ʍʋLongitudinal Equation of Motion

陽子シンクロトロンの高周波
加速ʍ Ļ ƣ YSeFòûBn w/ɚÒFȻ
5B[3]8*Ļƣǽ/ĝȑF-Ļƣ s Fn w/ɘȣžðn w/¡ȐĻƣ(x, y)0!C"CƳĶąǟžð/ĻƣFȥɘȣžðFð)ê³žF!C"CƳĶąǟžð/ plus +Bàreference orbit 0ĶăÚņ R [m] /¾ɉɜ+ȒB 0;. ȠĈǡƗƹƓ.ļȴC#ēŁĺɰėÏɖúƀȴJ-PARCʉJapan Proton Accelerator ResearchComplexʊ0źĘɕAƂƑ/9-@ǖFȥ
BɰėÏɖúƀȴ (Figure 10)* BÏɖú0400MeV ɰ ė ~ h I a S  3GeV Rapid cycling ʍ ȡ ɹ ǻ ė / ɚ Ò ž ǭ Ľ ɹǦċ EB /Fɖĺʜ*ɚÒBȡɹǻėsynchrotron (3GeV RCS) á 2 (50GeV) Mainʉɹȡeʊ0
Synchrotron (MR)*ơřCBɰė~hIaS+(2.2.1)
𝑑𝑑𝑝𝑝/𝑑𝑑𝑑𝑑 = 𝑒𝑒 𝐸𝐸 + 𝑣𝑣 × 𝐵𝐵
RCS025Hz/ƃȏAɑ*ɚɊCRCS@/3GeVɰėn w0ǊɀǑõǩĚƀȴʉMLFʊ
/ÍFåB*/ŔėɫėǑřá2ƒŻǈĥȵʆ.Çǒ*energy gain dE/dt FŊB#;.ɖĺCB8#!/ɡ0MR4©ȄCkf +Í/½ǰFȒBĞʆƀȴ+h{ e~jƀȴ*ÇǒB#;!
𝑑𝑑𝑝𝑝
𝑑𝑑𝑑𝑑
/NiR 030GeV8*ÏɖCBÏɖú/ɚ
=v∙= 𝑒𝑒𝑣𝑣 ∙ 𝐸𝐸 (2.2.2)
𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑
ɊWKS0kfĞʆƀȴ+h{ e~jƀ*ɹċ EC 0ɘȣžðʉlongitudinal:ȍ
ȴ.?AǙ-AǏĂkfĞʆƀȴ40 5.20
Ǫ*“əäAÂʉSXʊ”h{ e~jƀȴ40 žðʊ/ɹċ+Ǧċ/ƆɫďÖ.?BɁɹÍċ
2.48Ǫ/ȏAɑ*“ɖäAÂʉFXʊ”ȣEC EB /ö*ȥCB
𝐸𝐸 = 𝐸𝐸! + 𝐸𝐸!
ēŁĺɰėn wF©Ȅ)B
Ǵ 2 ʀ Electromotive force E B 0 Maxwell /ĽYSe/ȴȳ*0łƆ”Ē/100c}q @e”+ȲEC#m[ł#A10^14ppp/ēŁĺ
(2.2.3)𝑟𝑟𝑟𝑟𝑟𝑟 𝐸𝐸! = − 𝜕𝜕𝐵𝐵 𝑑𝑑𝑑𝑑
ɰėn wF[ǆÏɖB#;.γtp~ }cJ[AC}KɼïƐ^KvTY[cw=Ï*ȳǶCBɖǲȔY[cw.pM}KeǦŔ£Fǒ#ïȻ
ĆǲȔ*0-ɦĭǦŔ£Fǒ)ʇÏɖɹċÔ ʍ .3 ʇ ò Ʒ ɹ ċ E cɢFĞǏǆïȻÖ#+-,'/ũ*0ȡɹǻėʉ*0.ɰė: pʊ/śǜ-IKdIFäA¹C#[1]
ÏɖRzaqFɕɛBƆFȒBRzaqɹ
3GeV RCS*02015ķǨ/Ïɖȵʆ.)ȴ ā V [volt]RzaqɫɅɶ g [m]+B
(2.3.1)
𝑉𝑉 𝑡𝑡 = 𝑉𝑉! ∙ sin 𝜔𝜔𝜔𝜔ȳn wm 1MW/n wÏɖȻŹFĜ#žMR*0äAÂNiR
30GeV .   #  + <F50GeV@

Aȴȳn

wm

740kW/ĞǏFMRWKS/Ưɱǜ-ǢȎÖ*ɘȣžð/ɹċ Ec ă* B+B+ǻėÏɖRzaqFɕɛBƆ.ŊBNiR
ΔE(eV)0!

ĞǏBśǘ.žɧɊű)ɘ;)BǏĂ
MRWKS0ȴȳ±3.60Ǫ@2.48Ǫ.8*Ǣ
Ȏ500kWɚɊFĞǏ)B@-Bnmw

čŁ/#;.0MRWKSF1.3Ǫ.8*

ǢȎFÿ@-C1-@-[2]!/#;ʇȏAɑÖ.ģő#ɹǦǣɹǂ/Ȑű+RFY[cw/čŁFɘ;)B!
𝑒𝑒𝑉𝑉!
sin 𝜔𝜔𝜔𝜔 𝑑𝑑𝑑𝑑
Δ𝐸𝐸 =
𝑔𝑔 !!
!

(2.3.2)*s 0~Tɘȣžð/Ʌɶω 0ÏɖʇòƷ/ȰòƷŸ* Bǻė0ɫɲ g (m)/ÏɖRzaqFɕɛB/.Ǝɮ/Ɔɫ 𝛥𝛥𝛥𝛥(= 𝑡𝑡2 − 𝑡𝑡1) ŭBn w/
òûíòƷŸ𝜔𝜔! +B+𝜔𝜔𝜔𝜔 = 𝜔𝜔! /𝑣𝑣 + 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐.*
B@!CFĽ(2.3.2).¹)

３－ 1sin 𝑎𝑎 2(2.3.3)𝑎𝑎
2
 FŊBĽ/ a 0reference orbit /ō@ÏɖRzaq g FȬBōȰFȥ / Ľ (2.3.3)į ɍ / Ŧ ŀ [ ]* Ŧ & # ɡ Ã F
transit time factor +ô2n wÏɖǲȔF
ɕɛBɫ.åBĞÑǜ-NiR Fǧ¬Ÿ* BɰėYSe/ċîÏɖǲȔ/ɩ0 2m ǭĺ(ÏɖRzaq/ɩ*0Ȟ.ǲȔ0ȪŸ/ÏɖRzaq*ơřC)B)ĨĆ/ KEK-PS Booster [4]/ċî*<òɩ0 40m ǭĺ* B@ transit time factor> 0.995 .-Aƭ,/ċî1 +Ȓ)?
#&)ǻėÏɖRzaqFɕɛBƆ.ŊBNiR ΔE(eV)0(2.3.4)
Δ𝐸𝐸 = 𝑒𝑒𝑉𝑉! ∙ sin 𝜔𝜔𝜔𝜔*ȥCB
Δ𝐸𝐸 = 𝑒𝑒𝑉𝑉! ∙ sin 𝜔𝜔𝜔𝜔

ʍ .4 Ï ɖ ò Ʒ Ÿ + k y h a S Ÿǻėɖĺ v [m/s] *òɩ C [m= 2𝜋𝜋𝜋𝜋R0ĶăÚņ]/YSeʉ~TʊFòû)B+BòûòƐ To [sec]òûòƷŸ fo[Hz]0
òƐ 𝑇𝑇! = C/𝑣𝑣 = 1/f!

(2.4.1)* @ECBȡɹǻė~T/ÏɖǲȔFɕɛBĺ.ǛǑBʇòƷɹāFŗBȫ.C1?@ǛǑ BÏɖɹċ/òƷŸ+òûòƷŸ
/ɫ.0ŹŸÊǑB

/ŹŸｈFʇòƷ/k yhaSŸ
(harmonics or harmonic number)+ô3
Figure 1 .ǧ?.k yhaSŸ/Ÿ$ȍžð/ɚÒFìBʇòƷ¡ǠęĂB
n wF?AʇNiR 8*ÏɖB#;.YSeFP[U f.B+@>Bǅ*Ñǎǜ* B!/ċîʍ'/YSeɫ*n wåƾȣECBȍžð¡Ǡǲɫ(ňɒ)/ua`TFȒBŎȫBʍ'/~T/òɩ CÏɖòƷŸ frfkyhaS[Ÿ ℎ+!C"C.ʍ'/~TF×ÆB#;.ƼĘF'Bn wåƾƆ/ua`T/ƕ+)òƷŸǵ+
B+𝜔𝜔!"! = 𝜔𝜔!"!
→ ℎ! ∙ 𝜔𝜔!" = ℎ! ∙ 𝜔𝜔!"
→ ℎ! ∙ 𝑣𝑣 𝐶𝐶 = ℎ! ∙ 𝑣𝑣 𝐶𝐶!"!"*n w/ɖĺ v 0ï* B@ℎ!
ℎ!
→𝐶𝐶!" =𝐶𝐶!"
#&)!C"C/YSe/òɩ
Ʊ0!C"C/k yhaSŸ/Ʊ.ǵBŎȫ B
YSe~T/k yhaS[ŸF@.B0n wÇǒȓ,/?-ƆɫơɗFŧ&#n wFĞʆƏ8C)B
ɣȫ*Ûǿ*0-@.!/*k
fLMI ʉǲȔʇòƷǂQaP -,ʊĞǏ*B/ȺʂƮBǋ.100MeV /ɰėYSe*0ɖ v +ʇɖ c /Ʊβʔv/cʓʌ*NiR .ő)ďÖB
'8ANiR ʉɚÒɥʊ+».ÏɖòƷŸFďÖ -C1-ÏɖòƷŸèďFĞǏ
B#;/k fLMI /Éɮ<ʇòƷkyhaSŸ/ɠţ.ńɾF)BJ-PARC/ 2 '/YSe0òɩƱ 1 : 4.5 *Aƌȁǜ.!C"C/k yhaSŸ02 á
2 9 .ɠţC#[1]
ʍ .5 Ï ɖ . Ŏ ȫ - ɹ āȡɹǻė/¾ŃɚÒFìB/0²ðɹǦǣ/Ǧċ B [T] +Ǧǣ/ƈǎÚņ ρ[m] +B+ɚÒɥ p +/ɫ.
𝐺𝐺𝐺𝐺𝐺𝐺= 𝐵𝐵𝐵𝐵 (2.5.1)
𝑝𝑝 = −e𝐵𝐵𝐵𝐵 𝑜𝑜𝑜𝑜 3.3𝑝𝑝
𝑐𝑐

