#l" ZC && j+ j & ) 3+ L)) .
% c (
% c
(

!

ࡆࡿࡢࠉᣋᘴහࡡ➠ 2 㡧௧㜾࠿↋っ࡚ࡀࡾ᫤ࠉ
Z in " j#Ll!

ࡻࡽࠉ࢕ࣤࢱࢠࢰࣤࢪ࡛ぜ࡝ࡎࡾࠊ୕オࡡ᮪௲ࢅ(a)‫࠷࡝ࡈࡒ‮‬ሔྙࠉㄏᕣ࠿ኬࡀࡂ࡝ࡾࠊ
!

J-PARC RCS ࡡཱིฝࡊ࢞ࢴ࣭࢜ࡡౚࢅ⣺௒ࡌࡾࠊࡆࡡ࢞ࢴ࣭࢜ࡢఎ㏞⥲㊪ᆵ࡚࠵ࡾࠊ㏳ᖏࡡఎ㏞⥲㊪ᆵ࢞ࢴ࣭࢜ࡢࠉୌ᪁ࡡ❻㒂࠾ࡼ㞹Ὦࢅᑙථࡊࠉ
௙᪁ࡡ❻㒂࡚⤂❻ࢅࡌࡾࠊࡊ࠾ࡊᮇ࢞ࢴ࣭࢜ࡢ∞❻ࢨ࣭ࣘࢹ࡚㞹Ὦࢅಶቌࡈࡎࡾスゝ࡛࡝ࡖ࡙࠷
ࡾ[21]ࠊᅒࠉᅒ࡞ධ⣌࡛ᅂ㊪ᅒࢅ♟ࡌࠊࡆࡡ࢞ࢴ࣭࢜࡞ࡗ࠷࡙ࠉࢨ࣭ࣘࢹࡊࡒ࣋࢓ఎ㏞⥲㊪
࡚ᶋ᪁ྡྷ࢕ࣤࣅ࣭ࢱࣤࢪῼᏽࢅ࠽ࡆ࡝ࡖࡒ⤎ᯕࢅᅒ࡞♟ࡌ[22]ࠊ

(b)ᅒ㸯/5&7(7(8ཱིฝࡊ࢞ࢴ࣭࢜ࠊᶋ᪁ྡྷ㸝Ềᖲ᪁ྡྷ㸞࢕ࣤࣅ࣭ࢱࣤࢪ㸝ᐁᩐ㒂㸞ࠊ F ࢞ࢴ࣭࢜∞❻ࢨ࣭ࣘࢹ∞❻ࢢ࣭ࣇࣜ 㛜ᨲ ࠉ G ࢞ࢴ࣭࢜∞❻ࢨ࣭ࣘࢹ∞❻࢕ࣤࣅ࣭ࢱࣤࢪᩒྙࠊ

5. 䝗䞀䝤୘Ꮽᏽᛮ䛴ゝῼ୕オࡡࡻ࠹࡞㧏࿔ἴࡡᠺฦࢅࡵࡗ୘Ꮽᏽᛮࢅふῼࡌࡾࡒࡴ࡞ࡢࠉῼᏽჹࡡ࿔ἴᩐᖈᇡࢅヾㆉࡊ࡙

5. 1. ࣄ࣭࣑୘ Ꮽᏽᛮ ࡡふῼࣄ࣭࣑ࡡ㞗ᅆ୘Ꮽᏽᛮࡡฝ⌟ࡢࠉࡱࡍࣄ࣭࣑࣓ࢼ
ࢰ࣭࡚ࡡࢤࣃ࣭ࣝࣤࢹಘྒࡡቌኬ࡛ࡊ࡙ふῼࡈࡿࡾࠊࡆࡆ࡚ࡢࠉKEK-PS ࡚ࡡふῼࢅ୹࡝ౚ࡛ࡊ
࡙⣺௒ࡌࡾࠊ