Figure 1: RF harmonics
𝑓𝑓!" = ℎ ∙ 𝑓𝑓!

(2.4.2)/ɬ¬ B* e 0ɰėÏɖFŕĝ!/Ȁɹȡ+#
8#Ľ (2.2.2) @
1 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑=
2.5.2
𝑑𝑑𝑑𝑑 𝛽𝛽𝛽𝛽 𝑑𝑑𝑑𝑑
*𝐸𝐸 = 𝛾𝛾𝑚𝑚! 𝑐𝑐 ! 0ºNiR

３－ 2

*

Bmo

ɰė/ɻƪɀɥĨźĘ c 0¸ɖ+BžĽ (2.3.4) @ǻė~TFʌòBɫ.ŊBNiR 0Δ𝐸𝐸 = 𝑒𝑒𝑉𝑉! ∙ sin 𝜔𝜔𝜔𝜔 *!/Ķă±0
Δ𝐸𝐸 𝑒𝑒𝑉𝑉!
=∙ sin 𝜔𝜔𝜔𝜔
(2.5.3)
Δ𝑇𝑇 𝑇𝑇!"#ŉ&)Ľ (2.4.1) +Ľ (2.5.2)@𝑑𝑑 𝐵𝐵𝐵𝐵
(2.5.4)𝑉𝑉! ∙ sin 𝜔𝜔𝜔𝜔 = 2𝜋𝜋𝜋𝜋𝑑𝑑𝑑𝑑
Ǧċ/ƆɫďÖ.Ŏȫ-ÏɖɹāFŊBVo 0ǻė~TFòBɫ.åBƌē/Ïɖɹā [ volt/turn ] * B*Ïɖɹā~T/ē+Ǧċ/ƆɫďÖ/ƌē±/ǰ.Ʊ¨Ƶ8B+0ēďɣȫ* BŎȫÏɖɹāFŊB#;.ÏɖǲȔ/éŸFč=!
+œ&)<[r [-)č= -ċî
 B~TFē)ǲȔ/ċŜǥ­*
C1ȟ!$!/#;.Ïɖɹā@.
Ŏȫ.-B+FŐC)0-@-ʍ .6 Synchronous ǻ ė  Ë Ǹ / Ľ (2.5.4)  @ Ǧ ċ / Ɔ ɫ ď Ö ɥ
dB/dt +ƌēÏɖɹā Vo [volt/turn]Ƶ8C1
𝜙𝜙＝ sin!!

２𝜋𝜋𝑅𝑅 𝑑𝑑𝑑𝑑
𝜌𝜌
≡ 𝜙𝜙!
𝑒𝑒𝑉𝑉!
𝑑𝑑𝑑𝑑

(2.6.1)

Ľ 2.6.1 *Ƶ8BʇòƷ/¡Ǡ.Bǻė0ưûï¡Ǡ.Ǘ8B/¡ǠFïƐ¡Ǡsynchronous phase +/ïƐƕFǁ#)ɚÒBǻėF synchronous particleʉǻėʊà0reference particle +ô2/ǻė0Ĵ.ʇòƷ+ïƐ)B
ʍ .7 e }  Z Y |  N i  R ʒ γ t
YSg[ǻė0Ïɖú/dXKɕA.
ɚÒBǻė/+*Ǧċ/ōɉɜFɚÒ
Bƭ,/ǻė0YSg[ǻė/òA.ǙBɚÒɥFŧ&)ÃĲ)BɉɜFìB²ðɹǦǣɚÒɥÃŷFŧ'#;.!C@/ɉɜ
0ōɉɜ@CB
𝑑𝑑𝑑𝑑
𝛿𝛿𝛿𝛿
= 𝛼𝛼!
𝑝𝑝
𝐶𝐶

(2.7.1)Ľ (2.7.1).ǧ?.ɚÒɥ/C+ɉɜ/ɩ /  C F ȥ  ɥ +  ) momentum
compaction: αp ħ¹CB

𝛼𝛼! =𝑋𝑋! 𝑠𝑠 𝑑𝑑𝑑𝑑

(2.7.2)

𝑋𝑋! (𝑥𝑥)0}cJ[/ÃŷɬŸ𝛼𝛼! 0ÃŷɬŸ/~
Tò/ǰÃɥ/ōɉɜ.ģBÌî*ȥCBɚÒɥ+ɖĺ v /ɫ.0
𝑝𝑝 = 𝑚𝑚𝑚𝑚 = 𝑚𝑚! 𝛾𝛾𝛾𝛾ｃ𝛿𝛿𝛿𝛿 = 𝑚𝑚! 𝛽𝛽𝛽𝛽𝛽𝛽𝛽𝛽 + 𝑚𝑚! 𝛾𝛾𝛾𝛾 𝛿𝛿𝛿𝛿𝛿𝛿𝛿𝛿/𝑝𝑝 = 𝛿𝛿𝛿𝛿/𝛾𝛾 + 𝛿𝛿𝛿𝛿/𝛽𝛽＝1/𝛽𝛽 ! ∙ 𝛿𝛿𝛿𝛿/𝛾𝛾𝛿𝛿𝛿𝛿/𝛽𝛽＝1/𝛾𝛾 ! ∙ 𝛿𝛿𝛿𝛿/𝑝𝑝-Bɬ¬ Aǻė/Ȱɖĺ𝜔𝜔! +ɚÒɥ p /ɫ.0𝑇𝑇 = 𝐶𝐶/𝑣𝑣F§&)δω/ω! ʔ − δT/T! = −δC/C! + δ𝑣𝑣/𝑣𝑣!
1 𝛿𝛿𝛿𝛿𝛿𝛿𝛿𝛿= − 𝛼𝛼! − ! ∙(2.7.3)𝛾𝛾𝑝𝑝𝜔𝜔!

FŊB*ŦŀʉʊF slippage factor +ô2
1𝜂𝜂 = 𝛼𝛼! − !
(2.7.4)𝛾𝛾*ĝȑB
Slippage factor 0n w/NiR (2.7.5)𝛾𝛾 = 1𝛼𝛼!

*].-B/+/NiR FɟǬNiR ʉtransition energy: 𝛾𝛾! ʊ+ô3
n w0𝛾𝛾 = 𝛾𝛾! *WKSe/?.ǵ
ƆŔ­#CBÏɖʁĉ.ɟǬNiR ęĂB+nwěĝŔ=Ïɖ/#;/¡ǠÉŋ/#;ēŁĺÏɖú*0n wŲĕøʂ.-B!/#;ƌɐ/ȴȳ*0γt FÏɖĉ@Đ#A
αp FȽ/±.Bʉimaginary γtʊ-,/ĮĔ
C#J-PARC ēŁĺɰėÏɖúǯµĖ;#[4]
ʍ .8 ¡ Ǡ ě ĝ Ŕ / Þ ǐYSe*ÏɖCBǻė0NiR.1@'ƎAÏɖ.Ŏȫ-ʇòƷ.ģB¡Ǡ<ĝ*0-'8A2.6 *ɒ5#Y
Sg[ǻė/òA.ÃĲ)B!*ǺÛ/#;.NiR ÏɖòƷŸFĝ+)YSg[ǻė/òA.ǻė/ɚÒFȒ
B~Tɉɜ/ʌPŜ.ʇòƷÏɖǲȔ
AĽ(2.3.1)/Ïɖɹā V(t)３－ 3