࠽࠾࡝ࡄࡿࡣ࡝ࡼ࡝࠷ࠊࡆࡡౚ࡚ࡢ f㸱2GHz ࡚࠵ࡾࠊ࢕ࣤࣅ࣭ࢱࣤࢪ″࠿ㄢᰕࡈࡿࠉࣄ࣭࣑ࣂ࢕ࣈࡡ
ṹᕣ࡛ BPM ࡡᵋ㏸࠿୹ࡒࡾࡵࡡࡓ࡛ᛦࢂࡿࡒࠊ
MAFIA ࠽ࡻࡦྜྷ㍂⟮࡞ࡻࡾ࣊ࣤࢲῼᏽ࡞ࡻࡽࠉ
ࡐࡿࡑࡿࡡ⤎ྙ࢕ࣤࣅ࣭ࢱࣤࢪ࠿フ౮ࡈࡿࡒࠊ⾪5.1.1. ⦢᪁ྡྷ㸝longitudinal㸞⦢᪁ྡྷࡡࣄ࣭࣑୘Ꮽᏽᛮ࡞ࡢࠉ

㸦࡞ BPMࠉCVC㸝caviti-like vacuum chamber㸯ᤴẴ࣭࣎ࢹ௛ࡀࢲ࢘ࣤࣁ࣭㸞ࡡフ౮⤎ᯕࢅ♟ࡌࠊձࣞࣄࣤࢮࣤ୘Ꮽᏽᛮ
ղ⤎ྙࣁࣤࢲ୘Ꮽᏽᛮ
ճ࣏࢕ࢠ࣭ࣞࢗ࢘ࣇ୘Ꮽᏽᛮ
մࢾ࢝ࢷ࢔ࣇ࣏ࢪ୘Ꮽᏽᛮ
࡝࡜࠿࠵ࡾ࠿ࠉࡆࡆ࡚ࡢࠉKEK-PS MR ࡡన┞㐼⛛㸝phase-transition, ȚT㸞௛㎾࡚ぜࡼࡿࡒ࣏࢕
ࢠ࣭ࣞࢗ࢘ࣇ୘Ꮽᏽᛮࡡ⣺௒ࢅࡌࡾ[23]ࠊ
ࡐࡵࡐࡵࡡⓆ❻ࡢࠉన┞㐼⛛࡚ࡡ⃥ࡊ࠷ࣄ࣭࣑
ࣞࢪ࠿ࣄ࣭࣑ᙁᗐࡡቌᙁࢅโ㝀ࡌࡾさᅄࡡ㸦ࡗ࡚࠵ࡖࡒࡆ࡛࡚࠵ࡾ㸝ᅒ㸞ࠊ⾪㸯࢕ࣤࣅ࣭ࢱࣤࢪフ౮ࡡ⤎ᯕࠊࡆࡿࢅᇱ࡞ࡊࡒ୘Ꮽᏽᛮࡡࢨ࣐࣭ࣖࣝࢨࣘࣤ
࠿ࠉWCM ࢅ౐ࡖࡒ⦢᪁ྡྷ࢙࣐ࢴࢰࣤࢪῼᏽࡡ⤎ᯕ㸝ᅒ㸞ࢅⰃࡂ්⌟ࡊࡒࡆ࡛࠾ࡼࠉ୕オࡡࢪࢹ࣭࣭ࣛࡢḿࡊ࠷࡛⩻࠻ࡼࡿࡒࠊᅒ࡞ࡢࠉ⣑2/3 ࡡ BPM ࡛┷✭ᤴẴ࣭࣎ࢹࢅᑊ➿ࢅ᪃ࡊࡒࡵ
ࡡ࡞ஹᥦࡊࡒᚃࡡῼᏽ⤎ᯕࡵࣈࣞࢴࢹࡈࡿ࡙࠷
ࡾࠊṟࡖࡒ୘Ꮽᏽᛮࡢࠉ⦢᪁ྡྷ࢙࣐ࢴࢰࣤࢪࢅቌ
ኬࡈࡎࠉࣄ࣭࣑ࡡ⥲㞹Ⲭᐠᗐࢅ΅ࡼࡊ࡙ࠉ୘Ꮽᏽ
ᛮࢅᢒโࡊࡒ[24]ࠊ

(a)

(b)
ᅒ㸯0*05827 ࡚ふῼࡈࡿࡒ࣏࢕ࢠ࣭ࣞࢗ࢘ࣇ
୘Ꮽᏽᛮ㸯 F ࣆ࢔࣭ࢺࣁࢴࢠ௛ࡀ (9 ࡞ࡻࡾࣄ࣭
࣑ᙁᗐࡡῼᏽࠊ㉝⥲࠿ࣞࢪࡡ࠵ࡾࢨࣘࢴࢹࠊࣁࣚ
ࢵ࢞࠿ኬࡀ࠷ࠊ G <(2 ࡚ῼᏽࡊࡒࣁࣤࢲἴᙟࠊ