𝑉𝑉 𝑡𝑡 = 𝑉𝑉! 𝑠𝑠𝑠𝑠𝑠𝑠 𝜔𝜔𝜔𝜔ǛǑ)B

(2.8.1)8Ǧċĝ*Ïɖ-ƆYSg[ǻė0ÿ/ a ǅà0 b ǅ/ʇòƷɹā/,%@/ zero S[.Ǘ8B
!*YSg[ǻėʉa à0 bʊ+ɚÒ
ɥ/Ǚ-Bǻė/ɚÒFƩ.ȒBFigure 2 /Þǅ a /ʇòƷ¡Ǡ. Bǻė*ɚÒɥYS
g[ǻė?Aēǻė/ɚÒFȒBɚÒɥ/ēǻė0YSg[ǻė?A·.ǲȔ.ÈɝB#;ĩ'ƽɖC'YSg[ǻė/ɚÒɥ+ǵ-B!/Ɔǅ*YSg[ǻė+/¡Ǡİ0ƌē.-B@.ǻė0ƽɖCB#;ĺ0Y
Sg[ǻė?AəÏɖRzaq.ÈɝB+.-AYSg[ǻė+/¡Ǡİ0ɓ.Ȏ8&)/?.YSg[ǻė+ɚÒɥǙ-Bǻė0Ɔɫǜ.YSg[
ǻė/òAFȣ&#AƖ#AB+.-B
<'/ǅʉFigure 2 b ǅʊ/ǻė0,B* Dʖĺ0ɚÒɥ/ēǻė0@.ÏɖɹāFŗB#;?AɚÒɥ
-BɚÒɥ.Ĩǻė0òûBĺ.?
AɚÒɥĨ-B'8Ab ǅɐ/YSg[ǻė@ɚÒɥFǙ.Bǻė0Ĵ.
ěĝ-ŪBȜFB+.-BFigure 2 .
ǧ#ʇòƷ¡Ǡ/ŸĚǜ- 2 '/Òǅ/½  1 ' 0 stable fixed point  <   ' 
unstable fixed point +ô1CBʇòƷ.?BÏɖ*0ɚÒɥ=¡Ǡ/ďÖěĝ.-BƕęĂ/+FYSe.ſB¡ǠěĝŔ/Þǐ+)ǐȱC)B
ʍ .9 Y  S  e   ɚ Ò / ž ǭ ĽYSg[ǻė+ɚÒɥá2ÏɖǲȔ.È
ɝ#+/ʇòƷ¡ǠǙ-BǻėʉɼïƐǻėʊ/ɚÒ.')ĩȶȬBɼïƐǻė/m}x ^F.ĝȑB

Figure 2: ěĝ¡Ǡ+ěĝ¡Ǡ𝜔𝜔＝𝜔𝜔! ＋𝛥𝛥𝛥𝛥、𝑝𝑝＝𝑝𝑝! ＋𝛥𝛥𝛥𝛥、𝜙𝜙＝𝜙𝜙! ＋𝛥𝛥𝛥𝛥
𝐸𝐸＝𝐸𝐸! ＋𝛥𝛥𝛥𝛥、𝜃𝜃＝𝜃𝜃! ＋𝛥𝛥𝛥𝛥
*𝜔𝜔, 𝑝𝑝, 𝐸𝐸, 𝜙𝜙, 𝜃𝜃 0!C"CòûȰɖĺɚÒɥNiR ¡Ǡ~T/ǻė/ōȰʉɉɜȰʊFǧƼĘ “𝑠𝑠”, “𝑜𝑜” 0ōǦċFɚÒBǻėà0YSg[ǻė/ŧ'±FǧʇòƷ/¡Ǡ 𝜙𝜙+ɉɜȰ 𝜃𝜃 0ʇòƷ/k y
haS[F ℎ+)
𝛥𝛥𝛥𝛥＝𝜙𝜙 − 𝜙𝜙! ＝ − ℎ𝛥𝛥𝛥𝛥 +-BȰɖĺ ω 0𝜔𝜔＝𝑑𝑑𝑑𝑑/𝑑𝑑𝑑𝑑 -/*
1 𝑑𝑑𝑑𝑑𝑑𝑑
𝛥𝛥𝛥𝛥 = −𝛥𝛥𝛥𝛥 =ℎ 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑
Ľ (2.5.4) @ Vo→V +)òû #A.nwŊBNiR 0 𝑒𝑒𝑒𝑒 ∙ sin 𝜙𝜙 $@òûòƷŸ𝜔𝜔＝2𝜋𝜋𝜋𝜋F§&)energy gain
𝜔𝜔
𝑑𝑑𝑑𝑑
=
𝑒𝑒𝑒𝑒 sin 𝜙𝜙(2.9.1)
𝑑𝑑𝑑𝑑 2𝜋𝜋FŊBǋ.YSg[ǻė.ģBĽ(2.9.1)0
𝑑𝑑𝐸𝐸! 𝜔𝜔!
=𝑒𝑒𝑒𝑒 sin 𝜙𝜙!
𝑑𝑑𝑑𝑑
2𝜋𝜋+ƊB@
𝑑𝑑𝑑𝑑
𝑑𝑑𝐸𝐸!
1 𝑑𝑑
𝑑𝑑𝑑𝑑 Δ𝜔𝜔
𝑑𝑑 Δ𝐸𝐸
=
=
Δ𝐸𝐸 =
≈
!
𝜔𝜔𝜔𝜔𝜔𝜔 𝜔𝜔! 𝑑𝑑𝑑𝑑 𝜔𝜔! 𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑 𝜔𝜔! 𝑑𝑑𝑑𝑑 𝜔𝜔!

#&)YSg[ǻė+ɚÒɥ/ǙBǻė.ģ)
1
𝑑𝑑 Δ𝐸𝐸=𝑒𝑒𝑒𝑒 sin 𝜙𝜙 − sin 𝜙𝜙!
(2.9.2)2𝜋𝜋𝑑𝑑𝑑𝑑 𝜔𝜔!
FŊBž¡ǠȰ𝜙𝜙/ƆɫďÖ0
𝑑𝑑𝑑𝑑= −ℎΔ𝜔𝜔 (2.9.3)𝑑𝑑𝑑𝑑
𝜔𝜔＝𝛽𝛽𝛽𝛽/𝑅𝑅@Δ𝜔𝜔 𝜔𝜔 − 𝜔𝜔!
𝜔𝜔
𝛽𝛽𝑅𝑅!
=
=
−1=
− 1 (2.9.4)
𝜔𝜔!
𝛽𝛽! 𝑅𝑅𝜔𝜔!
𝜔𝜔!

ɚ Ò ɥ /  C + ò ɩ / ɬ ¬  @ momentum
compaction F§&)
𝑑𝑑𝑑𝑑
𝑅𝑅 = 𝑅𝑅! 1 + 𝛼𝛼! ∙𝑝𝑝!  F¹B+Ľ(2.7.3)𝛿𝛿𝛿𝛿
1 𝛿𝛿𝛿𝛿
= − 𝛼𝛼! − ! ∙
𝜔𝜔!
𝛾𝛾
𝑝𝑝
FŊB
(2.9.3)+ (2.7.3)@¡Ǡ.ɬ)

３－ 4

𝑑𝑑𝑑𝑑
Δ𝑝𝑝
ℎ𝜔𝜔!! Δ𝐸𝐸
= 𝜂𝜂ℎ𝜔𝜔!
= 𝜂𝜂 !
(2.9.5)𝛽𝛽 𝐸𝐸 𝜔𝜔!
𝑑𝑑𝑑𝑑𝑝𝑝!
/žǭĽFŊB ďŸ(𝜑𝜑、𝛥𝛥𝛥𝛥/𝜔𝜔𝑜𝑜)0YSeɚÒFȥ¡Ǡǲɫ/ďŸ* AĽ(2.9.2)+(2.9.5)0YSeɚÒFȥžǭĽ* AɼïƐǻė/¡Ǡ+NiR ïƐ¡Ǡ+ōNiR /òAFŪÒBɚÒʉYSeŪÒʊFǧ)B
ʍ ．.10 Sm all am plitude synchrotron
oscillation
 YSeɚÒ/žǭĽ.ſ)m}x^Ɔɫǜ.ēďÖ ¡ǠŪÒ/ŪĵĨċî𝑠𝑠𝑠𝑠𝑠𝑠(𝜙𝜙) − 𝑠𝑠𝑠𝑠𝑠𝑠(𝜙𝜙! )〜𝑐𝑐𝑐𝑐𝑐𝑐(𝜙𝜙! )𝛥𝛥𝛥𝛥
@.
𝑑𝑑𝑑𝑑 𝑑𝑑 𝜙𝜙! + Δ𝜙𝜙
𝑑𝑑 Δ𝜙𝜙
=
≈
𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑

F§&)Ľ(2.9.5)FĽ(2.9.2).¹B+
𝛥𝛥𝛥𝛥.')/ŪÒ/ŌÃžǭĽFŊBℎ𝑒𝑒𝑒𝑒Ω!! 𝜂𝜂 cos 𝜙𝜙!
𝑑𝑑 ! Δ𝜙𝜙=Δ𝜙𝜙 = 0 (2.10.1)
𝑑𝑑𝑡𝑡 !
2𝜋𝜋𝐸𝐸!
*𝛺𝛺! ＝𝑐𝑐/𝑅𝑅 = 𝜔𝜔! /𝛽𝛽! 0ɖĺ𝑣𝑣 = 𝑐𝑐ʉʇɖ)/Ɔ/ȰŪÒŸFȥĽ(2.10.1)?AYSeŪÒ/ěĝƕ:𝜂𝜂 cos 𝜙𝜙! < 0 McMillan + Veksler .?AǧC#[6]
nw/NiR e}ZY|NiR
𝛾𝛾! ?A¢à0ʇċî*ěĝ¡Ǡ/ƕďÖB   𝛾𝛾＜ 𝛾𝛾! → ０＜ 𝜙𝜙! ＜𝜋𝜋/2   𝛾𝛾＞ 𝛾𝛾! → 𝜋𝜋/2＜ 𝜙𝜙! ＜𝜋𝜋/žǭĽ/ŪÒŸΩ! = Ω!

ℎ𝑒𝑒𝑒𝑒 𝜂𝜂 cos 𝜙𝜙!
2𝜋𝜋𝐸𝐸!

FYSeŪÒŸ+ô2𝛺𝛺! F𝛺𝛺! *ɯ#ɥF synchrotron tune: 𝑄𝑄! +ô3
Q! =

Ω!
=
Ω!

ℎ𝑒𝑒𝑒𝑒 𝜂𝜂 cos 𝜙𝜙!
2𝜋𝜋𝐸𝐸!

.')¼Ćǜ-ɚɊǌŘ*/YSeɚÒ/ŪÒŸFȳǶTable-1 .ĊƑm}x^F8+;Bȥ/Ïɖɹā0¹ĥ?2Â
ĥƆ/±Fǧ)B
 J-PARC /¨*<ÃB?.YSe`{ 0 10-3 ǭĺ* BC0ǻėʌY
SeŪÒB/. 1000 ^ 6,ŭB+FŖó)B
Table  1 YSeĊƑm}x
NiR
ÏɖòƷŸ
Ω! !𝑐𝑐!𝑅𝑅 !
ÏɖɹāHarmonics
Transition γt
Slippage γt
Ωs/2π
Qs

^ (1)

RCS
400MeV~3GeV
1.23~1.67MHz
5.4E6 [1/sec]

MR
3GeV~30GeV
1.67~1.72MHz
1.2E6 [1/sec]

60kV/ 150kV
2
9.14-0.48/ -0.045
2.3kHz/630 Hz
0.004/ 0.0008

150kV/ 260kV
9j32.5-0.058/ -0.0013
340Hz/ 29Hz
0.002/ 0.00015

ʍ .11 Ham iltonian ƫ ǃ ď Ÿ . 𝜙𝜙、ｗʉ = 𝛥𝛥𝛥𝛥/𝜔𝜔! ʊ F ɠ 2 
Hamiltonian: H(φw)FȒB
𝑑𝑑𝑑𝑑 𝜕𝜕𝜕𝜕 𝜙𝜙, 𝑤𝑤
=
𝑑𝑑𝑑𝑑
𝜕𝜕𝜕𝜕
𝑑𝑑𝑑𝑑
𝜕𝜕 𝜙𝜙, 𝑤𝑤
=−
𝜕𝜕𝜕𝜕
𝑑𝑑𝑑𝑑

Ľ(2.9.2)+Ľ(2.9.5)@YSeŪÒ/
Hamiltonian 0
1 ℎ𝜂𝜂Ω!! !
𝑤𝑤
𝐻𝐻 𝜙𝜙, 𝑤𝑤 =
2 𝐸𝐸!
𝑒𝑒𝑒𝑒
+
cos 𝜙𝜙2𝜋𝜋+ 𝜙𝜙 sin 𝜙𝜙! (2.11.1)*ȥE+*B  ʇòƷɹā 𝑒𝑒𝑒𝑒 𝑠𝑠𝑠𝑠𝑠𝑠(𝜙𝜙)¦B potential á2YSeŪÒ/ɉɆ trajectory FȒB
Ľ(2.11.1)/êɍǴʍʀʇòƷ¡Ǡ𝜙𝜙.ģB
potential F ȥ   ¼ Ć ǜ - ï Ɛ ¡ Ǡ 𝜙𝜙! =0 𝑎𝑎𝑎𝑎𝑎𝑎 30∘ .ģ)ȳǶB+/ Fig. 3 /?
.-B

.J-PARC FơřB 2 '/YSe

３－ 5

Figure 3: ʇòƷɹā¦B Potential
Figure 3 /ȍɋ0 potential energy * B/*ȉ/ǅȊ+ɺ/ potential ƈȊ.þ8C# bucket
Ńǌ/ʁĉěĝʁĉ.-&)B+ǐȱ*?8#!/ŃʇòƷ/ěĝʁĉFl
Ub+ô3ǔƖ*< Bʍ .12 Separatrix  ʇòƷ/m}x ^* Bɹā:V =ïƐ¡Ǡ: 𝜙𝜙𝑠𝑠 Ĩ+n wòû #A.ŊB
NiR 0Ĩ-Aslippage factor: η /ďÖ<Ĩ/?-ƕ*0YSeŪÒ/ Hamiltonian <ƆɫªęŔĨżǉǜ-ɚÒ+)Ȓ@CB
1 𝑑𝑑𝑇𝑇!
1 𝑑𝑑𝜔𝜔!
=≪1
𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 = !
2𝜋𝜋 𝑑𝑑𝑑𝑑
𝜔𝜔! 𝑑𝑑𝑑𝑑
kvehI/ƆɫªęŔĨ+)Y
SeŪÒF<ĩȶȬ)
Ľ(2.11.1)F@.Ɗ#+
1 ℎ𝜂𝜂Ω!! ! 𝑒𝑒𝑒𝑒𝑤𝑤 +cos 𝜙𝜙 − cos 𝜙𝜙!
𝐻𝐻 =
2𝜋𝜋
2 𝐸𝐸!
+ 𝜙𝜙 − 𝜙𝜙! sin 𝜙𝜙!
(2.12.1)

unstable fixed point * A/òA/ɉɆâƈȊ.-B+@ hyperbolic fixed point +ô1CB8#unstable fixed point FɕBɉɆF Separatrix +ô2¡Ǡǲɫ.ſBěĝʁĉ/ČǖFȥěĝʁĉ½/ǻė0stable
fixed point /òAFYSeɚÒ
bunch +ô3ɵýFŃřB
𝑒𝑒𝑒𝑒
Γ 𝜙𝜙!
(2.12.2)
𝐻𝐻!" =
2𝜋𝜋
*
Γ 𝜙𝜙! = −2 cos 𝜙𝜙! + 𝜋𝜋 − 2𝜙𝜙! sin 𝜙𝜙!
+#
Separatrix / ɉ ɜ 0 𝐻𝐻 = 𝐻𝐻!" * B  @  Ľ(2.12.1)+(2.12.2)@𝑒𝑒𝑒𝑒𝐸𝐸!
𝑤𝑤 ! =− cos 𝜙𝜙 − cos 𝜙𝜙!
𝜂𝜂ℎ𝜋𝜋Ω!!
+ 𝜋𝜋 − 𝜙𝜙! − 𝜙𝜙 sin 𝜙𝜙!
FŊB/ Figure 4 .𝜙𝜙! = 0, 30⋄ , 60⋄ /Ɔ/ȭƜÖ# Separatrix FŰ
Separatrix 0ʍ'/ zero S[ǅF¡Ǡǲɫ/−𝜋𝜋 ≤ 𝜙𝜙 ≤ 𝜋𝜋 .ŧ'
ʌ'0·.ɒ5# unstable fixed point (π-φs, 0)

<'0 φU +)ƩĽcos 𝜙𝜙! + 𝜙𝜙! sin 𝜙𝜙!
= − cos 𝜙𝜙! + 𝜋𝜋 − 𝜙𝜙! sin 𝜙𝜙!
FǁɄB(φU, 0) B

 Figure 4 @<Ƅ@-?.𝜙𝜙! /čÏ+».Separatrix /ʇĵ+<.ēƽĩB
*

FŊB

/kvehI0¡Ǡǲɫ½.ʍ'/Òǅʉϕ! , 0ʊ(π 𝜙𝜙! , 0ʊFŧ' ʉ𝜙𝜙! , 0ʊ0stable fixed point +ǻė//òA/ɉɆ@ elliptical fixed point +<ô1CB<'/Òǅ π − ϕ! , 0 0

３－ 6

Figure 4: Synchronous phase andseparatrix

Bucket length: |𝜋𝜋 − 𝜙𝜙! − 𝜙𝜙! |Bucket height:.-B𝑤𝑤!!"!! =−𝑒𝑒𝑒𝑒𝐸𝐸!
Γ 𝜙𝜙!
𝜂𝜂 ℎ𝜋𝜋Ω!!