ᅒ㸯⦢᪁ྡྷ࢙࣐ࢴࢰࣤࢪቌኬࠊ

5.1.2. ᶋ᪁ྡྷ㸝transverse㸞մᶋ᪁ྡྷ࣓࣭ࢺ⤎ྙ୘Ꮽᏽᛮ㸝㛏ࣁࣤࢲ࡚ࡢ㉫ࡀᶋ᪁ྡྷࡡࣄ࣭࣑୘Ꮽᏽᛮ࡞ࡢࠉ

࡞ࡂ࠷㸞ձࣞࣄࣤࢮࣤ୘Ꮽᏽᛮ

յ㞹Ꮔ㞴୘Ꮽᏽᛮ

ղ⤎ྙࣁࣤࢲ୘Ꮽᏽᛮ

࡝࡜࠿࠵ࡾ࠿ࠉࡆࡆ࡚ࡢࠉKEK-PS MR ࡡථᑏࣆճࣉࢴࢺࢷ࢕ࣜ୘Ꮽᏽᛮ

ࣚࢴࢹ࣍ࢹ࣑࠽ࡻࡦຊ㏷㛜ጙ௛㎾࡚ぜࡼࡿࡒỀᖲ᪁ྡྷࣉࢴࢺࢷ࢕ࣜ୘Ꮽᏽᛮࡡ⣺௒ࢅࡌࡾ[25]ࠊ

ᅒ㸯0*05827 ࡡࣉࢴࢺࢷ࢕ࣜ୘Ꮽᏽᛮࠊ

ᅒ㸯ຊ㏷୯ࡡ '52 Ềᖲ᪁ྡྷǼಘྒࡡ࿔ἴᩐࢪ࣋ࢠࢹࣜࠊ㏭୯࣭࣊ࢰࢹࣞࣤࢦ࢕ࢺࣁࣤࢺ࠿Ⓠ⏍ࠊᅒࠉᅒ࡞♟ࡊࡒࡡࡢࠉKEK-PS MR ࡡථᑏ㸭ᴗ㞹☚▴ࠉࣆ࢔࣭ࢺࣁࢴࢠ࣬ࢱࣤࣂ࣭࡞ຊ࠻࡙ࠉ

ࣆࣚࢴࢹ࣍ࢹ࣑࠽ࡻࡦຊ㏷ิ᭿࡞ふῼࡈࡿࡒỀᖲxy ࢜ࢴࣈࣛࣤࢡࢅ౐ࡖ࡙ࠉỀᖲ᪁ྡྷࡡ୘Ꮽᏽᛮࢅࠉ

᪁ྡྷࣉࢴࢺࢷ࢕ࣜ୘Ꮽᏽᛮ࡚࠵ࡾࠊࣄ࣭࣑ᙁᗐࡡ

ᆱ├᪁ྡྷࡡᏭᏽᛮ࡚Ꮽᏽ໩ࡌࡾ◂✪ࡵ࡝ࡈࡿ࡙࠷

఩ୖ࡛୘ᏭᏽᛮࡡⓆ⏍࠿ྜྷ᫤࡞ふῼࡈࡿ࡙࠷ࡾࠊ

ࡾ[26]ࠊ

BPM ࡡỀᖲ᪁ྡྷǼಘྒࢅࠉRF ࢠࣞࢴࢠࢅฦ࿔ࡊ
ࡒࢹ࡚ࣛ࢝㔔ࡠ᭡ࡀࡊࡒࡵࡡࡢࠉࣉࢴࢺࢷ࢕ࣜ୘

6. ࣄ࣭࣑୘ᏭᏽᛮࡡᏭᏽ໩

Ꮽᏽᛮࡡ඼ᆵⓏ࡝ᣪິࢅ♟ࡊ࡙࠷ࡾࠊྎ࠾ࡼ⟿ࡡࡥ࡛ࡒࡦࣄ࣭࣑୘Ꮽᏽᛮ࠿⌟ࡿ࡙ࠉ࢙࣐ࢴࢰࣤ

ᩐ࠿㸥ࠉ㸦ࠉ㸧ࡡ࣓࣭ࢺ࡚࠵ࡾࠊ࣓࣭ࢺ㸥ࡢࠉనࢪ࠿ቌኬࡊࡒࡽࠉࣄ࣭࣑ࣞࢪ࠿Ⓠ⏍ࡌࡾ࡛ࠉຊ㏷┞㐼⛛㸝ȚT㸞௧ୖࡡ࢙ࢾ࣭࡚ࣜ࢟ࡢࠉࢠ࣏ࣞࢷ࢔

ჹࡡᛮ⬗ࡢⴥࡊࡂ຋໩ࡊ࡙ࡊࡱ࠹ࠊࡐࡡࡒࡴࠉ୘

ࢨࢷ࢔࠿ḿ࡚୘Ꮽᏽ࡞࡝ࡾࠊỀᖲ᪁ྡྷࢠ࣏ࣞࢷ࢔

ᏭᏽᛮࡡᏭᏽ໩ࡢ㟸ᖏ࡞㔔さ࡚࠵ࡾࠊ୕オ࡚ࡵ࠷ࢨࢷ࢔ࡡῼᏽࡡ⤎ᯕࠉࡔࡺ࠹࡜࣓࣭ࢺ㸥ࡡቌኬ࠿ࡂࡗ࠾ࡡᏭᏽ໩ࡡᡥἪ࡞りࡿࡒ࠿ࠉࡆࡆ࡚ࡢࠉᶋ

⃥ࡊࡂ࡝ࡾࢰ࢕࣐ࣤࢡ㸯ຊ㏷㛜ጙ㸝P2㸞࠾ࡼ 80ms

᪁ྡྷࣆ࢔࣭ࢺࣁࢴࢠ࣬ࢱࣤࣂ࣭ࢅ⣺௒ࡌࡾࠊ

௛㎾࡚ࠉḿೋ࡞࡝ࡖ࡙࠷ࡒࠊࢠ࣏ࣞࢷ࢔ࢨࢷ࢔࠿

୘Ꮽᏽᛮ࡞ࡻࡖ࡙㉫ࡆࡾࢤࣃ࣭ࣝࣤࢹ࣭࣬࣊ࢰ

ᅒࡡࡻ࠹࡞᫤㛣࡛භ࡞ኬࡀࡂን໩ࡌࡾࡡࡢ୹

ࢹࣞࣤᣲິࢅ BPM㸝ࣅࢴࢠ࢓ࢴࣈ㸞࡞ࡻࡖ࡙ῼᏽ

࡞೩ྡྷ㞹☚▴ࡡࣄ࣭࣑ࣂ࢕ࣈ࡞ㄇᑙࡈࡿࡒῸ㞹Ὦࡊࠉࡐࡡᣲິࢅ΅ࡼࡌࡻ࠹࡞࢞ࢴ࣭࡚࢜㋶ࡖ࡙ࡷ࡞ࡻࡾ㸫ᴗ☚ሔ࡞ࡻࡾࡆ࡛࠿ゝ⟤ࠉᐁῼ࡚ࢂ࠾ࡖ

ࡾࡵࡡ࡚࠵ࡾࠊࣆ࢔࣭ࢺࣁࢴࢠ⣌ࡡ࿔ἴᩐᖈᇡࡢࠉ

ࡒࠊ㸫ᴗ㞹☚▴࡞ࡻࡽỀᖲ᪁ྡྷࢠ࣏ࣞࢷ࢔ࢨࢷ࢔

B ࣆ࢒ࢠࢹ࣭ࣛࠉᨲᑏකࣛࣤࢡ࡝࡜࡚᤿⏕ࡈࡿ࡙ࢅㇿೋ࡞ࡵࡖ࡙࠷ࡖࡒ࡛ࡆࢀࠉ࠾࡝ࡽᏭᏽ໩ࡈࡿ

࠷ࡾࡵࡡࡢࠉ⤎ྙࣁࣤࢲ୘Ꮽᏽᛮࢅࢱࣤࣈࡌࡾࡒ

ࡒࠊࡱࡒ࢕ࣤࣅ࣭ࢱࣤࢪ࣓ࢸࣜ࡞ࡻࡾ୘Ꮽᏽᛮࡡ

ࡴ࡞ 250MHz ࡱ࡚ࡡᗀᖈᇡࡡࡵࡡ࡚࠵ࡾ[27, 28]ࠊቌኬ⋙ࡵᐁ㥺࡛Ⰳࡂୌ⮬ࡊࡒࠊࡊ࠾ࡊࠉථᑏࣆࣚ
ࢴࢹ࣍ࢹ࣑ࡡࣞࢪ࠿Ⓠ⏍ࡊࡒࡒࡴ㸭ᴗ㞹☚▴ࡡࣚࣤࢱࢗ΅⾮࡞ࡻࡽᏭᏽ໩ࡊࡒࠊ