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wFʇNiR 8*ÏɖBċî2 '/Ïɖú*ȍžð/ua`TF+B+ȟ+CBȞ.YSe A @YS
e B .n wFåƾċîYSe A /äAÂòƷŸ+YSe B /¹ ĥ ò Ʒ Ÿ F ï  . ɠ 2  𝜙𝜙! = 0 * / bucket
height  2 '/YSe*ï.-B?
.ua`T/ƕF+B+
𝑉𝑉

𝜂𝜂 ℎ

= constantFŊB J-PARC /ċîÏɖʇòƷĊƑƷ$/ċîua`Tƕ0MR /Ň%åɹāF
RCS äAÂɹā/ʐ°ǭĺ.ȴĝ-C1-@-

3ʋRF Y[cw  YSe/ʇòƷÏɖɹċ0ǻė/òûòƐ.ïƐ-C1-@-»ŪĆ/ǲȔ§ECBɰėYSe*0òƷŸ¢ïɋĆǲȔȞǜʉFigure 5ʊ* B

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LCR Äûɇ*ȥ#+/ÄK_S^[FȥǲȔ/K_S^[0ɔǦǎ/ēǦŔ£/ £ ǰ * Ƶ 8 B  @ ! / Ȫ Ȁ ɔ Ǧ ǎ : 𝜇𝜇! ＝𝜇𝜇’ −３－ 7

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ǦŔ£/K_S^[ L𝑏𝑏𝑏𝑏𝜇𝜇! 𝜇𝜇!
𝑙𝑙 ln= 2×10!! 𝜇𝜇! 𝑙𝑙! ln
𝐿𝐿 =
2𝜋𝜋 !
𝑎𝑎
𝑎𝑎
F§&)
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𝑏𝑏
𝐵𝐵 𝑟𝑟 𝑑𝑑𝑑𝑑 = 𝜔𝜔!" 𝑙𝑙! 𝐵𝐵! 𝑟𝑟! ln
𝑉𝑉!" = 𝜔𝜔!" 𝑙𝑙!
𝑎𝑎!
≈ 𝜔𝜔!" 𝐵𝐵!" 𝐴𝐴 (3.1.3)
*A 0ǦƔȾɽ.ąǟ-ǦŔ£/żɽ
ǰ* B

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Figure 7: ǦŔ£ µQ ǰ/òƷŸǋŔ+)ÇǒB+-,*ĞÑǜ- Q ±FďBèȕ* BC0ǦŔ£VI/Pae.?Aòžð/ǦƲŤŢFVe shunt ŤŢFďB+-K_S^[ LFèď*BŔɀF§&)BǏĂJ-PARC
RCS á2 MR 2 '/YSe/ÏɖǲȔ.ʇɹċÔɢFĞǏ#ĞǒƧ+)ǖ*Å
;)/ɚǒɪĖ#[8]
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4. n

w

dJT

n wɹƸʇòƷǲȔFɕɛB+ÏɖR
zaq. wakefield FȸɁBÏɖRzaq.
0ÏɖɹāFǛǑB#;/ generator current+ beam current ƸCɎ92 '/ɹƸ.?B
îřɹāǛǑB+.-B
4.1 n

w

dJT

  ǲȔɹā 𝑉𝑉 +ȢǰCBNiR F
𝑊𝑊 = 𝑎𝑎𝑉𝑉 ! FȒB*a 0Ŗ* BnwǲȔ.ȸɁBɹā Vb +n w.ń
ɾFBɹā 𝑉𝑉! = 𝑏𝑏𝑉𝑉! +B*ʙ0Ŗ/Ÿ0 < b < 1 * B

３－ 8 ʍ'/n w¡ǠȰ θ *ǲȔFɕɛBƆ
!C"CȸɁBɹāF𝑉𝑉!! , 𝑉𝑉!! +B
Vb1, Vb2 0ē Vb 0ǵ¡ǠȰ θ *ȸɁCB@ʍ'nBNiRw.?AǲȔ.Ȣ@C0

𝑊𝑊 = 𝑎𝑎 𝑉𝑉!! + 𝑉𝑉!!

!

= 𝑎𝑎 2𝑉𝑉! cos 𝜃𝜃 2

= 2𝑎𝑎𝑉𝑉!! 1 + cos 𝜃𝜃!

(4.1.1)* Bž!C"C/ǻė/NiR Ųĕ
0ǻė/ɹȡF q +)(4.1.2)Δ𝑈𝑈 = 𝑞𝑞𝑉𝑉! + 𝑞𝑞𝑉𝑉! + 𝑞𝑞𝑉𝑉! cos 𝜃𝜃FŊBNiR /­ę 𝑊𝑊 = Δ𝑈𝑈 @𝑞𝑞
1
1, 𝑉𝑉 = 𝑉𝑉 , b =(4.1.3)𝑉𝑉! =2𝑎𝑎 ! 2 !
2

'8Aòûn w0ȸɁ#ɹā Vb / 1/2 /ɹā Ve FȬBÏɖú/n w dJTFȒB*ɣȫ-Ȓ.-&)B[10]4.2 Shunt Im pedance + loaded Q L  ÏɖǲȔ.ǛǑBɹā V +ŲĕɹÍ Po
Shunt Impedance Rsh /ɫ.0𝑉𝑉 !
𝑅𝑅!! =(4.2.1)
2𝑃𝑃!
/ɬ¬ B ǲȔ/ Q ±0ǲȔFơřBǦŔƒŻ/ȪȀɔǦǎ/Ʊ𝑄𝑄 = µ’/µ” *Ƶ8BÏɖǲȔ.
0ÏɖɹāFǛǑ B#;/ʇòƷǂůȆC))ǲȔFɕɛBn w0!/ÂÍKo _[ Rg +ǲȔKo _[FȬBn wȬBKo _[ Rtot 0
𝑅𝑅!! 𝑅𝑅!
𝑅𝑅!!
𝑅𝑅 =
=
(4.2.2)
𝑅𝑅!! + 𝑅𝑅! 1 + 𝛽𝛽

Figure 8: ÏɖǲȔ/Ä»Ūûɇyd: Ïɖ ǲ Ȕ . 0 generator current + beam
current / vector îřɹƸƸCB.
𝑑𝑑𝑊𝑊!
𝜔𝜔𝑊𝑊!
= −𝑃𝑃! = −
𝑑𝑑𝑑𝑑
𝑄𝑄!

'8AȢǰNiR
𝑊𝑊! = 𝑊𝑊!! 𝑒𝑒(4.2.4)

Ws 0

!!"#

!!

(4.2.5)* Ɔ ɫ ǜ . ď Ö  B    *  filling time
𝑇𝑇 = 2𝑄𝑄! 𝜔𝜔 FĝȑB
4.3 Required RF source power  Ï ɖ ǲ Ȕ F LCR  Ä ǵ « û ɇ * ȥ Ǐ(Figure 8) #ƆǲȔ.ƸCBɹƸ0(4.3.1)
𝐼𝐼! = 𝐼𝐼! + 𝐼𝐼!

+-BòƐǜ.ǲȔFɕɛBn w/¦B
image current Ib 0Fourier Ĭɪ.?A bunchɩǢ+ǟƸřÃ 𝐼𝐼!" / 2 °/ŪĵFŧ'
8#n wǲȔ.ȸɁBɹā 𝑉𝑉! = 𝐼𝐼! 𝑅𝑅!!
0 ě ĝ - Ï ɖ . Ŏ ȫ - ï Ɛ ¡ Ǡ 𝜙𝜙! F beam+-B*𝛽𝛽 = 𝑅𝑅!! /𝑅𝑅! +#

 ʇòƷǂFñ:ǲȔY[cw/ Q ±0loaded
Q +ô2ǲȔ.ȢǰC#ɹÍ+Ųĕ/Ʊ*ĝȑCBòƷŸ ωȢǰNiR Ws +/ɫ.
0𝑃𝑃!"#$%& 𝜔𝜔𝑊𝑊!
𝑄𝑄! ==(4.2.3)𝑃𝑃!
𝑃𝑃!
/ɬ¬ B
NiR /­ę@３－ 9Figure 9: Vector Diagram (Phasor)

loading .?AďÖ )8[11][12]
(4.3.1)F vector diagram *ǧ+ Figure 9 /?
.ȥCBn wÏɖ.Ŏȫ- RF power 0
1
4.3.2
𝑃𝑃! = 𝑉𝑉 ∙ 𝐼𝐼!
2*ȥC
𝐼𝐼! = 𝐼𝐼! 𝑒𝑒 !" (4.3.3)?A
real part: 𝑃𝑃!.!"#$ = 1 2 𝑉𝑉𝐼𝐼! cos 𝜃𝜃imaginary part: 𝑃𝑃!.!"# = 1 2 𝑉𝑉𝐼𝐼! sin 𝜃𝜃FŊB
8#ĝĴǌŘ*/ vector diagram @𝑅𝑅!!
cos 𝜙𝜙!
tan 𝜙𝜙! = −𝐼𝐼!
1 + 𝛽𝛽 𝑉𝑉
𝐼𝐼!
− sin 𝜃𝜃
(4.3.4)𝐼𝐼!

8#loading factor YʔIb/I0 F§&)tan 𝜙𝜙! =tan 𝜃𝜃 − 𝑌𝑌 cos 𝜙𝜙!
1 + 𝑌𝑌 sin 𝜙𝜙!

𝐼𝐼! = 𝐼𝐼!