6. 1. ࣆ࢔࣭ࢺ ࣁࢴࢠ ࣬ࢱࣤ ࣂ࣭
BPM㸝ࣅࢴࢠ࢓ࢴࣈ㸞࡚ࡢࣄ࣭࣑㔔ᚨన⨠ xPUࢅ᳠ฝࡊࠉ࢞ࢴ࣭࡚࢜ࢣ࢕ࣤಶࡈࡿࡒಘྒ k xPU࡚࢞ࢴࢠǼx' = −k xPU ࢅ୙࠻ࡾࠊຝ⋙ࡻࡂࢱࣤࣈ
ࡌࡾ࡞ࡢࠉࡆࡡᥝാ࡚ࡢࠉࣅࢴࢠ࢓ࢴࣈ࠾ࡼࢱࣤ
ࣂ࣭ࡱ࡚࣭࣊ࢰࢹࣞࣤన┞ࡡ㐅ࡲ࠿ȧ/2 ࡡዃᩐಶ࡚࠵ࡾࡆ࡛࠿᮪௲࡛࡝ࡾࠊ㏳ᖏࡐࡡࡻ࠹࡞ཱིࡾ࡞
ࡢᅏ㞬࡝ሔྙ࠿࠵ࡾࠊࡐࡡሔྙࡢᅒࡡࡻ࠹࡞㸧ྋࡡࣅࢴࢠ࢓ࢴࣈࢅ౐࠹ࠊ

ᅒ㸯ࢠ࣏ࣞࢷ࢔ࢨࢷ࢔ࡡῼᏽࠊ
㯦୷㸯㸫ᴗን᭞๑ࠉⓉ୷㸯㸫ᴗን᭞ᚃࠉᐁ⥲㸯㸫ᴗን᭞๑ࡡゝ⟤ೋࠉ◒⥲㸯㸫ᴗን᭞ᚃࡡゝ⟤ೋ
CERN PS ࡚ ࡢ ࠉ ࡈ ࡼ ࡞ 㧏 ḗ ࡡ ࣓ ࣭ ࢺ
㸝4,5,6,7,8,10㸞ࡱ࡚ふῼࡈࡿ࡙࠷ࡾ[26]ࠊᏭᏽ໩ࡢࠉ

ᅒ㸯ᶋ᪁ྡྷࣆ࢔࣭ࢺࣁࢴࢠ࣬ࢱࣤࣂ࣭@Bࠊ࣓ࢸࣜࢅḗࡡࡻ࠹࡞⩻࠻ࡾࠊຊ㏷ჹ࡞Ἒࡖ࡙
sPU1ࠉsPU2ࠉsK ࡞ࠉ㡨࡞ࣅࢴࢠ࢓ࢴࣈ㸝BPM㸞"PU1",

"PU2", ࢞ࢴ࣭࢜"K"࠿⨠࠾ࡿ࡙࠷ࡾ࡛ࡌࡾࠊࣅࢴ
ࢠ࢓ࢴࣈࡡῼᏽన⨠ x1=x(sPU1)ࠉx2=x(sPU2)࠾ࡼ࢞

ࣄ࣭࣑㞹Ὦንన࠵ࡒࡽࡡಘྒฝງ㸝transverse
transfer impedance㸞ࢴ࣭࢜xK'=(sK)࡫ࡡࢣ࢕ࣤࢅࡐࡿࡑࡿ k1,k2 ࡛࠾ࡂ࡛ࠉ(x, x') ࡡንᥦࡢ㸝ຊ㏷ჹࡡ n ࿔ᅂ┘㸞ࠉ# x PU1 &%($ x"PU1 'n+1
= ) 0 M (sPU1 | sK )

!

!
!

IBx

,

#p$c
1
=
& e jks ( E" + v % B) K ds ,
eVK
VK s

࡛ࡡ㛣࡞ࡢࠉPanofsky-Wenzel ࡡᏽ⌦࡛ Lorentz ࡡ┞ཬᏽ⌦࡞ࡻࡽ
!

࡛࠽ࡂࠊs1 ࠾ࡼ s2 ࡡንᥦ⾔า㸯M (s2 | s1)'"2(cos #$ + %1 sin #$ )
)
"1
)
=
)
1
( (1+ %1% 2 ) sin #$ & (%1 & % 2 ) cos #$ ))&( "1" 2V࡛ࠉ࢞ࢴ࣭࡛࢜ࡊ࡙ࡡᛮ⬗㸝transverse kicker transfer
function㸞
K" =

,.
#x
& #
& 0.
0
*- M (sK | sPU1 )% PU1 ( + %( 1.
$ x"PU1 'n $ k1 x PU1 + k 2 x PU 2 'n 2.
/.

ࡆࡆ࡚ࠉ
x 2 = M (sPU 2 | sPU1 )11 x1 + M (sPU 2 | sPU1)12 x"1

ZP =ZP = "*,,,,"1!#$ ) ,
(cos #$ & % 2 sin"2+"1" 2 sin #$1
j k Z C K# ,2

ࡡ㛭౿࠿ᠺࡽ❟ࡗ[27,28]ࠊࡆࡿ࠾ࡼࠉ࿔ἴᩐ≁ᛮࡢᅒࡡࡻ࠹࡞࡝ࡾࡆ࡛࠿ࢂ࠾ࡾࠊ
|K⊥|

|Zp|

࠽ࡻࡦࠉȘ=0, șPU1=șPU2=șK=șࠉࣅࢴࢠ࢓ࢴࣈ
!

㸦࡛㸧ࡡ࣭࣊ࢰࢹࣞࣤన┞ᕣǼψ21=ȧ/2 ࢅ௫ᏽࡌࡾࠊࡈࡼ࡞ࠉ㸧ྋࡡࣅࢴࢠ࢓ࢴࣈ࠾ࡼ࢞ࢴ࣭࢜࡫
ࡡࣆ࢔࣭ࢺࣁࢴࢠ㔖ࢅ
k1 = k 0 cos " ,k 2 = k 0 sin " ,࡛᭡ࡂࠊࣆ࢔࣭ࢺࣁࢴࢠࡡ᭩㐲ೋࡢࠉୌ࿔ࡡንᥦ
!
!

⾔าࡡ≁ᛮ᪁⛤ᘟࡡึืᘟ=0 ࡡ࡛ࡀ࡚ࠉ
2sin "Ck0 ={ cos($"1K + % ) ± 1} .
# sin 2 ( $"1K + % )ࡐࡡ᫤ࡡ΅⾮⋙ࡢࣛࣤࢡ㸦࿔ᙔࡽ"+ = "# = 1 + $ k 0 sin(%& k1 # ' ) ,

!
!

0

ȧ

2ȧ 3ȧ
(a)

ȟ

0

ȧ

2ȧ 3ȧ
(b)