/ɬ¬FŊB

1 + 𝑌𝑌 sin 𝜙𝜙!
cos 𝜃𝜃

5 J-PARC ēŁĺɰėÏɖúƀȴ J-PARC YSe0ʇòƷÏɖǲȔ.
ɦĭǦŔ£(Magnetic Alloy: MA)F§&#Å;)/ēŁĺɰėYSe*

BMA ǲȔ0òƷŸïȻơɗFŧ#RCS/MR !C"C/Ï
ɖòƷŸ/ďÖǹþÕŦB Q ±ƌɞÖC)BǲȔ.§ǒBǦŔƒŻ.0ɦĭǦŔ£
FINEMET(FT-3M/FT-3L)§EC!/ʇʅöǦƔġĺ+Q{~ƿĺ¢ǦƬǋŔ@ǲ

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35kV/m)FĞǏ)BTable-2 . RCS + MRʇòƷY[cw/ĊƑm}x^F8+;B

(4.3.5)(4.3.6)

 (4.3.4), (4.3.6)@ generator current +ɹā/¡Ǡİ θ = 0 /ƆRF source power 0ƌĨ+A𝑉𝑉 !
+ 𝑉𝑉𝐼𝐼! sin 𝜙𝜙!
(4.3.7)
𝑃𝑃!.!"# =2𝑅𝑅
RF source +)Ïɖ.Ŏȫ-ɹāFǛǑBɹÍ(Ǵ 1 ʀ)+n wɹÍʉǴ 2 ʀʊF©ȄBȕÍƴ;@CB

J-PARC /ċîĹĳĉ-ǆïȻY[cw/#;RF carrier ȰòƷŸ ωr +ǲȔ/Ȱ»ŪòƷŸ(ωr)0ț-
(4.3.8)Δ𝜔𝜔 = 𝜔𝜔! − 𝜔𝜔!
Detuning angle 0
𝜙𝜙! = tan!! 2𝑄𝑄! ∙ Δ𝜔𝜔 𝜔𝜔! (4.3.9)
ÏɖòƷŸǲȔ/»ŪòƷŸá2ɹėǷF§&# RF source /ÂÍKo _[* dynamic.ďÖB

Figure 10:

J-PARC ēŁĺɰėÏɖúƀȴ

5.1 3GeV RCSKOǂ@/Ƚɰėnw0RFQ-Linac

324MHz *l`ÖCȆ SDTL (324MHz)
?2 ACS (972MHz)* 400MeV /NiR.ÏɖB~hIaSn

w0o

SɹƸ

50mA(ȴȳ±)m[ĵ 500µs /uSm[*AACS /ÏɖòƷŸ*

B 972MHz /ƆɫơɗFŧ&)BRCS /ÏɖòƷŸ.ɞî
#

B#;RCS / RF SaSF§&)K

Oǂ+ RFQ /ɫ.ȴ# RF `|am*ɫl`F¦&)BC.?ARCS /Ň%å RF ɹāFʇ*żǉǜūǍƶ(adiabatic
capture).Ʊ5)n

３－ 10wŲĕFš)ǲɫɹ

Table  2 YSeĊƑm}x
RCS
400MeV~3GeV
1.23~1.67MHz
348.333m
440kV
2
1 MW
9.14
8.33 x 1013ppp
11 A peak
25 Hz
0.02 sec
* ɝř±

NiR
ÏɖòƷŸ
òɩ
Ïɖɹā
Harmonics
Beam power*
Transition γt# of protons
Beam current
RepetitionAcc. Time

^ (2)

MR
3GeV~30GeV
1.67~1.72MHz
1567.5m
320kV
9500kW*
j32.5
2.61 x 1014ppp*
11.5 A peak
2.48 s/ 5.20 s
1.4 secȡÑƙFȋöB#;/ȍžðÃĲŴ¦Ġƅ.)BRCS ¹ĥƆ.ȡɹďűCɰėn Bmin ^KvT(t=0)/ 250µs ŝË@¹ĥw+)ÏɖCB

CBRCS ¹ĥ*0ɰėn ǲɫɹȡ.?B tune shift 0ƩĽ*Ŋ@CB
𝑟𝑟! 𝑁𝑁!
(5.1.1)𝛥𝛥𝛥𝛥 = −2𝜋𝜋𝛽𝛽 ! 𝛾𝛾 ! 𝜀𝜀𝐵𝐵!

1.6 µsec **rp 0ç¼ǜɹėÚņ(1.53 x 1018m)NT0ºɰėŸʉ8.3 x 1013 pppʊε 0na^[β + γ 0

w/Nv

b¬ŸBｆ0oSɹƸ+ĶăɹƸ/Ʊ*ȥCB bunching factor*

B RCS ¹ ĥ * 0 ¹ ĥ l  q Ʒ Ń F É ŋ  transverse paint Fèȕ.ƌēNva^[
216πmm mrad .8*Nva^[FĹB+*BĽ(5.1.1)@ Bf =0.25 /+ǲɫɹȡ.
?B tune shift 0 Δν

0.25 +-BRCS /¹ĥ

w/òûòƐ0ǾAm[ĵ 500µsec /~hIaSn

w0 RCS / longitudinal ¡Ǡǲɫ.Ǿ 300 ûƊCB~hIaSn

w0 RFQ /Âæ

/ RF chopper * RCS / 2 ' longitudinal bucket.îțB?.¹ĥCBRCS ɹǦǣ0ƫĿ
ǜ.ďÖ)B#;ʛʔ−250µs /^Kv
T / momentum rigidity 0 ɚ Ò ɥ ű Ƕ * Ǿ
0.12ʈ¹ĥn

w/ɚÒɥ?AʇÏɖʇòƷ/òƷŸm^

0Ɔɫǜ.ďÖBǦċ@ƴ;)B/*RCS .¹ĥCB~hIaSn
w0 RCS / longitudinal ¡Ǡǲɫ.șǇ.r
KeCB+.-B-@ē-*0ň.ɒ5Bȍžð/l`ÃĲŴ¦(longitudinal bunch manipulation) . ? A bunching factor Bｆ>= 0.4 FĞǏB[13]
5.1.1 Longitudinal bunch Manipulation  RCS /Ǧċ0ƫĿǜ.ÐǦďÖB(5.1.2)
ȏAɑòƐ fcycle 0 25 Hz'8A~hIaSnw0 400 MeV @ 3 GeV 8* 20 msec *ÏɖCBm[ĵ 500µs /~hIaSn

wm

[0𝐵𝐵 𝑡𝑡 = 𝐵𝐵!"# +𝐵𝐵!"# − 𝐵𝐵!"#
1
2(5.1.2)− 𝑐𝑐𝑐𝑐𝑐𝑐2𝜋𝜋𝑓𝑓!"#.

Figure 11: ɚÒɥOp\ae¹ĥ*/¡Ǡǲ
ɫ.n wrKeCBKx Zÿ

３－ 11Sg[¡Ǡ𝜙𝜙! 0 2 ƩʇȻƷ/¡ǠOp\aeFǧǺÛ/#;.𝜙𝜙! = 𝜙𝜙! = 0 /ċî/íɹāƷŃF Figure 12 .ǧYSg[¡Ǡɐ(𝜙𝜙! = 0) /ǻė0𝑉𝑉! 𝑉𝑉! = −0.5 /ċîɹāFå-ž𝑉𝑉! 𝑉𝑉! > 0.5 /ċîY
Sg[¡Ǡ/Ëň*Ïɖƽɖ/ɬ¬ĊƑ
Ʒ/9/ċî+ãɊB

RCS ¹ĥ/Ň%åÏɖɹā0 150kV *A!/Ɔ/YSeòƐ0 280µsec .-B #  & )  Linac  @ ¹ ĥ  C B n

w0500µsec /¹ĥƆɫ/ɫ. RCS /ȍžð¡Ǡǲ
ɫ½*/ŪÒ0ʇ 2 ûǭĺ++.-BÑƙǜ.ʇ𝐵𝐵! > 0.4FŊB#;ǻėe}aQ

Figure 12: Ïɖɹā+ 2 ƩʇȻƷɹā+!/îřv2/v1=0.5 (upper), v2/v1=0.8 (lower)

T.?BɚÒɥOp\ae Δp2 ƩʇȻƷɹāBｆFŊB.0ÙÃ*-ɕĴ/¹ĥžƶ*0
RF lUbō/ɹƸġĺĴ.ʇ-ABｆ0
Ĩ-B!* RCS *0 Bｆʕ0.4 Fɝř

B#;/ longitudinal Ŵ¦FȂ9îE)Ba) ɚÒɥOp\aeb) 2nd harmonic potential /ħ¹c) ¡Ǡ[Kq(2.9.3)Ľ@ RF òƷŸ.Op\ae( Δf ) FB+*¡Ǡǲɫ..ɚÒɥOp\aeFB+ÂƖBFigure 11 .ɚÒɥOp\ae¹ĥ#Ɔȍžð/¡Ǡǲɫ.nw

rKeCBƢėFƤĽǜ.ǧLinac/RCS/¹ĥ*0m[ĵƌē 500µs / linac beam 0
Figure 11 /?. longitudinal ¡Ǡǲɫ.rK
e¹ĥCBɚÒɥOp\ae$*0YSeŪÒŵɢǜ-#;ÃĲĶĄÖ
ÙÃ*0-
 2 ƩʇȻƷɹāFÏ#ċînwòû

ư.åBʇòƷɹāFĽ(5.1.3).ǧ
𝑉𝑉!" = 𝑉𝑉! sin ∅ + 𝑉𝑉! sin 2 ∅ − ∅! + ∅!

Figure 13: MR ȣn w/ RCS RF ɹām^  (upper) +òƷŸOp\aem^ (lower)/m^+!/¡Ǡ[Kqʉ𝜙𝜙! ʊ/ƕÂFȣ&)BFigure 13 0 MR ȣ RCS /Ïɖm^

/¼Ć¨*

B*0 0.2 %ɚ

ÒɥOp\aeF¹ĥ@ 250 µsec ȇŧ
Bmin /^KvT@ 250 µsec ŭ)ǟȊǜ.
ramping down Bm^FŮǒ2 ƩʇȻƷɹā0 V2/V1= −0.8 F¹ĥȁ8*ȇŧ5
msec *].Bm^+)B[14]8#òƷŸOp\aeFÏɖǝʉtʔ10 msecʊ

(5.1.3)

*V1V2 0ĊƑƷʉÏɖʇòƷʊ?2 2ƩʇȻƷ/Ūĵ, 𝜙𝜙, 𝜙𝜙! 0ÏɖʇòƷ/¡Ǡ+Y

ɐ@ȋ=.ÏǶ)B/0MR +/ɚÒɥua`T/#;*

BRCS .ſBlongitudinal paint I~gY*l`ƷŃ,/?.ŶùCB!/ƱɌF Figure 14 .ǧ
o

３－ 12

SɹƸš@CƷŃēĵ.Ŷù

Figure 14ʒRCS ¹ĥ.ſB longitudinal painting .?Bl`ƷŃʇ Bf ȇŧ*)B+8#Yv{
Y|/ȳǶȃƙ+ȟț)B+

5.2. M ain Synchrotron (M R)ÃB[13]

J-PARC MR Y  S  e   0  RCS  @  C8*pM}KeǦŔ£Fǒ#ïȻĆ

8.3×1013 pppʉ25Hz 1MW ǠłʊF 4 la`ÏɖǲȔY[cw/ċîÏɖǲȔ+ʇȻƷǲȔ

ʏ.3×1014 ppp / 3 GeV ɰėn

0!/ƧȕFÃɶ

NiR

ǒ[r

BFŊ-~T½/Ĥ[FȴBŎȫ&#-wm[Fƌʇ

50GeV ɏÏɖ3.6 Ǫ/òƐ*Ğʆ.ÇǒBǾ 740kW /ēŁĺYSe@J-PARC-RCS *0ǲȔǦŔ£.ɦĭǦŔ£+)ȴȳC#[5]

FǒòƷŸǋŔFĹĳĉÖB».¢ŲĕK J-PARC MR 02007 ķǨÏɖNiR

_S^[F*ÄůȆĊƑƷ+ 2 Ʃ

30GeV *ɚɊFɪĖ#łƆ50GeV m^

B Q ± = 2 /ǐ*ɹǦǣFɚɊB#;.0åɹ AC }Kŕǜ' passive -ǆïȻǲȔFĞǏ)B

/ɹāďÒFÙÃ.šB-,!/žĽ=ĞǏ

C.?AʇòƷÉŋY[cwʉLow Level RF

B#;/ŠȤǜȫȀ-,'/ȺʂFȱ

System: LLRFʊ*ǛǑƵBŎȫ

ʇȻƷ/òƷŸĳĉFPl# Ïɖɹā+ʍƩʇȻƷɹā/îřƷF 1 '/ǲȔ.ǛǑB

A 50GeV nw©Ȅ0 J-PARC

Phase-II *ɝřB+.-&)#ǏĂ MR+èȕ+-&##&)J-PARC RCS /*0ƌʇNiR

12 é/ÏɖǲȔ0º)n

MR WKSFǾ 1.3sec .Ȏ;)ȏAɑFwÏɖ+ȍžð/ÃĲŴ¦/ƧȕF´)B

0 30GeV *ȇŧ#88

B+*Ǟƣ+-B 750kW n

wm

FĞǏ

BȳǕɘ;@Cƌȁǜ.1.3MW n
m

３－ 13

w

FǞŨ#ŶɗȳǕɘȣ)B[2]

Table  3

MR YSe/ʇòƷY[cworiginal

1st Upgrade
1.67~1.72MHz
7
9
2
2
21
36
6
8
280kV
510kV
14
3.3 x 10 ppp
2.0 x 1014ppp
10 A
6A
740 kW
750 kW
50 GeV
30 GeV
3.60 s/ 6.00 s
1.32 s/ 3.92 s* h=9: ĊƑƷh=18: 2nd ʇȻƷÏɖòƷŸéŸʉh=9*ʊéŸʉh=18ʊ
RzaqŸ(h=9)
RzaqŸ(h=18)ƌēɹā(h=9)# of protons
Beam current (IDC)
Beam PowerBeam Energy
Repetition FX/SX

2nd Upgrade
11
2
44
8
600kV
3.3 x 1014ppp
10 A
1.3 MW
30 GeV
1.16 s/ 3.76 s!/88Çǒ)ǲȔ 1 é

5.2.1 750kW 4/äȂ9#A/ÏɖRzaq

ŸFč=+èȕ.-&#2018 ķ/ƆǅčŁ0ɰė*0FT-3L F§&# 4 RzaqǲȔ 9 é07

F 30GeV *ɝřBžɧ4+éFĊƑƷǲȔ+)2 éF 2 ƩʇȻƷǲȔ+žðɊű#!/#;.0ɹǦǣɹǂ/Ȉ¹)§ǒ)B2021 ķ.ĝ)BŽɹ

CƋ+ʇòƷÏɖɹā/°Ïʍ'/ē-Ǧǣɹǂ/¹CƋ/ň.[^ƚ+-&#

/ʇȏAɑɚɊ8*.Ž#. 2 ƩʇȻƷY[c MR YSe0RCS YSeï

w 2 éFŽȴBǃ´Fɘ;)B MR YSe/n/ƌʇNiR

Ƣ. 3 '/KWwm

eB 1.32 ǪY|FƎ3 ûģǫ/}cJ[*ơřC)1 '/KW

Y|.ʇ

òƷǲȔ 9 éɢȐC)B

5.2.2 MR YSe/ēŁĺɚɊ Table 3 .ǧ?“Original”*09 Ľ/Ïɖ

 MR ~  T / ò ɩ 0 RCS /  ĺ 4.5 ° /ǲȔȧȐ/½ʑéFn

1567.5 m *wÏɖ.ʍéFʍƩʇ

BÏɖʇòƷ/kyhaSŸȻƷY[cw+)§ǒ)#ÏɖǲȔ.0

0 h = 9 .ɠ2RCS äAÂ/ÏɖòƷŸ+pHKxae FT-3M /PaeVIF§ǒ

MR ¹ĥ/òƷŸFțǲȔ#A/ÏɖRzaqŸ0 3ƌē 45ʚV /ÏɖɹāFǛǑ)B

MR /¹ĥ*0RCS @ 4 la`/nw¹B+*BpHKxĥCB9 '/ MR RF lUb/½8 lUb

B FT-3L 0Cl`*ć;@CƮA 1 '/ǲlUb0äA

ae/}Kgaq/'*

8*/ FT-3M ƒ.Ʊ5ǲȔKo

_[F

ÂQaP

/ǳ%A.§ECBMR YBɔǦǎ+ Q ±/ǰʉµＱǰʊ 2 °ǭĺʇSe.)< bunching factor BｆFʇ

Ǧċ*ǉÁǐFȣ#;C8*Đņ

­'+ɣȫ+-BRCS /¹ĥ*ȣ&#?

800mm FɂBÏɖúǒēĆVI/ȩɗ0ȣE

.đ/l`Ŵ¦0*-MR *0

C-&#2010 ķJ-PARC *0ĦƖ.ða) 2nd harmonic potential /ħ¹#ɦĭǦŔ£/VI/ɪǛFxb) RF ¡ǠOp\ae¹ĥ

P+».ɘ;

FT-3L VI/Ïɖú4/ĞǒÖFřÎ
C.?AÏɖǲȔ/Rzaq
_[F

#[9]

/ 2 ŝƶ.?AFX *0 2nd harmonic potential#A/KoFSX *0 RF ¡ǠOp\aeʉglobal offsetʊB+*Ɓę/ɰƟɹǂF

.?A longitudinal emittance Fčē

３－ 14

òû

(a) 2nd off

Figure 15: ¼Ćǜ- RF m^
ĊƑƷ/9
n

w/o

SɹƸFBŴ¦Fȣ&)B

Figure 15 .h{
ēŁĺn

ʉFXʊ

e~jĞʆð FX ɚɊ*/wÏɖ*Ůǒ)B 2 ƩʇȻƷ.?

Bʗʘ¹ĥ.Bl`ƷŃ/ mountain plotFǧMR ¹ĥ* 2 ƩʇȻƷFħ¹-Ɔ(b) 2nd ON (V2 = 110 kV)

RCS @/l`ĵ0 160 nslongitudinalemittance 0 4.5 eVs  𝐵𝐵! ~ 0.2 ǭ ĺ * B

Figure 16: ¼Ćǜ- MR ¹ĥ/uLcqaeʉl`ƷŃʊ(ƥɋʒRF phase in nsec,
ȍɋʒslice number in turn(Figure16(a))žFigure 16(b)0 2 ƩʇȻƷ

V1/V2 = 70% / Ì î * ħ ¹  # ĝ Ĵ ɚ Ɋ * /mountain plot Fǧl`ĵÑǎȟĹ
Abunching factor F𝐵𝐵! ~ 0.3ǭĺ.ŶùB+*)B

𝑉𝑉!"# 𝜔𝜔! 𝑡𝑡= 𝑍𝑍!"# 𝜔𝜔! 𝑡𝑡∙ 𝐼𝐼!"#$ 𝜔𝜔! 𝑡𝑡

*𝜔𝜔! 𝑡𝑡 0òûíòƷŸ/k°/íòƷŸ*(5.3.1)yhaS hB/òûòƷŸ/±0Ïɖ+». MR *Ǿ 3%ďÖB#;Ɔɫ/ɬŸ.-

5.3 Beam Loading

B
 òûn

MR ÏɖǲȔ/Kow/ÏɖʇòƷřÃ0ĎɹƸƷŃ/_[0Ĺĳĉ*!/Fourier ȱƘ*Ŋ@C!/ĊƑƷòƷŸřÃ Ib

òƷŸǋŔ0 Fig. 18 .ǧɕA*

0  l  ` ɩ  Ǣ  Ɔ  ǟ Ƹ ř Ã 𝛪𝛪!" =𝑒𝑒𝛮𝛮! 𝜔𝜔!" /(2𝜋𝜋) / 2 °/𝐼𝐼! = 2 𝐼𝐼!" .-ARCSȸɁɹā/Ȩ¶Ŏȫ.-Bü9.RCS /ċBȪŸ/ʇȻƷřÃharmonic Ÿ h=8, 9, 10 .')