ȟ

ᅒ㸯ࢪࢹࣛࢴࣈࣚ࢕ࣤᆵࣅࢴࢠ࢓ࢴࣈࠊ F ࡛࢞ࢴ࣭࢜ G ࡡ࿔ἴᩐ≁ᛮࠊȟ"Ȱε [H ࠊ

7. ࠽ࢂࡽ࡞
ࡍ࠷ࡩࢆ೩ࡖ࡙⃨Ὲࡡ࠵ࡾහᐖ࡞࡝ࡖ࡙ࡊࡱࡖ

࡛࡝ࡾࠊψC ࡢࣛࣤࢡୌ࿔ࡡ࣭࣊ࢰࢹࣞࣤన┞ࡡ㐅ࡲࠉǼψK1 ࡢࣅࢴࢠ࢓ࢴࣈ࠾ࡼ࢞ࢴ࣭࢜࡫ࡡ࣭࣊ࢰ
ࢹࣞࣤన┞ࡡ㐅ࡲ࡚ࠉψC=Ǽψ1K+ǼψK1 ࡚࠵ࡾࠊࡆ
ࡡ⤎ᯕࡻࡽࠉࣅࢴࢠ࢓ࢴࣈ࡛࢞ࢴ࣭࢜ࡡన┞ᕣ࠿
ȧ/2 ࡚࡝ࡂࠉ࠷ࡂࡗ࡚࠵ࡖ࡙ࡵࠉțࠉࡌ࡝ࢂࡔ k1ࠉ
k2 ࡡ㔖ࢅㄢᩒࡌࡿࡣ᭩㐲࡝ࣆ࢔࣭ࢺࣁࢴࢠ㔖ࢅᐁ⌟࡚ࡀࡾࡆ࡛࠿ࢂ࠾ࡾࠊ᭩㎾ࡡࢸࢩࢰࣜ࣬ࣆ࢔࣭
ࢺࣁࢴࢠ࡚ࡢࠉన⨠ࡡᒓṌࡵ౐࠹ࡆ࡛࡞ࡻࡽ㸦ྋࡡࣅࢴࢠ࢓ࢴࣈ࡚ྜྷᵕࡡᛮ⬗ࢅᐁ⌟ࡊ࡙࠷ࡾ[27,28]ࠊ

ࡒࠊ௧ୖࡡᩝ⊡࡚⿭Ᏸࡊ࡙࠷ࡒࡓࡀࡒ࠷ࠊOHO ࢨࣅࢴࢠ࢓ࢴࣈ࡞ࡢࠉᚪさࡢ࿔ἴᩐᖈᇡ࡞ᚺࡋ࡙
㟴㞹ᆵࠉࢪࢹࣛࢴࣈࣚ࢕ࣤᆵ࡝࡜࠿౐ࢂࡿࡾࠊ࢞
ࢴ࣭࡛࢜ࡊ࡙ࡢࠉ50Ȑࡡ≁ᛮ࢕ࣤࣅ࣭ࢱࣤࢪ࡞ᩒ
ྙࢅ࡛ࡖࡒࢪࢹࣛࢴࣈࣚ࢕ࣤᆵ࠿ࡻࡂ౐ࢂࡿࡾࠊ
ࣅࢴࢠ࢓ࢴࣈ࡛ࡊ࡙ࡡࢪࢹࣛࢴࣈࣚ࢕ࣤࡡᛮ⬗㸯

ࡡ references ࡞ࡢࠉኣࡂࡡཋ඼࠿᥎㍍ࡈࡿ࡙࠷ࡾ

࣭ࣛࢫࢅጙࡴ࡛ࡊ࡙ࠉCERN Accelerator School(CAS), US Particle Accelerator School (USPAS,
AIP proc.)࡝࡜ࡡຊ㏷ჹࢪࢠ࣭ࣜࡡ proceedings,
BIW series, DIPAC series ࡝࡜ࡡࣄ࣭࣑ゝῼᑍ㛓ࡡ࣭࣠ࢠࢨࣘࢴࣈ proceedingsࠉAPAC, EPAC,
PAC,㸝ࡐࡊ࡙ 2010 ࠾ࡼጙࡱࡾ"IPAC"㸞, ຊ㏷ჹᏕఌ࡝࡜ࡡ proceedings ࡝࡜ࢅཤ↯ࡊ࡙࠷ࡒࡓࡀ
ࡒ࠷ࠊᩅ⛁᭡ࡵⰃ᭡࠿ฝ࡙࠷ࡾ[10, 30]ࠊ୕オᩝ⊡ࡡ࡚౼ฺ࡚࠵ࡾࠊ
ᮇㅦ⩇㘋࡚᥎㍍ࡊࡒᠺᯕࡢࠉཤ⩻ᩝ⊡࡞᥎㍍ࡈ
ࡿࡒ◂✪࠾ࡼࡡᘤ⏕ࠉ࠵ࡾ࠷ࡢභྜྷ◂✪࡚ᚋࡒࡵࡡ࡚࠵ࡾࠊࡆࡆ࡚វㅨࡌࡾࠊୌ᪁ࠉ㛣㐢࠻ࡷㄕ᪺୘㊂ࡢ㈈௴ࡢⴥ⩽࡞ᖉࡌࡾࠊ⮤ࡼࡇ☔ヾࡡ࠹࠻࡚౐⏕ࡊ࡙࠷ࡒࡓࡀࡒ࠷ࠊ
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㛏 ཿᒁ ㈏❮Ắࠉᖲᮿ ᠺ⠂Ắ࡞វㅨࡊࡱࡌࠊ

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ཤ ⩻ ᩝ ⊡

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