* 12AMR * 20A .ɝBJ-PARC *0îǲȔY[cw0 2 ƩʇȻƷ/òƷŸĳĉ<P

feedforward .?Abeam induced ɹā/Ȩ¶lFȣ&)BFigure 17 0feedforward Ȩ¶Ĺĳĉ*®ë/ƸCFǧnwɹƸ 𝑖𝑖!"#$ 0ǲȔFɕɛB+ȸɁɹā 𝑉𝑉!"# FǛǑB)B#;ǲȔKo_[0@.

Aharmonic Ÿ h=1~6 /ʇȻƷFȨ¶/ģȼ.)B[15]ÿ/ feedforward
module *0n
ƘCn

３－ 15

wɹƸ®ëkyhaSȱwǲȔɕɛƆ.ȸɁBɹā/ȊŃŔ­#CB+B+Feedforward Éŋ
FÒ¦

#Ɔn

wȬBKo!
𝑍𝑍!"#𝜔𝜔! 𝑡𝑡+ 𝑍𝑍!! 𝜔𝜔! 𝑡𝑡

_[

0

+-B+ÃB

'8AFeedforward /ȻŹ*0n
oFigure 17: Feedforward beam loading Ȩ¶/®ë/ƸC

_[F zero .B+𝑍𝑍!! 𝜔𝜔! 𝑡𝑡

𝑉𝑉!"#.!"#$ FŞ%ƻ?-ʇȻƷ®ëFÂÍ
B

wȬBK

!
= −𝑍𝑍!"#
𝜔𝜔! 𝑡𝑡(5.3.4)

FǁɄB?.!C"C/ harmonics .')gain + delay FƵĝB[16]

íǲȔ/Rzaqɹā0
𝑉𝑉!"# 𝜔𝜔! 𝑡𝑡= 𝑉𝑉!"#.!" 𝜔𝜔! 𝑡𝑡+ 𝑉𝑉!"#.!"#$ 𝜔𝜔! 𝑡𝑡+ 𝑉𝑉!"#.!! 𝜔𝜔! 𝑡𝑡(5.3.2)ʎǮʃ/ɹāƷŃ/îř*ȥCBǴʌʀ0ÏɖúŎȫ-ɹāǴʍʀ0nwȸɁBɹāǴʎʀfeedforward Y[cwǲȔ.
ǛǑBBɹā*B * Feedforward ǽ*!C"C/ɝɬŸF/?.ĝȑB
!"#𝐻𝐻!"
𝜔𝜔! 𝑡𝑡 : nā.ȚBɝɬŸ!
𝑍𝑍!"#
𝜔𝜔! 𝑡𝑡 : nwȸɁBɹā+n

@Ŋ@CBǲȔY[cw/Ko
𝑍𝑍!! 𝜔𝜔! 𝑡𝑡 : n

Figure 18: ¼Ćǜ- MR ÏɖǲȔ/Ko_[wǆƆLLRF @ǲȔɹ
wɹƸ

_[

w ɹ Ƹ Ɲ Â  @ Feedforward

module Fɕ)ǲȔɹā.ȚBɝɬŸ MR / Ï ɖ ǲ Ȕ Y [ c w * 0  Ï ɖ ò Ʒ Ÿ
ʉh=9ʊ+ɴ/ harmonics h=8, 10 /ʇȻƷ.')Ȩ¶ȣECB(Figure 18)Harmonics h=9.')0Ïɖºĉ.E#&)ÏɖɹƷ^ęĂŁŪĵÉŋ (Auto level control)

C@ 3 '/ɝɬŸF§&)Ľ(5.3.2)0/#;.Feedforward /𝑉𝑉!"# 𝜔𝜔! 𝑡𝑡!"#= 𝐻𝐻!"
𝜔𝜔! 𝑡𝑡!
+ 𝑍𝑍!"#
𝜔𝜔! 𝑡𝑡+ 𝑍𝑍!! 𝜔𝜔! 𝑡𝑡∙ 𝐼𝐼!"#$ 𝜔𝜔! 𝑡𝑡

𝑉𝑉!"#.!" 𝜔𝜔! 𝑡𝑡 .Ƅǥ-İ0Ȭ- Figure 19∙ 𝑉𝑉!" 𝜔𝜔! 𝑡𝑡ʉʊ-@harmonics h=8, 10 /n∙ 𝐼𝐼!"#$ 𝜔𝜔! 𝑡𝑡

wȸɁɹā0ÙÃ.šÉC)B+ÃB

(5.3.3)

 Feedforward Éŋ0®ë/ƸC open loop+ȥ+*B*𝑉𝑉!" 𝜔𝜔! 𝑡𝑡 0ÏɖRzaq.Ŏȫ-ɹā𝑉𝑉!"#.!" 𝜔𝜔! 𝑡𝑡 FÂ#;/ driving signal *A-*ɹāŪĵ

Bčĵúǽ/ɝɬŸ*Afeedback Éŋ*Ȭ@CBY[cw/ěĝŔ/ōɢ0--@Đȫü/ƝÂǼĺopen loop ǋŔ/ȊŃŔɣȫ.-A

３－ 16

ppp /ɰėn

wF 2.48 Ǫ/ȏAɑ*h{Fěĝ.)8J-PARC *Ůǒ#ɦĭe~jnw}K.©ȄB+.řÎǦŔ£F§&#ÏɖǲȔ*0C8*/ ferrite

#/n

wm

0 500 kW .ģőB

loaded system +Ǚ-AǲȔKo

470 kW FɂBn

wŁĺ*l`ȃî.?B!C@ť­C-C1ɓ.n

w/ɚÒ

_[

ʇòƷŪĵ=ƿĺ.ģ)åÒǜ* feedforward

dipole Ū Ò  Ï ɖ ň Ú * ȯ ǀ  C Ė ; # [18]žĽ+/ǠŔ0ɼĴ.ȟ@.0ʇǼĺ-d(Figure 20)Z^Éŋ.?A¿Ǐ<ǥ­CB-&)ɹ l`ȃîŪÒ0òûB 8 ¯/l`!CėǷF§&#ʇòƷčĵú/ÂÍǋŔ.ǏCB

"CÏɖǲȔFɕɛB+.ȸɁBîřɼȊŃŔnɹā/ńɾFå)ěĝ.-BǏȼ*@Ğɳ/ɚǒ*0nwɹƸ/čÏ.

wŁĺ/čÏ+».ǆȮ*-B-&)#!/#;Feedforward ǽ/ȊŃǋŔ/C.Ɂü#šÉȹİFȨ¶B#;ʇȻƷɹāF IQ-ȱƘ!C"CFěĝÖB
vector feedback /ħ¹Fɘ;)B[17]
5.4

Longitudinal

coupledbunch

instabilityMR ƃäAÂ*0 2018 ķ 5 ƍ2.6 × 10!"Figure 20: Mountain-view qae
MR ɖ  ä A Â   480kW ɚ Ɋ ʉ 2.8×
10!" 𝑝𝑝𝑝𝑝𝑝𝑝 × 8 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏ℎ𝑒𝑒𝑒𝑒ʊMR /ċîharmonics h=9 -/*M=9 ¯/l`ȃîyfęĂB òûl`/ 1 '/Ȋġĺ𝜆𝜆 𝑡𝑡 F 2 '/ġĺřÃ𝜆𝜆! 𝑡𝑡 , 𝜆𝜆! 𝑡𝑡 .Ãȱ)ȒB+ŪÒ/ĪŸm /Ÿ.ő)m = 1 – dipole, 2 – quadrupole, 3 –sextupole, , -,¡Ǡǲɫ*/l`ŪÒ/ƢėFÃʃB+*B[19]ʉFigure 21ʊ l`ȃîŪÒFòƷŸ[rSe}w0𝜔𝜔 = 𝑛𝑛 + 𝑙𝑙𝑙𝑙 𝜔𝜔! + 𝑚𝑚𝜔𝜔! (5.4.1)* ȥ  C B ò Ʒ Ÿ ř Ã F ŧ '  𝜔𝜔! = 2𝜋𝜋𝑓𝑓! +Figure 19:  dJTȨ¶/Ñƙ(upper) h=9 ʇòƷŪĵřÃ(lower) h=8, 10ʇòƷŪĵřÃ

𝜔𝜔! = 2𝜋𝜋𝑓𝑓! 0!C"CòûȰòƷŸ+YS

eȰòƷŸn 0𝑛𝑛 = 0, 1, 2, … , 𝑀𝑀 − 1 l`ȃîŪÒ/ mode Ÿ𝑙𝑙 = 0, ±1, ±2,∙∙∙∙/

±+-B

３－ 17

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Figure 21: l`ȃîŪÒ/ mode m^+!/¡Ǡǲɫ*/ŪÒ[19]&#!&#Ś;A&#ɦĭǦŔ£VI0Ǔǆ*/ɪǛǤǱFÓ(

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